An anisotropic linear thermo-viscoelastic constitutive law
An anisotropic linear thermo-viscoelastic constitutive law
Heinz E. Pettermann 0 1
0 Scuola Internazionale Superiore di Studi Avanzati - SISSA , Trieste , Italy
1 Institute of Lightweight Design and Structural Biomechanics, Vienna University of Technology , Vienna , Austria
A constitutive material law for linear thermo-viscoelasticity in the time domain is presented. The time-dependent relaxation formulation is given for full anisotropy, i.e., both the elastic and the viscous properties are anisotropic. Thereby, each element of the relaxation tensor is described by its own and independent Prony series expansion. Exceeding common viscoelasticity, time-dependent thermal expansion relaxation/creep is treated as inherent material behavior. The pertinent equations are derived and an incremental, implicit time integration scheme is presented. The developments are implemented into an implicit FEM software for orthotropic material symmetry under plane stress assumption. Even if this is a reduced problem, all essential features are present and allow for the entire verification and validation of the approach. Various simulations on isotropic and orthotropic problems are carried out to demonstrate the material behavior under investigation.
Viscoelastic; Anisotropic; Constitutive laws; Thermal expansion creep; Finite element method implementation
1 Introduction
The apparent effects of viscoelastic material behavior manifest themselves in a
timedependent response to loading and accompanying energy dissipation. Relaxation or creep
occurs when a viscoelastic material is exposed to quasi-static loads and load changes.
Under cyclic excitation damping is exhibited. These phenomena are treated typically in the
time and in the frequency domain, respectively.
Such viscoelastic effects are widespread in natural as well as in engineering materials.
Among them are almost all biological tissues and most polymers, in particular thermoplastic
materials. Biological materials are composites “by nature”, whereas engineering polymers
are often mixed with other constituents to improve their performance. These materials are
likely to exhibit thermal expansion relaxation/creep behavior, i.e. time-dependent changes
of unconstrained thermal strains as a response to temperature changes. Such composites,
natural and man made ones, often show direction dependent properties and the consideration
of anisotropy becomes inevitable.
A general introduction to viscoelasticity can be found, e.g. in
Lakes (2009)
,
Gurtin and
Sternberg (1962)
,
Schapery (2000)
, which focus predominantly on isotropic behavior and
treat the time as well as the frequency domain. Transversely isotropic (and orthotropic)
viscoelastic models have been presented, e.g. in
Holzapfel (2000)
,
Bergström and Boyce
(2001)
,
Pandolfi and Manganiello (2006)
,
Puso and Weiss (1998)
,
Taylor et al. (2009)
,
Nedjar (2007)
, based on invariants of the strain representation, which are commonly used for
biological soft tissues. Typically, they combine non-linear orthotropic elasticity with
linear isotropic viscosity. Linear viscoelasticity of orthotropic media is presented in
Carcione
(1995)
,
Dong and McMechan (1995)
in the context of geo-materials. Linear and non-linear
orthotropic viscoelasticity adopting not the full set of orthotropic viscous effects is presented
in
Melo and Radford (2004)
,
Kaliske (2000)
and applied to composites and foam materials,
respectively. Full elastic anisotropy is treated in
Lévesque et al. (2008)
, but with one single
relaxation function operating on the entire tensor. In
Schapery (1964)
full anisotropy is
modeled with various relaxation functions applied to elasticity tensor contributions. A review on
damping of composites is given in
Chandra et al. (1999)
. Anisotropic non-linear
viscoelastic constitutive laws have been introduced in
Schapery (1969)
,
Zocher et al. (1997)
, Sawant
and
Muliana (2008)
,
Poon and Ahmad (1998)
which found widespread application to study
composite materials. Implementation aspects are discussed in
Crochon et al. (2010)
,
Sorvari
and Hämäläinen (2010)
.
Relaxation type effects occurring for the thermal expansion behavior are mentioned in
Lakes (2009)
,
Zocher et al. (1997)
,
Schapery (1964)
, and observed experimentally for
polymers, e.g.
Spencer and Boyer (1946)
. A hygro-thermal expansion relaxation function is
presented in
Bažant (1970)
in the context of concrete behavior. Otherwise, to the knowledge
of the authors, there are no modeling attempts in the literature for thermal expansion creep.
It is noted that thermal expansion relaxation/creep, here, does not refer to secondary effects
arising, e.g. from transient thermal fields, but solely denotes the time-dependent response as
inherent material behavior.
In the present work linear thermo-viscoelasticity of anisotropic materials is treated in the
time domain. Thereby the thermo-elastic as well as the viscous res (...truncated)