PSI - Issue 52

Akihide Saimoto et al. / Procedia Structural Integrity 52 (2024) 323–339 A. Saimoto et al. / Structural Integrity Procedia 00 (2023) 000–000

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crack problems in anisotropic materials by referring about 400 research papers. As described above, elastic analysis of cracks in anisotropic materials has been carried out for about 60 years, but there are not many numerical solutions for an infinite anisotropic plate with cracks. That is, most of published numerical results are solution of a finite region problems such as cracks in a rectangular anisotropic plates. Regarding a crack to crack interference problem, it could be often important to ignore the size of the analysis domain or external boundary and simply focus on how cracks interact with each other depending on their relative positions, material constants, and the direction of the material principal axes. In this study, therefore, we deal with the problem of anisotropic infinite plates with multiple line cracks interfering with each other. In BFM, the crack problem is analyzed by solving an integral equation in which the relative displacement between the upper and the lower crack faces is an unknown function. In most numerical methods based on FEM and BEM described above, the SIF is often estimated from the value of an energy release rate as in a virtual crack closure integral technique by Rybiki and Kanninen (1977). However, in anisotropic materials, there is a nonlinear relationship between the SIF of each mode and the energy release rate as described in Sih et al. (1965), it is not easy to estimate them as in isotropic materials. On the other hand, in BFM, the SIF can be obtained directly from the solution of the simultaneous equations which is a discretized form of the integral equation without any post-processing. In this study, SIF at a crack tip was obtained as accurate as possible and presented in a form of a chart and numeric tables. In this section, following Lekhnitskii (1968), the relationship between components of stress, strain, displacement, and resultant force in the anisotropic elastic plate and the complex stress functions, as well as the fundamental solu tions of point force and its characteristics underlying the analysis are shown. Let us consider a 2 dimensional elastic plane of rectilinearly anisotropy as shown in Fig.1. x and y are in plane coordinates while z is a coordinate perpendic ular to the plane. In a case of plane stress, σ z = τ yz = τ zx = 0 is assumed then the constitutive relation regarding a direction of material principle axis x m , y m becomes    ε x m ε y m γ x m y m    =    1 E 1 − ν 21 E 2 0 − ν 12 E 1 1 E 2 0 0 0 1 G 12       σ x m σ y m τ x m y m    or ε m = C m σ m (1) Here, the strain and stress vectors regarding material principle axes ε m and σ m are interrelated by the compliance matrix C m , which is composed of 4 independent elastic constants E 1 , E 2 the Young’s moduli, ν 21 ( ν 12 = ν 21 × E 1 / E 2 ) the Poisson’s ratio and G 12 the shear modulus. Then the characteristic equation regarding material principle axes becomes 2. Plane elasticity of rectilinearly anisotropic body which has, in general, 4 mutually unequal complex roots as µ m 1 , µ m 2 , µ m 1 and µ m 2 where over bar denotes a complex conjugate. It is important to note that the imaginary part of µ m 1 and µ m 2 must be positive. If we introduce new real variables λ 1 and λ 2 as, λ 1 =  E 1 E 2 , λ 2 = E 1 2 G 12 − ν 21 E 1 E 2 (3) then both µ m 1 and µ m 2 are complex number and expressed as follows in which “ i ” is an imaginary unit ( i 2 = − 1). µ m 1 = i  λ 2 +  λ 2 2 − λ 2 1 , µ m 2 = i  λ 2 −  λ 2 2 − λ 2 1 (4) In case of an isotropic material, both λ 1 and λ 2 becomes unity at the same time leading to µ m 1 = µ m 2 = i . However in the following discussion, we assume that µ m 1 and µ m 2 are not equal. Combining the above relationship obtained for the material principal axis x m , y m and the transformation rule for stress and strain components, a new relationship can be obtained regarding another coordinate axis at an arbitrary angle to the principal axis direction. For example, if the µ 4 m +  E 1 G 12 − 2 ν 21 E 1 E 2  µ 2 m + E 1 E 2 = 0 (2)