linear_viscoelastic

Description

This behavior defines a generalized linear viscoelastic Maxwell model. The model defines the stress, \(\ten \sigma\), to the strain \(\ten \varepsilon\) by the following relation:

\[\ten \sigma(t) = \int_0^t 2G(t-\tau)\dot{\ten e}(\tau)\,{\rm d}\tau + \ten 1 \int_0^t K (t-\tau) \trace{\dot{\ten \varepsilon}} \,{\rm d}\tau\]

with \(\ten e\) the deviator of the strain tensor \(\ten \varepsilon\). The terms \(G\) and \(K\) are relaxation functions defined by Prony series:

(18)\[\begin{split}\begin{aligned} &G(\tau) = G_\infty -(G_\infty-G_0)\Psi_1(\tau) \quad \Psi_1(\tau)=\sum_{i=1}^{i=n_\alpha} \omega_{i} \exp(-\tau/\tau_{i}) \\ &K(\tau) = K_\infty -(K_\infty-K_0)\Psi_2(\tau) \quad \Psi_2(\tau)=\sum_{i=1}^{i=n_\beta} \omega_{i} \exp(-\tau/\tau_{i}) \end{aligned}\end{split}\]

\(G_0\) and \(G_\infty\) are shear modulus coefficients

\(K_0\) and \(K_\infty\) are bulk modulus coefficients

Remark

The sum of the coefficients \(\omega\) for modulus terms must equal one.

The implementation of the model is in differential form with the internal variables \(\ten \alpha\) and \(\beta\). The state equations are written in the following form:

\[\ten \sigma=\sum_{i=1}^{i=n_\alpha} \ten X_i + \sum_{i=1}^{i=n_\beta} \ten Y_i + 2G_\infty\ten e + K_\infty \trace{\ten \varepsilon}\ten one\]

with

\[\begin{split}\begin{aligned} &\ten X_i = -2(G_\infty-G_0)\omega_i(\ten e - \ten \alpha_i) \quad 1\le i\le n_\alpha \\ &\ten Y_i = -3(K_\infty-K_0)\omega_i(\trace(\ten \varepsilon)/3-\beta_i)\ten one \quad 1\le i\le n_\beta \end{aligned}\end{split}\]

The evolution equations for the internal variables are:

\[\begin{split}\begin{aligned} &\dot{\ten \alpha}_i={\frac{1}{\tau_i}}(\ten e-\ten \alpha_i) \quad 1\le i\le n_\alpha \\ &\dot{\beta}= {\frac{1}{\tau_i}}(\trace{\overline{\varepsilon}}/3-\beta_i) \quad 1\le i\le n_\beta \end{aligned}\end{split}\]

It is necessary to define a single time the coefficients \(K_0\), \(K_\infty\), \(G_0\) and \(G_\infty\) using the key words **K0, **K_inf, **G0 and **G_inf respectively. One may then define an arbitrary number of the variables \(\ten \alpha\) and \(\beta\) using the key words omega and tau. omega is a constant coefficient.

Syntax

The syntax for this behavior model is the following:

**K0 COEFFICIENT **K_inf COEFFICIENT **G0 COEFFICIENT **G_inf COEFFICIENT **shear … **volumic …

Options **shear represent the shear mechanisms and are defined by the following form:

**shear \(~\,~\,\) tau COEFFICIENT \(~\,~\,\) omega COEFFICIENT

where the coefficient tau corresponds to the material constant \(\tau\) and omega to \(\omega\) as in equation (18).

Similarly, the option *volumic represents the volumetric mechanisms using the same coefficient / material constant naming:

**volumic \(~\,~\,\) tau COEFFICIENT \(~\,~\,\) omega COEFFICIENT

Stored Variables

The internal variables stored for this model are the total strain (code etoxx), the tensorial variables \(\overline{\alpha_i}\) (code alpha#xx) and the \(\beta_i\) variables (code beta#).

prefix	size	description	default
eto	T-2	total (small deformation) strain	yes
sig	T-2	Cauchy stress	yes
alpha#	T-2	\(\ten \alpha\) variable	no
beta#	S	\(\beta\) variable	no

The code names will replace the # symbol with the sequential number of that variable type as given by the order of declaration, and the xx symbols will be replaced with the tensorial components. The default saving of variables in the output files are only eto and sig. Specify other saved variables in the .inp file.

Example

***behavior linear_viscoelastic
**K0    42261.904761
**K_inf 13500.0
**G0    29098.360655
**G_inf 0.
**shear
   tau   0.4321660
   omega 0.2324006
**shear
   tau   9.070154
   omega 0.1891879
**shear
   tau   27.61690
   omega 0.2665674
**shear
   tau   102.8596
   omega 0.3118441
**volumic
   tau  0.1000000E-01
   omega 0.29800651
**volumic
   tau   0.3096638
   omega 0.8500050E-01
**volumic
   tau   0.2696395
   omega 0.4522469E-01
**volumic
   tau   6.517014
   omega 0.5717688
***return