On Constitutive Equations For Anisotropic Nonlinearly Viscoelastic Solids
Jose Merodio1*
K R Rajagopal2
1 Department of Structural and Mechanical Engineering, E.T.S. Ing.
Industriales y Tel., University of Cantabria, 39005, Santander, Spain
2 Department of Mechanical Engineering, Texas A&M University, College
Station, TX 77845, USA
* To whom correspondence should be addressed.
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Abstract |
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Many biological, geological and synthetic bodies are anisotropic. In particular,
some of these bodies reflect the anisotropy due to fiber reinforcement along a direction
or more than one direction. In a recent paper Merodio developed constitutive equations
for fiber-reinforced transversely isotropic nonlinearly viscoelastic bodies where the
transverse isotropy was a consequence of the presence of a single family of
unidirectional reinforcements. It was shown that constitutive equations for such
materials can be expressed in terms of functions of 18 independent invariants, eight of
them associated with the deformation and the orientation of the fiber. Here we provide a
corresponding analysis for more general constitutive equations: anisotropic models with
two preferred fiber directions, i.e., two families of fiber reinforcements. In this
case, we show that the constitutive equations can be expressed in terms of functions of
37 independent invariants. These invariants are analyzed with regard to their
properties, e.g., their non-negativity, etc, which have a bearing on the response
characteristics of the body. There are 11 coupling invariants arising from the
simultaneous existence of both family of fibers. These invariants depend, among other
variables, on the relative fiber orientations, and the physical implications of these
invariants are discussed. We determine the invariants for two illustrative deformation
gradients: (i) a diagonal one aligned with one of the two fiber directions and
corresponding to a homogeneous deformation, and (ii) a simple shear deformation along
one of the two fiber directions in a plane containing both fibers. The physical
significance of all the invariants, in the two different cases, is discussed. The need
for simplifying these models to a form that is amenable to analysis and experimental
corroboration is also discussed.
Key Words:
Nonlinear viscoelasticity, fiber reinforcement, invariants, deformation indicators, rate of deformation indicators
