Wiener–Hopf method to solve the anti-plane problem of moving semi-infinite crack in orthotropic composite materials

Abstract

This paper contains the solution to the problem of a semi-infinite moving crack situated in an orthotropic strip bonded between two identical strips. The crack moves with a constant velocity, and the surface is under shear wave disturbance. We first examine the equations of elasticity, which include equilibrium equations and stress and displacement constitutive relations with the model-specific continuity and boundary conditions. Using the Fourier integral transform, the standard form for the Wiener–Hopf (W-H) equation is obtained, which is solved using the W-H method. The analytical expressions for the considered crack problem have been obtained for stress intensity factor (SIF), normalized stress intensity factor (NSIF), and stress magnification factor (SMF). The behavior of NSIF has been graphically presented for particular cases of composite materials for different crack propagation velocities and various depth ratios of the strips. The novelty of this paper lies in the analytic solutions using the W-H method for the semi-infinite moving crack problem under the influence of anti-plane shear waves. The pictorial presentations of normalized SIF clearly show their dependency on strip depths, crack propagation velocities and elastic constants.

Keywords

Stress intensity factor anti-plane traction analytic function composite materials Wiener–Hopf method

1. Introduction

A crack is a partial failure that can grow due to loadings and environmental conditions and become fatal. A critical crack leads to fracture failure, resulting in a material break. While developing a mechanical structure, engineers and scientists are constantly concerned about developing material cracks in the structure. The formation of cracks in the material is a common phenomenon, or we can also say that all materials have cracks. A material with multiple layers or strips is more susceptible to failure caused by fractures, as it is exposed to diverse external disturbances that might propagate cracks. Fractures that spread rapidly, particularly on the surface and in the contact region, can lead to material fractures and structural failures. Physical parameters like the stress intensity factor (SIF) are computed to analyze the behavior of crack propagation in engineering structures. A composite material exhibiting distinct mechanical properties in three mutually perpendicular directions is considered orthotropic. Orthotropic composites have distinct and frequently anisotropic properties along each principal axis, in contrast to isotropic materials, which have the same properties in all directions. This property renders them highly anisotropic, meaning that their mechanical behavior varies substantially depending on the direction in which a stimulus is applied. Orthotropic composites are synthetic materials manufactured by combining two or more constituents with distinct properties. Typically, they consist of high-strength fibers or reinforcements embedded in a polymer, metal, or ceramic matrix material. Composite materials are in high demand because of their durability, lightweight, resistance to high pressure, and versatility. The distinct mechanical properties of these substances in three orthogonal orientations are well known. The wide range of applications of composite materials has prompted researchers to conduct studies on their behavior and crack mechanics. Semi-infinite cracks under the action of anti-plane shear stress/strain are one of several cracks that may develop in composite materials. In fiber-reinforced composite materials, interfacial anti-plane shear fractures are particularly prone to develop since the contact is often where crack propagation and fracture initiation begin [1, 2].

A unique state of strain in a body is known as anti-plane shear or anti-plane strain [3]. When the displacements in the body are zero in the plane of interest but non-zero in the direction perpendicular to the plane, this condition of strain is reached. The strain tensor under anti-plane shear can be expressed for small strains as [2]:

ϵ = [\begin{matrix} 0 & 0 & ϵ_{xz} \\ 0 & 0 & ϵ_{yz} \\ ϵ_{zx} & ϵ_{zy} & 0 \end{matrix}] .

Due to the abrupt fracture in brittle materials under anti-plane shear strain, the cracks have been the subject of much interest up to this point [4]. Many types of research have been conducted on anti-plane problems over a period of time. In anti-plane shear, the material experiences shear stress in the direction perpendicular to the plane of loading. This type of deformation is frequently observed in thin structures, such as plates or sheets, where the deformation occurs predominantly in a single direction. Anti-plane shear boundary value problems have been (discussed in [5,6].)

Sih and Chen [7] have comprehensively described the idea of anti-plane shear stress. Das and Patra [8, 9] have explored the moving Griffith crack in dissimilar orthotropic materials’ interface. Yang et al. [10] and Bidadi et al. [11] have worked on the nanoscale mode-III interfacial crack in bimaterial, as well as the impact of thickness on anti-plane fracture mode crack propagation. Piccolroaz et al. [12] addressed the problem of Mode III interfacial crack that undergoes an anti-plane deformation. The results were then used to analyze the asymptotic behavior of stresses. Yoffee was the first to introduce the idea of a moving crack [13]. In [14], the problem of anti-plane shear in a bi-material, plane containing a semi-infinite crack on a soft, imperfect interface has been discussed using the Fourier transformation. Researchers have found the disturbances caused by shear waves on fractured surfaces intriguing because of their different impacts on crack propagation. The nature of SH waves has been the subject of several investigations and studies in fracture mechanics by many research authors. Trifunac [15] has done the plane shear wave scattering effect, while Yang and Qi [16] examined the steady-state shear wave effect in a bimaterial half-space. Diankui and Hong [17] found a solution to an interfacial linear fracture that interacts with a circular cavity. Investigation of how time-harmonic shear wave disturbance affected the edge crack problem has been addressed in [18]. Semi-infinite crack growth has been exhibited in [19] to show that edge wave phase speed exhibits dual dependency on current and frequency, which results in distinct asymptotic behaviors.

