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\boldsymbol {\tau} _ {i} = \frac {\bar {g} _ {i} ^ {P}}{\tau_ {i} ^ {G}} \int_ {0} ^ {t} e ^ {- s / \tau_ {i} ^ {G}} \bar {\mathbf {F}} _ {t} ^ {- 1} (t - s) \cdot \boldsymbol {\tau} _ {0} (t - s) \cdot \bar {\mathbf {F}} _ {t} (t - s) d s.
The \tau _ { i } are interpreted as state variables that control the stress relaxation.
For traction-separation elasticity, the Prony series expansion yields
\mathbf {t} = \left\{ \begin{array}{l} t _ {n} \\ t _ {s} \\ t _ {t} \end{array} \right\} = \left\{ \begin{array}{l} t _ {n} ^ {0} \\ t _ {s} ^ {0} \\ t _ {t} ^ {0} \end{array} \right\} - \sum_ {i = 1} ^ {N} \left\{ \begin{array}{l} t _ {n} ^ {i} \\ t _ {s} ^ {i} \\ t _ {t} ^ {i} \end{array} \right\} = \mathbf {t} ^ {0} - \sum_ {i = 1} ^ {N} \mathbf {t} ^ {i},
where
t _ {n} ^ {i} = \frac {\bar {k} _ {i} ^ {P}}{\tau_ {i} ^ {K}} \int_ {0} ^ {t} e ^ {- s / \tau_ {i} ^ {K}} t _ {n} ^ {0} (t - s) d s,
t _ {s} ^ {i} = \frac {\bar {g} _ {i} ^ {P}}{\tau_ {i} ^ {G}} \int_ {0} ^ {t} e ^ {- s / \tau_ {i} ^ {G}} t _ {s} ^ {0} (t - s) d s,
t _ {t} ^ {i} = \frac {\bar {g} _ {i} ^ {P}}{\tau_ {i} ^ {G}} \int_ {0} ^ {t} e ^ {- s / \tau_ {i} ^ {G}} t _ {t} ^ {0} (t - s) d s.
The \mathbf { t } ^ { i } are interpreted as state variables that control the relaxation of the traction stresses.
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 mandatory to use the same relaxation data for both behaviors. For linear anisotropic elasticity, the same relaxation data should be used for both behaviors when the elasticity definition is such that the deviatoric and volumetric response is coupled. Similarly, for coupled traction-separation elasticity you must use the same relaxation data for the normal and shear behaviors.
In all of the above expressions temperature dependence is readily introduced by replacing e ^ { - s / \tau _ { i } ^ { G } } by e ^ { - \xi ( s ) / \tau _ { i } ^ { G } } and e ^ { - s / \tau _ { i } ^ { K } } by e ^ { - \xi ( s ) / \tau _ { i } ^ { K } } .
Determination of viscoelastic material parameters
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 12.4.7 of the Abaqus/CAE User’s Guide.
Direct specification
The Prony series parameters \bar { g } _ { i } ^ { P } , \bar { k } _ { i } ^ { P } , and \tau _ { i } can be defined directly for each term in the Prony series. There is no restriction on the number of terms that can be used. If a relaxation time is associated with only one of the two moduli, leave the other one blank or enter a zero. The data should be given in ascending order of the relaxation time. The number of lines of data given defines the number of terms in the Prony series, N. If this model is used in conjunction with the hyperfoam material model, the two modulus ratios have to be the same ( \bar { g } _ { i } ^ { P } = \bar { k } _ { i } ^ { P } ) .
Input File Usage: *VISCOELASTIC, TIME=PRONY
The data line is repeated as often as needed to define the first, second, third, etc. terms in the Prony series.
Abaqus/CAE Usage: Property module: material editor: Mechanical→Elasticity→Viscoelastic: Domain: Time and Time: Prony
Enter as many rows of data in the table as needed to define the first, second, third, etc. terms in the Prony series.
