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Mathematical Problems in Engineering
VolumeΒ 2012, Article IDΒ 575891, 14 pages
http://dx.doi.org/10.1155/2012/575891
Research Article

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

Department of Mechanical Engineering, Kanagawa University, Rokkakubashi, Kanagawa-ku, Yokohama 221-8686, Japan

Received 8 February 2012; Accepted 22 March 2012

Academic Editor: Oleg V.Β Gendelman

Copyright Β© 2012 Shouetsu Itou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 𝑑π‘₯𝑑𝑦𝑑𝑧 with respect to the rectangular coordinates (π‘₯,𝑦,𝑧). Since the element 𝑑π‘₯𝑑𝑦𝑑𝑧 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/π‘Ÿ approaches zero, where 2π‘Ÿ 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 2π‘Ž 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/π‘Ž 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 (π‘₯,𝑦), the upper crack is located between βˆ’π‘Ž and π‘Žat 𝑦=0, and the lower crack is located between βˆ’π‘Ž and π‘Ž at 𝑦=βˆ’2β„Ž, as shown in Figure 1. Under plane strain conditions, the force stresses 𝜏π‘₯π‘₯,πœπ‘¦π‘¦,𝜏π‘₯𝑦,πœπ‘¦π‘₯ and the couple stresses πœ‡π‘₯,πœ‡π‘¦ are expressed as follows:𝜏π‘₯π‘₯=πœ•2πœ™πœ•π‘¦2βˆ’πœ•2πœ“πœ•π‘₯πœ•π‘¦,πœπ‘¦π‘¦=πœ•2πœ™πœ•π‘₯2+πœ•2πœ“,πœπœ•π‘₯πœ•π‘¦π‘₯π‘¦πœ•=βˆ’2πœ™βˆ’πœ•πœ•π‘₯πœ•π‘¦2πœ“πœ•π‘¦2,πœπ‘¦π‘₯πœ•=βˆ’2πœ™+πœ•πœ•π‘₯πœ•π‘¦2πœ“πœ•π‘₯2,πœ‡π‘₯=πœ•πœ“πœ•π‘₯,πœ‡π‘¦=πœ•πœ“,πœ•π‘¦(2.1) where πœ™ and πœ“ satisfy the following equations:βˆ‡4πœ™=0,βˆ‡2πœ“βˆ’π‘™2βˆ‡4πœ•πœ“=0,(2.2)ξ€·πœ•π‘₯πœ“βˆ’π‘™2βˆ‡2πœ“ξ€Έ=βˆ’2(1βˆ’πœˆ)𝑙2πœ•βˆ‡πœ•π‘¦2πœ•πœ™,ξ€·πœ•π‘¦πœ“βˆ’π‘™2βˆ‡2πœ“ξ€Έ=2(1βˆ’πœˆ)𝑙2πœ•βˆ‡πœ•π‘₯2πœ™,(2.3) where βˆ‡2 is the Laplacian operator, and 𝑙 is the new material constant. The rotation πœ”π‘§ and the strains πœ€π‘₯ and πœ€π‘¦ are given as follows:πœ”π‘§=12Γ—ξ‚΅πœ•πœˆβˆ’πœ•π‘₯πœ•π‘’ξ‚Ά,πœ•π‘¦2πΊπœ€π‘¦=2πΊπœ•πœˆπœ•π‘¦=(1βˆ’πœˆ)πœπ‘¦π‘¦βˆ’πœˆπœπ‘₯π‘₯,2πΊπœ€π‘₯=2πΊπœ•π‘’πœ•π‘₯=(1βˆ’πœˆ)𝜏π‘₯π‘₯βˆ’πœˆπœπ‘¦π‘¦,(2.4) where 𝑒 and 𝜈 are the π‘₯ and 𝑦 components of the displacement, and 𝐺 and 𝜈 are the shear modulus and Poisson’s ratio, respectively.

