Elastic constants of engineering materials are crucial for understanding the deformation and failure behaviors of structural components from both macroscopic and microscopic points of view. From microscopic perspective, their importance arises from their intimate relationship to such solid state phenomena as specific heat, Debye temperature and Grunelsen parameter [1].

Discovery of superconductivity in La-Ba-Cu-O system in the1980’s by Bednorz and Muller [2] has spurred an enormous amount of activities in search for high (i.e., above the boiling point of liquid nitrogen) T_c superconductors (HTS) such as mono-crystalline YBCO (yttrium barium copper oxide), which is also called 1-2-3 superconductor (because of the presence of one yttrium, two barium and three copper atoms in its unit cell). Modern applications include Josephson junctions, which can act as a switch for magnetic fields, or alternatively, perform the function of a magnetic detector, called the superconducting quantum interference device (SQUID) [3]. Debye temperature, Θ_D, of a superconductor can be determined from the knowledge of the elastic constants, c_ij, in a manner described by Eqs. (2) and (3) of Lei et al. [1], which, in combination with the electron-phonon coupling parameter, λ^∗, can be used to compute the superconducting transition temperature, T_c [4,5].

Practical applications of such mono-crystalline superconductors are, however, limited (at cryogenic temperature) by their poor fracture toughness [6,7,8,9,10,11,12,13,14]. Asymptotic behavior of two-dimensional stress fields at the tips of cracks and wedges, has been studied extensively in the literature [15,16,17,18,19,20]; see Nejati et al [21] for an extensive literature survey. Significant progress in research on a two-dimensional cracked anisotropic solid notwithstanding, the corresponding progress in its three-dimensional counterpart appears to, till recently [22], be marked by its complete absence. More important, the above studies employ the Lekhnitskii [15] and Stroh [16] type formulations, which are all based on complex variables-based methodology. Since the three-dimensional space is too small to accommodate the next higher dimensional analog of complex variables (for which at least a four-dimensional space will be required; e.g., quaternion [23]), these complex variables-based analyses are not by themselves adequate for analysis of three-dimensional cracked anisotropic solids.

The mathematical difficulties posed by the three-dimensional crack/anticrack type problems are substantially greater than their two-dimensional counterparts (to start with, the governing PDE’s are much more complicated). In the absence of the knowledge of the strength of singularity, in regions where the elastic stresses become unbounded, the majority of weighted residual type methods, e.g., the finite elements, finite difference and boundary elements, which are generally employed to solve fracture mechanics problems, encounter overwhelming numerical difficulties, such as lack of convergence, and oscillation resulting in poor accuracy [24]. There are several classes of problems pertaining to the issue of three-dimensional stress singularity [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49]: (i) through crack/anticrack as well as their bi- and tri-material interface counterparts, (ii) bi-material free edge, (iii) tri-material junction, (iv) penny shaped crack/anticrack and their bi-material counterparts, (iv) bi-material hole (and inclusion), (v) matrix cracking and fiber breaks in composites among others. Only the penny shaped crack/anticrack [50] (and their bi-material counterparts [51]) and the bimaterial hole [52,53] and inclusion problems [54] had earlier been adequately addressed in the literature. Earlier attempts to solve the three-dimensional through crack problem resulted in controversies that lasted for about a quarter century [25,55]. A unified three-dimensional eigenfunction approach has recently been developed by Chaudhuri and co-workers [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,55] to address the three-dimensional stress singularity problems covering all the aspects mentioned above.

The above separation of variables approach has recently been extended to the case of cracked/anticracked transversely isotropic (smeared-out composite) [56] as well as cubic/orthorhombic/diamond cubic mono-crystalline plates under mode I/II far-field loadings [22,57,58] and cubic/orthorhombic/monoclinic/diamond cubic mono/tri-crystalline plates subjected to mode III loading [22,58,59,60,61] by making use of the separation of the thickness variable technique in combination with an affine transformation, that is similar (but not identical) in spirit to that due to Eshelby et al. [62] and Stroh [16]. This eigenfunction expansion approach has also recently employed to obtain three-dimensional asymptotic stress fields in the vicinity of the front of the kinked carbon fiber-matrix junction [63] (see also Ref. [64] for its 2D counterpart with isotropic glass fibers).

