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A general unified treatment of lamellar inhomogeneities
Engineering Fracture Mechanics 74 (2007) 1499–1510 www.elsevier.com/locate/engfracmech
A general unified treatment of lamellar inhomogeneities Hossein M. Shodja
a,b,*
, Farzaneh Ojaghnezhad
a
a
b
Department of Civil Engineering, Center of Excellence in Structures and Earthquake Engineering, Sharif University of Technology, P.O. Box 11365-9313, Tehran, Iran Institute for Nanoscience and Nanotechnology, Sharif University of Technology, P.O. Box 11365-9161, Tehran, Iran Received 17 December 2005; received in revised form 3 August 2006; accepted 7 August 2006 Available online 27 October 2006
Abstract Consider a lamellar inhomogeneity embedded in an unbounded isotropic elastic medium. When the elastic moduli of the lamellar inhomogeneity are zero it is a crack, if its elastic moduli are infinite it is an anticrack, and when its elastic moduli are finite it is called a quasicrack. Based on the Eshelby’s equivalent inclusion method (EIM), the present paper develops a unified approach for determination of the exact closed-form expressions for modes I, II, and III stress intensity factors (SIFs) at the tips of lamellar inhomogeneities under a remote applied polynomial loading. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Lamellar inhomogeneity; Anticrack; Quasicrack; Equivalent inclusion method; Polynomial loadings
1. Introduction In the context of the present study, a lamellar inhomogeneity is deduced from an ellipsoidal inhomogeneity by letting one of its principal axes vanish. The aim of this paper is to determine modes I, II, and III stress intensity factors (SIFs) pertinent to a general class of lamellar shapes embedded in an infinite isotropic elastic medium under polynomial far-field applied loading in a unified manner. For a through finding on the SIFs of cracks in the literature, up to the year 2000, one should refer to the handbooks of Murakami [1] and Tada et al. [2]. A close scrutiny of the literature reveals that, except for a very few specific cases, the exact solution to the modes I, II, and III SIFs of three-dimensional penny shape and elliptic cracks under polynomial loading at infinity has not been obtained. Most of the closed-form solutions to the SIF pertinent to a penny shape crack in an infinite isotropic elastic body are devoted to at most linear far-field loading, [2–4]. The more general geometry of an elliptic crack under a uniform far-field tension was considered by Irwin [5] employing the stress function theory. Shibuya [6] also used stress function to study the elliptic crack under a linear far-field loading and evaluated only the mode I SIF within 1% accuracy [1]. As far
* Corresponding author. Address: Department of Civil Engineering, Center of Excellence in Structures and Earthquake Engineering, Sharif University of Technology, P.O. Box 11365-9313, Tehran, Iran. Tel.: +98 21 66164209; fax: +98 21 66014828. E-mail address: [email protected] (H.M. Shodja).
0013-7944/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2006.08.016
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as slit-like cracks are concerned Isida [7] has obtained the closed-form expression only for the mode I SIF at the tip of a two-dimensional crack under a polynomial loading at infinity by stress function method [1]. It is now more than three decades that scientists are concerned with the fiber–matrix force transfer in fiber reinforced composite materials. A well taken approach is to model the long thin fibers by line (lamellar) inhomogeneities [8–14], such that the fiber cross sectional area vanishes. In applying the proposed approach to real composites, if the microgeometries of the reinforcements are such that they can be approximated as limiting cases of an ellipsoid, reasonable estimation by the present method is expected. For example, often a short fiber can be represented by a prolate ellipsoid having the same aspect ratio [15]. Similarly, a long fiber can be modeled by an elliptic cylinder. Rigid inhomogeneities have important applications in materials science. Nowadays, the excellent technological applications of carbon nanofiber reinforced polymer composites have attracted the attentions of industry and numerous scientists. These composites are advantageous for their high tensile modulus, strength, and promising electrical and thermal properties. Vapor grown carbon nanofiber (VGCF) which is a new class of carbon fiber can be fabricated at high quantities and low cost. The fiber may have a diameter of about 150 nm and length of 10–20 lm [16]. The modulus of carbon nanofiber is normally in the range of 100–600 GPa and sometimes even higher, whereas the modulus of some polymers is usually 2–5 GPa [17]. Thus, for certain purposes the carbon nanofibers in the carbon nanofiber reinforced polymer composites may be considered as rigid fibers or anticracks in the context of the present study. A brief examination of typical properties of fibers and matrices which are now extensively used in composite materials reveals that, many types of fibers such as SiC, Al2O3, high modulus (HM) carbon, etc. (in the forms of platelets, whiskers, powder, etc.) have high stiffness in comparison to thermosets and thermoplastics matrices such as epoxy resins, polyesters, and polypropylene. Elastic deformation of these composites with different fiber architecture (long-fiber, short-fiber, ribbon-like fiber, etc.) can be drawn assuming the linear-elastic stress–strain behavior [15]. Moreover, non-metallic and very hard embrittlements in some metallic materials and alloys can be idealized as rigid imperfections relative to the matrix and linear-elastic analysis has proved to be useful in such situations, Hasebe et al. [18,19]. Same authors have considered an anticrack as a model for a thin rigid plate embedded in an elastic body. Atkinson [8], in an effort to determine the elastic fields of a metallic strain measuring device embedded in a rubber matrix, proposed ribbon-like inhomogeneity model. In this, Atkinson examines both cases of a rigid ribbon inhomogeneity (anticrack) and an elastic ribbon inhomogeneity (quasicrack). A considerable amount of literature is devoted to the two-dimensional problem of anticrack embedded in an isotropic elastic materials under uniform far-field loading, Hasebe et al. [18,19], Dundurs and Markenscoff [20], Markenscoff and Ni [21], Hurtado et al. [22], and Homentcovschi and Dascalu [23]. However, to the best of the authors’ knowledge, the three-dimensional cases of penny shape and elliptic anticracks under far-field polynomial loading have not been addressed in the literature. Also, to date, very little attention has been paid to the determination of the stress field of quasicracks. The existing theories devoted to this topic are tailored for the simpler two-dimensional and uniform loading conditions and have very limited ranges of applicability. For example, Hurtado et al. [22] who address quasicracks, consider the in-plane and out-of-plane cases under a uniform farfield loading. Depending on the dimension of the problem, loading condition, and the type of lamellar inhomogeneity: crack, quasicrack, or anticrack, various types of boundary value problems are encountered and a unified approach using the existing mathematical treatments is not possible, so that researchers have employed different methods. For the cases of in-plane and out-of-plane strain, and under a uniform applied far-field loading, Hurtado et al. [22] show that the lamellar inhomogeneity can equivalently be replaced by a suitable distribution of dislocations, which can be obtained by using the concept of surface dislocation density. Also, Homentcovschi and Dascalu [23] use the Muskhelishvili’s complex potentials to study the two-dimensional problems of lamellar inhomogeneity in unbounded isotropic elastic materials. The deficiency of the two latter approaches is the difficulty in their extension, not only to three-dimensional lamellar inhomogeneity, but also for remote polynomial loading. On the other hand, the beauty of the Eshelby’s [24–26] result is that there is an exact correspondence between the far-field loading and the form of the distribution of homogenizing eigenstrain inside the equivalent inclusion. This powerful theory implies many dramatic results. In the context of the present study, as one of its abundant applications, it enables us to find the exact SIFs for the two- and three-dimensional lamellar inhomogeneities due to a far-field applied polynomial loading.
