1. Introduction

2. Statement of the Problem

3. Singular Stress Fields in the Vicinity of a Crack Front Weakening a Bicrystalline Superlattice with Orthorhombic Phases under General Loading

4. Singular Stress Fields in the Vicinity of a (010)[001] Through-Thickness Crack Front Propagating under Mode I (Extension/Bending) and Mode II (Sliding Shear/Twisting) in [100] Direction

5. Satisfaction of Crack Face Boundary and Interfacial Contact Conditions

5.1. Both Crystal Layers with Complex Roots

For the case of both crystal layers with complex roots, substitution of Eqs. (A6a,b) as well as Eqs. (A2b,c) in conjunction with Eqs. (A7), into the left and right hand sides of Eqs. (4a,b) would yield four homogeneous equations. In addition, substitution of Eqs. (A2b,c) in conjunction with Eqs. (A7), into the left hand sides of Eqs. (5a,b) would yield another set of four homogeneous equations. These can be expressed in the compact form as follows:

[Δ(s)]{A_ij}=0.

(21a)

The existence of a nontrivial solution for A_ij Arequires vanishing of the coefficient determinant

|Δ(s)|=0.

(21b)

∆(s) is an 8×8 matrix involving s in a transcendental form. The solution that has physical meaning is given by 0< Re(s) < 1. Here s = 0.5 ± iε, in which e is obtained from the following relationships, obtained by equating the real and imaginary parts:

Real Part:

5.2. Both Crystal Layers with Imaginary Roots

Similarly, for the case of both crystal layers with imaginary roots, substitution of Eqs. (B5a,b) as well as Eqs. (B2b,c) in conjunction with Eqs. (A7), into the left and right hand sides of Eqs. (4a,b) would also yield four homogeneous equations. Additionally, substitution of Eqs. (B2b,c), into the left hand sides of (5a,b) would yield another set of four homogeneous equations. Proceeding in a similar manner, one obtains s = 0.5 ± iε′π, in which ε′ is obtained from the relationship:

5.3. Top Crystal (Layer 1) with Complex Roots and Bottom Crystal (Layer 2) with Imaginary Roots

For the case of top crystal (layer 1) with complex roots and bottom crystal (layer 2) with imaginary roots, substitution of Eqs. (A6a,b) as well as Eqs. (A2b,c) in conjunction with Eqs. (A7), into the left, and that of Eqs. (B5a,b) as well as Eqs. (B2b,c), in conjunction with Eqs. (A7) into the right hand sides of Eqs. (4a,b) would yield four homogeneous equations.

In addition, substitution of Eqs. (A2b,c) in conjunction with Eqs. (A7), into the left hand sides of Eqs. (5a) would yield another set of two homogeneous equations. Likewise, substitution of Eqs. (B2b,c), into the left hand sides of (5b) would yield the remaining set of two homogeneous equations. Proceeding in a similar manner as above, one obtains s = 0.5 ± iε¯π, in which ε¯ is obtained from the relationship:

in which K_I/IIj,2D and G_I/IIj,2D are available in the published literature; see e.g., Wu [10] and Wang et al. [13]. In what follows, the primary focus will be on three-dimensionality and lattice (more specifically super-lattice) fracture aspects of bi-crystals.

6. Boundary Conditions on the Bicrystalline Superlattice Plate Surfaces and Through-Thickness Distribution of Singular Stress Fields

6.1. General Distributed Far-Field Loading

By using the traction-free boundary conditions at the top and bottom surfaces, z* = z/h = +1, of the plate, given by Eqs. (3), the general form of Dbj(z), defined in Eq. (A3), can be obtained. The stress field that satisfies traction-free boundary conditions at z* = ± 1, in the vicinity of the front of a bi-material interface crack under extension/sliding shear, can be recovered if in Eqs. (A3):

D_bj(z) = D_{bs j}(z) = D_2jcos(αz), j = 1, 2

(23)

By substituting Eq. (A2d) or Eq. (B2d) into the boundary condition on the singular-stress-free surface of a cracked bicrystalline superlattice plate, given by Eq. (3), the general form of D_{bs j}(z) can be obtained as follows:

(24)

Hence, K_I = K_Is and K_II = K_IIs represent symmetric "stress intensity factors". If the odd functions are selected from D_b(z), that satisfies traction-free boundary condition at z^* = ± 1, it can yield the out-of-plane bending/twisting case given by

D_bj(z) = D_{ba j}(z) = D_1jsin(αz), j = 1, 2

(25)

D_{ba j}(z) that satisfies the singular-stress-free conditions on the cracked bicrystalline superlattice plate surfaces is given by

(26)

Here K_I = K_Ia and K_II = K_IIa are anti-symmetric stress intensity factors. Eqs. (25) and (26) are valid, provided the loading function vanishes at z^* = 0, thus eliminating the possibility of discontinuity of the function at z^* = 0. In the presence of discontinuity of the function at z^* = 0, can be written as follows:

(27)

If the singular-stress-free boundary conditions, given by Eqs. (3) on the cracked bicrystalline superlattice plate surfaces are satisfied, all the displacements also vanish on these surfaces in the vicinity of the front of a bicrystalline interface crack.

