Nonlinear Viscoelasticity

Abstract

Viscoelastic behavior may show physical and/or geometrical nonlinearity. Physical nonlinearity corresponds to situations in which the linear behavior described in Chap. 1 (Sect. 1.3.2) is not observed, even in small strain situations. Geometrical nonlinearity corresponds to situations of large deformations (large displacements and/or large strain). Both effects can appear combined in some problems (e.g. polymers, biomechanics). Alternative nonlinear or quasi-linear single integral representations have been proposed, some of which are described in Sect. 8.2. In Sect.8.3, a nonlinear state variables formulation proposed by Simo is described. The situation involving large displacements associated with small strains that is particularly important in the analyses of materials and structures is addressed in detail in Chap. 9.

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Viscoelastic behavior may show physical and/or geometrical nonlinearity. Physical nonlinearity corresponds to situations in which the linear behavior described in Chap. 1 (Sect. 1.3.2) is not observed, even in small strain situations. Geometrical nonlinearity corresponds to situations of large deformations (large displacements and/or large strain). Both effects can appear combined in some problems (e.g. polymers, biomechanics). Alternative nonlinear or quasi-linear single integral representations have been proposed, some of which are described in Sect. 8.2. In Sect. 8.3, a nonlinear state variables formulation proposed by Simo is described. The situation involving large displacements associated with small strains that is particularly important in the analyses of materials and structures is addressed in detail in Chap. 9.

8.1 Schapery Single Integral Non-Linear Viscoelasticity

The Schapery single integral constitutive equation of non-linear viscoelasticity was derived from fundamental principles utilizing the concepts of irreversible thermodynamics [11] in a small strain context. For isothermal condition and uniaxial stress, the creep constitutive equation proposed by Schapery can be written as

$$ \varepsilon (t) = {\text{g}}_{ 0} {\text{D(0)}}\sigma (t) + {\text{g}}_{ 1} \int\limits_{0}^{t} {C\left( {\psi - \psi^{\prime}} \right)\frac{\partial }{\partial \tau }} \left( {{\text{g}}_{ 2} \sigma (\tau )} \right)d\tau $$

(8.1)

where ${\text{D(0)}}$ and ${\text{C}}(\psi )$ are the instantaneous and transient components of the creep compliance in linear viscoelasticity, respectively. The arguments $\psi$and $\psi^{\prime}$ are reduced times that take into account simultaneously, temperature and stress effects and are given by

$$ \psi = \int_{0}^{t} {\frac{ds}{a(\theta )b(\sigma )}}\, {\text{and}}\, \psi^{\prime} = \int_{0}^{\tau } {\frac{ds}{a(\theta )b(\sigma )}} $$

(8.2)

in which $a(\theta )$ is the temperature shift factor used for thermorheologically simple materials (see Chap. 6) whereas $b(\sigma )$ is the stress shift factor. Then, $D(\psi - \psi^{\prime}) = D(0) + C(\psi - \psi^{\prime})$ is the creep compliance adjusted for stress and temperature variations.

The material properties $g_{0},$ $g_{1}$ and $g_{2}$ are nonlinear functions of stress. $g_{0} (\sigma )$ is related to the nonlinear instantaneous compliance. The transient creep parameter $g_{1} (\sigma )$ measures the nonlinear effect on creep and $g_{2}$ accounts for the load rate effect. When the applied stress is small, $g_{0} = g_{1} = g_{2} = b(\sigma ) = 1$ and (8.1) is reduced to the integral representation of linear viscoelastic behavior. Notice that when the mentioned stress functions are dependent on temperature, the material behavior is thermorheologically complex.

When strain is the independent variable, the corresponding relaxation constitutive equation for constant temperature is given by

$$ \sigma (t) = h_{\infty } {\text{E}}(\infty )\varepsilon (t) + h_{1} \int\limits_{0}^{t} {{\text{R}}(\zeta - \zeta^{\prime})\frac{\partial }{\partial \tau }\left( {h_{2} \varepsilon (\tau )} \right)d\tau } $$

(8.3)

where $h_{\infty },$ $h_{1}$ and $h_{2}$ are material nonlinear functions of the strain. $E(\infty )$ and $R(\zeta )$ indicate the asymptotic modulus at constant strain$(t \to \infty )$ and the transient component of the relaxation function, in linear viscoelasticity, respectively. The variables $\zeta$ and $\zeta^{\prime}$ stand for reduced times defined by

$$ \zeta = \int_{0}^{t} {\frac{ds}{a(\theta )c(\varepsilon )}} \qquad \zeta^{\prime} = \int_{0}^{\tau} {\frac{ds}{a(\theta )c(\varepsilon )}} $$