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2. Formulation an elastic medium X with elastic moduli C ijkl embedded in an infinite isotropic elastic matrix P Consider 3 ¼ R X with elastic moduli C ijkl . If the sub-domain X is made of the same material as the matrix, C ijkl ¼ C ijkl , and there is a prescribed eigenstrain in X, it is referred to as an inclusion. If C ijkl 6¼ C ijkl , and X does not contain any eigenstrain, X is called an inhomogeneity. Interesting results are inferred when X is an ellipsoidal inhomogeneity Fig. 1. The fact that a single ellipsoidal inhomogeneity embedded in an unbounded matrix under the remote applied loading oij ðxÞðroij ðxÞÞ can equivalently be substituted with an inclusion having properly chosen eigenstrains was first pointed out by Eshelby [24–26]. This method is called the equivalent inclusion method (EIM) and requires that the stress field of the ellipsoidal inhomogeneity X to be equivalent to the stress field of the equivalent inclusion. This equivalency gives rise to the following consistency equation C ijkl ½dkl ðxÞ þ okl ðxÞ ¼ C ijkl ½dkl ðxÞ þ okl ðxÞ kl ðxÞ;
x 2 X;
ð1Þ
where ij ðxÞ are the homogenizing eigenstrains to be determined from the consistency equation (1) and dij ðxÞ are the disturbance strains caused by the presence of the inhomogeneity or equivalently due to the homogenizing eigenstrains Z dij ðxÞ ¼ Cijkl ðx yÞkl ðyÞ dy; X
1 Cijkl ðx yÞ ¼ C mnkl ½Gim;nj ðx yÞ þ Gjm;ni ðx yÞ; 2
ð2Þ
where Gij(x y) is the associated Green’s function. For uniform far-field loading, Eshelby [24] has shown that the eigenstrain field is uniform inside X, and subsequently the disturbance strain is also uniform in X. Generally, when the far-field applied loading is a polynomial of degree n P 0, the homogenizing eigenstrain inside the ellipsoidal inhomogeneity is also a polynomial of degree n P 0. If the eigenstrain field in X is a homogeneous polynomial of degree n = 1, the disturbance strain for the points inside the ellipsoid X will also be a homogeneous polynomial of degree 1. If the eigenstrain is a homogeneous polynomial of degree n P 2, the associated disturbance strain field in X is an inhomogeneous polynomial in x whose terms are of degree n, (n 2), (n 4), . . . , Eshelby [26]. When the loading and subsequently the eigenstrain field is a polynomial of order n, there will be (n + 1)(n + 2)(n + 3) unknown coefficients in the expression of the eigenstrain field. Thus, for exact determination of the unknowns and the elastic fields for both interior and exterior points, it is necessary to write the consistency equation at (n + 1)(n + 2)(n + 3)/6 points in X. Suppose ap, p = 1, 2, 3 are the principal half axes of the ellipsoidal inhomogeneity ( ) 3 X x2p 3 X ¼ xjx 2 R ; 61 : a2p i¼1 A desired lamellar inhomogeneity is obtained by letting the corresponding principal half axis approach zero. In the process of deducing the eigenstrain field of the lamellar inhomogeneity by letting the desired dimension
Fig. 1. The ellipsoidal inhomogeneity, X embedded in an infinite isotropic elastic medium, R ¼ R3 X.
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approach zero, ap ! 0, some components of the homogenizing eigenstrains become infinite, however, on limap !0 ap ij ¼ finite. The associated closed-form expression for the SIF, Kij corresponding to a point x , such that the edge of X is obtained by finding the stress field at a point x 2 R in the vicinity of the point x j¼ r ! 0 jxx pffiffiffiffiffiffiffi ð3Þ K ij ¼ lim lim 2prrij ðrÞ; r!0 ap !0
where rij(r) is the stress field at the exterior point x due to the presence of the lamellar inhomogeneity. l, E, and m denote the Lame´ constants for the matrix. The Lame´ constants for the inhomogeneity are differentiated by using the superscript ‘‘*’’. 3. Results Throughout this section, a general class of lamellar inhomogeneities surrounded by an infinite body under a polynomial loading at infinity will be considered. For illustration, suppose that the remote applied stress is roij ðxÞ ¼ ruij þ rlij x1 þ rqij x21 þ ;
ð4Þ
where ruij , rlij and rqij are constants appearing in the uniform, linear, and quadratic terms, respectively. The results pertinent to slit-like, penny shape, and elliptic lamellar inhomogeneities are presented in Sections 3.1, 3.2 and 3.3, respectively. In Sections 3.1 and 3.2, the corresponding lamellar inhomogeneities, i.e. crack, anticrack, and quasicrack for all three modes of deformation are addressed. In Section 3.3, two special cases of elliptic crack and anticrack under the mode I homogeneous quadratic far-field loading are considered and other cases are eliminated for brevity. All numerical results presented in this paper are pertinent to m = m* = 0.3. 3.1. Slit-like lamellar inhomogeneity This section is concerned with the lamellar inhomogeneity whose shape is the infinitely extended elliptic cylinder with one of its principal axes approaching zero. Thus, the geometry of interest is obtained from Fig. 1 by letting a2 ! 0 and a3 ! 1. 3.1.1. Cracks Assume that the lamellar inhomogeneity X is a crack, i.e. C ijkl ! 0. If the far-field applied stress for modes I, II, and III is explained by (4), employment of the proposed approach and the definition of the SIF (3) yields pffiffiffiffiffiffiffi 1 1 pa1 ; K ij ¼ ruij rlij a1 þ rqij a21 þ 2 2 where ± denotes the right and left crack tips, respectively. Isida [7] reports the identical result for the mode I SIF of a slit-like crack under the remote polynomial loading. It is worthwhile to note that, even though r22(r), r21(r), and r23(r) have different mathematical expressions, lima2 !0 r22 ðrÞ ¼ lima2 !0 r21 ðrÞ ¼ lima2 !0 r23 ðrÞ. 3.1.2. Anticracks When the lamellar inhomogeneity is rigid, that is C ijkl ! 1, it is called an anticrack. This interesting physical problem can also be treated by the EIM, even for polynomial far-field loadings. Employing the proposed approach, the elastic field of an anticrack subjected to a polynomial far-field loading is evaluated and it is pffiffi observed that mode I anticracks, similar to cracks exhibit 1= r singularity and so the SIF of an anticrack is obtained via (3). In contrast to the crack problem, for an anticrack the normal stress applied parallel to the anticrack plane also contributes to the mode I SIF. For illustration, the problem of an anticrack subjected to both polynomial uniaxial loading and uniform biaxial loading will be considered in this section. The homogenizing eigenstrain fields associated with these types of loadings are briefly mentioned in Appendix A. It can be shown that the mode I SIF of an anticrack under the remote uniaxial applied loading of the form (4) is
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K 22
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pffiffiffiffiffiffiffi 1 2m 1 l 1 q 2 u mr22 mr22 a1 þ mr22 a1 þ ¼ pa1 : 3 4m 2 2
It should be emphasized that for anticracks, the mode I SIF depends upon the Poisson’s ratio of the matrix, and becomes equal to zero when the matrix is incompressible. Now, if the far-field applied stress is biaxial and uniform, ro11 ðxÞ ¼ ru11 and ro22 ðxÞ ¼ ru22 , the SIF for mode I will be pffiffiffiffiffiffiffi 1 2m u mr22 ð1 mÞru11 pa1 : K 22 ¼ 3 4m This result is also validated by the solution of Hurtado et al. [22] who considered only the uniform biaxial applied stress at infinity. The above relation reveals that ru11 also contributes to the mode I SIF through the Poisson’s ratio effect. Astonishingly, by examining the exact closed-form expression of the stress field, it can be inferred that rigid lamellar inhomogeneities do not disturb the modes II and III polynomial far-field loadings, and so the associated SIFs are zero. 3.1.3. Quasicracks This section considers lamellar inhomogeneities whose elastic moduli are finite, C ijkl ¼ finite referred to as quasicracks. It is noteworthy to mention that, cracks and anticracks with C ijkl ¼ 0 and 1, respectively, are the limiting cases of quasicracks. It is evidently observed that all the components of the homogenizing eigenstrain field for quasicrack are finite when a2 ! 0, thus unlike crack and anticrack, for which lima2 !0 a2 ij 6¼ 0, for the quasicrack lima2 !0 a2 ij ¼ 0. It implies that the SIF vanishes for quasicrack except for the limiting cases of K = l*/l = 0 and K ! 1. To study the mode I quasicrack, let jðrÞ ¼ r22 ðrÞ=ro22 ða1 ; 0; 0Þ denote the normalized stress outside the inhomogeneity on the x1-axis. For the sake of demonstration and comparison, variation of j(r) just outside the inhomogeneity with t = a2/a1 is plotted for different values of K under uniaxial uniform and quadratic remote loadings in Fig. 2(a) and (b), respectively.
Fig. 2. Variation of limr!0j(r) with t for the slit-like quasicrack, under (a) uniform far-field loading, ro22 ðxÞ ¼ ru22 and (b) quadratic far-field loading, ro22 ðxÞ ¼ rq22 x21 .
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From Fig. 2(a) for the special case of the uniform tensile stress disturbed by a circular hole, t = 1 and K = 0, the stress concentration factor equal to 3 prevails. By examination of the closed-form expressions of the stress field, it is proved that under uniform far-field loading limr!0j depends upon the aspect ratio t and is independent of a1, while for the homogeneous quadratic loading, it varies with a21 . As depicted in Fig. 2(a) and (b), limr!0j is negative for K > 1. Fig. 3(a) and (b) display the variation of j with r/a1, for K ! 1 and different values of t, under the remote uniform and quadratic loadings, respectively. Fig. 3(a) and (b) reveal that, if t is absolutely zero, j is positive everywhere for all values of r/a1, and for any other values of t > 0, there is a corresponding interval (0, r/a1) for which j is negative. The mode II quasicrack, for brevity, is studied by applying only a linear far-field stress, ro21 ðxÞ ¼ rl21 x1 . The ratio of the mode II SIF of quasicrack to that of crack is given by KQ r21 ðrÞ 2ðK 1Þa21 a2 21 ¼ ¼ lim : C 3 K 21 r!0 lim r21 ðrÞ Kðm 1Þða1 þ 3a1 a22 þ a32 Þ þ ðKð3m 1Þ 2Þa21 a2 K!0
Other cases of loadings can be treated in a similar manner. Note that by considering K = 0 and K ! 1, the C special cases of crack and anticrack for which K Q 21 =K 21 ¼ 1; 0 are recovered. Similarly, for the mode III quasicrack, only linear far-field loading is considered, ro23 ðxÞ ¼ rl23 x1 . In the same manner, for this mode the ratio of the SIFs simplifies to KQ r23 ðrÞ ð1 KÞa2 ð2a1 þ a2 Þ 23 ¼ ¼ lim : Ka21 þ 2a1 a2 þ a22 K C23 r!0 lim r23 ðrÞ K!0
The results pertaining to the limiting cases are readily inferred.