6.2. Hyperbolic Sine Distributed Far-Field Loading

Alternatively, since D_bs(z^*) has no discontinuity at z^* = 0,, it can also be derived as given below:

(32)

7. Singular Stress Fields in the Vicinity of a Through-Crack Front Propagating under Mode III (Anti-plane Shear) in [100] Direction

8. Crack Path Stability/Instability Criteria

8.1. Necessary Condition −− Griffith-Irwin theory-based crack deflection criterion

The important issue of a cleavage plane being deemed easy or difficult can be related to a crack deflection criterion, which is based on the relative fracture energy (or the energy release rate) available for possible fracture pathways [14]. The deflection or kinking of a crack from the cleavage system 1 to the cleavage system 2 is favored if (however. not iff, i.e., if and only if ):

(33)

in which G_j and G_j, j = 1, 2, are energy release rate (fracture energy) and surface energy, respectively, of the j^th cleavage system.

It may be noted here that experimental determination of critical surface energy, G_j, j = 1, 2, of the component phase or the corresponding interfacial energy, G_int, of a bicrystalline superlattice can sometimes be notoriously challenging, due to the presence of micro-to-nano scale defects, such as porosity, dislocation, twin boundaries, misalignment of bonds with respect to the loading axis, and the like.  In contrast, the bond shear strain at superlattice crack deflection, y_bd^s, and superlattice crack deflection (SCD) barrier, ΔK^s*, to be discussed below, are, relatively speaking, much easier in comparison to determination of the critical surface or interfacial energy.

8.2. Sufficient Condition

The above Griffith-Irwin theory-based crack deflection criterion condition (albeit being still very useful and widely employed) is not accepted as a sufficient condition for a cleavage system deemed to be easy or difficult for crack propagation in single crystals [60], the most enigmatic being the {100} cleavage of BCC transition metals, especially macroscopically isotropic tungsten (W). Although the close packed {110} surface has lower surface energy, the preferred cleavage plane has experimentally been observed by Hull et al. [61] to be {100} (see also Riedle et al. [60]). For W, {100}〈001〉 ({crack plane}<crack front>) is deemed to be the preferred cleavage system for reasons of having d²sp³ hybridized orbitals, while {-110}<001> is considered to be difficult for crack propagation.

Atomistic scale modeling of cracks requires consideration of both the long-range elastic interactions and the short-range chemical reactions. The Griffith theory does not take the latter into account [49]. Secondly and more importantly, fracture criteria derived from equilibrium theories such as the Griffith (thermodynamics-based) energy balance criterion is not equipped to meet the sufficiency condition, because of the prevailing non-equilibrium conditions such as physico-chemical reactions during crack propagation. Hence, such criteria can only be regarded as necessary conditions for fracture, but not as sufficient [60,62]. The effect of short-range chemical reactions can obviously be encapsulated by atomic scale simulations, such as the investigation of low-speed propagation instabilities in silicon using quantum-mechanical hybrid, multi-scale modelling due to Kermode et al. [62], which, however, entails extensive computational and other resources. Alternatively, and more importantly, such short-range interactions can also be captured by the elastic properties-based parameters (with a few exceptions), such as the planar anisotropic ratio, A_j, or equivalently, the normalized elastic parameter, k_j, j = 1, 2 [49]. General theory behind these characteristics pertaining to structural chemistry of crystals are available in well-known treatises (see e.g., [64,65,66]). More specifically, the elastic properties of superconducting YBa₂Cu₃O_7-d are strongly influenced by oxygen non-stoichiometry (as well as various structural defects).

A single dimensionless parameters, such as the planar anisotropic ratio, A_j, or equivalently, the normalized elastic parameter, k_j, j = 1, 2, can serve as the Holy Grail quantity for an a priori determination of the status of a cleavage system to be easy or difficult, very much akin to Reynold’s number for fluid flow problems, crossing a critical value of which signifies transition from one regime to another. Here, the planar anisotropic ratio, A_j, or equivalently, normalized elastic parameter, k_j, j = 1, 2, for a (010)[001]×[100] cleavage system, crossing the critical value of 1 or , j = 1, 2, respectively, signifies transitioning from self-similar crack growth or propagation to crack deflection or turning from a difficult cleavage system onto a nearby easy one. This is a significant qualitative as well as quantitative improvement over two-parameters based models, suggested by earlier researchers e.g. [13], in the context of two-dimensional anisotropic fracture mechanics.

Finally, just as the introduction of Reynold’s number facilitated design and setting up of experiments in addition to experimental verification of analytical and computational solutions in fluid dynamics, the accuracy and efficacy of the available experimental results on elastic constants of monocrystalline superconducting YBa₂Cu₃O_7-d, measured by modern experimental techniques with resolutions at the atomic scale or nearly so, such as X-Ray diffraction [67], ultrasound technique [68,69,70], neutron diffraction [71]/scattering [72], Brillouin spectroscopy [73,74]/scattering [75], resonant ultrasound spectroscopy [76,77] and the like, in a way best suited to preserve the characteristics associated with short-range interactions [59], is assessed with a powerful theoretical analysis on crack path stability/instability, in part based on a single dimensionless parameter, such as the planar anisotropic ratio, A_j, or equivalently, the normalized elastic parameter, k_j, j = 1, 2.

9. Numerical Results and Discussions

9.1. Structure-Fracture Property Relations for Certain Model Bicrystalline Superlattices

In what follows, 22 model bicrystal (superlattice) cleavage systems, each comprising a nano-film deposited on a substrate, are investigated. Table 1 lists the structures and elastic stiffness constants (with respect to <100> axes) of these mono-crystalline materials. Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9, Table 10, Table 11, Table 12, Table 13, Table 14, Table 15, Table 16, Table 17, Table 18 and Table 19 list the cleavage systems, (crack plane)[crack front]x[initial propagation direction], and elastic stiffness constants (with respect suitably rotated coordinates) of the component phases of the bicrystalline superlattice systems 1-22. For example, the hexagonal substrate or material 2 is rotated such that {110} (a prism plane) is parallel to the crack face. It may be noted that the rotated crystal displays tetragonal type symmetry.

