In materials science, plasticity is the ability of a solid material to undergo permanent deformation, a non-reversible change of shape in response to applied forces. For instance, a solid piece of metal being bent 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 known as yielding.

Plastic deformation is observed in most materials, especially in metals. In brittle materials such as rock, concrete, bone and so on, plasticity is caused mainly by slip at microcracks.

For ductile metals, each increment of load is accompanied by a proportional increment in extension under elastic behaviour. When the load is removed, the structure returns to its original size. However, once the load exceeds the yield strength of material, the extension increases more quickly. After elastic region reached, the extension would remain when the load is removed.

Plasticity in a crystal of pure metal is primarily caused by two modes of deformation in the crystal lattice:

  • Shear deformation
  • Plastic deformation

Plastic laws are all separated under two classification which are used in different applications;

  • Rate independent plasticity : This is used to model metal deformations at low temperatures at strain rates.
  • Rate dependent plasticity : This is used to model creep and metal deformations at high temperatures at strain rates.

Before jump in to plasticity in detail, let’s remember what the elasticity is. Elasticity can be summarises as, a material deforms under stress, then returns to its original shape when the stress is removed.

– Most metals have both elastic and plastic properties.
– Initially, materials show elastic behavior.
– After yielding, materials become plastic.
– By removing loading, materials become elastic again.

Elastic and Plastic Strain

One of the first steps in a plasticity algorithm is to find out that whether the material during a load increment is responding elastically or as elasto-plasticity. During the load step, the stress level and other parameters are required to compute the plastic strains if the material is not in the elastic range.

Elastic strain:

Elastic strain is temporal strain that does not remain after unloading.

Plastic strain:

Plastic strain is the permanent strain that remains after unloading. Total strain is the sum of elastic and plastic strains.

Yield Criterion

A yield criterion is used to mathematically identify when yielding happens in material.

There are many different yield criteria. The most common two yield criteria, which are mostly used in nonlinear material models in finite element analysis, are called as;

Material will undergo plastic (irreversible) deformation if the stress exceeds the critical value of material. This critical stress can be either of tensile or compressive. The Tresca and the von Mises criteria are commonly used to determine whether a material has yielded.

  • Von Mises yield criterion
  • Tresca’s yield criterion

Von Mises Yield Criterion

Von Mises yield criterion proves highly useful in cases of multi-axial loading. Von Mises yield criterion provide to obtain a equivalent scalar measure which can be compared with the yield stress of the material to predict that whether complex loading from the results at a point will cause material to yield or not.

In materials science and engineering, the Von Mises yield criterion can also be formulated in terms of the Von Mises stress or equivalent tensile stress. Material response can be a nonlinear elastic, viscoelastic, or linear elastic behavior. A material starts yielding when the Von Mises stress reaches a value which is known as yield strength.

Von Mises founds three principal stresses that can be calculated at any point in the x, y, z axis. The material can start yielding depending upon the combination of stresses.

Mathematical definition of Von-Mises yield criterion is defined as:

σy : The tensile yield strength of the material
k : The yield stress of the material
J2 : Plasticity / flow theory
s : The deviatoric stress

The Von Mises yield criterion is mathematically expressed as; J_{2}=k^{2}\,\!

At the onset of yielding, the magnitude of the shear yield stress in pure shear is √3 times lower than the tensile yield stress. In this case, yield stress is;

k={\frac  {\sigma _{y}}{{\sqrt  {3}}}}

If the equation is set the von Mises stress equal to the yield strength, the Von Mises yield criterion can be expressed as:

Substituting J2 with terms of the Cauchy stress tensor components;{\displaystyle \sigma _{\text{v}}^{2}={\frac {1}{2}}\left[(\sigma _{11}-\sigma _{22})^{2}+(\sigma _{22}-\sigma _{33})^{2}+(\sigma _{33}-\sigma _{11})^{2}+6\left(\sigma _{23}^{2}+\sigma _{31}^{2}+\sigma _{12}^{2}\right)\right]={\frac {3}{2}}s_{ij}s_{ij}}

Tresca’s Yield Criterion

The Tresca criterion is an equivalent calculation that yielding will occur at a critical value of the maximum shear stress when a material fails. The Tresca criterion is based on the notion of slip and dislocation motion in shear, which is a relatively good assumption when considering metals.

In the Tresca yield criterion states, a material yields if the critical shear stress is reached.

Mathematical definition of Tresca yield criterion is defined as:

σ1 : The maximum normal stress.
σ3 : The minimum normal stress.
σ0 : The stress of the material that fails in uniaxial loading.

The value of k can be achieved from an experiment. For instance, failure occurs when σ0 reaches Y. It becomes;
k = Y / 2

In a shear experiment, failure occurs when τ reaches τY.
σ1=τ, σ2=0, σ2=-τk = τy

Flow Plasticity

Flow plasticity is used to qualify the plastic behavior of materials. Flow plasticity theories are characterized to determine the amount of plastic deformation by the assumption of a flow rule.

In flow plasticity theory, it is assumed that the total strain in a structure can be dissolved iteratively into an elastic part and a plastic part. The elastic part of the strain state can be procured from a linear elastic or hyperelastic equation. The plastic part of the strain state requires a flow rule and a hardening model in order to get calculated.

The theory of dislocations could be used to explain the plastic deformation of ductile materials. The mathematical theory of flow plasticity uses a group of non-linear equations to describe the changes on strain and stress states.

Plasticity algorithms are often used in finite element analysis programs to model non-linear material behaviour. In situations such as this, the finite element analysis programs are incremental, and the plasticity algorithm is used at the element level in the finite element analysis program during each of these increments.

More detail about Flow Plasticity:

Material Hardening Laws