Frequency Domain Viscoelasticity

Consider a shear test at small strain, in which a harmonically varying shear strain $\gamma$ is applied:

where ${\gamma }_0$ is the amplitude, $i=\sqrt{-1}$ , $\omega$ is the circular frequency, and t is time. We assume that the specimen has been oscillating for a very long time so that a steady-state solution is obtained. The solution for the shear stress then has the form

where $G_s$ and $G_{\mathrm{\ell }}$ are the shear storage and loss moduli. These moduli can be expressed in terms of the (complex) Fourier transform $g^{*}\left(\omega \right)$ of the nondimensional shear relaxation function $g\left(t\right)=\frac{G_R\left(t\right)}{G_{\mathrm{\infty }}}-1$ :

where $G_R\left(t\right)$ is the time-dependent shear relaxation modulus, $\mathrm{\Re }\left(g^{*}\right)$ and $\mathrm{\Im }\left(g^{*}\right)$ are the real and imaginary parts of $g^{*}\left(\omega \right)$ , and $G_{\mathrm{\infty }}$ is the long-term shear modulus. See Frequency domain viscoelasticity for details.

The above equation states that the material responds to steady-state harmonic strain with a stress of magnitude $G_s{\gamma }_0$ that is in phase with the strain and a stress of magnitude $G_{\mathrm{\ell }}{\gamma }_0$ that lags the excitation by ${90}^{\circ }$ . Therefore, we can regard the factor

as the complex, frequency-dependent shear modulus of the steadily vibrating material. The absolute magnitude of the stress response is

and the phase lag of the stress response is

Measurements of $\left|\tau \right|$ and $\varphi$ as functions of frequency in an experiment can, therefore, be used to define $G_s$ and $G_{\mathrm{\ell }}$ and, thus, $\mathrm{\Re }\left(g^{*}\right)$ and $\mathrm{\Im }\left(g^{*}\right)$ as functions of frequency.

Unless stated otherwise explicitly, all modulus measurements are assumed to be “true” quantities.

For isotropic viscoelasticity, the frequency dependence of the shear (deviatoric) and volumetric behaviors are independent. The volumetric behavior is defined by the bulk storage and loss moduli ${\mathrm{K}}_s\left(\omega \right)$ and ${\mathrm{K}}_{\mathrm{\ell }}\left(\omega \right)$ . Similar to the shear moduli, these moduli can also be expressed in terms of the (complex) Fourier transform $k^{*}\left(\omega \right)$ of the nondimensional bulk relaxation function $k\left(t\right)$ :

where ${\mathrm{K}}_{\mathrm{\infty }}$ is the long-term elastic bulk modulus.

The linearized vibrations can also be associated with an elastomeric material whose long-term (elastic) response is nonlinear and involves finite strains (a hyperelastic material). We can retain the simplicity of the steady-state small-amplitude vibration response analysis in this case by assuming that the linear expression for the shear stress still governs the system, except that now the long-term shear modulus $G_{\mathrm{\infty }}$ can vary with the amount of static prestrain $\overline{\gamma }$ :

The essential simplification implied by this assumption is that the frequency-dependent part of the material's response, defined by the Fourier transform $g^{*}\left(\omega \right)$ of the relaxation function, is not affected by the magnitude of the prestrain. Therefore, strain and frequency effects are separated, which is a reasonable approximation for many materials.

Another implication of the above assumption is that the anisotropy of the viscoelastic moduli has the same strain dependence as the anisotropy of the long-term elastic moduli. Therefore, the viscoelastic behavior in all deformed states can be characterized by measuring the (isotropic) viscoelastic moduli in the undeformed state.

In situations where the above assumptions are not reasonable, the data can be specified based on measurements at the prestrain level about which the steady-state dynamic response is desired. In this case you must measure $G_s$ , $G_{\mathrm{\ell }}$ , and $G_{\mathrm{\infty }}$ (likewise $K_s$ , $K_{\mathrm{\ell }}$ , and $K_{\mathrm{\infty }}$ ) at the prestrain level of interest. Alternatively, the viscoelastic data can be given directly in terms of uniaxial and volumetric storage and loss moduli that might be specified as functions of frequency and prestrain (see Direct Specification of Storage and Loss Moduli for Large-Strain Isotropic Viscoelasticity below.)

The generalization of these concepts to arbitrary three-dimensional deformations is provided in Abaqus/Standard by assuming that the frequency-dependent material behavior has two independent components: one associated with shear (deviatoric) straining and the other associated with volumetric straining. In the general case of a compressible material, the model is, therefore, defined for kinematically small perturbations about a predeformed state as

and

where

$\mathbf{S}$

is the deviatoric stress, $\mathbf{S}=\mathit{\sigma }+p\mathbf{I}$ ;

p

is the equivalent pressure stress, $p=-\frac{1}{3}\mathrm{trace}\left(\mathit{\sigma }\right)$ ;

${\mathrm{\Delta }}^{▽}\left(J\mathbf{S}\right)$

is the part of the stress increment caused by incremental straining (as distinct from the part of the stress increment caused by incremental rotation of the preexisting stress with respect to the coordinate system);

