Frequency Domain Viscoelasticity

The frequency domain viscoelastic material model describes frequency-dependent material behavior in small steady-state harmonic oscillations for those materials in which dissipative losses caused by “viscous” (internal damping) effects must be modeled in the frequency domain.

The frequency domain viscoelastic material model:

assumes that the shear (deviatoric) and volumetric behaviors are independent in multiaxial stress states;
can be used in large-strain problems;
can be used only in conjunction with Linear Elastic Behavior, Hyperelastic Behavior of Rubberlike Materials, and Hyperelastic Behavior in Elastomeric Foams to define the long-term elastic material properties;
can be used in conjunction with the elastic-damage gasket behavior (Defining a Nonlinear Elastic Model with Damage ) to define the effective thickness-direction storage and loss moduli for gasket elements; and
is active only during the direct-solution steady-state dynamic (Direct-Solution Steady-State Dynamic Analysis), the subspace-based steady-state dynamic (Subspace-Based Steady-State Dynamic Analysis), the natural frequency extraction (Natural Frequency Extraction), and the complex eigenvalue extraction (Complex Eigenvalue Extraction) procedures.

This page discusses:

Defining the Shear Behavior
Defining the Volumetric Behavior
Large-Strain Viscoelasticity
Determination of Viscoelastic Material Parameters
Conversion of Frequency-Dependent Elastic Moduli
Direct Specification of Storage and Loss Moduli for Large-Strain Viscoelasticity
Defining the Rate-Independent Part of the Material Behavior
Material Options
Elements

Defining the Shear Behavior

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

γ (t) = γ_{0} \exp (i ω t),

where $γ_{0}$ is the amplitude, $i = \sqrt{- 1}$ , $ω$ 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

τ (t) = (G_{s} (ω) + i G_{ℓ} (ω)) γ_{0} \exp (i ω t),

where $G_{s}$ and $G_{ℓ}$ are the shear storage and loss moduli. These moduli can be expressed in terms of the (complex) Fourier transform $g^{*} (ω)$ of the nondimensional shear relaxation function $g (t) = \frac{G_{R} (t)}{G_{\infty}} - 1$ :

G_{s} (ω) = G_{\infty} (1 - ω ℑ (g^{*})), G_{ℓ} (ω) = G_{\infty} (ω ℜ (g^{*})),

where $G_{R} (t)$ is the time-dependent shear relaxation modulus, $ℜ (g^{*})$ and $ℑ (g^{*})$ are the real and imaginary parts of $g^{*} (ω)$ , and $G_{\infty}$ 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 $G_{s} γ_{0}$ that is in phase with the strain and a stress of magnitude $G_{ℓ} γ_{0}$ that lags the excitation by $90^{\circ}$ . Hence, we can regard the factor

G^{*} (ω) = G_{s} (ω) + i G_{ℓ} (ω)

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

| τ | = \sqrt{G_{s}^{2} (ω) + G_{ℓ}^{2} (ω}) | γ ​_{0} |,

and the phase lag of the stress response is

ϕ = \arctan (\frac{G_{ℓ} (ω)}{G_{s} (ω)}) .

Measurements of $| τ |$ and $ϕ$ as functions of frequency in an experiment can, thus, be used to define $G_{s}$ and $G_{ℓ}$ and, thus, $ℜ (g^{*})$ and $ℑ (g^{*})$ as functions of frequency.

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

Defining the Volumetric Behavior

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 $K_{s} (ω)$ and $K_{ℓ} (ω)$ . Similar to the shear moduli, these moduli can also be expressed in terms of the (complex) Fourier transform $k^{*} (ω)$ of the nondimensional bulk relaxation function $k (t)$ :

K_{s} (w) = K_{\infty} (1 - ω ℑ (k^{*})), K_{ℓ} (w) = K_{\infty} (ω ℜ (k^{*})),

where $K_{\infty}$ is the long-term elastic bulk modulus.