(8.4)

where $c(\varepsilon )$ is the strain shift factor. Hence, considering the above definitions, the relaxation function of the material, adjusted to take into account temperature and strain effects, can be expressed as $E(\zeta - \zeta^{\prime}) = E(\infty ) + R(\zeta - \zeta^{\prime})$

As seen in (8.1) and (8.2), the Schapery formulation involves five material functions and a material constant to be determined experimentally. This number of experimental parameters justifies the power of the model to fit the nonlinear behavior of many viscoelastic materials. A detailed explanation of the complex procedure needed to obtain these parameters may be found in the book by Brinson and Brinson [1].

Several authors extended the Schapery model to 3D situations and implemented it into finite element procedures [6, 7, 9].

8.2 Nonlinear Viscoelasticity at Large Strains in Integral Form

8.2.1 General Constitutive Relation

In the case of large strain problems (geometrical nonlinearity) appropriate measures for stress and strain and their functions have to be used (see Appendix B).

Constitutive relations in a large deformation context need to be objective, that is, independent of the presence of large displacements and rotations.

We begin with

$$ {\varvec{\sigma}}(t) = {\varvec{\mathcal{G}}}\mathop {({\mathbf{F}})}\limits_{\tau = 0}^{t} $$

(8.5)

(compare it with the small strain version (1.6)) and look for the conditions to make it objective. To do this, rigid body motions, characterized by translations ${\mathbf{d}}(\tau )$ and rotations ${\mathbf{Q}}(\tau ),$ $0 \le \tau \le t,$ are superposed on the body motion ${\mathbf{x}}(\tau ) = \chi ({\mathbf{X}},\tau ).$ For this case, the particle position and deformation gradient are given by ${\mathbf{x}}^{*} (\tau ) = {\mathbf{Q}}(\tau ){\mathbf{x}}(\tau ) + {\mathbf{d}}(\tau )$ and ${\mathbf{F}}^{*} (\tau ) = {\mathbf{Q}}(\tau ){\mathbf{F}},$ respectively. When the material is subjected to a rotation ${\mathbf{Q}}$ (${\mathbf{Q}}$ is an orthogonal tensor, ${\mathbf{QQ}}^{{\mathbf{T}}} {\mathbf{ = I}}$), the stress ${\varvec{\sigma}}$ transforms as a second order tensor while the deformation gradient ${\mathbf{F}}$ transforms as a vector. Then we have

$$ {\mathbf{Q\,\varvec{\sigma} Q}}^{T} = {\varvec{\mathcal{G}}}\mathop {({\mathbf{QF}})}\limits_{\tau = 0}^{t} = {\varvec{\mathcal{G}}}\mathop {({\mathbf{QRU}})}\limits_{\tau = 0}^{t} $$

(8.6)

To find a necessary condition we choose ${\mathbf{Q}} = {\mathbf{R}}^{T} = {\mathbf{R}}^{ - 1}$ and substitute it into (8.6) obtaining

$$ {\varvec{\sigma}} = {\varvec{\mathcal{G}}}\mathop {({\mathbf{F}})}\limits_{\tau = 0}^{t} = {\mathbf{R}}{\varvec{\mathcal{G}}}\mathop {({\mathbf{U}})}\limits_{\tau = 0}^{t} {\mathbf{R}}^{T} $$

(8.7)

and

$$ {\mathbf{F}}^{T} {\mathbf{\sigma F}} = {\mathbf{U}}F\mathop {({\mathbf{C}})}\limits_{\tau = 0}^{t} {\mathbf{U}} = {\tilde{\varvec{\mathcal{G}}}}\mathop {({\mathbf{C}})}\limits_{\tau = 0}^{t} $$

(8.8)

because ${\mathbf{F = RU}}$ and ${\mathbf{C}} = {\mathbf{F}}^{T} {\mathbf{F}} = {\mathbf{U}}^{2}$ (see Appendix B). Thus

$$ {\varvec{\sigma}}(t) = {\mathbf{F}}(t)\tilde{{\varvec{\mathcal{G}}}}\left[ {{\mathbf{C}}(t - \tau )} \right]_{\tau = 0}^{t} \,{\mathbf{F}}^{T} (t) $$

(8.9)

Using the relation between the Cauchy stress tensor ${\varvec{\sigma}}$ and the Second Piola–Kirchhoff stress tensor ${\mathbf{S}}$ (see Appendix B) in (8.9), we obtain

$$ {\mathbf{S}}(t) = J(t){\tilde{{{\varvec{\mathcal{G}}}}}}\mathop {\left\{ {{\mathbf{C}}(\tau )} \right\}}\limits_{\tau = 0}^{\tau = t} \, $$

(8.10)

The next problem is to describe the nonlinear functional.

