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

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In the present paper, we construct a numerical simulation of a model problem of junction of a thin elastic Timoshenko inclusion with a thin semirigid inclusion in a two-dimensional elastic body. The semirigid inclusion is rectilinear, the functions of vertical displacements are assumed to have a given structure. The inclusions delaminate from the surrounding elastic matrix e forming a crack, inequality-type boundary conditions are specified on the crack edges, as a result of which the problem is not linear. When developing an algorithm for the numerical implementation, all restrictions on the solutions specified in the posed problem were taken into account: the form of the displacement functions of the points of a thin semirigid inclusion, the conditions for non-penetration of the points of the crack faces, the junction conditions for the thin inclusions. For the numerical solution of the variational inequality, the domain decomposition method was applied and an Uzawa-type algorithm was constructed, which allows one to obtain solutions satisfying the given constraints. 2. Problem formulation Let an elastic body in an undeformed state occupy a bounded domain Ω ⊂ � � with a Lipschitz boundary Γ , and Γ � Γ� � ∪ Γ� � , meas�Γ � � � 0 , Γ � ∩ Γ � � � . The body is fixed on its part of boundary Γ � and is under external loads on Γ � . The unit normal vector to Γ is denoted by n . The elastic body contains two thin inclusions of the rectilinear shape: γ � � ��1,0� � �0� and γ � � �0,1� � �0� , wherein γ� � ∪ γ� � ⊂ Ω . The line γ � corresponds to a thin elastic inclusion modeled as a Timoshenko beam, and the γ � corresponds to a semirigid one. Denote by Σ the curve, which divides the domain Ω on two subdomains Ω � and Ω � with Lipschitz boundary such that meas�∂Ω � ∩ Γ � � � 0 , � � 1,� and Σ ⊃ γ , where γ � γ � ∪ γ � ∪ ��0,0�� . Also denote by ν and τ unit normal and tangential vectors to Σ respectively, note that ν � �0,1� and τ � �1,0� on γ . For the domains with a cut we will use the following notation: Ω � � Ω � γ� and Ω � � � Ω � γ� � ; to denote the cut faces, we will use the superscript corresponding to the subdomain: γ � , γ � � , γ � � ⊂ ∂Ω � , � � 1,� . Assume that ν coincides with the external normal to the boundary ∂Ω � . Hence, the boundary of each domain with a cut Ω � and Ω � � consists of an outer closed curve Γ and two faces of the corresponding cut. Let the vector function u � �u � , u � � specifies the displacements of elastic body points, where u � corresponds to displacements along the axis � � , � � �, � . For the components of the strain and stress tensors of an elastic body, we introduce the following formulas: � �� ��� � 0�� �� �,� � � �,� � ; � �� � � ���� � �� ; �, �, �, � � 1,�; � ,� � � � � � � , where the coefficients � ���� are the components of the elastic modulus tensor satisfying the conditions of symmetry and positive definiteness. Throughout what follows, we will assume summation over repeated indices. To describe a thin elastic Timoshenko inclusion, we put � � ��, �, �� , where � and � – inclusion points displacements along the axes � � and � � respectively, � – rotation angle of the cross-section of the beam. Timoshenko inclusion delaminates from the elastic matrix, forming a crack. We will assume that this inclusion is located on the crack face � �� . It means that � � � � , � � � � on � �� and the equilibrium equations of the Timoshenko beam are fulfilled on the � �� . The model of a thin semirigid inclusion in this study means that the functions of vertical displacements of body points on � � have a certain predetermined structure while the horizontal displacements satisfy the elasticity equation on � � . Namely, for all � � �� � , � � � � � � the function � of vertical displacements of points of the body coincides with some element ���� of the following space L�γ � � � �l | l�x � � � cx � � c � on γ � ; c, c � � ��� Let us give a differential formulation of the equilibrium problem for a two-dimensional elastic body containing delaminated thin semirigid and Timoshenko inclusion. We assume the junction with a break between inclusions. In this case, the elastic and semirigid inclusions have a contact at the point ���, ��� and the angle at this point before and after deformation not fixed. Let us formulate the