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
j _ {S} (t) = G _ {0} J _ {S} (t) \qquad \mathrm{and} \qquad j _ {K} (t) = K _ {0} J _ {K} (t),
where J _ { S } ( t ) = \gamma ( t ) / \tau _ { 0 } is the shear compliance, \gamma ( t ) is the total shear strain, and \tau _ { 0 } is the constant shear stress in a shear creep test; J _ { K } ( t ) = \varepsilon ^ { v o l } ( t ) / p _ { 0 } is the volumetric compliance, \varepsilon ^ { v o l } ( t ) is the total volumetric strain, and p _ { 0 } is the constant pressure in a volumetric creep test. At time t = 0 , j _ { S } ( 0 ) = j _ { K } ( 0 ) = 1 .
The creep data are converted to relaxation data through the convolution integrals
\int_ {0} ^ {t} g _ {R} (s) j _ {S} (t - s) d s = t \qquad \mathrm{and} \qquad \int_ {0} ^ {t} k _ {R} (s) j _ {K} (t - s) d s = t.
Abaqus then uses the normalized shear modulus g _ { R } ( t ) and normalized bulk modulus k _ { R } ( t ) 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, ( \overline { { g } } _ { i } ^ { P } , \tau _ { i } ^ { G } ) and ( \bar { k } _ { i } ^ { P } , \tau _ { i } ^ { K } ) , are merged into one set of parameters, ( \bar { g } _ { i } ^ { P } , \bar { k } _ { i } ^ { P } , \tau _ { i } ) .
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: Mechanical→Elasticity→Viscoelastic:
Domain: Time and Time: Creep test data
In addition, select one or both of the following:
Test Data→Shear Test Data
Test Data→Volumetric 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, ( \bar { g } _ { i } ^ { P } , \bar { k } _ { i } ^ { P } , \tau _ { i } ) .
Input File Usage: Use both of the following options:
*VISCOELASTIC, TIME=CREEP TEST DATA
*COMBINED TEST DATA
Abaqus/CAE Usage: Property module: material editor: Mechanical→Elasticity→Viscoelastic:
Domain: Time, Time: Creep test data, and
Test Data→Combined Test Data
Relaxation test data
As with creep test data, Abaqus will calculate the Prony series parameters automatically from relaxation test data.
Using the shear and volumetric test data consecutively
Again, the shear test data and volumetric test data can be used consecutively to define the relaxation moduli as functions of time. A separate nonlinear least-squares fit is performed on each data set; and the two derived sets of Prony series parameters, ( \bar { g } _ { i } ^ { P } , \tau _ { i } ^ { G } ) and ( \bar { k } _ { i } ^ { P } , \bar { \tau } _ { i } ^ { K } ) , are merged into one set of parameters, ( \bar { g } _ { i } ^ { P } , \bar { k } _ { i } ^ { P } , \tau _ { i } ) .
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=RELAXATION TEST DATA
*SHEAR TEST DATA
*VOLUMETRIC TEST DATA
Abaqus/CAE Usage: Property module: material editor: Mechanical→Elasticity→Viscoelastic:
Domain: Time and Time: Relaxation test data
In addition, select one or both of the following:
Test Data→Shear Test Data
Test Data→Volumetric Test Data
Using the combined test data
Alternatively, the combined test data can be used to specify the relaxation moduli 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, ( \bar { g } _ { i } ^ { P } , \bar { k } _ { i } ^ { P } , \tau _ { i } ) .