The bicrystalline superlattice systems investigated here are namely as follows: (i) Au (gold), nano-layer/film or material 1 deposited on Si₃N₄ (silicon nitride), substrate or material 2 (Tables 2–5), (ii) Au (nano-layer/film) deposited on substrate MgO (magnesium oxide) (Tables 6–9), (iii) YBa₂C₃O₇ (tetragonal/fully oxidized or non-superconducting YBa₂Cu₃O₇, in short YBCO^T) nano-layer/film, deposited on substrate Si₃N₄ (Table 10), (iv) YBa₂C₃O₇ (nano-layer/film) deposited on substrate, SrTiO₃ (Tables 11–14), (v) YBa₂C₃O_7-d (superconducting YBCO, in short YBCO) nano-layer/film, deposited on substrate Si₃N₄ (Table 15), (vi) YBa₂C₃O_7-d, nano-layer/film deposited on substrate MgO (Tables 16, 17), and (vii) YBa₂C₃O_7-d (nano-layer/film) deposited on substrate, SrTiO₃ (strontium titanate) (Tables 18–23),. For the bicrystalline superlattice systems under investigation, the computed mode I, II, or mixed mode I/II order of stress singularity, λ_i = 1-S_i, i = 1, 2, is found to be 0.5±ie or o.5.  In contrast, the computed mode III order of stress singularity, λ₃ = 1–s₃, is always equal to 0.5.

Table 20. Cleavage system (crack plane)[crack front]x[initial propagation direction], and elastic stiffness constants of the component phases of the bicrystalline superlattice system 19.

Table 20. Cleavage system (crack plane)[crack front]x[initial propagation direction], and elastic stiffness constants of the component phases of the bicrystalline superlattice system 19.

Material (j) #	Single Crystal Phase	Cleavage System	$C_{11}^{\prime }$ (GPa)	$C_{22}^{\prime }$ (GPa)	$C_{33}^{\prime }$ (GPa)	$C_{12}^{\prime }$ (GPa)	$C_{13}^{\prime }$ (GPa)	$C_{23}^{\prime }$ (GPa)	$C_{44}^{\prime }$ (GPa)	$C_{55}^{\prime }$ (GPa)	$C_{66}^{\prime }$ (GPa)
1	YBa2C3O7-d (Ortho-rhombic)	010)[001] x[100]	231.0	268.0	186.0	66.0	71.0	95.0	49.0	37.0	82.0
2	SrTiO3 (Simple Cubic)	(010)[001]x[100]	348.17	348.17	348.17	100.64	100.64	100.64	454.55	454.55	454.55

Table 21. Cleavage system (crack plane)[crack front]x[initial propagation direction], and elastic stiffness constants of the component phases of the bicrystalline superlattice system 20.

Table 21. Cleavage system (crack plane)[crack front]x[initial propagation direction], and elastic stiffness constants of the component phases of the bicrystalline superlattice system 20.

Material (j) #	Single Crystal Phase	Cleavage System	$C_{11}^{\prime }$ (GPa)	$C_{22}^{\prime }$ (GPa)	$C_{33}^{\prime }$ (GPa)	$C_{12}^{\prime }$ (GPa)	$C_{13}^{\prime }$ (GPa)	$C_{23}^{\prime }$ (GPa)	$C_{44}^{\prime }$ (GPa)	$C_{55}^{\prime }$ (GPa)	$C_{66}^{\prime }$ (GPa)
1	YBa2C3O7-d (Ortho-rhombic)	(010)[001] x[100]	231.0	268.0	186.0	66.0	71.0	95.0	49.0	37.0	82.0
2	SrTiO3 (Simple Cubic)	(10)[001] x[110]	678.96	678.96	348.17	-230.15	100.64	100.64	454.55	454.55	123.77

Table 22. Cleavage system (crack plane)[crack front]x[initial propagation direction], and elastic stiffness constants of the component phases of the bicrystalline superlattice system 21.

Table 22. Cleavage system (crack plane)[crack front]x[initial propagation direction], and elastic stiffness constants of the component phases of the bicrystalline superlattice system 21.

Material (j) #	Single Crystal Phase	Cleavage System	$C_{11}^{\prime }$ (GPa)	$C_{22}^{\prime }$ (GPa)	$C_{33}^{\prime }$ (GPa)	$C_{12}^{\prime }$ (GPa)	$C_{13}^{\prime }$ (GPa)	$C_{23}^{\prime }$ (GPa)	$C_{44}^{\prime }$ (GPa)	$C_{55}^{\prime }$ (GPa)	$C_{66}^{\prime }$ (GPa)
1	YBa2C3O7-d (Ortho-rhombic)	(00)[001] x[010]	268.0	231.0	186.0	66.0	95.0	71.0	37.0	49.0	82.0
2	SrTiO3 (Simple Cubic)	(010)[001]x[100]	348.17	348.17	348.17	100.64	100.64	100.64	454.55	454.55	454.55

Table 23. Cleavage system (crack plane)[crack front]x[initial propagation direction], and elastic stiffness constants of the component phases of the bicrystalline superlattice system 22.

Table 23. Cleavage system (crack plane)[crack front]x[initial propagation direction], and elastic stiffness constants of the component phases of the bicrystalline superlattice system 22.

