Damage Models for Soft Tissues: A Survey
---
paper_title: Introduction to continuum damage mechanics

paper_content:

1 - Introduction.- 2 - Creep and Fracture under Uniaxial Stress.- 3 - Creep and Fracture under Multiaxial Stress.- 4 - Crack Growth under Creep Conditions.- 5 - Damage Model for Ductile Fracture.- 6 - Fatigue Damage.- References.- Notations.
---
paper_title: Assessment of Tissue Damage due to Mechanical Stresses

paper_content:

While there are many benefits to minimally invasive surgery (MIS), force feedback or touch sensation is limited in the currently available MIS tools, such as surgical robots, creating the potential for excessive force application during surgery and unintended tissue injury. The goal of this work was to develop a methodology with which to identify stress magnitudes and durations that can be safely applied with a MIS grasper to di ferent tissues, potentially improving MIS device design and reducing potentially adverse clinically relevant consequences. Using the porcine model, stresses typically applied in MIS were applied to liver, ureter and small bowel using a motorized endoscopic grasper. Acute indicators of tissue damage including cellular death and infiltration of inflammatory cells were measured using histological and image analysis techniques. Finite element analysis was used to identify approximate stress distributions experienced by the tissues. Parameters used in these finite element models specifically reflected the properties of liver, which served as an initial proxy for all tissues, as stress distributions rather than absolute values were desired. Local regions predicted to have uniform stress by the computational models were mapped to and analyzed in the tissue samples for acute damage. Analysis of variance (ANOVA) and post-hoc analyses were used to detect stress magnitudes and durations that caused significantly increased tissue damage with the goal to ultimately identify safe stress `thresholds' during grasping of the studied tissues. Preliminary data suggests a graded non-linear response between applied stress magnitude and apoptosis in liver and small bowel as well as neutrophil infiltration in the small bowel. The ureter appeared to be more resistant to injury at the tested stress levels. By identifying stress magnitudes and durations within the range of grasping loads applied in MIS, it may be possible for researchers to create a `smart' surgical robot that can guide a surgeon to manipulate tissues with minimal resulting damage. In addition, surgical simulator design can be improved to reflect more realistic tissue responses and evaluate trainees' tissue handling skills.
---
paper_title: Fundamentals of Anatomy and Physiology

paper_content:

This text is designed specifically to meet the needs of a diverse student population. It offers three ways of covering real-world applications: Clinical Discussion; Clinical Boxes (which provide more in-depth coverage of major diseases); and the Applications Manual. The artwork is unusual in that it falls into three categories: Compound Art combines Bill Ober's award-winning illustrations with photographs, micrographs, X-rays, and MRI images; Process Art shows how the successive stages of dynamic physiology processes unfold in time; and Large Illustrations Viewed from a Macro-Micro Level, in context, help students better understand anatomical structures. As for the pedagogy, the book has four main features: Concept Checks, which are interspersed questions that encourage students to stop and check their understanding of material before they move on; Concept Links, that have page references to help students review concepts discussed earlier in the text; Figure Reference Locators, red squares next to call out figures in the text that function as placeholders to make it easier to toggle between illustrations and the running narrative; and Three-Level Chapter Ending Material - this moves students through memorizing facts, to understanding concepts, to applying what they learn to solve problems. Level One questions review basic facts and terms. Level Two questions test the student's ability to comprehend and integrate the chapter's key concepts. Level Three exercises are real world scenarios designed to develop student's problem solving skills.
---
paper_title: A pseudo–elastic model for the Mullins effect in filled rubber

paper_content:

When a rubber test piece is loaded in simple tension from its virgin state, unloaded and then reloaded, the stress required on reloading is less than that on the initial loading for stretches up to the maximum stretch achieved on the initial loading. This stress softening phenomenon is referred to as the Mullins effect . In this paper a simple phenomenological model is proposed to account for the Mullins effect observed in filled rubber elastomers. The model is based on the theory of incompressible isotropic elasticity amended by the incorporation of a single continuous parameter, interpreted as a damage parameter. This parameter controls the material properties in the sense that it enables the material response to be governed by a strain–energy function on unloading and subsequent submaximal loading different from that on the primary (initial) loading path from the virgin state. For this reason the model is referred to as pseudo-elastic } and a primary loading-unloading cycle involves energy dissipation. The dissipation is measured by a damage function which depends only on the damage parameter and on the point of the primary loading path from which unloading begins. A specific form of this function with two adjustable material constants, coupled with standard forms of the (incompressible, isotropic) strain–energy function, is used to illustrate the qualitative features of the Mullins effect in both simple tension and pure shear. For simple tension the model is then specialized further in order to fit Mullins effect data. It is emphasized that the model developed here is applicable to multiaxial states of stress and strain, not just the specific uniaxial tests highlighted.
---
paper_title: A constitutive model for the Mullins effect with permanent set in particle-reinforced rubber

paper_content:

Stress softening during initial loading cycles, known as the Mullins effect, and the residual strain upon unloading are not accounted for when the mechanical properties of rubber are represented in terms of a strain-energy function, i.e. if the material is modelled as hyperelastic. In this paper we first describe some experimental results that illustrate stress softening in particle-reinforced rubber together with associated residual strain effects. In particular, the results show how the stress softening and residual strain change with the magnitude of the applied strain. Then, on the basis of these data a constitutive model is derived to describe this behaviour. The theory of pseudo-elasticity is used for this model, the basis of which is the inclusion of two variables in the energy function in order separately to capture the stress softening and residual strain effects. The dissipation of energy, i.e. the difference between the energy input during loading and the energy returned on unloading is also accounted for in the model by the use of a dissipation function, which evolves with the deformation history.
---
paper_title: Introduction to continuum damage mechanics

paper_content:

1 - Introduction.- 2 - Creep and Fracture under Uniaxial Stress.- 3 - Creep and Fracture under Multiaxial Stress.- 4 - Crack Growth under Creep Conditions.- 5 - Damage Model for Ductile Fracture.- 6 - Fatigue Damage.- References.- Notations.
---
paper_title: Discontinuous and continuous damage evolution in Ogden-type large-strain elastic materials

paper_content:

The paper presents an application of continuum damage mechanics to large-strain Ogden-type elastic materials. A Eulerian setting of elasticity with isotropic damage-caused softening effect is discussed exclusively in terms of the Finger tensor as a strain measure. The local damage accumulation is related to two phenomenological variables, the maximum value and the arclength of effective free energy attained during a typical loading process. The frame is specified for a rigorously decoupled volumetric-isochoric response based on a multiplicative split of the Finger tensor into spherical and unimodular parts. In this context, the damage is related to the isochoric part of the deformation only, which is assumed to be governed by an Ogden-type effective free-energy function formulated in terms of the eigenvalues of the unimodular part of the Finger tensor. A constitutive algorithm for the computation of the nominal stresses and nominal tangent moduli for plane problems is developed. Finally, some fundamental finite-element simulations for rubber-like materials are presented.
---
paper_title: A pseudo–elastic model for the Mullins effect in filled rubber

paper_content:

When a rubber test piece is loaded in simple tension from its virgin state, unloaded and then reloaded, the stress required on reloading is less than that on the initial loading for stretches up to the maximum stretch achieved on the initial loading. This stress softening phenomenon is referred to as the Mullins effect . In this paper a simple phenomenological model is proposed to account for the Mullins effect observed in filled rubber elastomers. The model is based on the theory of incompressible isotropic elasticity amended by the incorporation of a single continuous parameter, interpreted as a damage parameter. This parameter controls the material properties in the sense that it enables the material response to be governed by a strain–energy function on unloading and subsequent submaximal loading different from that on the primary (initial) loading path from the virgin state. For this reason the model is referred to as pseudo-elastic } and a primary loading-unloading cycle involves energy dissipation. The dissipation is measured by a damage function which depends only on the damage parameter and on the point of the primary loading path from which unloading begins. A specific form of this function with two adjustable material constants, coupled with standard forms of the (incompressible, isotropic) strain–energy function, is used to illustrate the qualitative features of the Mullins effect in both simple tension and pure shear. For simple tension the model is then specialized further in order to fit Mullins effect data. It is emphasized that the model developed here is applicable to multiaxial states of stress and strain, not just the specific uniaxial tests highlighted.
---
paper_title: Computational modelling of the stress-softening phenomenon of rubber-like materials under cyclic loading

paper_content:

Abstract When a rubber specimen is subjected to cyclic loading, not only non-linear behaviour but also damage-induced stress-softening phenomena (the Mullins effect) have been observed. Applications of a continuum damage mechanics model and Ogden and Roxburgh's pseudo-elastic model to describe the Mullins effect in elastomers have been considered. Both models together with Gao's elastic law were implemented to describe the mechanical behaviour of rubber-like materials including the stress-softening phenomenon. Two sets of experimental data (a simple tension test and a simple tension and pure shear test) are used to validate the constitutive models. Model parameters are estimated via an inverse technique. Computational results show that both constitutive models together with Gao's elastic law can describe the typical Mullins effect. From engineering point of view, the pseudo-elastic model has the advantages that (i) the model is simple and practical, since it considers that the stress-softening function is only activated on unloading or reloading paths, (ii) the model with a slight modification of the damage variable is very stable in finite element calculations, and (iii) the numerical results agree very well with experimental data in both simple tension and pure shear deformation. Two applications illustrate the capability of combining the pseudo-elastic model with Gao's elastic law in describing the Mullins effect. It is emphasized that both models are applicable to multiaxial states of stress and strain because both models are energy-based and not strain-based.
---
paper_title: STRAIN AND STRESS BASED CONTINUUM DAMAGE MODELS-I. FORMULATION

paper_content:

Abstract Continuum elastoplastic damage models employing irreversible thermodynamics and internal state variables are developed within two alternative dual frameworks. In a strain [stress] -based formulation, damage is characterized through the effective stress [strain] concept together with the hypothesis of strain [stress] equivalence , and plastic flow is introduced by means of an additive split of the stress [strain] tensor . In a strain -based formulation we redefine the equivalent strain , usually defined as the J 2 -norm of the strain tensor, as the (undamaged) energy norm of the strain tensor. In a stress -based approach we employ the complementary energy norm of the stress tensor. These thermodynamically motivated definitions result, for ductile damage, in symmetric elastic-damage moduli. For brittle damage, a simple strain -based anisotropic characterization of damage is proposed that can predict crack development parallel to the axis of loading (splitting mode). The strain- and stress-based frameworks lead to dual but not equivalent formulations, neither physically nor computationally. A viscous regularization of strain-based, rate-independent damage models is also developed, with a structure analogous to viscoplasticity of the Perzyna type, which produces retardation of microcrack growth at higher strain rates. This regularization leads to well-posed initial value problems. Application is made to the cap model with an isotropic strain-based damage mechanism. Comparisons with experimental results and numerical simulations are undertaken in Part II of this work.
---
paper_title: Discontinuous and continuous damage evolution in Ogden-type large-strain elastic materials

paper_content:

The paper presents an application of continuum damage mechanics to large-strain Ogden-type elastic materials. A Eulerian setting of elasticity with isotropic damage-caused softening effect is discussed exclusively in terms of the Finger tensor as a strain measure. The local damage accumulation is related to two phenomenological variables, the maximum value and the arclength of effective free energy attained during a typical loading process. The frame is specified for a rigorously decoupled volumetric-isochoric response based on a multiplicative split of the Finger tensor into spherical and unimodular parts. In this context, the damage is related to the isochoric part of the deformation only, which is assumed to be governed by an Ogden-type effective free-energy function formulated in terms of the eigenvalues of the unimodular part of the Finger tensor. A constitutive algorithm for the computation of the nominal stresses and nominal tangent moduli for plane problems is developed. Finally, some fundamental finite-element simulations for rubber-like materials are presented.
---
paper_title: On a fully three-dimensional finite-strain viscoelastic damage model: Formulation and computational aspects

paper_content:

Abstract A fully three-dimensional finite-strain viscoelastic model is developed, characterized by: (i) general anisotropic response, (ii) uncoupled bulk and deviatoric response over any range of deformations, (iii) general relaxation functions, and (iv) recovery of finite elasticity for very fast or very slow processes; in particular, classical models of rubber elasticity (e.g. Mooney-Rivlin). Continuum damage mechanics is employed to develop a simple isotropic damage mechanism, which incorporates softening behavior under deformation, and leads to progressive degradation of the storage modulus in a cyclic test with increasing amplitude (Mullins' effect). A numerical integration procedure is proposed which trivially satisfies objectivity and bypasses the use of midpoint configurations. The resulting algorithm can be exactly linearized in closed form, and leads to symmetric tangent moduli. Quasi-incompressible response is accounted for within the context of a three-field variational formulation of the Hu-Washizu type.
---
paper_title: A pseudo–elastic model for the Mullins effect in filled rubber

paper_content:

When a rubber test piece is loaded in simple tension from its virgin state, unloaded and then reloaded, the stress required on reloading is less than that on the initial loading for stretches up to the maximum stretch achieved on the initial loading. This stress softening phenomenon is referred to as the Mullins effect . In this paper a simple phenomenological model is proposed to account for the Mullins effect observed in filled rubber elastomers. The model is based on the theory of incompressible isotropic elasticity amended by the incorporation of a single continuous parameter, interpreted as a damage parameter. This parameter controls the material properties in the sense that it enables the material response to be governed by a strain–energy function on unloading and subsequent submaximal loading different from that on the primary (initial) loading path from the virgin state. For this reason the model is referred to as pseudo-elastic } and a primary loading-unloading cycle involves energy dissipation. The dissipation is measured by a damage function which depends only on the damage parameter and on the point of the primary loading path from which unloading begins. A specific form of this function with two adjustable material constants, coupled with standard forms of the (incompressible, isotropic) strain–energy function, is used to illustrate the qualitative features of the Mullins effect in both simple tension and pure shear. For simple tension the model is then specialized further in order to fit Mullins effect data. It is emphasized that the model developed here is applicable to multiaxial states of stress and strain, not just the specific uniaxial tests highlighted.
---
paper_title: Computational modelling of the stress-softening phenomenon of rubber-like materials under cyclic loading

paper_content:

Abstract When a rubber specimen is subjected to cyclic loading, not only non-linear behaviour but also damage-induced stress-softening phenomena (the Mullins effect) have been observed. Applications of a continuum damage mechanics model and Ogden and Roxburgh's pseudo-elastic model to describe the Mullins effect in elastomers have been considered. Both models together with Gao's elastic law were implemented to describe the mechanical behaviour of rubber-like materials including the stress-softening phenomenon. Two sets of experimental data (a simple tension test and a simple tension and pure shear test) are used to validate the constitutive models. Model parameters are estimated via an inverse technique. Computational results show that both constitutive models together with Gao's elastic law can describe the typical Mullins effect. From engineering point of view, the pseudo-elastic model has the advantages that (i) the model is simple and practical, since it considers that the stress-softening function is only activated on unloading or reloading paths, (ii) the model with a slight modification of the damage variable is very stable in finite element calculations, and (iii) the numerical results agree very well with experimental data in both simple tension and pure shear deformation. Two applications illustrate the capability of combining the pseudo-elastic model with Gao's elastic law in describing the Mullins effect. It is emphasized that both models are applicable to multiaxial states of stress and strain because both models are energy-based and not strain-based.
---
paper_title: Mechanical Properties and Microstructure of Intraluminal Thrombus From Abdominal Aortic Aneurysm

paper_content:

Accurate estimation of the wall stress distribution in an abdominal aortic aneurysm (AAA) may prove clinically useful by predicting when a particular aneurysm will rupture. Appropriate constitutive models for both the wall and the intraluminal thrombus (ILT) found in most AAA are necessary for this task. The purpose of this work was to determine the mechanical properties of ILT within AAA and to derive a more suitable constitutive model for this material. Uniaxial tensile testing was carried out on 50 specimens, including 14 longitudinally oriented and 14 circumferentially oriented specimens from the luminal region of the ILT, and 11 longitudinally oriented and 11 circumferentially oriented specimens from the medial region. A two-parameter, large-strain, hyperelastic constitutive model was developed and used to fit the uniaxial tensile testing data for determination of the material parameters. Maximum stiffness and strength were also determined from the data for each specimen, Scanning electron microscopy (SEM) was conducted to study the regional microstructural difference. Our results indicate that the microstructure of ILT differs between the luminal, medial, and abluminal regions, with the luminal region stronger and stiffer than the medial region. In all cases, the constitutive model fit the experimental data very well (R 2 >0.98). No significant difference was found for either of the two material parameters between longitudinal and circumferential directions, but a significant difference in material parameters, stiffness, and strength between the luminal and medial regions was determined (p<0.01). Therefore, our results suggest that ILT is an inhomogeneous and possibly isotropic material. The two-parameter, hyperelastic, isotropic, incompressible material model derived here for ILT can be easily incorporated into finite element models for simulation of wall stress distribution in AAA.
---
paper_title: Mechanical properties of abdominal aortic aneurysm wall.

paper_content:

There is a need to understand why and where the abdominal aortic aneurysm may rupture. Our goal therefore is to investigate whether the mechanical properties are different in different regions of the aneurysm. Aorta samples from five freshly excised whole aneurysms, U 5 cm in diameter, from five patients, average age 71 - 10 years, were subjected to uniaxial testing. We report the wall thickness, yield stress and strain, and parameters that describe nonlinear stress-strain curves for the anterior, lateral and posterior regions of the aneurysm. The posterior region was thicker than the anterior region ( 2.73 - 0.46 mm versus 2.09 - 0.51 mm). The stress-strain curves were described by † = a k b, where † is true stress and k is engineering strain. In the circumferential direction, the wall stiffness increased from posterior to anterior to lateral. In the longitudinal direction, the lateral and anterior regions showed greater wall stiffness than the posterior region. The wall stiff...
---
paper_title: The effects of aneurysm on the biaxial mechanical behavior of human abdominal aorta

paper_content:

Abstract The biomechanical response of normal and pathologic human abdominal aortic tissue to uniaxial loading conditions is insufficient for the characterization of its three-dimensional (3D) mechanical behavior. Planar biaxial mechanical evaluation allows for 3D constitutive modeling of nearly incompressible tissues, as well as the investigation of the nature of mechanical anisotropy. In the current study, 26 abdominal aortic aneurysm (AAA) tissue samples and 8 age-matched (>60 years of age) nonaneurysmal abdominal aortic (AA) tissue samples were obtained and tested using a tension-controlled biaxial testing protocol. Graphical response functions (Sun et al., 2003. J. Biomech. Eng. 125, 372–380) were used as a guide to describe the pseudo-elastic response of AA and AAA. Based on the observed pseudo-elastic response, a four-parameter exponential strain energy function developed by Vito (1990. J. Biomech. Eng. 112, 153–159) was used from which both an individual specimen and group material parameter sets were determined for both AA and AAA. Peak Green strain values in the circumferential ( E θ θ , max ) and longitudinal ( E LL , max ) directions under an equibiaxial tension of 120 N/m were also compared. The strain energy function fit all of the individual specimens well with an average R 2 of 0.95±0.02 and 0.90±0.02 (mean±SEM) for the AA and AAA groups, respectively. The average E θ θ , max at 200 N/m equibiaxial tension was found to be significantly smaller for AAAs as compared to AAs (0.07±0.01 versus 0.13±0.03, respectively; p 0.01 ). There was also a pronounced increase in the circumferential stiffness for AAA tissue as compared to AA tissue, indicating a larger degree of anisotropy for this tissue as compared to age-matched AA tissue. We also observed that the four-parameter Fung-elastic model was not able to fit the AAA tissue mechanical response using physically realistic material parameter values. It was concluded that aneurysmal degeneration of the abdominal aorta is associated with an increase in mechanical anisotropy, with preferential stiffening in the circumferential direction.
---
paper_title: A continuum description of the damage process in the arterial wall of abdominal aortic aneurysms

paper_content:

In the present work, we develop a three-dimensional isotropic finite-strain damage model for abdominal aortic aneurysm (AAA) wall that considers both the characteristic softening of the material caused by damage and the spatial variation of the material properties. A strain energy function is formulated that accounts for a hyperelastic, slightly compressible, isotropic material behavior during the elastic phase, ::: whereas the damage process only contributes to the material response when the elastic limit of the AAA wall is exceeded. Material and damage parameters are obtained by fitting the strain energy function to the experimental data obtained by uniaxial tensile tests of freshly harvested AAA wall samples. The damage model extends the validity of the material law to a strain range of up to 50%. Purely elastic material ::: laws for AAA wall are only valid for a strain range of up to 17%. In a series of finite element simulations of patient-specific AAAs, serving as numerical examples, we investigate the applicability of the damage model. The use of the damage model does not yield a more distinct identification of ruptureprone AAAs than other computational-based risk indices. However, the benefit of the finite-strain damage ::: model is the potential capability to trigger growth and remodeling processes in mechanobiological models.
---
paper_title: REVIEW OF THE ENERGY LIMITERS APPROACH TO MODELING FAILURE OF RUBBER

paper_content:

ABSTRACT Nonlinear theories of elasticity describe rubber deformation but not failure; however, in reality, rubbers do fail. In the present work, we review a new approach of energy limiters that al...
---
paper_title: Cavitation instability as a trigger of aneurysm rupture

paper_content:

Aneurysm formation and growth is accompanied by microstructural alterations in the arterial wall. Particularly, the loss of elastin may lead to tissue disintegration and appearance of voids or cavities at the micron scale. Unstable growth and coalescence of voids may be a predecessor and trigger for the onset of macroscopic cracks. In the present work, we analyze the instability of membrane (2D) and bulk (3D) voids under hydrostatic tension by using two experimentally calibrated constitutive models of abdominal aortic aneurysm enhanced with energy limiters. The limiters provide the saturation value for the strain energy, which indicates the maximum energy that can be stored and dissipated by an infinitesimal material volume. We find that the unstable growth of voids can start when the critical stress is considerably less than the aneurysm strength. Moreover, this critical stress may even approach the arterial wall stress in the physiological range. This finding suggests that cavitation instability can be a rational indicator of the aneurysm rupture.
---
paper_title: Computational modelling of the stress-softening phenomenon of rubber-like materials under cyclic loading

paper_content:

Abstract When a rubber specimen is subjected to cyclic loading, not only non-linear behaviour but also damage-induced stress-softening phenomena (the Mullins effect) have been observed. Applications of a continuum damage mechanics model and Ogden and Roxburgh's pseudo-elastic model to describe the Mullins effect in elastomers have been considered. Both models together with Gao's elastic law were implemented to describe the mechanical behaviour of rubber-like materials including the stress-softening phenomenon. Two sets of experimental data (a simple tension test and a simple tension and pure shear test) are used to validate the constitutive models. Model parameters are estimated via an inverse technique. Computational results show that both constitutive models together with Gao's elastic law can describe the typical Mullins effect. From engineering point of view, the pseudo-elastic model has the advantages that (i) the model is simple and practical, since it considers that the stress-softening function is only activated on unloading or reloading paths, (ii) the model with a slight modification of the damage variable is very stable in finite element calculations, and (iii) the numerical results agree very well with experimental data in both simple tension and pure shear deformation. Two applications illustrate the capability of combining the pseudo-elastic model with Gao's elastic law in describing the Mullins effect. It is emphasized that both models are applicable to multiaxial states of stress and strain because both models are energy-based and not strain-based.
---
paper_title: A new Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models

paper_content:

In this paper we develop a new constitutive law for the description of the (passive) mechanical response of arterial tissue. The artery is modeled as a thick-walled nonlinearly elastic circular cylindrical tube consisting of two layers corresponding to the media and adventitia (the solid mechanically relevant layers in healthy tissue). Each layer is treated as a fiber-reinforced material with the fibers corresponding to the collagenous component of the material and symmetrically disposed with respect to the cylinder axis. The resulting constitutive law is orthotropic in each layer. Fiber orientations obtained from a statistical analysis of histological sections from each arterial layer are used. A specific form of the law, which requires only three material parameters for each layer, is used to study the response of an artery under combined axial extension, inflation and torsion. The characteristic and very important residual stress in an artery in vitro is accounted for by assuming that the natural (unstressed and unstrained) configuration of the material corresponds to an open sector of a tube, which is then closed by an initial bending to form a load-free, but stressed, circular cylindrical configuration prior to application of the extension, inflation and torsion. The effect of residual stress on the stress distribution through the deformed arterial wall in the physiological state is examined.
---
paper_title: Anisotropic microsphere-based approach to damage in soft fibered tissue

paper_content:

An anisotropic damage model for soft fibered tissue is presented in this paper, using a multi-scale scheme and focusing on the directionally dependent behavior of these materials. For this purpose, a micro-structural or, more precisely, a microsphere-based approach is used to model the contribution of the fibers. The link between micro-structural contribution and macroscopic response is achieved by means of computational homogenization, involving numerical integration over the surface of the unit sphere. In order to deal with the distribution of the fibrils within the fiber, a von Mises probability function is incorporated, and the mechanical (phenomenological) behavior of the fibrils is defined by an exponential-type model. We will restrict ourselves to affine deformations of the network, neglecting any cross-link between fibrils and sliding between fibers and the surrounding ground matrix. Damage in the fiber bundles is introduced through a thermodynamic formulation, which is directly included in the hyperelastic model. When the fibers are stretched far from their natural state, they become damaged. The damage increases gradually due to the progressive failure of the fibrils that make up such a structure. This model has been implemented in a finite element code, and different boundary value problems are solved and discussed herein in order to test the model features. Finally, a clinical application with the material behavior obtained from actual experimental data is also presented.
---
paper_title: An anisotropic inelastic constitutive model to describe stress softening and permanent deformation in arterial tissue.

paper_content:

Abstract Inelastic phenomena such as softening and unrecoverable inelastic strains induced by loading have been observed experimentally in soft tissues such as arteries. These phenomena need to be accounted for in constitutive models of arterial tissue so that computational models can accurately predict the outcomes of interventional procedures such as balloon angioplasty and stenting that involve non-physiological loading of the tissue. In this study, a novel constitutive model is described that accounts for inelastic effects such as Mullins-type softening and permanent set in a fibre reinforced tissue. The evolution of inelasticity is governed by a set of internal variables. Softening is introduced through a typical continuum damage mechanics approach, while the inelastic residual strains are introduced through an additive split in the stress tensor. Numerical simulations of aorta and carotid arterial tissue subjected to uniaxial testing in the longitudinal, circumferential and axial directions are used to demonstrate the model’s ability to reproduce the anisotropic inelastic behaviour of the tissue. Material parameters derived from best-fits to experimental data are provided to describe these inelastic effects for both aortic and carotid tissue.
---
paper_title: An anisotropic pseudo-elastic approach for modelling Mullins effect in fibrous biological materials

paper_content:

Abstract This paper presents a fully three-dimensional finite-strain formulation with a simple pseudo-elastic anisotropic model to reproduce the softening behavior that appears in soft biological fibred tissues (Mullins effect). This is an anisotropic extension of the pseudo-elastic model of Ogden and Roxburgh which was earlier proposed to describe the softening behavior of rubbers under large deformation. The model makes use of the concept of internal variables which provides a very general description of materials involving irreversible effects. Numerical simulations of uniaxial tests of several soft tissues are used to show the performance of the model in realistic situations. The computed results indicate that the proposed model is able to describe with enough accuracy the softening behavior of real soft biological tissues.
---
paper_title: A constitutive model for the Mullins effect with permanent set in particle-reinforced rubber

paper_content:

Stress softening during initial loading cycles, known as the Mullins effect, and the residual strain upon unloading are not accounted for when the mechanical properties of rubber are represented in terms of a strain-energy function, i.e. if the material is modelled as hyperelastic. In this paper we first describe some experimental results that illustrate stress softening in particle-reinforced rubber together with associated residual strain effects. In particular, the results show how the stress softening and residual strain change with the magnitude of the applied strain. Then, on the basis of these data a constitutive model is derived to describe this behaviour. The theory of pseudo-elasticity is used for this model, the basis of which is the inclusion of two variables in the energy function in order separately to capture the stress softening and residual strain effects. The dissipation of energy, i.e. the difference between the energy input during loading and the energy returned on unloading is also accounted for in the model by the use of a dissipation function, which evolves with the deformation history.
---
paper_title: Modeling failure of soft anisotropic materials with application to arteries

paper_content:

Abstract The arterial wall is a composite where the preferred orientation of collagen fibers induces anisotropy. Though the hyperelastic theories of fiber-reinforced composites reached a high level of sophistication and showed a reasonable correspondence with the available experimental data they are short of the failure description. Following the tradition of strength of materials the failure criteria are usually separated from stress analysis. In the present work we incorporate a failure description in the hyperelastic models of soft anisotropic materials by introducing energy limiters in the strain energy functions. The limiters provide the saturation value for the strain energy which indicates the maximum energy that can be stored and dissipated by an infinitesimal material volume. By using some popular constitutive models enhanced with the energy limiters we analyze rupture of a sheet of arterial material under the plane stress state varying from the uniaxial to equal biaxial tension. We calculate the local failure criteria including the maximum principal stress, the maximum principal stretch, the von Mises stress, and the strain energy at the moment of the sheet rupture. We find that the local failure criterion in the form of the critical strain energy is the most robust among the considered ones. We also find that the tensile strength–the maximum principal stress–that is usually obtained in uniaxial tension tests might not be appropriate as a failure indicator in the cases of the developed biaxiality of the stress–strain state.
---
paper_title: Statistical approach for a continuum description of damage evolution in soft collagenous tissues

paper_content:

Abstract We propose a statistical approach to describe microscopic damage evolution in soft collagenous tissues such as arterial walls. The damage model extends a framework published by Balzani et al. (2012), Comput. Methods Appl. Mech. Engrg., 213–216:139–151, by postulating specific damage functions that result from the fibers’ microstructure. Statistical distributions of three different microscopic quantities such as proteoglycan orientation, fibril length parameters and ultimate proteoglycan stretch are considered. The resulting stress–stretch response is compared with experimental data obtained from uniaxial tension tests given in the literature. In particular, the individual statistical distributions are analyzed in regard to their ability to capture the distinct softening hysteresis observed when subjecting soft tissues to cyclic loading in the supra-physiological domain. Details regarding the algorithmic implementation are provided, and the applicability of the model within a finite element framework is shown by simulating the overexpansion of simplified atherosclerotic arteries.
---
paper_title: Determination and modeling of the inelasticity over the length of the porcine carotid artery.

paper_content:

The study of the mechanical properties of swine carotids has clinical relevance because it is important for the appropriate design of intravascular devices in the animal trial phases. The inelastic properties of porcine carotid tissue were investigated. Experimental uniaxial cyclic tests were performed along the longitudinal and circumferential directions of vessels. The work focused on the determination, comparison, and constitutive modeling of the softening properties and residual stretch set of the swine carotid artery over long stretches and stress levels in both proximal and distal regions. It was observed that the residual strain depends on the maximum stretch in the previous load cycle. The strain was higher for distal than for proximal samples and for circumferential than for longitudinal samples. In addition, a pseudoelastic model was used to reproduce the residual stretch and softening behavior of the carotid artery. The model presented a good approximation of the experimental data. The results demonstrate that the final results in animal trial studies could be affected by the location studied along the length of the porcine carotid.
---
paper_title: Discontinuous and continuous damage evolution in Ogden-type large-strain elastic materials

paper_content:

The paper presents an application of continuum damage mechanics to large-strain Ogden-type elastic materials. A Eulerian setting of elasticity with isotropic damage-caused softening effect is discussed exclusively in terms of the Finger tensor as a strain measure. The local damage accumulation is related to two phenomenological variables, the maximum value and the arclength of effective free energy attained during a typical loading process. The frame is specified for a rigorously decoupled volumetric-isochoric response based on a multiplicative split of the Finger tensor into spherical and unimodular parts. In this context, the damage is related to the isochoric part of the deformation only, which is assumed to be governed by an Ogden-type effective free-energy function formulated in terms of the eigenvalues of the unimodular part of the Finger tensor. A constitutive algorithm for the computation of the nominal stresses and nominal tangent moduli for plane problems is developed. Finally, some fundamental finite-element simulations for rubber-like materials are presented.
---
paper_title: Biomechanical behavior and histological organization of the three-layered passive esophagus as a function of topography

paper_content:

The zero-stress state of the mucosa-submucosa and two muscle esophageal layers has been delineated, but their multi-axial response has not, because muscle dissection may not leave tubular specimens intact for inflation/extension testing. The histomechanical behavior of the three-layered porcine esophagus was investigated in this study, through light microscopic examination and uniaxial tension, with two-dimensional strain measurement in pairs of orthogonally oriented specimens. The two-dimensional Fung-type strain–energy function described suitably the pseudo-elastic tissue response, affording faithful simulations to our data. Differences in the scleroprotein content and configuration were identified as a function of layer, topography, and orientation, substantiating the macromechanical differences found. In view of the failure and optimized material parameters, the mucosa-submucosa was stronger and stiffer than muscle, associating it with a higher collagen content. A notable topographical distribution wa...
---
paper_title: Determination of layer-specific mechanical properties of human coronary arteries with nonatherosclerotic intimal thickening and related constitutive modeling

paper_content:

At autopsy, 13 nonstenotic human left anterior descending coronary arteries [71.5 +/- 7.3 (mean +/- SD) yr old] were harvested, and related anamnesis was documented. Preconditioned prepared strips (n = 78) of segments from the midregion of the left anterior descending coronary artery from the individual layers in axial and circumferential directions were subjected to cyclic quasi-static uniaxial tension tests, and ultimate tensile stresses and stretches were documented. The ratio of outer diameter to total wall thickness was 0.189 +/- 0.014; ratios of adventitia, media, and intima thickness to total wall thickness were 0.4 +/- 0.03, 0.36 +/- 0.03, and 0.27 +/- 0.02, respectively; axial in situ stretch of 1.044 +/- 0.06 decreased with age. Stress-stretch responses for the individual tissues showed pronounced mechanical heterogeneity. The intima is the stiffest layer over the whole deformation domain, whereas the media in the longitudinal direction is the softest. All specimens exhibited small hysteresis and anisotropic and strong nonlinear behavior in both loading directions. The media and intima showed similar ultimate tensile stresses, which are on average three times smaller than ultimate tensile stresses in the adventitia (1,430 +/- 604 kPa circumferential and 1,300 +/- 692 kPa longitudinal). The ultimate tensile stretches are similar for all tissue layers. A recently proposed constitutive model was extended and used to represent the deformation behavior for each tissue type over the entire loading range. The study showed the need to model nonstenotic human coronary arteries with nonatherosclerotic intimal thickening as a composite structure composed of three solid mechanically relevant layers with different mechanical properties. The intima showed significant thickness, load-bearing capacity, and mechanical strength compared with the media and adventitia.
---
paper_title: Comparison between mechanical properties of human saphenous vein and umbilical vein

paper_content:

BackgroundAs a main cause of mortality in developed countries, Coronary Artery Disease (CAD) is known as silent killer with a considerable cost to be dedicated for its treatment. Coronary Artery Bypass Graft (CABG) is a common remedy for CAD for which different blood vessels are used as a detour. There is a lack of knowledge about mechanical properties of human blood vessels used for CABG, and while these properties have a great impact on long-term patency of a CABG. Thus, studying these properties, especially those of human umbilical veins which have not been considered yet, looks utterly necessary.MethodsUmbilical vein, as well as human Saphenous vein, are respectively obtained after cesarean and CABG. First, histological tests were performed to investigate different fiber contents of the samples. Having prepared samples carefully, force-displacement results of samples were rendered to real stress–strain measurements and then a fourth-order polynomial was used to prove the non-linear behavior of these two vessels.ResultsResults were analyzed in two directions, i.e. circumferentially and longitudinally, which then were compared with each other. The comparison between stiffness and elasticity of these veins showed that Saphenous vein’s stiffness is much higher than that of umbilical vein and also, it is less stretchable. Furthermore, for both vessels, longitudinal stiffness was higher than that of circumferential and in stark contrast, stretch ratio in circumferential direction came much higher than longitudinal orientation.ConclusionBlood pressure is very high in the region of aorta, so there should be a stiff blood vessel in this area and previous investigations showed that stiffer vessels would have a better influence on the flow of bypass. To this end, the current study has made an attempt to compare these two blood vessels’ stiffness, finding that Saphenous vein is stiffer than umbilical vein which is somehow as stiff as rat aortic vessels. As blood vessel’s stiffness is directly related to elastin and mainly collagen content, results showed the lower amount of these two contents in umbilical vein regarding Saphenous vein.
---
paper_title: Mechanical characterization and constitutive modelling of the damage process in rectus sheath

paper_content:

Abstract The aim of this study is to characterize and model the damage process in the anterior rectus abdominal aponeurosis (anterior rectus sheath) undergoing finite deformations. The resistance of the anterolateral abdominal aponeuroses is important when planning the surgical repair of incisional hernias, among other medical procedures. Previous experiments in prolapsed vaginal tissue revealed that a softening process occurs before tissue rupture. This nonlinear damage behaviour requires a continuum damage theory commonly used to describe the softening behaviour of soft tissues under large deformations. The structural model presented here was built within the framework of non-linear continuum mechanics. Tissue damage was simulated considering different damage behaviours for the matrix and the collagen fibres. The model parameters were fit to the experimental data obtained from anterior rectus sheath samples undergoing finite deformations in uniaxial tension tests. The tests were carried out with samples cut along the direction of the collagen fibres, and transversal to the fibres. Longitudinal and transverse mechanical properties of human anterior rectus sheath are significantly different. The damage model was able to predict the stress–strain behaviour and the damage process accurately. The error estimations pointed to an excellent agreement between experimental results and model fittings. For all the fitted data, the normalized RMS error e presented very low values and the coefficient of determination R 2 was close to 1. The present work constitutes the first attempt (as far as the authors know) to present a damage model for the human rectus sheath.
---
paper_title: Progressive structural and biomechanical changes in elastin degraded aorta

paper_content:

Aortic aneurysm is an important clinical condition characterized by common structural changes such as the degradation of elastin, loss of smooth muscle cells, and increased deposition of fibrillary collagen. With the goal of investigating the relationship between the mechanical behavior and the structural/biochemical composition of an artery, this study used a simple chemical degradation model of aneurysm and investigated the progressive changes in mechanical properties. Porcine thoracic aortas were digested in a mild solution of purified elastase (5 U/mL) for 6, 12, 24, 48, and 96 h. Initial size measurements show that disruption of the elastin structure leads to increased artery dilation in the absence of periodic loading. The mechanical properties of the digested arteries, measured with a biaxial tensile testing device, progress through four distinct stages termed (1) initial-softening, (2) elastomer-like, (3) extensible-but-stiff, and (4) collagen-scaffold-like. While stages 1, 3, and 4 are expected as a result of elastin degradation, the S-shaped stress versus strain behavior of the aorta resulting from enzyme digestion has not been reported previously. Our results suggest that gradual changes in the structure of elastin in the artery can lead to a progression through different mechanical properties and thus reveal the potential existence of an important transition stage that could contribute to artery dilation during aneurysm formation.
---
paper_title: A pseudo–elastic model for the Mullins effect in filled rubber

paper_content:

When a rubber test piece is loaded in simple tension from its virgin state, unloaded and then reloaded, the stress required on reloading is less than that on the initial loading for stretches up to the maximum stretch achieved on the initial loading. This stress softening phenomenon is referred to as the Mullins effect . In this paper a simple phenomenological model is proposed to account for the Mullins effect observed in filled rubber elastomers. The model is based on the theory of incompressible isotropic elasticity amended by the incorporation of a single continuous parameter, interpreted as a damage parameter. This parameter controls the material properties in the sense that it enables the material response to be governed by a strain–energy function on unloading and subsequent submaximal loading different from that on the primary (initial) loading path from the virgin state. For this reason the model is referred to as pseudo-elastic } and a primary loading-unloading cycle involves energy dissipation. The dissipation is measured by a damage function which depends only on the damage parameter and on the point of the primary loading path from which unloading begins. A specific form of this function with two adjustable material constants, coupled with standard forms of the (incompressible, isotropic) strain–energy function, is used to illustrate the qualitative features of the Mullins effect in both simple tension and pure shear. For simple tension the model is then specialized further in order to fit Mullins effect data. It is emphasized that the model developed here is applicable to multiaxial states of stress and strain, not just the specific uniaxial tests highlighted.
---
paper_title: Layer-specific damage experiments and modeling of human thoracic and abdominal aortas with non-atherosclerotic intimal thickening.

paper_content:

Many treatments for cardiovascular diseases include an endovascular insertion of stents or stent grafts into arteries, a procedure which may cause high tissue stresses and even damage in the arterial wall. In order to study such problems by using finite element methods, both appropriate constitutive models and experimental data on human tissue samples are required. Layer-specific experimental data for human tissue tested up to the supra-physiological loading range are rare in the literature. In this study, intact and layer-separated experimental data from uniaxial extension tests are presented for human thoracic and abdominal aortas with non-atherosclerotic intimal thickening undergoing supra-physiological loading. A novel pseudo-elastic damage model, proposed to describe discontinuous softening in aortic arterial tissues, is fit to the obtained experimental data. Fitting of the model with and without consideration of damage accumulation in the non-collagenous matrix material reveals that tissue damage is primarily related to the collagen fiber fabric. By employing the fit model, the effect of aortic tissue pre-conditioning on the material parameters from the resulting data fits is evaluated. Histological examination of the collagen fibers under different applied stretches is used to gain more insights into the structural changes of the tissue under supra-physiological loading.
---
paper_title: Hyperelastic modelling of arterial layers with distributed collagen fibre orientations

paper_content:

Constitutive relations are fundamental to the solution of problems in continuum mechanics, and are required in the study of, for example, mechanically dominated clinical interventions involving soft biological tissues. Structural continuum constitutive models of arterial layers integrate information about the tissue morphology and therefore allow investigation of the interrelation between structure and function in response to mechanical loading. Collagen fibres are key ingredients in the structure of arteries. In the media (the middle layer of the artery wall) they are arranged in two helically distributed families with a small pitch and very little dispersion in their orientation (i.e. they are aligned quite close to the circumferential direction). By contrast, in the adventitial and intimal layers, the orientation of the collagen fibres is dispersed, as shown by polarized light microscopy of stained arterial tissue. As a result, continuum models that do not account for the dispersion are not able to capture accurately the stress–strain response of these layers. The purpose of this paper, therefore, is to develop a structural continuum framework that is able to represent the dispersion of the collagen fibre orientation. This then allows the development of a new hyperelastic free-energy function that is particularly suited for representing the anisotropic elastic properties of adventitial and intimal layers of arterial walls, and is a generalization of the fibre-reinforced structural model introduced by Holzapfel & Gasser (Holzapfel & Gasser 2001 Comput. Meth. Appl. Mech. Eng. 190, 4379–4403) and Holzapfel et al. (Holzapfel et al. 2000 J. Elast. 61, 1–48). The model incorporates an additional scalar structure parameter that characterizes the dispersed collagen orientation. An efficient finite element implementation of the model is then presented and numerical examples show that the dispersion of the orientation of collagen fibres in the adventitia of human iliac arteries has a significant effect on their mechanical response.
---
paper_title: Prediction of arterial failure based on a microstructural bi-layer fiber-matrix model with softening.

paper_content:

Abstract Two approaches to predict failure of soft tissue are available. The first is based on a pointwise criticality condition, e.g. von Mises maximum stress, which is restrictive because only local state of deformation is considered to be critical and the failure criterion is separated from stress analysis. The second is based on damage mechanics where internal (unobservable) variables are introduced which make the experimental calibration of the theory complex. As an alternative to the local failure criteria and damage mechanics we present a softening hyperelasticity approach, where the constitutive description of soft tissue is enhanced with strain softening, which is controlled by material constants. This approach is attractive because the new material constants can be readily calibrated in experiments on the one hand and the failure criteria are global on the other hand. We illustrate the efficiency of the softening hyperelasticity approach on the problem of prediction of arterial failure. For this purpose, we enhance a bi-layer fiber–matrix microstructural arterial model with softening and analyze the arterial failure under internal pressure. We show that the overall arterial strength is (a) dominated by the media layer, (b) controlled by microfibers and (c) increased by residual stresses.
---
paper_title: Human thoracic and abdominal aortic aneurysmal tissues: Damage experiments, statistical analysis and constitutive modeling.

paper_content:

