13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- 2 0 2 2 2 6 1 a L x x E F b t k x E th h (9) The shear stress in the adhesive layer can be given as 0 0 · a a G z E x (10) with 2 0 2 2 cosh , sinh x F L bL (11) 2 2 2 2 1 3 1 , 2 1 3 1 . t t k h h (12) As typical for weak interface models, it is assumed that crack advancement corresponds to a shortening of the overlap length. For the case of cracks emerging from one reentrant corner of the adherends and the adhesive layer the following relationship of the differential energy release rate to the peak stresses at the end of the overlap can be derived from the energy stored in the adhesive layer at the end of the overlap: 2 2 0 0,max 0,max 2 2 /2 12 1 1 2 2 a x a a L a G t E x t t G E (13) As outlined previously the incremental energy release rate is to be obtained from the differential energy release rate by integration. Of course, it must be considered that the peak stresses change when the overlap length decreases with higher crack lengths. 0 0 1 ( a da a L L a (14) This integral cannot be solved in closed-form analytical manner but must be solved with a numerical integration scheme. 3. The optimization problem As discussed previously it is necessary to solve the optimization problem (3) posed by the coupled stress and energy criterion (1) to identify the failure load of the joint. For the stress function f the maximum principal stress criterion is used in this work. In the case of the presently used simplified model of the single lap joint it reads: 2 2 0 0 ( ) ( / 2 2 2 ij c f z t (15) As the incremental energy release rate has to be obtained by numerical integration and the stresses exhibit a non-linear dependence on the acting forces the coupled criterion cannot be solved analytically. The more general approach of solving the optimization problem with the two variables
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