pyoptmat.models: structural material models

This module contains objects to define and integrate full material models.

In our context a material model is a ODE that defines the stress rate and an associated set of internal variables. Mathematically, we can define this as two ODEs:

\[ \begin{align}\begin{aligned}\dot{\sigma} = f(\sigma, h, T, \dot{\varepsilon}, t)\\\dot{h} = g(\sigma, h, T, \dot{\varepsilon}, t)\end{aligned}\end{align} \]

where \(\sigma\) is the uniaxial stress, \(h\) is some arbitrary set of internal variables, \(T\) is the temperature, \(\dot{\varepsilon}\) is the strain rate, and \(t\) is the time. Note then that we mathematically define models as strain controlled: the input is the strain rate and the output is the stress rate. Currently there is only one implemented full material model: pyoptmat.models.InelasticModel, which is a standard viscoplastic formulation. However other types of models, including rate-independent plasticity, could be defined with the same basic form.

The model itself just defines a system of ODEs. To solve for stress or strain as a function of the experimental conditions we need to integrate this model using the methods in pyoptmat.ode. We could do this in two ways, in strain control where we provide the strains and temperatures as a function of time and integrate for the stress or provide the stresses and temperatures as a function of time and integrate for the strains. The pyoptmat.models.ModelIntegrator provides both options, where each experiment can either be strain or stress controlled.

The basic process of setting up a material model capable of simulating experimental tests is to define the model form mathematically, using a Model class and wrap that Model with an Integrator to provide actual time series of stress or strain. As the integrator class uses the methods in pyoptmat.ode to actually do the integration, the results (and subsequent mathematical operations on the results) can be differentiated using either PyTorch backpropogation AD or the adjoint method.

class pyoptmat.models.BothBasedModel(model, rate_fn, base_fn, T_fn, control, *args, **kwargs)

Bases: Module

Provides both the strain rate and stress rate form at once, for better vectorization

Parameters:

  • model – base InelasticModel

  • rate_fn – controlled quantity rate interpolator

  • base_fn – controlled quantity base interpolator

  • T_fn – temperature interpolator

  • indices – split into strain and stress control

forward(t, y)

Evaluate both strain and stress control and paste into the right locations.

Parameters:

  • t – input times

  • y – input state

class pyoptmat.models.DamagedInelasticModel(E, flowrule, *args, dmodel=NoDamage(), **kwargs)

Bases: Module

This object provides the standard strain-based rate form of a constitutive model

\[\dot{\sigma} = E \left(\dot{\varepsilon} - (1-d) \dot{\varepsilon}_{in} \right)\]
Parameters:
forward(t, y, erate, T)

Return the rate equations for the strain-based version of the model

Parameters:

  • t – (nbatch,) times

  • y – (nbatch,1+nhist+1) [stress, history, damage]

  • erate – (nbatch,) strain rates

  • T – (nbatch,) temperatures

Returns:

(nbatch,1+nhist+1) state rate d_y_dot_d_y: (nbatch,1+nhist+1,1+nhist+1) Jacobian wrt the state d_y_dot_d_erate:(nbatch,1+nhist+1) Jacobian wrt the strain rate d_y_dot_d_T: (nbatch,1+nhist+1) derivative wrt temperature (unused)

Return type:

y_dot

property nhist

Number of internal variables

class pyoptmat.models.InelasticModel(E, flowrule, *args, **kwargs)

Bases: Module

This object provides the standard strain-based rate form of a constitutive model

\[\dot{\sigma} = E \left(\dot{\varepsilon} - \dot{\varepsilon}_{in} \right)\]
Parameters:
forward(t, y, erate, T)

Return the rate equations for the strain-based version of the model

Parameters:

  • t – (nbatch,) times

  • y – (nbatch,1+nhist) [stress, history]

  • erate – (nbatch,) strain rates

  • T – (nbatch,) temperatures

Returns:

(nbatch,1+nhist) state rate d_y_dot_d_y: (nbatch,1+nhist,1+nhist) Jacobian wrt the state d_y_dot_d_erate:(nbatch,1+nhist) Jacobian wrt the strain rate d_y_dot_d_T: (nbatch,1+nhist) derivative wrt temperature (unused)

Return type:

y_dot

property nhist

Number of internal variables

class pyoptmat.models.ModelIntegrator(model, *args, use_adjoint=True, **kwargs)

Bases: Module

This class provides infrastructure for integrating constitutive models in either strain or stress control.

Parameters:

  • model – base strain-controlled model

  • method (optional) – integrate method used to solve the equations, defaults to “backward-euler”

  • use_adjoint (optional) – if True use the adjoint approach to

  • **kwargs – passed on to the odeint method

forward(t, y)

Evaluate both strain and stress control and paste into the right locations.

Parameters:

  • t – input times

  • y – input state

solve_both(times, temperatures, idata, control)

Solve for either strain or stress control at once

Parameters:

  • times – input times, (ntime,nexp)

  • temperatures – input temperatures (ntime,nexp)

  • idata – input data (ntime,nexp)

  • control – signal for stress/strain control (nexp,)

solve_strain(times, strains, temperatures)

Basic model definition: take time and strain rate and return stress

Parameters:

  • times – input times, shape (ntime)

  • strains – input strains, shape (ntime, nbatch)

  • temperatures – input temperatures, shape (ntime, nbatch)

Returns:

stacked [stress, history, damage] vector of shape

(ntime,nbatch,1+nhist+1)

Return type:

y

solve_stress(times, stresses, temperatures)

Inverse model definition: take time and stress rate and return strain

Parameters:

  • times – input times, shape (ntime,)

  • stresses – input stresses, shape (ntime, nbatch)

  • temperatures – input temperatures, shape (ntime, nbatch)

Returns:

stack [strain, history, damage] vector

of shape (ntime,nbatch,2+nhist)

Return type:

y

class pyoptmat.models.StrainBasedModel(model, erate_fn, T_fn, *args, **kwargs)

Bases: Module

Provides the strain rate form

Parameters:

  • model – base InelasticModel

  • erate_fn – erate(t)

  • T_fn – T(t)

forward(t, y)

Strain rate as a function of t and state

Parameters:

  • t – input times

  • y – input state

class pyoptmat.models.StressBasedModel(model, srate_fn, stress_fn, T_fn, *args, **kwargs)

Bases: Module

Provides the stress rate form

Parameters:

  • model – base InelasticModel

  • srate_fn – srate(t)

  • T_fn – T(t)

forward(t, y)

Stress rate as a function of t and state

Parameters:

  • t – input times

  • y – input state