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 ${90}^{\circ }$. Hence, 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, thus, 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.

In multiaxial stress states Abaqus/Standard assumes that 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. Thus, 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. Hence, 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 may be specified as functions of frequency and prestrain (see Direct specification of storage and loss moduli for large-strain 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 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.

For some cases of small straining of isotropic viscoelastic materials, the material data are provided as frequency-dependent uniaxial storage and loss moduli, $E_s\left(\omega \right)$ and $E_{\mathrm{\ell }}\left(\omega \right)$, and bulk moduli, ${\mathrm{K}}_s\left(\omega \right)$ and ${\mathrm{K}}_{\mathrm{\ell }}\left(\omega \right)$. In that case the data must be converted to obtain the frequency-dependent shear storage and loss moduli $G_s\left(\omega \right)$ and $G_{\mathrm{\ell }}\left(\omega \right)$.

The complex shear modulus is obtained as a function of the complex uniaxial and bulk moduli with the expression

Replacing the complex moduli by the appropriate storage and loss moduli, this expression transforms into

After some algebra one obtains

For large-strain viscoelasticity Abaqus allows direct specification of storage and loss moduli from uniaxial and volumetric tests. This approach can be used when the assumption of the independence of viscoelastic properties on the pre-strain level is too restrictive.

You specify the storage and loss moduli directly as tabular functions of frequency, and you specify the level of pre-strain at the base state about which the steady-state dynamic response is desired. For uniaxial test data the measure of pre-strain is the uniaxial nominal strain; for volumetric test data the measure of pre-strain is the volume ratio. Abaqus internally converts the data that you specify to ratios of shear/bulk storage and loss moduli to the corresponding long-term elastic moduli. Subsequently, the basic formulation described in Large-strain viscoelasticity above is used.

For a general three-dimensional stress state it is assumed that the deviatoric part of the viscoelastic response depends on the level of pre-strain through the first invariant of the deviatoric left Cauchy-Green strain tensor (see Hyperelastic material behavior for a definition of this quantity), while the volumetric part depends on the pre-strain through the volume ratio. A consequence of these assumptions is that for the uniaxial case, data can be specified from a uniaxial-tension preload state or from a uniaxial-compression preload state but not both.

The storage and loss moduli that you specify are assumed to be nominal quantities.

VISCOELASTIC, PRELOAD=UNIAXIAL

VISCOELASTIC, PRELOAD=VOLUMETRIC

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

Type: Isotropic or Traction: Preload: Uniaxial

Type: Isotropic: Preload: Volumetric

Type: Isotropic: Preload: Uniaxial and Volumetric

In all cases elastic moduli must be specified to define the rate-independent part of the material behavior. The elastic behavior is defined by an elastic, hyperelastic, or hyperfoam material model. Since the frequency domain viscoelastic material model is developed around the long-term elastic moduli, the rate-independent elasticity must be defined in terms of long-term elastic moduli. This implies that the response in any analysis procedure other than a direct-solution steady-state dynamic analysis (such as a static preloading analysis) corresponds to the fully relaxed long-term elastic solution.