pyoptmat.flowrules: inelastic flow rules for structural materials

Module containing inelastic flow rules. These provide the rate of the viscoplastic strain and internal variables as a function of the stress, the current values of the internal variables, the time, and the temperature.

So these objects define two functions

\[ \begin{align}\begin{aligned}\sigma, h, t, T \rightarrow \dot{\varepsilon}_{in}\\\sigma, h, t, T \rightarrow \dot{h}\end{aligned}\end{align} \]

In addition, the object needs to define the derivative of the inelastic strain rate and internal variable evolution rates with respect to the current values of stress and the current values of the internal variables. The objects return “self” derivatives (i.e. the derivative of the inelastic strain rate with respect to stress and the derivative of the internal variable rate with respect to the internal variables) along with the rates themselves. The “cross” derivatives are defined with separate methods.

class pyoptmat.flowrules.FlowRule

Bases: Module

Superclass for flow rule models

This implementation provides default zero cross derivatives and that’s it.

dflow_derate(s, h, t, T, e)

The derivative of the flow rate with respect to the total strain rate

The superclass implementation provides a default of zero with the right shape.

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – internal variables

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

derivative of flow rate with respect to the

internal variables

Return type:

torch.tensor

dflow_dhist(s, h, t, T, e)

The derivative of the flow rate with respect to the internal variables

The superclass implementation provides a default of zero with the right shape.

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – internal variables

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

derivative of flow rate with respect to the

internal variables

Return type:

torch.tensor

dhist_derate(s, h, t, T, e)

The derivative of the flow rate with respect to the total strain rate

The superclass implementation provides a default of zero with the right shape.

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – internal variables

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

derivative of flow rate with respect to the strain rate

Return type:

torch.tensor

dhist_dstress(s, h, t, T, e)

The derivative of the flow rate with respect to the stress

The superclass implementation provides a default of zero with the right shape.

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – internal variables

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

derivative of flow rate with respect to the stress

Return type:

torch.tensor

class pyoptmat.flowrules.IsoKinViscoplasticity(n, eta, s0, isotropic, kinematic)

Bases: FlowRule

Viscoplasticity with isotropic and kinematic hardening, defined as

\[\dot{\varepsilon}_{in}=\left\langle \frac{\left|\sigma-x\right|-s_{0}-k}{\eta}\right\rangle ^{n}\operatorname{sign}\left(\sigma-X\right)\]

and where the pyoptmat.hardening.IsotropicHardeningModel and pyoptmat.hardening.KinematicHardeningModel objects determine the history rate.

The pyoptmat.hardening.IsotropicHardeningModel and pyoptmat.hardening.KinematicHardeningModel objects each define both a set of internal variables, including the corresponding rate forms and Jacobians, but also a map from those internal variables to the isotropic hardening value \(k\) (for pyoptmat.hardening.IsotropicHardeningModel) and the kinematic hardening value \(x\) (for pyoptmat.hardening.KinematicHardeningModel), along with the derivatives of those maps. All this information is required to assemble the information this class needs to provide.

Parameters:
dflow_dhist(s, h, t, T, e)

The derivative of the flow rate with respect to the internal variables

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the derivative of the flow rate

Return type:

torch.tensor

dflow_diso(s, h, t, T, e)

The derivative of the flow rate with respect to the isotropic hardening

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the derivative of the flow rate

Return type:

torch.tensor

dflow_dkin(s, h, t, T, e)

The derivative of the flow rate with respect to the kinematic hardening

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the derivative of the flow rate

Return type:

torch.tensor

dhist_derate(s, h, t, T, e)

The derivative of the flow rate with respect to the total strain rate

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – internal variables

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

derivative of flow rate with respect to the strain rate

Return type:

torch.tensor

dhist_dstress(s, h, t, T, e)

The derivative of the history rate with respect to the stress

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the derivative of the flow rate

Return type:

torch.tensor

flow_rate(s, h, t, T, e)

The flow rate itself and the derivative with respect to stress

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the flow rate and the derivative of the flow rate with

respect to stress

Return type:

tuple(torch.tensor, torch.tensor)

history_rate(s, h, t, T, e)

The vector of the rates of the internal variables split into portions defined by each hardening model

The first chunk of entries is for the isotropic hardening, the second for the kinematic hardening.

