Effect of Couple Stresses on the Stress Intensity Factors for Two Parallel Cracks in an Infinite Elastic Medium under Tension

Abstract

Stresses around two parallel cracks of equal length in an infinite elastic medium are evaluated based on the linearized couple-stress theory under uniform tension normal to the cracks. Fourier transformations are used to reduce the boundary conditions with respect to the upper crack to dual integral equations. In order to solve these equations, the differences in the displacements and in the rotation at the upper crack are expanded through a series of functions that are zero valued outside the crack. The unknown coefficients in each series are solved in order to satisfy the boundary conditions inside the crack using the Schmidt method. The stresses are expressed in terms of infinite integrals, and the stress intensity factors can be determined using the characteristics of the integrands for an infinite value of the variable of integration. Numerical calculations are carried out for selected crack configurations, and the effect of the couple stresses on the stress intensity factors is revealed.

1. Introduction

In the classical theory of elasticity, the differential equations of equilibrium are derived from the equilibrium of the forces for the rectangular parallelepiped element 𝑑𝑥𝑑𝑦𝑑𝑧 with respect to the rectangular coordinates (𝑥,𝑦,𝑧). Since the element 𝑑𝑥𝑑𝑦𝑑𝑧 is infinitesimal, the normal stresses and shearing stresses are considered to act only on the surfaces of the element. The classical theory of elasticity is valid for a homogeneous material. In contrast, for materials with microstructures, such as porous materials and discrete materials, the differential equations of equilibrium may be derived from a parallelepiped element, which, although very small, is not infinitesimal. This produces additional stresses, called couple stresses, on the surfaces of the parallelepiped element. In the linearized couple stress theory (also referred to as the Cosserat theory with constrained rotations), the couple stresses are assumed to be proportional to the curvature, which yields a new material constant 𝑙, the dimension of which is length [1].

Mindlin evaluated the effect of couple stresses on the stress concentration around a circular hole in an infinite medium under tension [1], and Itou examined the effect of the couple stresses on the stress concentration around a circular hole in an infinite elastic layer under tension [2]. A similar problem has been solved for an infinite medium containing an infinite row of equally spaced holes of equal diameter under tension in the linearized couple-stress theory [3]. In these studies [1–3], the values of the stress concentration are shown to approach those for the corresponding classical solutions as 𝑙/𝑟 approaches zero, where 2𝑟 is the diameter of the holes.

Sternberg and Muki solved the stress intensity factor around a finite crack in an infinite Cosserat medium under tension and revealed that the Mode I stress intensity factor is always larger than the corresponding value for the classical theory of elasticity [4]. Yoffe assumed that a crack propagates only to the right, maintaining its length 2𝑎 to be constant and solved the stress intensity factor for the propagating crack [5]. For the Yoffe model, Itou evaluated the stress intensity factor in the linearized couple-stress theory [6]. For the crack problems, it has been shown that the values of the stress intensity factors are always larger than those for the classical theory of elasticity and that the values increase as 𝑙/𝑎 approaches zero.

Savin et al. determined the characteristic length 𝑙 by measuring the velocity of the transverse ultrasonic wave for brass, bronze, duralumin, and aluminum [7], and the material constant 𝑙 has been shown to be approximately one order of magnitude smaller than the mean grain size [7]. Thus, the effect of the couple stresses does not significantly affect the stress concentrations caused by the existence of circular holes, circular inclusions, and notches, whereas the effect of the couple stresses on the stress intensity factor around a crack is always larger than the corresponding value in the classical theory of elasticity.

If the weight of airplanes, high-speed trains, and automobiles can be reduced, fuel consumption can be curtailed considerably. This may be accomplished by using polycarbonate honeycomb materials and metal foam materials when designing machine elements. Mora and Waas performed a compression test of honeycomb materials with a rigid circular inclusion and estimated that 𝑙/𝑑 falls in the range 10.0 to 15.0, where 𝑑 is the diameter of each cell of the honeycomb [8]. Although no experiment has been performed to determine the value of the characteristic length 𝑙 for metal foam materials, 𝑙 may be expected to have a value on the order of the mean average value of the diameter of the foam. The metal foam materials and the polycarbonate honeycomb materials reduce the need for landfills because these materials are reusable. As such, these materials are increasingly being used for structural components in airplanes, high-speed trains, and automobiles. As a result, the couple-stress theory has been used increasingly for the evaluation of the stresses produced in such materials.

