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# Plasticity (physics)

Continuum mechanics
Conservation of mass
Conservation of momentum
Navier-Stokes equations
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In physics and materials science, plasticity is a property of a material to undergo a non-reversible change of shape in response to an applied force. 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.

## Explanation

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.

Plasticity is a property of materials to undergo large deformation without fracture. This is found in most metals, and in general is a good description of a large class of materials. Perfect plasticity is a property of materials to undergo large shear deformation without any increase of (shear) stress. Plastic materials that are not perfectly plastic are visco-plastic.

Microscopically, plasticity in metals is a consequence of dislocations.

## 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.

## Elastic vs plastic failure

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.

### 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:

σ1 - σ3 ≥ σ0

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. See Henri Tresca.

### 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.

σeffective² = 1/2 ((σ11 – σ22)² + (σ22 – σ33)² + (σ11 – σ33)²) + 3 (σ12² + σ13² + σ23²)

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. See Von Mises stress

## 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.

### Crazing

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 mater 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.