Abstract Development of aortic aneurysms includes significant morphological changes within the tissue: collagen content increases, elastin content reduces and smooth muscle cells degenerate. We seek to quantify the impact of these changes on the passive mechanical response of aneurysms in the supra-physiological loading range via mechanical testing and constitutive modeling. We perform uniaxial extension tests on circumferentially and axially oriented strips from five thoracic (65.6 years±13.4, mean±SD) and eight abdominal (63.9 years±11.4) aortic fusiform aneurysms to investigate both continuous and discontinuous softening during supra-physiological loading. We determine the significance of the differences between the fitted model parameters: diseased thoracic versus abdominal tissues, and healthy ( Weisbecker et al., J. Mech. Behav. Biomed. Mater. 12, 93–106, 2012 ) versus diseased tissues. We also test correlations among these parameters and age, Body Mass Index (BMI) and preoperative aneurysm diameter, and investigate histological cuts. Tissue response is anisotropic for all tests and the anisotropic pseudo-elastic damage model fits the data well for both primary loading and discontinuous softening which we interpret as damage. We found statistically relevant differences between model parameters fitted to diseased thoracic versus abdominal tissues, as well as between those fitted to healthy versus diseased tissues. Only BMI correlated with fitted model parameters in abdominal aortic aneurysmal tissues.
---
paper_title: On the Mullins effect and hysteresis of fibered biological materials: A comparison between continuous and discontinuous damage models

paper_content:

Deformation induced softening is an inelastic phenomenon frequently accompanying mechanical behavior of soft biological tissues. This paper presents and compare continuous and discontinuous damage approaches to model softening effects in fibered materials such as soft biological tissues. The structural model is formulated using the concept of internal variables that provides a very general description of materials involving irreversible effects. We consider the internal variables associated to damage to correspond to separated contributions of the matrix and the fibers. Local damage accumulation is related to two phenomenological variables, the maximum value and the arclength of the effective free energy attained during the loading process, respectively. A local multiplicative decomposition of the deformation gradient into volume-preserving and dilatational parts is used that permits to model the incompressibility property of most types of soft biological tissues. In this context, damage is related only to the isochoric part of the deformation. Finally, simulations of biaxial and uniaxial tests in two directions are used to compare the performance of both models. Numerical simulations indicate that only a mixed model that consider both, continuous and discontinuous, damage models is able to capture the softening phenomenon of soft biological tissues.
---
paper_title: Finite element modeling of cerebral angioplasty using a structural multi-mechanism anisotropic damage model

paper_content:

SUMMARY ::: Arterial progressive damage, softening and non-recoverable deformation are typical effects of percutaneous transluminal angioplasty (PTA), induced mechanically by the overstretching and widening of arterial walls during balloon dilatation. In this paper, we model this ‘controlled vessel injury’ in cerebral balloon angioplasty by extending a recently developed structural multi-mechanism damage model for cerebral arterial tissue in early stage aneurysms. The current constitutive model focuses on arterial mechanical damage and can characterize the wall anisotropy and subfailure damage of the elastin, ground matrix and collagen, represented by the recruitment of collagen fibers, gradual degeneration, failure of arterial components and changes in the tissue mechanical properties. Cyclic uniaxial response of homogeneous artery models is used to validate the numerical implementation of the constitutive model. Cerebral arteries are modeled as a multi-layer heterogeneous wall model composed of the internal elastic lamina, media and adventitia. Cerebral angioplasty is simulated by contact between the heterogeneous artery model and a simplified balloon model. The states of arterial deformation, tissue damage and wall stress during angioplasty are analyzed, which reproduces the qualitative features of cerebral PTA observed experimentally. The constitutive model and computational methods are demonstrated to be robust in simulating complex tissue response caused by the mechanical intervention of angioplasty, and are applied in more realistic artery-plaque-balloon-stent models. Copyright © 2012 John Wiley & Sons, Ltd.
---
paper_title: A Structural Multi-Mechanism Damage Model for Cerebral Arterial Tissue

paper_content:

Early stage cerebral aneurysms are characterized by the disruption of the internal elastic lamina. The cause of this breakdown is still not understood, but it has been conjectured to be due to fatigue failure and/or by a breakdown in homeostatic mechanisms in the wall arising from some aspect of the local hemodynamics and wall tension. We propose to model this disruption using a structural damage model. It is built on a previously introduced nonlinear, inelastic multi-mechanism model for cerebral arteries (2005, "An Inelastic Multi-Mechanism Constitutive Equation for Cerebral Arterial Tissue, " Biomech. Model. Mechanobiol., 4(4), pp. 235-248), as well as a recent generalization to include the wall anisotropy (2009, "A Structural Multi-Mechanism Constitutive Equation for Cerebral Arterial Tissue, " Int. J. Solids Struct., 46(14-15), pp. 2920-2928). The current model includes subfailure damage of the elastin, represented by changes in the tissue mechanical properties and unloaded reference length. A structural model is used to characterize the gradual degradation, failure of elastin, and recruitment of anisotropic collagen fibers. The collagen fibers are arranged in two helically oriented families with dispersion in their orientation. Available inelastic experimental data for cerebral arteries are used to evaluate the constitutive model. It is then implemented in a commercial finite element analysis package and validated using analytical solutions with representative values,for cerebral arterial tissue.
---
paper_title: Modeling of anisotropic softening phenomena: Application to soft biological tissues

paper_content:

A thermodynamically consistent dissipative model is proposed to describe softening phenomena in anisotropic materials. The model is based on a generalized polyconvex anisotropic strain energy function represented by a series. Anisotropic softening is considered by evolution of internal variables governing the anisotropic properties of the material. Accordingly, evolution equations are formulated and anisotropic conditions for the onset of softening are defined. In numerical examples, the model is applied to simulate the preconditioning behavior of soft biological tissues subjected to cyclic loading experiments. The results suggest that the general characteristics of preconditioning with different upper load limits are well captured including hysteresis and residual deformations. A model for the Mullins effect is obtained as a special case and shows very good agreement with experimental data on mouse skin.
---
paper_title: On a Class of Continuum Damage Mechanics Theories

paper_content:

A thermodynamically consistent continuum damage mechanics formula tion in the stress space is outlined. Damage is reflected in the fourth-order material com pliance tensor involving a damage parameter whose increment is obtained from the consis tency condition of the damage surface. A new damage function and an alternate damage response tensor in mode I are formulated by means of which a smooth transition from the cleavage damage mode to the compressive damage mode is achieved and nonlinear re sponses in the lateral directions are predicted. Furthermore, the permanent deformation in damage mode I is predicted by using a simple evolutionary equation. The tangent compli ance tensor is also given in terms of the response tensors and the first variation of the damage function with respect to state variables. The model is illustrated for different stress paths in mortar and concrete.
---
paper_title: On modelling damage process in vaginal tissue.

paper_content:

Abstract The goal of this study was to characterize and model the damage process in prolapsed vaginal tissue undergoing finite deformations. Experiments in prolapsed vaginal tissue revealed that a softening process occurs before tissue's rupture. This nonlinear damage behavior requires a continuum damage theory commonly used to describe the softening behavior of soft tissues under large deformations. The structural model here presented was built within the framework of nonlinear continuum mechanics. Tissue damage was simulated considering different damage behaviors for the matrix and the fibers. The model parameters were fit to the experimental data obtained from prolapsed vaginal tissue undergoing finite deformations in uniaxial tension tests. The tests were developed with samples cut along the longitudinal axis of the vagina. The damage model was able to predict the stress–strain behavior and the damage process accurately. The error estimations pointed to an excellent agreement between experimental results and model fittings. For all the fitted data, the normalized RMS error ɛ presented very low values and the coefficient of determination R 2 was close to 1.
---
paper_title: A gradient-enhanced large-deformation continuum damage model for fibre-reinforced materials

paper_content:

A non-local gradient-based damage formulation within a geometrically non-linear setting is presented. The hyperelastic constitutive response at local material point level is governed by a strain energy which is additively composed of an isotropic matrix and of an anisotropic fibre-reinforced material, respectively. The inelastic constitutive response is governed by a scalar [1–d]-type damage formulation, where only the anisotropic elastic part is assumed to be affected by the damage. Following the concept in Dimitrijevic and Hackl [28], the local free energy function is enhanced by a gradient-term. This term essentially contains the gradient of the non-local damage variable which, itself, is introduced as an additional independent variable. In order to guarantee the equivalence between the local and non-local damage variable, a penalisation term is incorporated within the free energy function. Based on the principle of minimum total potential energy, a coupled system of Euler–Lagrange equations, i.e., the balance of linear momentum and the balance of the non-local damage field, is obtained and solved in weak form. The resulting coupled, highly non-linear system of equations is symmetric and can conveniently be solved by a standard incremental-iterative Newton–Raphson-type solution scheme. Several three-dimensional displacement- and force-driven boundary value problems—partially motivated by biomechanical application—highlight the mesh-objective characteristics and constitutive properties of the model and illustratively underline the capabilities of the formulation proposed.
---
paper_title: Structural damage models for fibrous biological soft tissues

paper_content:

This paper presents a comparison between deterministic and stochastically based three-dimensional finite-strain damage models for fibrous biological soft tissues, accounting for separate contributions on damage for the matrix and the fibers. Both models are compared in terms of their numerical performance and qualitative predictions under different loading conditions. Continuum damage mechanics is used to describe the softening behavior of soft tissues under large deformation, making use of the concept of internal variables which provides a very general description of materials involving irreversible effects. In the stochastic model, statistical aspects related to the distribution of fiber length lead to the strain-driven damage model for the fibrous part. Simulations of a uniaxial test, a hollowed plate under biaxial displacement control, and a 3D simulation of a coronary artery undergoing balloon angioplasty are used to compare the performance of both models. Numerical simulations indicate that both models provide similar predictions of damage.
---
paper_title: Finite element implementation of a stochastic three dimensional finite-strain damage model for fibrous soft tissue

paper_content:

Abstract A fully three dimensional finite-strain damage model for fibrous soft tissue is here presented. The model assumes uncoupled contributions for the matrix and collagen fibers, and uncoupled bulk and deviatoric response over any range of deformations. A simple isotropic damage mechanism within the framework of continuum damage mechanics has been used to describe the softening behavior under deformation for the matrix. On the other hand, statistical aspects related to the length distribution of the reinforcing fibers lead to a damage model for the reinforcing material. As a result, a general theoretical framework for constitutive modeling of biological soft tissue with continuum damage is obtained. The formulation has been implemented in a finite element framework and the capabilities of the model tested in three computer simulations of inhomogeneous boundary value problems. The results show the quadratic rate of convergence of the implementation and the ability to describe typical stress–strain behavior of fiber reinforced soft tissues.
---
paper_title: A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials

paper_content:

Aconstitutive model is proposed for the deformation of rubber materials which is shown to represent successfully the response of these materials in uniaxial extension, biaxial extension, uniaxial compression, plane strain compression and pure shear. The developed constitutive relation is based on an eight chain representation of the underlying macromolecular network structure of the rubber and the non-Gaussian behavior of the individual chains in the proposed network. The eight chain model accurately captures the cooperative nature of network deformation while requiring only two material parameters, an initial modulus and a limiting chain extensibility. Since these two parameters are mechanistically linked to the physics of molecular chain orientation involved in the deformation of rubber, the proposed model represents a simple and accurate constitutive model of rubber deformation. The chain extension in this network model reduces to a function of the root-mean-square of the principal applied stretches as a result of effectively sampling eight orientations of principal stretch space. The results of the proposed eight chain model as well as those of several prominent models are compared with experimental data of Treloar (1944, Trans. Faraday Soc. 40, 59) illustrating the superiority, simplicity and predictive ability of the proposed model. Additionally, a new set of experiments which captures the state of deformation dependence of rubber is described and conducted on three rubber materials. The eight chain model is found to model and predict accurately the behavior of the three tested materials further confirming its superiority and effectiveness over earlier models.
---
paper_title: A stochastic-structurally based three dimensional finite-strain damage model for fibrous soft tissue

paper_content:

A fully three dimensional finite-strain damage model for fibrous soft tissue is developed. The model assumes uncoupled contributions for the matrix and collagen fibers, and uncoupled bulk and deviatoric response over any range of deformations. A simple isotropic damage mechanism within the framework of continuum damage mechanics has been used to describe the softening behavior under deformation for the matrix. On the other hand, statistical aspects related to the length distribution of the reinforcing fibers lead to a damage model for the reinforcing material. As a result, a general theoretical framework for constitutive modeling of biological soft tissue with continuum damage is obtained. A theoretical example consisting of a biaxial test of a soft tissue reinforced with two families of collagen fibers has been considered to demonstrate the capabilities of the proposed model and to study the sensitivity to changes in the statistical parameters associated with the reinforcing material. Also, a preliminary numerical example is included to demonstrate the model on a inhomogeneous boundary value problem. Results show that the model is able to capture the typical stress–strain behavior observed in fibrous soft tissue and seems to confirm the soundness of the proposed formulation.
---
paper_title: Cumulative microdamage model to describe the hysteresis of living tissue

paper_content:

A hysteresis model based on a cumulative microfailure criterion has been proposed to describe the stress-stretch behavior in response to external mechanical loading in strip biaxial deformation of living animal tissue (cat's mesentery). The strain energy density representation used in the hysteresis model appears to be adequately versatile to describe the tissue response over a wide range of stretch. The model is thus useful in interpreting the influence of the material parameters on the birefringence of the tissue as well as on the magnitude and shape of the stress response. Photomicrographs of the tissue have been correlated with the cumulative microfailure model.
---
paper_title: Structural damage models for fibrous biological soft tissues

paper_content:

This paper presents a comparison between deterministic and stochastically based three-dimensional finite-strain damage models for fibrous biological soft tissues, accounting for separate contributions on damage for the matrix and the fibers. Both models are compared in terms of their numerical performance and qualitative predictions under different loading conditions. Continuum damage mechanics is used to describe the softening behavior of soft tissues under large deformation, making use of the concept of internal variables which provides a very general description of materials involving irreversible effects. In the stochastic model, statistical aspects related to the distribution of fiber length lead to the strain-driven damage model for the fibrous part. Simulations of a uniaxial test, a hollowed plate under biaxial displacement control, and a 3D simulation of a coronary artery undergoing balloon angioplasty are used to compare the performance of both models. Numerical simulations indicate that both models provide similar predictions of damage.
---
paper_title: A new Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models

paper_content:

In this paper we develop a new constitutive law for the description of the (passive) mechanical response of arterial tissue. The artery is modeled as a thick-walled nonlinearly elastic circular cylindrical tube consisting of two layers corresponding to the media and adventitia (the solid mechanically relevant layers in healthy tissue). Each layer is treated as a fiber-reinforced material with the fibers corresponding to the collagenous component of the material and symmetrically disposed with respect to the cylinder axis. The resulting constitutive law is orthotropic in each layer. Fiber orientations obtained from a statistical analysis of histological sections from each arterial layer are used. A specific form of the law, which requires only three material parameters for each layer, is used to study the response of an artery under combined axial extension, inflation and torsion. The characteristic and very important residual stress in an artery in vitro is accounted for by assuming that the natural (unstressed and unstrained) configuration of the material corresponds to an open sector of a tube, which is then closed by an initial bending to form a load-free, but stressed, circular cylindrical configuration prior to application of the extension, inflation and torsion. The effect of residual stress on the stress distribution through the deformed arterial wall in the physiological state is examined.
---
paper_title: Probabilistic constitutive law for damage in ligaments.

