Nondestructive testing of defects in adhesive joints

ABSTRACT
Finite element analysis of hyperelastic material
1 R.Sujithra and 2 R.Dhanaraj
1. Department of Rubber Tech., 2.Department of Aerospace Engg.,
Madras Institute of Technology,
Anna University, Chennai-600044
E-mail Id: suji_forever@rediffmail.com
Due to the increasing demands of the rubber components used in the automotive and aerospace
industries, the development of computational methods for elastomer analysis has attracted extensive
research attention. This research work is focused on developing a finite element program code for the
analysis of hyperelastic material restricted to plane stress case.
Hyperelastic refers to materials which can experience large elastic strain that is recoverable.
Rubber like materials falls in this category. The behavior of hyperelastic material is described based on
Mooney-Rivilin and Neo-Heokean material models. The constitutive theories for a large elastic
deformation is based on the strain energy density function which is coupled with finite element method,
can be used effectively to analyze and design elastomer.
Due to non-linear stress-strain relation, the hyperelastic materials will cause a structure stiffness
to change at different load levels. In finite element formulation, the modeling of hyperelastic material
should incorporate both geometric non-linearity due to large deformation as well as material nonlinearity.
Because of nonlinear relationship the finite element equations are nonlinear in terms of
displacement. An iterative Newton-Raphson method was employed for the solution of non-linear
governing equations. At each finite element solution step we obtained strain or strain increment, used to
compute the stress which is needed to evaluate the internal force as well as the tangent stiffness matrix.
The computed stress and deformation results obtained from finite element program code shows a good
agreement with simple benchmark problem and as well as with FEA package (ANSYS).
I. INTRODUCTION
Hyperelastic is the capability of the material to undergo large elastic strain due to small forces,
without losing its original properties. Rubber-like materials exhibit a highly non-linear behavior
characterized by hyperelastic deformability and incompressibility or nearly incompressibility. But in
general, all real materials are compressible to a certain degree even if the bulk modulus of the rubberlike
material is larger than the shear modulus. Rubbers are widely used in tires, door seals, o-rings,
conveyor belts, bearings, shock absorbers etc. An increase of applications of rubber-like materials
requires better understanding of mechanical behavior which cannot be described by a simple stressstrain
relation, but by the strain energy function. A hyperelastic material is also defined as a material
whose stresses can be defined by a strain energy function.
Two sources of non-linearity exist in the analysis of hyperelastic materials namely, material and
geometric non-linearity. The former occurs when the stress-strain behavior given by the constitutive
relation is non-linear, whereas latter is important, when changes in geometry, however large or small
have a significant effect on the load deformation behavior. The basic features of stress-strain behavior
have been well modeled by invariant based or stretch based continuum mechanics theories. An
important problem in non-linear elasticity theory is to come up with a reasonable and applicable elastic
law which is the key to the development of reliable analysis tools. Many strain energy functions are
proposed for rubber-like materials by Mooney (1940), Treloar (1944), Ogden (1972) etc. The simplest
one parameter model is the Neo-Hookean model. In this present work, hyperelastic constitutive
equations are discussed based on compressible Neo-Hookean model.
Finite element method is a procedure whereby the continuum behavior described at infinity of
points is approximated in terms of finite number of points called nodes located at specific points in the
continuum. These nodes are used to define regions, called finite elements over which both the geometry