Material (j) #	Single Crystal Phase	Cleavage System	$C_{11}^{\prime }$ (GPa)	$C_{22}^{\prime }$ (GPa)	$C_{33}^{\prime }$ (GPa)	$C_{12}^{\prime }$ (GPa)	$C_{13}^{\prime }$ (GPa)	$C_{23}^{\prime }$ (GPa)	$C_{44}^{\prime }$ (GPa)	$C_{55}^{\prime }$ (GPa)	$C_{66}^{\prime }$ (GPa)
1	YBa2C3O7-d (Ortho-rhombic)	(00)[001] x[010]	268.0	231.0	186.0	66.0	95.0	71.0	37.0	49.0	82.0
2	SrTiO3 (Simple Cubic)	(10)[001] x[110]	678.96	678.96	348.17	-230.15	100.64	100.64	454.55	454.55	123.77

In the FCC metal nano-film, Au, listed in Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9 and Table 24, Table 25, Table 26, Table 27, Table 28, Table 29, Table 30 and Table 31, all the orbitals belong to the d-block with partially filled d-shells [66]. In a monocrystalline FCC metal, the bonds are oriented along the face diagonals, <110>. Such a metal contains linear chains of near-neighbor bonds in these directions, resulting in higher elastic stiffness constants along them. As shown in Tables 20, 25, 28 and 29, A = 2.8522 > 1, and k = 4.9777 > $\sqrt{c_{22}/c_{11}}$ = 1 for Au, giving rise to complex roots for a {010}<001>x<100> through-crack. Similar calculations yield A′ = 0.3494 < 1, and k′ = 0.2487 < $\sqrt{c_{22}^{\prime }/c_{11}^{\prime }}$ = 1 for Au, giving rise to imaginary roots for the {10}<001>x<110> through-crack [49], as can be seen from Tables 26, 27, 30 and 31. It can then be inferred that {10}<001>x<110> would constitute an easy cleavage system, while {010}<001>x<100> would be deemed difficult.

The ionic crystal MgO (an alkaline earth metal oxide), used as a substrate, which is listed in Tables 6–9, 16 and 17, in addition to in Tables 28–31, 38 and 39,  is a structurally of rock salt type, but is an exception to the general rule for ionic crystals, such as  alkali halides (e.g., NaCl and KCl), with the rock salt structure [49,64]. The reason is, as explained by Newnham [64], due to Mg2+ (and also Li+) being small cations, which permit the anions to be in contact with one another and consequently, restrict bending actions. As a result, elastic stiffness coefficients in the <110> and <111> directions become larger than their <100> counterparts. This is in contrast to NaCl and KCl, wherein Cl- anions are not in contact. Additionally, the importance of anion-anion forces were pointed out by Weidner and Simmons [79]. These researchers have found, in connection with the computation of elastic properties of several alkali halides from a two-body central force model, the necessity to include anion-anion interactions in addition to cation-anion forces. As shown in Tables 29, 31 and 39, A′ = 0.6513 and k′ = 0.6297 are both less than unity  (A′ < 1, k′ < $\sqrt{c_{22}^{\prime }/c_{11}^{\prime }}$ = 1) giving rise to imaginary roots for the ({10)[001]x[110] through-crack. This is in contrast to A =  1.5354 and k = 1.8329 being both larger than unity  (A > 1,  k > $\sqrt{c_{22}/c_{11}}$ = 1), giving rise to complex roots for the (010)[001]x[100] through-crack, as shown in Tables 28, 30 and 38. It can then be inferred that {10}<001>x<110> would constitute an easy cleavage system, while {010}<001>x<100> would be deemed difficult.

The next substrate, perovskite, Strontium titanate (SrTiO₃), is listed in Tables 11–14  and 18–23 as well as in Tables 33–36 and 40-45. SrTiO₃ has at room temperature, an ideal cubic perovskite structure with TiO₆ octahedra being connected by straight chains [63]. Tables 34, 36 and 41 show that A′ = 0.2723 and k′ = 0.3614 are both less than unity (A′ < 1, k′ < $\sqrt{c_{11}/c_{11}^{\prime }}$ = 1), giving rise to imaginary roots for the (10)[001]x[110] through-crack. This is in contrast to A = 3.6727  and k = 16.1473 being both larger than unity  (A > 1,  k > $\sqrt{c_{22}/c_{11}}$  = 1), giving rise to complex roots for the (010)[001]x[100] through-crack, as shown in Tables 33, 35 and 40. It can then be inferred that {10}<001>x<110> would constitute an easy cleavage system, while {010}<001>x<100> would be deemed difficult.

The third and last substrate studied here is hexagonal close-packed (HCP) Si₃N₄, which is listed in Table 2, Table 3, Table 4 and Table 5, 10 and 15 as well as in Table 24, Table 25, Table 26 and Table 27, 32 and 37. Tables 24, 26 and 32 show that A′ = 0.7433 < 1 and k′ = 0.9206 < $\sqrt{c_{22}^{\prime }/c_{11}^{\prime }}$ = 1.3226, giving rise to imaginary roots for the (001)[00] x[00] through-crack. In a similar vein, A′ = 0.7433 < 1 and k′ = 0.5263 < $\sqrt{c_{22}^{\prime }/c_{11}^{\prime }}$ = 0.5717, giving rise to imaginary roots for the (00)[00]x[001] through-crack, as shown in Tables 25, 27 and 37. It can then be inferred that {001}<00>x<00> and {00}<00>x<001> would both constitute easy cleavage systems.

