| J | is the ratio of current to original volume; |
| Δe | is the (small) incremental deviatoric strain, Δe = Δε - 1/3 ΔεvolI; |
| è | is the deviatoric strain rate, è = è - 1/3 εvolI; |
| Δεvol | is the (small) incremental volumetric strain, Δεvol = trace(Δε); |
| εvol | is the rate of volumetric strain, εvol = trace(ε); |
| CS|0 | is the deviatoric tangent elasticity matrix of the material in its predeformed state (for example, C1212 is the tangent shear modulus of the prestrained material); |
| Q|0 | is the volumetric strain-rate/deviatoric stress-rate tangent elasticity matrix of the material in its predeformed state; and |
| K|0 | is the tangent bulk modulus of the predeformed material. |
For a fully incompressible material only the deviatoric terms in the first constitutive equation above remain and the viscoelastic behavior is completely defined by $g ^ { * } ( \omega )$ .
# Determination of viscoelastic material parameters
The dissipative part of the material behavior is defined by giving the real and imaginary parts of $g ^ { * }$ and $k ^ { * }$ (for compressible materials) as functions of frequency. The moduli can be defined as functions of the frequency in one of three ways: by a power law, by tabular input, or by a Prony series expression for the shear and bulk relaxation moduli.
# Power law frequency dependence
The frequency dependence can be defined by the power law formulæ
$$
g ^ {*} (\omega) = g _ {1} ^ {*} f ^ {- a} \qquad \text {and} \qquad k ^ {*} (\omega) = k _ {1} ^ {*} f ^ {- b},
$$
where a and b are real constants, $g _ { 1 } ^ { * }$ and $k _ { 1 } ^ { * }$ are complex constants, and $f = \omega / 2 \pi$ is the frequency in cycles per time.
Input File Usage: \*VISCOELASTIC, FREQUENCY=FORMULA
Abaqus/CAE Usage: Property module: material editor: Mechanical→Elasticity→Viscoelastic: Domain: Frequency and Frequency: Formula
# Tabular frequency dependence
The frequency domain response can alternatively be 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. 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, FREQUENCY=TABULAR
Abaqus/CAE Usage: Property module: material editor: Mechanical→Elasticity→Viscoelastic: Domain: Frequency and Frequency: Tabular
Abaqus provides an alternative approach for specifying the viscoelastic properties of hyperelastic and hyperfoam materials. This approach involves the direct (tabular) specification of storage and loss moduli from uniaxial and volumetric tests, as functions of excitation frequency and a measure of the level of pre-strain. The level of pre-strain refers to the level of elastic deformation at the base state about which the steady-state harmonic response is desired. This approach is discussed in “Direct specification of storage and loss moduli for large-strain viscoelasticity” below.
# Prony series parameters
The frequency dependence can also be obtained from a time domain Prony series description of the dimensionless shear and bulk relaxation moduli:
$$
g _ {R} (t) = 1 - \sum_ {i = 1} ^ {N} \bar {g} _ {i} ^ {P} (1 - e ^ {- t / \tau_ {i}}),
$$
$$
k _ {R} (t) = 1 - \sum_ {i = 1} ^ {N} \bar {k} _ {i} ^ {P} (1 - e ^ {- t / \tau_ {i}}),
$$
where $N , \ \bar { g } _ { i } ^ { P } , \ \bar { k } _ { i } ^ { P }$ , and $\tau _ { i } , i \ = \ 1 , 2 , . . . , N$ , are material constants. Using Fourier transforms, the expression for the time-dependent shear modulus can be written in the frequency domain 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. The expressions for the bulk moduli, $\mathrm { K } _ { s } ( \omega )$ and $\mathrm { K } _ { \ell } ( \omega )$ , are written analogously. Abaqus/Standard will automatically perform the conversion from the time domain to the frequency domain. The Prony series parameters $\bar { g } _ { i } ^ { P } , \bar { k } _ { i } ^ { P } , \tau _ { i }$ can be defined in one of three ways: direct specification of the Prony series parameters, inclusion of creep test data, or inclusion of relaxation test data. If creep test data or relaxation test data are specified, Abaqus/Standard will determine the Prony series parameters in a nonlinear least-squares fit. A detailed description of the calibration of Prony series terms is provided in “Time domain viscoelasticity,” Section 22.7.1.
For the test data you can specify the normalized shear and bulk data separately as functions of time or specify the normalized shear and bulk data simultaneously. A nonlinear least-squares fit is performed to determine the Prony series parameters, $( \bar { g } _ { i } ^ { P } , \bar { k } _ { i } ^ { P } , \tau _ { i } )$ .