The above-mentioned importance notwithstanding, relatively fewer attempts at experimental determination of elastic constants of monocrystalline YBCO have been reported in the solid state physics literature [1,65,66,67,68,69,70,71,72,73]; these are summarized in Table 1 of Lei et al. [1]. Golding et al. [66] and Saint-Paul and co-workers [70,71] have reported experimental results on c₁₁ and c₃₃, and c₃₃, c₄₄, c₆₆ and c₁₂, respectively, by employing the ultrasound technique, while Baumgart et al. [68,69] and Zouboulis et al. [72] have resorted to Brillouin spectroscopy/scattering to determine c₁₁, c₃₃ and c₄₄, and c₄₄, c₅₅ and c₆₆, respectively. Only three investigations [1,67,73] report complete sets of elastic constants, accessible to the present author. Worse still, those reported by Ledbetter and Lei [73] are just estimates (marked ** in Table 1(a)), while their experimental counterparts due to Reichard et al. [67] are based on the assumption of tetragonal symmetry; see Table 1(b). This only leaves the experimental measurements (by resonant ultrasound spectroscopy, described in detail by Migliori et al. [74,75]) due to Lei et al. [1], marked * in Table 1. However, their $c_{12}$ value appears to be excessively high. This is because, according to these authors themselves, "no wave speed in the crystal depends only upon $c_{12}$, it is no way to estimate it directly." It also has been known for some times that while $c_{12}$ and $c_{66}$ can be measured independently by static tests [76], these constants are always coupled in vibrations-based measurements [77,78].

The above literature review reveals an absence of reliable and accurate experimentally measured complete sets of nine elastic constants needed for characterization of the deformation/fracture as well as other solid state (e.g., Debye temperature, T_c, etc.) behaviors of superconducting (orthorhombic) YBCO single crystals. This calls for a reliable criterion for assessment of the measured data that would allow us to come up with a reasonably accurate complete set of nine elastic constants, which is the primary objective of the present investigation. It is proposed here that one way to address this important issue is to analytically examine the effect of elastic constants on crack deflection in monocrystalline YBCO (YBa₂Cu₃O_7−δ) and compare with the experimental results for easy cleavage planes, reported by Cook et al. [6], Raynes [9], and Granozio and di Uccio [14] among others. In what follows, the above-mentioned modified eigenfunction expansion technique, based in part on separation of the thickness-variable and partly on the Eshelby-Stroh type affine transformation, is developed to derive three-dimensional asymptotic stress field in the vicinity of the front of a semi-infinite through-thickness crack weakening an infinite orthorhombic mono-crystalline plate, of finite thickness and subjected to far-field mode I/II loadings. Crack-face boundary conditions and those that are prescribed on the top and bottom (free or fixed) surfaces of the plate are exactly satisfied. The present investigation considers six through-crack systems ― (010)[001] with the [100] length direction, (0$\overline{1}$0)[100] with the [001] length direction, ($\overline{1}$00)[001] with the [010] length direction, (100)[010] with the [001] length direction, (001)[0$\overline{1}$0] with the [100] length direction, (001)[100] with the [010] length direction ― weakening orthorhombic mono-crystalline plates. Explicit expressions for the singular stresses in the vicinity of the front of a through-thickness crack weakening an orthorhombic mono-crystalline plate, subjected to far-field mode I/II loadings, are presented. In addition, through-thickness distribution of the stress intensity factors and energy release rates are also presented. Next, the important issue of easy or difficult cleavage plane and the related question of crack deflection criterion is discussed. The latter is based on the relative fracture energy (or the energy release rate) available for possible “fracture paths” [18]. This said, it is noteworthy that any fracture criterion derived from equilibrium theories such as the Griffith criterion can only be regarded as a necessary condition but not as sufficient [79]. This calls for establishment of a sufficient condition for determination of easy or difficult cleavage plane, and the associated question of the crack deflection criterion, which is the second objective of the present study.

The present study, although to a smaller extent a review on earlier work on this topic, is largely based on original research on this subject. The topic, which covers mathematics (e.g., asymptotic, solution to 3D mixed boundary-value problem, and necessary and sufficient condition for fracture), solid state physics (e.g., superconductivity, single crystal cleavage) and engineering (e.g., 3D fracture mechanics), has, so far, remained largely unexplored in the literature.

The Cartesian coordinate system (x, y, z) is convenient to describe the deformation behavior in the vicinity of a semi-infinite through-thickness crack, weakening an infinite orthotropic plate of thickness, 2h (Figure 1 and Figure 2). Here, the z-axis is placed along the straight crack front, while the coordinates x, y, are used to define the directions along the length of the crack and transverse to it, respectively, in the plane of the plate. u, v and w represent the components of the displacements in the x, y and z directions, respectively.

