Flow plasticity theory

However, determination of the plastic part of the strain requires a flow rule and a hardening model.

Instead, it is typically assumed that the plastic strain increment and the normal to the pressure-dependent yield surface have the same direction, i.e., where

The above flow rule is easily justified for perfectly plastic deformations for which

, i.e., the yield surface remains constant under increasing plastic deformation.

For a work hardening material, the yield surface can expand with increasing stress.

Examination of the work done over a cycle of plastic loading-unloading can be used to justify the validity of the associated flow rule.

[2] The Prager consistency condition is needed to close the set of constitutive equations and to eliminate the unknown parameter

, and hence Large deformation flow theories of plasticity typically start with one of the following assumptions: The first assumption was widely used for numerical simulations of metals but has gradually been superseded by the multiplicative theory.

The concept of multiplicative decomposition of the deformation gradient into elastic and plastic parts was first proposed independently by B.

The spatial velocity gradient is given by where a superposed dot indicates a time derivative.

We can write the above as The quantity is called a plastic velocity gradient and is defined in an intermediate (incompatible) stress-free configuration.

The elastic behavior in the finite strain regime is typically described by a hyperelastic material model.

The elastic strain can be measured using an elastic right Cauchy-Green deformation tensor defined as: The logarithmic or Hencky strain tensor may then be defined as The symmetrized Mandel stress tensor is a convenient stress measure for finite plasticity and is defined as where S is the second Piola-Kirchhoff stress.

Plastic deformation of a thin metal sheet.
Stress-strain curve showing typical plastic behavior of materials in uniaxial compression. The strain can be decomposed into a recoverable elastic strain ( ) and an inelastic strain ( ). The stress at initial yield is . For strain hardening materials (as shown in the figure) the yield stress increases with increasing plastic deformation to a value of .