The fully oxidized (non-superconducting) tetragonal YBa₂C₃O₇, the second nano-film investigated here is listed in Table 10, Table 11, Table 12, Table 13 and Table 14 and also in Table 32, Table 33, Table 34, Table 35 and Table 36. Granozio and di Uccio [83] have also presented approximate theoretical results of fully oxidized YBCO’s (d = 0, 1), and concluded that the three lowest surface energies follow the inequality:  (001) < g (100) < g (010). Furthermore, based on experimental results from transmission electron microscopy [84], X-ray photo-emission microscopy [85], low-energy ion scattering spectroscopy [86], and surface polarity [87] analyses performed on fully oxidized YBa₂C₃O₇ crystals, these authors [83] have shown that the low energy cut is between the Ba=O and Cu=O planes. Tables 32–34 show that A = 1.9077 and k = 3.1514 are both larger than unity (A > 1, k > $\sqrt{c_{22}/c_{11}}$ = 1), giving rise to complex roots for the (010)[001]x[100] through-crack. This is in contrast to A′ = 0.7647 and k′ = 0.7112 being both less than unity (A′ < 1, k′ < $\sqrt{c_{11}/c_{11}^{\prime }}$ = 1), giving rise to imaginary roots for the (10)[001]x[110] through-crack. as shown in Table 35 and Table 36. It can then be inferred that {10}<001>x<110> would constitute an easy cleavage system, while {010}<001>x<100> would be deemed difficult.

Finally, the third nano-film, YBa₂C₃O_7-d (orthorhombic), investigated here is a high TC superconductor, and is listed in Table 15, Table 16, Table 17, Table 18, Table 19, Table 20, Table 21, Table 22 and Table 23, in addition to in Table 37, Table 38, Table 39, Table 40, Table 41, Table 42, Table 43, Table 44 and Table 45. As can be seen from Table 2, Table 3, Table 4, Table 5, Table 6 and Table 7 of Chaudhuri [59], all the cleavage systems are predicted to be easy, which are in agreement with the experimentally observed fracture characteristics of YBa₂C₃O_7-d due to Cook et al. [80], Raynes et al. [81] and Goyal et al. [82] among others; see also Granozio and di Uccio [83] for a summary of the available experimental results. Here, A′ = 0.764 < 1 and k′ = 0.5784 < $\sqrt{c_{22}^{\prime }/c_{11}^{\prime }}$ = 0.8331, giving rise to imaginary roots for the (001)[100]x[010] through-crack, as shown in Tables 37–41; see also Table 6 of Chaudhuri [59]. Similarly, A = 0.8971 < 1 and k = 0.9406 < $\sqrt{c_{22}^{\prime }/c_{11}^{\prime }}$ = 1.0771, giving rise to imaginary roots for the (010)[001]x[100] through-crack, as shown in Table 42 and Table 43; see also Table 2 of Chaudhuri [59]. Likewise, A′ = 0.8971 < 1 and k′ = 0.817 < $\sqrt{c_{22}^{\prime }/c_{11}^{\prime }}$ = 0.9284, giving rise to imaginary roots for the (00)[001]x[010] through-crack, as shown in Table 44 and Table 45; see also Table 4 of Chaudhuri [59]. It can then be inferred that {001}<100>x<010> would constitute an easy cleavage system.

Table 46 summarizes the nature (easy or difficult) of the cleavage system in component phases of the afore-mentioned bicrystalline superlattice systems. It also lists the real or complex eigenvalues, s = 0.5±ie, of these bicrystalline superlattices. These results suggest that the interfacial cracks would propagate in mixed (I/II) mode, primarily when both the component phases are characterized by difficult cleavage systems (complex roots), the exception being perovskite SrTiO₃ serving as the substrate (for YBa₂C₃O_7-d or YBa₂C₃O₇ nano-films). A plausible reason for this exceptional behavior of SrTiO₃ may lie in its unusually high shear stiffness, c₆₆, which is substantially greater than its longitudinal stiffness, c₁₁, in combination with easiest cleavage system of YBa₂C₃O_7-d (or YBa₂C₃O₇). This is in contrast with other cubic mono-crystals, such as FCC rock salt MgO. This fact results in negative Poisson’s ratio effect, when rotated about the [001] axis by 45^o, which is not generally encountered in cubic crystal elasticity. Additionally, this behavior is also in contrast with other easy cleavage systems of YBa₂C₃O_7-d deposited on the same 45^o rotated SrTiO₃, as shown in Table 43, Table 45 and Table 46.

.

9.2. Superlattice Trapping and Superlattice Crack Deflection (SCD)

The theory of lattice crack deflection (LCD) is discussed in Chaudhuri [49]. Table 47 displays the structures and elastic compliance constants of mono-crystalline FCC transition metal Au, FCC rock salt MgO, cubic perovskite SrTiO₃, HCP ceramic Si₃N₄, fully oxidized tetragonal YBa₂C₃O₇, and orthorhombic (superconducting) YBa₂C₃O_7-d [49,59,78]. Table 48 shows the results for computed lattice crack deflection/deviation (LCD) parameters (energy barrier) and associated bond shear strains at crack deviation from a difficult cleavage system to an easy one, and their correlations with the anisotropic ratios relating to the difficult cleavage system along with Bravais lattice and structure. Only two crack systems are considered: {010}〈001〉x<100]> and {10}<001> x<110]>.