The W-H method is a mathematical technique used to solve certain integral equations. It was developed jointly by Norbert Wiener and Eberhard Hopf in the early 20th century. The method is particularly useful for solving problems involving wave propagation, diffraction, and scattering phenomena. In dynamic fracture mechanics, the W-H equation may be used to solve a number of crack problems in a strip in the complex transform plane. Noble and Wiess [20] first introduced the W-H technique approach employed in their article in 1959, which other scientists and engineers have afterward researched. The hardest part of this method is factorizing the kernel. Nilsson [21,22] has provided a method for determining the asymptotic expression of SIF using the W-H methodology without the kernel’s visible factorization. Achenbach and Gautesen [23] have explored the elastodynamic SIF for semi-infinite cracks. Abrahams [24] has looked into a few issues involving various forms of cracks in dynamic elasticity utilizing the W-H approach. Maurya and Sharma [25] have solved the problem of two semi-infinite cracks on a square lattice. Gourgiotis et al. [26] deal with the plane–strain problem of semi-infinite crack under concentrated loading, using the W-H method for coupled stress components. Thus, the literature review reveals that the majority of publications pertaining to semi-infinite cracks are of the embedded kind. Determination of the expressions of SIF, normalized stress intensity factor (NSIF), and stress magnification factor (SMF) for moving semi-infinite cracks in a sandwiched orthotropic strip under anti-plane loading is not yet been explored to that extent to the best of the authors’ knowledge.

The moving crack problem refers to the study of crack propagation in materials under applied loads. It involves analyzing the behavior of a crack as it grows or moves through a material over time. When a material contains a crack or a pre-existing flaw, the application of external loads can cause stress concentrations at the crack tip. These high stresses can lead to crack growth, which can be detrimental to the integrity and functionality of the material. The moving crack problem aims to understand how crack propagates and how its behavior affects the overall structural response. The authors typically address this problem using fracture mechanics principles, which provide a framework for studying crack growth and failure.

This paper considers the problem of a semi-infinite moving crack in an orthotropic strip bonded between two identical strips from top and bottom. The crack is moving with a constant velocity at a given time, and the surface is under shear wave disturbance. The authors first analytically examine the governing equations of motion and stress–displacement constitutive relations with the model-specific continuity and boundary conditions. Using the Fourier integral transform on equations and boundary conditions, the standard form for the Wiener-Hopf (W-H) equation is obtained, which is then solved using the W-H method. The analytical expressions for the considered crack problem have been obtained for the SIF, NSIF, and the SMF as a particular case of the problem. The behavior of NSIF has been graphically represented for particular cases of composite materials, for different crack propagation velocities, and for various depth ratios of semi-infinite strips. Validation of the present results is shown graphically by comparison with an existing study. The novelty of this paper lies in the analytic solutions using the W-H method for the semi-infinite moving crack in a semi-infinite orthotropic strip of depth $h_{1}$ bonded between strips of depth $h_{2}$ under the influence of anti-plane shear waves. The pictorial presentations of NSIF clearly represent their dependency on strip depths, crack propagation velocities, and elastic constants.

2. Governing equations and problem formulation

Consider a semi-infinite moving crack present in a semi-infinite orthotropic strip of depth $2 h_{1}$ sandwiched between two identical strips of depth $h_{2}$ from both the upper and lower sides. Let the crack be propagating toward the positive $X$ - direction with a constant velocity $V$ . We consider $(X, Y, Z)$ to be a Cartesian coordinate system such that it aligns with the crack geometry. The crack is positioned at $- \infty < X < 0, Y = 0$ as shown in Figure 1. At a time instant $t$ , crack is positioned at $Y = 0, - \infty < X < Vt$ . The crack surface is under anti-plane shear loading.

Figure 1.

Geometry of the problem.

The governing equations for motion in anti-plane are given by Horgan [2]:

C_{55}^{(j)} \frac{\partial^{2} W^{(j)}}{\partial X^{2}} + C_{44}^{(j)} \frac{\partial^{2} W^{(j)}}{\partial Y^{2}} = ρ^{(j)} \frac{\partial^{2} W^{(j)}}{\partial t^{2}},

(1)

where $j = 1$ and $j = 2$ represent the orthotropic strip 1 and the strip 2, and $ρ^{(j)}$ represent the material densities. Rewriting equation (3) as:

\frac{\partial^{2} W^{(j)}}{\partial X^{2}} + (α^{(j)})^{2} \frac{\partial^{2} W^{(j)}}{\partial Y^{2}} = \frac{1}{{(C_{s}^{(j)})}^{2}} \frac{\partial^{2} W^{(j)}}{\partial t^{2}},

(2)

where $α^{(j)}$ = $\sqrt{C_{44}^{(j)} / C_{55}^{(j)}}$ . Here, $C_{s}^{(j)} = \sqrt{C_{55}^{(j)} / ρ^{(j)}}$ are the shear wave velocities. The displacement component $W^{(j)} = W^{(j)} (X, Y, t)$ for the problem is found in the $Z$ -direction only, i.e., out-of-plane direction. Applying the Galilean transformation by making the substitutions $x = X - Vt$ , $y = Y$ in equation (2). The problem is converted from time-dependent to free from time $t$ . The new equation is:

H^{{(j)}^{2}} \frac{\partial^{2} ω^{(j)}}{\partial x^{2}} + \frac{\partial^{2} ω^{(j)}}{\partial y^{2}} = 0,