575891.fig.001
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=βˆ’β„Ž, and it is sufficient to solve the problem for βˆ’β„Ž<𝑦<0 only. For convenience, we refer to the layer βˆ’β„Ž<𝑦<∞ as layer (1) and to the upper half-plane 0<𝑦<∞ as half plane (2). The boundary conditions can be expressed as follows:𝜏0𝑦𝑦1𝜏=βˆ’π‘,for|π‘₯|<π‘Ž,𝑦=0,(3.1)0𝑦π‘₯1πœ‡=0,for|π‘₯|<π‘Ž,𝑦=0,(3.2)0𝑦1𝑒=0,for|π‘₯|<π‘Ž,𝑦=0,(3.3)01=𝑒02𝑣,forπ‘Ž<|π‘₯|,𝑦=0,(3.4)01=𝑣02πœ”,forπ‘Ž<|π‘₯|,𝑦=0,(3.5)0𝑧1=πœ”0𝑧2𝜏,forπ‘Ž<|π‘₯|,𝑦=0,(3.6)0𝑦𝑦1=𝜏0𝑦𝑦2𝜏,for|π‘₯|<∞,𝑦=0,(3.7)0𝑦π‘₯1=𝜏0𝑦π‘₯2πœ‡,for|π‘₯|<∞,𝑦=0,(3.8)0𝑦1=πœ‡0𝑦2𝜏,for|π‘₯|<∞,𝑦=0,(3.9)βˆ’β„Žπ‘¦π‘₯1𝑣=0,for|π‘₯|<∞,𝑦=βˆ’β„Ž,(3.10)1βˆ’β„Žπœ”=0,for|π‘₯|<∞,𝑦=βˆ’β„Ž,(3.11)βˆ’β„Žπ‘§1=0,for|π‘₯|<∞,𝑦=βˆ’β„Ž,(3.12) where subscripts 1 and 2 indicate layer (1) and half plane (2), respectively, and superscripts 0 and βˆ’β„Ž indicate the values at 𝑦=0 and 𝑦=βˆ’β„Ž, respectively.

4. Analysis

In order to find the solution, the Fourier transforms are introduced as follows:ξ€œπ‘“(πœ‰)=βˆžβˆ’βˆž1𝑓(π‘₯)exp(π‘–πœ‰π‘₯)𝑑π‘₯,(4.1)𝑓(π‘₯)=ξ€œ2πœ‹βˆžβˆ’βˆžπ‘“(πœ‰)exp(βˆ’π‘–πœ‰π‘₯)π‘‘πœ‰.(4.2)

Applying (4.1) to (2.2) yields the following:𝑑4πœ™π‘‘π‘¦4βˆ’2πœ‰2𝑑2πœ™π‘‘π‘¦2+πœ‰4π‘™πœ™=0,2𝑑4πœ“π‘‘π‘¦4βˆ’ξ€·2πœ‰2𝑙2𝑑+12πœ“π‘‘π‘¦2+πœ‰2ξ€·πœ‰2𝑙2ξ€Έ+1πœ“=0.(4.3)

The solutions for (4.3) take the following forms for 𝑖=1 and 2:πœ™1=𝐴1cosh(πœ‰π‘¦)+𝐡1𝑦cosh(πœ‰π‘¦)+𝐢1sinh(πœ‰π‘¦)+𝐷1𝑦sinh(πœ‰π‘¦),πœ“1=𝐸1cosh(π‘˜π‘¦)+𝐹1cosh(πœ‰π‘¦)+𝐻1sinh(π‘˜π‘¦)+𝐼1sinh(πœ‰π‘¦),πœ™2=𝐴2+𝐡2π‘¦ξ€Έξ€·βˆ’||πœ‰||𝑦,expπœ“2=𝐸2ξ€·βˆ’||πœ‰||𝑦exp+𝐹2exp(βˆ’π‘˜π‘¦),(4.4) where 𝐴1, 𝐡1, 𝐢1, 𝐷1, 𝐸1, 𝐹1, 𝐻1, 𝐼1, 𝐴2, 𝐡2, 𝐸2, and 𝐹2 are unknown coefficients, and π‘˜ is given by:ξƒŽπ‘˜=ξ€·πœ‰2𝑙2ξ€Έ+1𝑙2.(4.5)

