PSI - Issue 30

116 Tatiana S. Popova et al. / Procedia Structural Integrity 30 (2020) 113–119 Tatiana S. Popova / Structural Integrity Procedia 00 (2020) 000–000 problem as follows: for a given on � � functions � � �� � � � � � of external loads find in � � the displacements field � � �� � � � � � of elastic body and stress tensor � � �� �� � , �� � � �� � , also, find the displacements � on � , the displacements � , the rotation angle � on � � and an element ���� of the space ��� � � such that �� ���� ��� � 0 in Ω � , � � 1�2� (1) ���� ��� � �� � � on � , ����� ��� � � �� � � �� � � , ���� ��� � ���� �� � �� � 0 on � � , (2) � � � 0 on Γ � , � � 1�2� ���0�� � 0� (3) � �� ���� � � � � on Γ � , � � 1�2� (4) � � � on � � , � � ��� � �� � � ��� � � on � , (5) �� � � � 0� � � ��� � 0 , � � ��� � 0 , � � ��� �� � � � 0 on � , (6) � �� � 0 for � � � �1� � �� � � � 0 for � � � �1� � �� � 0 for � � � �1�0 , (7) � �� � � � � �̅ � ����� �� � ���̅��0� � 0� ��̅ � ��� � �. (8) Here � � � � �� � � � � , � � � ����� ; � � � �� . Square brackets denote jump of function on the crack faces, i.e. ��� � � ��� � � ��� , where upper subscript in parentheses fits for the function value on the corresponding crack face. By E, G, I we denote the Young modulus, the shear modulus, and the inertia moment of the cross section of the beam, respectively. By h and S we denote the thickness of the beam and the area of its cross section, respectively: � � � 2�1 � �� ⁄ ; � � � � ⁄12 ; � � � � ; � – Poisson ratio. Relations (1) are the equilibrium equations for the elastic body, and (2) are equilibrium equations for the inclusions � � and � � . The righthand parts of two first relations in (2) describe forces acting on � and � � from the surrounding elastic media. Relations (4) determine the surface forces exerted on Γ � . In accordance with the relations (5), the vertical displacements of elastic body coincide on � � with an element of the space ��� � � , the vertical and horizontal displacements of the beam coincide with the normal and tangential displacements of the body on � � . Relations (6) are the standard system of boundary conditions describing possible contact of the crack faces on � , where first of (6) stands for the non-penetration condition [Khludnev and Popova (2017)] and the last one provides the relation between the contact of crack faces and the normal stresses at given point. If there is no contact at a given point then �� � � � 0 and the normal stress on the crack face is zero: � � ��� � 0 . On the other hand, if the normal stress at some point is negative � � ��� � 0 , then the crack faces are in contact �� � � � 0 . The condition (8) is a part of the equilibrium equations for the semirigid inclusion. This nonlocal relation can be rewritten in equivalent form. For doing this we use decomposition of the vector �� �� � � � in normal and tangential components and substitute �̅ � �̅� � � �̅ � in (8). Now we have � �� � � � ���� �� � ���0� � 0� � �� � �� � � 0. � � � � (9) Obtained relations (9) provide zero forces and a zero moment acting on γ � . 3. Numerical modeling For the numerical solution of problem (1)-(8), it is necessary to construct an algorithm that allows finding solutions that satisfy all equations and conditions. The numerical computations were performed using the finite element method. 4