PSI - Issue 72

Thomas Steffen Methfessel et al. / Procedia Structural Integrity 72 (2025) 105–112

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Fig. 2. Sandwich-type model configuration.

In the following model it is assumed that the deformations of the adherends follow standard Mindlin kinematics whereas for the adhesive layer a third-order deformation approach is chosen. This leads to the following kind of deformation representation: u (i) (x,z i )=u 0 (i) (x)+z i  (i) (x), w (i) (x,z i )=w 0 (i) (x), i=1,2, (1) u (a) (x,z) = 1 2 ( ̂ (1) + ̂ (2) )+ ( ̂ (2) – ̂ (1) ) +  u (1-4 2 2 ) +  u (z-4 3 2 ), (2) w (a) (x,z) = 1 2 ( ̂ (1) + ̂ (2) )+ ( ̂ (2) – ̂ (1) ) +  w (1-4 2 2 ) +  w (z-4 3 2 ). (3) Herein the quantities ̂ (1) , ̂ (2) and ̂ (1) , ̂ (2) are the displacements at the upper and lower adherend-adhesive interface so that the representations (2) and (3) start with a linear interpolation of these displacements into the adhesive layer. This is then supplemented by the given quadratic and cubic extensions, where the quantities φ u , φ w , χ u and χ w are newly introduced deformation functions depending on the coordinate x. From the given deformation representation (1) – (3), of course, the corresponding strains can be calculated by standard linear kinematics and through Hooke’s law also the respective stresses can be obtained. In total, the deformation representation contains 10 unknown deformation functions and their derivatives, which can be put together in a summarized manner in the following array  :   u 0 (1) ,u 0 (1) ’,w 0 (1) ,w 0 (1) ’,     ’,u 0 (2) ,u 0 (2) ’,w 0 (2) ,w 0 (2) ’,  (2) ,  (2) ’,  u ,  u ’,  u ,  u ’,  w ,  w ’,  w ,  w ’   (4) For the actual determination of the introduced deformation functions use is made of the principle of minimum total potential energy, which means that the sum of the internal potential (i.e. the elastically stored strain energy) and the external potential (resulting from given external forces, as N 11 , V 11 etc. indicated in Fig. 2)  (6) When the actual variation process is performed this leads to a set of 10 linear differential equations of second order or, equivalently, to a set of 20 differential equations of first order. In a condensed manner this can be written down as (7) where the quantities A and B are known matrices, the right-hand side d is an array resulting from thermal loading (if given) and prime means the derivative with respect to the coordinate x . This system of differential equations can be solved by a standard exponential solution approach leading to a linear combination of exponential functions and free constants. The free constants are to be determined from given boundary conditions respectively the given loading. A  ‘ + B  = d  int +  ext , (5) has to become stationary for the equilibrium state: