## Viscoelastic Material Models for Stress Analysis (B2000++ Pro)

The visco-elastic material model is provided with B2000++ Pro and represents an elastic isotropic or orthotropic material with dynamic material damping according to the Generalised Maxwell Model (GMM). This material model is supported by shell and 3D elements and assumes small strains.

## 1. Viscoelastic Isotropic Material

material id type viscoelastic_isotropic   e v   nu v   g v   relative_moduli flist   relaxation_times flist   [volumetric yes|no]   [instantaneous_elasticity yes|no]   [alpha v]   [density v] end

The following parameters are specified:

e v

Specifies the modulus of elasticity.

nu v

Specifies Poisson's ratio. Default is 0.

relative_moduli flist

Specifies the values of the relative moduli of the GMM elements. The number of values must be identical to those in relaxation_times.

relaxation_times flist

Specifies the relaxation times for the GMM elements. The number of values must be identical to those in relative_moduli.

volumetric yes|no

Specifies if the components refer to volumetric tensor or not. Default is yes. If no is specified, the volumetric-deviatoric split is performed, and the viscous damping is performed only on the deviatoric components.

instantaneous_elasticity yes|no

Specifies if the relaxation function assumes instantaneous elasticity. Default is yes (creep). If no is specified, viscous damping is performed.

alpha v

Specifies the thermal expansion coefficient. Default is 0.

failure ... end

Specifies a failure criterion (optional).

density v

Specifies the material density. Default is 0.

## 2. Viscoelastic Orthotropic Material

material id type viscoelastic_orthotropic   e1|e2|e3 v   nu12|nu13|nu23 v   g12|g13|g23 v   relative_moduli_ij flist   relaxation_times_ij flist   [volumetric yes|no]   [instantaneous_elasticity yes|no]   [alpha11|alpha22|alpha33|alpha12|alpha13|alpha23 v]   [density v] end

The following parameters are specified:

e1|e2|e3 v

Specifies the modulus of elasticity in the material directions. If e is specified

nu12|nu13|nu23 v

Specifies Poisson's ratio relating the different material directions. Shell elements make use of nu13 and nu23 for the plane-stress condition.

g12|g13|g23 v

Specifies the shear modulus.

relative_moduli_ij flist

Specifies the values of the relative moduli of the GMM elements in the material direction ij (11, 22, 33, 12, 13, 23). The number of values must be identical to those in relaxation_times.

relaxation_times_ij flist

Specifies the relaxation times for the GMM elements in the material direction ij (11, 22, 33, 12, 13, 23). The number of values must be identical to those in relative_moduli_ij.

volumetric yes|no

Specifies if the components refer to volumetric tensor or not. Default is yes. If no is specified, the volumetric-deviatoric split is performed, and the viscous damping is performed only on the deviatoric components.

instantaneous_elasticity yes|no

Specifies if the relaxation function assumes instantaneous elasticity. Default is yes (creep). If no is specified, viscous damping is performed.

alpha11|alpha22|alpha33|alpha12|alpha13|alpha23 v

Specifies the thermal expansion coefficients. Default is 0.

failure ... end

Specifies a failure criterion (optional).

density v

Specifies the material density. Default is 0.

## 3. Theoretical Background

The convolution integral of the constitutive equation of a generalized Maxwell element for a linear initial material and infinitesimal ('small') strain is

σ ij t = u = - t g ij t - u σ . ij 0 d u

and the normalized relaxation functions are defined as

g ij t = γ ij + k= 1 N γ ij k e -t τ i

γ ij = 1 - k = 1 N γ ij k

τ i is the relaxation time and γ i the relative modulus of the ith out of N Maxwell elements.

This model is sometimes extended with an additive split of the strain as a deviatoric and a volumetric strain with a different constitutive equation for each part of the strain (option volumetric no). This is justified for nearly incompressible materials like some polymeric materials where the bulk response is only elastic and the deviatoric response is viscoelastic. The material model assumes ‘small’ strains, which is justified for metallic or composite materials.

The relaxation function as reported above assumes instantaneous elasticity. If not (parameter instantaneous_elasticity set to no), then the relaxation function g is defined as:

g ij t = 1 + i = 1 N γ ij k e -t τ i

Staverman and Schwarz [Staverman52] give the expression of the free energy for a generalized Maxwell element in one dimension. Piero and Deseri [Piero96] generalize this expression to three-dimensional linear viscoelasticity. The free energy F of the generalized Maxwell element is

F = u,v = - t g 2 t - u - v ε . u C 0 ε . v d u d v

Using the thermodynamic equation

σ ε . = F + T 0 Θ

T 0 is the constant temperature and Θ the entropy production. By differentiating with respect to the time we obtain the rate of dissipated energy D = T 0 Θ

D = - u,v = - t