(3)

where $H^{(j)}$ = $\sqrt{\frac{1}{α^{(j) 2}} (1 - \frac{V^{2}}{C_{s}^{(j) 2}})}$ and $ω^{(j)} = ω^{(j)} (x, y)$ . Taking into consideration the symmetric nature of the geometric model with respect to the $y$ -axis, from further on in the problem, we only consider the $y > 0$ plane. Thus, the continuity and boundary conditions for the considered mathematical problem are given by:

σ_{yz}^{(1)} (x, 0) = σ_{0}, - \infty < x < 0,

(4a)

ω^{(1)} (x, 0) = 0, x > 0,

(4b)

σ_{yz}^{(1)} (x, h_{1}) = σ_{yz}^{(2)} (x, h_{1}), - \infty < x < \infty,

(4c)

ω^{(1)} (x, h_{1}) = ω^{(2)} (x, h_{1}), - \infty < x < \infty,

(4d)

ω^{(2)} (x, h_{1} + h_{2}) = 0, - \infty < x < \infty,

(4e)

where $ω^{(j)} and σ_{yz}^{(j)} for j = 1, 2$ are the displacements and stress components for respective materials 1 and 2. The conditions mentioned above have been modified to employ the W-H technique by imposing a constant shear load, say $σ_{0}$ . Now, using the Fourier transform on equation (3) as:

\begin{matrix} {\bar{ω}}^{(j)} (s, y) = F [ω^{(j)} (x, y) : x \to s], \\ = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} ω^{(j)} (x, y) e^{isx} dx, j = 1, 2, \end{matrix}

(5)

where s is the complex variable, $s = α + i γ$ . The solution is assumed in the form of:

{\bar{ω}}^{(1)} (s, y) = A_{1} (s) e^{- H^{(1)} sy} + B_{1} (s) e^{H^{(1)} sy},

(6)

for strip 1.

\begin{matrix} {\bar{ω}}^{(2)} (s, y) = A_{2} (s) e^{- s H^{(2)} (h_{1} + h_{2})} \\ [e^{- s H^{(2)} (y - h_{1} - h_{2})} - e^{s H^{(2)} (y - h_{1} - h_{2})}], \end{matrix}

(7)

for strip 2. Using the relation:

σ_{yz}^{(j)} = C_{44}^{(j)} \frac{\partial ω^{(j)}}{\partial y}, for j = 1, 2,

(8)

we obtain the anti-plane stress components as:

{\bar{σ}}_{yz}^{(1)} (s, y) = - C_{44}^{(1)} H^{(1)} s (A_{1} (s) e^{- H^{(1)} sy} - B_{1} (s) e^{H^{(1)} sy}),

(9)

for strip 1 and:

\begin{matrix} {\bar{σ}}_{yz}^{(2)} (s, y) = - C_{44}^{(2)} H^{(2)} s e^{- s H^{(2)} (h_{1} + h_{2})} \\ [e^{- s H^{(2)} (y - h_{1} - h_{2})} + e^{s H^{(2)} (y - h_{1} - h_{2})}] A_{2} (s), \end{matrix}

(10)

for strip 2, where $A_{1} (s), A_{2} (s)$ , and $B_{1} (s)$ are the unknown coefficients that have been solved in section 3.

To employ the W-H technique, the modified boundary condition of equation (4a) is written as:

σ_{yz}^{(1)} (x, 0) = σ_{0} e^{ϵ x}, - \infty < x < 0,

(11)

where $ϵ \to 0$ .

Applying equation (5), in the transformed domain, the continuity and boundary conditions, in equation (4) becomes:

{\bar{σ}}_{yz}^{(1)} (s, 0) = \frac{σ_{0}}{\sqrt{2 π} (ϵ + is)}, - \infty < s < 0,

(12a)

{\bar{ω}}^{(1)} (s, 0) = 0, 0 < s < \infty,

(12b)

{\bar{σ}}_{yz}^{(1)} (s, h_{1}) = {\bar{σ}}_{yz}^{(2)} (s, h_{1}), - \infty < s < \infty,

(12c)

{\bar{ω}}^{(1)} (s, h_{1}) = {\bar{ω}}^{(2)} (s, h_{1}), - \infty < s < \infty .

(12d)

From the interfacial condition (12c) with the aid of equations (9) and (10), we get:

\begin{matrix} C_{44}^{(1)} H^{(1)} s (A_{1} (s) e^{- s H^{(1)} h_{1}} - B_{1} (s) e^{s H^{(1)} h_{1}}) = \\ C_{44}^{(2)} H^{(2)} s [e^{- s H^{(2)} h_{2}} + e^{s H^{(2)} h_{2}}] A_{2} (s) . \end{matrix}

(13)

The other interfacial condition (12d) with the aid of equations (6) and (7) gives:

A_{1} (s) e^{- s H^{(1)} h_{1}} + B_{1} (s) e^{s H^{(1)} h_{1}} = e^{- s H^{(2)} (h_{1} + h_{2})} [e^{s H^{(2)} (h_{2})} - e^{- s H^{(2)} (h_{2})}] A_{2} (s) .

(14)

Equations (13) and (14) give rise to:

B_{1} (s) = [\frac{η_{1} e^{s H^{(2)} h_{2}} - η_{2} e^{- s H^{(2)} h_{2}}}{η_{2} e^{s H^{(2)} h_{2}} - η_{1} e^{- s H^{(2)} h_{2}}}] e^{- 2 s H^{(1)} h_{1}} A_{1} (s),

(15)

where $η_{1} = C_{44}^{(1)} H^{(1)} - C_{44}^{(2)} H^{(2)}$ and $η_{2} = C_{44}^{(1)} H^{(1)} + C_{44}^{(2)} H^{(2)}$ .