Using the Fourier transformed expressions of (2.3), the coefficients 𝐼1,𝐹1, and 𝐸2 can be represented by the coefficients 𝐷1,𝐡1, and 𝐡2 as follows:𝐼1=βˆ’4𝑖(1βˆ’πœˆ)𝑙2πœ‰π·1,𝐹1=βˆ’4𝑖(1βˆ’πœˆ)𝑙2πœ‰π΅1,𝐸2=βˆ’4𝑖(1βˆ’πœˆ)𝑙2πœ‰π΅2.(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 βˆ’βˆž<π‘₯<+∞, unknowns 𝐢1, 𝐷1, and 𝐸1 are given as follows:𝐢1=𝐴1𝑐11+𝐡1𝑐12+𝑖𝐻1𝑐13,𝐷1=𝐴1𝑐21+𝐡1𝑐22+𝑖𝐻1𝑐23,𝑖𝐸1=𝐴1𝑐31+𝐡1𝑐32+𝑖𝐻1𝑐33,(4.7) where the expressions of the known functions 𝑐𝑖𝑗(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,𝐡1, and 𝐻1. Thus, the displacements 2𝐺𝑒01 and 2𝐺𝜈01 at 𝑦=0 and the rotation 4πΊπœ”0𝑧1 at 𝑦=0 can be expressed in terms of three unknown coefficients, that are, 𝐴1,𝐡1, and 𝐻1. In contrast, coefficients 𝐴1,𝐡1, and 𝐻1 can be expressed as 2𝐺𝑒01,2𝐺𝜈01, and 4πΊπœ”0𝑧1, and then the stresses, the displacements, and the rotation can be expressed in terms of 2𝐺𝑒01,2𝐺𝑣01, and 4πΊπœ”0𝑧1 in the Fourier domain. For examples, 𝜏0𝑦𝑦1,𝜏0𝑦π‘₯1, and πœ‡0𝑦1 have the following forms:𝜏0𝑦𝑦1=ξ‚€βˆ’π‘–π‘’01ξ‚π‘˜11(πœ‰)+𝜈01π‘˜12ξ‚€(πœ‰)+βˆ’π‘–πœ”0𝑧1ξ‚π‘˜13(πœ‰),𝜏0𝑦π‘₯1=ξ‚€βˆ’π‘–π‘’01ξ‚π‘–π‘˜14(πœ‰)+𝜈01π‘–π‘˜15ξ‚€(πœ‰)+βˆ’π‘–πœ”0𝑧1ξ‚π‘–π‘˜16(πœ‰),πœ‡0𝑦1=ξ‚€βˆ’π‘–π‘’01ξ‚π‘–π‘˜17(πœ‰)+𝜈01π‘–π‘˜18ξ‚€(πœ‰)+βˆ’π‘–πœ”0𝑧1ξ‚π‘–π‘˜19(πœ‰),(4.8) where the expressions of the known functions π‘˜1𝑖(πœ‰)(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,𝐡2, and 𝐹2. Thus, the displacements 2𝐺𝑒02 and 2𝐺𝜈02 at 𝑦=0 and the rotation 4πΊπœ”0𝑧2 at 𝑦=0 can be described by three unknown coefficients 𝐴2,𝐡2, and 𝐹2. In a similar manner to the case for layer (1), the unknown coefficients 𝐴2,𝐡2, and 𝐹2 are represented by 2𝐺𝑒02,2𝐺𝜈02, and 4πΊπœ”0𝑧2. Then, the stresses, the displacements, and the rotation can be expressed in terms of 2𝐺𝑒02,2𝐺𝜈02, and 4πΊπœ”0𝑧2 in the Fourier domain. For examples, 𝜏0𝑦𝑦2,𝜏0𝑦π‘₯2, and πœ‡0𝑦2 have following forms:𝜏0𝑦𝑦2=ξ‚€βˆ’π‘–π‘’02ξ‚π‘˜21(πœ‰)+𝜈02π‘˜22ξ‚€(πœ‰)+βˆ’π‘–πœ”0𝑧2ξ‚π‘˜23(πœ‰),𝜏0𝑦π‘₯2=ξ‚€βˆ’π‘–π‘’02ξ‚π‘–π‘˜24(πœ‰)+𝜈02π‘–π‘˜25ξ‚€(πœ‰)+βˆ’π‘–πœ”0𝑧2ξ‚π‘–π‘˜26(πœ‰),πœ‡0𝑦2=ξ‚€βˆ’π‘–π‘’02ξ‚π‘–π‘˜27(πœ‰)+𝜈02π‘–π‘˜28ξ‚€(πœ‰)+βˆ’π‘–πœ”0𝑧2ξ‚π‘–π‘˜29(πœ‰),(4.9) where the expressions of the known functions π‘˜2𝑖(πœ‰)(𝑖=1,2,3,…,9) are omitted.

