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Isotropic Hardening and Kinetic Hardening

Introduction

There are two types of hardening for elasto-plastic materials: isotropic and kinetic.

The yield surface uniformly yields in case of isotropic hardening, whereas it shifts when the loading is reversed in the case of kinetic hardening.

 

Yield Surface in Deviatoric Stress Space

The figures below show the yield surfaces of isotropic and kinetic hardening in the deviatoric stress space.

 

Subtract the mean stress from the stress tensor to get the deviatoric stress.

The material is in the region of elastic deformation if the norm of deviatoric stress is less than "square root of 2/3" times the yield stress.

The material starts to yield when the norm of deviatoric stress reaches "square root of 2/3" times the yield stress.

 

 

In the case of isotropic hardening, the yield stress increases with the yield progress.

It means the increase of radius in the figure above. The yield surface's centroid remains at the origin of the deviatoric stress space.

 

In the case of kinetic hardening, the norm of deviatoric stress increases with the yield progress,

but the radius doesn't. As such, the yield surface's centroid shifts from the origin.

The magnitude of this shift is called back stress.

Stress-Strain Curves of Isotropic and Kinetic Hardening

Isotropic and kinetic hardening are compared where stress is gradually loaded until the yield occurs, then the stress is loaded reversely.

 

Figures below shows the stress-strain curves of isotropic and kinetic hardening. The stress on the vertical axis is equivalent to "square root of 3/2" times the norm of deviatoric stress.

The stress is positive initially and negative when reversed. The absolute value is von Mises equivalent stress.

 

 

The object is loaded beyond the yield stress. It begins to deform plastically. It is further loaded up to State 1.

 

Then it is unloaded and reversely loaded.

The deformation is elastic around State 2.

 

It begins to deform plastically again. (State 3)

 

With isotropic hardening, the stress change from State 1 to State 3 is twice the maximum stress.

That means the von Mises equivalent stress of State 3 is equal to that of State 1.

 

With kinetic hardening, the stress change from State 1 to State 3 is twice the yield stress.

That means the von Mises equivalent stress of State 3 is less than that of State 1.

It is called the Bauschinger effect.

 

For the cyclic loading, the accumulated equivalent plastic strain is greater with kinetic hardening than with isotropic hardening.

 

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