Nonlinear Viscoelasticity

The nonlinear viscoelastic material model:

consists of multiple viscoelastic networks and, optionally, an elastoplastic network in parallel;
uses a hyperelastic material model to specify the elastic response;
can be combined with Mullins effect;
bases the elastoplastic response on multiplicative split of the deformation gradient and the theory of incompressible isotropic hardening plasticity;
an include nonlinear kinematic hardening with multiple backstresses in the elastoplastic response in implicit analysis; and
uses multiplicative split of the deformation gradient and a flow rule derived from a creep potential to specify the viscous behavior.

Parallel rheological framework

The parallel rheological framework allows definition of a nonlinear viscoelastic-elastoplastic model consisting of multiple networks connected in parallel. The number of viscoelastic networks, $N$ , can be arbitrary; however, at most one equilibrium network is allowed in the model. The equilibrium network response might be purely elastic or elastoplastic. In addition, it might include Mullins effect to predict material softening. The definition of the equilibrium network is optional. If it is not defined, the stress in the material will relax completely over time.

Elastic Behavior

The elastic part of the response for all the networks is specified using the hyperelastic material model. Any available hyperelastic model (see Isotropic Hyperelasticity) can be used. The same hyperelastic material definition is used for all the networks, scaled by a stiffness ratio specific to each network. Consequently, only one hyperelastic material definition is required by the model along with the stiffness ratio for each network. The elastic response can be specified by defining either the instantaneous response or the long-term response.

In addition to the elastic response described above, the response of the equilibrium network can include plasticity with isotropic hardening (see Plastic Options) and Mullins effect (see Mullins Effect) to predict material softening, and thermorheologically simple temperature effects (see Thermorheologically Simple Temperature Effects). In an implicit analysis the nonlinear kinematic hardening model with multiple backstresses can be specified in addition to isotropic plastic hardening.

Viscous Behavior

Viscous behavior must be defined for each viscoelastic network. It is modeled by assuming the multiplicative split of the deformation gradient and the existence of the creep potential, $G^{c r}$ , from which the flow rule is derived. In the multiplicative split the deformation gradient is expressed as

F = F^{e} \cdot F^{c r}

where

F^{e}

is the elastic part of the deformation gradient (representing the hyperelastic behavior) and

F^{c r}

the creep part of the deformation gradient (representing the stress-free intermediate configuration). The creep potential is assumed to have the general form

G^{c r} = G^{c r} (σ)

where

σ

is the Cauchy stress. If the potential is specified, the flow rule can be obtained from

D^{c r} = \dot{λ} \frac{\partial G^{c r} (σ)}{\partial σ}

where

D^{c r}

is the symmetric part of the velocity gradient,

L^{c r}

, expressed in the current configuration and

\dot{λ}

is the proportionality factor. In this model the creep potential is given by

G^{c r} = \bar{q}

and the proportionality factor is taken as

\dot{λ} = {\dot{\bar{ε}}}^{c r}

, where

\bar{q}

is the equivalent deviatoric Cauchy stress and

{\dot{\bar{ε}}}^{c r}

is the equivalent creep stain rate. In this case the flow rule has the form

D^{c r} = \frac{3}{2 \bar{q}} {\dot{\bar{ε}}}^{c r} \bar{σ}

or, equivalently

D^{c r} = \frac{3}{2 \tilde{q}} {\dot{\bar{ε}}}^{c r} \bar{τ}

where

τ = J σ

is the Kirchhoff stress,

J

is the determinant of

F

\bar{σ}

is the deviatoric Cauchy stress,

\bar{τ}

is the deviatoric Kirchhoff stress, and

\tilde{q} = J \bar{q}

. To complete the derivation, the evolution law for

{\dot{\bar{ε}}}^{c r}

must be provided. In this model

{\dot{\bar{ε}}}^{c r}

can be defined by the power law model, the hyperbolic-sine law model, or the Bergstrom-Boyce Power Law model.

