Issue 49

N. S. Popova et alii, Frattura ed Integrità Strutturale, 49 (2019) 267-271; DOI: 10.3221/IGF-ESIS.49.26 268 on the stress and strain field of the uncracked solid and search functions. In this case, the crack path on the surface is described by means of search functions. The present paper deals with a variational principle of fracture mechanics to predict the crack path in a wedge under a concentrated tensile force. T HE VARIATIONAL PRINCIPLE IN A SEARCH OF THE CRACK PATH he crack tip is assumed to be a material particle which has some effective mass [7]. In this case, the crack propagates due to movement of this effective mass at the crack tip. The above-mentioned assumption leads to the variational problem for predicting crack paths and can be given by the following equation   L 0 . (1) The functional L should be determined to solve the variational problem for predicting the crack path in a solid under considered loading conditions. The functional L in the case of a flat plate can be written as follows      B A L x, y ds ,    2 ds 1 y dx . (2) The weight function in Eq. (1) is denoted as Φ( x,y ). This function is dependent on stresses (or strains) in the uncracked solid. The crack path on the solid surface can be described by the following equation r=r(x,y) , where r is the radius-vector. A point belonging to the crack path has Cartesian coordinates x and y . Equation of the crack path is assumed to be an extremal that should be calculated from basic Eqn. (1). A formulation of the variational problem, taking into account the Euler-Lagrange equation, the minimum of the functional L and the transversally condition, gives well-defined boundary conditions for ends of the crack path, i.e. the crack path should be perpendicular to the free surface of a solid [9]. According to the variational principle, the crack path is interpreted as a geodesic line which is the shortest possible line between two points A and B on the surface of a solid. In this case, the crack propagation line has to satisfy the following condition    B A ds 0 . The crack increment is also assumed to be deviated from the initial direction of crack growth by the stress state. Therefore, a metric of the generalized geodesic line is also determined by the stress state of the untracked body and can be written by the following expression   * ds ds , i.e.    B * A ds 0 [7, 8]. It was shown [7-9] that the function  in brittle fracture mechanics is connected with the maximum principal stress or strain in solids without cracks. To illustrate predicting the crack path by means of variational principle, an infinite wedge under concentrated tensile force P is analyzed (Fig. 1). The angle between the wedge axis and the lateral surface is marked as α. In this case, the function  in Eq. (2) is assumed to be proportional to the maximum strain  1 [9]          1 2 2 2P x x, y E x y (3) where ν is Poisson’s ratio and E is Young’s modulus. Taking into account the above-mentioned statement, the minimum of the functional L can be written in the following form    2 1 x 2 2 2 x 2νP x δ 1+ y' dx = 0 πE x + y (4) This relationship leads to equation for solving the variational problem in a search of the crack path T