Large-Strain Viscoelasticity

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 $G_{\infty}$ can vary with the amount of static prestrain $\bar{γ}$ :

G_{\infty} = G_{\infty} (\bar{γ}) .

The essential simplification implied by this assumption is that the frequency-dependent part of the material's response, defined by the Fourier transform $g^{*} (ω)$ 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 $G_{s}$ , $G_{ℓ}$ , and $G_{\infty}$ (likewise $K_{s}$ , $K_{ℓ}$ , and $K_{\infty}$ ) 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 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

\frac{1}{J} Δ^{▽} (J S) = {(1 + i ω g^{*}) C^{S} |}_{0} : Δ e + {Q |}_{0} Δ ε^{vol},

and

Δ p = - {Q |}_{0} : Δ e - {(1 + i ω k^{*}) K |}_{0} Δ ε^{vol},

where

$S$: is the deviatoric stress, $S = σ + p I$ ;
p: is the equivalent pressure stress, $p = - \frac{1}{3} trace (σ)$ ;
$Δ^{▽} (J S)$: 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;
$Δ e$: is the (small) incremental deviatoric strain, $Δ e = Δ ε - \frac{1}{3} Δ ε^{vol} I$ ;
$\dot{e}$: is the deviatoric strain rate, $\dot{e} = \dot{ε} - \frac{1}{3} {\dot{ε}}^{vol} I$ ;
$Δ ε^{vol}$: is the (small) incremental volumetric strain, $Δ ε^{vol} = trace (Δ ε)$ ;
${\dot{ε}}^{vol}$: is the rate of volumetric strain, ${\dot{ε}}^{vol} = trace (\dot{ε})$ ;
${C^{S} |}_{0}$: 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);
${Q |}_{0}$: 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 $g^{*} (ω)$ .

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^{*} (ω) = g_{1}^{*} f^{- a} and k^{*} (ω) = k_{1}^{*} f^{- b},

where a and b are real constants, $g_{1}^{*}$ and $k_{1}^{*}$ are complex constants, and $f = ω / 2 π$ is the frequency in cycles per time.

Tabular Frequency Dependence

The frequency domain response can alternatively be defined in tabular form by giving the real and imaginary parts of $ω g^{*}$ and $ω k^{*}$ —where $ω$ is the circular frequency—as functions of frequency in cycles per time. Given the frequency-dependent storage and loss moduli $G_{s} (ω)$ , $G_{ℓ} (ω)$ , $K_{s} (ω)$ , and $K_{ℓ} (ω)$ , the real and imaginary parts of $ω g^{*}$ and $ω k^{*}$ are then given as

ω ℜ (g^{*}) = G_{ℓ} / G_{\infty}, ω ℑ (g *) = 1 - G_{s} / G_{\infty}, ω ℜ (k^{*}) = K_{ℓ} / K_{\infty}, ω ℑ (k^{*}) = 1 - K_{s} / K_{\infty},

where $G_{\infty}$ and $K_{\infty}$ 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 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 - \overset{N}{\sum_{i = 1}} {\bar{g}}_{i}^{P} (1 - e^{- t / τ_{i}}),

k_{R} (t) = 1 - \overset{N}{\sum_{i = 1}} {\bar{k}}_{i}^{P} (1 - e^{- t / τ_{i}}),

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

G_{s} (ω) = G_{0} [1 - \overset{N}{\sum_{i = 1}} {\bar{g}}_{i}^{P}] + G_{0} \overset{N}{\sum_{i = 1}} \frac{{\bar{g}}_{i}^{P} τ_{i}^{2} ω^{2}}{1 + τ_{i}^{2} ω^{2}},

G_{ℓ} (ω) = G_{0} \overset{N}{\sum_{i = 1}} \frac{{\bar{g}}_{i}^{P} τ_{i} ω}{1 + τ_{i}^{2} ω^{2}},

where $G_{s} (ω)$ is the storage modulus, $G_{ℓ} (ω)$ is the loss modulus, $ω$ is the angular frequency, and N is the number of terms in the Prony series. The expressions for the bulk moduli, $K_{s} (ω)$ and $K_{ℓ} (ω)$ , are written analogously. Abaqus/Standard will automatically perform the conversion from the time domain to the frequency domain. The Prony series parameters ${\bar{g}}_{i}^{P}, {\bar{k}}_{i}^{P}, τ_{i}$ 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.

