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Continuum mechanics

In physics and materials science, plasticity describes the deformation of a material undergoing non-reversible changes of shape in response to applied forces. For example, a solid piece of metal or plastic being bent or pounded into a new shape displays plasticity as permanent changes occur within the material itself. By contrast, a permanent crease in a sheet of paper or a re-shaping of wet clay is due to a rearrangement of separate fibers or particles. In engineering, the transition from elastic behavior to plastic behavior is called yield.



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, and when the load is removed, the piece returns exactly to its original size. However, once the load exceeds some threshold (the yield strength), the extension increases more rapidly than in the elastic region, and when the load is removed, some amount of the extension remains. A generic graph displaying this behavior is below.

It must be noted however that elastic deformation is an approximation and its quality depends on the considered time frame and loading speed. If the deformation behavior includes elastic deformation as indicated in the graph below it is also often referred to elastic-plastic or elasto-plastic deformation.

Plasticity describes the behavior of materials which undergo irreversible deformation without fracture or damage. This is found in most metals, and in general is a good description for a large class of materials. 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. usually higher stresses have to be applied to increase the rate of deformation and such materials are said to deform visco-plastically.

Microscopically at the crystal level, plasticity in metals is usually a consequence of dislocations. In most crystalline materials such defects are a rare exception on the rule presented by unit cell of the crystal. However, there are also materials where defects are very numerous and are part of the very crystal structure, in such cases plastic crystallinity can result.

Mathematical descriptions of plasticity


Deformation theory

There are several mathematical descriptions of Plasticity. One is deformation theory (see e.g. Hooke's law) where the 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.


The image above represents a shear stress component with respect to a shear strain component, under increasing strain loading.

Ductile materials can sustain large plastic deformations without fracture. However, even ductile metals will fracture when the strain becomes large enough - this is as a result of work-hardening of the material, which causes it to become brittle. Heat treatment such as annealing can restore the ductility of a worked piece, so that shaping can continue.

Flow plasticity theory

In 1934, Egon Orowan, Michael Polanyi and Geoffrey Ingram Taylor, roughly simultaneously, realized that the plastic deformation of ductile materials could be explained in terms of the theory of dislocations. The more correct mathematical theory of plasticity, flow plasticity theory, uses a set of non-linear, non-integrable equations to describe the set of changes on strain and stress with respect to a previous state and a small increase of deformation.

Yield criteria

Comparison of Tresca criterion to Von Mises criterion.

If the stress exceeds a critical value, as was mentioned above, the material will undergo plastic, or irreversible, deformation. This critical stress can be tensile or compressive. The Tresca and the von Mises criteria are commonly used to determine whether a material has yielded. However, these criteria have proved inadequate for a large range of materials and several other yield criteria are in widespread use.

Tresca criterion

This criterion is based on the notion that when a material fails, it does so in shear, which is a relatively good assumption when considering metals. Given the principal stress state, we can use Mohr’s circle to solve for the maximum shear stresses our material will experience and conclude that the material will fail if:

\sigma_1 - \sigma_3 \ge \sigma_0

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. Outside of the surface, deformation is plastic.

Von Mises criterion

This criterion is based on the Tresca criterion but takes into account the assumption that hydrostatic stresses do not contribute to material failure. 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.

\sigma_{\mathrm{effective}}^2 = \frac{1}{2} \left [ \left (\sigma_{11} - \sigma_{22} \right )^2 + \left (\sigma_{22} - \sigma_{33} \right )^2 + \left (\sigma_{11} - \sigma_{33} \right)^2 \right ] + 3 \left(\sigma_{12}^2 + \sigma_{13}^2 + \sigma_{23}^2 \right )

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. Outside of the surface they undergo plastic deformation.

Atomic mechanisms

Slip systems

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 wiki 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.

Shear banding

The presence of other defects within a crystal may entangle dislocations or otherwise prevent them from gliding. When this happens, plasticity is localized to particular regions in the material. For crystals, these regions of localized plasticity are called shear bands.


In amorphous materials, the discussion of “dislocations” is inapplicable, since the entire material lacks long range order. These materials can still undergo plastic deformation. Since amorphous materials, like polymers, are not well-ordered, they contain a large amount of free volume, or wasted space. Pulling these materials in tension opens up these regions and can give materials a hazy appearance. This haziness is the result of crazing, where fibrils are formed within the material in regions of high hydrostatic stress. The material may go from an ordered appearance to a "crazy" pattern of strain and stretch marks.

Martensitic materials

Some materials, especially those prone to Martensitic transformations, deform in ways that are not well described by the classic theories of plasticity and elasticity. One of the best-known examples of this is nitinol, which exhibits pseudoelasticity: deformations which are reversible in the context of mechanical design, but irreversible in terms of thermodynamics.

Cellular materials

These materials plastically deform when the bending moment exceeds the fully plastic moment. This applies to open cell foams where the bending moment is exerted on the cell walls. The foams can be made of any material with a plastic yield point which includes rigid polymers and metals. This method of modeling the foam as beams is only valid if the ratio of the density of the foam to the density of the matter is less than 0.3. This is because beams yield axially instead of bending. In closed cell foams, the yield strength is increased if the material is under tension because of the membrane that spans the face of the cells.

See also


  • R. Hill, The Mathematical Theory of Plasticity, Oxford University Press (1998).
  • Jacob Lubliner, Plasticity Theory, Macmillan Publishing, New York (1990).
  • L. M. Kachanov, Fundamentals of the Theory of Plasticity, Dover Books.
  • A.S. Khan and S. Huang, Continuum Theory of Plasticity, Wiley (1995).
  • J. C. Simo, T. J. Hughes, Computational Inelasticity, Springer.
  • M. F. Ashby. Plastic Deformation of Cellular Materials. Encyclopedia of Materials: Science and Technology, Elsevier, Oxford, 2001, Pages 7068-7071.
  • Van Vliet, K. J., 3.032 Mechanical Behavior of Materials, MIT (2006)
  • International Journal of Plasticity, Elsevier Science.


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