3. Solution procedure

In order to proceed further in the problem, we first consider two unknown functions $f_{+} (x)$ and $g_{-} (x)$ given by:

σ_{yz}^{(1)} (x, 0) = f_{+} (x), 0 < x < \infty,

(16a)

ω^{(1)} (x, 0) = g_{-} (x), - \infty < x < 0 .

(16b)

Applying the Fourier transformation given in equation (5) on both of the functions in equation (16), we obtain:

C_{44}^{(1)} H^{(1)} s η (s) A_{1} (s) = {\bar{f}}_{+} (s) + \frac{σ_{0}}{\sqrt{2 π} (ϵ + is)},

(17)

and

μ (s) A_{1} (s) = {\bar{g}}_{-} (s),

(18)

where:

{\bar{f}}_{+} (s) = \frac{1}{\sqrt{2 π}} \int_{0}^{\infty} f_{+} (x) e^{isx} dx,

(19a)

{\bar{g}}_{-} (s) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{0} g_{-} (x) e^{isx} dx,

(19b)

with $η (s) = 1 - [\frac{η_{1} e^{s H^{(2)} h_{2}} - η_{2} e^{- s H^{(2)} h_{2}}}{η_{2} e^{s H^{(2)} h_{2}} - η_{1} e^{- s H^{(2)} h_{2}}}] e^{- 2 s H^{(1)} h_{1}}$ and $μ (s) = 1 + [\frac{η_{1} e^{s H^{(2)} h_{2}} - η_{2} e^{- s H^{(2)} h_{2}}}{η_{2} e^{s H^{(2)} h_{2}} - η_{1} e^{- s H^{(2)} h_{2}}}] e^{- 2 s H^{(1)} h_{1}}$ .

For the above transform given in equation (17) to be valid in $(- \infty, 0)$ , it is necessary that $Im (s) = γ < 0$ .

Justified by the behavior of the problem that the disturbances vanish at points far away from the crack tip and based on the wave propagation field, the functions $f_{+} (s)$ and $g_{-} (s)$ are bounded at infinity, and their upper bounds are assumed as:

\begin{matrix} | f_{+} (s) | < M e^{l_{f} x}, l_{f} < 0 as x \to \infty, \\ > l_{f}, \\ | g_{-} (s) | < N e^{l_{g} x}, as x \to - \infty, \\ < l_{g}, \end{matrix}

where $M > 0$ and $N > 0$ are finite. Thus, ${\bar{f}}_{+} (s)$ and ${\bar{g}}_{-} (s)$ are analytic in $γ > l_{f}$ and $γ < l_{g}$ , respectively. It is assumed that $l_{g} > l_{f}$ and region of analyticity overlap between $l_{f} < γ < l_{g} = 0$ . Equations (17) and (18) together give:

{\bar{f}}_{+} (s) = K (s) {\bar{g}}_{-} (s) - \frac{σ_{0}}{\sqrt{2 π} (ϵ + i s)},

(20)

where $K (s) = \frac{- C_{44}^{(1)} H^{(1)} s η (s)}{μ (s)}$ .

The problem now contains two unknown functions $f_{+} (s)$ and $g_{-} (s)$ .

3.1. W-H method for the solution of the problem

Let us factorize the kernel $K (s)$ into the following form as suggested in Noble and Weiss [20]:

K (s) = K_{+} (s) K_{-} (s),

(21)

where the functions $K_{+} (s)$ and $K_{-} (s)$ are non-zero analytic functions for $γ > a_{1} (a_{1} < 0)$ and $γ < a_{2} (a_{2} > 0)$ , respectively. The way to factorize $K (s) = K_{1} (s) K_{2} (s)$ is by choosing a $K_{2} (s)$ that can be factored by inspection where $li m_{| s | \to \infty} K_{2} (s) \to \infty$ and $li m_{| s | \to 0} K_{2} (s) \to 0$ . Using the conditions that $K_{1} (s)$ is analytic and free from zeroes in strip $- d < γ < d, d > 0$ and $K_{1} (\infty) = 1$ , the function can be split using Cauchy’s integral theorem. As a result:

K_{1} (s) = K_{1}^{+} (s) K_{1}^{-} (s),

where $K_{1}^{\pm} (\infty) = 1$ and:

\ln (K_{1}^{+} (s)) = \frac{1}{2 π i} \int_{- \infty + id}^{\infty + id} \frac{\ln (K_{1} (ξ))}{ξ - s} d ξ

\ln (K_{1}^{-} (s)) = \frac{- 1}{2 π i} \int_{- \infty + id}^{\infty + id} \frac{\ln (K_{1} (ξ))}{ξ - s} d ξ,

and hence:

K_{-} (s) = - C_{44}^{(1)} H^{(1)} \sqrt{s - id} K_{1}^{-} (s)

K_{+} (s) = \sqrt{s + id} K_{1}^{+} (s) .

Furthermore, equation (20) with the help of equation (21) becomes:

\frac{{\bar{f}}_{+} (s)}{K_{+} (s)} = K_{-} (s) {\bar{g}}_{-} (s) - \frac{σ_{0}}{\sqrt{2 π} (ϵ + i s) K_{+} (s)} .