For mono-crystalline FCC transition metals, {010}〈001〉x<100]> is deemed to be a difficult cleavage system for reasons explained above, while {10}<001>x<110]> is considered to be the preferred one for crack propagation. This is illustrated in Figure 26(a) of Chaudhuri [49]. Nonvanishing lattice crack deflection (LCD) energy barrier implies that a {010}〈001〉x<100]> through-crack in such single crystals would not deflect right at the appropriate Griffith/Irwin critical stress intensity factor (K_c) for mixed mode propagation because of the lattice effect, but would require additional bond shear strains for Au (Table 48). In the case of nonvanishing lattice crack deflection (LCD) barrier, e.g., in Au with moderately high anisotropic ratio, A = 2.8481 > 1, the difficult {010}〈001〉x<100]> crack may initially get lattice trapped and/or propagate in a “difficult” manner till an applied load somewhat higher than its Griffith mixed mode counterpart is reached, and then only deflect into the easy cleavage system, {10}lt;001>x<110]>. In addition, the bond breaking would not be continuous but abrupt. In contrast, for the same crystal with the very low modified anisotropic ratio, A‣ = 0.3494 < 1, lattice crack deflection (LCD) barrier vanishes and the easy {10}<001>x<110]> crack would begin to propagate right at the Griffith/Irwin critical stress intensity factor. There would be no crack turning.

Bicrystals form superlattices, which has not been discussed, to the author’s knowledge, in the anisotropic fracture mechanics literature [88]. However, the geometric mean of the two constituent phases would serve as a reasonably accurate procedure for computation of bond shear strain at superlattice crack deviation $y_{bd}^s$, and superlattice crack deflection (SCD) barrier, ∆K^S*; see also Chaudhuri [88]. A rigorous proof of this is currently being worked out, and will be presented in the near future. The numerical results are shown in Table 49. For a bicrystalline superlattice, e.g., Au/MgO (respectively, YBa₂C₃O₇/SrTiO₃), with both difficult cleavage systems, (010)[001]x[100]/(010)[001]x[100], serving as the interface, with the SCD barrier, ∆K^S*, value of 0.7324 (resp. 0.6240), the interfacial crack would encounter a tough interface, and would initially be superlattice-trapped and/or experience a mixed mode propagation in a “difficult” manner till an applied load somewhat higher than its Griffith/Irwin mixed mode interfacial fracture toughness counterpart −− quantified by ∆K^S* −− is reached, and thence deflect into the available easier cleavage system, {10}<001>x<110]>, of the component phase with the lower LCD barrier, ∆K^* = 0.6414 for MgO (resp. 0.5114 for SrTiO₃). In addition, the bond breaking would not be continuous but abrupt. In contrast, for the same bicrystalline superlattice, Au/MgO, with both easy (10)[001]x[110]/(10)[001]x[110] cleavage systems serving as the interface, the SCD barrier, ∆K^S*, vanishes, and the easy interfacial crack would begin to propagate (in the absence of mode mixity) in a self-similar manner right at the Griffith/Irwin critical stress intensity factor. The bond breaking would be smooth and continuous. Interestingly, for the Au/MgO or YBa₂C₃O₇/SrTiO₃ superlattice, with one easy and the second one difficult, either (010)[001]x[100]/(10)[001]x[110] or (10)[001]x[110]/(010)[001]x[100] cleavage systems serving as the interface, the SCD barrier, ∆K^S*, also vanishes, and the interfacial crack would begin to propagate (in the absence of mode mixity) on the easier side of and parallel to the interface at the Griffith/Irwin critical stress intensity factor. The bond breaking would be smooth and continuous on the easier side, but discontinuous and abrupt on the tougher side of the interface.

For the Au/Si₃N₄ superlattice, with both easy cleavage systems, either (10)[001]x[110]/(001)[00]x[00] or (10)[001]x[110]/(00)[00]x[001], serving as the interface, the SCD barrier, ∆K^S*, again vanishes, and the easy interfacial crack would begin to propagate (in the absence of mode mixity) in a self-similar manner right at the Griffith/Irwin critical stress intensity factor. The bond breaking would be smooth and continuous. As before, the same superlattice, with one easy and the second one difficult, either (010)[001]x[100]/(001)[00]x[00] or (010)[001]x[100]/(00)[00]x[001] cleavage systems serving as the interface, ∆K^S* also vanishes, and the interfacial crack would begin to propagate (in the absence of mode mixity) on the easier side of and parallel to the interface at the Griffith/Irwin critical stress intensity factor. The bond breaking would be smooth and continuous on the easier side, but discontinuous and abrupt on the tougher side of the interface.

Similar situation prevails for YBa₂C₃O₇/Si₃N₄ superlattice, with one difficult and the second one easy, (010)[001]x[100]/(00)[00]x[001], serving as the interface.

For the orthorhombic perovskite/HCP YBa₂C₃O_7-d/Si₃N₄ superlattice, with both easy (001)[100]x[010]/(001)[00]x[00] cleavage systems serving as the interface, the SCD barrier, ∆K^S*, again vanishes, and the easy interfacial crack would begin to propagate (in the absence of mode mixity) in a self-similar manner right at the Griffith/Irwin critical stress intensity factor. Similar results follow for the orthorhombic perovskite/FCC rock salt YBa₂C₃O_7-d/MgO superlattice, with both easy (001)[100]x[010]/(10)[001]x[110] cleavage systems serving as the interface.

For the perovskite orthorhombic/FCC rock salt MgO bicrystalline superlattice, with both easy (001)[100]x[010]/(10)[001]x[110] cleavage systems serving as the interface, ∆K^S* would again vanish, and the resulting easy interfacial crack would begin to propagate (in the absence of mode mixity) in a self-similar manner right at the Griffith/Irwin critical stress intensity factor. The bond breaking would be smooth and continuous. As before ∆K^S* for the same superlattice, with one easy and the second one difficult, (001)[100]x[010]/(010)[001]x[100] would also vanish, and the interfacial crack would begin to propagate (in the absence of mode mixity) on the easier side of and parallel to the interface at the Griffith/Irwin critical stress intensity factor. The bond breaking would be smooth and continuous on the easier side, but discontinuous and abrupt on the tougher side of the interface.