Fig. 3. Distribution of the normalized stress component, j(r) for the slit-like quasicrack, under (a) uniform, ro22 ðxÞ ¼ ru22 and (b) quadratic, ro22 ðxÞ ¼ rq22 x21 far-field loadings.
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3.2. Penny shape lamellar inhomogeneity Throughout this section, the three-dimensional case of penny shape lamellar inhomogeneity constructed by letting a3 ! 0 and a1 = a2 = a, will be studied, Fig. 1. 3.2.1. Penny shape cracks To evaluate the mode I SIF, it is supposed that a penny shape crack is subjected to the far-field applied stress ro33 ðxÞ given by Eq. (4). Using the proposed approach, the closed-form expression of the mode I SIF is determined as below 2 pffiffiffi 2 l 4 q 2 u pa r33 þ r33 a cos h þ r33 a ð3 þ 2 cos 2hÞ þ ; K 33 ðhÞ ¼ p 3 33 where h is the angle measured counterclockwise from the positive direction of the x1-axis. In the literature, only the results up to the linear far-field loading is available, Tada et al. [2]. For a penny shape crack under a shear mode far-field loading, however, it is assumed that the only nonzero component of the far-field applied stress is ro31 ðxÞ defined in the form of Eq. (4). For this type of loading, except for the points at h = kp/2, k = 0, 1, . . . , all the other points along the edge of the crack undergo both modes II and III fracture. pffiffiffi 3 2 aru31 ð2 m þ m cos 2hÞ 8a2 rl31 cos hð2 m þ m cos 2hÞ pffiffiffi pffiffiffi K 31 ðhÞ ¼ þ pð2 mÞ 3 pð4 mÞ 5
4a2 rq31 fð16 4m þ m2 Þ cos 2h þ ð2 mÞ½12 m þ 2m cos 4hg pffiffiffi þ ; 3 pð44 28m þ 3m2 Þ pffiffiffi 3 2 2 amru31 sin 2h 16a2 mrl31 ðcos hÞ sin h pffiffiffi K 32 ðhÞ ¼ pffiffiffi þ pð2 mÞ 3 pð4 mÞ þ
5
þ
4a2 mrq31 ½3ð4 mÞ þ 4ð2 mÞ cos 2h sin 2h pffiffiffi þ : 3 pð44 28m þ 3m2 Þ
In the handbook of Tada et al. [2], only the results for the uniform far-field loading is available. 3.2.2. Penny shape anticracks Similar to the slit-like anticrack, in addition to the opening type of loading, the normal stress applied parallel to the plane of the penny shape anticrack at infinity contributes to the mode I SIF, so, for demonstration, the penny shape anticrack is considered to be subjected to both uniaxial opening and triaxial remote stress in this section. Let a penny shape anticrack be subjected to a uniaxial far-field loading of the form (4), then the mode I SIF becomes pffiffiffi 3 4 amð1 2mÞru33 16a2 mð1 2mÞ cos h½7 8m ð3 4mÞ cos 2h l pffiffiffi K 33 ðhÞ ¼ pffiffiffi þ r33 þ : 3 pð33 47m 32m2 þ 48m3 Þ pð3 m 4m2 Þ Now, it is assumed that the uniform and triaxial loading, ro11 ðxÞ ¼ ru11 , ro22 ðxÞ ¼ ru22 , and ro33 ðxÞ ¼ ru33 , is applied to the penny shape anticrack at infinity. Then, the SIF of mode I is obtained as below pffiffiffi pffiffiffi 4 amð1 2mÞru33 2 að1 2mÞð13 17mÞðru11 þ ru22 Þ K 33 ðhÞ ¼ pffiffiffi pffiffiffi : pð21 31m 20m2 þ 32m3 Þ pð3 m 4m2 Þ Note that, similar to the two-dimensional case, the mode I SIF for anticracks depends on the Poisson’s ratio of the matrix and is equal to zero when the matrix is incompressible, m = 1/2. Also, calculations of the stress field show that the shear mode SIFs for anticracks always vanish. 3.2.3. Penny shape quasicracks For brevity, the exact solution to the penny shape quasicrack subjected to a linear far-field applied stress is given here, the other cases of loadings can be treated in a similar manner.
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In the case of mode I quasicrack, it is assumed that j(r, h) is the normalized stress component r33 ðr; hÞ=ro33 ða; 0; 0Þ, where r is the normal distance from the edge of the penny shape quasicrack. For this problem, except for the extreme cases of crack and anticrack, no matter how small a3 becomes, limr!0j(r, h) is bounded for all the values of t = a3/a. The variations of limr!0j(r, h) with h for different values of K = l*/l and t are plotted in Fig. 4. Also, for K ! 1, j(r, h) is plotted for two different values of t as r/a and h vary, Fig. 5. By examination of Fig. 5(a) and (b), an interesting phenomenon pertinent to the behavior of the stress component r33(r, h) near a penny shape rigid inhomogeneity (K ! 1) is revealed. For very small values of t, as the edge of the rigid inhomogeneity is approached radially for p/2 < h < 3p/2, j(r, h) takes on a very large negative value followed by an abrupt reversal and becomes very large positive quantity, Fig. 5(a). This trend is reversed when p/2 < h < p/2. When t is absolute zero, limr!0j(r, h) ! 1 for p/2 < h < 3p/2, and limr!0j(r, h) ! + 1 for p/2 < h < p/2, Fig. 5(b). For the shear mode case of a quasicrack, assume that the far-field applied loading is described via ro31 ðxÞ ¼ rl31 x1 . Similar to the two-dimensional quasicrack, the exact closed-form expressions for the ratios of the SIFs of quasicrack to that of crack are obtained n pffiffiffiffiffiffiffiffiffiffiffiffi KQ r31 ðr; hÞ 31 ¼ tð1 KÞ t 1 t2 ½t2 ð1 7mÞ þ 2t4 ðm 1Þ ¼ lim C K 31 r!0 lim r31 ðr; hÞ K!0 o.npffiffiffiffiffiffiffiffiffiffiffiffi þ 5m 44 þ 3½4 m þ t2 ð11 þ mÞ arccos t 1 t2 ½t4 ð1 þ Kð23 17mÞ 7mÞ þ ðm 1Þð2t6 ð1 þ 3KÞ 8KÞ þ t2 ð5m þ Kð20 þ 19mÞ 44Þ o 3tðK 1Þð4 m þ t2 ð11 þ mÞÞ arccos t ;
Fig. 4. Variation of limr!0j(r, h) with h for the penny shape quasicrack, under the linear far-field loading, ro33 ðxÞ ¼ rl33 x1 .
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Fig. 5. Distribution of the normalized stress component, j(r, h) around the penny shape lamellar inhomogeneity under the remote linear loading, ro33 ðxÞ ¼ rl33 x1 .
