Rate-dependent plasticity: creep and swelling

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 typically recommended only in cases 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. In the case where the stress is constant and there are no temperature and/or field dependencies, the time-hardening and strain-hardening versions of the power-creep law are equivalent. 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, $\sigma$, at high stress levels ($\sigma /{\sigma }^0\gg 1$, where ${\sigma }^0$ is the yield stress) and reduces to the power-law at low stress levels (with no explicit time dependence).

The double power, Anand, and Darveaux models are particularly well suited for modeling the behavior of solder alloys used in electronic packaging and have been shown to produce accurate results for a wide range of temperatures and strain rates.

None of the above models is suitable for modeling creep under cyclic loading. The ORNL model (ORNL – Oak Ridge National Laboratory constitutive model) 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).

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

CREEP, LAW=TIME

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

CREEP, TIME=TOTAL (default)
CREEP, TIME=CREEP

CREEP, LAW=STRAIN

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

The hyperbolic-sine law is available in the form

where

${\dot{\overline{\epsilon }}}^{cr}$ and $\tilde{q}$

are defined above,

$\theta$

is the temperature,

${\theta }^Z$

is the user-defined value of absolute zero on the temperature scale used,

$\mathrm{△}H$

is the activation energy,

R

is the universal gas constant, and

A, B, and n

are other material parameters.

This model includes temperature dependence, which is apparent in the above expression; however, the parameters A, B, n, $\mathrm{△}H$, and R cannot be defined as functions of temperature.

CREEP, LAW=HYPERB
PHYSICAL CONSTANTS, ABSOLUTE ZERO=${\theta }^Z$

Property module: material editor: MechanicalPlasticityCreep: Law: Hyperbolic-Sine

Any module: ModelEdit Attributesmodel_name: Absolute zero temperature

The Anand model is available in the form

where

${\dot{\overline{\epsilon }}}^{cr}$, $\tilde{q}$, R, $\theta$, and ${\theta }^Z$

are defined above,

$Q$

is the activation energy,

$s$

is the deformation resistance, and

$A$, $m$, and $\xi$

are material parameters.

The evolution equation for the deformation resistance, $s$ (initially $s=s_0$), is

with

where

and $a$, $n$, $\widehat{s}$, $A_0$, $A_1$, $A_2$, $A_3$, and $A_4$ are material parameters.

In addition, the initial deformation resistance is a function of temperature of the form

where $S_1$, $S_2$, and $S_3$ are material parameters.

CREEP, LAW=ANAND
PHYSICAL CONSTANTS, ABSOLUTE ZERO=${\theta }^Z$

The Darveau model involves both primary and secondary creep. The secondary creep (steady-state) component is described by a standard hyperbolic sine law

The steady-state law is modified to include the primary creep effects through

where

${\dot{\overline{\epsilon }}}^{cr}$, $\tilde{q}$, R, Q, $\theta$, and ${\theta }^Z$

are defined above,

$C_{ss}$

is the steady-state creep prefactor,

$\alpha$

is the steady-state creep power law breakdown, and

$n$, ${ϵ}_T$, and $B$

are other material parameters.

CREEP, LAW=DARVEAUX
PHYSICAL CONSTANTS, ABSOLUTE ZERO=${\theta }^Z$

The double power law is available in the form

where

${\dot{\overline{\epsilon }}}^{cr}$, $\tilde{q}$, $\theta$, and ${\theta }^Z$

are defined above,

${\sigma }_0$

is the normalized stress, and

$A_1$, $A_2$, $B_1$, $B_2$, $C_1$, and $C_2$

are other material parameters.

CREEP, LAW=DOUBLE POWER
PHYSICAL CONSTANTS, ABSOLUTE ZERO=${\theta }^Z$

Anisotropic creep can be defined to specify the stress ratios that appear in Hill's function. You must define the ratios $R_{ij}$ 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). Further details are provided in Anisotropic yield/creep. 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

Property module: material editor: MechanicalPlasticityCreep: SuboptionsPotential

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

where ${\dot{\overline{\epsilon }}}^{sw}$ is the volumetric swelling strain rate that you define either directly (discussed above) or in user subroutine CREEP. The ratios $r_{11}$, $r_{22}$, and $r_{33}$ are also user-defined. The directions of the components of the swelling strain rate are defined by the local material directions, which can be either user-defined or the default directions (see Orientations).