Finally, for the perovskite orthorhombic/cubic perovskite YBa₂C₃O_7-d/SrTiO₃ (respectively, YBa₂C₃O₇/SrTiO₃) bicrystalline superlattice, with both easy (001)[100]x[010]/(10)[001]x[110] (resp., (10)[001]x[110]/(10)[001]x[110]) cleavage systems serving as the interface, ∆K^S* would again vanish, and the resulting easy interfacial crack would experience a mixed mode propagation/growth right at the Griffith/Irwin critical complex stress intensity factor. Rice [9] has discussed the computation and interpretation of the resulting complex stress intensity factor (S.I.F.) for an isotropic bimaterial interface crack; see Sec. 9.3 below. The bond breaking is expected to be smooth and continuous. The reason for this exceptional behavior of SrTiO₃ lies, as has been explained above, in its unusually high shear stiffness, c₆₆, which is substantially greater than its longitudinal stiffness, c₁₁. However, ∆K^S* for the same superlattices, with one easy and the second one difficult, (001)[100]x[010]/(010)[001]x[100] for YBa₂C₃O_7-d/SrTiO₃, or ((10)[001]x[110]/(010)[001]x[100] and (010)[001]x[100]/((10)[001]x[110] for YBa₂C₃O₇/SrTiO₃ also vanish, and the interfacial crack would begin to propagate (in the absence of mode mixity) on the easier side of and parallel to the interface at the Griffith/Irwin critical stress intensity factor. The bond breaking would be smooth and continuous on the easier side, but discontinuous and abrupt on the tougher side of the interface.

9.3. Complex Stress Intensity Factor (S.I.F.) and Raman Spectroscopic Surface Measurement

9.3.1. Complex Stress Intensity Factor (S.I.F.)

Rice [9] has discussed the validity of the complex stress intensity factor, K = K_I + iK_II, as a characterizing parameter in the context of a two-dimensional bi-material interface crack for material systems (of isotropic phases), for small scale yielding and small-scale contact zones at the crack tip. The proportion of opening and shearing modes at the interfacial crack tip is characterized by means of the phase angle, given by ψ = Arctan(K_II/K_I). Wang et al. [13] and Wu [90] have extended Rice’s [9] analysis, and presented corresponding relations for interface cracks in monoclinic composites.

9.3.2. Raman Spectroscopic Surface Measurement of Carbon/Graphite Fiber-Epoxy Interfacial Bond

It has been argued by Chaudhuri et al. [36], in connection with Raman spectroscopic study of carbon-fiber epoxy interfacial debond, that in real-life composite materials, the fiber-matrix interface (a sharp material discontinuity) is replaced by an interphase region (of the order of 0.5 mm thickness), which permits us to ignore the contact of the Comninou [90] type (e = 0). Various factors that contribute to the formation of this interphase region are molecular entanglement following interdiffusion, electrostatic attraction, cationic groups at the ends of molecules attracted to an anionic surface that result in polymer matrix orientation at the fiber surface, chemical reaction, mechanical keying, etc. [91]. Additionally, the residual stress and coupling agents (e.g., silanes on glass fibers) applied to the fiber surface have pronounced effects on the interfacial bond strength. Under these circumstances, the macroscopic (mean field, e.g., linear elasticity) analysis presented above may be considered to be only a first approximation to the detailed microscopic (fluctuating) state of stress at the carbon fiber-epoxy debond tip. It may further be noted that the oscillatory behavior characterizing interpenetration of the component phases is not that unrealistic after all, but is rather a first-order approximation of the interdiffusion followed by molecular entanglement, chemical bonding and other similar microscopic (kinetic) phenomena studied by materials scientists.

A close scrutiny of two asymptotic solutions for bimaterials involving (i) the complex eigenvalues (e > 0), and (ii) their real counterpart (e = 0), in light of the Raman spectroscopic measurements, reveals that they represent two extreme cases of the interfacial (interphase) region of the carbon/graphite fibers with a polymeric matrix such as epoxy [36]. While the case (i) corresponds to an idealized carbon fiber with the surface layer(s) being comprised of completely disordered (amorphous) carbon resembling activated charcoal (R = I_A1g/I_E2g = 1), the case (ii) represents the idealized version of a graphite fiber with the surface layer(s) consisting of completely ordered (crystalline) carbon lattice resembling stress annealed pyrolite graphite (R = 0). Actual carbon/graphite fibers, both commercial and experimental, fall somewhere in between the two idealized extremes with 0 < R < 1. Therefore, the interfacial debond nucleation and propagation in an actual carbon/graphite fiber/epoxy matrix composite will be governed by solution with e replaced by [36]:

(36)

where I_A1g and I_E2g represent the relative (Raman) intensities of the peaks corresponding to the A_1g (weak) and E_2g (strong) modes, respectively. R represents the degree of chemical functionality (potential for strong or covalent bond formation). R ranges from 0.22 (Morganite I) to 0.85 (Thornel 10), with the industry standard AS4 being 0.811, as reported by Tuinstra and Koenig [92]. For the experimental fibers (grown in a carefully controlled environment to mimic their commercial counterparts) investigated by Chaudhuri et al. [36], R ranges from 0.279 to 0.785.

