In most cases, Fcr1 and Fcr3 depend on stress, temperature and time, while secondary creep, Fcr2, depends only on stress level and temperature. Normally, secondary creep is the dominant process. Tertiary creep is seldom important because it only accounts for a small fraction of the total lifetime of a structure.

All creep models are contributing subnodes to a basic material model like Linear Elastic Material and they can be combined with any other subnodes, such as Plasticity or Thermal Expansion to create more advanced models. They can also be combined with each other to model several creep mechanisms acting at the same time.

Here, Qcr is a user defined creep potential, which is normally written in terms of invariants of the stress tensor.

Volumetric creep is obtained when the creep potential depends only on the first invariant of Cauchy stress tensor, I1(σ), since

This is equivalent to that the creep potential would depend on the pressure p = −I1/3.

When the creep potential depends only on the second deviatoric invariant of Cauchy stress tensor, J2(σ), the deviatoric creep model is obtained since

This is equivalent to that the creep potential would depend on the equivalent stress σe = √3J2.

When (in SI units) the creep potential, Qcr, is given in units of Pa, the rate multiplier η is given in units of 1/s.

so that the creep rate tensor is a diagonal tensor. The trace of the creep strain rate tensor, the volumetric creep strain rate, equals the user input

The volumetric creep strain rate usually depends on the first invariant of Cauchy stress I1(σ) or the pressure p = −I1/3, in addition to the temperature and other material parameters.

Here, nD is a deviatoric tensor coaxial to the stress tensor.

The effective creep strain rate, , normally depends on the second deviatoric invariant of the stress J2(σ) or the effective or von Mises (effective) stress σe, in addition to the temperature and other material parameters.

The deviatoric tensor nD is defined as

the effective creep strain rate equals the absolute value of the user input

The most common model for secondary creep is the Norton equation where the creep strain rate is proportional to a power of the equivalent stress, σe:

This is normally true at intermediate to high stress levels and at absolute temperatures of T/Tm > 0.5, where Tm is the melting temperature (that is, the temperature in the solid is at least as high as half the melting temperature Tm). An “Arrhenius type” temperature dependency can also be included. It is defined by

where Q is the activation energy (SI unit: J/mol), R is the gas constant, and T is the absolute temperature (SI unit: K).

Here, A is the creep rate coefficient (SI unit: 1/s), n is the stress exponent (dimensionless), σref a reference stress level (SI unit: Pa), and nD is a deviatoric tensor coaxial to the stress tensor as defined in Equation 3-56.

where nD is a deviatoric tensor coaxial to the stress tensor as defined in Equation 3-56, and Fcr is expressed as in the Norton model:

Here, A is the creep rate coefficient (SI unit: 1/s), n is the stress exponent (dimensionless), σref is a reference stress level (SI unit: Pa), tref and tshift are the reference and shift times (SI unit: s), and m is the time-hardening exponent (dimensionless).

where εcr,e is the effective creep strain, and εshift is the effective creep strain shift.

The time and frequency shifts in Equation 3-58 and Equation 3-60 serve two purposes:

Garofalo showed (Ref. 8, Ref. 9) that the power-law and exponential creep are limiting cases for the general empirical expression

This equation reduces to a power law (the Norton law) for ασe < 0.8 and approaches exponential creep for ασe > 1.2, where 1/α is a reference equivalent stress level.

where Q is the activation energy (SI unit: J/mol), R is the gas constant, and T is the absolute temperature (SI unit: K). The complete creep rate equation as used in COMSOL Multiphysics then reads

where, A is the creep rate (SI unit: 1/s), n is the stress exponent (dimensionless), and σref a reference equivalent stress level (SI unit: Pa). nD is a deviatoric tensor coaxial to the stress tensor as defined in Equation 3-56.

At low stress levels and high temperatures, Nabarro and Herring (Ref. 6, Ref. 7) independently derived an expression for the creep rate as a function of atomic diffusion

Here, d is the grain diameter, Dv is the volume diffusivity through the grain interior, b is Burgers vector, kB is the Boltzmann’s constant, and T is the absolute temperature. nD is a deviatoric tensor coaxial to the stress tensor as defined in Equation 3-56.