n pffiffiffiffiffiffiffiffiffiffiffiffi KQ r32 ðr; hÞ 32 ¼ tð1 KÞ t 1 t2 ½t2 ð1 7mÞ 44 þ 5m þ 2t4 ðm 1Þ ¼ lim K C32 r!0 lim r32 ðr; hÞ K!0 o.npffiffiffiffiffiffiffiffiffiffiffiffi þ 3½4 m þ t2 ð11 þ mÞ arccos t 1 t2 ½t4 ð1 þ Kð23 17mÞ 7mÞ þ ðm 1Þð2t6 ð1 þ 3KÞ 8KÞ
o þ t2 ð5m þ Kð20 þ 19mÞ 44Þ 3tðK 1Þ½4 m þ t2 ð11 þ mÞ arccos t :
3.3. Elliptic lamellar inhomogeneity The general shape of an elliptic lamellar inhomogeneity, a1 5 a2 and a3 ! 0, can be treated in a similar manner. In this case, utilization of the Eshelby’s [24] EIM, the expressions of the disturbance strain (2) are obtained in terms of elliptic integrals which can exactly be calculated and the SIFs of modes I, II, and III along the edge of the elliptic lamellar inhomogeneity under a polynomial far-field loading are evaluated via (3). For Table 1 q 2 o The exact normalized values of mode I SIF, K 33 =ðrq33 a5=2 1 Þ, for the elliptic crack under the quadratic loading r33 ðxÞ ¼ r33 x1 a2 a1
0.5 1 2 5 10
h 0
p 12
p 6
p 4
p 3
5p 12
p 2
p
0.526 0.684 0.792 0.860 0.878
0.480 0.647 0.780 0.858 0.877
0.334 0.547 0.737 0.850 0.875
0.209 0.410 0.650 0.830 0.870
0.134 0.274 0.494 0.776 0.854
0.097 0.173 0.276 0.585 0.780
0.086 0.137 0.151 0.121 0.091
0.526 0.684 0.792 0.860 0.878
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Table 2 q 2 o The exact normalized values of mode I SIF, K 33 =ðrq33 a5=2 1 Þ, for the elliptic anticrack under the uniaxial quadratic loading r33 ðxÞ ¼ r33 x1 a2 a1
h
0.5 1 2 5 10
0
p 12
p 6
p 4
p 3
5p 12
p 2
p
0.060 0.057 0.055 0.055 0.055
0.056 0.060 0.056 0.055 0.055
0.035 0.062 0.058 0.055 0.055
0.016 0.052 0.061 0.056 0.055
0.005 0.028 0.059 0.057 0.056
0.001 0.000 0.025 0.058 0.057
0.003 0.012 0.050 0.306 1.077
0.060 0.057 0.055 0.055 0.055
5=2
illustration, the exact normalized mode I SIF, K 33 =ðrq33 a1 Þ along the edge of the elliptic crack and anticrack under the remote homogeneous quadratic loading ro33 ðxÞ ¼ rq33 x21 are given in Tables 1 and 2, respectively. Other cases of lamellar inhomogeneity and loading conditions can similarly be considered which for brevity are not given here. Acknowledgements This work has been supported by the center of excellence in structures and earthquake engineering at Sharif University of Technology. Appendix A. The expressions of the eigenstrain fields pertinent to the slit-like anticrack under remote uniaxial and biaxial loadings Assume that the homogenizing eigenstrain field in X is defined as ij ¼ cij þ cijk xk þ cijkl xk xl þ : Let E* ! 1 and m* = 0, solving the consistency equation (1), the non-zero components of the eigenstrain field are derived in terms of t = a2/a1, for the following loading conditions. A.1. Biaxial uniform loading ro11 ðxÞ ¼ ru11 and ro22 ðxÞ ¼ ru22 c11 ¼ ð1 mÞfru11 ½2ð1 mÞ þ tð3 3m þ 2m2 Þ þ 2m2 t2 þ ru22 ½tð1 5m þ 2m2 Þ 2m þ 2mt2 ðm 1Þg=½Eð4m 3Þt; c22 ¼ ð1 mÞfru11 ½tð1 5m þ 2m2 Þ 2mð1 m þ t2 Þ þ ru22 ½tð3 3m þ 2m2 Þ þ 2m2 þ 2t2 ð1 mÞg=½Eð4m 3Þt; c33 ¼
2ð1 mÞmf½ð1 þ 2tÞðm 1Þ þ mt2 ru11 þ ½m þ ðm 1Þtð2 þ tÞru22 g : Eð4m 3Þt
A.2. Homogeneous uniaxial linear loading ro22 ðxÞ ¼ rl22 x1 ð1 4m þ 2m2 Þt þ ðm 1Þmt2 ð3 þ t2 Þ m ; Et½ð4m 3Þ þ 2ðm 1Þt ð3 3m þ m2 Þt ðm 1Þt2 ð3 þ tÞ þ m2 ; ¼ ð1 mÞrl22 ð1 þ tÞ Et½ð4m 3Þ þ 2ðm 1Þt ð3m 4Þt þ ðm 1Þt2 ð3 þ tÞ þ m : ¼ ð1 mÞmrl22 ð1 þ tÞ Et½ð4m 3Þ þ 2ðm 1Þt
c111 ¼ ð1 mÞrl22 ð1 þ tÞ c221 c331
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A.3. Homogeneous uniaxial quadratic loading ro22 ðxÞ ¼ rq22 x21 c11 ¼ rq22 a21 ð1 þ tÞ½mð3 7m þ 4m2 Þ þ mð5 9m þ 4m2 Þt þ ð3 15m þ 35m2 35m3 þ 12m4 Þt2 þ ðm 1Þ2 ðð1 15m þ 24m2 Þt3 þ mð16m 11Þt4 Þ þ ðm 1Þ2 mð4m 3Þt5 =fEð4m 3Þt½3ð4m 3Þ þ 12ðm 1Þt þ ð4m 5Þt2 g; c22 ¼ rq22 a21 ð1 þ tÞ½m2 ð3 7m þ 4m2 Þ þ m2 ð5 9m þ 4m2 Þt þ ð35m 9 41m2 þ 15m3 Þt2 þ 5ðm 1Þ2 ð5m 3Þt3 þ ðm 1Þ2 ð16m 11Þt4 2
þ ðm 1Þ ð4m 3Þt5 =fEð4m 3Þt½3ð4m 3Þ þ 12ðm 1Þt þ ð4m 5Þt2 g; c33 ¼ mrq22 a21 ð1 þ tÞ½mð3 7m þ 4m2 Þ þ mð5 9m þ 4m2 Þt þ 2ð6 19m þ 19m2 6m3 Þt2 2
2
ðm 1Þ t3 ð3ð3m 2Þ þ ð16m 11ÞtÞ ðm 1Þ ð4m 3Þt5 =fEð4m 3Þt½3ð4m 3Þ þ 12ðm 1Þt þ ð4m 5Þt2 g; 2 c1111 ¼ rq22 ð1 þ tÞ ð3 14m þ 17m2 6m3 Þt þ 2ðm 1Þmð1 6ðm 1Þt2 Þ 2
2ðm 1Þ mt3 ð4 þ tÞ=fEt½3ð4m 3Þ þ 12ðm 1Þt þ ð4m 5Þt2 g; c2211 ¼ rq22 ð1 þ tÞ2 ½2ð1 mÞ½m2 þ ð1 mÞt2 ð6 þ 4t þ t2 Þ þ ð9 18m þ 11m2 2m3 Þt=fEt½3ð4m 3Þ þ 12ðm 1Þt þ ð4m 5Þt2 g; 2
c3311 ¼ 2ð1 mÞm rq22 ð1 þ tÞ ½m þ 2ð2m 3Þt þ ðm 1Þt2 ð6 þ 4t þ t2 Þ=fEt½3ð4m 3Þ þ 12ðm 1Þt þ ð4m 5Þt2 g: It should be noted that, if one assumes m* 5 0, the components of the eigenstrain field turn out to be quite lengthy functions of m*. Nevertheless, m* will be eliminated in the process of derivation of the SIFs, as if m* is taken to be zero at the beginning. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
Murakami Y, editor. Stress intensity factors handbook. New York: Pergamon Press; 1986. Tada H, Paris P, Irwin G. The stress analysis of cracks handbook. St. Louis (MO): Paris Production Inc.; 2000. Sack RA. Extension of Griffith theory of rupture to three dimensions. Proc Phys Soc Lond 1946;58:729–36. Sneddon IN. The distribution of stress in the neighborhood of a crack in an elastic solid. Proc Royal Soc Lond Ser A 1946;187:229–60. Irwin GR. Crack-extension force for a part-through crack in a plate. Trans ASME Ser E J Appl Mech 1962;29:651–4. Shibuya T. Some mixed boundary value problems for an infinite solid containing a flat elliptical crack. Trans Japan Soc Mech Engng 1976;42(364):3718–25. Isida M. Elastic analysis of cracks and stress intensity factors (in Japanese). Fracture mechanics and strength of materials 2. Baifuukan. 1976. Atkinson C. Some ribbon-like inclusion problems. Int J Engng Sci 1973;11:243–66. Brussat TR, Westmann RA. A Westergaard-type stress function for line inclusion problems. Int J Solids Struct 1975;11:665–77. Sih GC. Plane extension of rigidly embedded line inclusions. In: Huang TC, Johnson MW, editors. Developments in mechanics, vol. 3. New York: Wiley; 1965. p. 61–79. Chang CS, Conway HD. Bond stresses in fiber reinforced composites subjected to uniform tension. J Compos Mater 1968;2:168. Chu WWL, Conway HD. Bond stresses in composites with overlapping fibers. Int J Mech Sci 1970;12:761. Bu¨rgel B, Perry AJ, Schneider WR. On the theory of fiber strengthening. J Mech Phys Solids 1970;18:101. Erdogan F, Gupta GD. Stresses near a flat inclusion in bonded dissimilar materials. Int J Solids Struct 1972;8:533. Hull D, Clyne TW. An introduction to composite materials. 2nd ed. Cambridge: Cambridge University Press; 1996. Choi Y, Sugimoto K, Song S, Gotoh Y, Ohkoshi Y, Endo M. Mechanical and physical properties of epoxy composites reinforced by vapor grown carbon nanofibers. Carbon 2005;43:2199–208. Zeng J, Saltysiak B, Johnson WS, Schiraldi A, Kumar S. Processing and properties of poly(methyl methacrylate)/carbon nanofiber composites. Compos Part B: Engng 2004;35:245–9. Hasebe N, Keer LM, Nemat-Nasser S. Stress analysis of a kinked crack initiating from a rigid line inclusion. Part I: formulation. Mech Mater 1984;3:131–45.