Input File Usage: Use both of the following options:
*VISCOELASTIC, TIME=RELAXATION TEST DATA
*COMBINED TEST DATA
Abaqus/CAE Usage: Property module: material editor: Mechanical→Elasticity→Viscoelastic:
Domain: Time, Time: Relaxation test data, and
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:
G _ {s} (\omega) = G _ {0} [ 1 - \sum_ {i = 1} ^ {N} \bar {g} _ {i} ^ {P} ] + G _ {0} \sum_ {i = 1} ^ {N} \frac {\bar {g} _ {i} ^ {P} \tau_ {i} ^ {2} \omega^ {2}}{1 + \tau_ {i} ^ {2} \omega^ {2}},
G _ {\ell} (\omega) = G _ {0} \sum_ {i = 1} ^ {N} \frac {\bar {g} _ {i} ^ {P} \tau_ {i} \omega}{1 + \tau_ {i} ^ {2} \omega^ {2}},
where G _ { s } ( \omega ) is the storage modulus, G _ { \ell } ( \omega ) 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 \chi ^ { 2 } :
\chi^ {2} = \sum_ {i = 1} ^ {M} \frac {1}{G _ {\infty} ^ {2}} [ (G _ {s} - \bar {G} _ {s}) _ {i} ^ {2} + (G _ {\ell} - \bar {G} _ {\ell}) _ {i} ^ {2} ],
where \bar { G } _ { s } and \bar { G } _ { \ell } are the test data and G _ { 0 } and G _ { \infty } , respectively, are the instantaneous and long-term shear moduli. The expressions for the bulk moduli, \mathrm { K } _ { s } ( \omega ) and \mathrm { K } _ { \ell } ( \omega ) , are written analogously.
The frequency domain data are defined in tabular form by giving the real and imaginary parts of \omega g ^ { * } and \omega k ^ { * } —where is the circular frequency—as functions of frequency in cycles per time. g ^ { * } ( \omega ) is the Fourier transform of the nondimensional shear relaxation function \begin{array} { r } { g ( \dot { t } ) = \dot { \frac { G _ { R } ( t ) } { G _ { \infty } } } - 1 } \end{array} . Given the
frequency-dependent storage and loss moduli G _ { s } ( \omega ) , G _ { \ell } ( \omega ) , \mathrm { K } _ { s } ( \omega ) , and \mathrm { K } _ { \ell } ( \omega ) , the real and imaginary parts of \omega g ^ { * } and \omega k ^ { * } are then given as
\omega \Re (g ^ {*}) = G _ {\ell} / G _ {\infty}, \omega \Im (g *) = 1 - G _ {s} / G _ {\infty}, \omega \Re (k ^ {*}) = \mathrm{K} _ {\ell} / \mathrm{K} _ {\infty}, \omega \Im (k ^ {*}) = 1 - \mathrm{K} _ {s} / \mathrm{K} _ {\infty},
where G _ { \infty } and \mathrm { K } _ { \infty } 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: Mechanical→Elasticity→Viscoelastic: Domain: Time and Time: Frequency data
Calibrating the Prony series parameters
You can specify two optional parameters related to the calibration of Prony series parameters for viscoelastic materials: the error tolerance and N _ { m a x } . The error tolerance is the allowable average root-mean-square error of data points in the least-squares fit, and its default value is 0.01. N _ { m a x } is the maximum number of terms N in the Prony series, and its default (and maximum) value is 13. Abaqus will perform the least-squares fit from N \ : = \ : 1 \mathrm { t o } \ N \ : = \ : N _ { m a x } until convergence is achieved for the lowest N with respect to the error tolerance.
The following are some guidelines for determining the number of terms in the Prony series from test data. Based on these guidelines, you can choose N _ { m a x } .
• There should be enough data pairs for determining all the parameters in the Prony series terms. Thus, assuming that N is the number of Prony series terms, there should be a total of at least data points in shear test data, data points in volumetric test data, data points in combined test data, and data points in the frequency domain.
• The number of terms in the Prony series should be typically not more than the number of logarithmic “decades” spanned by the test data. The number of logarithmic “decades” is defined as \log _ { 1 0 } ( t _ { m a x } / t _ { m i n } ) , where t _ { m a x } and t _ { m i n } are the maximum and minimum time in the test data, respectively.