paper_content:

A new constitutive equation is presented to describe the damage evolution process in parallel-fibered collagenous tissues such as ligaments. The model is formulated by accounting for the fibrous structure of the tissues. The tissue's stress is defined as the average of the collagen fiber's stresses. The fibers are assumed to be undulated and straightened out at different stretches that are randomly defined according to a Weibull distribution. After becoming straight, each collagen fiber is assumed to be linear elastic. Damage is defined as a reduction in collagen fiber's stiffness and occurs at different stretches that are also randomly defined by a Weibull distribution. Due to the lack of experimental data, the predictions of the constitutive equation are analyzed by varying the values of its structural parameters. Moreover, the results are compared with the available stress-strain data in the biomechanics literature that evaluate damage produced by subfailure stretches in rat medial collateral ligaments.
---
paper_title: Tendons and ligaments: A morphological and biochemical comparison

paper_content:

The purpose of this study was to compare selected rabbit tendons and ligaments morphologically and biochemically. Five representative structures from each of six age- and sex-matched rabbits were compared. Biochemical analyses included total collagen, reducible collagen cross-links, quantitative collagen typing, DNA, and glycosaminoglycans. Histological and chemical differences were demonstrated between the tendons and the ligaments. Smaller differences were also found between the individual ligaments (collateral and cruciate) and between the two tendons (patellar and Achilles) that were examined. These findings suggest that ligaments are more metabolically active than tendons, having more plump cellular nuclei, higher DNA content, larger amounts of reducible cross-links, and the presence of more type III collagen, as compared with tendons. They also contain slightly less total collagen than tendons and more glycosaminoglycans. We conclude that the tendons and ligaments studied have unique histological and biochemical characteristics, despite their gross similarities. Relatively increased metabolic activity in ligaments, implied by our findings, may be species specific, age related (transient), or may truly represent a structural expression of functional need for more rapid adaptation. Further investigation of other similarities or differences between particular ligaments (or tendons) is indicated, and attention is directed toward the importance of such variables in development of models for tendon and ligament studies.
---
paper_title: A constitutive law for the failure behavior of medial collateral ligaments.

paper_content:

A constitutive model is proposed for the description of the tensile properties of medial collateral ligaments (MCLs). The model can reproduce the three regions – the toe region, the linear region, and the failure region – of the stress–stretch curve of ligamentous tissues. The collagen fibers are assumed to be the only load-bearing component of the tissues. They are all oriented along the physiological loading direction of the ligament. They are crimped in the slack configuration and are unable to sustain load. After becoming taut and before failing, each collagen fiber exhibits a linear elastic behavior. The fiber straightening and failure processes are defined stochastically by means of Weibull distributions. Published experimental data for the MCLs are employed to validate the constitutive relationship. Finally, the constitutive model is generalized in order to describe the three-dimensional mechanical behavior of the ligaments by following he structural approach.
---
paper_title: A FAILURE MODEL FOR LIGAMENTS

paper_content:

Abstract We propose a failure model for ligament which assumes that sequential uncrimping and stretching of collagen fibers is responsible for the mechanical response of ligament. We further assume that the fibers rupture sequentially and in a brittle, strain-limited manner. The model was fit to stress–strain curves obtained from medial collateral ligaments of New Zealand White rabbits from two age groups (4 and 7 months). The model indicated that collagen modulus values ranged from 300 to 680 MPa and that fiber failure strains ranged from 6 to 22%. The model provides a convenient means of describing the elastic and failure response of ligament using four structurally based parameters.
---
paper_title: Fatigue damage of human tendons

paper_content:

Abstract The study was designed to examine the effects of partial fatigue on specific mechanical parameters which characterise human tendons in vitro. Specimens prepared from 12 intact Extensor digitorum longus tendons of the foot were subjected to partial fatigue, equivalent to 25% of the median fatigue life, by a cyclic square tension–tension stress waveform at the physiological frequency of 4 Hz. The maximum stress was set at a value corresponding to 20% of the calculated ultimate tensile strength (UTS) of 100 MPa. The minimum stress was set at 1% of the UTS. Dynamic characterisation was performed at stress levels of 10% and 20% of the UTS prior to and following partial mechanical fatigue. Subsequent quasi-static tests were performed on some of the specimens. Comparative analysis of the damage ratios (DRs) of quasi-static and dynamic mechanical parameters suggested the use of the DR for dynamic tensile modulus as a good indicator of damage inflicted by mechanical fatigue. Such an approach might be used for an in vivo assessment of tendon damage.
---
paper_title: Anisotropic elasto-damage constitutive model for the biomechanical analysis of tendons.

paper_content:

The aim of this work is to investigate the instantaneous mechanical response of tendons by the use of an anisotropic elasto-damage constitutive model. This study addresses the analysis of the mechanical behaviour of healthy tendons during physiological loading and to degeneration phenomena. These are correlated with aging or traumatic events such as chronic or acute overloading during sporting activities. Histo-morphometric considerations suggest the adoption of a transversally isotropic constitutive model that describes the anisotropy of the material. The non-linearity of its overall mechanical response is taken into account by using a hyperelastic approach and also evaluates softening behaviour related to damage phenomena. The values of the parameters adopted within the analytical model are estimated both for human tendons previously subjected to cyclical loading and for specimens not subjected to cyclical loading. The results obtained by adopting this analytical model are compared with the experimental data in order to evaluate the capability of the model to describe the mechanical response of the tissue.
---
paper_title: Subfailure damage in ligament: a structural and cellular evaluation

paper_content:

Subfailure damage in ligaments was evaluated macroscopically from a structural perspective (referring to the entire ligament as a structure) and microscopically from a cellular perspective. Freshly harvested rat medial collateral ligaments (MCLs) were used as a model in ex vivo experiments. Ligaments were preloaded with 0.1 N to establish a consistent point of reference for length (and strain) measurements. Ligament structural damage was characterized by nonrecoverable difference in tissue length after a subfailure stretch. The tissue's mechanical properties (via stress vs. strain curves measured from a preloaded state) after a single subfailure stretch were also evaluated (n = 6 pairs with a different stretch magnitude applied to each stretched ligament). Regions containing necrotic cells were used to characterize cellular damage after a single stretch. It should be noted that the number of damaged cells was not quantified and the difference between cellular area and area of fluorescence is not known. Structural and cellular damage were represented and compared as functions of subfailure MCL strains. Statistical analysis indicated that the onset of structural damage occurs at 5.14% strain (referenced from a preloaded length). Subfailure strains above the damage threshold changed the shape of the MCL stress-strain curve by elongating the toe region (i.e., increasing laxity) as well as decreasing the tangential modulus and ultimate stress. Cellular damage was induced at ligament strains significantly below the structural damage threshold. This cellular damage is likely to be part of the natural healing process in mildly sprained ligaments.
---
paper_title: Ligament structure, physiology and function.

paper_content:

Ligaments are specialized connective tissues with very interesting biomechanical properties. They have the ability to adapt to the complex functions that each are required to perform. While ligaments were once thought to be inert, they are in fact responsive to many local and systemic factors that influence their function within the organism. Injury to a ligament results in a drastic change in its structure and physiology and creates a situation where ligament function is restored by the formation of scar tissue that is biologically and biomechanically inferior to the tissue it replaces. This article will briefly review the basic structure, physiology and function of normal versus healing knee ligaments, referring specifically to what is known about two of the most extensively studied and clinically relevant knee ligaments, the anterior cruciate (ACL) and medial collateral (MCL) ligaments of the knee. Those readers wishing for more comprehensive sources of information on ligament biology and biomechanics are referred to many excellent reviews on these topics.
---
paper_title: A structural model used to evaluate the changing microstructure of maturing rat skin.

paper_content:

A microstructurally based model for soft biological tissues was adapted to model the non-linear response of the collagenous network in skin. Quasi-static uniaxial tensile tests were conducted on dorsal skin of rats aged 1-4 months. Specimens were taken longitudinally and transversely (laterally) to the spine and tested at 1.5% s-1. The model showed fiber stiffness to increase laterally during maturation and to be greater laterally than longitudinally. It also indicated the migration of the 'heel' portion of the response curve toward the origin during maturation to be due to less crimping in the fibers. These results were qualitatively supported by microscopical observations. The model was found to be well suited to fit response curves exhibiting a well-defined 'heel' region. For curves exhibiting a more flat, featureless 'heel' region (typical of 1 month longitudinal specimens) the model was not well suited. The results of this study indicate that this simple three-parameter model might be a useful tool in future studies to analyze pathological changes in the collagenous microstructure.
---
paper_title: Failure mechanisms in fibrous scaffolds.

paper_content:

Polymeric fibrous scaffolds have been considered as replacements for load-bearing soft tissues, because of their ability to mimic the microstructure of natural tissues. Poor toughness of fibrous materials results in failure, which is an issue of importance to both engineering and medical practice. The toughness of fibrous materials depends on the ability of the microstructure to develop toughening mechanisms. However, such toughening mechanisms are still not well understood, because the detailed evolution at the microscopic level is difficult to visualize. A novel and simple method was developed, namely, a sample-taping technique, to examine the detailed failure mechanisms of fibrous microstructures. This technique was compared with in situ fracture testing by scanning electron microscopy. Examination of three types of fibrous networks showed that two different failure modes occurred in fibrous scaffolds. For brittle cracking in gelatin electrospun scaffolds, the random network morphology around the crack tip remained during crack propagation. For ductile failure in polycaprolactone electrospun scaffolds and nonwoven fabrics, the random network deformed via fiber rearrangement, and a large number of fiber bundles formed across the region in front of the notch tip. These fiber bundles not only accommodated mechanical strain, but also resisted crack propagation and thus toughened the fibrous scaffolds. Such understanding provides insight for the production of fibrous materials with enhanced toughness.
---
paper_title: In vitro fatigue of human tendons.

paper_content:

The purpose of this study was to determine the fatigue behaviour of human tendons in vitro. The testing was accomplished with the use of specially designed grips and the local measurement of tendon cross-sectional area. Ninety specimens prepared from Extensor digitorum longus (EDL) tendons of the foot were subjected to a cyclic square tension-tension stress waveform at physiological frequencies. The maximum tensile stress was normalised to values corresponding to prescribed levels between 10% and 90% of the calculated ultimate tensile strength (UTS) of 100 MPa. The minimum stress was set at 1% of the UTS. A replication of 10 specimens per stress level allowed the use of statistical models for the distribution of fatigue life. Results followed a linear model, of form S = 101.3 - 14.8 log(N), relating the normalised stress to the median number of cycles to failure, therefore suggesting the absence of an endurance limit. The Weibull distribution was found to describe adequately the probability of failure at each stress level. A model which takes into account in vivo healing was proposed. This model was able to explain the presence of intact tendons throughout the lifetime of an individual.
---
paper_title: An elastic stress-strain relation for soft biological tissues based on a structural model

paper_content:

Abstract We propose a mechanical model to describe the “quasi-elastic” stress-strain relation for soft biological tissues, based upon their fibre structured design. The corresponding elastic constitutive equation is derived and we show how the model parameters may be calculated to fit the model to experimental data. Some examples of such fittings are given for different materials such as human vein, fascia, tympanic membrane and rabbit papillary muscle. The good quality of these adaptations illustrates the multiple application possibility of the proposed model.
---
paper_title: An irreversible constitutive model for fibrous soft biological tissue: A 3-D microfiber approach with demonstrative application to abdominal aortic aneurysms

paper_content:

Abstract Understanding the failure and damage mechanisms of soft biological tissue is critical to a sensitive and specific characterization of tissue injury tolerance and its relation to biological responses. Despite increasing experimental and analytical efforts, failure-related irreversible effects of soft biological tissue are still poorly understood. There is still no clear definition of what “damage” of a soft biological material is, and conventional macroscopic indicators, as known from damage of engineering materials for example, may not identify the tissue’s tolerance to injury appropriately. To account for the complex three-dimensional arrangement of collagen, a microfiber model approach is applied, where constitutive relations for collagen fibers are integrated over the unit sphere, which in turn defines the tissue’s macroscopic properties. A collagen fiber is represented by a bundle of proteoglycan cross-linked collagen fibrils that undergoes irreversible deformations when exceeding its elastic tensile limit. The proposed constitutive model is able to predict strain stiffening at physiological strain levels and does not exhibit a clear macroscopic elastic limit, two typical features known from soft biological tissue testing. An elastic-predictor/plastic-corrector implementation of the model is followed and constitutive parameters are estimated from in vitro test data from a particular abdominal aortic aneurysm (AAA). Damage-based structural instabilities of the AAA under different inflation conditions are investigated, where the collagen orientation density has been estimated from its in vivo stress state.
---
paper_title: Elasticity in extracellular matrix ‘ shape modules ’ of tendon , cartilage , etc . A sliding proteoglycan-filament model

paper_content:

Connective tissues (CTs), which define bodily shape, must respond quickly, robustly and reversibly to deformations caused by internal and external stresses. Fibrillar (elastin, collagen) elasticity under tension depends on molecular and supramolecular mechanisms. A second intra-/inter-molecular pair, involving proteoglycans (PGs), is proposed to cope with compressive stresses. PG interfibrillar bridges ('shape modules'), supramolecular structures ubiquitously distributed throughout CT extracellular matrices (ECMs), are examined for potential elastic properties. L-iduronate residues in shape module decoran PGs are suggested to be molecular springs, cycling through alternative conformations. On a larger scale, anionic glycosaminoglycan (AGAG) interfibrillar bridges in shape modules are postulated to take part in a sliding filament (dashpot-like) process, which converts local compressions into disseminated tensile strains. The elasticity of fibrils and AGAGs, manifest at molecular and larger-scale levels, provides a graduated and smooth response to stresses of varying degrees. NMR and rheo NMR, computer modelling, electron histochemical, biophysical and chemical morphological evidence for the proposals is reviewed.
---
paper_title: The structure of interfibrillar proteoglycan bridges ('shape modules') in extracellular matrix of fibrous connective tissues and their stability in various chemical environments

paper_content:

Collagen fibrils in extracellular matrices of connective tissues (tendon, cornea, etc.) are bridged and linked by the anionic glycosaminoglycans (AGAGs) of the small proteoglycans (decoron, etc.). It was proposed that these bridges and ties maintain the collagen fibril dispositions in relation to each other, helping to define tissue shape, and hence called shape modules. This investigation describes chemical and physicochemical conditions in which these structures are stable and what treatments cause their disruption. The effects on fixed and unfixed sections of tendon, cornea, lung and ear from rat, mouse and rabbit of pH, electrolyte concentration, EDTA, mercaptoethanol, hydrogen peroxide, free radicals, periodate, acetylation, urea, nonionic detergent and organic solvents were assessed by staining with Cupromeronic blue or Alcec blue in CEC techniques to localise AGAG bridges or their disintegration products. Ca2+ was not involved in the structures, oxidation/reduction had no effect and Triton X100, a nonionic detergent did not damage them. They were stable between pH 4.5 and 9.5. Periodate as a glycol-cleaving reagent did not affect them. High concentrations of urea (> 2.0 M) and MgCl2 (0.5 M) disrupted the tissues. The combination of Triton and urea at concentrations too low to cause damage separately was disruptive. Free radicals in periodate solutions were damaging. Organic solvents caused collapse and rearrangements of the AGAG filaments. Acetylation caused considerable disruption of shape modules. Dermochondan but not keratan sulphate AGAGs were removed by treatment with NaOH. After fixing with glutaraldehyde only free radical and NaOH treatments were severely disruptive of shape modules. The results are compatible with a previously proposed structure for the shape modules, stabilised by hydrophobic and hydrogen bonding.
---
paper_title: Proteoglycan:collagen interactions and subfibrillar structure in collagen fibrils. Implications in the development and ageing of connective tissues.