8.2.2 Multiple Integral Representations

A formulation to describe nonlinear viscoelastic functionals was given by Volterra using an earlier representation developed by Frechet in the early 1900’s. This formulation was forgotten until the procedure was generalized to three dimensions by Rivlin and Green.

Assuming that the response functional obeys the continuity condition required by the weak principle of fading memory [14], Green and Rivlin [5] derived an approximate integral constitutive relation. Considering that the functional ${\tilde{\text{G}}}$ in (8.9) is continuous in ${\mathbf{C}}(\tau )$, $0 \le \tau \le t$, they used the Stone-Weierstrass theorem and the Fourier expansion of polynomials by integrals to derive the multiple integral representation

$$ {\varvec{\sigma}}(t) = {\mathbf{F}}(t)\left[ {\int_{0}^{t} {K_{1} } \left( {t - \tau_{1} } \right){\mathbf{C}}(\tau_{1} )d\tau_{1} + \int_{0}^{t} {\int_{0}^{t} {{\mathbf{K}}_{2} \left( {t - \tau_{1} ,t - \tau_{2} } \right){\mathbf{C}}(\tau_{1} ){\mathbf{C}}(\tau_{2} )d\tau_{1} } } d\tau_{2} + \ldots } \right]{\mathbf{F}}^{T} (t) $$

(8.11)

where the memory kernels ${\mathbf{K}}_{k}$, $k = 1,2, \ldots ,n$, are positive, continuous and monotonically decreasing tensor-valued functions of time. The number of terms required in (8.11) to obtain an adequate approximation depends on the characteristics of the strain history. Findley et al. [3] describes the procedure both theoretically and experimentally. Because of the difficulty to evaluate experimentally a large number of functions and because of stability problems, expansions are limited to the third order. The experimental determination of parameters [15] is difficult and the numerical computations are time consuming. Thus, this representation is seldom applied to the solution of practical problems [12].

Another multiple integral representation has been proposed by Coleman and Noll [14]

8.2.3 Pipkin–Rogers Model

The nonlinear viscoelastic constitutive theory presented by Pipkin and Rogers [10] is based on the analysis of the response of the material to step strain histories. According to this model, the functional in (8.9) can be expanded in a series whose first term provides the best approximation to measured mechanical behavior using single step strain histories. This leading term is given by

$$ {\varvec{\sigma}}(t) = {\mathbf{F}}(t)\left\{ {{\mathbf{K}}\left[ {{\mathbf{C}}(t),0} \right] + \int\limits_{0}^{t} {\frac{\partial }{\partial \tau }{\mathbf{K}}\left[ {{\mathbf{C}}(\tau ),t - \tau } \right]d\tau } } \right\}{\mathbf{F}}^{T} (t) $$

(8.12)

${\mathbf{K}}({\mathbf{C}},t)$ is the strain dependent relaxation tensor induced by a single step strain history and has the form ${\mathbf{K}} = \Upphi_{0} {\mathbf{I}} + \Upphi_{1} {\mathbf{C}} + \Upphi_{2} {\mathbf{C}}^{2}$, where $\Upphi_{0}$, $\Upphi_{1}$ and $\Upphi_{2}$ are scalar functions of t and the invariants of C [2, 16].

