Fully anisotropic finite strain viscoelasticity based on a reverse multiplicative decomposition and logarithmic strains

doi:10.1016/j.compstruc.2015.09.001

Computers & Structures

Volume 163, 15 January 2016, Pages 56-70

https://doi.org/10.1016/j.compstruc.2015.09.001 Get rights and content

Highlights

•
We present a novel formulation for fully anisotropic finite strain viscoelasticity.
•
It is based on a reversed multiplicative decomposition respect to Sidoroff’s proposal.
•
The stored energy includes anisotropic equilibrated and non-equilibrated addends.
•
The viscosities employed in the formulation may also be anisotropic.
•
Examples show applicability of the model to finite element simulations.

Abstract

In this paper we present a novel formulation for phenomenological anisotropic finite visco-hyperelasticity. The formulation is based on a multiplicative decomposition of the equilibrated deformation gradient into nonequilibrated elastic and viscous contributions. The proposal in this paper is a decomposition reversed respect to that from Sidoroff allowing for anisotropic viscous contributions. Independent anisotropic stored energies are employed for equilibrated and non-equilibrated parts. The formulation uses logarithmic strain measures in order to be teamed with spline-based hyperelasticity. Some examples compare the results with formulations that use the Sidoroff decomposition and also show the enhanced capabilities of the present model.

Introduction

Rubberlike materials and biological tissues are capable of sustaining large strains and are frequently considered quasi-incompressible and hyperelastic in finite element analyses, see for example [1], [2], [3], [4], [5], [6], [7]. In the observed behavior of these materials, specially in biological tissues, there is frequently a relevant viscous component [3], [4]. Hence, visco-hyperelastic models are very important in both the engineering and biomechanics fields.

Among the many types of formulations proposed for isochoric viscoelasticity, two approaches stand out in finite element simulations. The first one was advocated by Simo and Hughes [6] and Simo [8] and successfully used by other researchers, see [3], [9], [10], [11], [12], among others. This formulation is based on stress-like internal variables and allows for anisotropic stored energies. However, this formulation is not adequate for large deviations from thermodynamical equilibrium [13], [14] (i.e. finite linear viscoelasticity). Furthermore, the instantaneous and relaxed stored energies are usually proportional [6], [8].

The second approach has been proposed by Reese and Govindjee [13] and used also in Refs. [15], [16] among others. In this approach the Sidoroff multiplicative decomposition [17] is employed and the stored energy is separated into equilibrated and nonequilibrated parts following the framework introduced by Lubliner [18]. The main advantage of this formulation is that it is valid for deformations away from thermodynamical equilibrium and that distinct instantaneous and relaxed stored energies may be considered. As a drawback, the phenomenological formulation is only valid for isotropy, although anisotropic formulations are possible following these ideas and modelling the microstructure [19].

Recently we have developed a formulation following the ideas from Reese and Govindjee which is valid for anisotropic hyperelasticity and for deformations arbitrarily away from thermodynamic equilibrium (i.e. finite nonlinear viscoelasticity) [20]. This formulation uses the Sidoroff multiplicative decomposition of the total (equilibrated) deformation gradient into non-equilibrated elastic and viscous parts. The equilibrated and non-equilibrated stored energies are formulated in terms of logarithmic strains. These strain measures are intuitive [21], [22], [23] and allow for simple formulations in large strain elasto-plasticity [26], [24], [25]. Furthermore, they are employed in spline-based hyperelasticity [27], [28], [29]. Spline-based hyperelasticity introduced by Sussman and Bathe permits the exact (in practice) replication of experimental data and also facilitates the interpretation of the material behavior [20] in visco-hyperelasticity. Furthermore, it may be formulated as to preserve both theoretical and numerical material symmetries consistency [30]. However, the inconvenience of the formulation of Ref. [20] based on Sidoroff’s decomposition is that whereas the stored energies may be anisotropic, the viscous component should arguably be isotropic. This is due to the intermediate configuration imposed by the Sidoroff multiplicative decomposition. Hence, only one relaxation time can be considered as an independent parameter (i.e. obtained from an experiment). The relaxation times for the remaining components are given by the prescribed stored energies [20].