10. Summary and Conclusions

(iii) A single dimensionless parameters, such as the planar anisotropic ratio, A_j, or equivalently, the normalized elastic parameter, k_j, j = 1, 2, can serve as the Holy Grail quantity for an a priori determination of the status of a cleavage system to be easy or difficult, very much akin to Reynold’s number for fluid flow problems, crossing a critical value of which signifies transition from one regime to another. Here, the planar anisotropic ratio, A_j, or equivalently, normalized elastic parameter, k_j, j = 1, 2, for a (010)[001]×[100] cleavage system, crossing the critical value of 1 or , j = 1, 2, respectively, signifies transitioning from self-similar crack growth or propagation to crack deflection or turning from a difficult cleavage system onto a nearby easy one. This is a significant qualitative as well as quantitative improvement over two-parameters based models, suggested by earlier researchers e.g. [13], in the context of two-dimensional anisotropic fracture mechanics.

(vi) Computed complex eigenvalues, s = 0.5±ie, for Au/MgO, and YBa₂C₃O₇/SrTiO₃, bicrystalline superlattices, with (010)[001]x[100]/(010)[001]x[100] serving as the interface suggest that the corresponding interfacial cracks would propagate in a mixed (I/II) mode. Likewise, for the bicrystalline superlattice, YBa₂C₃O_7-d/SrTiO₃ (resp. YBa₂C₃O₇/SrTiO₃) with (001)[100]x[010]/(10)[001]x[110] (resp. (10)[001]x[110]/(10)[001]x[110]) cleavage systems serving as the interface, the computed eigenvalues are also complex, resulting in a mixed (I/II) mode interfacial crack growth.

(vii) For the bicrystalline superlattice, Au/MgO (respectively, YBa₂C₃O₇/SrTiO₃), with both difficult cleavage systems, (010)[001]x[100]/(010)[001]x[100], serving as the interface, with the bond shear strain at superlattice crack deflection, y_bd^S, value of 0.4710 (resp. 0.5202) and superlattice crack deflection (SCD) barrier ΔK^S*, value of 0.7324 (resp. 0.6240), the interfacial crack would encounter a tough interface, and would initially be superlattice-trapped and/or experience a mixed mode propagation in a “difficult” manner till an applied load somewhat higher than its Griffith/Irwin mixed mode interfacial fracture toughness counterpart −− quantified by ΔK^S* −− is reached, and thence deflect onto the available easier cleavage system, {10}<001>x<110]>, of the component phase with the lower LCD barrier, ΔK^* = 0.6414 for MgO (resp. 0.5114 for SrTiO₃). In addition, the bond breaking would not be continuous but abrupt.

(viii) In contrast, for the perovskite orthorhombic/cubic perovskite YBa₂C₃O_7-d/SrTiO₃ (respectively, YBa₂C₃O₇/SrTiO₃) bicrystalline superlattice, with both easy (001)[100]x[010]/(10)[001]x[110] (resp., (10)[001]x[110]/(10)[001]x[110]) cleavage systems serving as the interface, both bond shear strain at superlattice crack deflection, y_bd^S, and superlattice crack deflection (SCD) barrier, ΔK^S*, vanish, and the resulting easy interfacial crack would experience a mixed mode propagation/growth right at the Griffith/Irwin-based critical complex stress intensity factor (S.I.F.), the computation and interpretation of which is expounded by Rice’s [9] extension from a two-dimensional isotropic bimaterial interface crack to its to anisotropic counterpart. The bond breaking is expected to be smooth and continuous.

(ix) For the Au/Si₃N₄ or YBa₂C₃O₇/Si₃N₄ superlattice, with both easy cleavage systems, either (10)[001]x[110]/(001)[00]x[00] or (10)[001]x[110]/(00)[00]x[001], serving as the interface, both bond shear strain at superlattice crack deflection, y_bd^S, and superlattice crack deflection (SCD) barrier, ΔK^S*, vanish, and the easy interfacial crack would begin to propagate (in the absence of mode mixity) in a self-similar manner right at the Griffith/Irwin critical stress intensity factor. Likewise, a YBa₂C₃O_7-d/Si₃N₄ superlattice, with both easy cleavage systems, either {001}<100>x<010>/(001)[00]x[00] or {001}<100>x<010>/(00)[00]x[001], serving as the interface, would elicit a similar behavior. Other examples include Au/MgO (resp. YBa₂C₃O_7-d/MgO) bicrystalline superlattice with (10)[001]x[110/(10)[001]x[110] (resp. {001}<100>x<010>/(10)[001]x[110] cleavage systems serving as the interface. Similar response also ensues for the YBa₂C₃O_7-d/SrTiO₃ bicrystalline superlattice with (010)[001]x[100]/(10)[001]x[110] or (00)[001]x[010]/(10)[001]x[110] cleavage systems serving as the interface. The bond breaking would be smooth and continuous.

(x) Interestingly, for the Au/MgO or YBa₂C₃O₇/SrTiO₃ superlattice, with one easy and the second one difficult, either (010)[001]x[100]/(10)[001]x[110] or (10)[001]x[110]/(010)[001]x[100] cleavage systems serving as the interface, the SCD barrier, ΔK^S*, also vanishes, and the interfacial crack would begin to propagate (in the absence of mode mixity) on the easier side of and parallel to the interface at the Griffith/Irwin critical stress intensity factor. Similar response ensues for the YBa₂C₃O_7-d/SrTiO₃ bicrystalline superlattice with {001}<100>x<010>/(010)[001]x[100] cleavage systems serving as the interface. Likewise, the Au/Si₃N₄ or YBa₂C₃O₇/Si₃N₄ superlattice with one easy and the second one difficult, either (010)[001]x[100]/(001)[00]x[00] or (010)[001]x[100]/(00)[00]x[001] cleavage systems serving as the interface, produces the same outcome. The bond breaking would be smooth and continuous on the easier side, but discontinuous and abrupt on the tougher side of the interface.

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