8.2.4 Quasi-Linear Viscoelastic Model

When the constitutive tensor ${\mathbf{K}}({\mathbf{C}}(\tau ),t - \tau )$ appearing in (8.12) can be decomposed in the form

$$ {\mathbf{K}}\left[ {{\mathbf{C}},t - \tau } \right] = {\mathbf{K}}_{e} \left[ {\mathbf{C}} \right]F(t - \tau ) $$

(8.13)

with $F(0) = 1,$ the formulation is known as quasi-linear viscoelasticity. For this case, (8.12) can be expressed as

$$ {\varvec{\sigma}}(t) = {\mathbf{F}}(t)\left\{ {{\mathbf{K}}_{e} \left[ {{\mathbf{C}}(t)} \right] + \int\limits_{0}^{t} {{\mathbf{K}}_{e} \left[ {{\mathbf{C}}(\tau )} \right]\frac{\partial F(t - \tau )}{\partial (t - \tau )}d\tau } } \right\}{\mathbf{F}}^{T} (t) $$

(8.14)

This constitutive relation was proposed by Fung [4] and used for modelling the mechanical behavior of biological tissues. The terminology “quasi-linear viscoelasticity” is used because ${\mathbf{K}}\left[ {\mathbf{C}} \right]$ can be thought of as a nonlinear measure of strain. The expression in braces in (8.14) is linear in this nonlinear strain measure.

8.3 Nonlinear Viscoelasticity at Large Strains Using State Variables

Simo and co-workers [13] developed a constitutive model for nonlinear viscoelasticity based on state variables. It is particularly addressed to materials like polymers and rubbers that behave as hyperelastic in short time loading situations and uses the concepts of deviatoric-volumetric split that is also convenient from the computational point of view. This model assumes that viscoelastic behavior is restricted to shear and that bulk strain is purely elastic. According to Simo the formulation has the following attractive features:

• 1. It uses the numerical implementation of incremental integration as described in Chap. 3.

• 2. It is a description of time-dependent behavior that contains hyperelasticity as a particular case.

• 3. It allows a separation of volume preserving and dilatational responses.

The development follows the pattern of linear viscoelasticity: the time-dependent behavior reduces to the corresponding hyperelastic behavior for very fast or very slow processes and the state variables formulation is an extension of the same formulation for small strain. Thus, we will describe first the hiperelastic formulation, then a state variable formulation for small strains recast in a slightly different way to allow for the introduction of the strain energy and finally the finite strain formulation.

8.3.1 Hyperelastic Formulation

A hyperelastic material is characterized by a strain energy function

$$ \Uppsi = \Uppsi ({\mathbf{F}}) = \Uppsi ({\mathbf{C}}) = \Uppsi ({\mathbf{E}}) \ge 0 \, {\text{with}}\, \Uppsi ({\mathbf{F}} = {\mathbf{I}}) = 0 $$

(8.15)

The constitutive relation for a hyperelastic material is by definition

$$ {\mathbf{S}} = 2\frac{{\partial \Uppsi ({\mathbf{C}})}}{{\partial {\mathbf{C}}}} = \frac{{\partial \Uppsi ({\mathbf{E}})}}{{\partial {\mathbf{E}}}} $$

(8.16)

Volumetric-shear split

Some materials (e.g., polymers) behave quite differently in bulk and in shear. Then it may be convenient to split the deformation into a volumetric part and an isochoric part, as it is done in the small strain case (Sect. 4.2.1). This split has advantages also from the computational point of view. Thus, we use the multiplicative decomposition [8]

$$ {\bar{\mathbf{F}}} = J^{{ - {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}} {\mathbf{F}} $$

(8.17)

where $J = \det {\mathbf{F}}$ is related to the volume changes, i.e., to the dilatational part of ${\mathbf{F}},$ and ${\bar{\mathbf{F}}}$ is associated to the volume-preserving or isochoric part of ${\mathbf{F}}.$ Notice that, from (8.17), $\bar{J} = \det {\bar{\mathbf{F}}} = 1.$

Introducing (8.17) into the definition of the right Cauchy-Green tensor ${\mathbf{C}} = {\mathbf{F}}^{T} {\mathbf{F}},$ we have

$$ {\mathbf{C}} = J^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}}} {\bar{\mathbf{C}}} $$

(8.18)

being ${\bar{\mathbf{C}}} = {\bar{\mathbf{F}}}^{T} {\bar{\mathbf{F}}}$ the isochoric part of ${\mathbf{C}}.$ Similarly, the strain energy is divided into volumetric and isochoric parts

$$ \Uppsi ({\mathbf{C}}) = \Uppsi_{vol} (J) + \Uppsi_{iso} ({\bar{\mathbf{C}}}) $$

(8.19)

Thus,

$$ \dot{\Uppsi } = \frac{{\partial \Uppsi_{vol} (J)}}{\partial J}\dot{J} + \frac{{\partial \Uppsi_{iso} ({\bar{\mathbf{C}}})}}{{\partial {\bar{\mathbf{C}}}}}:{\mathbf{\dot{\bar{C}}}} = p\dot{J} + \frac{1}{2}{\mathbf{\bar{S}:\dot{\bar{C}}}} $$