Gourgiotis and Georgiadis solved the Mode II and Mode III stress intensity factors for a crack in an infinite medium using the couple-stress theory and the distributed dislocation technique [9]. A Mode I crack problem was later solved by Gourgiotis and Georgiadis for a crack in an infinite medium [10]. Recently, Gourgiotis and Georgiadis evaluated the stress field in the vicinity of a sharp notch in an infinite medium under tension and searing stress using the couple-stress theory [11]. To the author’s knowledge, the stress intensity factors have been only evaluated for a crack in an infinite medium. In the present paper, stresses are solved for two equal parallel cracks in an infinite medium under tension using the couple-stress theory. The Fourier transform technique is used to reduce the boundary conditions with respect to the upper crack to dual integral equations. The differences in the displacements and in the rotation at the upper crack are expanded through a series of functions that vanish outside the crack. The unknown coefficients in each series are solved using the Schmidt method [12]. The stress intensity factors and the couple-stress intensity factor are calculated numerically for several crack configurations.

2. Fundamental Equations

In Cartesian coordinates (𝑥,𝑦), the upper crack is located between −𝑎 and 𝑎at 𝑦=0, and the lower crack is located between −𝑎 and 𝑎 at 𝑦=−2ℎ, as shown in Figure 1. Under plane strain conditions, the force stresses 𝜏_𝑥𝑥,𝜏_𝑦𝑦,𝜏_𝑥𝑦,𝜏_𝑦𝑥 and the couple stresses 𝜇_𝑥,𝜇_𝑦 are expressed as follows:𝜏_𝑥𝑥=𝜕²𝜙𝜕𝑦²−𝜕²𝜓𝜕𝑥𝜕𝑦,𝜏^𝑦𝑦=𝜕²𝜙𝜕𝑥²+𝜕²𝜓𝜏𝜕𝑥𝜕𝑦_𝑥𝑦,=−𝜕²𝜙𝜕𝑥𝜕𝑦−𝜕²𝜓𝜕𝑦²,𝜏_𝑦𝑥=−𝜕²𝜙𝜕𝑥𝜕𝑦+𝜕²𝜓𝜕𝑥²𝜇_𝑥,=𝜕𝜓𝜕𝑥,𝜇^𝑦=𝜕𝜓𝜕𝑦,(2.1) where 𝜙 and 𝜓 satisfy the following equations:𝜕∇⁴𝜙=0,∇²𝜓−𝑙²∇⁴𝜓=0,(2.2)𝜕𝑥𝜓−𝑙²∇²𝜓=−2(1−𝜈)𝑙²𝜕𝜕𝑦∇²𝜙,𝜕𝜕𝑦𝜓−𝑙²∇²𝜓=2(1−𝜈)𝑙²𝜕𝜕𝑥∇²𝜙,(2.3) where ∇² is the Laplacian operator, and 𝑙 is the new material constant. The rotation 𝜔_𝑧 and the strains 𝜀_𝑥 and 𝜀_𝑦 are given as follows:𝜔_𝑧=12𝐺𝜀_𝑦2×𝜕𝜕𝑥𝜈−𝜕𝜕𝑦𝑢,=2𝐺𝜕𝜈𝜕𝑦=(1−𝜈)𝜏^𝑦𝑦−𝜈𝜏_𝑥𝑥,2𝐺𝜀_𝑥=2𝐺𝜕𝑢𝜕𝑥=(1−𝜈)𝜏^𝑥𝑥−𝜈𝜏_𝑦𝑦,(2.4) where 𝑢 and 𝜈 are the 𝑥 and 𝑦 components of the displacement, and 𝐺 and 𝜈 are the shear modulus and Poisson’s ratio, respectively.

Figure 1 

Geometry and coordinate system.

3. Boundary Conditions

If we assume that a tensile stress 𝑝 is applied perpendicular to the two cracks, the stress field is symmetric with respect to the plane 𝑦=−ℎ, and it is sufficient to solve the problem for −ℎ<𝑦<0 only. For convenience, we refer to the layer −ℎ<𝑦<∞ as layer (1) and to the upper half-plane 0<𝑦<∞ as half plane (2). The boundary conditions can be expressed as follows:𝜏𝜏^0𝑦𝑦1=−𝑝,for|𝑥|<𝑎,𝑦=0,(3.1)_0𝑦𝑥1𝜇=0,for|𝑥|<𝑎,𝑦=0,(3.2)_0𝑦1𝑢=0,for|𝑥|<𝑎,𝑦=0,(3.3)₀₁𝑣=𝑢⁰²,for𝑎<|𝑥|,𝑦=0,(3.4)₀₁𝜔=𝑣⁰²,for𝑎<|𝑥|,𝑦=0,(3.5)_0𝑧1𝜏=𝜔^0𝑧2,for𝑎<|𝑥|,𝑦=0,(3.6)_0𝑦𝑦1𝜏=𝜏^0𝑦𝑦2,for|𝑥|<∞,𝑦=0,(3.7)_0𝑦𝑥1𝜇=𝜏^0𝑦𝑥2,for|𝑥|<∞,𝑦=0,(3.8)_0𝑦1𝜏=𝜇^0𝑦2,for|𝑥|<∞,𝑦=0,(3.9)_{−ℎ𝑦𝑥1}𝑣=0,for|𝑥|<∞,𝑦=−ℎ,(3.10)_1−ℎ𝜔=0,for|𝑥|<∞,𝑦=−ℎ,(3.11)_−ℎ𝑧1=0,for|𝑥|<∞,𝑦=−ℎ,(3.12) where subscripts 1 and 2 indicate layer (1) and half plane (2), respectively, and superscripts 0 and −ℎ indicate the values at 𝑦=0 and 𝑦=−ℎ, respectively.

