Nondestructive testing of defects in adhesive joints

and the primary variables in the governing equations are approximated. The governing equations
describing the nonlinear behavior of the solid are recast in a weak integral form using principal of
virtual work. The finite element approximations are then introduced into these integral equations which
yields a finite set of non-linear algebraic equations in the primary variable. These equations are usually
solved using Newton-Raphson iterative technique.
Based on finite element formulation, finite element program code is developed in C for the
analysis of hyperelastic material for computing stresses and deformation. For simple benchmark
problems, the result obtained from the coding is validated with ANSYS packages.
II. NON-LINEAR CONTINUUM MECHANICS
Continuum mechanics [1, 2] is an essential building block of nonlinear finite element analysis.
We will confine our attention to the strain and stress measures which are most frequently employed in
nonlinear finite element program. Two approaches are used to describe the deformation and response of
a continuum. Consider a solid that is subjected to some forces and displacements such that its
configuration changes from the initial to the current. In finite deformation analysis, any quantity can be
described either in terms of the undeformed state or in a deformed state; the former is called material or
Langrangian description while latter is called spatial or Eulerian description. In solids, the stresses
generally depend on the history of deformation and an undeformed configuration must be specified to
define the strain. So Langrangian descriptions are prevalent in solid mechanics.
A point is identified by a vector X in the initial configuration which moves to a new point
identified by a vector x in the current configuration the mapping Ө of the motion is expressed as
X = Ө(X, t) (2.1)
In similar way, consider a line segment dx in material configuration is deformed to dX after the motion.
The deformation gradient is defined by, F = ∂x / ∂X (2.2)
The displacement of point is given by the vector u as a function of X, the differential displacement is
given as du = H dX (2.3)
The relation between deformation gradient and displacement gradient is F = H + I
To calculate the change in length dx after deformation using (2.2)
dx.dx = dX. F T F .dX (2.4)
At this point, we define the right Cauchy-green deformation tensor where c operates on material
element and hence it is a Lagranian tensor. c=F T F (2.5)
Similarly, we define the left Cauchy-green deformation tensor b is an Eulerian tensor which operates on
spatial element dx b -1 dx = dX dX (2.6)
b = FF T
.The green (or Lagragian) strain tensor is defined as E = ½ (c-I) (2.8)
And corresponding spatial spatial tensor, called Almansi strain tensor e = ½ (I-b -1 ) (2.9)
Also the volume dv and area da after deformation can be related to the initial volume dV and area dA
using the deformation gradient F (det F=J) and is given as
(2.7)
dv = JdV (2.10)
da = JF -T .dA (2.11)
Cauchy stress (Eulerian quantity) which relate the force dh acting on a differential area da is given as
dh = σ .da (2.12)
From equ, shifting area to its initial configuration, dh = J F -T dA = P.dA (2.13)
Where P is the first piola-Kirchhoff stress defned as P = JF -T
(2.14)