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the history rate and the

derivative of the history rate with respect to history

Return type:

tuple(torch.tensor, torch.tensor)

property nhist

The number of internal variables, here the sum from the isotropic and kinematic hardening models

class pyoptmat.flowrules.KocksMeckingRegimeFlowRule(model1, model2, g0, mu, b, eps0, k, eps=tensor(1.0000e-20), g0_scale=<function KocksMeckingRegimeFlowRule.<lambda>>)

Bases: FlowRule

Switches between two different flow rules depending on the value of the Kocks-Mecking normalized activation energy

\[g = \frac{kT}{\mu b^3} \log{\frac{\dot{\varepsilon}_0}{\dot{\varepsilon}}}\]

with \(k\) the Boltzmann constant, \(T\) temperature, \(\mu\) the shear modulus, \(b\) a representative Burgers vector, \(\dot{\varepsilon}_0\) a reference strain rate, and \(\dot{\varepsilon}\) the current applied strain rate.

If the activation energy is less than or equal to a threshold \(g_0\) the flow rate is equal to that of the first model. If the activation energy is greater than the threshold then the model switches to the second flow rule.

The two models should have compatible internal history vectors.

Parameters:
Keyword Arguments:

  • eps (float) – default 1e-20, offset to avoid divide-by-zero

  • g0_scale (function) – scaling function for g0, defaults to no scaling

dflow_derate(s, h, t, T, e)

The derivative of the flow rate with respect to the total strain rate

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – internal variables

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

derivative of flow rate with respect to the

internal variables

Return type:

torch.tensor

dflow_dhist(s, h, t, T, e)

The derivative of the flow rate with respect to the internal variables

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the derivative of the flow rate

Return type:

torch.tensor

dhist_derate(s, h, t, T, e)

The derivative of the history rate with respect to the stress

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the derivative of the flow rate

Return type:

torch.tensor

dhist_dstress(s, h, t, T, e)

The derivative of the history rate with respect to the stress

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the derivative of the flow rate

Return type:

torch.tensor

flow_rate(s, h, t, T, e)

The uniaxial flow rate itself and the derivative with respect to stress

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the flow rate and the derivative

of the flow rate with respect to stress

Return type:

tuple(torch.tensor, torch.tensor)

g(T, e)

The current value of activation energy

Parameters:

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

value of the activation energy

Return type:

torch.tensor

history_rate(s, h, t, T, e)

The history rate and the derivative of the history rate with respect to the current history

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the history rate and the

derivative of the history rate with respect to history

Return type:

tuple(torch.tensor, torch.tensor)

property nhist

The number of internal variables

switch_values(vals1, vals2, T, e)

Switch between the two model results

Parameters:

  • vals1 (torch.tensor) – values from first model

  • vals2 (torch.tensor) – values from second model

  • T (torch.tensor) – temperatures

  • e (torch.tensor) – strain rates

class pyoptmat.flowrules.PerfectViscoplasticity(n, eta)

Bases: FlowRule

Perfect viscoplasticity defined as

\[ \begin{align}\begin{aligned}\dot{\varepsilon}_{in}=\left(\frac{\left|\sigma\right|}{\eta}\right)^{n}\operatorname{sign}\left(\sigma\right)\\\dot{h} = \emptyset\end{aligned}\end{align} \]
Parameters:
flow_rate(s, h, t, T, e)

The uniaxial flow rate itself and the derivative with respect to stress

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the flow rate and the derivative

of the flow rate with respect to stress

Return type:

tuple(torch.tensor, torch.tensor)

history_rate(s, h, t, T, e)

The history rate and the derivative of the history rate with respect to the current history

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

Returns:

the history rate and the

derivative of the history rate with respect to history

Return type:

tuple(torch.tensor, torch.tensor)

property nhist

The number of internal variables

Here 0…

class pyoptmat.flowrules.RateIndependentFlowRuleWrapper(base, lmbda, eps_ref)

Bases: FlowRule

Wraps another flow rule using Walker’s time dilation trick to make it behave as if it was rate-independent.