4. Analysis

In order to find the solution, the Fourier transforms are introduced as follows:𝑓(𝑓𝑥)(𝜉=)=1^∞−∞𝑓(𝑥)exp(𝑖𝜉𝑥)𝑑𝑥,(4.1)2𝜋^∞−∞𝑓(𝜉)exp(−𝑖𝜉𝑥)𝑑𝜉.(4.2)

Applying (4.1) to (2.2) yields the following:𝑑⁴𝜙𝑑𝑦⁴−2𝜉²𝑑²𝜙𝑑𝑦²𝑙+𝜉⁴𝜙=0,₂𝑑⁴𝜓𝑑𝑦⁴−2𝜉²𝑙²+1𝑑²𝜓𝑑𝑦²+𝜉²𝜉²𝑙²+1𝜓=0.(4.3)

The solutions for (4.3) take the following forms for 𝑖=1 and 2:𝜙₁=𝐴₁cosh(𝜉𝑦)+𝐵₁𝑦cosh(𝜉𝑦)+𝐶₁sinh(𝜉𝑦)+𝐷₁𝑦sinh(𝜉𝑦),𝜓₁=𝐸₁cosh(𝑘𝑦)+𝐹₁cosh(𝜉𝑦)+𝐻₁sinh(𝑘𝑦)+𝐼₁sinh(𝜉𝑦),𝜙₂=𝐴₂+𝐵₂𝑦exp−||𝜉||𝑦,𝜓₂=𝐸₂exp−||𝜉||𝑦+𝐹₂exp(−𝑘𝑦),(4.4) where 𝐴₁, 𝐵₁, 𝐶₁, 𝐷₁, 𝐸₁, 𝐹₁, 𝐻₁, 𝐼₁, 𝐴₂, 𝐵₂, 𝐸₂, and 𝐹₂ are unknown coefficients, and 𝑘 is given by:𝑘=𝜉²𝑙²+1𝑙².(4.5)

Using the Fourier transformed expressions of (2.3), the coefficients 𝐼₁,𝐹₁, and 𝐸₂ can be represented by the coefficients 𝐷₁,𝐵₁, and 𝐵₂ as follows:𝐼₁=−4𝑖(1−𝜈)𝑙²𝜉𝐷₁,𝐹₁=−4𝑖(1−𝜈)𝑙²𝜉𝐵₁,𝐸₂=−4𝑖(1−𝜈)𝑙²𝜉𝐵₂.(4.6) Thus, the stresses, the displacements, and the rotation can be expressed by nine unknown coefficients in the Fourier domain.

Using (3.10) through (3.12), which are valid for −∞<𝑥<+∞, unknowns 𝐶₁, 𝐷₁, and 𝐸₁ are given as follows:𝐶₁=𝐴₁𝑐₁₁+𝐵₁𝑐𝑖𝐸₁₂₁+𝑖𝐻₁𝑐₁₃,𝐷₁=𝐴₁𝑐₂₁+𝐵₁𝑐₂₂+𝑖𝐻₁𝑐₂₃,=𝐴₁𝑐₃₁+𝐵₁𝑐₃₂+𝑖𝐻₁𝑐₃₃,(4.7) where the expressions of the known functions 𝑐_𝑖𝑗(𝑖,𝑗=1,2,3) are omitted. Then, the Fourier-transformed expressions of the stresses, the displacements, and the rotation in layer (1) can be expressed in terms of only three unknown coefficients, that are, 𝐴₁,𝐵₁, and 𝐻₁. Thus, the displacements 2𝐺𝑢⁰¹ and 2𝐺𝜈⁰¹ at 𝑦=0 and the rotation 4𝐺𝜔^0𝑧1 at 𝑦=0 can be expressed in terms of three unknown coefficients, that are, 𝐴₁,𝐵₁, and 𝐻₁. In contrast, coefficients 𝐴₁,𝐵₁, and 𝐻₁ can be expressed as 2𝐺𝑢⁰¹,2𝐺𝜈⁰¹, and 4𝐺𝜔^0𝑧1, and then the stresses, the displacements, and the rotation can be expressed in terms of 2𝐺𝑢⁰¹,2𝐺𝑣⁰¹, and 4𝐺𝜔^0𝑧1 in the Fourier domain. For examples, 𝜏^0𝑦𝑦1,𝜏^0𝑦𝑥1, and 𝜇^0𝑦1 have the following forms:𝜏^0𝑦𝑦1=−𝑖𝑢⁰¹𝑘¹¹(𝜉)+𝜈⁰¹𝑘¹²(𝜉)+−𝑖𝜔^0𝑧1𝑘¹³(𝜉),𝜏^0𝑦𝑥1=−𝑖𝑢⁰¹𝑖𝑘¹⁴(𝜉)+𝜈⁰¹𝑖𝑘¹⁵(𝜉)+−𝑖𝜔^0𝑧1𝑖𝑘¹⁶(𝜉),𝜇^0𝑦1=−𝑖𝑢⁰¹𝑖𝑘¹⁷(𝜉)+𝜈⁰¹𝑖𝑘¹⁸(𝜉)+−𝑖𝜔^0𝑧1𝑖𝑘¹⁹(𝜉),(4.8) where the expressions of the known functions 𝑘^1𝑖(𝜉)(𝑖=1,2,3,…,9) are omitted.

