Objective Stress Rates

Overview

A constitutive model supplies a stress measure and its algorithmic tangent to the stress-divergence kernels. A model can be written directly in a large-deformation stress measure – the Cauchy stress (ComputeLagrangianStressCauchy), the first Piola-Kirchhoff stress (ComputeLagrangianStressPK1), or the second Piola-Kirchhoff stress (ComputeLagrangianStressPK2) – and the material system converts it to whichever measure each kernel needs.

Alternatively, a model can be written in the small (engineering) stress and strain and converted to a large-deformation response by integrating an objective stress rate**. This is the subject of this page, implemented by ComputeLagrangianObjectiveStress. It lets an existing small-deformation constitutive model – including many of the module's existing stress models – run in a large-deformation simulation without being rewritten.

Why an Objective Rate

A constitutive law written in the small (engineering) stress and strain is not frame indifferent: applied directly in a large-deformation setting, a rigid rotation of the body would spuriously change the stress. An objective stress rate removes the rotational part of the stress evolution so that the small-deformation law integrates correctly as the body rotates and deforms. The conversion is only needed for large deformation kinematics; under small kinematics the small stress is used directly, controlled by the large_kinematics option.

Integration

The conversion integrates an objective rate of the Cauchy stress Simo and Hughes (2006). Most supported rates take the form where is the small stress supplied by the constitutive model and is a kinematic measure whose choice defines the particular rate. The kinematic increments that drive the integration – the spatial velocity gradient increment and the vorticity increment – come from the strain calculator and depend on the selected kinematic approximation. See ComputeLagrangianObjectiveStress for the incremental stress update and the corresponding algorithmic tangent.

Available Rates

Four objective rates are available, selected with the objective_rate parameter:

See ComputeLagrangianObjectiveStress for the defining kinematic tensor and algorithmic tangent of each rate.

Limitations

Integrating an objective rate of a small-stress model has well-known limitations Simo and Hughes (2006) and Bažant and Vorel (2014):

• Integrated rotations. For the Truesdell, Jaumann, and Green-Naghdi rates the rotational kinematics are integrated in time, so large rotations are captured accurately only in the limit of small time steps. The rashid rate integrates the rotation in closed form and is exact for any step size.

• Large shears. Some rates (notably Jaumann) produce a non-physical, oscillatory stress under very large shear.

For problems with very large stretches, a hyperelastic model defined directly in a large-deformation measure (ComputeLagrangianStressPK1 or ComputeLagrangianStressPK2) is preferable. See ComputeLagrangianObjectiveStress for a fuller discussion with illustrative examples.

Implementation

Objective-rate integration is provided by ComputeLagrangianObjectiveStress, with ComputeLagrangianObjectiveCustomStress available for models that supply their own stress form. Select the rate with the objective_rate parameter; the default is truesdell.

References