The power-law creep model is attractive for its simplicity. However, it is limited in its range of application. The time-hardening version of the power-law creep model is most suitable when the stress state remains essentially constant. The strain-hardening version of power-law creep should be used when the stress state varies during an analysis. For either version of the power law, the stresses should be relatively low.

In regions of high stress, such as around a crack tip, the creep strain rates frequently show an exponential dependence of stress. The hyperbolic-sine creep law shows exponential dependence on the stress, , at high stress levels (, where is the yield stress) and reduces to the power-law at low stress levels (with no explicit time dependence).

None of the above models is suitable for modeling creep under cyclic loading. The ORNL model (“ORNL – Oak Ridge National Laboratory constitutive model,” Section 18.2.12) is an empirical model for stainless steel that gives approximate results for cyclic loading without having to perform the cyclic loading numerically. Generally, creep models for cyclic loading are complicated and must be added to a model with user subroutine CREEP or with user subroutine UMAT.

If creep and plasticity occur simultaneously and implicit creep integration is in effect, both behaviors may interact and a coupled system of constitutive equations needs to be solved. If creep and plasticity are isotropic, ABAQUS/Standard properly takes into account such coupled behavior, even if the elasticity is anisotropic. However, if creep and plasticity are anisotropic, ABAQUS/Standard integrates the creep equations without taking plasticity into account, which may lead to substantial errors in the creep strains. This situation develops only if plasticity and creep are active at the same time, such as would occur during a long-term load increase; one would not expect to have a problem if there is a short-term preloading phase in which plasticity dominates, followed by a creeping phase in which no further yielding occurs. Integration of the creep laws and rate-dependent plasticity are discussed in “Rate-dependent metal plasticity (creep),” Section 4.3.4 of the ABAQUS Theory Manual.

The power-law model can be used in its “time hardening” form or in the corresponding “strain hardening” form.

Time hardening form

The “time hardening” form is the simpler of the two forms of the power-law model:

where

is the uniaxial equivalent creep strain rate,

is the uniaxial equivalent deviatoric stress,

t

is the total time, and

A, n, and m

are defined by you as functions of temperature.

is Mises equivalent stress or Hill's anisotropic equivalent deviatoric stress according to whether isotropic or anisotropic creep behavior is defined (discussed below). For physically reasonable behavior A and n must be positive and . Since total time is used in the expression, such reasonable behavior also typically requires that small step times compared to the creep time be used for any steps for which creep is not active in an analysis; this is necessary to avoid changes in hardening behavior in subsequent steps.

Input File Usage:

*CREEP, LAW=TIME

ABAQUS/CAE Usage:

Property module: material editor: MechanicalPlasticityCreep: Law: Time-Hardening

Strain hardening form

The “strain hardening” form of the power law is

where and are defined above and is the equivalent creep strain.

Input File Usage:

*CREEP, LAW=STRAIN

ABAQUS/CAE Usage:

Property module: material editor: MechanicalPlasticityCreep: Law: Strain-Hardening

The hyperbolic-sine law is available in the form

*CREEP, LAW=HYPERBOLIC
*PHYSICAL CONSTANTS, ABSOLUTE ZERO=

Anisotropic creep can be defined to specify the stress ratios that appear in Hill's function. You must define the ratios in each direction that will be used to scale the stress value when the creep strain rate is calculated. The ratios can be defined as constant or dependent on temperature and other predefined field variables. The ratios are defined with respect to the user-defined local material directions or the default directions (see “Orientations,” Section 2.2.5). Further details are provided in “Anisotropic yield/creep,” Section 18.2.6. Anisotropic creep is not available when creep is used to define a rate-dependent gasket behavior since only the gasket thickness-direction behavior can have rate-dependent behavior.

*POTENTIAL

Anisotropy can easily be included in the swelling behavior. If anisotropic swelling behavior is defined, the swelling strain rate in each material direction is expressed as

*SWELLING
*RATIOS

Nonlinear creep problems are often solved efficiently by forward-difference integration of the inelastic strains (the “initial strain” method). This explicit method is computationally efficient because, unlike implicit methods, iteration is not required. Although this method is only conditionally stable, the numerical stability limit of the explicit operator is usually sufficiently large to allow the solution to be developed in a small number of time increments.

ABAQUS/Standard uses either an explicit or an implicit integration scheme or switches from explicit to implicit in the same step. These schemes are outlined first, followed by a description of which procedures use these integration schemes.

The use of the above integration schemes is determined by the procedure type, your choice of the integration type to be used, as well as whether or not geometric nonlinearity is requested. For quasi-static and coupled temperature-displacement procedures, if you do not choose an integration type, integration scheme 1 is used for a geometrically linear analysis and integration scheme 3 is used for a geometrically nonlinear analysis. You can force ABAQUS/Standard to use explicit integration for creep and swelling effects in coupled temperature-displacement or quasi-static procedures, when plasticity is not active throughout the step (integration scheme 2). Explicit integration can be used regardless of whether or not geometric nonlinearity has been requested (see “General and linear perturbation procedures,” Section 6.1.2).

For a transient soils consolidation procedure, the implicit integration scheme (integration scheme 3) is always used, irrespective of whether a geometrically linear or nonlinear analysis is performed.

*VISCO, CREEP=EXPLICIT
*COUPLED TEMPERATURE-DISPLACEMENT, CREEP=EXPLICIT

ABAQUS/Standard monitors the stability limit automatically during explicit integration. If, at any point in the model, the creep strain increment is larger than the total elastic strain, the problem will become unstable. Therefore, a stable time step, , is calculated every increment by

At every increment for which explicit integration is performed, the stable time increment, , is compared to the critical time increment, , which is calculated as follows:

The integration tolerance must be chosen so that increments in stress, , are calculated accurately. Consider a one-dimensional example. The stress increment, , is

*VISCO, CETOL=errtol 
*COUPLED TEMPERATURE-DISPLACEMENT, CETOL=errtol 
*SOILS, CONSOLIDATION, CETOL=errtol