Elastic-Plastic Adhesive Analysis

CHAPTER 3
THEORETICAL MODEL

There are numerous models available for isotropic materials bonded adhesively by scarf joints. However fewer models exist for adhesive bonds joining composite materials and most of these are restricted to adhesives which are treated as linearly elastic (Vinnson, 1975). In many practical situations, the adhesive does not behave in a linearly elastic manner.
The objective of this chapter is to develop a theoretical model which can be used to calculate the mechanical properties of laminates repaired by the scarf method. This is achieved by applying the elastic-plastic adhesive analysis techniques developed by Hart-Smith (Hart-Smith, Adhesive-Bonded Scarf and Stepped-Lap Joints, 1973). The approach used is that of continuum

Equations [6] and [10] yields γ_p^n=(〖(λ〗^n )^2 γ_ef)/2 x^2+r^n x+s^n (plastic region) [11]
It is important to mention that in equations [8] and [11], ‘p^n,q^n,r^n,and s^n’ are constants which must be determined from boundary and continuity conditions.
3.1.2 Boundary and continuity conditions
It is apparent from figure 3.8 that there are three regions that needs to be defined. These are at the edges (x=0 and x=k), at the transition points (from plastic to elastic) ‘Xp1 and Xp2’ and at the edge of each overlap segment (at x = xi). Hence the boundary and continuity conditions corresponding to equations [8] and [11] are as follows Figure 3.8 Boundary conditions for the in-plane loads in the base laminate and in the repair patch At X=0, the axial load (per unit width) is zero in the base laminate and is equal to the applied load ‘P’ in the repair patch. At x=xk, the axial load is zero in the repair patch and P in the base laminate.
N_L^1=0 (at x=0) [12]
N_R^1=P (at x=0) [12]
N_L^n=P (at x=x_k ) [13]
N_R^n=0 (at x=x_k ) [13]

Combining equation [5], [12] and [13], the boundary conditions can be re-written