J

is the ratio of current to original volume;

$\mathrm{\Delta }\mathbf{e}$

is the (small) incremental deviatoric strain, $\mathrm{\Delta }\mathbf{e}=\mathrm{\Delta }\mathit{\epsilon }-\frac{1}{3}\mathrm{\Delta }{\epsilon }^{\mathrm{vol}}\mathbf{I}$ ;

$\dot{\mathbf{e}}$

is the deviatoric strain rate, $\dot{\mathbf{e}}=\dot{\mathit{\epsilon }}-\frac{1}{3}{\dot{\epsilon }}^{\mathrm{vol}}\mathbf{I}$ ;

$\mathrm{\Delta }{\epsilon }^{\mathrm{vol}}$

is the (small) incremental volumetric strain, $\mathrm{\Delta }{\epsilon }^{\mathrm{vol}}=\mathrm{trace}\left(\mathrm{\Delta }\mathit{\epsilon }\right)$ ;

${\dot{\epsilon }}^{\mathrm{vol}}$

is the rate of volumetric strain, ${\dot{\epsilon }}^{\mathrm{vol}}=\mathrm{trace}\left(\dot{\mathit{\epsilon }}\right)$ ;

${{\mathbf{C}}^S|}_0$

is the deviatoric tangent elasticity matrix of the material in its predeformed state (for example, $C_{1212}$ is the tangent shear modulus of the prestrained material);

${\mathbf{Q}|}_0$

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 $g^{*}\left(\omega \right)$ .

The frequency dependence can be defined by the power law formulas

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.

VISCOELASTIC, FREQUENCY=FORMULA

Property module: material editor: MechanicalElasticityViscoelastic: 
Domain: Frequency and Frequency: Formula

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 $\omega$ is the circular frequency—as functions of frequency in cycles per time. Given the frequency-dependent storage and loss moduli $G_s\left(\omega \right)$ , $G_{\mathrm{\ell }}\left(\omega \right)$ , ${\mathrm{K}}_s\left(\omega \right)$ , and ${\mathrm{K}}_{\mathrm{\ell }}\left(\omega \right)$ , the real and imaginary parts of $\omega g^{*}$ and $\omega k^{*}$ are then given as

where $G_{\mathrm{\infty }}$ and ${\mathrm{K}}_{\mathrm{\infty }}$ are the long-term shear and bulk moduli determined from the elastic or hyperelastic properties.

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 prestrain. The level of prestrain 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 Isotropic Viscoelasticity below.

VISCOELASTIC, FREQUENCY=TABULAR

Property module: material editor: MechanicalElasticityViscoelastic: 
Domain: Frequency and Frequency: Tabular

The frequency dependence can also be obtained from a time domain Prony series description of the dimensionless shear and bulk relaxation moduli:

where N, ${\overline{g}}_i^P$ , ${\overline{k}}_i^P$ , and ${\tau }_i$ , $i=1,2,\mathrm{\dots },N$ , are material constants. Using Fourier transforms, the expression for the time-dependent shear modulus can be written in the frequency domain as follows:

where $G_s\left(\omega \right)$ is the storage modulus, $G_{\mathrm{\ell }}\left(\omega \right)$ is the loss modulus, $\omega$ is the angular frequency, and N is the number of terms in the Prony series. The expressions for the bulk moduli, ${\mathrm{K}}_s\left(\omega \right)$ and ${\mathrm{K}}_{\mathrm{\ell }}\left(\omega \right)$ , are written analogously. Abaqus/Standard automatically performs the conversion from the time domain to the frequency domain. The Prony series parameters ${\overline{g}}_i^P,{\overline{k}}_i^P,{\tau }_i$ 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 you specify creep test data or relaxation test data, Abaqus/Standard determines 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.

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, $\left({\overline{g}}_i^P,{\overline{k}}_i^P,{\tau }_i\right)$ .

VISCOELASTIC, FREQUENCY=PRONY
VISCOELASTIC, FREQUENCY=CREEP TEST DATA
VISCOELASTIC, FREQUENCY=RELAXATION TEST DATA

SHEAR TEST DATA
VOLUMETRIC TEST DATA

COMBINED TEST DATA

Property module: material editor: MechanicalElasticityViscoelastic: 
Domain: Frequency and Frequency: Prony, Creep test data, or 
Relaxation test data

Test DataShear Test Data
Test DataVolumetric Test Data

Test DataCombined Test Data

You can include thermorheologically simple temperature effects in frequency domain viscoelasticity. In this case the reduced angular frequency, ${\omega }_r$ , is used to obtain the frequency-dependent material moduli. The reduced angular frequency is computed as

In many cases the viscous behavior is associated only with deviatoric straining, so that the bulk modulus is real and constant: ${\mathrm{K}}_s={\mathrm{K}}_{\mathrm{\infty }}$ and ${\mathrm{K}}_{\mathrm{\ell }}=0$ . For this case, the expressions for the shear moduli simplify to

If the bulk modulus is very large compared to the shear modulus, the material can be considered to be incompressible, and the expressions simplify further to