Strain Rate Sensitivity for Linear Mobility Model

Linear Mobility

The question has been rased: "What are the consequences in assuming a linear mobility law?" Wei suggested that, under the assumption of linear mobility, the stress strain curve produced would be subjected to unrealistic constraints. It was implied that this was linked to the strain rate dependence, but he could not remember exactly what the claim was.

Elastoplastic Response in One Dimension

In order to answer the above questions, we will start with a simple analytical model in one dimension. A one dimensional model will avoid the complications that arise using tensors but will still capture the important points of the analysis.

From an experimental point of view, we generally impose a constant strain rate and measure the materials stress response. The strain rate can generally be decomposed into elastic and plastic parts

${\dot {e}}={\dot {e}}^{e}+{\dot {e}}^{p}$

The elastic strain rate must be related to the stress rate through Hooke's law

${\dot {e}}^{e}={\frac {\dot {\sigma }}{E}}$

and the plastic strain rate can be approximated as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{e}^{p} = \rho v b }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho} is the dislocation density, v is the average dislocation velocity, and b is the burger's vector. If we now assume a linear mobility, then the average dislocation velocity can be related to the applied stress as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v = M \sigma b }

and the resulting stress strain relationship becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{e} = \frac{\dot{\sigma}}{E} + M \rho b^{2} \sigma }