Using (3.7), (3.8) and (3.9) the following relations are obtained:ξ‚€βˆ’π‘–π‘’01ξ‚π‘˜11(πœ‰)+𝜈01π‘˜12ξ‚€(πœ‰)+βˆ’π‘–πœ”0𝑧1ξ‚π‘˜13=ξ‚€(πœ‰)βˆ’π‘–π‘’02ξ‚π‘˜21(πœ‰)+𝜈02π‘˜22ξ‚€(πœ‰)+βˆ’π‘–πœ”0𝑧2ξ‚π‘˜23ξ‚€(πœ‰),βˆ’π‘–π‘’01ξ‚π‘–π‘˜14(πœ‰)+𝜈01π‘–π‘˜15ξ‚€(πœ‰)+βˆ’π‘–πœ”0𝑧1ξ‚π‘–π‘˜16=ξ‚€(πœ‰)βˆ’π‘–π‘’02ξ‚π‘–π‘˜24(πœ‰)+𝜈02π‘–π‘˜25ξ‚€(πœ‰)+βˆ’π‘–πœ”0𝑧2ξ‚π‘–π‘˜26ξ‚€(πœ‰),βˆ’π‘–π‘’01ξ‚π‘–π‘˜17(πœ‰)+𝜈01π‘–π‘˜18ξ‚€(πœ‰)+βˆ’π‘–πœ”0𝑧1ξ‚π‘–π‘˜19=ξ‚€(πœ‰)βˆ’π‘–π‘’02ξ‚π‘–π‘˜27(πœ‰)+𝜈02π‘–π‘˜28ξ‚€(πœ‰)+βˆ’π‘–πœ”0𝑧2ξ‚π‘–π‘˜29(πœ‰).(4.10) Equation (4.10) can be solved for (βˆ’π‘–π‘’01),𝜈01 and (βˆ’π‘–πœ”0𝑧1) as follows:ξ‚€βˆ’π‘–π‘’01=ξ‚€βˆ’π‘–π‘’02𝑙1(πœ‰)+𝜈02𝑙2ξ‚€(πœ‰)+βˆ’π‘–πœ”0𝑧2𝑙3(πœ‰),𝜈01=ξ‚€βˆ’π‘–π‘’02𝑙4(πœ‰)+𝜈02𝑙5ξ‚€(πœ‰)+βˆ’π‘–πœ”0𝑧2𝑙6ξ‚€(πœ‰),βˆ’π‘–πœ”01=ξ‚€βˆ’π‘–π‘’02𝑙7(πœ‰)+𝜈02𝑙8ξ‚€(πœ‰)+βˆ’π‘–πœ”0𝑧2𝑙9(πœ‰),(4.11) where the expressions of the known functions 𝑙𝑖(πœ‰)(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œ‹ξ€·πœˆ01βˆ’πœˆ02ξ€Έ=βˆžξ“π‘›=1𝑐𝑛(cos2π‘›βˆ’1)sinβˆ’1ξ‚€π‘₯π‘Žπœ‹ξ€·π‘’ξ‚ξ‚„for|π‘₯|β‰€π‘Ž=0forπ‘Žβ‰€|π‘₯|β‰€βˆž,(4.12)01βˆ’π‘’02ξ€Έ=βˆžξ“π‘›=1𝑑𝑛sin2𝑛sinβˆ’1ξ‚€π‘₯π‘Žπœ‹ξ€·πœ”ξ‚ξ‚„for|π‘₯|β‰€π‘Ž=0forπ‘Žβ‰€|π‘₯|β‰€βˆž,(4.13)0𝑧1βˆ’πœ”0𝑧2ξ€Έ=βˆžξ“π‘›=1𝑒𝑛sin2𝑛sinβˆ’1ξ‚€π‘₯π‘Žξ‚ξ‚„for|π‘₯|β‰€π‘Ž=0forπ‘Žβ‰€|π‘₯|β‰€βˆž,(4.14) where 𝑐𝑛, 𝑑𝑛, and 𝑒𝑛 are the unknown coefficients that are to be determined. The Fourier transforms of (4.12) through (4.14) areξ‚€πœˆ01βˆ’πœˆ02=βˆžξ“π‘›=1𝑐𝑛(2π‘›βˆ’1)πœ‰π½2π‘›βˆ’1(ξ‚€π‘Žπœ‰)𝑒01βˆ’π‘’02=π‘–βˆžξ“π‘›=1𝑑𝑛2π‘›πœ‰π½2𝑛(π‘Žπœ‰)πœ”0𝑧1βˆ’πœ”0𝑧2=π‘–βˆžξ“π‘›=1𝑒𝑛2π‘›πœ‰π½2𝑛(π‘Žπœ‰)(4.15) where 𝐽𝑛(πœ‰) is the Bessel function.