As for the upper half plane (2), the stresses, the displacements, and the rotation are shown by the three unknown coefficients 𝐴₂,𝐵₂, and 𝐹₂. Thus, the displacements 2𝐺𝑢⁰² and 2𝐺𝜈⁰² at 𝑦=0 and the rotation 4𝐺𝜔^0𝑧2 at 𝑦=0 can be described by three unknown coefficients 𝐴₂,𝐵₂, and 𝐹₂. In a similar manner to the case for layer (1), the unknown coefficients 𝐴₂,𝐵₂, and 𝐹₂ are represented by 2𝐺𝑢⁰²,2𝐺𝜈⁰², and 4𝐺𝜔^0𝑧2. Then, the stresses, the displacements, and the rotation can be expressed in terms of 2𝐺𝑢⁰²,2𝐺𝜈⁰², and 4𝐺𝜔^0𝑧2 in the Fourier domain. For examples, 𝜏^0𝑦𝑦2,𝜏^0𝑦𝑥2, and 𝜇^0𝑦2 have following forms:𝜏^0𝑦𝑦2=−𝑖𝑢⁰²𝑘²¹(𝜉)+𝜈⁰²𝑘²²(𝜉)+−𝑖𝜔^0𝑧2𝑘²³(𝜉),𝜏^0𝑦𝑥2=−𝑖𝑢⁰²𝑖𝑘²⁴(𝜉)+𝜈⁰²𝑖𝑘²⁵(𝜉)+−𝑖𝜔^0𝑧2𝑖𝑘²⁶(𝜉),𝜇^0𝑦2=−𝑖𝑢⁰²𝑖𝑘²⁷(𝜉)+𝜈⁰²𝑖𝑘²⁸(𝜉)+−𝑖𝜔^0𝑧2𝑖𝑘²⁹(𝜉),(4.9) where the expressions of the known functions 𝑘^2𝑖(𝜉)(𝑖=1,2,3,…,9) are omitted.

Using (3.7), (3.8) and (3.9) the following relations are obtained:−𝑖𝑢=⁰¹𝑘−¹¹𝑖(𝜉)+𝜈⁰¹𝑘¹²(𝜉)+−𝑖𝜔^0𝑧1𝑘¹³(𝜉)−𝑖𝑢⁰²𝑘²¹(𝜉)+𝜈⁰²𝑘²²(𝜉)+−𝑖𝜔^0𝑧2𝑘²³(𝜉),𝑢=⁰¹𝑖−𝑘𝑖¹⁴(𝜉)+𝜈⁰¹𝑖𝑘¹⁵(𝜉)+−𝑖𝜔^0𝑧1𝑖𝑘¹⁶(𝜉)−𝑖𝑢⁰²𝑖𝑘²⁴(𝜉)+𝜈⁰²𝑖𝑘²⁵(𝜉)+−𝑖𝜔^0𝑧2𝑖𝑘²⁶(𝜉),𝑢=⁰¹𝑖−𝑘𝑖¹⁷(𝜉)+𝜈⁰¹𝑖𝑘¹⁸(𝜉)+−𝑖𝜔^0𝑧1𝑖𝑘¹⁹(𝜉)𝑢⁰²𝑖𝑘²⁷(𝜉)+𝜈⁰²𝑖𝑘²⁸(𝜉)+−𝑖𝜔^0𝑧2𝑖𝑘²⁹(𝜉).(4.10) Equation (4.10) can be solved for (−𝑖𝑢⁰¹),𝜈⁰¹ and (−𝑖𝜔^0𝑧1) as follows:−𝑖𝑢⁰¹=−𝑖𝑢⁰²𝑙¹(𝜉)+𝜈⁰²𝑙₂(𝜉)+−𝑖𝜔^0𝑧2𝑙³(𝜉),𝜈⁰¹−=𝑖−𝑖𝑢⁰²𝑙⁴(𝜉)+𝜈⁰²𝑙₅(𝜉)+−𝑖𝜔^0𝑧2𝑙⁶(𝜉),𝜔⁰¹=−𝑖𝑢⁰²𝑙⁷(𝜉)+𝜈⁰²𝑙₈(𝜉)+−𝑖𝜔^0𝑧2𝑙⁹(𝜉),(4.11) where the expressions of the known functions 𝑙_𝑖(𝜉)(𝑖=1,2,3,…,9) are omitted.

