Plasticity (physics)

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"Plastic material" redirects here. For the material used in manufacturing, see Plastic.
Stress–strain curve showing typical yield behavior for nonferrous alloys. (Stress () shown as a function of strain ().)
A stress–strain curve typical of structural steel.

In physics and materials science, plasticity describes the deformation of a (solid) material undergoing non-reversible changes of shape in response to applied forces.[1][2] For example, a solid piece of metal being bent or pounded into a new shape displays plasticity as permanent changes occur within the material itself. In engineering, the transition from elastic behavior to plastic behavior is called yield.

Plastic deformation is observed in most materials, particularly metals, soils, rocks, concrete, foams, bone and skin.[3][4][5][6][7][8] However, the physical mechanisms that cause plastic deformation can vary widely. At a crystalline scale, plasticity in metals is usually a consequence of dislocations. Such defects are relatively rare in most crystalline materials, but are numerous in some and part of their crystal structure; in such cases, plastic crystallinity can result. In brittle materials such as rock, concrete and bone, plasticity is caused predominantly by slip at microcracks.

For many ductile metals, tensile loading applied to a sample will cause it to behave in an elastic manner. Each increment of load is accompanied by a proportional increment in extension. When the load is removed, the piece returns to its original size. However, once the load exceeds a threshold – the yield strength – the extension increases more rapidly than in the elastic region; now when the load is removed, some degree of extension will remain.

Elastic deformation, however, is an approximation and its quality depends on the time frame considered and loading speed. If, as indicated in the graph opposite, the deformation includes elastic deformation, it is also often referred to as "elasto-plastic deformation" or "elastic-plastic deformation".

Perfect plasticity is a property of materials to undergo irreversible deformation without any increase in stresses or loads. Plastic materials with hardening necessitate increasingly higher stresses to result in further plastic deformation. Generally, plastic deformation is also dependent on the deformation speed, i.e. higher stresses usually have to be applied to increase the rate of deformation. Such materials are said to deform visco-plastically.

Contributing properties[edit]

The plasticity of a material is directly proportional to the ductility and malleability of the material.

Physical mechanisms[edit]

Plasticity under a spherical Nanoindenter in (111) Copper. All particles in ideal lattice positions are omitted and the color code refers to the von Mises stress field.

Plasticity in metals[edit]

Plasticity in a crystal of pure metal is primarily caused by two modes of deformation in the crystal lattice, slip and twinning. Slip is a shear deformation which moves the atoms through many interatomic distances relative to their initial positions. Twinning is the plastic deformation which takes place along two planes due to a set of forces applied to a given metal piece. Most metals show more plasticity when hot than when cold. Lead shows sufficient plasticity at room temperature, while cast iron does not possess sufficient plasticity for any forging operation even when hot. This property is of importance in forming, shaping and extruding operations on metals. Most metals are rendered plastic by heating and hence shaped hot.

Slip systems[edit]

Crystalline materials contain uniform planes of atoms organized with long-range order. Planes may slip past each other along their close-packed directions, as is shown on the slip systems page. The result is a permanent change of shape within the crystal and plastic deformation. The presence of dislocations increases the likelihood of planes slipping.

Reversible plasticity[edit]

On the nanoscale the primary plastic deformation in simple face centered cubic metals is reversible, as long as there is no material transport in form of cross-glide.[9]

Shear banding[edit]


Plasticity in amorphous materials[edit]



Plasticity in martensitic materials[edit]


Plasticity in cellular materials[edit]


Plasticity in soils and sand[edit]

Soils, particularly clays, display a significant amount of inelasticity under load. The causes of plasticity in soils can be quite complex and are strongly dependent on the microstructure, chemical composition, and water content. Plastic behavior in soils is caused primarily by the rearrangement of clusters of adjacent grains.

Plasticity in rocks and concrete[edit]

Main article: rock mass plasticity

Inelastic deformations of rocks and concrete are primarily caused by the formation of microcracks and sliding motions relative to these cracks. At high temperatures and pressures, plastic behavior can also be affected by the motion of dislocations in individual grains in the microstructure.

Mathematical descriptions of plasticity[edit]

Deformation theory[edit]

An idealized uniaxial stress-strain curve showing elastic and plastic deformation regimes for the deformation theory of plasticity

There are several mathematical descriptions of plasticity.[10] One is deformation theory (see e.g. Hooke's law) where the Cauchy stress tensor (of order d in d dimensions) is a function of the strain tensor. Although this description is accurate when a small part of matter is subjected to increasing loading (such as strain loading), this theory cannot account for irreversibility.


Flow plasticity theory[edit]


Yield criteria[edit]

Comparison of Tresca criterion to Von Mises criterion
Main article: Yield (engineering)


Tresca criterion[edit]


Where σ1 is the maximum normal stress, σ3 is the minimum normal stress, and σ0 is the stress under which the material fails in uniaxial loading. A yield surface may be constructed, which provides a visual representation of this concept. Inside of the yield surface, deformation is elastic. On the surface, deformation is plastic. It is impossible for a material to have stress states outside its yield surface.

Huber-von Mises criterion[edit]

The von Mises yield surfaces in principal stress coordinates circumscribes a cylinder around the hydrostatic axis. Also shown is Tresca's hexagonal yield surface.

This criterion[11] is based on the Tresca criterion but takes into account the assumption that hydrostatic stresses do not contribute to material failure. M.T. Huber was the first[12] who proposed the criterion of shear energy (see S. P. Timoshenko,p. 369 [13]). Von Mises solves for an effective stress under uniaxial loading, subtracting out hydrostatic stresses, and claims that all effective stresses greater than that which causes material failure in uniaxial loading will result in plastic deformation.

Again, a visual representation of the yield surface may be constructed using the above equation, which takes the shape of an ellipse. Inside the surface, materials undergo elastic deformation. Reaching the surface means the material undergoes plastic deformations. It is physically impossible for a material to go beyond its yield surface.

See also[edit]


  3. ^ M. Jirasek and Z. P. Bazant, 2002, Inelastic analysis of structures, John Wiley and Sons.
  4. ^ W-F. Chen, 2008, Limit Analysis and Soil Plasticity, J. Ross Publishing
  5. ^ M-H. Yu, G-W. Ma, H-F. Qiang, Y-Q. Zhang, 2006, Generalized Plasticity, Springer.
  6. ^ W-F. Chen, 2007, Plasticity in Reinforced Concrete, J. Ross Publishing
  7. ^ J. A. Ogden, 2000, Skeletal Injury in the Child, Springer.
  8. ^ J-L. Leveque and P. Agache, ed., 1993, Aging skin:Properties and Functional Changes, Marcel Dekker.
  9. ^ Gerolf Ziegenhain and Herbert M. Urbassek: Reversible Plasticity in fcc metals. In: Philosophical Magazine Letters. 89(11):717-723, 2009 DOI
  10. ^ R. Hill, 1998, The Mathematical Theory of Plasticity, Oxford University Press.
  11. ^ von Mises, R. (1913). Mechanik der Festen Korper im plastisch deformablen Zustand. Göttin. Nachr. Math. Phys., vol. 1, pp. 582–592.
  12. ^ Huber, M. T. The Specific Shear Strain Work as Criterion of material strength. Czasopismo Techniczne, Lwów (1904).
  13. ^ S. P. Timoshenko, History of Strength of Materials, New York, Toronto, London, McGraw-Hill Book Company,Inc., 1953.

Further reading[edit]

  • International Journal of Plasticity, Elsevier Science.
  • Han W and Reddy BD, Plasticity: Mathematical Theory and Numerical Analysis. 2nd edition, Springer, New York (2013).