Input File Usage: *VISCOELASTIC, ERRTOL=error_tolerance, \mathrm { N M A X } { = } N _ { m a x }
Abaqus/CAE Usage: Property module: material editor: Mechanical→Elasticity→Viscoelastic: Domain: Time; Time: Creep test data, Relaxation test data, or Frequency data; Maximum number of terms in the Prony series:
N _ { m a x } ; ; and Allowable average root-mean-square error: error_tolerance
Thermo-rheologically simple temperature effects
Regardless of the method used to define the viscoelastic behavior, thermo-rheologically simple temperature effects can be included by specifying the method used to define the shift function. Abaqus supports the following forms of the shift function: the Williams-Landel-Ferry (WLF) form, the Arrhenius form, and user-defined forms.
Thermo-rheologically simple temperature effects can also be included in the definition of equation of state models with viscous shear behavior (see “Viscous shear behavior” in “Equation of state,” Section 25.2.1).
Williams-Landel-Ferry (WLF) form
The shift function can be defined by the Williams-Landel-Ferry (WLF) approximation, which takes the form:
\log_ {1 0} (A) = - \frac {C _ {1} (\theta - \theta_ {0})}{C _ {2} + (\theta - \theta_ {0})},
where \theta _ { 0 } is the reference temperature at which the relaxation data are given; is the temperature of interest; and C _ { 1 } , C _ { 2 } are calibration constants obtained at this temperature. If \theta \leq \theta _ { 0 } - C _ { 2 } , deformation changes will be elastic, based on the instantaneous moduli.
For additional information on the WLF equation, see “Viscoelasticity,” Section 4.8.1 of the Abaqus Theory Guide.
Input File Usage: *TRS, DEFINITION=WLF
Abaqus/CAE Usage: Property module: material editor: Mechanical→Elasticity→Viscoelastic: Domain: Time, Time: any method, and Suboptions→Trs: Shift function: WLF
Arrhenius form
The Arrhenius shift function is commonly used for semi-crystalline polymers. It takes the form
\ln (A) = \frac {E _ {0}}{R} \left(\frac {1}{\theta - \theta^ {Z}} - \frac {1}{\theta_ {0} - \theta^ {Z}}\right),
where E _ { 0 } is the activation energy, is the universal gas constant, \theta ^ { Z } is the absolute zero in the temperature scale being used, \theta _ { 0 } is the reference temperature at which the relaxation data are given, and is the temperature of interest.
Input File Usage: Use the following option to define the Arrhenius shift function:
*TRS, DEFINITION=ARRHENIUS
In addition, use the *PHYSICAL CONSTANTS option to specify the universal gas constant and absolute zero.
Abaqus/CAE Usage: The Arrhenius shift function is not supported in Abaqus/CAE.
User-defined form
The shift function can be specified alternatively in user subroutines UTRS in Abaqus/Standard and VUTRS in Abaqus/Explicit.
Input File Usage: *TRS, DEFINITION=USER
Abaqus/CAE Usage: Property module: material editor: Mechanical→Elasticity→Viscoelastic: Domain: Time, Time: any method, and Suboptions→Trs: Shift function: User subroutine UTRS
Defining the rate-independent part of the material response
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 22.2.1), and large-deformation behavior is defined by a hyperelastic (“Hyperelastic behavior of rubberlike materials,” Section 22.5.1) or hyperfoam (“Hyperelastic behavior in elastomeric foams,” Section 22.5.2) 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 isotropic materials
For linear elastic isotropic material behavior
G _ {R} (t) = G _ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {g} _ {k} ^ {P} (1 - e ^ {- t / \tau_ {k}})\right)
and
\mathrm{K} _ {R} (t) = \mathrm{K} _ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {k} _ {k} ^ {P} (1 - e ^ {- t / \tau_ {k}})\right),
where G _ { 0 } and \mathrm { K } _ { 0 } are the instantaneous shear and bulk moduli determined from the values of the userdefined instantaneous elastic moduli E _ { 0 } and \nu _ { 0 } \colon G _ { 0 } = E _ { 0 } / 2 ( 1 + \nu _ { 0 } ) and \mathrm { K } _ { 0 } = E _ { 0 } / 3 ( 1 - 2 \nu _ { 0 } ) .