(8.20)

with $p = \frac{{\partial \Uppsi_{vol} (J)}}{\partial J}$ and ${\bar{\mathbf{S}}} = 2\frac{{\partial \Uppsi_{iso} ({\bar{\mathbf{C}}})}}{{\partial {\bar{\mathbf{C}}}}}$

The Second Piola–Kirchhoff stress S is also divided into isochoric and volumetric parts

$$ {\mathbf{S}} = 2\frac{{\partial \Uppsi ({\mathbf{C}})}}{{\partial {\mathbf{C}}}} = {\mathbf{S}}_{iso} + {\mathbf{S}}_{vol} $$

(8.21)

where

$$ {\mathbf{S}}_{vol} = 2\frac{{\partial \Uppsi_{vol} (J)}}{{\partial {\mathbf{C}}}} = J\frac{{\partial \Uppsi_{vol} (J)}}{\partial J}{\mathbf{C}}^{ - 1} = Jp{\mathbf{C}}^{ - 1} $$

(8.22)

$$ {\mathbf{S}}_{iso} = 2\frac{{\partial \Uppsi_{iso} ({\bar{\mathbf{C}}})}}{{\partial {\mathbf{C}}}} = J^{{ - {2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}}} DEV{\bar{\mathbf{S}}} $$

with $DEV{\bar{\mathbf{S}}} = \left( {\mathbb{I} - \frac{1}{3}{\mathbf{C}} \otimes {\mathbf{C}}^{ - 1} } \right):{\bar{\mathbf{S}}}.$ In this relation, $\mathbb{I}$ is the fourth-order unit tensor and $\otimes$ denotes a tensor product.

Example 1: Derive the relations (8.22).

Applying the chain rule to (8.22)₁ and using the relation $\frac{\partial J}{{\partial {\mathbf{C}}}} = \frac{J}{2}{\mathbf{C}}^{ - 1},$ we have

$$ {\mathbf{S}}_{vol} = 2\frac{{\partial \Uppsi_{vol} (J)}}{{\partial {\mathbf{C}}}} = 2\frac{{\partial \Uppsi_{vol} (J)}}{\partial J}\frac{\partial J}{{\partial {\mathbf{C}}}} = 2\frac{{\partial \Uppsi_{vol} (J)}}{\partial J}\frac{J}{2}{\mathbf{C}}^{ - 1} = Jp{\mathbf{C}}^{ - 1} $$

Now, we apply the chain rule to (8.22)₂ and use the definition of ${\bar{\mathbf{S}}}$ to obtain

$$ {\mathbf{S}}_{iso} = 2\frac{{\partial \Uppsi_{iso} ({\bar{\mathbf{C}}})}}{{\partial {\mathbf{C}}}} = 2\frac{{\partial \Uppsi_{iso} ({\bar{\mathbf{C}}})}}{{\partial {\bar{\mathbf{C}}}}}\frac{{\partial {\bar{\mathbf{C}}}}}{{\partial {\mathbf{C}}}} = {\bar{\mathbf{S}}}:\frac{{\partial {\bar{\mathbf{C}}}}}{{\partial {\mathbf{C}}}} $$

where, from (8.18),

$$ \frac{{\partial {\bar{\mathbf{C}}}}}{{\partial {\mathbf{C}}}} = \frac{{\partial \left( {J^{{ - {2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}}} {\mathbf{C}}} \right)}}{{\partial {\mathbf{C}}}} = J^{{ - {2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}}} \left( {\mathbb{I} - \frac{2}{3}\frac{1}{J}\frac{\partial J}{{\partial {\mathbf{C}}}} \otimes {\mathbf{C}}} \right) = J^{{ - {2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}}} \left( {\mathbb{I}- \frac{1}{3}{\mathbf{C}} \otimes {\mathbf{C}}^{ - 1} } \right) $$

and therefore, ${\mathbf{S}}_{iso} = J^{{ - {2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}}} DEV{\bar{\mathbf{S}}},$ where the DEV operator delivers the deviatoric part of the stress tensor.

8.3.2 Viscoelastic Small Strain Relations

Simo proposes a visco-hyperelastic model which is an extension of the linear viscoelastic formulation described here in Chap. 3, recast in a convenient format.