In order to satisfy (3.4), (3.5), and (3.6) the differences in the displacements and in the rotation at 𝑦=0 are expanded by the following series:𝜋𝜈⁰¹−𝜈⁰²=^∞_𝑛=1𝑐_𝑛cos(2𝑛−1)sin⁻¹𝑥𝜋𝑢=0for𝑎≤|𝑥|≤∞,𝑎for|𝑥|≤𝑎(4.12)₀₁−𝑢⁰²=^∞_𝑛=1𝑑_𝑛sin2𝑛sin⁻¹𝑥𝜋𝜔=0for𝑎≤|𝑥|≤∞,𝑎for|𝑥|≤𝑎(4.13)_0𝑧1−𝜔^0𝑧2=^∞_𝑛=1𝑒_𝑛sin2𝑛sin⁻¹𝑥=0for𝑎≤|𝑥|≤∞,𝑎for|𝑥|≤𝑎(4.14) where 𝑐_𝑛, 𝑑_𝑛, and 𝑒_𝑛 are the unknown coefficients that are to be determined. The Fourier transforms of (4.12) through (4.14) are𝜈⁰¹−𝜈⁰²=^∞_𝑛=1𝑐_𝑛(2𝑛−1)𝜉𝐽^2𝑛−1(𝑎𝜉)𝑢⁰¹−𝑢⁰²=𝑖^∞_𝑛=1𝑑_𝑛2𝑛𝜉𝐽^2𝑛(𝑎𝜉)𝜔^0𝑧1−𝜔^0𝑧2=𝑖^∞_𝑛=1𝑒_𝑛2𝑛𝜉𝐽^2𝑛(𝑎𝜉)(4.15) where 𝐽_𝑛(𝜉) is the Bessel function.