If long-term elastic moduli are defined, the instantaneous moduli are determined from
G _ {\infty} = G _ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {g} _ {k} ^ {P}\right) \qquad \text {and} \qquad K _ {\infty} = K _ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {k} _ {k} ^ {P}\right).
Linear elastic anisotropic materials
For linear elastic anisotropic material behavior the relaxation coefficients are applied to the elastic moduli as
\overline {{\mathbf {D}}} _ {R} (t) = \overline {{\mathbf {D}}} _ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {g} _ {k} ^ {P} (1 - e ^ {- t / \tau_ {k}})\right)
and
\mathrm{K} _ {R} (t) = \mathrm{K} _ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {k} _ {k} ^ {P} (1 - e ^ {- t / \tau_ {k}})\right),
where \overline { { \mathbf { D } } } _ { 0 } and \mathrm { K } _ { 0 } are the instantaneous deviatoric elasticity tensor and bulk moduli determined from the values of the user-defined instantaneous elastic moduli \mathbf { D } _ { 0 } . If both shear and bulk relaxation coefficients are specified and they are unequal, Abaqus issues an error message if the elastic moduli \mathbf { D } _ { 0 } is such that the deviatoric and volumetric response is coupled.
If long-term elastic moduli are defined, the instantaneous moduli are determined from
\overline {{\mathbf {D}}} _ {\infty} = \overline {{\mathbf {D}}} _ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {g} _ {k} ^ {P}\right) \qquad \text {and} \qquad K _ {\infty} = K _ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {k} _ {k} ^ {P}\right).
Hyperelastic materials
For hyperelastic material behavior the relaxation coefficients are applied either to the constants that define the energy function or directly to the energy function. For the polynomial function and its particular cases (reduced polynomial, Mooney-Rivlin, neo-Hookean, and Yeoh)
C _ {i j} ^ {R} (t) = C _ {i j} ^ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {g} _ {k} ^ {P} (1 - e ^ {- t / \tau_ {k}})\right),
for the Ogden function
\mu_ {i} ^ {R} (t) = \mu_ {i} ^ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {g} _ {k} ^ {P} \left(1 - e ^ {- t / \tau_ {k}}\right)\right),
for the Arruda-Boyce and Van der Waals functions
\mu^ {R} (t) = \mu^ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {g} _ {k} ^ {P} (1 - e ^ {- t / \tau_ {k}})\right),
and for the Marlow function
U _ {d e v} ^ {R} (t) = U _ {d e v} ^ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {g} _ {k} ^ {P} (1 - e ^ {- t / \tau_ {k}})\right).
For the coefficients governing the compressible behavior of the polynomial models and the Ogden model
D _ {i} ^ {R} (t) = D _ {i} ^ {0} / \left(1 - \sum_ {k = 1} ^ {N} \bar {k} _ {k} ^ {P} (1 - e ^ {- t / \tau_ {k}})\right),
for the Arruda-Boyce and Van der Waals functions
D ^ {R} (t) = D ^ {0} / \left(1 - \sum_ {k = 1} ^ {N} \bar {k} _ {k} ^ {P} (1 - e ^ {- t / \tau_ {k}})\right),
and for the Marlow function
U _ {v o l} ^ {R} (t) = U _ {v o l} ^ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {k} _ {k} ^ {P} (1 - e ^ {- t / \tau_ {k}})\right).