Substituting (4.11) into (4.15), we obtain the following equations: ξ‚€βˆ’π‘–π‘’02𝑙4(πœ‰)+𝜈02𝑙5(ξ€»+ξ‚€πœ‰)βˆ’1βˆ’π‘–πœ”0𝑧2𝑙6(πœ‰)=βˆžξ“π‘›=1𝑐𝑛(2π‘›βˆ’1)πœ‰π½2π‘›βˆ’1(ξ‚€π‘Žπœ‰),βˆ’π‘–π‘’02𝑙1ξ€»+(πœ‰)βˆ’1𝜈02𝑙2ξ‚€(πœ‰)+βˆ’π‘–πœ”0𝑧2𝑙3(πœ‰)=βˆžξ“π‘›=1𝑑𝑛2π‘›πœ‰π½2𝑛(π‘Žπœ‰),βˆ’π‘–π‘’02𝑙7(πœ‰)+𝜈02𝑙8ξ‚€(πœ‰)+βˆ’π‘–πœ”0𝑧2𝑙9ξ€»=(πœ‰)βˆ’1βˆžξ“π‘›=1𝑒𝑛2π‘›πœ‰π½2𝑛(π‘Žπœ‰).(4.16) Equations (4.16) can be solved for (βˆ’π‘–π‘’02),𝜈02 and (βˆ’π‘–πœ”0𝑧2) as follows:ξ‚€βˆ’π‘–π‘’02=βˆžξ“π‘›=1π‘π‘›π‘š1(πœ‰)(2π‘›βˆ’1)πœ‰π½2π‘›βˆ’1(π‘Žπœ‰)+βˆžξ“π‘›=1π‘‘π‘›π‘š2(πœ‰)Γ—2π‘›πœ‰π½2𝑛(+π‘Žπœ‰)βˆžξ“π‘›=1π‘’π‘›π‘š3(πœ‰)Γ—2π‘›πœ‰π½2𝑛(π‘Žπœ‰),𝜈02=βˆžξ“π‘›=1π‘π‘›π‘š4(πœ‰)(2π‘›βˆ’1)πœ‰π½2π‘›βˆ’1(π‘Žπœ‰)+βˆžξ“π‘›=1π‘‘π‘›π‘š5(πœ‰)Γ—2π‘›πœ‰π½2𝑛+(π‘Žπœ‰)βˆžξ“π‘›=1π‘’π‘›π‘š6(πœ‰)Γ—2π‘›πœ‰π½2𝑛(ξ‚€π‘Žπœ‰),βˆ’π‘–πœ”0𝑧2=βˆžξ“π‘›=1π‘π‘›π‘š7(πœ‰)(2π‘›βˆ’1)πœ‰π½2π‘›βˆ’1(π‘Žπœ‰)+βˆžξ“π‘›=1π‘‘π‘›π‘š8(πœ‰)Γ—2π‘›πœ‰π½2𝑛+(π‘Žπœ‰)βˆžξ“π‘›=1π‘’π‘›π‘š9(πœ‰)Γ—2π‘›πœ‰π½2𝑛(π‘Žπœ‰),(4.17) where the expressions of the known functions π‘šπ‘–(πœ‰)(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, 𝑐𝑛, 𝑑𝑛, and 𝑒𝑛. For example, stresses 𝜏0𝑦𝑦2,𝜏0𝑦π‘₯2, and πœ‡0𝑦2 are expressed