The stress-strain relations for an orthorhombic single crystal are given as follows: where c_ij, i, j = 1,...,6, denotes the elastic stiffness constants of an orthorhombic mono-crystalline plate. σ_x, σ_y, σ_z represent the normal stresses, and τ_xy, τ_xz, τ_yz denote the shear stresses, while ε_x, ε_y, ε_z denote normal strains, and γ_xy, γ_xz, γ_yz represent the shear strains. For the special case of a tetragonal single crystal,

The three equilibrium equations for a linear elastic solid, made of an orthotropic/ orthorhombic material, can be expressed in terms of the displacement components, u, v, and w, as follows:

The boundary conditions include those at the plate faces and crack-side surfaces. The boundary conditions on the plate faces, z = +h, are given by [22,25]

while those at the crack-side surfaces are more conveniently expressed in local cylindrical polar coordinates (Figure 1), which are given as follows: where σ_r, σ_θ, σ_z represent the normal stresses, and τ_rθ, τ_rz, τ_θz are the shear stresses, while ε_r, ε_θ, ε_z denote the normal strains, and γ_rθ, γ_rz, γ_θz are the shear strains in the cylindrical polar coordinate system (r, θ, z). u_r and u_θ represent the components of the displacement in r and θ directions, respectively.

The assumed displacement functions for the three-dimensional crack problem under consideration are selected on the basis of separation of z-variables. These are as given below [22,56,57,58,64]:

It may be noted that since the z-dependent term and its first partial derivative can either be bounded and integrable at most admitting ordinary discontinuities, or the first partial derivative at worst be square integrable (in the sense of Labesgue integration) in its interval z ∈ [-h, h], i.e., admitting singularities weaker than square root (i.e., z ^(-1/2+ε), ε > 0), it can be best represented by Fourier series [22,25,58]. The latter case is justified by Parseval’s theorem [80], and its physical implication is that of satisfying the criterion of finiteness of local strain energy and path independence [81]. Substitution of Eqs. (4) into Eqs. (2) yields the following system of coupled partial differential equations (PDE’s): where

The solution to the system of coupled partial differential equations (5) subjected to the most general loading, can now be sought in the form of the following modified Frobenius type series in terms of the variable x1+py1 as follows:

Out of the various combinations, such as (a′, b′, c′), (a, b, c), (a′, b, c), (a, b′, c), (a, b, c′), (a′, b′, c), (a′, b, c′), and (a, b′, c′), only the first two groupings can produce meaningful solutions, for the mode I/II and mode III loading cases, respectively. This step permits separation of the mode III from the modes I/II. The first grouping is described below, while the second one has already been employed for the antiplane shear case [59,60,61].

The solution to the system of coupled partial differential equations (5), subjected to the far-field mode I/II loading, can now be sought in the form of the following modified Frobenius type series in terms of the variable x1+py1 as follows [22,53,54,60], although unlike in the case of isotropic materials [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,55], the x₁ and y₁ variables are no longer separable:

Here, the combined variable x1+py1 represents an affine transformation in the same spirit as that by Eshelby et al. [62], Stroh [16] and Shih et al. [17], although these authors have employed completely different techniques. On substitution of Eqs. (8) into Eqs. (5), and equating the coefficients of ${(x_1+p{y}_1)}^{s+2n-2}$, the following set of recurrent relationships can be derived:

which for n = 0, reduces to

The characteristic equation for the coupled partial differential equations (2) or (5) can now be written as follows: in which the normalized elastic parameter, $\chi$, is given by in which E₁ is Young’s modulus in the x direction, G₁₂ is the shear modulus in the x-y plane, while ν₁₂ is the major Poisson’s ratio in the x-y plane. ν₁₃ and ν₃₁ denote the major and minor Poisson’s ratios in the x-z plane, while ν₃₂ represents the minor Poisson’s ratio in the y-z plane. $\chi$ can also be expressed in terms of the inverse anisotropic ratio (in the x-y plane), $\lambda$, as follows: where $\lambda$, is defined as

Eq. (11) has either (a) four complex or (b) four imaginary roots, depending on whether: (a) λ < 1 or equivalently, or

(b) λ > 1 or equivalently,

λ= 1 or χ = 1 represents the degenerate isotropic material case, for which the solution is available in Chaudhuri and Xie [25].