(22)

Now, decomposing the last term of equation (22), will have:

\frac{σ_{0}}{\sqrt{2 π} (ϵ + i s) K_{+} (s)} = L_{+} (s) + L_{-} (s),

(23)

where:

L_{+} (s) = \frac{σ_{0}}{\sqrt{2 π} (ϵ + is)} [\frac{1}{K_{+} (s)} - \frac{1}{K_{+} (i ϵ)}],

(24a)

L_{-} (s) = \frac{σ_{0}}{\sqrt{2 π} (ϵ + i s) K_{+} (i ϵ)} .

(24b)

The functions $L_{+} (s)$ and $L_{-} (s)$ are non-zero analytic functions in $γ > a_{1} (a_{1} < 0)$ and $γ < ϵ,$ respectively. Rewriting equation (22) using equation (23), we get:

\frac{{\bar{f}}_{+} (s)}{K_{+} (s)} + L_{+} (s) = K_{-} (s) {\bar{g}}_{-} (s) - L_{-} (s) .

(25)

Equation (25) makes use of the fact that the functions ${\bar{f}}_{+} (s), K_{+} (s), L_{+} (s), {\bar{g}}_{-} (s), K_{-} (s)$ , and $L_{-} (s)$ are all analytic functions in the region for $γ > l_{f} (l_{f} < 0), γ > a_{1} (a_{1} < 0), γ > a_{1} (a_{1} < 0), γ < 0, γ < a_{2} (a_{2} > 0), and γ < ϵ (ϵ > 0)$ , respectively. Since the right side of equation (25) and the left side is analytic in the shared region of analyticity, the function in equation (25) is an entire function which can be successfully stated using the concept of analytic continuation. For significantly large values of $s$ , the functions ${\bar{f}}_{+} (s)$ and ${\bar{g}}_{-} (s)$ will be bounded, and the functions $K_{+} (s)$ and $K_{-} (s)$ tend to $s^{1 / 2}$ . The right side of equation (25) tends to $s^{1 / 2}$ while the left side of the same equation tends to $s^{- 1 / 2}$ for very large values of $| s |$ on $γ \geq 0$ . Based on the Extended Liouville’s theorem, we may conclude that the right-hand term of the equation (25) is a constant function as $| s | \to \infty$ . While the left-hand term tends to zero as $| s | \to \infty$ . Therefore, the only entire function $F (s)$ which satisfies the aforementioned conditions must be single-valued, and hence:

F (s) = 0 .

(26)

Using equations (24)–(25), with the help of equation (26), we get:

{\bar{f}}_{+} (s) = \frac{σ_{0}}{\sqrt{2 π} (ϵ + i s)} (\frac{K_{+} (s)}{K_{+} (i ϵ)} - 1),

(27a)

{\bar{g}}_{-} (s) = \frac{σ_{0}}{\sqrt{2 π} (ϵ + i s) K_{+} (i ϵ) K_{-} (s)} .

(27b)

Taking $ϵ \to 0$ , the above relations become:

{\bar{f}}_{+} (s) = \frac{σ_{0}}{\sqrt{2 π} (is)} (\frac{K_{+} (s)}{K_{+} (0)} - 1),

(28a)

and

{\bar{g}}_{-} (s) = \frac{σ_{0}}{\sqrt{2 π} (is) K_{+} (0) K_{-} (s)} .

(28b)

The approach described [21, 22] is used to get the expression of SIF, which is also analytic without factorizing the kernel $K (s)$ . The expression of SIF may be determined from the expression of $K (s)$ at small and large values of $s$ . Consequently, the asymptotic values of $K (s)$ are:

lim_{s \to \infty} \frac{K (s)}{s} = β_{1} = - C_{44}^{(1)} H^{(1)},

(29)

lim_{s \to 0} K (s) = β_{2} = - \frac{C_{44}^{(1)} H^{(1)}}{H^{(1)} h_{1} + H^{(2)} h_{2} (\frac{η_{2} + η_{1}}{η_{2} - η_{1}})} .

(30)

For a large value of $s$ , equation (27a) can be rewritten as:

lim_{s \to \infty} {\bar{f}}_{+} (s) = lim_{s \to \infty} \frac{σ_{0}}{\sqrt{2 π} (is)} (\frac{K_{+} (s)}{K_{+} (0)}) .

(31)

Thus, we have:

{\bar{f}}_{+} (s) = s^{- \frac{1}{2}} \frac{σ_{0} \sqrt{β_{1}}}{i \sqrt{2 π β_{2}}} as s \to \infty .

(32)

Applying the Inverse Fourier transform in equation (32), the expression of the above function is found to be:

f (x) = \frac{- σ_{0}}{2} \sqrt{\frac{β_{1}}{2 π β_{2}}} x^{- \frac{1}{2}} .

(33)

In this case, equation (33) displays the shear component and its distribution on the considered crack’s exterior along the direction of $x$ -axis. In addition, the nature of the problem clearly shows the presence of square root singularity at the origin or crack tip.

3.2. Expressions of SIF and NSIF

The SIF $(K_{III})$ for considered model is found as:

K_{III} = lim_{x \to 0 +} \sqrt{2 π x} σ_{yz}^{(1)} (x, 0) = - \frac{σ_{0}}{2} \sqrt{\frac{β_{1}}{β_{2}}} .