as follows:𝜏0𝑦𝑦2=βˆžξ“π‘›=1𝑐𝑛(2π‘›βˆ’1)πœ‹Γ—ξ€œβˆž0π‘ž1(πœ‰)πœ‰π½2π‘›βˆ’1(+π‘Žπœ‰)cos(πœ‰π‘₯)π‘‘πœ‰βˆžξ“π‘›=1𝑑𝑛2π‘›πœ‹Γ—ξ€œβˆž0π‘ž2(πœ‰)πœ‰π½2𝑛+(π‘Žπœ‰)cos(πœ‰π‘₯)π‘‘πœ‰βˆžξ“π‘›=1𝑒𝑛2π‘›πœ‹Γ—ξ€œβˆž0π‘ž3(πœ‰)πœ‰π½2π‘›πœ(π‘Žπœ‰)cos(πœ‰π‘₯)π‘‘πœ‰,(4.18)0𝑦π‘₯2=βˆžξ“π‘›=1𝑐𝑛(2π‘›βˆ’1)πœ‹Γ—ξ€œβˆž0π‘ž4(πœ‰)πœ‰π½2π‘›βˆ’1(+π‘Žπœ‰)sin(πœ‰π‘₯)π‘‘πœ‰βˆžξ“π‘›=1𝑑𝑛2π‘›πœ‹Γ—ξ€œβˆž0π‘ž5(πœ‰)πœ‰π½2𝑛+(π‘Žπœ‰)sin(πœ‰π‘₯)π‘‘πœ‰βˆžξ“π‘›=1𝑒𝑛2π‘›πœ‹Γ—ξ€œβˆž0π‘ž6(πœ‰)πœ‰π½2π‘›πœ‡(π‘Žπœ‰)sin(πœ‰π‘₯)π‘‘πœ‰,(4.19)0𝑦2=βˆžξ“π‘›=1𝑐𝑛(2π‘›βˆ’1)πœ‹Γ—ξ€œβˆž0π‘ž7(πœ‰)πœ‰π½2π‘›βˆ’1+(π‘Žπœ‰)sin(πœ‰π‘₯)π‘‘πœ‰βˆžξ“π‘›=1𝑑𝑛2π‘›πœ‹Γ—ξ€œβˆž0π‘ž8(πœ‰)πœ‰π½2𝑛+(π‘Žπœ‰)sin(πœ‰π‘₯)π‘‘πœ‰βˆžξ“π‘›=1𝑒𝑛2π‘›πœ‹Γ—ξ€œβˆž0π‘ž9(πœ‰)πœ‰π½2𝑛(π‘Žπœ‰)sin(πœ‰π‘₯)π‘‘πœ‰,(4.20) where the known expressions π‘ž1(πœ‰),π‘ž2(πœ‰),…,π‘ž8(πœ‰), and π‘ž9(πœ‰) are omitted. Functions π‘žπ‘–(πœ‰)/πœ‰(𝑖=2,3,4,6,7,8) decrease rapidly as πœ‰ increases. Functions π‘žπ‘–(πœ‰)/πœ‰(𝑖=1,5,9) have the following property when πœ‰ increases:π‘žπ‘–(πœ‰)πœ‰βŸΆπ‘žπΏπ‘–,(4.21) where constants π‘žπΏπ‘–(𝑖=1,5,9) can be calculated asπ‘žπΏπ‘–=π‘žπ‘–ξ€·πœ‰πΏξ€Έπœ‰πΏ,(4.22) with πœ‰πΏ being a large value of πœ‰.