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[19] Hasebe N, Nemat-Nasser S, Keer LM. Stress analysis of a kinked crack initiating from a rigid line inclusion. Part II: direction of propagation. Mech Mater 1984;3:147–56. [20] Dundurs J, Markenscoff X. A Green’s function formulation of anticracks and their interaction with load-induced singularities. J Appl Mech 1989;56:550–5. [21] Markenscoff X, Ni L. The debonded interface anticrack. J Appl Mech 1996;63:621–7. [22] Hurtado JA, Dundurs J, Mura T. Lamellar inhomogeneities in a uniform stress field. J Mech Phys Solids 1996;44(1):1–21. [23] Homentcovschi D, Dascalu C. Uniform asymptotic solutions for lamellar inhomogeneities in plane elasticity. J Mech Phys Solids 2000;48:153–73. [24] Eshelby JD. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc A 1957;241:376–96. [25] Eshelby JD. The elastic field outside an ellipsoidal inclusion. Proc R Soc Lond Ser A 1959;A252:561–9. [26] Eshelby JD. Elastic inclusions and inhomogeneities. Progress in solid mechanics, vol. 2. Amsterdam: North-Holland; 1961. p. 89–140.
A general unified treatment of lamellar inhomogeneities Hossein M. Shodja
a,b,*
, Farzaneh Ojaghnezhad
a
a
b
Department of Civil Engineering, Center of Excellence in Structures and Earthquake Engineering, Sharif University of Technology, P.O. Box 11365-9313, Tehran, Iran Institute for Nanoscience and Nanotechnology, Sharif University of Technology, P.O. Box 11365-9161, Tehran, Iran Received 17 December 2005; received in revised form 3 August 2006; accepted 7 August 2006 Available online 27 October 2006
Abstract Consider a lamellar inhomogeneity embedded in an unbounded isotropic elastic medium. When the elastic moduli of the lamellar inhomogeneity are zero it is a crack, if its elastic moduli are infinite it is an anticrack, and when its elastic moduli are finite it is called a quasicrack. Based on the Eshelby’s equivalent inclusion method (EIM), the present paper develops a unified approach for determination of the exact closed-form expressions for modes I, II, and III stress intensity factors (SIFs) at the tips of lamellar inhomogeneities under a remote applied polynomial loading. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Lamellar inhomogeneity; Anticrack; Quasicrack; Equivalent inclusion method; Polynomial loadings
1. Introduction In the context of the present study, a lamellar inhomogeneity is deduced from an ellipsoidal inhomogeneity by letting one of its principal axes vanish. The aim of this paper is to determine modes I, II, and III stress intensity factors (SIFs) pertinent to a general class of lamellar shapes embedded in an infinite isotropic elastic medium under polynomial far-field applied loading in a unified manner. For a through finding on the SIFs of cracks in the literature, up to the year 2000, one should refer to the handbooks of Murakami [1] and Tada et al. [2]. A close scrutiny of the literature reveals that, except for a very few specific cases, the exact solution to the modes I, II, and III SIFs of three-dimensional penny shape and elliptic cracks under polynomial loading at infinity has not been obtained. Most of the closed-form solutions to the SIF pertinent to a penny shape crack in an infinite isotropic elastic body are devoted to at most linear far-field loading, [2–4]. The more general geometry of an elliptic crack under a uniform far-field tension was considered by Irwin [5] employing the stress function theory. Shibuya [6] also used stress function to study the elliptic crack under a linear far-field loading and evaluated only the mode I SIF within 1% accuracy [1]. As far
* Corresponding author. Address: Department of Civil Engineering, Center of Excellence in Structures and Earthquake Engineering, Sharif University of Technology, P.O. Box 11365-9313, Tehran, Iran. Tel.: +98 21 66164209; fax: +98 21 66014828. E-mail address: [email protected] (H.M. Shodja).
0013-7944/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2006.08.016
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as slit-like cracks are concerned Isida [7] has obtained the closed-form expression only for the mode I SIF at the tip of a two-dimensional crack under a polynomial loading at infinity by stress function method [1]. It is now more than three decades that scientists are concerned with the fiber–matrix force transfer in fiber reinforced composite materials. A well taken approach is to model the long thin fibers by line (lamellar) inhomogeneities [8–14], such that the fiber cross sectional area vanishes. In applying the proposed approach to real composites, if the microgeometries of the reinforcements are such that they can be approximated as limiting cases of an ellipsoid, reasonable estimation by the present method is expected. For example, often a short fiber can be represented by a prolate ellipsoid having the same aspect ratio [15]. Similarly, a long fiber can be modeled by an elliptic cylinder. Rigid inhomogeneities have important applications in materials science. Nowadays, the excellent technological applications of carbon nanofiber reinforced polymer composites have attracted the attentions of industry and numerous scientists. These composites are advantageous for their high tensile modulus, strength, and promising electrical and thermal properties. Vapor grown carbon nanofiber (VGCF) which is a new class of carbon fiber can be fabricated at high quantities and low cost. The fiber may have a diameter of about 150 nm and length of 10–20 lm [16]. The modulus of carbon nanofiber is normally in the range of 100–600 GPa and sometimes even higher, whereas the modulus of some polymers is usually 2–5 GPa [17]. Thus, for certain purposes the carbon nanofibers in the carbon nanofiber reinforced polymer composites may be considered as rigid fibers or anticracks in the context of the present study. A brief examination of typical properties of fibers and matrices which are now extensively used in composite materials reveals that, many types of fibers such as SiC, Al2O3, high modulus (HM) carbon, etc. (in the forms of platelets, whiskers, powder, etc.) have high stiffness in comparison to thermosets and thermoplastics matrices such as epoxy resins, polyesters, and polypropylene. Elastic deformation of these composites with different fiber architecture (long-fiber, short-fiber, ribbon-like fiber, etc.) can be drawn assuming the linear-elastic stress–strain behavior [15]. Moreover, non-metallic and very hard embrittlements in some metallic materials and alloys can be idealized as rigid imperfections relative to the matrix and linear-elastic analysis has proved to be useful in such situations, Hasebe et al. [18,19]. Same authors have considered an anticrack as a model for a thin rigid plate embedded in an elastic body. Atkinson [8], in an effort to determine the elastic fields of a metallic strain measuring device embedded in a rubber matrix, proposed ribbon-like inhomogeneity model. In this, Atkinson examines both cases of a rigid ribbon inhomogeneity (anticrack) and an elastic ribbon inhomogeneity (quasicrack). A considerable amount of literature is devoted to the two-dimensional problem of anticrack embedded in an isotropic elastic materials under uniform far-field loading, Hasebe et al. [18,19], Dundurs and Markenscoff [20], Markenscoff and Ni [21], Hurtado et al. [22], and Homentcovschi and Dascalu [23]. However, to the best of the authors’ knowledge, the three-dimensional cases of penny shape and elliptic anticracks under far-field polynomial loading have not been addressed in the literature. Also, to date, very little attention has been paid to the determination of the stress field of quasicracks. The existing theories devoted to this topic are tailored for the simpler two-dimensional and uniform loading conditions and have very limited ranges of applicability. For example, Hurtado et al. [22] who address quasicracks, consider the in-plane and out-of-plane cases under a uniform farfield loading. Depending on the dimension of the problem, loading condition, and the type of lamellar inhomogeneity: crack, quasicrack, or anticrack, various types of boundary value problems are encountered and a unified approach using the existing mathematical treatments is not possible, so that researchers have employed different methods. For the cases of in-plane and out-of-plane strain, and under a uniform applied far-field loading, Hurtado et al. [22] show that the lamellar inhomogeneity can equivalently be replaced by a suitable distribution of dislocations, which can be obtained by using the concept of surface dislocation density. Also, Homentcovschi and Dascalu [23] use the Muskhelishvili’s complex potentials to study the two-dimensional problems of lamellar inhomogeneity in unbounded isotropic elastic materials. The deficiency of the two latter approaches is the difficulty in their extension, not only to three-dimensional lamellar inhomogeneity, but also for remote polynomial loading. On the other hand, the beauty of the Eshelby’s [24–26] result is that there is an exact correspondence between the far-field loading and the form of the distribution of homogenizing eigenstrain inside the equivalent inclusion. This powerful theory implies many dramatic results. In the context of the present study, as one of its abundant applications, it enables us to find the exact SIFs for the two- and three-dimensional lamellar inhomogeneities due to a far-field applied polynomial loading.