The purpose of this paper is to present a formulation for anisotropic visco-hyperelasticity in which both the stored energies and the viscous contribution are anisotropic. Therefore, in orthotropy up to six independent relaxation times may be prescribed, i.e. obtained from six different experiments as for the case of isochoric spline-based orthotropic equilibrated and nonequilibrated stored energies [29]. The procedure employs a reversed multiplicative decomposition from that used by Sidoroff. The intermediate configuration from this decomposition allows for the formulation of the stored energies using the same structural tensors and, hence, facilitates the use of anisotropic viscosity tensors. The algorithm is introduced using a special co-rotational formulation in order to facilitate a parallelism with the formulation introduced in Ref. [20]. As an inconvenience of the present formulation when compared to the one presented in [20], the resulting non-equilibrated consistent tangent moduli tensor is slightly non-symmetric for off-axis nonproportional loading. However, for the numerical nonproportional examples presented in this paper typically only one additional iteration is employed when using a symmetrized tensor. For the case of nonproportional off-axes loading, the observed behavior is also slightly different due to the also different multiplicative decomposition employed. Therefore, if the viscosity is considered isotropic, the formulation given in [20] may be preferred, but for more general anisotropic viscosities, the present formulation is clearly better.

In this paper we focus mainly on the large strain formulation using the reversed decomposition. For a detailed small strains motivation and for some concepts used in the kinematics of the multiplicative decomposition, the reader can refer to Ref. [20].

Section snippets

Sidoroff’s and reverse multiplicative decompositions

Unidimensional viscoelasticity is motivated by the standard solid rheological model [6], see Fig. 1, where the small elongations of the springs and the viscous dashpot per unit device-length (i.e. infinitesimal strains) are related through $ε = ε_{e} + ε_{v}$

Within the context of three-dimensional large deformations, a generalization of this additive decomposition in terms of some finite deformation measure is needed as point of departure in order to formulate strain-based constitutive viscoelastic models.

Finite strain viscoelasticity based on the reversed decomposition

In Ref. [20] we derived a computational model for finite fully non-linear anisotropic visco-hyperelasticity based on the Sidoroff’s multiplicative decomposition of the deformation gradient given in Eq. (2). Departing from this kinematic hypothesis, we show that the material formulation of the finite theory can be derived following analogous steps to those followed in the infinitesimal case. This is possible due to the fact that the second-order tensor on which depends $Ψ_{neq}$ in Eq. (4), i.e. the

Finite strain viscoelasticity based on logarithmic strain measures

In the preceding section we have obtained all the required tensors needed to properly formulate a finite fully nonlinear visco-hyperelastic model based on the reversed decomposition defined in terms of Green–Lagrange measures. This continuum formulation is valid for anisotropic compressible materials. However, we are mostly interested in formulating a model for nearly-incompressible materials using logarithmic strains because of both their special properties [21] and the possibility of using

Constitutive equation for the viscous flow

In order to enforce the physical restriction given in Eq. (24)₂ in the logarithmic strain space, we rewrite it attending to the purely kinematic power-conjugacy equivalence between the stress power given by the elastic deformation rate tensor $d_{e}$ and the stress power given by the material rate of the elastic logarithmic strains ${\dot{E}}_{e}$ —recall Eq. (33) $- {{\dot{W}}_{neq}|}_{d = 0} = - τ_{neq}^{| e} : {d_{e}|}_{d = 0} = - T_{neq}^{| e} : {{\dot{E}}_{e}|}_{\dot{E} = 0} = - {{\dot{W}}_{neq}|}_{\dot{E} = 0} ⩾ 0$ where we define the non-equilibrated purely deviatoric generalized Kirchhoff stresses as $T_{neq}^{| e} ≔ d W_{neq}$