Finally, the remaining boundary conditions (3.1), (3.2), and (3.3) can be reduced to the following equations:βˆžξ“π‘›=1𝑐𝑛𝐾𝑛(π‘₯)+βˆžξ“π‘›=1𝑑𝑛𝐿𝑛(π‘₯)+βˆžξ“π‘›=1𝑒𝑛𝑀𝑛(π‘₯)=βˆ’π‘’(π‘₯),βˆžξ“π‘›=1𝑐𝑛𝑂𝑛(π‘₯)+βˆžξ“π‘›=1𝑑𝑛𝑃𝑛(π‘₯)+βˆžξ“π‘›=1𝑒𝑛𝑄𝑛(π‘₯)=βˆ’πœˆ(π‘₯),βˆžξ“π‘›=1𝑐𝑛𝑅𝑛(π‘₯)+βˆžξ“π‘›=1𝑑𝑛𝑆𝑛(π‘₯)+βˆžξ“π‘›=1𝑒𝑛𝑇𝑛(π‘₯)=βˆ’π‘€(π‘₯)for|π‘₯|β‰€π‘Ž,(4.23) with𝐾𝑛(π‘₯)=(2π‘›βˆ’1)πœ‹Γ—ξƒ―ξ€œβˆž0ξ‚Έπ‘ž1(πœ‰)πœ‰βˆ’π‘žπΏ1𝐽2π‘›βˆ’1+π‘ž(π‘Žπœ‰)cos(πœ‰π‘₯)π‘‘πœ‰πΏ1ξ€Ίcos(2π‘›βˆ’1)sinβˆ’1ξ€»(π‘₯/π‘Ž)ξ€·π‘Ž2βˆ’π‘₯2ξ€Έ1/2ξƒ°,𝐿𝑛(π‘₯)=2π‘›πœ‹ξ€œβˆž0π‘ž2(πœ‰)πœ‰π½2𝑛𝑀(π‘Žπœ‰)cos(πœ‰π‘₯)π‘‘πœ‰,𝑛(π‘₯)=2π‘›πœ‹ξ€œβˆž0π‘ž3(πœ‰)πœ‰π½2𝑛𝑂(π‘Žπœ‰)cos(πœ‰π‘₯)π‘‘πœ‰,𝑛(π‘₯)=(2π‘›βˆ’1)πœ‹ξ€œβˆž0π‘ž4(πœ‰)πœ‰π½2π‘›βˆ’1𝑃(π‘Žπœ‰)sin(πœ‰π‘₯)π‘‘πœ‰,𝑛(π‘₯)=2π‘›πœ‹Γ—ξƒ―ξ€œβˆž0ξ‚Έπ‘ž5(πœ‰)πœ‰βˆ’π‘žπΏ5𝐽2𝑛+π‘ž(π‘Žπœ‰)sin(πœ‰π‘₯)π‘‘πœ‰πΏ5ξ€Ίsin2𝑛sinβˆ’1ξ€»(π‘₯/π‘Ž)ξ€·π‘Ž2βˆ’π‘₯2ξ€Έ1/2ξƒ°,𝑄𝑛(π‘₯)=2π‘›πœ‹ξ€œβˆž0π‘ž6(πœ‰)πœ‰π½2𝑛𝑅(π‘Žπœ‰)sin(πœ‰π‘₯)π‘‘πœ‰,𝑛(π‘₯)=(2π‘›βˆ’1)πœ‹ξ€œβˆž0π‘ž7(πœ‰)πœ‰π½2π‘›βˆ’1𝑆(π‘Žπœ‰)sin(πœ‰π‘₯)π‘‘πœ‰,𝑛(π‘₯)=2π‘›πœ‹ξ€œβˆž0π‘ž8(πœ‰)πœ‰π½2𝑛𝑇(π‘Žπœ‰)sin(πœ‰π‘₯)π‘‘πœ‰,𝑛(π‘₯)=2π‘›πœ‹Γ—ξƒ―ξ€œβˆž0ξ‚Έπ‘ž9(πœ‰)πœ‰βˆ’π‘žπΏ9𝐽2𝑛+π‘ž(π‘Žπœ‰)sin(πœ‰π‘₯)π‘‘πœ‰πΏ9ξ€Ίsin2𝑛sinβˆ’1ξ€»(π‘₯/π‘Ž)ξ€·π‘Ž2βˆ’π‘₯2ξ€Έ1/2ξƒ°,(4.24)𝑒(π‘₯)=𝑝,𝜈(π‘₯)=0,𝑀(π‘₯)=0.(4.25) The unknown coefficients 𝑐𝑛, 𝑑𝑛, and𝑒𝑛 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 relationsξ€œβˆž0𝐽𝑛[]=(π‘Žπœ‰)cos(πœ‰π‘₯),sin(πœ‰π‘₯)π‘‘πœ‰βˆ’π‘Žπ‘›ξ€·π‘₯2βˆ’π‘Ž2ξ€Έβˆ’1/2π‘₯π‘₯+2βˆ’π‘Ž2ξ€Έβˆ’1/2ξ‚„βˆ’π‘›ξ‚€sinπ‘›πœ‹2,π‘Žπ‘›ξ€·π‘₯2βˆ’π‘Ž2ξ€Έβˆ’1/2π‘₯π‘₯+2βˆ’π‘Ž2ξ€Έβˆ’1/2ξ‚„βˆ’π‘›ξ‚€cosπ‘›πœ‹2,forπ‘Ž<π‘₯,(5.1) the stress intensity factors and the couple-stress intensity factor can be determined as follows:𝐾𝐼=[]2πœ‹(π‘₯βˆ’π‘Ž)1/2𝜏0𝑦𝑦2||π‘₯β†’π‘Ž+=βˆžξ“π‘›=1𝑐𝑛(2π‘›βˆ’1)(βˆ’1)π‘›π‘žπΏ1(πœ‹π‘Ž)1/2,𝐾𝐼𝐼=[]2πœ‹(π‘₯βˆ’π‘Ž)1/2𝜏0𝑦π‘₯2||π‘₯β†’π‘Ž+=βˆžξ“π‘›=1𝑑𝑛(2𝑛)(βˆ’1)π‘›π‘žπΏ5(πœ‹π‘Ž)1/2,𝑀0=[]2πœ‹(π‘₯βˆ’π‘Ž)1/2πœ‡0𝑦2||π‘₯β†’π‘Ž+=βˆžξ“π‘›=1𝑒𝑛(2𝑛)(βˆ’1)π‘›π‘žπΏ9(πœ‹π‘Ž)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 10βˆ’5500 to 10+5500. The stress intensity factors and the couple-stress intensity factor are calculated for β„Ž/π‘Ž=5.0,0.5, and 0.1 with a Poisson’s ratio of 𝜈=0.25.

