viscoelastic_spectral

Description

This behavior defines a spectral viscoelastic model. The model defines the stress, \(\bf sig\), to the strain \(\bf eps\) by the following differential equations system:

\[\ten \sigma = \tenf{C}_0(T) : (\ten \epsilon - \ten \epsilon_{a} - \ten \epsilon_{th})\]

\[\ten \epsilon_{th} = \ten \alpha (T - T_0)\]

\[\dot{\ten\epsilon}_a = g(\ten \sigma) \sum_{i=1}^{i=n_t}\dot{X}_i\]

\[\dot{X}_i = \frac{1}{\tau_i}(\mu_i(T)*g(\sigma)C_R^{-1}(T):\sigma-X_i)\]

\(C_0\) is the elastic matrix. Note that \(S_0\) used latter is defined by : \(S_0=C_0^{-1}\).

\(g(\sigma)\) is the viscous non linear function

\(\tau_i\) and \(\mu_i\) define the spectrum. Each mechanism \(X_i\) is associated with a relaxation time \(\tau_i\) weighted by \(\mu_i\). The whole strain family (\(X_i\)) describes a Gaussian continuous spectrum.

Note

For the moment, the thermal strains are not taken into account.

Syntax

The material file structure for the viscoelastic_spectral model consists of an elasticity object, the definition of spectrum, of the viscous_effects tensor and of the asymptote. The syntax for this behavior model is the following:

***behavior viscoelastic_spectral [ modifier ] \(~\,\) **elasticity <ELASTICITY> \(~\,\) **spectrum … \(~\,\) **viscous_effects … \(~\,\) **reversible_asymptote …

Options **spectrum represent the spectrum which defined \(\mu_i\) and \(\tau_i\) (see next figure).

\[\mu_i=\frac{1}{n_o\pi}exp(-(\frac{i-n_c}{n_0})^2)\]

The spectrum is defined by the following form:

**spectrum [ *limit double] [ *n1 double] [ *n2 double] \(~\,\) *nt interger \(~\,\) *nc COEFFICIENT \(~\,\) *n0 COEFFICIENT

See the figure for the meaning of nt, nc and n0. n1 and n2 can be read in the material file (*n1 and *n2) or calculated such as \(\mu(n1)>limit\) and \(\mu(n2)<limit\).

Note

By default limit is equal to 1.e-4 and nt to 30. The option *viscous_effects defined the viscous effects tensor \(C_R\).

**viscous_effects \(~\,\) *lrt COEFFICIENT \(~\,\) *lrc COEFFICIENT

\(C_R\) is defined such as :

for i equal 1 to 3 :

\[C_R(i,i)=lrt*S_0(i,i)\]

for i equal 4 to 6 :

\[C_R(i,i)=lrc*S_0(i,i)\]

and if \(i{\neq}j\) for i and j equal 1 to 3 (isotropic condition):

\[C_R(i,j)=lrn*S_0(i,j)\]

with \(lrn=(lrt*S_0(1,1)-lrc*S_0(2,2))/S_0(1,2)\).

Finally the option *reversible_asymptote is used to define the non linear function \(g(\sigma)\).

**reversible_asymptote \(~\,\) *beta COEFFICIENT \(~\,\) *p COEFFICIENT

\[g(\sigma)=1+beta(\sqrt(\sigma:C_R^{-1}:\sigma))^p\]

Remark

beta=0 is equivalent to a classical linear asymptote.

Note

Another definition of the asymptote (**plastic_asymptote) is also implemented but not documented here because still under development.

Stored Variables

The internal variables stored for this model are the total strain (code etoxx), the tensorial variables \(X_i\) (code kip#xx).

prefix	size	description	default
eto	T-2	total (small deformation) strain	yes
sig	T-2	Cauchy stress	yes
eel	T-2	elastic strain	yes
ean	T-2	anelastic strain	yes
kip#	T-2	\(X_i\) variable	no

The code names will replace the # symbol with a sequential number from 1. to nt.

The default saving of variables in the output files are only those marked by yes in the previous table.

Example

***behavior viscoelastic_spectral
  **elasticity isotropic
    young 2800.
    poisson 0.3
  **spectrum
          *n1 -30.
          *n2  30.
          *nt  50
          *nc  7.
          *n0  3.
  **viscous_effects
          *lrt 0.6
          *lrc 0.6
  **reversible_asymptote
          *beta 1.
          *p    1.
***return