Equilibrium problem of an elastic plate with an oblique crack

Abstract

Equilibrium problems of elastic plates having vertical cracks (cuts) have been thus far studied in sufficient detail [1][2][3]. In the present paper, the conditions of mutual impenetrability of the sides of an oblique crack in the Kirchhoff-Love plate are obtained, equilibrium problems of a plate are formulated, and the main difficulties that arise in studying similar problems are discussed. As it turns out, the impenetrability condition for an oblique crack is of a nonlocal character in the sense that its expression for a given fixed point contains the values of plate displacements both at this point and at a point on the opposite side of the crack. This circumstance makes the impenetrability condition obtained substantially different from the corresponding condition for vertical cracks. In particular, unlike vertical cracks where the equilibrium conditions are satisfied at all points of the middle surface, the equilibrium conditions here are satisfied only in a region that is external relative to the projection of the crack surface onto the middle plate surface. This property of nonlocality is new in the theory of plates and can serve as the source of a number of new mathematical formulations of boundary-value problems. As for conventional approa~:hes to the description of cracks in elastic (and inelastic) bodies, the literature is extensive.In the present paper, we shall not discuss these approaches but simply note that they admit the possibility of mutual penetration of the crack sides [4][5][6]. The properties of boundary-value problems that arise in such cases were analyzed, for example, by Grisvard [7], Ohtsuka [8], Kondrat'ev et al. [9], Oleinik et al. [10], and Nicaise [11]. From the viewpoint of boundary-value problems, it is important to emphasize that, in any case, the presence of cracks (cuts) in a plate leads to the appearance of nonsmooth components of the boundary. The approach considered in [1][2][3] differs from the traditional ones by the fact that it is characterized by boundary conditions in the form of inequalities at the crack sides. For an oblique crack, the situation becomes even more complicated, because the equilibrium conditions are affected by the opposite crack sides. In reality, this means that additional terms which incorporate the response of opposite sides appear in the equilibrium equations. These terms can be found only after the problem is solved as a whole.

Equilibrium problem for a Timoshenko plate with an oblique crack

“…Pates and shells containing cracks have been studied in many papers (see, e.g., [1][2][3][4][5][6][7][8][9][10][11]), which mainly focused on vertical cracks defined by surfaces the normals to which (at each point of the surface) are parallel to the middle plane of the plate. As is known, the classical approach to the modeling of cracks in deformable bodies uses linear boundary conditions formulated in the form of equalities on the curve describing the crack [1][2][3].…”

“…Note that this approach allows for the possibility of mutual penetration of the crack faces [4]. Variational methods have been used to study the large class of mathematical problems of cracks with nonlinear nonpenetration conditions (in the form of systems of equalities and inequalities) defined on the curve (surface) which describes cracks in elastic bodies [6][7][8][9][10][11][12].…”

“…Nonlinear problems of the equilibrium of Kirchhoff-Love plates with the condition of mutual nonpenetration of the faces of an oblique crack were studied in [7,8]. In [7], the nonpenetration condition is given for cracks defined by a smooth surface z = F (x 1 , x 2 ), where F (x 1 , x 2 ) is a function defined in the middle plane (x 1 , x 2 ).…”

“…In [7], the nonpenetration condition is given for cracks defined by a smooth surface z = F (x 1 , x 2 ), where F (x 1 , x 2 ) is a function defined in the middle plane (x 1 , x 2 ). Kovtunenko et al [8] derived a nonpenetration condition for a crack whose position is slightly different from the vertical position.…”

Quasistatic delamination models for Kirchhoff‐Love plates

“…A similar model has been obtained also in [13] as a limit of a dimension reduction problem involving a Barenblatt-like cohesive crack surface energy. For a static delamination on a generally-positioned delamination surface in Kirchhoff-Love plates, we refer to [15,16].…”