(34)

The Various graphs for NSIF are presented in the article for different considered cases and discussed in detail in the section numerical results and discussion. The expression of NSIF is required to make SIF dimensionless and is obtained by dividing the equation (34) by the expression of SIF for a static semi-infinite crack between two semi-infinite orthotropic planes i.e., when strip width $h_{1} \to \infty$ . The expression of NSIF for the problem becomes:

NSIF = \frac{1}{2} \sqrt{\frac{β_{1}}{h_{1} β_{2}}} .

(35)

4. Particular case

If the elastic constants of strip 1 are identical to strip 2, i.e., $C_{55}^{(1)} = C_{55}^{(2)}$ , $C_{44}^{(1)} = C_{44}^{(2)}$ , $ρ^{(1)} = ρ^{(2)}$ , then our considered model is reduced to the problem of a semi-infinite crack situated in a strip 1 of depth $2 h_{1}$ . The corresponding boundary conditions will be reduced to:

σ_{yz}^{(1)} (x, 0) = σ_{0}, - \infty < x < 0,

(36a)

ω^{(1)} (x, 0) = 0, 0 < x < \infty,

(36b)

ω^{(1)} (x, h_{1}) = 0, - \infty < x < \infty .

(36c)

Here, the parameters’ values will be $β_{1} = - C_{44}^{(1)} H^{(1)}$ and $β_{2} = - C_{44}^{(1)} / h_{1}$ . Therefore, the expression of SIF becomes:

K_{III}^{*} = - σ_{0} \sqrt{π H^{(1)} h_{1} / 2} .

(37)

5. Numerical results and discussion

5.1. Validation

The results of this paper are validated with a previous study [27]. Figure 2 depicts the similarity in both studies by presenting a particular case. Graphs of SIF versus crack velocity have been plotted for the problem when the semi-infinite crack is embedded in an orthotropic half-plane. The same material choice is taken for both problems, and material parameters are set such that $C_{44}^{(1)} = C_{55}^{(1)}$ . Similarities can be seen in the results of both studies under similar conditions. In Figure 2, when $V = 0$ , SIF becomes 1, it is because the problem is converted into a static problem for the said conditions, which is also the normalizing factor used to find NSIF.

Figure 2.

Plots of stress intensity factor versus $V$ for the present result and the existing result [27].

5.2. Numerical results

This paper contains the analytical calculations for SIFs, NSIF, and SMF. The SIF expression is derived successfully using the W-H method for the considered composite materials prepreg, graphite epoxy, and carbon fiber, whose elastic constants are given in Table 1. First, in the material choice, we chose such two materials for which $C_{(55)}^{(1)} > C_{(55)}^{(2)}$ . In this case, graphs have been plotted for NSIF in Figures 2 –4.

Table 1.

Material parameters for the considered orthotropic materials.

	$C_{44}$ (GPa)	$C_{55}$ (GPa)	$ρ (g / c m^{3})$
Prepreg	7.8	7.8	1.595
Carbon fiber	6.15	6.15	1.5
Graphite epoxy	5.40	5.50	1.60

Figure 3.

Plots of normalized stress intensity factor versus $h_{1} / h_{2}$ for $V / C_{s}^{(1)} = 0.1, 0.3, and 0.45$ for material choice of prepeg as strip 1 and graphite epoxy as strip 2.

Figure 4.

Plots of normalized stress intensity factor versus $V / C_{s}^{(1)}$ for $h_{1} / h_{2} = 0.25, 0.5, 1.0, 2.0, and 4.0$ .

In Figure 3, we present the results indicating the NSIF versus ratio of strip 1 and strip 2 widths, i.e., $h_{1} / h_{2}$ for different $V / C_{s}^{(1)}$ . We graphically represent the values of NSIF at the crack tip, which are essential for determining crack growth. The variations in NSIF as the crack propagates with certain velocities $0.1, 0.3, and 0.45$ , considering the ratio of depths of the strips ranging from $0.0$ to $1.0$ , are displayed pictorially. In Figure 4, the results indicate NSIF versus ratio of crack velocity and shear wave velocity $V / C_{s}^{(1)}$ are presented at different ratios of strip width values. We graphically represent the variations in NSIF as the crack propagates, considering the depth of the strip as $0.25, 0.5, 1.0, 2.0, and 4.0$ , as $V / C_{s}^{(1)}$ ranges between $0.0$ and $1.0$ are displayed pictorially. Figure 5 shows a three-dimensional plot for NSIF versus $h_{1} / h_{2}$ and $V / C_{s}^{(1)}$ for the same material choices.

Figure 5.

Plot of normalized stress intensity factor versus $V / C_{s}^{(1)}$ and $h_{1} / h_{2}$ for material choice of prepeg as strip 1 and graphite epoxy as strip 2.

In the second case, we chose such two materials for which $C_{(55)}^{(1)} < C_{(55)}^{(2)}$ . In this case, graphs have been plotted for NSIF. In Figure 6, we present the results indicating NSIF versus $h_{1} / h_{2}$ for different $V / C_{s}^{(1)}$ . We graphically represent the values of NSIF at the crack tip, which are essential for determining crack growth. The variations in NSIF as the crack propagates with certain velocities $0.1, 0.3, and 0.45$ , considering the ratios of strips’ depths as $h_{1} / h_{2}$ ranging from $0.0$ to $1.0$ , are displayed in pictorially. In Figure 7, the results indicate the NSIF versus ratio of crack velocity and shear velocity $V / C_{s}^{(1)}$ for different values of $h_{1} / h_{2}$ . The variations in NSIF have been presented, considering $h_{1} / h_{2}$ as $0.25, 0.5, 1.0, 2.0, and 4.0$ , when $V / C_{s}^{(1)}$ ranges between $0.0$ and $1.0$ through Figure 8, in a three-dimensional plot for NSIF versus $h_{1} / h_{2}$ and $V / C_{s}^{(1)}$ for the same material choices.