Substituting (4.11) into (4.15), we obtain the following equations: −𝑖𝑢⁰²𝑙⁴(𝜉)+𝜈⁰²𝑙₅(𝜉)−1+−𝑖𝜔^0𝑧2𝑙⁶(𝜉)=^∞_𝑛=1𝑐_𝑛(2𝑛−1)𝜉𝐽^2𝑛−1−𝑖(𝑎𝜉),𝑢⁰²𝑙¹(𝜉)−1+𝜈⁰²𝑙₂(𝜉)+−𝑖𝜔^0𝑧2𝑙³(𝜉)=^∞_𝑛=1𝑑_𝑛2𝑛𝜉𝐽^2𝑛−𝑖(𝑎𝜉),𝑢⁰²𝑙⁷(𝜉)+𝜈⁰²𝑙₈(𝜉)+−𝑖𝜔^0𝑧2𝑙⁹(𝜉)−1=^∞_𝑛=1𝑒_𝑛2𝑛𝜉𝐽^2𝑛(𝑎𝜉).(4.16) Equations (4.16) can be solved for (−𝑖𝑢⁰²),𝜈⁰² and (−𝑖𝜔^0𝑧2) as follows:−𝑖𝑢⁰²=^∞_𝑛=1𝑐_𝑛𝑚₁(𝜉)(2𝑛−1)𝜉𝐽^2𝑛−1(𝑎𝜉)+^∞_𝑛=1𝑑_𝑛𝑚₂(𝜉)×2𝑛𝜉𝐽^2𝑛+(𝑎𝜉)_∞_𝑛=1𝑒_𝑛𝑚₃(𝜉)×2𝑛𝜉𝐽^2𝑛(𝑎𝜉),𝜈⁰²=^∞_𝑛=1𝑐_𝑛𝑚₄(𝜉)(2𝑛−1)𝜉𝐽^2𝑛−1(𝑎𝜉)+^∞_𝑛=1𝑑_𝑛𝑚₅(𝜉)×2𝑛𝜉𝐽^2𝑛+(𝑎𝜉)_∞_𝑛=1𝑒_𝑛𝑚₆(𝜉)×2𝑛𝜉𝐽^2𝑛−𝑖(𝑎𝜉),𝜔^0𝑧2=^∞_𝑛=1𝑐_𝑛𝑚₇(𝜉)(2𝑛−1)𝜉𝐽^2𝑛−1(𝑎𝜉)+^∞_𝑛=1𝑑_𝑛𝑚₈(𝜉)×2𝑛𝜉𝐽^2𝑛+(𝑎𝜉)_∞_𝑛=1𝑒_𝑛𝑚₉(𝜉)×2𝑛𝜉𝐽^2𝑛(𝑎𝜉),(4.17) where the expressions of the known functions 𝑚_𝑖(𝜉)(𝑖=1,2,3,…,9) are omitted. At this stage, the stresses, the displacements, and the rotation are represented by only three unknown coefficients, that are, 𝑐_𝑛, 𝑑_𝑛, and 𝑒_𝑛. For example, stresses 𝜏^0𝑦𝑦2,𝜏^0𝑦𝑥2, and 𝜇^0𝑦2 are expressed as follows:𝜏^0𝑦𝑦2=^∞_𝑛=1𝑐_𝑛(2𝑛−1)𝜋×^∞0𝑞₁(𝜉)𝜉𝐽^2𝑛−1+(𝑎𝜉)cos(𝜉𝑥)𝑑𝜉_∞_𝑛=1𝑑_𝑛2𝑛𝜋×^∞0𝑞₂(𝜉)𝜉𝐽^2𝑛+(𝑎𝜉)cos(𝜉𝑥)𝑑𝜉_∞_𝑛=1𝑒_𝑛2𝑛𝜋×^∞0𝑞₃(𝜉)𝜉𝐽^2𝑛𝜏(𝑎𝜉)cos(𝜉𝑥)𝑑𝜉,(4.18)_0𝑦𝑥2=^∞_𝑛=1𝑐_𝑛(2𝑛−1)𝜋×^∞0𝑞₄(𝜉)𝜉𝐽^2𝑛−1+(𝑎𝜉)sin(𝜉𝑥)𝑑𝜉_∞_𝑛=1𝑑_𝑛2𝑛𝜋×^∞0𝑞₅(𝜉)𝜉𝐽^2𝑛+(𝑎𝜉)sin(𝜉𝑥)𝑑𝜉_∞_𝑛=1𝑒_𝑛2𝑛𝜋×^∞0𝑞₆(𝜉)𝜉𝐽^2𝑛𝜇(𝑎𝜉)sin(𝜉𝑥)𝑑𝜉,(4.19)_0𝑦2=^∞_𝑛=1𝑐_𝑛(2𝑛−1)𝜋×^∞0𝑞₇(𝜉)𝜉𝐽^2𝑛−1+(𝑎𝜉)sin(𝜉𝑥)𝑑𝜉_∞_𝑛=1𝑑_𝑛2𝑛𝜋×^∞0𝑞₈(𝜉)𝜉𝐽^2𝑛+(𝑎𝜉)sin(𝜉𝑥)𝑑𝜉_∞_𝑛=1𝑒_𝑛2𝑛𝜋×^∞0𝑞₉(𝜉)𝜉𝐽^2𝑛(𝑎𝜉)sin(𝜉𝑥)𝑑𝜉,(4.20) where the known expressions 𝑞₁(𝜉),𝑞₂(𝜉),…,𝑞₈(𝜉), and 𝑞₉(𝜉) are omitted. Functions 𝑞_𝑖(𝜉)/𝜉(𝑖=2,3,4,6,7,8) decrease rapidly as 𝜉 increases. Functions 𝑞_𝑖(𝜉)/𝜉(𝑖=1,5,9) have the following property when 𝜉 increases:𝑞_𝑖(𝜉)𝜉⟶𝑞^𝐿𝑖,(4.21) where constants 𝑞^𝐿𝑖(𝑖=1,5,9) can be calculated as𝑞^𝐿𝑖=𝑞^𝑖𝜉_𝐿𝜉_𝐿,(4.22) with 𝜉_𝐿 being a large value of 𝜉.

