Time domain viscoelasticity is available in ABAQUS for small-strain applications where the rate-independent elastic response can be defined with a linear elastic material model and for large-strain applications where the rate-independent elastic response must be defined with a hyperelastic or hyperfoam material model.

Small strain

Consider a shear test at small strain, in which a time varying shear strain, , is applied to the material. The response is the shear stress . The viscoelastic material model defines as

where is the time-dependent “shear relaxation modulus” that characterizes the material's response. This constitutive behavior can be illustrated by considering a relaxation test in which a strain is suddenly applied to a specimen and then held constant for a long time. The beginning of the experiment, when the strain is suddenly applied, is taken as zero time, so that

where is the fixed strain. The viscoelastic material model is “long-term elastic” in the sense that, after having been subjected to a constant strain for a very long time, the response settles down to a constant stress; i.e., as .

The shear relaxation modulus can be written in dimensionless form:

where is the instantaneous shear modulus, so that the expression for the stress takes the form

The dimensionless relaxation function has the limiting values and .

Large strain

The equation for the stress can be transformed by using integration by parts:

It is convenient to write this equation in the form

where is the instantaneous shear stress at time t. This form allows a straightforward generalization to nonlinear elastic deformations by replacing the linear elastic relation with the nonlinear elasticity relation . This generalization yields a linear viscoelasticity model, in the sense that the dimensionless stress relaxation function is independent of the magnitude of the deformation.

In the above equation the instantaneous stress, , applied at time influences the stress, , at time t. Therefore, to create a proper finite-strain formulation, it is necessary to map the stress that existed in the configuration at time into the configuration at time t. In ABAQUS this is done by means of a mixed “push-forward” transformation with the relative deformation gradient :

To ensure that the stress remains symmetric, ABAQUS uses the integral form:

where is the deviatoric part of the Kirchhoff stress.

The finite-strain theory is described in more detail in “Finite-strain viscoelasticity,” Section 4.8.2 of the ABAQUS Theory Manual.

The volumetric behavior can be written in a form that is similar to the shear behavior:

The above expansion applies to small as well as finite strain since the volume strains are generally small and there is no need to map the pressure from time to time t.

The effect of temperature, , on the material behavior is introduced through the dependence of the instantaneous stress, , on temperature and through a reduced time concept. The expression for the linear-elastic shear stress is rewritten as

The reduced time concept is also used for the volumetric behavior and the large-strain formulation. For additional information on the WLF equation, see “Viscoelasticity,” Section 4.8.1 of the ABAQUS Theory Manual.

ABAQUS assumes that the viscoelastic material is defined by a Prony series expansion of the dimensionless relaxation modulus:

A similar Prony series expansion is used for the volumetric response, which is valid for both small- and finite-strain applications:

The Prony series expansion, in combination with the finite-strain expression for the shear stress, produces the following large-strain shear model:

If the instantaneous material behavior is defined by linear elasticity or hyperelasticity, the bulk and shear behavior can be defined independently. However, if the instantaneous behavior is defined by the hyperfoam model, the deviatoric and volumetric constitutive behavior are coupled and it is necessary to use the same relaxation data for both behaviors.

In all of the above expressions temperature dependence is readily introduced by replacing by and by .

The above equations are used to model the time-dependent shear and volumetric behavior of a viscoelastic material. The relaxation parameters can be defined in one of four ways: direct specification of the Prony series parameters, inclusion of creep test data, inclusion of relaxation test data, or inclusion of frequency-dependent data obtained from sinusoidal oscillation experiments. Temperature effects are included in the same manner regardless of the method used to define the viscoelastic material.

ABAQUS/CAE allows you to evaluate the behavior of viscoelastic materials by automatically creating response curves based on experimental test data or coefficients. A viscoelastic material can be evaluated only if it is defined in the time domain and includes hyperelastic and/or elastic material data. See “Evaluating hyperelastic and viscoelastic material behavior,” Section 12.4.6 of the ABAQUS/CAE User's Manual.

*VISCOELASTIC, TIME=PRONY

Creep test data

If creep test data are specified, ABAQUS will calculate the terms in the Prony series automatically. The normalized shear and bulk compliances are defined as

where is the shear compliance, is the total shear strain, and is the constant shear stress in a shear creep test; is the volumetric compliance, is the total volumetric strain, and is the constant pressure in a volumetric creep test. At time , .

The creep data are converted to relaxation data through the convolution integrals

ABAQUS then uses the normalized shear modulus and normalized bulk modulus in a nonlinear least-squares fit to determine the Prony series parameters.

Using the shear and volumetric test data consecutively

The shear test data and volumetric test data can be used consecutively to define the normalized shear and bulk compliances as functions of time. A separate least-squares fit is performed on each data set; and the two derived sets of Prony series parameters, and , are merged into one set of parameters, .