Specifically, the model dilates time with the formula:

\[\Delta t \rightarrow \kappa \Delta t\]

with

\[\kappa = 1 - \lambda + \frac{\lambda \left| \dot{\varepsilon} \right|}{\dot{\varepsilon}_{ref}}\]

where \(\lambda\) is a parameter where \(\lambda = 0\) gives the original, rate dependent response and \(\lambda \approx 1\) gives an approximately rate independent response, \(\dot{\varepsilon}_{ref}\) is a reference strain rate which should be several orders of magnitude smaller than the applied strain rate, and \(\dot{\varepsilon}\) is the current, transient strain rate applied to the model.

Parameters:

  • base (flowrules.FlowRule) – the base model

  • lmbda (scalar) – the tuning parameter \(\lambda\)

  • eps_ref (scalar) – the reference strain rate \(\dot{\varepsilon}_{ref}\)

dflow_derate(s, h, t, T, e)

The derivative of the flow rate with respect to the total strain rate

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – internal variables

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

derivative of flow rate with respect to the

internal variables

Return type:

torch.tensor

dflow_dhist(s, h, t, T, e)

The derivative of the flow rate with respect to the internal variables

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the derivative of the flow rate

Return type:

torch.tensor

dhist_derate(s, h, t, T, e)

The derivative of the history rate with respect to the stress

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the derivative of the flow rate

Return type:

torch.tensor

dhist_dstress(s, h, t, T, e)

The derivative of the history rate with respect to the stress

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the derivative of the flow rate

Return type:

torch.tensor

dscale(e)

The derivative of the current value of the time dilation factor with respect to the total strain rate

Parameters:

e (torch.tensor) – total strain rate

Returns:

derivative of the scale factor

with respect to the total strain rate

Return type:

torch.tensor

flow_rate(s, h, t, T, e)

The uniaxial flow rate itself and the derivative with respect to stress

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the flow rate and the derivative

of the flow rate with respect to stress

Return type:

tuple(torch.tensor, torch.tensor)

history_rate(s, h, t, T, e)

The history rate and the derivative of the history rate with respect to the current history

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the history rate and the

derivative of the history rate with respect to history

Return type:

tuple(torch.tensor, torch.tensor)

property nhist

The number of internal variables

scale(e)

The current value of the time dilation factor

Parameters:

e (torch.tensor) – total strain rate

Returns:

current scale factor

Return type:

torch.tensor

class pyoptmat.flowrules.SoftKocksMeckingRegimeFlowRule(model1, model2, g0, mu, b, eps0, k, sf, eps=tensor(1.0000e-20), g0_scale=<function SoftKocksMeckingRegimeFlowRule.<lambda>>)

Bases: FlowRule

Switches between two different flow rules depending on the value of the Kocks-Mecking normalized activation energy

\[g = \frac{kT}{\mu b^3} \log{\frac{\dot{\varepsilon}_0}{\dot{\varepsilon}}}\]

with \(k\) the Boltzmann constant, \(T\) temperature, \(\mu\) the shear modulus, \(b\) a representative Burgers vector, \(\dot{\varepsilon}_0\) a reference strain rate, and \(\dot{\varepsilon}\) the current applied strain rate.

If the activation energy is less than or equal to a threshold \(g_0\) the flow rate is equal to that of the first model. If the activation energy is greater than the threshold then the model switches to the second flow rule.

The two models should have compatible internal history vectors.

