PSI - Issue 23

Vera Petrova et al. / Procedia Structural Integrity 23 (2019) 407–412 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

410

4

Here the unknown functions g n  ( x ) are the derivatives of displacement jumps on the crack lines; [ u n ] and [ v n ] are shear and vertical displacement jumps, respectively, on the n -th crack line, μ = E /2(1+ ν ) is the shear modulus, E - Young’s modulus, ν - Poisson’s ratio, κ = 3 - 4 ν for the plane strain state and κ = (3 - ν )/(1+ ν ) for the plane stress state. In Eq. (3) an overbar denotes the complex conjugate. The regular kernels R nk and S nk determine the geometry of the problem, and on the right side of equations (3) the functions p n are known functions determined by the load on the crack lines. For internal cracks, the condition of displacement continuity at the crack tips has to be taken into account. The solution of the equations is obtained by the method of mechanical quadratures, Erdogan and Gupta (1972). Applying quadrature formulas based on Chebyshev polynomials, the integral equations (3) reduce to systems of algebraic equations. In Petrova and Schmauder (2017), a scheme for solving this method is given. After determining the unknowns, the main characteristics of the fracture mechanics are calculated, e.g. the stress intensity factors (SIFs) at the crack tips where the upper part of the “  ” or “ ” signs concerns to the right tip and the lower to the left tip of the crack. Experimental and theoretical studies show that with a mixed-mode type of loading, the crack propagation deviates from its original one. In FGMs, a mixed stress-strain state can also arise due to the heterogeneity of the material. Applying a fracture criterion, it is possible to determine the angles of the deflection of the propagation of cracks from their original direction and the critical loads at which this propagation occurs. For example, according to the criterion of maximum normal stresses (Erdogan and Sih, 1962), the crack deflection angle (or the fracture angle) and the critical stresses are calculated as   2 2 0 2arctg 8 / 4 I I II II k k k k           ,   3 , cos ( / 2) 3 tan( / 2) / I II Ic tip k k K      , (5) where k I and k II are the SIFs, and K Ic,tip is the fracture toughness of the material near the crack tip. First, the angle of the crack propagation is obtained using the results for the calculation of SIFs. Then, the local fracture stability is evaluated. This criterion is the simplest one to employ. As a result (by using this criterion) the fracture angles are derived as functions of the geometry of the problem and of the non-homogeneity parameters of the FGC with additional parameters due to thermal loadings. Then, the critical stresses are obtained near cracks (for different crack locations). 2 2 lim ( ) / n n  ( ) n n nI k ik  nII   n n n a x a g x  x a   , (4) The presented model allows to analyze the influence of geometric parameters (such as the size of the cracks, the coordinates of their centers and the inclination angles, and the thickness of the FGM layer), and the parameters of material inhomogeneity (determined by formulas (1) and (2)) on the main characteristics of the problem. A series of numerical calculations was performed to study the interaction of edge cracks as well as edge cracks and internal cracks in FGC/H structures. SIFs are normalized, i.e. k I,II = K I,II / K 0 , where K 0 is the SIF for a single crack. Illustrative examples for the geometry shown in Fig. 1c are presented in Figs. 2 and 3 for SIFs and fracture angles as functions of inclination angles β . The inhomogeneity parameters are εa = 0.3, and ωa = 0, and other parameters are h/a = 4, d/a = (2, 4, 6), a = max a k , a = 1mm, a 2 = a 3 = a 4 = 0.5 a 1 ; the coordinate of the center of the internal crack is y/a = – 3.3. In Figs. 2 and 3 the non-dimensional distances are denoted by d for simplicity. 4. Results and discussion