The inelastic behavior of materials is often described by the rate of the inelastic quantity, especially if the behavior is time dependent and nonlinear. For example, Elastoplastic Materials,
Linear Viscoelasticity,
Creep and Viscoplasticity all describe material models that are implemented in dedicated features. The
Inelastic Strain Rate node provides functionality to specify a general expression for the rate of inelastic deformation, entered either as strain tensor
εie; deformation gradient
Fie; deformation gradient inverse
; or stretch
λie. The user defined expression can be a function of any variable in the model.
Given a generic inelastic quantity Uie, the feature sets up and solves the following type of distributed ODE
where t is the time,
u is the displacement, and
f(
t,
u,
Uie,...) is a user defined expression.
Equation 3-143 is also subjected to the initial condition
The distributed ODE in Equation 3-143 is integrated in time to compute the value of the inelastic quantity
Uie at each time step. This can be done using any of the following methods:
The Backward Euler method is used to discretize
Equation 3-143 as
where n+1 indicates the current time step, and
Δt is the time step. The inelastic quantity at the previous time step
is stored as internal state variables, their initial value is given by
.
Equation 3-144 defines a system of nonlinear equations that is solved locally at each Gauss point for
using Newton’s method.
For Domain ODEs,
Equation 3-143 is converted to weak-form and solved as part of the general initial-boundary value problem. The components of the inelastic quantity
Uie are here treated as a degrees-of-freedom of the model.