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2. Formulation an elastic medium X with elastic moduli C ijkl embedded in an infinite isotropic elastic matrix P Consider 3 ¼ R X with elastic moduli C ijkl . If the sub-domain X is made of the same material as the matrix, C ijkl ¼ C ijkl , and there is a prescribed eigenstrain in X, it is referred to as an inclusion. If C ijkl 6¼ C ijkl , and X does not contain any eigenstrain, X is called an inhomogeneity. Interesting results are inferred when X is an ellipsoidal inhomogeneity Fig. 1. The fact that a single ellipsoidal inhomogeneity embedded in an unbounded matrix under the remote applied loading oij ðxÞðroij ðxÞÞ can equivalently be substituted with an inclusion having properly chosen eigenstrains was first pointed out by Eshelby [24–26]. This method is called the equivalent inclusion method (EIM) and requires that the stress field of the ellipsoidal inhomogeneity X to be equivalent to the stress field of the equivalent inclusion. This equivalency gives rise to the following consistency equation C ijkl ½dkl ðxÞ þ okl ðxÞ ¼ C ijkl ½dkl ðxÞ þ okl ðxÞ kl ðxÞ;
x 2 X;
ð1Þ
where ij ðxÞ are the homogenizing eigenstrains to be determined from the consistency equation (1) and dij ðxÞ are the disturbance strains caused by the presence of the inhomogeneity or equivalently due to the homogenizing eigenstrains Z dij ðxÞ ¼ Cijkl ðx yÞkl ðyÞ dy; X
1 Cijkl ðx yÞ ¼ C mnkl ½Gim;nj ðx yÞ þ Gjm;ni ðx yÞ; 2
ð2Þ
where Gij(x y) is the associated Green’s function. For uniform far-field loading, Eshelby [24] has shown that the eigenstrain field is uniform inside X, and subsequently the disturbance strain is also uniform in X. Generally, when the far-field applied loading is a polynomial of degree n P 0, the homogenizing eigenstrain inside the ellipsoidal inhomogeneity is also a polynomial of degree n P 0. If the eigenstrain field in X is a homogeneous polynomial of degree n = 1, the disturbance strain for the points inside the ellipsoid X will also be a homogeneous polynomial of degree 1. If the eigenstrain is a homogeneous polynomial of degree n P 2, the associated disturbance strain field in X is an inhomogeneous polynomial in x whose terms are of degree n, (n 2), (n 4), . . . , Eshelby [26]. When the loading and subsequently the eigenstrain field is a polynomial of order n, there will be (n + 1)(n + 2)(n + 3) unknown coefficients in the expression of the eigenstrain field. Thus, for exact determination of the unknowns and the elastic fields for both interior and exterior points, it is necessary to write the consistency equation at (n + 1)(n + 2)(n + 3)/6 points in X. Suppose ap, p = 1, 2, 3 are the principal half axes of the ellipsoidal inhomogeneity ( ) 3 X x2p 3 X ¼ xjx 2 R ; 61 : a2p i¼1 A desired lamellar inhomogeneity is obtained by letting the corresponding principal half axis approach zero. In the process of deducing the eigenstrain field of the lamellar inhomogeneity by letting the desired dimension
Fig. 1. The ellipsoidal inhomogeneity, X embedded in an infinite isotropic elastic medium, R ¼ R3 X.
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approach zero, ap ! 0, some components of the homogenizing eigenstrains become infinite, however, on limap !0 ap ij ¼ finite. The associated closed-form expression for the SIF, Kij corresponding to a point x , such that the edge of X is obtained by finding the stress field at a point x 2 R in the vicinity of the point x j¼ r ! 0 jxx pffiffiffiffiffiffiffi ð3Þ K ij ¼ lim lim 2prrij ðrÞ; r!0 ap !0
where rij(r) is the stress field at the exterior point x due to the presence of the lamellar inhomogeneity. l, E, and m denote the Lame´ constants for the matrix. The Lame´ constants for the inhomogeneity are differentiated by using the superscript ‘‘*’’. 3. Results Throughout this section, a general class of lamellar inhomogeneities surrounded by an infinite body under a polynomial loading at infinity will be considered. For illustration, suppose that the remote applied stress is roij ðxÞ ¼ ruij þ rlij x1 þ rqij x21 þ ;
ð4Þ
where ruij , rlij and rqij are constants appearing in the uniform, linear, and quadratic terms, respectively. The results pertinent to slit-like, penny shape, and elliptic lamellar inhomogeneities are presented in Sections 3.1, 3.2 and 3.3, respectively. In Sections 3.1 and 3.2, the corresponding lamellar inhomogeneities, i.e. crack, anticrack, and quasicrack for all three modes of deformation are addressed. In Section 3.3, two special cases of elliptic crack and anticrack under the mode I homogeneous quadratic far-field loading are considered and other cases are eliminated for brevity. All numerical results presented in this paper are pertinent to m = m* = 0.3. 3.1. Slit-like lamellar inhomogeneity This section is concerned with the lamellar inhomogeneity whose shape is the infinitely extended elliptic cylinder with one of its principal axes approaching zero. Thus, the geometry of interest is obtained from Fig. 1 by letting a2 ! 0 and a3 ! 1. 3.1.1. Cracks Assume that the lamellar inhomogeneity X is a crack, i.e. C ijkl ! 0. If the far-field applied stress for modes I, II, and III is explained by (4), employment of the proposed approach and the definition of the SIF (3) yields pffiffiffiffiffiffiffi 1 1 pa1 ; K ij ¼ ruij rlij a1 þ rqij a21 þ 2 2 where ± denotes the right and left crack tips, respectively. Isida [7] reports the identical result for the mode I SIF of a slit-like crack under the remote polynomial loading. It is worthwhile to note that, even though r22(r), r21(r), and r23(r) have different mathematical expressions, lima2 !0 r22 ðrÞ ¼ lima2 !0 r21 ðrÞ ¼ lima2 !0 r23 ðrÞ. 3.1.2. Anticracks When the lamellar inhomogeneity is rigid, that is C ijkl ! 1, it is called an anticrack. This interesting physical problem can also be treated by the EIM, even for polynomial far-field loadings. Employing the proposed approach, the elastic field of an anticrack subjected to a polynomial far-field loading is evaluated and it is pffiffi observed that mode I anticracks, similar to cracks exhibit 1= r singularity and so the SIF of an anticrack is obtained via (3). In contrast to the crack problem, for an anticrack the normal stress applied parallel to the anticrack plane also contributes to the mode I SIF. For illustration, the problem of an anticrack subjected to both polynomial uniaxial loading and uniform biaxial loading will be considered in this section. The homogenizing eigenstrain fields associated with these types of loadings are briefly mentioned in Appendix A. It can be shown that the mode I SIF of an anticrack under the remote uniaxial applied loading of the form (4) is
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K 22
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pffiffiffiffiffiffiffi 1 2m 1 l 1 q 2 u mr22 mr22 a1 þ mr22 a1 þ ¼ pa1 : 3 4m 2 2
It should be emphasized that for anticracks, the mode I SIF depends upon the Poisson’s ratio of the matrix, and becomes equal to zero when the matrix is incompressible. Now, if the far-field applied stress is biaxial and uniform, ro11 ðxÞ ¼ ru11 and ro22 ðxÞ ¼ ru22 , the SIF for mode I will be pffiffiffiffiffiffiffi 1 2m u mr22 ð1 mÞru11 pa1 : K 22 ¼ 3 4m This result is also validated by the solution of Hurtado et al. [22] who considered only the uniform biaxial applied stress at infinity. The above relation reveals that ru11 also contributes to the mode I SIF through the Poisson’s ratio effect. Astonishingly, by examining the exact closed-form expression of the stress field, it can be inferred that rigid lamellar inhomogeneities do not disturb the modes II and III polynomial far-field loadings, and so the associated SIFs are zero. 3.1.3. Quasicracks This section considers lamellar inhomogeneities whose elastic moduli are finite, C ijkl ¼ finite referred to as quasicracks. It is noteworthy to mention that, cracks and anticracks with C ijkl ¼ 0 and 1, respectively, are the limiting cases of quasicracks. It is evidently observed that all the components of the homogenizing eigenstrain field for quasicrack are finite when a2 ! 0, thus unlike crack and anticrack, for which lima2 !0 a2 ij 6¼ 0, for the quasicrack lima2 !0 a2 ij ¼ 0. It implies that the SIF vanishes for quasicrack except for the limiting cases of K = l*/l = 0 and K ! 1. To study the mode I quasicrack, let jðrÞ ¼ r22 ðrÞ=ro22 ða1 ; 0; 0Þ denote the normalized stress outside the inhomogeneity on the x1-axis. For the sake of demonstration and comparison, variation of j(r) just outside the inhomogeneity with t = a2/a1 is plotted for different values of K under uniaxial uniform and quadratic remote loadings in Fig. 2(a) and (b), respectively.
