Closed form solution for a semi-infinite crack moving in an infinite orthotropic material with a circular crack breaker under antiplane strain

This study investigates the influence of a circular crack breaker on mode-III deformation behavior of a semi-infinite crack in a homogeneous, elastic orthotropic material subjected to longitudinal shear loads. The Galilean transformation is employed to convert the governing wave equation to Laplace’s equation which is time independent, rendering the problem amenable to analysis within the realm of the classical theory of two-dimensional elasticity. Considering the geometrical configuration of the problem, the analytical solution of the problem is possible if the problem is transformed using the appropriate mapping function. Our construction of a holomorphic function that maps the circular hole into a straight line with the edge terminating at the origin is a novelty which enables the use of integral transform method to obtain an analytic solution of the displacement, leading to closed-form expression for mode-III stress intensity factor, K111\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{111}$$\end{document}. The asymptotic values of the fields are obtained and shown to depend on the radius of the crack breaker. A parametric study shows that, for a fixed loading interval, a crack breaker of larger radius leads to increased stress intensity factor.

, Fanni et al. [11], and Chen [12] used the finite element method to obtain an optimum crack breaker that gives maximum fatigue crack initiation life. The endeavour in this paper is to obtain a closed form solution for the elastic fields in an orthotropic material containing a semi-infinite crack in the presence of a stop hole.

Formulation
Consider the problem of an infinite elastic orthotropic material containing a semi-infinite crack the tip of which is referred to a moving coordinate system ( x ′ , y ′ , z ′ ) space. The semi-infinite crack occupies the region defined by −∞ < x ′ < 0 . A pair of longitudinal shear loads of magnitude Q is applied along the crack surface on an interval [−a , −d] of length L . A circular crack breaker (stop hole) of radius b is introduced at the center of the orthotropic material which is at the origin of a fixed coordinate system x, y, z . Suppose that, at time t = 0 , the crack tip starts to move with constant velocity v along the x ′ -direction and ends up at the crack breaker, attaining a displacement v t . Suppose, also, that the disturbance due to the load is anti-plane so that it creates only an out of plane displacement and stresses in the z-direction. The problem is to investigate the influence of the circular stop hole on the stress intensity factor at the crack tip. This necessitates investigating the elastic fields at the point (x,0,0) on the boundary of the crack breaker.
Under anti-plane strain condition, the displacement components (u, v, w) reduce to (0, 0, w) where w = w x ′ , y ′ , t . Consequently, the only nonzero stress components σ x ′ z and σ y ′ z are given by where c 44 = µ xz and c 55 = µ yz are the shear moduli in the x ′ and y ′ directions.
Accordingly, the equation of motion takes the form  Let Be a holomorphic function which may be expressed in terms a new set of polar coordinates (ϒ, ∅) as Then, using the holomorphic function defined in Eqs. (13), (10), (11) and (12) transforms to Subject to the boundary condition

Solution of the transformed problem
sin φs s cos π s where Evaluating the integrals in (21) term by term using the convergent series, Nnadi [13] where the coefficients are defined by   (29), we obtain where The first part of Eq. (31) represents the displacement field for ϒ < β and the second part of the same equation represents the displacement field for ϒ < α.

Stress field near the crack-tip
We investigate the tip of the crack for the stress field, using the solution obtained above.
The tip is at the origin and is approached as ϒ → 0 . The asymptotic value of the displacement field is obtained as We introduce local polar coordinates (R, ψ) at the intersection of the crack breaker boundary and the x-axis that are related to the coordinates (ϒ, ∅) through Then, by the conformality condition W (ρ, φ) = w(R, ψ) , the asymptotic displacement becomes where Now, Hence, the near crack-tip stress field is given by Equation (46) shows that the near crack-tip stress field is directly proportional to mode III stress intensity factor, K III and inversely proportional to the radius of the circular hole, b.

Effect of size of the circular hole on the stress field
The result (47) means that the larger the radius of the circular hole, the less the stress concentration factor. In effect, crack growth would be delayed, or even prevented. The relationship between the stress field and the radius of the crack breaker is illustrated graphically in Fig. 2. The graph shows that the larger the radius of the crack breaker, the lower the stress values at the crack tip. The physical implication is that placing a large circular hole at the crack tip would contribute to an extension of the life of the material. This result agrees with the numerical results found in [10,11] and [12].

Effect of load application interval length on the stress intensity factor
A parametric study is introduced to investigate the effect of the length L of interval of application of the load and the radius of the crack breaker, on the stress intensity factor. Using Eqs. (41) and (45), we obtain the normalized stress intensity factor As we vary the length,L , of the interval [−α, −β] and substituting the corresponding values of β , α in Eq. (47), plotting k 111 Q with b L to have.

Conclusion
In this paper, the influence of a circular crack breaker on mode III deformation behaviour of a semi-infinite crack in a homogeneous elastic orthotropic material subjected to longitudinal shear loads is studied. The novelty of constructing a holomorphic function that mapped the circular hole into a straight line with the edge terminating at the origin as shown in Fig. 3, enabled the use of integral transform method to obtain an analytic solution of the displacement, leading to closed-form expression for the mode-III stress intensity factor,K 111 .
The results show that the larger the radius of the circular hole, the less the stress concentration factor. In effect, crack growth would be delayed, or even prevented and, thus, the life of the material would be extended. Further, the stress intensity factor (47) was shown in Fig. 4 to increase with increasing length of interval of load application. Our analytical results agree with numerical results found in the literature. As analytical solutions in closed form are desirable for accurate analysis and design due to their many advantages over numerical solutions, this study significantly contributes to the theory of antiplane crack propagation and arrest. It provides a theoretical basis for considering the use of a circular crack breaker for crack arrest [14], and the results serve as a benchmark for the purpose of judging the accuracy and efficiency of various numerical methods [15]. This study will be immensely useful to industries where orthotropic material such as fibre-reinforced composites have many applications in technology [16,17].