Creep Laws for Nonlinear Viscoelasticity

Nonlinear viscoelasticity supports three creep laws: the power law model, the hyperbolic-sine law model, and Bergstrom-Boyce model.

The power law model is available in the form

{\dot{\bar{ε}}}^{c r} = {{\dot{ε}}_{0} [{{(\frac{\tilde{q}}{q_{0} + a < p >})}^{n} [(m + 1) {\bar{ε}}^{c r}]}^{m})]}^{\frac{1}{m + 1}}

where

{\dot{\bar{ε}}}^{c r}

, is the equivalent creep strain rate,

\sqrt{\frac{2}{3} {\dot{ε}}^{c r} : {\dot{ε}}^{c r}}

{\bar{ε}}^{c r}

is the equivalent creep strain,

\tilde{q}

is the equivalent Kirchhoff stress,

p

is the equivalent Kirchhoff pressure, and

q_{0}

m

n

a

, and

{\overset{\cdot}{ε}}_{0}

are material parameters.

The hyperbolic-sine law is available in the form

{\dot{\bar{ε}}}^{c r} = A {(\sinh B \tilde{q})}^{n}

where

{\dot{\bar{ε}}}^{c r}

and

\tilde{q}

are defined above and

A

B

, and

n

are material parameters.

The Bergstrom-Boyce power law model has the form

{\dot{\bar{ε}}}^{c r} = {\dot{ε}}_{0} {(λ^{c r} - 1 + E)}^{C} {(\frac{\tilde{q}}{q_{0}})}^{m}

where

λ^{c r} = \sqrt{\frac{1}{3} I : C^{c r}}

{\dot{\bar{ε}}}^{c r}

and

\tilde{q}

are defined above and

q_{0}

m

C

E

and

{\overset{\cdot}{ε}}_{0}

are material parameters.

Number of Viscoelastic Networks

The nonlinear viscoelastic response is defined by specifying the number of networks and the stiffness ratio and creep law for each network.

You must define the number of viscoelastic networks in the material model.


Input Data	Description
Number of networks	Specify the number of networks, $N$ .

The contribution of each network to the overall response of the material is determined by the value of the stiffness ratio, $r$ , which is used to scale the elastic response of the network material. The sum of the stiffness ratios of the viscoelastic networks must be smaller than or equal to 1. If the sum of the ratios is equal to 1, the purely elastic equilibrium network is not created. If the sum of the ratios is smaller than 1, the equilibrium network is created with a stiffness ratio, $r_{0}$ , equal to

r_{0} = 1 - \sum_{k = 1}^{N} r_{k}

where

N

denotes the number of viscoelastic networks and

r_{k}

is the stiffness ratio of network

k


Input Data	Description
Stiffness Ratio	Set this parameter equal to the stiffness ratio for the current network, $r_{k}$ .

The definition of the viscoelastic network is completed by specifying the creep law.

Parameters

Table 1. Law=Hyperbolic Sine
Input Data	Description
A	Material parameter, $A$
B	Material parameter, $B$
n	Material parameter, $n$
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specify material parameters that depend on field variables. Field columns appear in the data table for each field variable you add. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Table 2. Law=Power Law
Input Data	Description
Reference stress	Reference stress, $q_{0}$
n	Material parameter, $n$
m	Material parameter, $m$
a	Material parameter, $a$
Reference strain rate	Reference strain rate, ${\overset{\cdot}{ε}}_{0}$
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specify material parameters that depend on field variables. Field columns appear in the data table for each field variable you add. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Table 3. Law=Bergstrom-Boyce Power Law
Input Data	Description
Reference stress	Reference stress, $q_{0}$
m	Material parameter, $m$
C	Material parameter, $C$
E	Material parameter, $E$
Initial strain rate	Initial strain rate, ${\overset{\cdot}{ε}}_{0}$
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specify material parameters that depend on field variables. Field columns appear in the data table for each field variable you add. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.