Fig. 2. Variation of limr!0j(r) with t for the slit-like quasicrack, under (a) uniform far-field loading, ro22 ðxÞ ¼ ru22 and (b) quadratic far-field loading, ro22 ðxÞ ¼ rq22 x21 .
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From Fig. 2(a) for the special case of the uniform tensile stress disturbed by a circular hole, t = 1 and K = 0, the stress concentration factor equal to 3 prevails. By examination of the closed-form expressions of the stress field, it is proved that under uniform far-field loading limr!0j depends upon the aspect ratio t and is independent of a1, while for the homogeneous quadratic loading, it varies with a21 . As depicted in Fig. 2(a) and (b), limr!0j is negative for K > 1. Fig. 3(a) and (b) display the variation of j with r/a1, for K ! 1 and different values of t, under the remote uniform and quadratic loadings, respectively. Fig. 3(a) and (b) reveal that, if t is absolutely zero, j is positive everywhere for all values of r/a1, and for any other values of t > 0, there is a corresponding interval (0, r/a1) for which j is negative. The mode II quasicrack, for brevity, is studied by applying only a linear far-field stress, ro21 ðxÞ ¼ rl21 x1 . The ratio of the mode II SIF of quasicrack to that of crack is given by KQ r21 ðrÞ 2ðK 1Þa21 a2 21 ¼ ¼ lim : C 3 K 21 r!0 lim r21 ðrÞ Kðm 1Þða1 þ 3a1 a22 þ a32 Þ þ ðKð3m 1Þ 2Þa21 a2 K!0
Other cases of loadings can be treated in a similar manner. Note that by considering K = 0 and K ! 1, the C special cases of crack and anticrack for which K Q 21 =K 21 ¼ 1; 0 are recovered. Similarly, for the mode III quasicrack, only linear far-field loading is considered, ro23 ðxÞ ¼ rl23 x1 . In the same manner, for this mode the ratio of the SIFs simplifies to KQ r23 ðrÞ ð1 KÞa2 ð2a1 þ a2 Þ 23 ¼ ¼ lim : Ka21 þ 2a1 a2 þ a22 K C23 r!0 lim r23 ðrÞ K!0
The results pertaining to the limiting cases are readily inferred.
Fig. 3. Distribution of the normalized stress component, j(r) for the slit-like quasicrack, under (a) uniform, ro22 ðxÞ ¼ ru22 and (b) quadratic, ro22 ðxÞ ¼ rq22 x21 far-field loadings.
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3.2. Penny shape lamellar inhomogeneity Throughout this section, the three-dimensional case of penny shape lamellar inhomogeneity constructed by letting a3 ! 0 and a1 = a2 = a, will be studied, Fig. 1. 3.2.1. Penny shape cracks To evaluate the mode I SIF, it is supposed that a penny shape crack is subjected to the far-field applied stress ro33 ðxÞ given by Eq. (4). Using the proposed approach, the closed-form expression of the mode I SIF is determined as below 2 pffiffiffi 2 l 4 q 2 u pa r33 þ r33 a cos h þ r33 a ð3 þ 2 cos 2hÞ þ ; K 33 ðhÞ ¼ p 3 33 where h is the angle measured counterclockwise from the positive direction of the x1-axis. In the literature, only the results up to the linear far-field loading is available, Tada et al. [2]. For a penny shape crack under a shear mode far-field loading, however, it is assumed that the only nonzero component of the far-field applied stress is ro31 ðxÞ defined in the form of Eq. (4). For this type of loading, except for the points at h = kp/2, k = 0, 1, . . . , all the other points along the edge of the crack undergo both modes II and III fracture. pffiffiffi 3 2 aru31 ð2 m þ m cos 2hÞ 8a2 rl31 cos hð2 m þ m cos 2hÞ pffiffiffi pffiffiffi K 31 ðhÞ ¼ þ pð2 mÞ 3 pð4 mÞ 5
4a2 rq31 fð16 4m þ m2 Þ cos 2h þ ð2 mÞ½12 m þ 2m cos 4hg pffiffiffi þ ; 3 pð44 28m þ 3m2 Þ pffiffiffi 3 2 2 amru31 sin 2h 16a2 mrl31 ðcos hÞ sin h pffiffiffi K 32 ðhÞ ¼ pffiffiffi þ pð2 mÞ 3 pð4 mÞ þ
5
þ
4a2 mrq31 ½3ð4 mÞ þ 4ð2 mÞ cos 2h sin 2h pffiffiffi þ : 3 pð44 28m þ 3m2 Þ
In the handbook of Tada et al. [2], only the results for the uniform far-field loading is available. 3.2.2. Penny shape anticracks Similar to the slit-like anticrack, in addition to the opening type of loading, the normal stress applied parallel to the plane of the penny shape anticrack at infinity contributes to the mode I SIF, so, for demonstration, the penny shape anticrack is considered to be subjected to both uniaxial opening and triaxial remote stress in this section. Let a penny shape anticrack be subjected to a uniaxial far-field loading of the form (4), then the mode I SIF becomes pffiffiffi 3 4 amð1 2mÞru33 16a2 mð1 2mÞ cos h½7 8m ð3 4mÞ cos 2h l pffiffiffi K 33 ðhÞ ¼ pffiffiffi þ r33 þ : 3 pð33 47m 32m2 þ 48m3 Þ pð3 m 4m2 Þ Now, it is assumed that the uniform and triaxial loading, ro11 ðxÞ ¼ ru11 , ro22 ðxÞ ¼ ru22 , and ro33 ðxÞ ¼ ru33 , is applied to the penny shape anticrack at infinity. Then, the SIF of mode I is obtained as below pffiffiffi pffiffiffi 4 amð1 2mÞru33 2 að1 2mÞð13 17mÞðru11 þ ru22 Þ K 33 ðhÞ ¼ pffiffiffi pffiffiffi : pð21 31m 20m2 þ 32m3 Þ pð3 m 4m2 Þ Note that, similar to the two-dimensional case, the mode I SIF for anticracks depends on the Poisson’s ratio of the matrix and is equal to zero when the matrix is incompressible, m = 1/2. Also, calculations of the stress field show that the shear mode SIFs for anticracks always vanish. 3.2.3. Penny shape quasicracks For brevity, the exact solution to the penny shape quasicrack subjected to a linear far-field applied stress is given here, the other cases of loadings can be treated in a similar manner.
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In the case of mode I quasicrack, it is assumed that j(r, h) is the normalized stress component r33 ðr; hÞ=ro33 ða; 0; 0Þ, where r is the normal distance from the edge of the penny shape quasicrack. For this problem, except for the extreme cases of crack and anticrack, no matter how small a3 becomes, limr!0j(r, h) is bounded for all the values of t = a3/a. The variations of limr!0j(r, h) with h for different values of K = l*/l and t are plotted in Fig. 4. Also, for K ! 1, j(r, h) is plotted for two different values of t as r/a and h vary, Fig. 5. By examination of Fig. 5(a) and (b), an interesting phenomenon pertinent to the behavior of the stress component r33(r, h) near a penny shape rigid inhomogeneity (K ! 1) is revealed. For very small values of t, as the edge of the rigid inhomogeneity is approached radially for p/2 < h < 3p/2, j(r, h) takes on a very large negative value followed by an abrupt reversal and becomes very large positive quantity, Fig. 5(a). This trend is reversed when p/2 < h < p/2. When t is absolute zero, limr!0j(r, h) ! 1 for p/2 < h < 3p/2, and limr!0j(r, h) ! + 1 for p/2 < h < p/2, Fig. 5(b). For the shear mode case of a quasicrack, assume that the far-field applied loading is described via ro31 ðxÞ ¼ rl31 x1 . Similar to the two-dimensional quasicrack, the exact closed-form expressions for the ratios of the SIFs of quasicrack to that of crack are obtained n pffiffiffiffiffiffiffiffiffiffiffiffi KQ r31 ðr; hÞ 31 ¼ tð1 KÞ t 1 t2 ½t2 ð1 7mÞ þ 2t4 ðm 1Þ ¼ lim C K 31 r!0 lim r31 ðr; hÞ K!0 o.npffiffiffiffiffiffiffiffiffiffiffiffi þ 5m 44 þ 3½4 m þ t2 ð11 þ mÞ arccos t 1 t2 ½t4 ð1 þ Kð23 17mÞ 7mÞ þ ðm 1Þð2t6 ð1 þ 3KÞ 8KÞ þ t2 ð5m þ Kð20 þ 19mÞ 44Þ o 3tðK 1Þð4 m þ t2 ð11 þ mÞÞ arccos t ;
Fig. 4. Variation of limr!0j(r, h) with h for the penny shape quasicrack, under the linear far-field loading, ro33 ðxÞ ¼ rl33 x1 .