The values of the functions π‘žπ‘–(πœ‰π‘Ž)/(πœ‰π‘Ž)(𝑖=2,3,4,6,7,8) are verified to decay rapidly as (πœ‰π‘Ž) increases, and the values of the functions π‘žπ‘–(πœ‰π‘Ž)/(πœ‰π‘Ž)(𝑖=1,5,9)are verified to rapidly approach constants π‘žπΏπ‘–(𝑖=1,5,9) as (πœ‰π‘Ž) 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 𝑐𝑛,𝑑𝑛, and 𝑒𝑛 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 𝑐𝑛, 𝑑𝑛, and 𝑒𝑛 are verified to be successfully determined by the Schmidt method.

The values of the Mode I stress intensity 𝐾𝐼√/(π‘πœ‹π‘Ž) are shown for β„Ž/π‘Ž=5.0 in Table 1, in which the values in parentheses are obtained from the diagram in [4]. For β„Ž/π‘Ž=5.0, the distance between two cracks is 10π‘Ž, 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 𝐾𝐼,𝐾𝐼𝐼, and M0 are plotted with respect to 𝑙/π‘Ž for β„Ž/π‘Ž=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/π‘Ž=0.0.

tab1
Table 1: Values of 𝐾𝐼√/(π‘πœ‹π‘Ž) for β„Ž/π‘Ž=5.0. (Values in parentheses were obtained from the diagram in [4]).
575891.fig.002
Figure 2: Values of 𝐾𝐼, 𝐾𝐼𝐼, and 𝑀0 plotted with respect to 𝑙/π‘Ž for β„Ž/π‘Ž=5.0 (broken lines show the values for 𝑙/π‘Ž=0.0).
575891.fig.003
Figure 3: Values of 𝐾𝐼,𝐾𝐼𝐼, and 𝑀0 plotted with respect to 𝑙/π‘Ž for β„Ž/π‘Ž=0.5 (broken lines show the values for 𝑙/π‘Ž=0.0).
575891.fig.004
Figure 4: Values of 𝐾𝐼,𝐾𝐼𝐼, and 𝑀0 plotted with respect to 𝑙/π‘Ž for β„Ž/π‘Ž=0.1 (broken lines show the values for 𝑙/π‘Ž=0.0).

7. Conclusion

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

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