Input File Usage:	Use the following three options. The first option is required. Only one of the second and third options is required.
	*VISCOELASTIC, TIME=CREEP TEST DATA *SHEAR TEST DATA *VOLUMETRIC TEST DATA

ABAQUS/CAE Usage:	Property module: material editor: MechanicalElasticityViscoelastic: Domain: Time and Time: Creep test data
	In addition, select one or both of the following: Test DataShear Test Data Test DataVolumetric Test Data

Using the combined test data

Alternatively, the combined test data can be used to specify the normalized shear and bulk compliances simultaneously as functions of time. A single least-squares fit is performed on the combined set of test data to determine one set of Prony series parameters, .

Input File Usage:	Use both of the following options:
	*VISCOELASTIC, TIME=CREEP TEST DATA *COMBINED TEST DATA

ABAQUS/CAE Usage:

Property module: material editor: MechanicalElasticityViscoelastic: Domain: Time, Time: Creep test data, and Test DataCombined Test Data

*VISCOELASTIC, TIME=RELAXATION TEST DATA
*SHEAR TEST DATA
*VOLUMETRIC TEST DATA

*VISCOELASTIC, TIME=RELAXATION TEST DATA 
*COMBINED TEST DATA

Frequency-dependent test data

The Prony series terms can also be calibrated using frequency-dependent test data. In this case ABAQUS uses analytical expressions that relate the Prony series relaxation functions to the storage and loss moduli. The expressions for the shear moduli, obtained by converting the Prony series terms from the time domain to the frequency domain by making use of Fourier transforms, can be written as follows:

where is the storage modulus, is the loss modulus, is the angular frequency, and N is the number of terms in the Prony series. These expressions are used in a nonlinear least-squares fit to determine the Prony series parameters from the storage and loss moduli cyclic test data obtained at M frequencies by minimizing the error function :

where and are the test data and and , respectively, are the instantaneous and long-term shear moduli. The expressions for the bulk moduli, and , are written analogously.

The frequency domain data are defined in tabular form by giving the real and imaginary parts of and —where is the circular frequency—as functions of frequency in cycles per time. is the Fourier transform of the nondimensional shear relaxation function . Given the frequency-dependent storage and loss moduli , , , and , the real and imaginary parts of and are then given as

where and are the long-term shear and bulk moduli determined from the elastic or hyperelastic properties.

Input File Usage:

*VISCOELASTIC, TIME=FREQUENCY DATA

ABAQUS/CAE Usage:

Property module: material editor: MechanicalElasticityViscoelastic: Domain: Time and Time: Frequency data

*VISCOELASTIC, ERRTOL=error_tolerance, NMAX=

*TRS, DEFINITION=WLF

*TRS

*TRS, DEFINITION=USER

In all cases elastic moduli must be specified to define the rate-independent part of the material behavior. Small-strain linear elastic behavior is defined by an elastic material model (“Linear elastic behavior,” Section 17.2.1), and large-deformation behavior is defined by a hyperelastic (“Hyperelastic behavior of rubberlike materials,” Section 17.5.1) or hyperfoam (“Hyperelastic behavior in elastomeric foams,” Section 17.5.2) material model. The rate-independent elasticity for any of these models can be defined in terms of either instantaneous elastic moduli or long-term elastic moduli. The choice of defining the elasticity in terms of instantaneous or long-term moduli is a matter of convenience only; it does not have an effect on the solution.

The effective relaxation moduli are obtained by multiplying the instantaneous elastic moduli with the dimensionless relaxation functions as described below.

Linear elastic materials

For linear elastic material behavior

and

where and are the instantaneous shear and bulk moduli determined from the values of the user-defined instantaneous elastic moduli and : and .

If long-term elastic moduli are defined, the instantaneous moduli are determined from

The time-domain viscoelastic material model is active during the following procedures:

Viscoelastic material response is always ignored in a static analysis. It can also be ignored in a coupled temperature-displacement analysis or in a soils consolidation analysis by specifying that no creep or viscoelastic response is occurring during the step even if creep or viscoelastic material properties are defined (see “Fully coupled thermal-stress analysis,” Section 6.5.4, or “Coupled pore fluid diffusion and stress analysis,” Section 6.7.1). In these cases it is assumed that the loading is applied instantaneously, so that the resulting response corresponds to an elastic solution based on instantaneous elastic moduli.

ABAQUS/Standard also provides the option to obtain the fully relaxed long-term elastic solution directly in a static or steady-state transport analysis without having to perform a transient analysis. The long-term value is used for this purpose. The viscous damping stresses (the internal stresses associated with each of the Prony-series terms) are increased gradually from their values at the beginning of the step to their long-term values at the end of the step if the long-term value is specified.