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Fig. 5. Distribution of the normalized stress component, j(r, h) around the penny shape lamellar inhomogeneity under the remote linear loading, ro33 ðxÞ ¼ rl33 x1 .
n pffiffiffiffiffiffiffiffiffiffiffiffi KQ r32 ðr; hÞ 32 ¼ tð1 KÞ t 1 t2 ½t2 ð1 7mÞ 44 þ 5m þ 2t4 ðm 1Þ ¼ lim K C32 r!0 lim r32 ðr; hÞ K!0 o.npffiffiffiffiffiffiffiffiffiffiffiffi þ 3½4 m þ t2 ð11 þ mÞ arccos t 1 t2 ½t4 ð1 þ Kð23 17mÞ 7mÞ þ ðm 1Þð2t6 ð1 þ 3KÞ 8KÞ
o þ t2 ð5m þ Kð20 þ 19mÞ 44Þ 3tðK 1Þ½4 m þ t2 ð11 þ mÞ arccos t :
3.3. Elliptic lamellar inhomogeneity The general shape of an elliptic lamellar inhomogeneity, a1 5 a2 and a3 ! 0, can be treated in a similar manner. In this case, utilization of the Eshelby’s [24] EIM, the expressions of the disturbance strain (2) are obtained in terms of elliptic integrals which can exactly be calculated and the SIFs of modes I, II, and III along the edge of the elliptic lamellar inhomogeneity under a polynomial far-field loading are evaluated via (3). For Table 1 q 2 o The exact normalized values of mode I SIF, K 33 =ðrq33 a5=2 1 Þ, for the elliptic crack under the quadratic loading r33 ðxÞ ¼ r33 x1 a2 a1
0.5 1 2 5 10
h 0
p 12
p 6
p 4
p 3
5p 12
p 2
p
0.526 0.684 0.792 0.860 0.878
0.480 0.647 0.780 0.858 0.877
0.334 0.547 0.737 0.850 0.875
0.209 0.410 0.650 0.830 0.870
0.134 0.274 0.494 0.776 0.854
0.097 0.173 0.276 0.585 0.780
0.086 0.137 0.151 0.121 0.091
0.526 0.684 0.792 0.860 0.878
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Table 2 q 2 o The exact normalized values of mode I SIF, K 33 =ðrq33 a5=2 1 Þ, for the elliptic anticrack under the uniaxial quadratic loading r33 ðxÞ ¼ r33 x1 a2 a1
h
0.5 1 2 5 10
0
p 12
p 6
p 4
p 3
5p 12
p 2
p
0.060 0.057 0.055 0.055 0.055
0.056 0.060 0.056 0.055 0.055
0.035 0.062 0.058 0.055 0.055
0.016 0.052 0.061 0.056 0.055
0.005 0.028 0.059 0.057 0.056
0.001 0.000 0.025 0.058 0.057
0.003 0.012 0.050 0.306 1.077
0.060 0.057 0.055 0.055 0.055
5=2
illustration, the exact normalized mode I SIF, K 33 =ðrq33 a1 Þ along the edge of the elliptic crack and anticrack under the remote homogeneous quadratic loading ro33 ðxÞ ¼ rq33 x21 are given in Tables 1 and 2, respectively. Other cases of lamellar inhomogeneity and loading conditions can similarly be considered which for brevity are not given here. Acknowledgements This work has been supported by the center of excellence in structures and earthquake engineering at Sharif University of Technology. Appendix A. The expressions of the eigenstrain fields pertinent to the slit-like anticrack under remote uniaxial and biaxial loadings Assume that the homogenizing eigenstrain field in X is defined as ij ¼ cij þ cijk xk þ cijkl xk xl þ : Let E* ! 1 and m* = 0, solving the consistency equation (1), the non-zero components of the eigenstrain field are derived in terms of t = a2/a1, for the following loading conditions. A.1. Biaxial uniform loading ro11 ðxÞ ¼ ru11 and ro22 ðxÞ ¼ ru22 c11 ¼ ð1 mÞfru11 ½2ð1 mÞ þ tð3 3m þ 2m2 Þ þ 2m2 t2 þ ru22 ½tð1 5m þ 2m2 Þ 2m þ 2mt2 ðm 1Þg=½Eð4m 3Þt; c22 ¼ ð1 mÞfru11 ½tð1 5m þ 2m2 Þ 2mð1 m þ t2 Þ þ ru22 ½tð3 3m þ 2m2 Þ þ 2m2 þ 2t2 ð1 mÞg=½Eð4m 3Þt; c33 ¼
2ð1 mÞmf½ð1 þ 2tÞðm 1Þ þ mt2 ru11 þ ½m þ ðm 1Þtð2 þ tÞru22 g : Eð4m 3Þt
A.2. Homogeneous uniaxial linear loading ro22 ðxÞ ¼ rl22 x1 ð1 4m þ 2m2 Þt þ ðm 1Þmt2 ð3 þ t2 Þ m ; Et½ð4m 3Þ þ 2ðm 1Þt ð3 3m þ m2 Þt ðm 1Þt2 ð3 þ tÞ þ m2 ; ¼ ð1 mÞrl22 ð1 þ tÞ Et½ð4m 3Þ þ 2ðm 1Þt ð3m 4Þt þ ðm 1Þt2 ð3 þ tÞ þ m : ¼ ð1 mÞmrl22 ð1 þ tÞ Et½ð4m 3Þ þ 2ðm 1Þt
c111 ¼ ð1 mÞrl22 ð1 þ tÞ c221 c331
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A.3. Homogeneous uniaxial quadratic loading ro22 ðxÞ ¼ rq22 x21 c11 ¼ rq22 a21 ð1 þ tÞ½mð3 7m þ 4m2 Þ þ mð5 9m þ 4m2 Þt þ ð3 15m þ 35m2 35m3 þ 12m4 Þt2 þ ðm 1Þ2 ðð1 15m þ 24m2 Þt3 þ mð16m 11Þt4 Þ þ ðm 1Þ2 mð4m 3Þt5 =fEð4m 3Þt½3ð4m 3Þ þ 12ðm 1Þt þ ð4m 5Þt2 g; c22 ¼ rq22 a21 ð1 þ tÞ½m2 ð3 7m þ 4m2 Þ þ m2 ð5 9m þ 4m2 Þt þ ð35m 9 41m2 þ 15m3 Þt2 þ 5ðm 1Þ2 ð5m 3Þt3 þ ðm 1Þ2 ð16m 11Þt4 2
þ ðm 1Þ ð4m 3Þt5 =fEð4m 3Þt½3ð4m 3Þ þ 12ðm 1Þt þ ð4m 5Þt2 g; c33 ¼ mrq22 a21 ð1 þ tÞ½mð3 7m þ 4m2 Þ þ mð5 9m þ 4m2 Þt þ 2ð6 19m þ 19m2 6m3 Þt2 2
2
ðm 1Þ t3 ð3ð3m 2Þ þ ð16m 11ÞtÞ ðm 1Þ ð4m 3Þt5 =fEð4m 3Þt½3ð4m 3Þ þ 12ðm 1Þt þ ð4m 5Þt2 g; 2 c1111 ¼ rq22 ð1 þ tÞ ð3 14m þ 17m2 6m3 Þt þ 2ðm 1Þmð1 6ðm 1Þt2 Þ 2
2ðm 1Þ mt3 ð4 þ tÞ=fEt½3ð4m 3Þ þ 12ðm 1Þt þ ð4m 5Þt2 g; c2211 ¼ rq22 ð1 þ tÞ2 ½2ð1 mÞ½m2 þ ð1 mÞt2 ð6 þ 4t þ t2 Þ þ ð9 18m þ 11m2 2m3 Þt=fEt½3ð4m 3Þ þ 12ðm 1Þt þ ð4m 5Þt2 g; 2
c3311 ¼ 2ð1 mÞm rq22 ð1 þ tÞ ½m þ 2ð2m 3Þt þ ðm 1Þt2 ð6 þ 4t þ t2 Þ=fEt½3ð4m 3Þ þ 12ðm 1Þt þ ð4m 5Þt2 g: It should be noted that, if one assumes m* 5 0, the components of the eigenstrain field turn out to be quite lengthy functions of m*. Nevertheless, m* will be eliminated in the process of derivation of the SIFs, as if m* is taken to be zero at the beginning. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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