Finally, the remaining boundary conditions (3.1), (3.2), and (3.3) can be reduced to the following equations:^∞_𝑛=1𝑐_𝑛𝐾_𝑛(𝑥)+^∞_𝑛=1𝑑_𝑛𝐿_𝑛(𝑥)+^∞_𝑛=1𝑒_𝑛𝑀_𝑛(𝑥)=−𝑢(𝑥),_∞_𝑛=1𝑐_𝑛𝑂_𝑛(𝑥)+^∞_𝑛=1𝑑_𝑛𝑃_𝑛(𝑥)+^∞_𝑛=1𝑒_𝑛𝑄_𝑛(𝑥)=−𝜈(𝑥),_∞_𝑛=1𝑐_𝑛𝑅_𝑛(𝑥)+^∞_𝑛=1𝑑_𝑛𝑆_𝑛(𝑥)+^∞_𝑛=1𝑒_𝑛𝑇_𝑛(𝑥)=−𝑤(𝑥)for|𝑥|≤𝑎,(4.23) with𝐾_𝑛(𝑥)=(2𝑛−1)𝜋×^∞0𝑞¹(𝜉)𝜉−𝑞^𝐿1𝐽^2𝑛−1+𝑞(𝑎𝜉)cos(𝜉𝑥)𝑑𝜉_𝐿1cos(2𝑛−1)sin⁻¹(𝑥/𝑎)𝐿_𝑛𝑎²−𝑥²^1/2,(𝑥)=2𝑛𝜋^∞0𝑞₂(𝜉)𝜉𝐽^2𝑛𝑀(𝑎𝜉)cos(𝜉𝑥)𝑑𝜉,_𝑛(𝑥)=2𝑛𝜋^∞0𝑞₃(𝜉)𝜉𝐽^2𝑛𝑂(𝑎𝜉)cos(𝜉𝑥)𝑑𝜉,_𝑛(𝑥)=(2𝑛−1)𝜋^∞0𝑞₄(𝜉)𝜉𝐽^2𝑛−1𝑃(𝑎𝜉)sin(𝜉𝑥)𝑑𝜉,_𝑛(𝑥)=2𝑛𝜋×^∞0𝑞⁵(𝜉)𝜉−𝑞^𝐿5𝐽^2𝑛+𝑞(𝑎𝜉)sin(𝜉𝑥)𝑑𝜉_𝐿5sin2𝑛sin⁻¹(𝑥/𝑎)𝑄_𝑛𝑎²−𝑥²^1/2,(𝑥)=2𝑛𝜋^∞0𝑞₆(𝜉)𝜉𝐽^2𝑛𝑅(𝑎𝜉)sin(𝜉𝑥)𝑑𝜉,_𝑛(𝑥)=(2𝑛−1)𝜋^∞0𝑞₇(𝜉)𝜉𝐽^2𝑛−1𝑆(𝑎𝜉)sin(𝜉𝑥)𝑑𝜉,_𝑛(𝑥)=2𝑛𝜋^∞0𝑞₈(𝜉)𝜉𝐽^2𝑛𝑇(𝑎𝜉)sin(𝜉𝑥)𝑑𝜉,_𝑛(𝑥)=2𝑛𝜋×^∞0𝑞⁹(𝜉)𝜉−𝑞^𝐿9𝐽^2𝑛+𝑞(𝑎𝜉)sin(𝜉𝑥)𝑑𝜉_𝐿9sin2𝑛sin⁻¹(𝑥/𝑎)𝑢(𝑥)=𝑝,𝜈(𝑥)𝑎=₂0−,𝑥²𝑤^1/2(𝑥)=0.,((44..2245)) The unknown coefficients 𝑐_𝑛, 𝑑_𝑛, and𝑒_𝑛 in (4.23) can be solved by the Schmidt method [12].

5. Stress Intensity Factors

If we slightly modify the integrands in (4.18) through (4.20) and use the relations^∞0𝐽=_𝑛(𝑎𝜉−)𝑎[_𝑛cos(𝜉𝑥),sin(𝜉𝑥)]𝑑𝜉𝑎_𝑛𝑥²−𝑎²^−1/2𝑥+𝑥²−𝑎²^−1/2^−𝑛sin𝑛2𝜋,𝑥²−𝑎²^−1/2𝑥+𝑥²−𝑎²^−1/2^−𝑛cos𝑛2𝜋,for𝑎<𝑥,(5.1) the stress intensity factors and the couple-stress intensity factor can be determined as follows:𝐾_𝐼=[2𝜋(𝑥−𝑎)]^1/2𝜏^0𝑦𝑦2||_𝑥→𝑎+=^∞_𝑛=1𝑐_𝑛(2𝑛−1)(−1)^𝑛𝑞^𝐿1(𝜋𝑎)^1/2𝐾_𝐼𝐼,=[2𝜋(𝑥−𝑎)]^1/2𝜏^0𝑦𝑥2||_𝑥→𝑎+=^∞_𝑛=1𝑑_𝑛(2𝑛)(−1)^𝑛𝑞^𝐿5(𝜋𝑎)^1/2𝑀₀,=[2𝜋(𝑥−𝑎)]^1/2𝜇^0𝑦2||_𝑥→𝑎+=^∞_𝑛=1𝑒_𝑛(2𝑛)(−1)^𝑛𝑞^𝐿9(𝜋𝑎)^1/2.(5.2)

6. Numerical Examples

Numerical calculations are performed with quadruple precision using a Fortran program for which overflow and underflow do not occur within the range of from 10⁻⁵⁵⁰⁰ to 10⁺⁵⁵⁰⁰. The stress intensity factors and the couple-stress intensity factor are calculated for ℎ/𝑎=5.0,0.5, and 0.1 with a Poisson’s ratio of 𝜈=0.25.