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, $({\bar{g}}_{i}^{P}, {\bar{k}}_{i}^{P}, τ_{i})$ .

Thermorheologically Simple Temperature Effects in Frequency Domain Viscoelasticity

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

ω_{r} = A (θ) ω,

where $A (θ)$ and $θ$ denote the shift function and temperature, respectively. Abaqus/Standard supports the following forms of the shift function: the Williams-Landel-Ferry (WLF) form, the Arrhenius form, the tabular form, and user-defined forms (see Thermorheologically Simple Temperature Effects).

Conversion of Frequency-Dependent Elastic Moduli

For some cases of small straining of isotropic viscoelastic materials, the material data are provided as frequency-dependent uniaxial storage and loss moduli, $E_{s} (ω)$ and $E_{ℓ} (ω)$ , and bulk moduli, $K_{s} (ω)$ and $K_{ℓ} (ω)$ . In that case the data must be converted to obtain the frequency-dependent shear storage and loss moduli $G_{s} (ω)$ and $G_{ℓ} (ω)$ .

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

G^{*} = \frac{3 K^{*} E^{*}}{9 K^{*} - E^{*}} .

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

G_{s} + i G_{ℓ} = \frac{3 (K_{s} + i K_{ℓ}) (E_{s} + i E_{ℓ})}{9 (K_{s} + i K_{ℓ}) - (E_{s} + i E_{ℓ})} .

After some algebra one obtains

G_{s} = 3 \frac{9 E_{s} (K_{s}^{2} + K_{ℓ}^{2}) - K_{s} (E_{s}^{2} + E_{ℓ}^{2})}{{(9 K_{s} - E_{s})}^{2} + {(9 K_{ℓ} - E_{ℓ})}^{2}}, G_{ℓ} = 3 \frac{9 E_{ℓ} (K_{s}^{2} + K_{ℓ}^{2}) - K_{ℓ} (E_{s}^{2} + E_{ℓ}^{2})}{{(9 K_{s} - E_{s})}^{2} + {(9 K_{ℓ} - E_{ℓ})}^{2}} .

Shear Strain Only

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

G_{s} = 3 K_{\infty} \frac{9 E_{s} K_{\infty} - E_{s}^{2} - E_{ℓ}^{2}}{{(9 K_{\infty} - E_{s})}^{2} + E_{ℓ}^{2}}, G_{ℓ} = 3 K_{\infty} \frac{9 E_{ℓ} K_{\infty}}{{(9 K_{\infty} - E_{s})}^{2} + E_{ℓ}^{2}} .

Incompressible Materials

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

G_{s} = E_{s} / 3, G_{ℓ} = E_{ℓ} / 3.

Direct Specification of Storage and Loss Moduli for Large-Strain Viscoelasticity

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.

Defining the Rate-Independent Part of the Material Behavior

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.

Material Options

The viscoelastic material model must be combined with the isotropic linear elasticity model to define classical, linear, small-strain, viscoelastic behavior. It is combined with the hyperelastic or hyperfoam model to define large-deformation, nonlinear, viscoelastic behavior. The long-term elastic properties defined for these models can be temperature dependent.

Viscoelasticity cannot be combined with any of the plasticity models. See Combining Material Behaviors for more details.

Elements

The frequency domain viscoelastic material model can be used with any stress/displacement element in Abaqus/Standard.