If long-term elastic moduli are defined, the instantaneous moduli are determined from
C _ {i j} ^ {\infty} = C _ {i j} ^ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {g} _ {k} ^ {P}\right), \mathrm{or} \mu_ {i} ^ {\infty} = \mu_ {i} ^ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {g} _ {k} ^ {P}\right), \mathrm{or} \mu^ {\infty} = \mu^ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {g} _ {k} ^ {P}\right),
while the instantaneous bulk compliance moduli are obtained from
D _ {i} ^ {\infty} = D _ {i} ^ {0} / \left(1 - \sum_ {k = 1} ^ {N} \bar {k} _ {k} ^ {P}\right), \text {or} D ^ {\infty} = D ^ {0} / \left(1 - \sum_ {k = 1} ^ {N} \bar {k} _ {k} ^ {P}\right);
for the Marlow functions we have
U _ {d e v} ^ {\infty} = U _ {d e v} ^ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {g} _ {k} ^ {P}\right) \text {and} U _ {v o l} ^ {\infty} = U _ {v o l} ^ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {k} _ {k} ^ {P}\right).
Mullins effect
If long-term moduli are defined for the underlying hyperelastic behavior, the instantaneous value of the parameter in Mullins effect is determined from
m ^ {\infty} = m ^ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {g} _ {k} ^ {P}\right).
Elastomeric foams
For elastomeric foam material behavior the instantaneous shear and bulk relaxation coefficients are assumed to be equal and are applied to the material constants \mu _ { i } in the energy function:
\mu_ {i} ^ {R} (t) = \mu_ {i} ^ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {g} _ {k} ^ {P} (1 - e ^ {- t / \tau_ {k}})\right) = \mu_ {i} ^ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {k} _ {k} ^ {P} (1 - e ^ {- t / \tau_ {k}})\right).
If only the shear relaxation coefficients are specified, the bulk relaxation coefficients are set equal to the shear relaxation coefficients and vice versa. If both shear and bulk relaxation coefficients are specified and they are unequal, Abaqus issues an error message.
If long-term elastic moduli are defined, the instantaneous moduli are determined from
\mu_ {i} ^ {\infty} = \mu_ {i} ^ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {g} _ {k} ^ {P}\right) = \mu_ {i} ^ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {k} _ {k} ^ {P}\right).
Traction-separation elasticity
For cohesive elements with uncoupled traction-separation elastic behavior:
K _ {n n} (t) = K _ {n n} ^ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {k} _ {k} ^ {P} (1 - e ^ {- t / \tau_ {k}})\right),
K _ {s s} (t) = K _ {s s} ^ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {g} _ {k} ^ {P} (1 - e ^ {- t / \tau_ {k}})\right),
and
K _ {t t} (t) = K _ {t t} ^ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {g} _ {k} ^ {P} (1 - e ^ {- t / \tau_ {k}})\right),
where K _ { n n } ^ { 0 } is the instantaneous normal modulus and K _ { s s } ^ { 0 } and K _ { t t } ^ { 0 } are the instantaneous shear moduli. If long-term elastic moduli are defined, the instantaneous moduli are determined from
K _ {n n} ^ {\infty} / K _ {n n} ^ {0} = \left(1 - \sum_ {k = 1} ^ {N} \bar {k} _ {k} ^ {P}\right), \qquad \mathrm{and} \qquad K _ {s s} ^ {\infty} / K _ {s s} ^ {0} = K _ {t t} ^ {\infty} / K _ {t t} ^ {0} = \left(1 - \sum_ {k = 1} ^ {N} \bar {g} _ {k} ^ {P}\right).
For cohesive elements with coupled traction-separation elastic behavior the shear and bulk relaxation coefficients must be equal:
\mathbf {K} (t) = \mathbf {K} ^ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {k} _ {k} ^ {P} (1 - e ^ {- t / \tau_ {k}})\right) = \mathbf {K} ^ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {g} _ {k} ^ {P} (1 - e ^ {- t / \tau_ {k}})\right),
where \mathbf { K } ^ { 0 } is the user-defined instantaneous elasticity matrix. If long-term elastic moduli are defined, the instantaneous moduli are determined from
\mathbf {K} ^ {\infty} = \mathbf {K} ^ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {k} _ {k} ^ {P}\right) = \mathbf {K} ^ {0} \left(1 - \sum_ {k = 1} ^ {N} \bar {g} _ {k} ^ {P}\right).