Nondestructive testing of defects in adhesive joints

The equ (3.2) is linearized and we get element tangent matrix equations as follows
∫ ∫ ∫ [BŜBT + BF T C F B T ]dV o Δd = - ∫ ∫ ∫B F T S dV o + ∫ ∫ N q o dA o + ∫ ∫ ∫N b o dV o
V o V o A o V o
Thus, ( KS+KC)Δd = r i + r q + r b (3.4)
Where Kc is the current stiffness matrix, Ks is the geometric stiffness matrix, r i , r q , r b are the
equivalent nodal load vector due to stresses in the current known configuration, due to surface forces
and due to body forces. At the current state, the linearized finite element equation is calculated for each
element and assembled. The equations are assembled and solved in the usual manner to obtain the
increments in the displacements. The new deformed configuration can be obtained by adding these
displacements to the initial coordinates using constitutive equations, we first calculate Cauchy stresses
in deformed configuration. If the computed stresses are correct, the equilibrium equations must be
satisfied in the configuration
Equivalent load vector due to stresses rI=∫∫∫BBLσdV (3.5)
The unbalance force is calculated as follows r = rE-rI (3.6)
The load vector from each element is assembled to form global rI vector. This vector is
compared against the applied nodal load vector. If the difference between two is large, a new iteration is
carried out to establish a new deformed configuration. This process is repeated until a desired level of
convergence tolerance is achieved.
IV. IMPLEMENTATION OF FEM IN PROGRAMMING
A C program is written based on the above finite element formulation using Four Noded
Isoparametric element for the analysis of hyperelastic material restricted to plane stress case. Flow
diagram for the program is shown in Fig: 1.
A computational procedures for programming is given briefly as
1. In the first step, get the nodal datas and assume initial displacement is zero.
2. Calculate bandwidth for global stiffness matrix.
3. Begin of iteration loop
3.1 compute the geometric and current stiffness matrix
3.2 Assemble global stiffness matrix from individual element tangent matrix
3.3 By gauss elimination, solving {Δd} = [kt] -1 {F}
3.4 By solving, we get incremental solution of displacement.di+1=di+Δd
3.4 Update the coordinates
3.5 Compute stress from the new deformation gradient.
3.6 Compute updated thickness
3.7 Calculate internal load vector
3.8 Unbalance force=external load – internal load.
3.9 Check for convergence parameter.
4. If no, go for next iteration step No.3 .If yes, terminate the program.
Cooks Problem [8]:
A rubber tapered panel is clamped at one end and loaded by nodal forces on the top of the other end.
The material properties used are C1=0.5 and bulk modulus1500. From the material parameter we
determine lames constant which is the input for coding. We consider the problem to be nearly
incompressibility and the poissons ratio is taken as 0.499. The geometry is shown in Fig.2. The nodal
coordinates and result obtained from coding is shown in Table: 1. The problem is solved in ANSYS
using Neo-Hookean material model and deformed shape is shown in Fig: 3. The Computed results from
coding shows a good agreement with ANSYS shown in Table 2.
V.CONCLUSION
The analysis of hyperelastic material behavior by finite element method has been described in this
study. Rubber like materials are incompressible but still its nearly incompressibility is considered as a
special feature of compressible material. The Neo-Hookean material model is discussed based on
compressible strain energy function by combining both volumetric and deviatoric term. Based on this
approach, a finite element program code is implemented. A simple rubber tapered panel is solved by
(3.3)