He begins with a state variables formulation similar to that introduced in Sect. 3.1 for the generalized Maxwell model, modified to make the extension to finite strains easier. First, a new internal variable $\hat{q}_{i}$ is introduced, so that

$$ \hat{q}_{i} = E\left( {\varepsilon - q_{i} } \right) $$

(8.23)

where $q_{i}$ $(i = 1, \ldots ,n)$ are the state variables defined for the generalized Maxwell model in Chap. 3. Thus, (3.9) and (3.10) become

$$ \dot{\hat{q}}_{i} + \frac{{\hat{q}_{i} }}{{T_{i} }} = \frac{{\gamma_{i} }}{{T_{i} }}E_{0} \varepsilon $$

$$ \sigma = E_{0} \varepsilon - \sum\limits_{i = 1}^{n} {\hat{q}_{i} } $$

(8.24)

where $E_{0} = E(0) = E_{\infty } + \sum\limits_{i = 1}^{n} {E_{i} }$ and $\gamma_{i} = E_{i} /E_{0} .$

In the small strain three-dimensional context the strain energy can be decomposed in isochoric and volumetric parts

$$ \Uppsi^{0} ({\mathbf{\varvec\varepsilon }}) = \Uppsi_{iso}^{0} ({\mathbf{e}}) + \Uppsi_{vol}^{0} (tr{\mathbf{\varvec\varepsilon }}) $$

(8.25)

Considering that bulk deformation is elastic (8.24) may be written

$$ {\dot{\hat{\mathbf{q}}}}_{{\mathbf{i}}} + \frac{{{\hat{\mathbf{q}}}_{{\mathbf{i}}} }}{{T_{i} }} = \frac{{\gamma_{i} }}{{T_{i} }}\frac{{\partial \Uppsi^{0} ({\mathbf{e}})}}{{\partial {\mathbf{e}}}} $$

(8.26)

$$ {\varvec{\sigma}} = \frac{{\partial \Uppsi^{0} ({\mathbf{\varepsilon }})}}{{\partial {\mathbf{\varepsilon }}}} - \sum\limits_{i = 1}^{n} {{\hat{\mathbf{q}}}_{{\mathbf{i}}} } $$

8.3.3 Formulation of the Nonlinear Viscoelastic Model

The generalization of (8.24)₂ to the finite deformation regime is

$$ {\mathbf{S}}(t) = {\mathbf{S}}^{0} (t) - J^{ - 2/3} DEV\left[ {\sum\limits_{i = 1}^{n} {{\mathbf{Q}}_{i} (t)} } \right] $$

(8.27)

where ${\mathbf{S}}^{0} (t)$ is given by (8.21) with $\Uppsi$ being the total initial stored-energy function $\Uppsi^{0} = \Uppsi_{iso}^{0} + \Uppsi_{vol}^{0}$ and ${\mathbf{C}}(t),$ being a function of time.

The growth law for the internal state variables is written, following (8.26)

$$ {\dot{\mathbf{Q}}}_{i} (t) + \frac{1}{{T_{i} }}{\mathbf{Q}}_{i} (t) = \frac{{\gamma_{i} }}{{T_{i} }}DEV\left( {2\frac{{\partial \Uppsi_{iso}^{0} ({\bar{\mathbf{C}}}(t)}}{{\partial {\bar{\mathbf{C}}}}}} \right) $$

(8.28)

with ${\mathbf{Q}}_{i} (t \le \tau_{0} ) = {\mathbf{0}} .$ $\Uppsi_{iso}^{0}$ denotes the volume-preserving contribution to the stored-energy function.

The solution of the differential equation (8.28) has the integral representation

$$ {\mathbf{Q}}_{i} (t) = \frac{{\gamma_{i} }}{{T_{i} }}\int\limits_{ - \infty }^{t} {e^{{ - (t - \tau )/T_{i} }} DEV\left( {2\frac{{\partial \Uppsi_{iso}^{0} ({\bar{\mathbf{C}}}(\tau )}}{{\partial {\bar{\mathbf{C}}}}}} \right)d\tau } $$

(8.29)

These expressions are formally similar to the corresponding expressions in Chap. 3. The recurrence formula for determination of ${\mathbf{Q}}_{i}$is also similar to that shown in Sect. 3.2.

