# Plasticity (physics)

Stress–strain curve showing typical yield behavior for nonferrous alloys. (Stress${\displaystyle \sigma }$, shown as a function of strain${\displaystyle \epsilon }$.)
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. In cellular materials such as liquid foams or biological tissues, plasticity is mainly a consequence of bubble or cell rearrangements, notably T1 processes.

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 propertiesEdit

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

## Physical mechanismsEdit

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 metalsEdit

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 systemsEdit

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.

#### Reversible plasticityEdit

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 bandingEdit

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.

### Plasticity in amorphous materialsEdit

#### CrazingEdit

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.

### Plasticity in martensitic materialsEdit

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. In the case of iron, the martensitic phase transformation from bcc to hcp phases induces significant work hardening.[10]

### Plasticity in cellular materialsEdit

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.

### Plasticity in soils and sandEdit

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 concreteEdit

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 plasticityEdit

### Deformation theoryEdit

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.[11] One is deformation theory (see e.g. Hooke's law) where the Cauchy stress tensor (of order d-1 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.

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 theoryEdit

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 criteriaEdit

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 also in widespread use.

### Tresca criterionEdit

The Tresca 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:

${\displaystyle \sigma _{1}-\sigma _{3}\geq \sigma _{0}}$

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. On the surface, deformation is plastic. It is impossible for a material to have stress states outside its yield surface.

### Huber-von Mises criterionEdit

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

The Huber-von Mises criterion[12] 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[13] who proposed the criterion of shear energy (see S. P. Timoshenko, p. 369.[14]). Von Mises solves for an effective stress under uniaxial loading, subtracting out hydrostatic stresses, and states that all effective stresses greater than that which causes material failure in uniaxial loading will result in plastic deformation.

${\displaystyle \sigma _{v}^{2}={\tfrac {1}{2}}[(\sigma _{11}-\sigma _{22})^{2}+(\sigma _{22}-\sigma _{33})^{2}+(\sigma _{11}-\sigma _{33})^{2}+6(\sigma _{23}^{2}+\sigma _{31}^{2}+\sigma _{12}^{2})]}$

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.

## ReferencesEdit

1. ^ J. Lubliner, 2008, Plasticity theory, Dover, ISBN 0-486-46290-0, ISBN 978-0-486-46290-5.
2. ^ Bigoni, D. Nonlinear Solid Mechanics: Bifurcation Theory and Material Instability. Cambridge University Press, 2012 . ISBN 9781107025417.
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. ^ Belof JL, Cavallo RM, Olson RT, King RS, Gray GT, Holtkamp D, Chen S, Rudd RE, Barton N, Arsenlis A, Remington B, Park H, Prisbrey ST, Vitello P, Bazan G, Mikaelian KO, Comley AJ, Maddox B, May MJ (2012). "Rayleigh-Taylor strength experiments of the pressure-induced α→ε→α′ phase transition in iron". AIP Conf. Proc. 1426: 1521–1524. doi:10.1063/1.3686572.
11. ^ R. Hill, 1998, The Mathematical Theory of Plasticity, Oxford University Press.
12. ^ von Mises, R. (1913). Mechanik der Festen Korper im plastisch deformablen Zustand. Göttin. Nachr. Math. Phys., vol. 1, pp. 582–592.
13. ^ Huber, M. T. The Specific Shear Strain Work as Criterion of material strength. Czasopismo Techniczne, Lwów (1904).
14. ^ S. P. Timoshenko, History of Strength of Materials, New York, Toronto, London, McGraw-Hill Book Company, Inc., 1953.