PSI - Issue 5

N.A. Kosheleva et al. / Procedia Structural Integrity 5 (2017) 99–106 V.P. Matveenko et al. / Structural Integrity Procedia 00 (2017) 000 – 000

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concentration both in the adhesive layer and in the adherends. In this case, only one shape of spew fillets is usually considered. An exception is Ref. [11], where various shapes of spew fillets are subjected to a comparative analysis. Note that the conclusions in [11] about the impact of the spew fillet shape on the stress level in adhesive joints have information on the decisive role of the angle of attachment of the free surface of the spew fillet to the adherend. At the same time, there is no conclusion about the optimality of the determined spew fillet shape and the effectiveness of the proposed variants for adhesive joints made of other adherends and adhesives.

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Fig. 1. Variants of adhesive lap joints of materials: (a) without and (b) with a spew: 1 − material; 2 − adhesive. In the numerical analysis of the stress-strain state of the considered adhesive joints, it is difficult to estimate the accuracy and convergence of solutions in the vicinity of singular points. If the finite element method is used, one of the ways to solve this problem is to refine the finite element mesh. In this case, we can achieve the required accuracy of stress calculations outside a certain vicinity of the singular point. The size of this vicinity depends on the degree of clustering of the finite element mesh. An effective estimation of the accuracy of results obtained by the numerical solution of the problem on the basis of the principle of virtual displacements is the accuracy of satisfaction of natural boundary conditions on the load-free external surface of the samples and the contact surface of two different materials. In the present study, we used sampling that ensures the fulfillment of natural boundary conditions outside three or four elements adjacent to a singular point with an error less than 1%. Various shapes of spew fillets were analyzed in [6 – 9, 11, 12]. Most of the authors considered the triangular geometry of spew fillets with the angle of coupling of the external surface of the adhesive with the adherend surface usually equal to 45°. Lang and Mallick [11] studied eight spew geometries, among which the most significant decrease in the stress peaks is provided by a shape formed of a circle arc with a center and a radius at which the coupling angles are g А = g B = 0° (see Fig. 2). It should be noted that this shape is sufficiently producible. In this work, the shape of the free surface of the spew fillet is also chosen as a circle arc with a center, which makes it possible to obtain different angles g А and g B of coupling of the external surface of the adhesive with the surface of the adherends at the points A and B . When estimating the stress states obtained by the finite element method, we are interested in data on the singular nature of the stresses at singular points. These data may be obtained by analyzing eigensolutions for composite wedges formed by tangents at singular points to the surfaces of the material, adhesive, and contact surface (see Fig. 2). In a polar coordinate system with the origin at the wedge apex, the eigensolutions have the form [3]:             , , , , k k k k r r u r r u r r             where r is the distance from the singular point,  k are the eigenvalues,     k r   and     k    are the eigenforms, and     , , , r u r u r    are the displacement components. The procedure of constructing eigensolutions for a composite wedge [13] includes finding the eigenvalues of transcendental equations:                             2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 1 sin sin 1 sin sin 1 sin sin 1 sin sin 2 1 sin sin cos sin sin cos 0. m m a a m m a a m a a m m a a m pg p g pg p g pg p g pg p g pg p p g g p g g g                               (1)