Yield criterion

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Yield criterion

Yield criterion is an expression constraining the components of stress in a point and determining the transition to a plastic state. Usually a plastic state is preceded by an elastic state. Elastic deformation is fully recovered after the removal of applied loads. Plastic deformations after removal of all the external loads remain in the deformed material and cause the distortion of deformed body initial shape. Thus, the new workpiece shape obtained during the forming operation is the consequence of plastic deformations.

Click to show/hide hidden textTransition to plastic state at uniaxial tension

For uniaxial tension the plastic state occurs when the tensile stress σ1 attains a certain threshold value σY known as yield stress. On the graph of stress against strain along the specimen axis this point corresponds to the moment of occurrence of nonlinear dependence of stress on strain.

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If we unload the specimen that was previously elongated for more, then yield stress up to the value σS, then at its next loading the transition to a plastic state will occur at

This expression is the yield criterion at uniaxial tension.

Alternating stress σS acting along the axis at uniaxial tension of specimen is called the flow stress or the resistance to plastic deformation. It depends on the cumulative plastic deformation, temperature and strain rate. It depends on the cumulative plastic deformation, temperature and strain rate.
Click to show/hide hidden textThe transition to a plastic state in general

In general terms the yield criterion can be written down in a form:

Here  

- stress tensor components

- flow stress or material resistance to plastic deformation, which changes in the process of plastic deformation

Geometrically the yield criterion represents the surface in the stress space (one of the stress components is drawn along every axis of stress space). In the course of plastic deformation the plastic surface in the general case changes its size and location. If the plastic surface expands uniformly in all directions, then an isotropic hardening takes place.

InQForm UK von Mises yield criterion is used for compact (continuum) material and Green yield criterion is used for non-compact (porous) materials.

According to von Mises yield criterion in a plastic state, the effective stress is equal to the flow stress of the material

For uniaxial tension it is true that

then von Mises criterion also holds to the full extent for uniaxial tension.

The explanation of physical meaning of von Mises criterion was made by Hencky (1924). He showed that the transition to the plastic state according to von Mises criterion occurs at reaching, by the deformation specific energy, its limit value that is independent on the stress state mechanism and is the characteristic of material.

It follows from von Mises criterion that at plastic deformation the maximum shear stresses depends on the metal flow stress and can not exceed the following values

Here k is the maximum shear stress at shear deformation.

Shown dependence leads to specific features of accounting for friction at plastic deformation.

Noncompact material is considered as isotropic continuum medium possessing the incompressibility properties. Yield criterion is accepted in a form suggested by R. Green,

where A and B  are the functions of material relative density, and B=1 - 1/3A, σR is the reduced yield stress for porous material with relative density R at uniaxial tension, J2 is the second invariant of stress deviator, I1 is the first invariant of stress deviator.

Here

Reduced effective strain rate for non-compact material is determined from the condition of powers equality

and is equal to

Here

- strain rate deviator

- volumetric strain rate

Yield stress for porous material σR is related to yield stress of base material (material flow stress with relative density of 100%) σS by the relation

where η is the function of relative density.

Porous material's relative density is determined with the ratio of current density of representative volume of porous body to base material density:

The dependence of functions A and h on relative density  R for R>0.7 is determined with correlations suggested by Shima S, Oyane M.

The magnitude of material initial relative density is to be set while preparing simulation data

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