## History

The isostrain assumption states that all crystals of the body deform exactly as the entire body. It is often also referred to as »full constraints (FC) Taylor« assumption as Taylor (1938) first applied it for the prediction of the deformation behavior of polycrystals.

## Deformation partitioning

In the framework of finite strain, the isostrain assumption can formally be written as equality of the deformation gradient of each grain $g$ with the average deformation gradient of the body $\cal{B}$ $$\label{eq: isostrain} \tnsr F^{g} = \bar{\tnsr F}\quad \forall \; g \in \cal{B}$$ or in rate form $$\label{eq: isostrain rate} \dot{\tnsr F^{g}} = \dot{\bar{\tnsr F}}\quad \forall \; g \in \cal{B}$$ at any time.

## Stress response

Typically, a different stress $\tnsr P^g$ will be required in each grain $g$ to obtain the prescribed deformation. This, for instance, can be due to anisotropy or different constitutive behavior (strength) of the grains. The isostrain scheme, therefore, usually violates stress equilibrium among the constituent grains.

### grain average (default)

The stress at the material point is calculated as the average of the stresses of all grains: $$\label{eq: stress average} \bar{\tnsr P}= \sum_{g=1}^{N}\nu^g\tnsr P^g$$ with $\nu^g = 1/N$ the (constant) volume fraction of grain $g$. This is the default behavior for the isostrain homogenization scheme.

### grains in parallel

In case all grains act in parallel, the stress at the material point is taken as the sum of the stresses of all grains: $$\label{eq: stress sum} \bar{\tnsr P}= \sum_{g=1}^{N}\tnsr P^g.$$

## Material configuration

### Parameters

To select the isostrain homogenization scheme and set the above parameters use the following (case-insensitive) naming scheme in a material.config file:

key value comment
type isostrain
Ngrains $N$ number of grains (of equal volume) at material point
mapping avg
mean
average
stress calculation according to \eqref{eq: stress average}
sum
parallel
stress calculation according to \eqref{eq: stress sum}

### Outputs

key output
(output) Ngrains report $N$ at material point

### Boolean flags

flag comment
/echo/ copy whole section to output log

## References

[1]
Taylor, G. I.
Plastic strain in metals
J. Inst. Metals 62 (1938) 307324

This topic: Documentation > Background > HomogenizationScheme > IsoStrain
Topic revision: 14 Dec 2015, PhilipEisenlohr
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