Issue 48

T. Profant et alii, Frattura ed Integrità Strutturale, 48 (2019) 503-512; DOI: 10.3221/IGF-ESIS.48.48 510 E NERGY RELEASE RATE EVALUATION or the purpose of the assessment of the crack path direction, the knowledge of the stress intensity factors at the tips of the just existing crack is insufficient and the change of the energy state of the unit crack advance is necessary. The shape optimization methods tender an interesting way to eliminate the problem of the complexity of the geometry of the presented problem. The shape functional ( ) U e W plays an important role in the following method, because represents the potential energy of the fracture problem (3). The change of the shape functional with respect to the small perturbation de of the crack e g at its tip ˆ x e = can be expressed by its asymptotic expansion, see [3], [8] and [9], ( ) ( ) ( ) ( ) ( ) ˆ ˆ , x U U g J x o e de e e de de + = W = W + + (29) where J is the topological derivative of U on the perturbed domain e de + W . It is supposed that ( ) g e is the positive function such that ( ) 0 g de  with 0 de  . The additional requirements about the remainder of (29) can be found in [8]. In [8] it is also shown the relation between the shape derivative of   U   and the topological one in the sense to the domain variation produced by an uniform expansion of some perturbation. This method, the so-called topological-shape sensitivity method, uses the concept of shape sensitivity analysis as an intermediate step in the topological derivative calculation. From the combination of the topological and shape sensitivity one gets the expression of the energy release rate, [3], [9], ( ) ( ) ( ) 2 2 ˆ ˆ ˆ 0 ˆ ˆ lim d x I II x x B G x J x S K K E Σ n e de e e e e de e = =  ¶ é ù = =- ⋅ ⋅ = + ë û ò  (30) where 1 1 /(1 ) E E n = - for plane strain, 1 E E = for plane stress, ( ) ˆ J x  is a derivative of   ˆ J x with respect to de , Σ e is the Eshelby’s momentum tensor [7], ˆ x e is a basis vector of ˆ x -axis and n is an unit normal to the boundary B de ¶ of the stress-free circular shaped perturbation B de at the crack tip ˆ x e = . N UMERICAL RESULTS correct description of the stress field surrounding the inclusion 3 W with the interfacial zone 2 W situated in the matrix 1 W is conditioned by the fundamental solution (1) behavior at the inclusion, interfacial zone and matrix interfaces. The Figs. 3-5 show a numerical example of the convergence and the fulfilling of the compatibility conditions of the stress component xx T , xy T and yy T of the stress tensor   * , ; T x   along the positive x -axis ( [ ] , 0 x = x , 0 x > ) generated by the fundamental solution (1) for the domain 1 W and its additional forms for the domain 2 W and 3 W along their interfaces. The applied values of elastic moduli, geometry and degree of Laurent series are also added. The unit point force f is parallel with respect to y h º -axis and situated at a point [ ] , x h . The graphs show the matching of the Laurent approximation of the fundamental solution and the FEM solution along the positive x -axis for series degree  10 n (especially in the interval    10,11 x mm, where the interfacial zone is situated). In the next numerical example, see Fig. 6, the case of the assessment of the crack path direction 0 q of the crack initiated from the point on the interface between the matrix and interfacial zone corresponding to 0 a =  is considered. The elastic moduli of the used materials ( ) 1 1 , E n , ( ) 2 2 , E n , ( ) 3 3 , E n , the finite crack size e , geometry 1 R , 2 R and degree of Laurent series are also given in the graph. The graph shows the dependence of the corresponding energy release rate G e on q , where the external load 1 T varies from zero value to the negative value of 2 T to model the shear loading of the cracked matrix. The crack initiation direction corresponding to the local maximum value of G e is of course fairly F A