On the interlaminar stresses of a composite plate around the neighborhood of a hole

doi:10.1016/0020-7683(89)90076-0

International Journal of Solids and Structures

Volume 25, Issue 10, 1989, Pages 1193-1200

https://doi.org/10.1016/0020-7683(89)90076-0 Get rights and content

Abstract

The author in this paper investigates the 3-D stress field in the immediate vicinity of a bonded interface and the free edge of a hole in a laminated composite plate. The laminates are assumed to be of homogeneous and Isotropic materials, but of different elastic properties. As to loading, a uniform tensile load is applied in the plane of the plate and at points far remote from the hole (shown in Fig. 1).

In constructing the local asymptotic solution, the author assumes the 3-D field in a certain form which then permits a straightforward Williams approach for the determination of the stress singularities. The displacement and stress fields are recovered explicitly and a stress singularity is shown to exist for certain shear moduli ratios of G₂/G₁ In general, the stress singularity is shown to be a function of the respective ratios of the shear moduli and Poisson's ratio. Moreover, the presence of a second singularity is observed which has significant implications for the problem of adhesion. An extension of the results to anisotropic layers is also discussed.

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Nonapplicability of Saint–Venant’s principle is essentially a result of this path dependence of particular solutions of nonhomogeneous singular boundary value problems. There are several classes of problems pertaining to the issue of three-dimensional stress singularity: (i) through-thickness crack [22,23,37–41] and their bimaterial interface [24] as well as trimaterial counterparts [24,28], (ii) through-thickness notches/wedges [12,25–27,31], (iii) trimaterial junction [29,48], (iv) penny shaped crack/anticrack [49–51] and their bimaterial counterparts [52–54], (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 [59–61], 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 [32–36], would result in violation of Saint–Venant’s principle.
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Three-dimensional asymptotic stress fields at the front of a trimaterial junction
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The mathematical difficulties posed by the three-dimensional wedge/crack type problems are substantially greater than their two-dimensional counterparts (to start with the governing PDE’s are much more complicated). There are several classes of problems pertaining to the issue of three-dimensional stress singularity: (i) through-thickness crack [10]/anticrack [11] and their bimaterial interface counterparts [12] as well as the corresponding notches/wedges [13–15], (ii) bimaterial free/fixed straight face (free/fixed edge) [16–18], (iii) penny shaped crack/anticrack [19,20] and their bimaterial counterparts [21–25], (iv) fiber–matrix interfacial debond [26–28], (v) through/part-through hole/rigid inclusion [29,30] as well as their bimaterial counterparts [31,32] and elastic inclusion [33,34] and (vi) matrix cracking and fiber breaks in composites [35]. Only the penny shaped crack/anticrack (and their bimaterial counterparts) and the hole, bimaterial hole and elastic inclusion problems have been adequately addressed in the literature.
A recently developed eigenfunction expansion method is employed for obtaining three-dimensional asymptotic displacement and stress fields in the vicinity of the junction corner front of an infinite pie-shaped trimaterial wedge, of finite thickness, formed as a result of bimaterial (matrix plus reaction product or contaminant) deposit over a substrate or reinforcement. The wedge is subjected to extension/bending (mode I), inplane shear/twisting (mode II) and antiplane shear (mode III) far field loading. Each material is isotropic and elastic, but with different material properties. The material 2 (substrate) is always taken to be a half-space, while the wedge aperture angle of the material 1 is varied to represent varying composition of the bimaterial deposit. Numerical results pertaining to the variation of the mode I, II, III eigenvalues (or stress singularities) with various moduli ratios as well as the wedge aperture angle of the material 1 (reaction product/contaminant), are also presented. Hitherto unavailable results, pertaining to the through-thickness variations of stress intensity factors for symmetric exponentially decaying distributed load and its skew-symmetric counterpart that also satisfy the boundary conditions on the top and bottom surfaces of the trimaterial plates under investigation, bridge a longstanding gap in the stress singularity/interfacial fracture mechanics literature.
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In addition, the three-dimensional singular stress fields near a partially debonded cylindrical rigid fiber [26], and in the vicinity of the circumferential tip of a fiber–matrix interfacial debond [27,28] have also been derived using the same afore-mentioned three-dimensional eigenfunction expansion technique. Finally, the asymptotic solutions pertaining to the stress fields in the neighborhoods of holes [29], elastic inclusions [30] and bimaterial holes [31] have also been shown to be in agreement with their counterparts derived by Folias [32–34], using Lure’s symbolic method. These facts not only lend credence to the validity of the afore-mentioned three-dimensional eigenfunction expansion approach, but also reinforce the afore-mentioned conceptual as well as mathematical similarity of and linkages among the afore-cited classes of three-dimensional stress singularity problems.
An eigenfuntion expansion method is employed for obtaining three-dimensional asymptotic displacement and stress fields in the vicinity of the front of a crack/anticrack discontinuity weakening/reinforcing an infinite pie-shaped trimaterial plate, of finite thickness, formed as a result of bimaterial (matrix/ARC plus reaction product/scatterer) deposit over a substrate (fiber/semiconductor). The wedge is subjected to mode I/II far field loading. Each material is isotropic and elastic, but with different material properties. The material 2 or the substrate is always taken to be a half-space, while the wedge aperture angle of the material 1 is varied to represent varying composition of the bimaterial deposit. Numerical results pertaining to the variation of the mode I/II eigenvalues (or stress singularities) with Young’s moduli ratio, as well as with the wedge aperture angle of the material 1 (reaction product/scatterer) are presented. Hitherto generally unavailable results, pertaining to the through-thickness variations of stress intensity factors or stress singularity coefficients for symmetric exponentially growing distributed load and its skew-symmetric counterpart that also satisfy the boundary conditions on the top and bottom surfaces of the trimaterial plates under investigation, bridge a longstanding gap in the stress singularity/interfacial fracture mechanics literature.
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On the interlaminar stresses of a composite plate around the neighborhood of a hole

Abstract

Int. J. Engng. Sci.

Int. J. Solids Structures

Int. J. Solids Structures

Int. J. Engng. Sci.

Theorie Van de Driedimensionale Spanningstoestand in een Doorboorde Plaat

Edge-bonded dissimilar orthogonal elastic wedges under normal and shear loading

J. Appl. Mech.

Two edge-bonded elastic wedges of different materials and wedge angles under surface Tractions

J. Appl. Mech.

On the three-dimensional theory of cracked plates

J. Appl. Mech.