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ANSYS 10.0:
Formulation of Finite Nonlinear Anisotropic Elasticity
This introduction presents a constitutive formulation for
material that is considered as aniso tropic. The approach
covers material, for example, with a fi brous microstructure
like bio logical tissues and all kinds of unidirectional fi bre
matrix combinations. The present description focuses on the
homo ge neous macroscopic material level in order to provide
a formulation suitable for structural fi nite element simulations
and it is available in ANSYS 10.0. The material’s structure
is characterized by two constitutive vectors. The formulation
is physical and geometrical nonlinear for the isotropic and
the anisotropic part of the hyperelastic function. Incompressibility
and nearly incompressibility can be taken into
account because of a consequent volumetric and isochoric
split of the approach.
Introduction
The origin of the anisotropic structure in materials can be
diverse. Finite anisotropy may be used to characterize fi bre
reinforced elastomeric components like tires or air springs.
The production process also can induce direction depending
properties through, for example, the extrusion of the
material. Moreover, elastomeric material is given some kind
of adaptive properties through the introduction of fi ller particles
which are direction depending. If a material exhibits
any kind of geometric structure on a length scale that is not
modelled explicitly on the discretization scale, the material
cannot be con sidered as isotropic because of its microstructure
and an anisotropic constitutive formulation is required
for the description. Another very large class of nonlinear
anisotropic materials is formed by bio materials, which often
show a fi brous structure. Biological tissues are in many cases
deformed at large strains as can be found for muscles and
for arteries.
In many fi elds like reinforced elastomers or biomaterials,
apart from direction depending features, nearly incompressibility
of the material has to be taken into account. This
mechanical property is considered for the development of
the formulation. The basis is the strict separation of volumetric
and isochoric deformations. With this split at hand, the
approach introduced fi ts into the concept of known element
formulations valid for incompressible or nearly incompressible
situations. The shown material approach is a new feature
of the current ANSYS 10.0 release.
Continuum Mechanical Formulation
The constitutive strain energy density function �� of a
nonlinear elastic material with respect to the reference
confi guration, which forms the basis for the subsequent
Applications and Technology
CADFEM GmbH INFOPLANER 2/2005
developments, is a function of the right Cauchy Green deformation
tensor and the two constitutive material directions,
where, (see Figure 1).
,
Fig. 1: Reinforcing fi bres
and material directions.
A lot of materials show
different behaviour for
volumetric J det F and
T
for isochoric C : F F
states of deformation.
Biomaterials as well as
elastomers are incompressible
or nearly incompressible
with respect to hydrostatic pressure loading. In order
to take this obser vation into account, an additive division of the
constitutive strain energy function
� � U(J) � w( C,
A, B)
2 3
is introduced, where C � J C,
detC
� 1 . A separate determination
of the volumetric term and the incompressible part
of the strain energy is carried out. This constitutive repre
sen tation is based on the multiplicative split of the deformation
gradient
�
U(J)
w( C,
A, B)
1 3
into a volumetric and an incompressible component,
where . Subsequently, the irre ducible basis of invariants
is computed analogously to the isotropic case, however, the
incompressible right Cauchy Green tensor is C
employed. A similar
defi nition of the invariants has been introduced by Spencer
[1] for the compressible approach.
A specifi c structure of the isochoric term of the constitutive strain
energy density function is employed subsequently. It is assumed,
that no coupling of the different invariants occurs and the invariants
form separate terms in series. Thus, the energy function yields
(1)
(2)
(3)
(4)