paper_content:

Evidence is presented confirming that collagen fibrils are aggregates of subfibrils. Rat tail tendon fibrils swollen in acetate buffer, pH 5.8, (with or without MgCl2 or Cupromeronic blue, a proteoglycan (PG) precipitant used to locate PGs by electron histochemistry) showed two characteristic populations of subfibrils, of 10-15 nm, and approximately 25 nm thickness respectively. The smaller protofibrils, helically orientated, were derived from, or coalesced into, the larger subfibrils. The results prove that cross-links are not regularly distributed throughout the tissue but are limited to intra-protofibrillar collagen. Cupromeronic blue-stained PGs were seen in thick fibrils from cartilage and tendons, axially orientated in bands across the fibrils in longitudinal sections, or chordally in fibril cross-sections. Based on the intrafibrillar location of PGs vis-à-vis the fibril a-e banding pattern, the relative orientation of PGs and protofibrils and PG distribution in very young, as compared with mature or old, tissue, it is suggested that intrafibrillar PG is originally associated with protofibrils, which coalesce to larger fibrils, e.g. during development. The function of the large amount of PG and hyaluronan in young tissues is seen to keep the collagen protofibrils and subfibrils from coalescing. Disaggregation to protofibrils was demonstrated in conditions not very different from certain in vivo situations. It is suggested that this phenomenon is potentially important during tissue modelling and remodelling, allowing rapid access of reactants to the interior of thick fibrils and recycling of component protofibrils.
---
paper_title: An anisotropic pseudo-elastic approach for modelling Mullins effect in fibrous biological materials

paper_content:

Abstract This paper presents a fully three-dimensional finite-strain formulation with a simple pseudo-elastic anisotropic model to reproduce the softening behavior that appears in soft biological fibred tissues (Mullins effect). This is an anisotropic extension of the pseudo-elastic model of Ogden and Roxburgh which was earlier proposed to describe the softening behavior of rubbers under large deformation. The model makes use of the concept of internal variables which provides a very general description of materials involving irreversible effects. Numerical simulations of uniaxial tests of several soft tissues are used to show the performance of the model in realistic situations. The computed results indicate that the proposed model is able to describe with enough accuracy the softening behavior of real soft biological tissues.
---
paper_title: Probabilistic constitutive law for damage in ligaments.

paper_content:

A new constitutive equation is presented to describe the damage evolution process in parallel-fibered collagenous tissues such as ligaments. The model is formulated by accounting for the fibrous structure of the tissues. The tissue's stress is defined as the average of the collagen fiber's stresses. The fibers are assumed to be undulated and straightened out at different stretches that are randomly defined according to a Weibull distribution. After becoming straight, each collagen fiber is assumed to be linear elastic. Damage is defined as a reduction in collagen fiber's stiffness and occurs at different stretches that are also randomly defined by a Weibull distribution. Due to the lack of experimental data, the predictions of the constitutive equation are analyzed by varying the values of its structural parameters. Moreover, the results are compared with the available stress-strain data in the biomechanics literature that evaluate damage produced by subfailure stretches in rat medial collateral ligaments.
---
paper_title: A constitutive law for the failure behavior of medial collateral ligaments.

paper_content:

A constitutive model is proposed for the description of the tensile properties of medial collateral ligaments (MCLs). The model can reproduce the three regions – the toe region, the linear region, and the failure region – of the stress–stretch curve of ligamentous tissues. The collagen fibers are assumed to be the only load-bearing component of the tissues. They are all oriented along the physiological loading direction of the ligament. They are crimped in the slack configuration and are unable to sustain load. After becoming taut and before failing, each collagen fiber exhibits a linear elastic behavior. The fiber straightening and failure processes are defined stochastically by means of Weibull distributions. Published experimental data for the MCLs are employed to validate the constitutive relationship. Finally, the constitutive model is generalized in order to describe the three-dimensional mechanical behavior of the ligaments by following he structural approach.
---
paper_title: A FAILURE MODEL FOR LIGAMENTS

paper_content:

Abstract We propose a failure model for ligament which assumes that sequential uncrimping and stretching of collagen fibers is responsible for the mechanical response of ligament. We further assume that the fibers rupture sequentially and in a brittle, strain-limited manner. The model was fit to stress–strain curves obtained from medial collateral ligaments of New Zealand White rabbits from two age groups (4 and 7 months). The model indicated that collagen modulus values ranged from 300 to 680 MPa and that fiber failure strains ranged from 6 to 22%. The model provides a convenient means of describing the elastic and failure response of ligament using four structurally based parameters.
---
paper_title: Anisotropic elasto-damage constitutive model for the biomechanical analysis of tendons.

paper_content:

The aim of this work is to investigate the instantaneous mechanical response of tendons by the use of an anisotropic elasto-damage constitutive model. This study addresses the analysis of the mechanical behaviour of healthy tendons during physiological loading and to degeneration phenomena. These are correlated with aging or traumatic events such as chronic or acute overloading during sporting activities. Histo-morphometric considerations suggest the adoption of a transversally isotropic constitutive model that describes the anisotropy of the material. The non-linearity of its overall mechanical response is taken into account by using a hyperelastic approach and also evaluates softening behaviour related to damage phenomena. The values of the parameters adopted within the analytical model are estimated both for human tendons previously subjected to cyclical loading and for specimens not subjected to cyclical loading. The results obtained by adopting this analytical model are compared with the experimental data in order to evaluate the capability of the model to describe the mechanical response of the tissue.
---
paper_title: A pseudo–elastic model for the Mullins effect in filled rubber

paper_content:

When a rubber test piece is loaded in simple tension from its virgin state, unloaded and then reloaded, the stress required on reloading is less than that on the initial loading for stretches up to the maximum stretch achieved on the initial loading. This stress softening phenomenon is referred to as the Mullins effect . In this paper a simple phenomenological model is proposed to account for the Mullins effect observed in filled rubber elastomers. The model is based on the theory of incompressible isotropic elasticity amended by the incorporation of a single continuous parameter, interpreted as a damage parameter. This parameter controls the material properties in the sense that it enables the material response to be governed by a strain–energy function on unloading and subsequent submaximal loading different from that on the primary (initial) loading path from the virgin state. For this reason the model is referred to as pseudo-elastic } and a primary loading-unloading cycle involves energy dissipation. The dissipation is measured by a damage function which depends only on the damage parameter and on the point of the primary loading path from which unloading begins. A specific form of this function with two adjustable material constants, coupled with standard forms of the (incompressible, isotropic) strain–energy function, is used to illustrate the qualitative features of the Mullins effect in both simple tension and pure shear. For simple tension the model is then specialized further in order to fit Mullins effect data. It is emphasized that the model developed here is applicable to multiaxial states of stress and strain, not just the specific uniaxial tests highlighted.
---
paper_title: Layer-specific damage experiments and modeling of human thoracic and abdominal aortas with non-atherosclerotic intimal thickening.

paper_content:

Many treatments for cardiovascular diseases include an endovascular insertion of stents or stent grafts into arteries, a procedure which may cause high tissue stresses and even damage in the arterial wall. In order to study such problems by using finite element methods, both appropriate constitutive models and experimental data on human tissue samples are required. Layer-specific experimental data for human tissue tested up to the supra-physiological loading range are rare in the literature. In this study, intact and layer-separated experimental data from uniaxial extension tests are presented for human thoracic and abdominal aortas with non-atherosclerotic intimal thickening undergoing supra-physiological loading. A novel pseudo-elastic damage model, proposed to describe discontinuous softening in aortic arterial tissues, is fit to the obtained experimental data. Fitting of the model with and without consideration of damage accumulation in the non-collagenous matrix material reveals that tissue damage is primarily related to the collagen fiber fabric. By employing the fit model, the effect of aortic tissue pre-conditioning on the material parameters from the resulting data fits is evaluated. Histological examination of the collagen fibers under different applied stretches is used to gain more insights into the structural changes of the tissue under supra-physiological loading.
---
paper_title: Human thoracic and abdominal aortic aneurysmal tissues: Damage experiments, statistical analysis and constitutive modeling.

paper_content:

Abstract Development of aortic aneurysms includes significant morphological changes within the tissue: collagen content increases, elastin content reduces and smooth muscle cells degenerate. We seek to quantify the impact of these changes on the passive mechanical response of aneurysms in the supra-physiological loading range via mechanical testing and constitutive modeling. We perform uniaxial extension tests on circumferentially and axially oriented strips from five thoracic (65.6 years±13.4, mean±SD) and eight abdominal (63.9 years±11.4) aortic fusiform aneurysms to investigate both continuous and discontinuous softening during supra-physiological loading. We determine the significance of the differences between the fitted model parameters: diseased thoracic versus abdominal tissues, and healthy ( Weisbecker et al., J. Mech. Behav. Biomed. Mater. 12, 93–106, 2012 ) versus diseased tissues. We also test correlations among these parameters and age, Body Mass Index (BMI) and preoperative aneurysm diameter, and investigate histological cuts. Tissue response is anisotropic for all tests and the anisotropic pseudo-elastic damage model fits the data well for both primary loading and discontinuous softening which we interpret as damage. We found statistically relevant differences between model parameters fitted to diseased thoracic versus abdominal tissues, as well as between those fitted to healthy versus diseased tissues. Only BMI correlated with fitted model parameters in abdominal aortic aneurysmal tissues.
---
paper_title: On the Mullins effect and hysteresis of fibered biological materials: A comparison between continuous and discontinuous damage models

paper_content:

Deformation induced softening is an inelastic phenomenon frequently accompanying mechanical behavior of soft biological tissues. This paper presents and compare continuous and discontinuous damage approaches to model softening effects in fibered materials such as soft biological tissues. The structural model is formulated using the concept of internal variables that provides a very general description of materials involving irreversible effects. We consider the internal variables associated to damage to correspond to separated contributions of the matrix and the fibers. Local damage accumulation is related to two phenomenological variables, the maximum value and the arclength of the effective free energy attained during the loading process, respectively. A local multiplicative decomposition of the deformation gradient into volume-preserving and dilatational parts is used that permits to model the incompressibility property of most types of soft biological tissues. In this context, damage is related only to the isochoric part of the deformation. Finally, simulations of biaxial and uniaxial tests in two directions are used to compare the performance of both models. Numerical simulations indicate that only a mixed model that consider both, continuous and discontinuous, damage models is able to capture the softening phenomenon of soft biological tissues.
---
paper_title: A new Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models

paper_content:

In this paper we develop a new constitutive law for the description of the (passive) mechanical response of arterial tissue. The artery is modeled as a thick-walled nonlinearly elastic circular cylindrical tube consisting of two layers corresponding to the media and adventitia (the solid mechanically relevant layers in healthy tissue). Each layer is treated as a fiber-reinforced material with the fibers corresponding to the collagenous component of the material and symmetrically disposed with respect to the cylinder axis. The resulting constitutive law is orthotropic in each layer. Fiber orientations obtained from a statistical analysis of histological sections from each arterial layer are used. A specific form of the law, which requires only three material parameters for each layer, is used to study the response of an artery under combined axial extension, inflation and torsion. The characteristic and very important residual stress in an artery in vitro is accounted for by assuming that the natural (unstressed and unstrained) configuration of the material corresponds to an open sector of a tube, which is then closed by an initial bending to form a load-free, but stressed, circular cylindrical configuration prior to application of the extension, inflation and torsion. The effect of residual stress on the stress distribution through the deformed arterial wall in the physiological state is examined.
---
paper_title: A constitutive model for the Mullins effect with permanent set in particle-reinforced rubber

paper_content:

Stress softening during initial loading cycles, known as the Mullins effect, and the residual strain upon unloading are not accounted for when the mechanical properties of rubber are represented in terms of a strain-energy function, i.e. if the material is modelled as hyperelastic. In this paper we first describe some experimental results that illustrate stress softening in particle-reinforced rubber together with associated residual strain effects. In particular, the results show how the stress softening and residual strain change with the magnitude of the applied strain. Then, on the basis of these data a constitutive model is derived to describe this behaviour. The theory of pseudo-elasticity is used for this model, the basis of which is the inclusion of two variables in the energy function in order separately to capture the stress softening and residual strain effects. The dissipation of energy, i.e. the difference between the energy input during loading and the energy returned on unloading is also accounted for in the model by the use of a dissipation function, which evolves with the deformation history.
---
paper_title: Modeling failure of soft anisotropic materials with application to arteries

paper_content:

Abstract The arterial wall is a composite where the preferred orientation of collagen fibers induces anisotropy. Though the hyperelastic theories of fiber-reinforced composites reached a high level of sophistication and showed a reasonable correspondence with the available experimental data they are short of the failure description. Following the tradition of strength of materials the failure criteria are usually separated from stress analysis. In the present work we incorporate a failure description in the hyperelastic models of soft anisotropic materials by introducing energy limiters in the strain energy functions. The limiters provide the saturation value for the strain energy which indicates the maximum energy that can be stored and dissipated by an infinitesimal material volume. By using some popular constitutive models enhanced with the energy limiters we analyze rupture of a sheet of arterial material under the plane stress state varying from the uniaxial to equal biaxial tension. We calculate the local failure criteria including the maximum principal stress, the maximum principal stretch, the von Mises stress, and the strain energy at the moment of the sheet rupture. We find that the local failure criterion in the form of the critical strain energy is the most robust among the considered ones. We also find that the tensile strength–the maximum principal stress–that is usually obtained in uniaxial tension tests might not be appropriate as a failure indicator in the cases of the developed biaxiality of the stress–strain state.
---
paper_title: Prediction of arterial failure based on a microstructural bi-layer fiber-matrix model with softening.

paper_content:

Abstract Two approaches to predict failure of soft tissue are available. The first is based on a pointwise criticality condition, e.g. von Mises maximum stress, which is restrictive because only local state of deformation is considered to be critical and the failure criterion is separated from stress analysis. The second is based on damage mechanics where internal (unobservable) variables are introduced which make the experimental calibration of the theory complex. As an alternative to the local failure criteria and damage mechanics we present a softening hyperelasticity approach, where the constitutive description of soft tissue is enhanced with strain softening, which is controlled by material constants. This approach is attractive because the new material constants can be readily calibrated in experiments on the one hand and the failure criteria are global on the other hand. We illustrate the efficiency of the softening hyperelasticity approach on the problem of prediction of arterial failure. For this purpose, we enhance a bi-layer fiber–matrix microstructural arterial model with softening and analyze the arterial failure under internal pressure. We show that the overall arterial strength is (a) dominated by the media layer, (b) controlled by microfibers and (c) increased by residual stresses.
---
paper_title: Probabilistic constitutive law for damage in ligaments.

