Viscoelasticity

Frequency domain viscoelastic response describes frequency-dependent material behavior in small, steady-state harmonic oscillations. Time domain response describes the isotropic rate-dependent behavior of materials for which the dissipative losses caused by internal damping effects must be modeled in the time domain.

Note: Viscoelasticity must be defined in conjunction with either elasticity or hyperelasticity.


Input Data	Description
Domain	Frequency or Time
Formula (Freq.)	Define the frequency dependence by the power law formulae.
Prony (Freq.)	Calculate the frequency dependence from a time domain Prony series description of the dimensionless shear and bulk relaxation moduli.
Tabular (Freq.)	Define the frequency response by providing tabular entries of the real and imaginary parts of the circular frequency as functions of frequency in cycles per time.
Prony (Time)	Enter the Prony series parameters for each term directly.
Frequency Data (Time)	Calibrate the Prony series parameters using frequency-dependent test data.

Formula (Frequency domain)

Frequency domain viscoelasticity can be defined by entering the values for the power law formulae.

g^{*} (ω) = g_{1}^{*} f^{- a}

and

k^{*} (ω) = k_{1}^{*} f^{- b},

where $f = \frac{ω}{2 π}$ is the frequency in cycles per time.


Input Data	Description
g1*real	Real part of $g_{1}^{*}$
g1*imag	Imaginary part of $g_{1}^{*}$
a	Real constant
k1*real	Real part of $k_{1}^{*}$
k1*imag	Imaginary part of $k_{1}^{*}$
b	Real constant

Prony (Frequency domain)

Frequency domain viscoelasticity can be defined by using a Prony series expansion of the shear and bulk relaxation moduli in the material. Data are entered in a table in which each row represents a set of constants in one term of the Prony series; the order of the rows corresponds to the order of the terms in the series. The input data are the same as those used for a time domain Prony viscoelastic definition, but the data are transformed internally to obtain the frequency domain behavior.


Input Data	Description
Gi Prony	The shear relaxation or shear traction relaxation modulus ratio, ${\bar{g}}_{i}^{P}$ , in the Prony series expansion.
Ki Prony	The bulk relaxation or normal traction relaxation modulus ratio, ${\bar{k}}_{i}^{P}$ , in the Prony series expansion.
Tau_i Prony	The relaxation time $τ_{i}$ , in units of seconds, in the Prony series expansion.

Tabular (Frequency domain)

Frequency domain viscoelasticity can be defined by tabular entries. The required entries depend on the Type and Preload options. In most cases, the loss and storage moduli are required as functions of frequency. However, isotropic data with no preload requires data for the real and imaginary parts of the circular frequency as functions of frequency in cycles per time.


Input Data	Description
Omega g1*real	The real part of the circular frequency $ω g_{1}^{*}$ .
Omega g1*imag	The imaginary part of the circular frequency $ω g_{1}^{*}$ .
Omega k1*real	The real part of the circular frequency $ω k_{1}^{*}$ .
Omega k1*imag	The imaginary part of the circular frequency $ω k_{1}^{*}$ .
Loss Modulus	The lost energy, representing the viscous portion of the material response (dissipated as heat).
Storage Modulus	The stored energy, representing the elastic portion of material response.
Normalized Loss Modulus	For traction type with no preload (gaskets).
Normalized Storage Modulus	For traction type with no preload (gaskets).
Uniaxial Strain	For isotropic type with uniaxial preload.
Volume Ratio	For isotropic type with volumetric preload.
Closure	For traction type with uniaxial preload (gaskets).
Frequency	The frequency, in cycles per unit time.

Prony (Time domain)

Time domain viscoelasticity is defined numerically using a Prony series expansion of the shear and bulk relaxation moduli in the material. Data are entered in a table in which each row represents a set of constants in one term of the Prony series; the order of the rows corresponds to the order of the terms in the series.


Input Data	Description
Gi Prony	The shear relaxation or shear traction relaxation modulus ratio, ${\bar{g}}_{i}^{P}$ , in the Prony series expansion.
Ki Prony	The bulk relaxation or normal traction relaxation modulus ratio, ${\bar{k}}_{i}^{P}$ , in the Prony series expansion.
Tau_i Prony	The relaxation time $τ_{i}$ , in units of seconds, in the Prony series expansion.

Frequency Data (Time domain)

Time domain viscoelasticity is defined by calibrating the Prony series parameters using frequency-dependent test data. In this case analytical expressions are used that relate the Prony series relaxation functions to the storage and loss moduli.


Input Data	Description
Maximum number of terms in the Prony series	The maximum number of terms ( $N$ ) in the Prony series. A least-squares fit is performed from $N = 1$ to $N =$ NMAX until convergence is achieved for the lowest $N$ with respect to the error tolerance.
Allowable average root-mean-square error	Error tolerance for the data points in the least-squares fit.
Omega g real	Real part of $ω g^{*}$ .
Omega g imag	Imaginary part of $ω g^{*}$ .
Omega k real	Real part of $ω k^{*}$ .
Omega k imag	Imaginary part of $ω k^{*}$ .
Frequency	Frequency, $f$ , in cycles per time.