Figure 6.

Plots of normalized stress intensity factor versus $h_{1} / h_{2}$ for different $V / C_{s}^{(1)} = 0.1, 0.3, 0.45$ for material choice of graphite epoxy as strip 1 and carbon fiber as strip 2.

Figure 7.

Plots of normalized stress intensity factor versus $V / C_{s}^{(1)}$ for $h_{1} / h_{2} = 0.25, 0.5, 1.0, 2.0, 4.0$ for material choice of graphite epoxy as strip 1 and carbon fiber as strip 2.

Figure 8.

Plot of normalized stress intensity factor versus $V / C_{s}^{(1)}$ and $h_{1} / h_{2}$ for material choice of graphite epoxy as strip 1 and carbon fiber as strip 2.

Figure 9 depicts SMF versus $h_{1} / h_{2}$ plots for $C_{(55)}^{(1)} < C_{(55)}^{(2)}$ and $C_{(55)}^{(1)} > C_{(55)}^{(2)}$ . The graphs for different material choices are plotted, and their behavior is depicted pictorially as $h_{1} / h_{2}$ ranges between $0$ and $10$ . SMF is obtained by normalizing the SIF, $K_{III}$ by the expression of $K_{III}^{*}$ , which is obtained for the particular case considered in the previous section.

Figure 9.

Plots of SMF $K_{III} / K_{III}^{*}$ versus $h_{1} / h_{2}$ for $C_{55}^{(1)} / C_{55}^{(2)} > 1$ for material choice of graphite epoxy as strip 1 and carbon fiber as strip 2 and $C_{55}^{(1)} / C_{55}^{(2)} < 1$ for material choice of prepeg as strip 1 and graphite epoxy as strip 2.

5.3. Discussion

The analytical formulations of SIF and NSIF depend on elastic constants, crack propagation velocity, and strip widths, as equations (34)–(35) show. In Figures 3 and 6 of NSIF versus $h_{1} / h_{2}$ , the values of SIF decrease with increasing values of $h_{1} / h_{2}$ . This decreasing behavior of SIF is justified. As the strip width increases, the impact of strips on the crack decreases, and hence, the stress field becomes less. The NSIF approaches zero asymptotically as $h_{1} / h_{2}$ increases. Similar behavior is shown in Figures 4 and 7, whereas the values of $h_{1} / h_{2}$ increase, the graphs of NSIFs are decreasing further. From the numerical results, it can be concluded that as the strip with $h_{2}$ increases, it puts more impact over the strip $h_{1}$ and hence more impact over the crack. In the same way, as the strip width $h_{1}$ decreases, its impact on the crack keeps on decreasing, which results in a lower value of SIF. In Figures 4 and 7 of NSIF versus $V / C_{s}^{(1)}$ , as the velocity ratio increases, NSIF decreases exponentially and becomes nearly zero as $V / C_{s}^{(1)} \to 1.0$ . As it can be seen through graphically for increasing the value of velocity $V$ , NSIF decreases and vanishes as $V \to C_{s}$ . This phenomenon is justified theoretically from the expressions of $H^{(j)}$ that the results are not valid for values beyond $V / C_{s}^{(1)} > 1$ . So, theoretically, it is clear that shear velocity is the limiting value of crack propagation velocity. This similar result is visible for all cases presented in this paper, and a similar phenomenon can be found in different articles [18,21,22] . Similar behavior is depicted in Figures 3 and 6, where for increasing values of $V / C_{s}^{(1)}$ , NSIF decreases. The NSIF is shown to be approaching zero as $V$ approaches $C_{s}^{(1)}$ , i.e., when the crack velocity is equal to shear wave velocity, is justified because the $V < C_{s}^{(1)}$ is the threshold point for the limit of SIF. In Figure 9, the graphs of SMF for two different material choices depict that the value is always more than one, which shows an amplification effect. SMF for $C_{55}^{(1)} > C_{55}^{(2)}$ shows more amplification as compared to $C_{55}^{(1)} < C_{55}^{(2)}$ . These results are important in fracture mechanics to arrest the crack by limiting the value of SIF. The crack growth can be controlled by controlling the parameters of crack propagation velocity and strip widths.

6. Conclusion

The W-H method involves transforming the integral equation into a pair of integral equations, one of which is known as the W-H equation, and the other one is the dual W-H equation. These equations are derived by splitting the integral into two parts using a contour in the complex plane and applying appropriate transformations. Anti-plane shear is commonly encountered in solid mechanics, especially in the study of elastic waves and fracture mechanics. It is often analyzed using simplified models, assuming plane stress or strain conditions. In anti-plane shear, the material experiences shear stress in the direction perpendicular to the plane of loading. This type of deformation is frequently observed in thin structures, such as plates or sheets, where the deformation occurs predominantly in a single direction. In this paper, moving semi-infinite crack in a semi-infinite orthotropic strip of depth $2 h_{1}$ sandwiched between two identical strips of width $h_{2}$ has been studied. This paper has achieved the following goals successfully:

The analytic determination of asymptotic expression of SIF using the W-H method and Fourier integral transformation technique.