This version uses a soft blending of the two models rather than a hard switch. Specifically,

\[M = f M_1 + (1-f) M_2\]

with

\[f = \frac{\tanh \left[s_f \left(g - g_0 \right) \right] + 1}{2}\]

where \(s_f\) is some scaling function, with values around 100 providing a fairly hard switch between the two models

Parameters:

  • model1 (flowrules.FlowRule) – first flow rule

  • model2 (flowrules.FlowRule) – second flow rule

  • g0 (torch.tensor) – activation energy threshold

  • mu (temperature.TemperatureParameter) – shear modulus

  • b (torch.tensor) – burgers vector

  • eps0 (torch.tensor) – reference strain rate

  • k (torch.tensor) – Boltzmann constant

  • sf (torch.tensor) – sharpness parameter

Keyword Arguments:

  • eps (float) – default 1e-20, offset to avoid divide-by-zero

  • g0_scale (function) – scaling function for g0, defaults to no scaling

blend_values(vals1, vals2, T, e)

Switch between the two model results

Parameters:

  • vals1 (torch.tensor) – values from first model

  • vals2 (torch.tensor) – values from second model

  • T (torch.tensor) – temperatures

  • e (torch.tensor) – strain rates

df_e(T, e)

The derivative of the weight function with respect to the strain rate

Parameters:

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

derivative of the weight function

Return type:

torch.tensor

dflow_derate(s, h, t, T, e)

The derivative of the flow rate with respect to the total strain rate

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – internal variables

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

derivative of flow rate with respect to the

internal variables

Return type:

torch.tensor

dflow_dhist(s, h, t, T, e)

The derivative of the flow rate with respect to the internal variables

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the derivative of the flow rate

Return type:

torch.tensor

dg_e(T, e)

Derivative of the activation energy with respect to the strain rate

Parameters:

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

derivative of the activation energy

Return type:

torch.tensor

dhist_derate(s, h, t, T, e)

The derivative of the history rate with respect to the stress

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the derivative of the flow rate

Return type:

torch.tensor

dhist_dstress(s, h, t, T, e)

The derivative of the history rate with respect to the stress

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the derivative of the flow rate

Return type:

torch.tensor

f(T, e)

The weight function value

Parameters:

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

value of the weighting function

Return type:

torch.tensor

flow_rate(s, h, t, T, e)

The uniaxial flow rate itself and the derivative with respect to stress

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the flow rate and the derivative

of the flow rate with respect to stress

Return type:

tuple(torch.tensor, torch.tensor)

g(T, e)

The current value of activation energy

Parameters:

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

value of the activation energy

Return type:

torch.tensor

history_rate(s, h, t, T, e)

The history rate and the derivative of the history rate with respect to the current history

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the history rate and the

derivative of the history rate with respect to history

Return type:

tuple(torch.tensor, torch.tensor)

property nhist

The number of internal variables

class pyoptmat.flowrules.SuperimposedFlowRule(models)

Bases: FlowRule

Superimpose multiple flow rules with

\[\dot{\varepsilon}_{in}=\sum_i \dot{\varepsilon}_{in,i}\]

and the history the union of all the individual models

Parameters:

models (list) – list of the individual models

dflow_derate(s, h, t, T, e)

The derivative of the flow rate with respect to the total strain rate

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – internal variables

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

derivative of flow rate with respect to the

internal variables

Return type:

torch.tensor

dflow_dhist(s, h, t, T, e)

The derivative of the flow rate with respect to the internal variables

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the derivative of the flow rate

Return type:

torch.tensor

dhist_derate(s, h, t, T, e)

The derivative of the history rate with respect to the stress

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the derivative of the flow rate

Return type:

torch.tensor

dhist_dstress(s, h, t, T, e)

The derivative of the history rate with respect to the stress

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the derivative of the flow rate

Return type:

torch.tensor

flow_rate(s, h, t, T, e)

The uniaxial flow rate itself and the derivative with respect to stress

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the flow rate and the derivative

of the flow rate with respect to stress

Return type:

tuple(torch.tensor, torch.tensor)

history_rate(s, h, t, T, e)

The history rate and the derivative of the history rate with respect to the current history

Parameters:

  • s (torch.tensor) – stress

  • h (torch.tensor) – history

  • t (torch.tensor) – time

  • T (torch.tensor) – temperature

  • e (torch.tensor) – total strain rate

Returns:

the history rate and the

derivative of the history rate with respect to history

Return type:

tuple(torch.tensor, torch.tensor)

property nhist

The number of internal variables