The values of the functions 𝑞_𝑖(𝜉𝑎)/(𝜉𝑎)(𝑖=2,3,4,6,7,8) are verified to decay rapidly as (𝜉𝑎) increases, and the values of the functions 𝑞_𝑖(𝜉𝑎)/(𝜉𝑎)(𝑖=1,5,9)are verified to rapidly approach constants 𝑞^𝐿𝑖(𝑖=1,5,9) as (𝜉𝑎) increases. Thus, the semi-infinite integrals in (4.24) can be evaluated numerically. The Schmidt method, truncated to 12 terms for an infinite series, was then applied to solve for coefficients 𝑐_𝑛,𝑑_𝑛, and 𝑒_𝑛 in (4.23). The values of the left-hand side of (4.23) are verified to coincide with those of the right-hand side of (4.23). Then, coefficients 𝑐_𝑛, 𝑑_𝑛, and 𝑒_𝑛 are verified to be successfully determined by the Schmidt method.

The values of the Mode I stress intensity 𝐾_𝐼/(𝑝√𝜋𝑎) are shown for ℎ/𝑎=5.0 in Table 1, in which the values in parentheses are obtained from the diagram in [4]. For ℎ/𝑎=5.0, the distance between two cracks is 10𝑎, and the lower crack is considered not to affect the stress field around the upper crack and vice versa. Both values in Table 1 are well coincident with each other, and the accuracy of the method described in the present paper is considered to be superior. The values of 𝐾_𝐼,𝐾_𝐼𝐼, and M₀ are plotted with respect to 𝑙/𝑎 for ℎ/𝑎=5.0,0.5, and 0.1, respectively, in Figures 2, 3, and 4, in which the straight broken lines indicate the corresponding value for 𝑙/𝑎=0.0.

Table 1 

Values of 𝐾_𝐼/(𝑝√𝜋𝑎) for ℎ/𝑎=5.0. (Values in parentheses were obtained from the diagram in [4]).

Figure 2 

Values of 𝐾𝐼, 𝐾𝐼𝐼, and 𝑀0 plotted with respect to 𝑙/𝑎 for ℎ/𝑎=5.0 (broken lines show the values for 𝑙/𝑎=0.0).

Figure 3 

Values of 𝐾𝐼,𝐾𝐼𝐼, and 𝑀0 plotted with respect to 𝑙/𝑎 for ℎ/𝑎=0.5 (broken lines show the values for 𝑙/𝑎=0.0).

Figure 4 

Values of 𝐾𝐼,𝐾𝐼𝐼, and 𝑀0 plotted with respect to 𝑙/𝑎 for ℎ/𝑎=0.1 (broken lines show the values for 𝑙/𝑎=0.0).

7. Conclusion

Based on the numerical calculations described above, the following conclusions are obtained.(i)The values of 𝐾_𝐼/(𝑝√𝜋𝑎) for ℎ/𝑎=5.0 are considered to be approximately coincident with those for a crack in an infinite medium, and the values of 𝐾_𝐼𝐼/(𝑝√𝜋𝑎) are considered to be approximately equal to zero.(ii)As 𝑙/𝑎 approaches zero, 𝐾_𝐼/(𝑝√𝜋𝑎)and 𝐾_𝐼𝐼/(𝑝√𝜋𝑎) do not approach the corresponding values calculated using the classical theory of elasticity, whereas the values of 𝑀₀/(𝑝𝑎√𝜋𝑎) approach zero, which is the value calculated by the classical theory of elasticity.(iii)The values of 𝐾_𝐼/(𝑝√𝜋𝑎) decrease as ℎ/𝑎 decreases, and the same behavior is observed for the absolute values of 𝑀₀/(𝑝𝑎√𝜋𝑎).(iv)The new material constant 𝑙 may be comparatively small, even for materials with microstructures. Therefore, the key value is 𝐾_𝐼/(𝑝√𝜋𝑎), even for materials with microstructures, because the values of 𝐾_𝐼𝐼/(𝑝√𝜋𝑎) and 𝑀₀/(𝑝𝑎√𝜋𝑎) are considerably smaller than the value of 𝐾_𝐼/(𝑝√𝜋𝑎).