Equilibrated contribution

If the total gradient ${}_{0}^{t + Δ t}X$ is known at time step $t + Δ t$ , then the equilibrated contributions $^{t + Δ t} S_{eq}$ and $^{t + Δ t} C_{eq}$ are just obtained from $Ψ_{eq} (E) = W_{eq} (E^{d}) + U_{eq} (J)$ as usual uncoupled deviatoric–volumetric hyperelastic calculations, i.e. $S_{eq} = \frac{d Ψ_{eq}}{d A} = \frac{d Ψ_{eq}}{d E} : \frac{d E}{d A} = T_{eq} : \frac{d E}{d A}$ $C_{eq} = \frac{d S_{eq}}{d A} = \frac{d E}{d A} : \frac{d T_{eq}}{d E} : \frac{d E}{d A} + T_{eq} : \frac{d^{2} E}{d A d A}$ For detailed formulae to compute these contributions for an incompressible orthotropic material, we refer to Ref. [29], Section 2.5.

Determination of the viscosity parameters of the orthotropic model

Consider a small strains uniaxial relaxation test performed about the preferred material direction $a_{1}$ of an incompressible material. Eq. (59) represented in preferred material axes and specialized at $t = 0^{+}$ (just after the total deformation in direction $a_{1}$ is applied and retained) reads—note that shear terms are not needed and that $ε_{e}^{0} = ε_{e} (t = 0^{+}) = ε (t = 0^{+}) = ε^{0}$ are isochoric (traceless) $- {[\begin{matrix} {\dot{ε}}_{e 11}^{0} \\ {\dot{ε}}_{e 22}^{0} \\ {\dot{ε}}_{e 33}^{0} \end{matrix}]}_{\dot{ε} = 0} = \frac{ε_{11}^{0}}{9} [\begin{matrix} 2 \frac{E_{11}^{neq}}{η_{11}^{d}} + \frac{H_{12}^{neq}}{η_{22}^{d}} + \frac{H_{13}^{neq}}{η_{33}^{d}} \\ - \frac{E_{11}^{neq}}{η_{11}^{d}} - 2 \frac{H_{12}^{neq}}{η_{22}^{d}} + \frac{H_{13}^{neq}}{η_{33}^{d}} \\ - \frac{E_{11}^{neq}}{η_{11}^{d}} + \frac{H_{12}^{neq}}{η_{22}^{d}} - \end{matrix}]$

Examples

The following examples are designed to compare the obtained behavior against models based on the Sidoroff decomposition [13], [20] and to highlight the enhanced capabilities of the present anisotropic visco-hyperelasticity formulation based on the reverse multiplicative decomposition.

Conclusions

In this paper we present a phenomenological formulation and numerical algorithm for anisotropic visco-hyperelasticity. The formulation is based on a reverse multiplicative decomposition and on a split of the stored energy into distinct anisotropic equilibrated and nonequilibrated addends. The formulation is valid for large deviations from thermodynamic equilibrium. The procedure may employ anisotropic stored energies and anisotropic viscosities. For the orthotropic case, six relaxation

Acknowledgement

Partial financial support for this work has been given by grant DPI2011-26635 from the Dirección General de Proyectos de Investigación of the Ministerio de Economía y Competitividad of Spain.

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Fully anisotropic finite strain viscoelasticity based on a reverse multiplicative decomposition and logarithmic strains

Highlights

Abstract

Introduction

Section snippets

Sidoroff’s and reverse multiplicative decompositions

Finite strain viscoelasticity based on the reversed decomposition

Finite strain viscoelasticity based on logarithmic strain measures

Constitutive equation for the viscous flow

Equilibrated contribution

Determination of the viscosity parameters of the orthotropic model

Examples

Conclusions

Acknowledgement

Comput Methods Appl Mech Eng

Eur J Mech-A/Sol

J Biomech

Int J Sol Struct

Mech Res Commun

Int J Sol Struct

Int J Sol Struct

Int J Sol Struct

Int J Sol Struct

Comput Struct

Mech Res Commun

Comput Struct

Eur J Mech-A/Sol

J Mech Phys Sol

Finite element procedures

Nonlinear elastic deformations