Coble creep (Ref. 6, Ref. 7) is closely related to Nabarro-Herring creep but takes into account the grain boundary diffusivity, Dgb

At intermediate to high stress levels and temperatures T/Tm > 0.5, the creep mechanism is assumed to be diffusion-controlled movements of dislocations in the crystal lattices (Ref. 7)

where and nD is a deviatoric tensor coaxial to the stress tensor as defined in Equation 3-56. Generally, the stress exponent n takes values between 3 and 5.

A general relation between creep rate and several material parameters is the Mukherjee-Bird-Dorn equation (Ref. 6)

Here, T is the temperature, d is the grain size, b is the Burgers vector, D is the self diffusion coefficient, G is the shear modulus, and e−Q/RT is an “Arrhenius” type of temperature dependency.

For high temperatures, Mukherjee-Bird-Dorn equation describes Weertman creep when setting p = 0. Setting n = 0 and p = 2 describes Nabarro-Herring, and setting n = 0 and p = 3 describes Coble creep. Harper-Dorn creep is obtained by setting n = 1 and p = 0.

The Anand viscoplasticity (Ref. 9) is a deviatoric creep model suitable for large, isotropic, viscoplastic deformations in combination with small elastic deformations.

where nD is a deviatoric tensor coaxial to the stress tensor as defined in Equation 3-56, and the creep rate is calculated from

Here, A is the creep rate coefficient (SI unit: s−1), Q is the activation energy (SI unit: J/mol), m is the stress sensitivity, ξ is the multiplier of stress, R is the gas constant, and T is the absolute temperature (SI unit: K).

The internal variable, sa, is called deformation resistance (SI unit: Pa) and is calculated from the rate equation

with the initial condition sa(0) = sinit. Here, h0 is the hardening constant (SI unit: Pa), and a is the hardening sensitivity.

The variable sa* is the saturation value of the deformation resistance sa, which is calculated from the expression

where s0 is the deformation resistance saturation coefficient (SI unit: Pa), and n is the deformation resistance sensitivity.

Here, A is the viscoplastic rate coefficient (SI unit: 1/s), n is the stress exponent (dimensionless), σref a reference stress level (SI unit: Pa), and nD is a deviatoric tensor coaxial to the stress tensor. The Macaulay brackets are applied on the yield function, which is defined as done for plasticity

The equivalent stress is either the von Mises, Tresca, or Hill stress; or a user defined expression, and σys is the yield stress (which may include a linear or nonlinear Isotropic Hardening model). The stress tensor used in the equivalent stress is shifted by what is usually called the back stress, σback when Kinematic Hardening is included.

The deviatoric tensor nD is computed from the plastic potential Qp

When von Mises equivalent stress is used, the associated flow rule reads Qp = Fy, and the deviatoric tensor nD is defined as done for deviatoric creep

Some authors denote the viscosity as the quotient η = σref/A.

Since creep and viscoplasticity are inelastic processes, the dissipated energy density can be calculated by integrating the creep dissipation rate density (SI unit: W/m3) given by

The total energy dissipated in a given volume can be calculated by a volume integration of the dissipated creep energy density Wc (SI unit: J/m3).

The equation forms described for the different creep models above differ from the forms most commonly found in the literature. The difference lies in the introduction of normalizing reference values such as the reference stress σref and reference time tref. These values are in a sense superfluous and can in principle be chosen arbitrarily. The choice of reference values, however, affects the numerical values to be entered for the material data. This system has two advantages:

Material data for a Norton law is often available as the parameters AN and n in the equation

The coefficient AN has a physical dimension that depends on the value of n, and the unit has an implicit dependence on the stress and time units. Converting the data to the form used in COMSOL Multiphysics (Equation 3-57) requires the introduction of the reference stress σref. It is here convenient to use the implicit stress unit for which AN is given as reference stress. The creep rate coefficient A will then have the same numerical value as AN, and you do not need to do any conversions.

The physical dimension of A is, however, (time)−1, whereas the physical dimension of AN is (stress)−n(time)−1.

Another popular way of representing creep data is to supply the stress giving a certain creep rate. As an example, σc7 is the stress at which the creep strain rate is 10−7/h. Data on this form is also easy to enter: You set the reference stress to the value of σc7 and enter the creep rate coefficient as 1e-7[1/h].

In this case, there is no arbitrariness in the choice of σref, since α is an actual material parameter.