A drive has been taken to convert the original mixed boundary value problem into a standard form of the W-H equation with the help of Fourier transformation. Furthermore, we are able to derive an asymptotic expression of NSIF at the tip of the crack.

A drive has also been taken toward damping of the SIFs of the considered model combined with a problem of a semi-infinite crack in a strip through finding SMF.

Variations of SIF, NSIF, and SMF values have been displayed graphically for different depths of the strip and various crack propagation velocities for different particular cases of composite materials.

Footnotes

Acknowledgements

The authors extend their heartfelt thanks to the revered reviewers for their suggestions toward the improvement of the article.

Funding

The author(s) disclosed the following financial support for the research, authorship, and/or publication of this article: One of the authors (S.D.) acknowledges the project grant provided by the National Board for Higher Mathematics (NBHM), Department of Atomic Energy, Government of India (File no. 02011/2022 NBHM (R.P.)/R&D II / 2171).

ORCID iD

Eduard-Marius Craciun

References

Timoshenko

Woinowsky-Krieger

. Theory of plates and shells, vol. 2. New York: McGraw Hill, 1959.

Horgan

. Anti-plane shear deformations in linear and nonlinear solid mechanics. SIAM Rev 1995; 37(1): 53–81.

Slaughter

. The linearized theory of elasticity. Berlin: Springer Science & Business Media, 2012.

Interaction among interfacial offset cracks in composite materials under the anti-plane shear loading. Z Angew Math Mech 2023; 103: e202300081.

Lioubimova

Schiavone

. Integral solutions of boundary value problems of anti-plane piezoelectricity. Math Mech Solids 2007; 12(1): 24–39.

Shmoylova

Potapenko

. Stress distribution around a crack in anti-plane micropolar elasticity. Math Mech Solids 2008; 13(2): 148–171.

Sih

Chen

. Dynamic response of dissimilar materials with cracks. In: Sih

Chen

(eds) Cracks in composite materials: a compilation of stress solutions for composite systems with cracks. Berlin: Springer, 1981, pp. 277–440.

Das

Patra

. Stress intensity factors for moving interfacial crack between bonded dissimilar fixed orthotropic layers. Comput Struct 1998; 69(4): 459–472.

Das

Patra

. Moving Griffith crack at the interface of two dissimilar orthotropic half planes. Eng Fract Mech 1996; 54(4): 523–531.

10.

Yang

. Nanoscale mode-III interface crack in a bimaterial with surface elasticity. Mech Mater 2020; 140: 103246.

11.

Bidadi

Akbardoost

Aliha

MRM

. Thickness effect on the mode III fracture resistance and fracture path of rock using ENDB specimens. Fatigue Fract Eng M 2020; 43(2): 277–291.

12.

Piccolroaz

Mishuris

Radi

. Mode III interfacial crack in the presence of couple-stress elastic materials. Eng Fract Mech 2012; 80: 60–71.

13.

Yoffe

. Lxxv. The moving Griffith crack. Lond Edinb Dublin Philos Mag J Sci 1951; 42(330): 739–750.

14.

Vellender

Mishuris

Piccolroaz

. Perturbation analysis for an imperfect interface crack problem using weight function techniques. Int J Solids Struct 2013; 50(24): 4098–4107.

15.

Trifunac

. Scattering of plane SH-waves by a semi-cylindrical canyon. Earthquake Eng Struc 1972; 1(3): 267–281.

16.

Yang

. The scattering of steady-state SH-waves in a bi-material half space with multiple cylindrical elastic inclusions. Wave Random Complex 2019; 29(1): 162–177.

17.

Diankui

Hong

. Scattering of SH-waves by an interacting interface linear crack and a circular cavity near bimaterial interface. Acta Mech Sinica 2004; 20(3): 317–326.

18.

Trivedi

Das

Craciun

. The mathematical study of an edge crack in two different specified models under time-harmonic wave disturbance. Mech Compos Mater 2022; 58(1): 1–14.

19.

Abrahams

Lawrie

. Scattering of flexural waves by a semi-infinite crack in an elastic plate carrying an electric current. Math Mech Solids 2012; 17(1): 43–58.

20.

Noble

Weiss

. Methods based on the Wiener–Hopf technique for the solution of partial differential equations. Phys Today 1959; 12(9): 50.

21.

Nilsson

. Dynamic stress-intensity factors for finite strip problems. Int J Fract Mech 1972; 8: 403–411.

22.

Nilson

. Dynamic stress-intensity factors for finite strip. Int J Fracture 1973; 9(4): 477.

23.

Achenbach

Gautesen

. Elastodynamic stress-intensity factors for a semi-infinite crack under 3-D loading. J Appl Mech 1977; 44(2): 243–249.

24.

Abrahams

. On the application of the Wiener–Hopf technique to problems in dynamic elasticity. Wave Motion 2002; 36(4): 311–333.

25.

Maurya

Sharma

. Scattering by two staggered semi-infinite cracks on the square lattice: an application of asymptotic Wiener–Hopf factorization. Z Angew Math Phys 2019; 70(5): 133.

26.

Gourgiotis

Georgiadis

Sifnaiou

. Couple-stress effects for the problem of a crack under concentrated shear loading. Math Mech Solids 2012; 17(5): 433–459.

27.

Karan

Mandal

Basu

, et al. Interaction of shear waves with semi-infinite moving crack inside of a orthotropic media. Waves Random Complex Media 2021; 19: 1–7.