Time integration algorithm

At time $t_{n}$ we assume to know the displacement field ${\mathbf{u}}_{n},$ its dependent variables ${\mathbf{F}}_{n} = {\mathbf{I}} + {{\partial {\mathbf{u}}_{n} } \mathord{\left/ {\vphantom {{\partial {\mathbf{u}}_{n} } {\partial {\mathbf{X}}}}} \right. \kern-\nulldelimiterspace} {\partial {\mathbf{X}}}},$ $J_{n} = \det {\mathbf{F}}_{n},$ ${\mathbf{C}}_{n} = {\mathbf{F}}_{n}^{T} {\mathbf{F}}_{n},$ ${\bar{\mathbf{C}}}_{n} = J^{{ - {2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}}} {\mathbf{C}}_{n}$ and the stress ${\mathbf{S}}_{n}$ satisfying the equilibrium conditions. We need to determine the updated values of these variables for a new displacement field ${\mathbf{u}}_{n + 1} ,$ at time $t_{n + 1} = t_{n} + \Updelta t ,$ which is corrected iteratively until the balance equations are satisfied within the given tolerance. For this, we can use the following time integration algorithm:

• (1) Given initial information at time $t_{n} :$ ${\bar{\mathbf{S}}}_{n}^{0},$ ${\mathbf{C}}_{n}$ and $\left( {{\mathbf{Q}}_{i} } \right)_{n}$ with $i = 1, \ldots ,m ;$

• (2) For a given trial solution ${\mathbf{u}}_{n + 1}$ at time $t_{n + 1},$ compute ${\mathbf{F}}_{n + 1} = {\mathbf{I}} + {{\partial {\mathbf{u}}_{n + 1} } \mathord{\left/ {\vphantom {{\partial {\mathbf{u}}_{n + 1} } {\partial {\mathbf{X}}}}} \right. \kern-\nulldelimiterspace} {\partial {\mathbf{X}}}},$ $J_{n + 1} = \det {\mathbf{F}}_{n + 1},$ ${\mathbf{C}}_{n + 1} = {\mathbf{F}}_{n + 1}^{T} {\mathbf{F}}_{n + 1},$ ${\bar{\mathbf{C}}}_{n + 1} = J_{n + 1}^{{ - {2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}}} {\mathbf{C}}_{n + 1} ;$

• (3) Evaluate ${\bar{\mathbf{S}}}_{n + 1}^{0} = \left[ {2\frac{{\partial \Uppsi^{0} ({\bar{\mathbf{C}}})}}{{\partial {\bar{\mathbf{C}}}}}} \right]_{n + 1},$ $\left( {{\mathbf{S}}_{iso}^{0} } \right)_{n + 1} = J_{n + 1}^{{ - {2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}}} DEV{\bar{\mathbf{S}}}_{n + 1}^{0},$ $p_{n + 1}^{0} = \left( {\frac{{\partial \Uppsi_{vol}^{0} }}{\partial J}} \right)_{n + 1},$ $\left( {{\mathbf{S}}_{vol}^{0} } \right)_{n + 1} = J_{n + 1} p_{n + 1}^{0} {\mathbf{C}}_{n + 1}^{ - 1} ,$ ${\mathbf{S}}_{n + 1}^{0} = \left( {{\mathbf{S}}_{iso}^{0} } \right)_{n + 1} + \left( {{\mathbf{S}}_{vol}^{0} } \right)_{n + 1};$

• (4) Update state variables and stresses: $\left( {{\mathbf{Q}}_{i} } \right)_{n + 1} = e^{{ - {{\Updelta t} \mathord{\left/ {\vphantom {{\Updelta t} {T_{i} }}} \right. \kern-\nulldelimiterspace} {T_{i} }}}} \left( {{\mathbf{Q}}_{i} } \right)_{n} + \frac{{\gamma_{i} }}{2}\left( {DEV{\bar{\mathbf{S}}}_{n + 1}^{0} + DEV{\bar{\mathbf{S}}}_{n}^{0} } \right)\left( {1 - e^{{ - {{\Updelta t} \mathord{\left/ {\vphantom {{\Updelta t} {T_{i} }}} \right. \kern-\nulldelimiterspace} {T_{i} }}}} } \right)$ and ${\mathbf{S}}_{n + 1} = {\mathbf{S}}_{n + 1}^{0} - J_{n + 1}^{{ - {2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}}} DEV\left[ {\sum\limits_{i = 1}^{m} {\left( {Q_{i} } \right)_{n + 1} } } \right]$

An alternative time integration algorithm can be found in Holtzapfel [8].