On three-dimensional singular stress field at the front of a planar rigid inclusion (anticrack) in an orthorhombic mono-crystalline plate

Abstract

A novel eigenfunction expansion technique, based in part on separation of the thicknessvariable 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 anticrack reinforcing an infinite orthorhombic single crystal plate, of finite thickness and subjected to far-field mode I/II loadings. Anticrack-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-anticrack systems reinforcing orthorhombic single crystal plates. Explicit expressions for the singular stresses in the vicinity of the front of a through-thickness anticrack reinforcing an orthorhombic plate, subjected to farfield mode I/II loadings, are presented. Finally, hitherto largely unavailable results, pertaining to the throughthickness variations of stress singularity coefficients corresponding to symmetric and skew-symmetric sinusoidal loads that also satisfy the boundary conditions on the top and bottom surfaces of an orthorhombic mono-crystalline plate under investigation, bridge a Electronic supplementary material The online version of this article (longstanding gap in the stress singularity/fracture mechanics literature.

Effects of thickness and fibre misalignment on compression fracture in cross-ply (very) long cylindrical shells under external pressure

“…Chaudhuri et al [12] and Chaudhuri [6] have investigated kink propagation in glass fibre reinforced unidirectional composites, using eigenfunction expansion techniques [13][14][15][16] applied to fully bonded fibrematrix kink-wedges. A three-dimensional eigenfunction expansion technique, based in part on separation of the thickness-variable and partly using a modified Frobenius-type series expansion [17][18][19][20][21][22][23] in conjunction with the Eshelby-Stroh formalism, has been used to compute the local stress singularity, in the vicinity of a kinked-fibre/matrix trimaterial junction front, representing a measure of the degree of inherent flaw sensitivity of unidirectional carbon fibre reinforced composites under compression [6]. In a polyacrylonitrile (PAN) based carbon fibre such as AS4, micro/nano-kinking, more prominent in the fibre-misaligned regions, is primarily caused by crystallite disorientations [24] and the presence of crystal boundaries, as detected Figure 1.…”

On three-dimensional asymptotic solution, and applicability of Saint–Venant’s principle to pie-shaped wedge and end face (of a semi-infinite plate) boundary value problems

“…It may be noted that since the separated 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 Lebesgue integration) in its interval z 2 ½Àh; h, i.e., admitting singularities weaker than square root (i.e., z ðÀ1=2þeÞ , e > 0 being a very small number), it can be best represented by Fourier series [36][37][38][39][40][41][42]. The latter case is justified by the Riesz-Fischer theorem [43], and its physical implication is that of satisfying the criterion of finiteness of local strain energy and path independence [44].…”

“…There are several classes of problems pertaining to the issue of three-dimensional stress singularity: (i) through-thickness crack [22,23,[37][38][39][40][41] and their bimaterial interface [24] as well as trimaterial counterparts [24,28], (ii) through-thickness notches/wedges [12,[25][26][27]31], (iii) trimaterial junction [29,48], (iv) penny shaped crack/anticrack [49][50][51] and their bimaterial counterparts [52][53][54], (v) through/part-through hole/rigid inclusion [55] as well as their bimaterial counterparts [56,57], (vi) elastic inclusion [58], (vii) fiber-matrix interfacial debond [59][60][61][62], and (viii) island/substrate system [62]. If any of these stress singularities occurs in the gage section of a tensile (or compressive) test specimen, the interaction of such a singularity with its edge face (free edge in two dimension) counterpart at the plate boundary [32][33][34][35][36], would result in violation of Saint-Venant's principle.…”

A combined FEM and three-dimensional asymptotic solution based analysis on stress concentration/intensity around elliptical/circular cylinder shaped surface flaws in stretched plates

“…Only recently, three-dimensional stress singularity problems [16][17][18][19][20][21][22][23] have been solved by introducing a novel eigenfunction expansion technique, which has recently been extended to cubic and orthorhombic/orthotropic and monoclinic/anisotropic materials [24][25][26][27][28]. The same technique has also been utilized to compute the asymptotic stress fields in the neighborhoods of holes [12], elastic inclusions [29], bimaterial holes [9] and island-substrate interface [30].…”