SWELLING
RATIOS

Property module: material editor: MechanicalPlasticitySwelling: SuboptionsRatios

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.

1. Integration scheme 1: Starts with explicit integration and switches to implicit integration based on either stability or if plasticity is active. The stability limit used in explicit integration is discussed in the next section.

2. Integration scheme 2: Starts with explicit integration and switches to implicit integration when plasticity is active. The stability criterion does not play a role here.

3. Integration scheme 3: Always uses implicit integration.

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 perturbation procedures).

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

Step module: Create Step: 
Visco: Incrementation: Creep/swelling/viscoelastic integration: 
Explicit

Coupled temp-displacement: toggle on Include creep/swelling/
viscoelastic behavior: Incrementation: Creep/swelling/viscoelastic integration: Explicit

Abaqus/Standard monitors the stability limit automatically during explicit integration. If, at any point in the model, the creep strain increment $\left({{\dot{\overline{\epsilon }}}^{cr}|}_t\mathrm{\Delta }t\right)$ is larger than the total elastic strain, the problem will become unstable. Therefore, a stable time step, $\mathrm{\Delta }{t}_s$, is calculated every increment by

where ${{\epsilon }^{el}|}_t$ is the equivalent total elastic strain at time t, the beginning of the increment, and ${{\dot{\overline{\epsilon }}}^{cr}|}_t$ is the equivalent creep strain rate at time t. Furthermore,

where ${\tilde{q}|}_t$ is the Mises stress at time t, and

where

$\mathbf{n}={\partial \tilde{q}|}_t/\partial \mathit{\sigma }$

is the gradient of the deviatoric stress potential,

${\mathbf{D}}^{el}$

is the elasticity matrix, and

$\overline{E}$

is an effective elastic modulus—for isotropic elasticity $\overline{E}$ can be approximated by Young's modulus.

At every increment for which explicit integration is performed, the stable time increment, $\mathrm{\Delta }{t}_s$, is compared to the critical time increment, $\mathrm{\Delta }{t}_c$, which is calculated as follows:

The quantity errtol is an error tolerance that you define as discussed below. If $\mathrm{\Delta }{t}_s$ is less than $\mathrm{\Delta }{t}_c$, $\mathrm{\Delta }{t}_s$ is used as the time increment, which would mean that the stability criterion was limiting the size of the time step further than required by accuracy considerations. Abaqus/Standard will automatically switch to the backward difference operator (the implicit method, which is unconditionally stable) if $\mathrm{\Delta }{t}_s$ is less than $\mathrm{\Delta }{t}_c$ for nine consecutive increments, you have not restricted Abaqus/Standard to explicit integration as discussed above, and there is sufficient time left in the analysis (time left $\ge 50\mathrm{\Delta }t$). The stiffness matrix will be reformed at every iteration if the implicit algorithm is used.

The integration tolerance must be chosen so that increments in stress, $\mathrm{△}\mathit{\sigma }$, are calculated accurately. Consider a one-dimensional example. The stress increment, $\mathrm{△}\sigma$, is

where $\mathrm{△}{\epsilon }^{el}$, $\mathrm{△}\epsilon$, and $\mathrm{△}{\epsilon }^{cr}$ are the uniaxial elastic, total, and creep strain increments, respectively, and E is the elastic modulus. For $\mathrm{△}\sigma$ to be calculated accurately, the error in the creep strain increment, $\mathrm{△}{\epsilon }_{err}^{cr}$, must be small compared to $\mathrm{△}{\epsilon }^{el}$; that is,

Measuring the error in $\mathrm{△}{\epsilon }^{cr}$ as

leads to

You define errtol for the applicable procedure by choosing an acceptable stress error tolerance and dividing this by a typical elastic modulus; therefore, it should be a small fraction of the ratio of the typical stress and the effective elastic modulus in a problem. It is important to recognize that this approach for selecting a value for errtol is often very conservative, and acceptable solutions can usually be obtained with higher values.

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

Step module: Create Step: 
Visco: Incrementation: toggle on Creep/swelling/viscoelastic strain error
tolerance, and enter a value

Coupled temp-displacement: toggle on Include creep/swelling/
viscoelastic behavior: Incrementation: toggle on Creep/swelling/
viscoelastic strain error tolerance, and enter a value

Soils: Pore fluid response: Transient consolidation: toggle on Include creep/swelling/viscoelastic behavior: Incrementation: toggle on Creep/swelling/viscoelastic strain error tolerance, and enter a value