paper_content:

A new constitutive equation is presented to describe the damage evolution process in parallel-fibered collagenous tissues such as ligaments. The model is formulated by accounting for the fibrous structure of the tissues. The tissue's stress is defined as the average of the collagen fiber's stresses. The fibers are assumed to be undulated and straightened out at different stretches that are randomly defined according to a Weibull distribution. After becoming straight, each collagen fiber is assumed to be linear elastic. Damage is defined as a reduction in collagen fiber's stiffness and occurs at different stretches that are also randomly defined by a Weibull distribution. Due to the lack of experimental data, the predictions of the constitutive equation are analyzed by varying the values of its structural parameters. Moreover, the results are compared with the available stress-strain data in the biomechanics literature that evaluate damage produced by subfailure stretches in rat medial collateral ligaments.
---
paper_title: A constitutive law for the failure behavior of medial collateral ligaments.

paper_content:

A constitutive model is proposed for the description of the tensile properties of medial collateral ligaments (MCLs). The model can reproduce the three regions – the toe region, the linear region, and the failure region – of the stress–stretch curve of ligamentous tissues. The collagen fibers are assumed to be the only load-bearing component of the tissues. They are all oriented along the physiological loading direction of the ligament. They are crimped in the slack configuration and are unable to sustain load. After becoming taut and before failing, each collagen fiber exhibits a linear elastic behavior. The fiber straightening and failure processes are defined stochastically by means of Weibull distributions. Published experimental data for the MCLs are employed to validate the constitutive relationship. Finally, the constitutive model is generalized in order to describe the three-dimensional mechanical behavior of the ligaments by following he structural approach.
---
paper_title: Statistical approach for a continuum description of damage evolution in soft collagenous tissues

paper_content:

Abstract We propose a statistical approach to describe microscopic damage evolution in soft collagenous tissues such as arterial walls. The damage model extends a framework published by Balzani et al. (2012), Comput. Methods Appl. Mech. Engrg., 213–216:139–151, by postulating specific damage functions that result from the fibers’ microstructure. Statistical distributions of three different microscopic quantities such as proteoglycan orientation, fibril length parameters and ultimate proteoglycan stretch are considered. The resulting stress–stretch response is compared with experimental data obtained from uniaxial tension tests given in the literature. In particular, the individual statistical distributions are analyzed in regard to their ability to capture the distinct softening hysteresis observed when subjecting soft tissues to cyclic loading in the supra-physiological domain. Details regarding the algorithmic implementation are provided, and the applicability of the model within a finite element framework is shown by simulating the overexpansion of simplified atherosclerotic arteries.
---
paper_title: A FAILURE MODEL FOR LIGAMENTS

paper_content:

Abstract We propose a failure model for ligament which assumes that sequential uncrimping and stretching of collagen fibers is responsible for the mechanical response of ligament. We further assume that the fibers rupture sequentially and in a brittle, strain-limited manner. The model was fit to stress–strain curves obtained from medial collateral ligaments of New Zealand White rabbits from two age groups (4 and 7 months). The model indicated that collagen modulus values ranged from 300 to 680 MPa and that fiber failure strains ranged from 6 to 22%. The model provides a convenient means of describing the elastic and failure response of ligament using four structurally based parameters.
---
paper_title: A continuum description of the damage process in the arterial wall of abdominal aortic aneurysms

paper_content:

In the present work, we develop a three-dimensional isotropic finite-strain damage model for abdominal aortic aneurysm (AAA) wall that considers both the characteristic softening of the material caused by damage and the spatial variation of the material properties. A strain energy function is formulated that accounts for a hyperelastic, slightly compressible, isotropic material behavior during the elastic phase, ::: whereas the damage process only contributes to the material response when the elastic limit of the AAA wall is exceeded. Material and damage parameters are obtained by fitting the strain energy function to the experimental data obtained by uniaxial tensile tests of freshly harvested AAA wall samples. The damage model extends the validity of the material law to a strain range of up to 50%. Purely elastic material ::: laws for AAA wall are only valid for a strain range of up to 17%. In a series of finite element simulations of patient-specific AAAs, serving as numerical examples, we investigate the applicability of the damage model. The use of the damage model does not yield a more distinct identification of ruptureprone AAAs than other computational-based risk indices. However, the benefit of the finite-strain damage ::: model is the potential capability to trigger growth and remodeling processes in mechanobiological models.
---
paper_title: Anisotropic elasto-damage constitutive model for the biomechanical analysis of tendons.

paper_content:

The aim of this work is to investigate the instantaneous mechanical response of tendons by the use of an anisotropic elasto-damage constitutive model. This study addresses the analysis of the mechanical behaviour of healthy tendons during physiological loading and to degeneration phenomena. These are correlated with aging or traumatic events such as chronic or acute overloading during sporting activities. Histo-morphometric considerations suggest the adoption of a transversally isotropic constitutive model that describes the anisotropy of the material. The non-linearity of its overall mechanical response is taken into account by using a hyperelastic approach and also evaluates softening behaviour related to damage phenomena. The values of the parameters adopted within the analytical model are estimated both for human tendons previously subjected to cyclical loading and for specimens not subjected to cyclical loading. The results obtained by adopting this analytical model are compared with the experimental data in order to evaluate the capability of the model to describe the mechanical response of the tissue.
---
paper_title: Layer-specific damage experiments and modeling of human thoracic and abdominal aortas with non-atherosclerotic intimal thickening.

paper_content:

Many treatments for cardiovascular diseases include an endovascular insertion of stents or stent grafts into arteries, a procedure which may cause high tissue stresses and even damage in the arterial wall. In order to study such problems by using finite element methods, both appropriate constitutive models and experimental data on human tissue samples are required. Layer-specific experimental data for human tissue tested up to the supra-physiological loading range are rare in the literature. In this study, intact and layer-separated experimental data from uniaxial extension tests are presented for human thoracic and abdominal aortas with non-atherosclerotic intimal thickening undergoing supra-physiological loading. A novel pseudo-elastic damage model, proposed to describe discontinuous softening in aortic arterial tissues, is fit to the obtained experimental data. Fitting of the model with and without consideration of damage accumulation in the non-collagenous matrix material reveals that tissue damage is primarily related to the collagen fiber fabric. By employing the fit model, the effect of aortic tissue pre-conditioning on the material parameters from the resulting data fits is evaluated. Histological examination of the collagen fibers under different applied stretches is used to gain more insights into the structural changes of the tissue under supra-physiological loading.
---
paper_title: Modeling the propagation of arterial dissection

paper_content:

Arterial dissections are frequently observed in clinical practice and during road traffic accidents. In particular, the lamellarly arrangement of elastin, collagen, in addition to smooth muscle cells in the middle arterial layer, the media, favors dissection failure. Experimental studies and related biomechanical models are rare in the literature. Finite strain kinematics is employed, and the discontinuity in the displacement field accounts for tissue separation. Dissection is regarded as a gradual process in which separation between incipient material surfaces is resisted by cohesive traction. Two variational statements together with their consistent linearizations form the basis for a finite element implementation. We combine the cohesive crack concept with the partition of unity finite element method, where nodal degrees of freedom adjacent to the discontinuity are enhanced. The developed continuum mechanical and numerical frameworks allow the analysis of the propagation of dissections within general nonlinear boundary-value problems, where the constitutive description for the continuous and the cohesive material is considered independent from each other. The continuous material is modeled as a fiber-reinforced composite with the fibers corresponding to the collagenous component which are assumed to be embedded in a non-collagenous isotropic groundmatrix. Dispersion of the collagen fiber orientation is considered in a continuum sense by one structure parameter. A novel cohesive potential per unit undeformed area is used to derive a traction separation law appropriate for the description of the mechanical properties of medial dissection. The cohesive stiffness contribution to the element stiffness matrix is explicitly derived. In particular, the dissection propagation of a rectangular strip of a human aortic media is investigated. Cohesive material properties are quantified by comparing the experimentally measured load with the computed dissection load.
---
paper_title: Simulation of dynamic fracture with the Material Point Method using a mixed J-integral and cohesive law approach

paper_content:

A new approach to simulating fracture, in which toughness is partitioned between the crack tip and, optionally, a process zone, is applied to dynamic fracture processes. In this approach, classical fracture mechanics determines crack tip propagation, and cohesive laws characterize process zone response and determine crack root and process zone propagation. The approach is implemented in the Material Point Method, a particle method in which the fracture path is unconstrained by a body-fitted mesh. The approach is found suitable for modeling a range of dynamic fracture processes, from brittle fracture to fracture with crack bridging. A variety of ways of partitioning toughness are explored with the aim of distinguishing model parameters via experimental measurements, particularly R curves. While no unique relationship exists, R curves, or effective R curves, on a suite of materials would provide substantial insight into model parameters. Advantages to the approach are identified, both in versatility and in regards to practical matters associated with implementing numerical fracture algorithms. It is found to perform well in dynamic fracture scenarios.
---
paper_title: Introduction to continuum damage mechanics

paper_content:

1 - Introduction.- 2 - Creep and Fracture under Uniaxial Stress.- 3 - Creep and Fracture under Multiaxial Stress.- 4 - Crack Growth under Creep Conditions.- 5 - Damage Model for Ductile Fracture.- 6 - Fatigue Damage.- References.- Notations.
---
paper_title: Simulation of Soft Tissue Failure Using the Material Point Method

paper_content:

Understanding the factors that control the extent of tissue damage as a result of material failure in soft tissues may provide means to improve diagnosis and treatment of soft tissue injuries. The objective of this research was to develop and test a computational framework for the study of the failure of anisotropic soft tissues subjected to finite deformation. An anisotropic constitutive model incorporating strain-based failure criteria was implemented in an existing computational solid mechanics software based on the material point method (MPM), a quasi-meshless particle method for simulations in computational mechanics. The constitutive model and the strain-based failure formulations were tested using simulations of simple shear and tensile mechanical tests. The model was then applied to investigate a scenario of a penetrating injury: a low-speed projectile was released through a myocardial material slab. Sensitivity studies were performed to establish the necessary grid resolution and time-step size. Results of the simple shear and tensile test simulations demonstrated the correct implementation of the constitutive model and the influence of both fiber family and matrix failure on predictions of overall tissue failure. The slab penetration simulations produced physically realistic wound tracts, exhibiting diameter increase from entrance to exit. Simulations examining the effect of bullet initial velocity showed that the anisotropy influenced the shape and size of the exit wound more at lower velocities. Furthermore, the size and taper of the wound cavity was smaller for the higher bullet velocity. It was concluded that these effects were due to the amount of momentum transfer. The results demonstrate the feasibility of using MPM and the associated failure model for large-scale numerical simulations of soft tissue failure.
---
paper_title: Failure mechanisms in fibrous scaffolds.

paper_content:

Polymeric fibrous scaffolds have been considered as replacements for load-bearing soft tissues, because of their ability to mimic the microstructure of natural tissues. Poor toughness of fibrous materials results in failure, which is an issue of importance to both engineering and medical practice. The toughness of fibrous materials depends on the ability of the microstructure to develop toughening mechanisms. However, such toughening mechanisms are still not well understood, because the detailed evolution at the microscopic level is difficult to visualize. A novel and simple method was developed, namely, a sample-taping technique, to examine the detailed failure mechanisms of fibrous microstructures. This technique was compared with in situ fracture testing by scanning electron microscopy. Examination of three types of fibrous networks showed that two different failure modes occurred in fibrous scaffolds. For brittle cracking in gelatin electrospun scaffolds, the random network morphology around the crack tip remained during crack propagation. For ductile failure in polycaprolactone electrospun scaffolds and nonwoven fabrics, the random network deformed via fiber rearrangement, and a large number of fiber bundles formed across the region in front of the notch tip. These fiber bundles not only accommodated mechanical strain, but also resisted crack propagation and thus toughened the fibrous scaffolds. Such understanding provides insight for the production of fibrous materials with enhanced toughness.
---
paper_title: Three Dimensional cohesive modeling of dynamic mixed-mode fracture

paper_content:

A cohesive formulation of fracture is taken as a basis for the simulation of processes of combined tension-shear damage and mixed-mode fracture in specimens subjected to dynamic loading. Our three-dimensional finite-element calculations account explicitly for crack nucleation, microcracking, the development of macroscopic cracks and inertia. In particular, a tension-shear damage coupling arises as a direct consequence of slanted microcrack formation in the process zone. We validate the model against the three-point-bend concrete beam experiments of Guo et al. (International Journal of Solids and Structures 1995; 32(17/18):2951–2607), John (PhD Thesis, Northwestern University, 1988), and John and Shah (Journal of Structural Engineering 1990; 116(3):585–602) in which a pre-crack is shifted from the central cross-section, leading to asymmetric loading conditions and the development of a mixed-mode process zone. The model accurately captures the experimentally observed fracture patterns and displacement fields, as well as crack paths and crack-tip velocities, as a function of pre-crack geometry and loading conditions. In particular, it correctly accounts for the competition between crack-growth and nucleation mechanisms.
---
paper_title: Strain Softening in Rat Left Ventricular Myocardium

paper_content:

We investigated whether strain softening (or the Mullins effect) may explain the reduced left ventricular stiffness previously associated with the strain-history-dependent preconditioning phenomenon. Passive pressure-volume relations were measured in the isolated, arrested rat heart during LV balloon inflation and deflation cycles. With inflation to a new higher maximum pressure, the pressure-volume relation became less stiff, particularly in the low (diastolic) pressure range, without a significant change in unloaded ventricular volume. In five different loading protocols in which the maximum passive cycle pressure ranged from 10 to 120 mmHg. we measured increases at 10 mmHg in LV volume up to 350 percent of unloaded volume that depended significantly on the history (p < 0.05) and magnitude (p < 0.01) of maximum previous pressure. Although a quasi-linear viscoelastic model based on the pressure-relaxation response could produce a nonlinear pressure-volume relation with hysteresis, it was unable to show any significant change in ventricular stiffness with new maximum pressure. We incorporated a strain softening theory proposed by Johnson and Beatty (1992) into the model by modifying the elastic response with a volume-amplification factor that depended on the maximum previous pressure. This model more accurately reproduced the experimentally observed behavior. Thus, the preconditioning behavior of the myocardium is better explained by strain softening rather than viscoelasticity and may be due to damage to elastic components, rather than the effects of viscous tissue components.
---
paper_title: Experimental study and constitutive modelling of the passive mechanical properties of the ovine infrarenal vena cava tissue.

paper_content:

The passive mechanical properties of the ovine infrarenal vena cava are analysed in this paper. In vivo stretch from 15 venae was measured in order to get data from the physiological situation before harvesting the vessel. Vena cava strips (n=64) both in longitudinal and circumferential directions were cut and subjected to simple tension tests. Results showed the strongly marked anisotropic character of the caval tissue. The maximal stretch ranges reached in both directions were very different, with the longitudinal range being much higher than the circumferential range in all cases. Three anisotropic constitutive models were used to fit the data obtained from the experiments. Advantages and drawbacks of each of these models are also discussed.
---
paper_title: Statistical approach for a continuum description of damage evolution in soft collagenous tissues

paper_content:

Abstract We propose a statistical approach to describe microscopic damage evolution in soft collagenous tissues such as arterial walls. The damage model extends a framework published by Balzani et al. (2012), Comput. Methods Appl. Mech. Engrg., 213–216:139–151, by postulating specific damage functions that result from the fibers’ microstructure. Statistical distributions of three different microscopic quantities such as proteoglycan orientation, fibril length parameters and ultimate proteoglycan stretch are considered. The resulting stress–stretch response is compared with experimental data obtained from uniaxial tension tests given in the literature. In particular, the individual statistical distributions are analyzed in regard to their ability to capture the distinct softening hysteresis observed when subjecting soft tissues to cyclic loading in the supra-physiological domain. Details regarding the algorithmic implementation are provided, and the applicability of the model within a finite element framework is shown by simulating the overexpansion of simplified atherosclerotic arteries.
---
paper_title: On damage modelling for elastic and viscoelastic materials at large strain

paper_content:

Abstract Geometrical and different physical nonlinearities have to be taken into account in order to model elastomeric material. In the present contribution, a novel approach of rubber elasticity is introduced. The approach considers the topological constraints as well as the limited extensibility of network chains in filled rubber. Subsequently, a formulation of finite viscoelastic damage is derived. With this description at hand, softening effects as well as time and frequency dependency of the material can be characterized. The goal of the present paper is twofold. Firstly, the constitutive formulations are introduced. Moreover, computational aspects are stressed. The shown approaches are presented in a format ready for a finite element implementation.
---
paper_title: Assessment of Tissue Damage due to Mechanical Stresses

paper_content:

While there are many benefits to minimally invasive surgery (MIS), force feedback or touch sensation is limited in the currently available MIS tools, such as surgical robots, creating the potential for excessive force application during surgery and unintended tissue injury. The goal of this work was to develop a methodology with which to identify stress magnitudes and durations that can be safely applied with a MIS grasper to di ferent tissues, potentially improving MIS device design and reducing potentially adverse clinically relevant consequences. Using the porcine model, stresses typically applied in MIS were applied to liver, ureter and small bowel using a motorized endoscopic grasper. Acute indicators of tissue damage including cellular death and infiltration of inflammatory cells were measured using histological and image analysis techniques. Finite element analysis was used to identify approximate stress distributions experienced by the tissues. Parameters used in these finite element models specifically reflected the properties of liver, which served as an initial proxy for all tissues, as stress distributions rather than absolute values were desired. Local regions predicted to have uniform stress by the computational models were mapped to and analyzed in the tissue samples for acute damage. Analysis of variance (ANOVA) and post-hoc analyses were used to detect stress magnitudes and durations that caused significantly increased tissue damage with the goal to ultimately identify safe stress `thresholds' during grasping of the studied tissues. Preliminary data suggests a graded non-linear response between applied stress magnitude and apoptosis in liver and small bowel as well as neutrophil infiltration in the small bowel. The ureter appeared to be more resistant to injury at the tested stress levels. By identifying stress magnitudes and durations within the range of grasping loads applied in MIS, it may be possible for researchers to create a `smart' surgical robot that can guide a surgeon to manipulate tissues with minimal resulting damage. In addition, surgical simulator design can be improved to reflect more realistic tissue responses and evaluate trainees' tissue handling skills.
---
paper_title: Strain softening behaviour in nonviable rat right-ventricular trabeculae, in the presence and the absence of butanedione monoxime.

paper_content:

Strain softening is commonly reported during mechanical testing of passive whole hearts. It is typically manifested as a stiffer force-extension relationship in the first deformation cycle relative to subsequent cycles and is distinguished from viscoelasticity by a lack of recovery of stiffness, even after several hours of rest. The cause of this behaviour is presently unknown. In order to investigate its origins, we have subjected trabeculae to physiologically realistic extensions (5-15% of muscle length at 26 degrees C and 0.5 mm Ca(2+)), while measuring passive force and dynamic stiffness. While we did not observe strain softening in viable trabeculae, we found that it was readily apparent in nonviable (electrically inexcitable) trabeculae undergoing the same extensions. This result was obtained in both the presence and absence of 2,3-butanedione monoxime (BDM). Furthermore, BDM had no effect on the passive compliance of viable specimens, while its presence partly inhibited, but could not prevent, stiffening of nonviable specimens. Loss of viability was accompanied by a uniform increase of dynamic stiffness over all frequencies examined (0.2-100 Hz). The presence of strain softening during length extensions of nonviable tissue resulted in a comparable uniform decrease of dynamic stiffness. It is therefore concluded that strain softening is neither intrinsic to viable rat right ventricular trabeculae nor influenced by BDM but, rather, reflects irreversible damage of tissue in partial, or full, rigor.
---
paper_title: An experimental study of the mouse skin behaviour : Damage and inelastic aspects

paper_content:

Abstract Samples of male and female mice skin were tested under monotonic and cyclic loading to mechanically characterize the tissue for large deformations. Cyclic tests have shown a typical Mullins effect widely known for elastomers and other soft tissues. No statistical difference was found in the maximum stretch of the sample after the fifth loading cycle for male ( 1.26 ± 0.035 ) and female ( 1.18 ± 0.083 ) . However, larger dispersion was obtained for the maximum stress for both genders, 0.61 ± 0.16 MPa for male and 0.78 ± 0.32 MPa for female. Results show the presence of inelastic strain and stress softening in the skin at large deformations. They also have shown how stress softening and residual strain change with the magnitude of the applied load. Good correlation was observed between the residual strain and the maximum strain previously attained by the sample during loading for all samples. However, the correlation was different between genders.
---
paper_title: Transversely isotropic properties of porcine liver tissue: experiments and constitutive modelling

paper_content:

Knowledge of the biomechanical properties of soft tissue, such as liver, is important in modelling computer aided surgical procedures. Liver tissue does not bear mechanical loads, and, in numerical simulation research, is typically assumed to be isotropic. Nevertheless, a typical biological soft tissue is anisotropic. In vitro uniaxial tension and compression experiments were conducted on porcine cylindrical and cubical liver tissue samples respectively assuming a simplistic architecture of liver tissue with its constituent lobule and connective tissues components. With the primary axis perpendicular to the cross sectional surface of samples, the tissue is stiffer with tensile or compressive force in the axial direction compared to that of the transverse direction. At 20% strain, about twice as much force is required to elongate a longitudinal tissue sample than that of a transverse sample. Results of the study suggest that liver tissue is transversely isotropic. A combined strain energy based constitutive equation for transversely isotropic material is proposed. The improved capability of this equation to model the experimental data compared to its previously disclosed isotropic version suggests that the assumption on the fourth invariant in the constitutive equation is probably correct and that anisotropy properties of liver tissue should be considered in surgical simulation.
---
paper_title: On the Mullins effect and hysteresis of fibered biological materials: A comparison between continuous and discontinuous damage models

paper_content:

Deformation induced softening is an inelastic phenomenon frequently accompanying mechanical behavior of soft biological tissues. This paper presents and compare continuous and discontinuous damage approaches to model softening effects in fibered materials such as soft biological tissues. The structural model is formulated using the concept of internal variables that provides a very general description of materials involving irreversible effects. We consider the internal variables associated to damage to correspond to separated contributions of the matrix and the fibers. Local damage accumulation is related to two phenomenological variables, the maximum value and the arclength of the effective free energy attained during the loading process, respectively. A local multiplicative decomposition of the deformation gradient into volume-preserving and dilatational parts is used that permits to model the incompressibility property of most types of soft biological tissues. In this context, damage is related only to the isochoric part of the deformation. Finally, simulations of biaxial and uniaxial tests in two directions are used to compare the performance of both models. Numerical simulations indicate that only a mixed model that consider both, continuous and discontinuous, damage models is able to capture the softening phenomenon of soft biological tissues.
---
paper_title: Constitutive modelling of brain tissue; experiment and theory

paper_content:

Recent developments in computer-integrated and robot-aided surgery--in particular, the emergence of automatic surgical tools and robots--as well as advances in virtual reality techniques, call for closer examination of the mechanical properties of very soft tissues (such as brain, liver, kidney, etc.). The ultimate goal of our research into the biomechanics of these tissues is the development of corresponding, realistic mathematical models. This paper contains experimental results of in vitro, uniaxial, unconfined compression of swine brain tissue and discusses a single-phase, non-linear, viscoelastic tissue model. The experimental results obtained for three loading velocities, ranging over five orders of magnitude, are presented. The applied strain rates have been much lower than those applied in previous studies, focused on injury modelling. The stress-strain curves are concave upward for all compression rates containing no linear portion from which a meaningful elastic modulus might be determined. The tissue response stiffened as the loading speed increased, indicating a strong stress-strain rate dependence. The use of the single-phase model is recommended for applications in registration, surgical operation planning and training systems as well as a control system of an image-guided surgical robot. The material constants for the brain tissue are evaluated. Agreement between the proposed theoretical model and experiment is good for compression levels reaching 30% and for loading velocities varying over five orders of magnitude.
---
paper_title: Viscoelastic properties of human skin and processed dermis.

paper_content:

BACKGROUND/AIMS ::: The purpose of this work is to attempt to determine the elastic spring constant for collagen and elastic fibers (elastin) in skin and to determine if the values of these elastic constants are similar to those reported for other tissues. ::: ::: ::: METHODS ::: We studied the viscoelastic mechanical properties of human skin and dermis by measuring the incremental stress-strain behavior. Elastic stress-strain curves were used to obtain the elastic spring constant of elastin and collagen while the collagen fibril length was obtained from the slope of viscous stress-strain curves. ::: ::: ::: RESULTS ::: Our results suggest that the elastic spring constant for elastin is about 4.0 MPa while that for collagen is about 4.4 GPa. The former value is similar to that calculated for ligamentum nuchae while the latter value is about 70% of the value found for tendon and self-assembled type I collagen fibers. The differences between the elastic constants for collagen molecules in tendon and skin is hypothesized to reflect the higher molecular tilt angle and lower D period found in skin compared to tendon as well as a shorter fibril length. ::: ::: ::: CONCLUSION ::: The differences in the collagen types present in skin and tendon may influence collagen self-assembly and the resulting viscoelastic properties.
---
paper_title: Viscoelastic properties of pig kidney in shear, experimental results and modelling

paper_content:

In this paper, the results from a series of rheological tests of fresh pig kidney have been reported. Using a standard strain-controlled rheometer, the oscillation strain sweep experiment showed a linear viscoelastic strain limit of the order of 0.2% strain. To determine the components of dynamic moduli in terms of frequency, shear oscillation tests were done at strain 0.2% using a stress-controlled rheometer. Shear stress relaxation tests were carried out with a fixed strain of 0.2% and 0.02 s rise time. The model we have developed uses a multi-mode upper convected Maxwell (UCM) model with variable viscosities and time constants, to which we have added a Mooney hyper-elastic response, both multiplied by a damping function. Different forms of damping functions that control the non-linearity of strain-stress profile have been tested. The model was fitted to our experimental data, and matched the entire test data reasonably well with a single set of parameters.
---
paper_title: A simple constitutive model for the Scapholunate ligament.

paper_content:

A simple constitutive equation is presented to describe the non-linear load-displacement behavior of the Scapholunate ligament. The model is based on an elastic large strain probabilistic constitutive equation and has been formulated offering three characteristic constants. Experimental data from five human cadaveric Scapholunate ligaments were used to corroborate the accuracy of the model. Good correlation (r(2)=0.90-0.98) was found with the experimental data. Representative characteristic constants for the human Scapholunate ligament based on the model were determined.
---
paper_title: An elastic stress-strain relation for soft biological tissues based on a structural model

paper_content:

Abstract We propose a mechanical model to describe the “quasi-elastic” stress-strain relation for soft biological tissues, based upon their fibre structured design. The corresponding elastic constitutive equation is derived and we show how the model parameters may be calculated to fit the model to experimental data. Some examples of such fittings are given for different materials such as human vein, fascia, tympanic membrane and rabbit papillary muscle. The good quality of these adaptations illustrates the multiple application possibility of the proposed model.
---

Title: Damage Models for Soft Tissues: A Survey

Section 1: Introduction
Description 1: In this section, provide an overview of what soft tissues are, their physiological functions, the types of soft tissue injuries, and the significance of damage modeling. Summarize the types of damage models and the organization of the paper.

Section 2: Rubber-Like Materials
Description 2: Discuss the application of continuum damage mechanics to isotropic soft tissues similar to rubber, the Mullins effect, and specific damage models developed for these materials.

Section 3: Continuum Damage Mechanics Method for Isotropic Materials
Description 3: Explain the fundamental principles of continuum damage mechanics applied to isotropic materials, including strain energy functions, damage variables, and stress-strain relationships.

Section 4: Damage Model for Abdominal Aortic Aneurysms
Description 4: Highlight the specific damage modeling approaches used for abdominal aortic aneurysms, and compare various strain energy functions and damage variables employed in such models.

Section 5: Deterministic Damage Models
Description 5: Review deterministic damage models for anisotropic soft tissues like arteries, ligaments, and tendons, detailing how these models account for the fibrous structure and softening effects.

Section 6: Probabilistic Damage Models of Fibers
Description 6: Summarize probabilistic approaches to damage modeling in fibrous tissues, including the statistical distribution of damage across fibers and the impact on the stress-stretch relationship.

Section 7: Microstructure-Based Damage Model of Fibers
Description 7: Describe microstructure-based models that focus on the behavior and damage of collagen fibrils, incorporating the effects of proteoglycan bridges and fibril orientation on tissue mechanics.

Section 8: Discussion and Challenges
Description 8: Discuss the key challenges in existing damage models, the complexity of parameter determination, and the computational effort required. Explore the limitations and potential improvements in these models.

Section 9: Experimental Data Curve-Fitting
Description 9: Provide examples of how damage models are validated against experimental data, including the process of fitting theoretical curves to experimental stress-stretch data.

Section 10: Computational Effort
Description 10: Discuss the computational demands of various damage models, highlighting the use of numerical methods and software tools for finite element analysis.

Section 11: Application of Damage Models
Description 11: Examine the practical applications of damage models in medical diagnostics, surgery, and the design of artificial soft tissues. Include examples of damage variable distributions in clinical scenarios.

Section 12: Connection Between Damage and Fracture
Description 12: Investigate the relationship between damage models and fracture mechanics, including the need for linking damage variables to crack propagation in soft tissues.

Section 13: Other Issues
Description 13: Address additional considerations such as viscoelastic effects, histological changes, active stress in smooth muscle, and the need for damage models for very soft tissues.

Section 14: Conclusion
Description 14: Summarize the key findings of the survey, emphasizing the strengths and limitations of existing damage models, and suggest future directions for research and application in biomedical engineering.