PSI - Issue 30

Tatiana S. Popova et al. / Procedia Structural Integrity 30 (2020) 113–119 Tatiana S. Popova / Structural Integrity Procedia 00 (2020) 000–000 5 С onsider the domain Ω � ���,�� � ���,�� and the curve � � ���,�� � ��� which divides Ω on two subdomains Ω � � ���,�� � ���,�� and Ω � � ���,�� � ��,�� , see Fig.2. Also take Γ � � ���� � ���,�� and Γ � � ���,�� � ���� . The plane-stress Lamé model of an isotropic solid is given in terms of stress tensor: � �� ��� � ��� � ��� �� ��� � �� �� ���, � �� ��� � � �� ��� � ��� �� ���, (10) σ �� �u� � �� �� �u� � ��� � ��� �� �u�,�������� � ��� � ��� ,�� � � � � � � � � .� ����� The body is fixed at lateral borders Γ � and subject to tensile loads: the follow surface forces are given on Γ � : � � ��, ��.������ on lower part of boundary ���,�� � ���� and � � ��, �.����� ) the upper one ���,�� � ��� . We take the following values of material parameters of the elastic matrix: � � �.�� , � � �� � �� � . Since the numerical experiment in this study is used as an example to illustrate the variational method used, the physical characteristics for an elastic inclusion are taken in the form �� � �� � �� � �� � �� � . To solve the problem, the domain decomposition method and the Uzawa algorithm were used, one of examples was found by Rudoy and Lazarev (2018). Fig. 3 shows the graphs of functions of horizontal and vertical displacements of points located on the abscissa axis and corresponding to the boundary of the subdomain � � . As can be seen from the graphs, the vertical displacements on � � � ��, �� � ��� has a form ��� � � � �� � � � � satisfying the restrictions on solution. Comparison of Fig. 4 and Fig. 5 show that there is no contact of crack faces at points from -1 to 1, i.e. �� � � � � and there is a crack opening, at the same points the normal stresses � � are equal to zero. Thus, conditions (6) in this model are satisfied. Outside the interval (-1,1), condition (6) is not specified, so the normal stresses have different signs but singularities at the crack tips occurs. 117

a b Fig. 2. (a) Computational grid � � ���, �� � ���, �� ; (b) Deformed configuration