Three-dimensional Asymptotic Stress Field Research Articles

On through-thickness distribution of stress intensity factors and energy release rates in the vicinity of crack fronts

First, an eigenfunction expansion method is presented to obtain three-dimensional asymptotic stress fields in the vicinity of the front of a semi-infinite crack, subjected to the far-field extension/bending, in-plane shear/twisting and antiplane shear loadings, distributed through the plate thickness. A complex variable approach in conjunction with this eigenfunction expansion technique is employed here to derive the heretofore-unavailable through-thickness distribution of stress intensity factors and energy release rates for a center-crack weakening an infinite plate. Interesting numerical results for three-dimensional stress intensity factors and energy release rates for five different far-field loading scenarios are also presented.

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

Three-dimensional asymptotic singular stress fields near the fronts of infinite wedges are presented. This investigation is devoted to establishment of rigorous conditions as to whether the presence of wedge or end face stress singularity, which depends on the prescribed boundary condition, can validate or invalidate the heuristic assumption implicit in Saint–Venant’s principle. This is accomplished by applying a general approach to the solution of canonically singular problems, based on the concept of proper boundary-value problem, the theorem of homogeneous solutions, and classification of boundary value problems (BVP) of three-dimensional elasticity theory into class S (Saint–Venant) or class N (non-Saint–Venant).

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

A novel 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 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 far-field mode I/II loadings, are presented. Finally, hitherto largely unavailable results, pertaining to the through-thickness 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 longstanding gap in the stress singularity/fracture mechanics literature.

Three-dimensional singular stress/residual stress fields at crack/anticrack fronts in monoclinic plates under antiplane shear loading

A recently developed eigenfunction expansion technique, based in part on separation of the thickness-variable, is first developed to derive three-dimensional asymptotic stress field in the vicinity of the front of a semi-infinite through-thickness crack/anticrack weakening/reinforcing an infinite monoclinic plate, of finite thickness and subjected to far-field antiplane shear loading. Crack/anticrack-face boundary conditions and those that are prescribed on the top and bottom (free or fixed) surfaces of the anisotropic (monoclinic) plate are exactly satisfied. Five different through-thickness crack/anticrack-face boundary conditions are considered: (i) slit crack, (ii) anticrack or perfectly bonded rigid inclusion, (iii) transversely rigid inclusion (longitudinal slip permitted), (iv) rigid inclusion in part perfectly bonded, the remainder with slip, and (v) rigid inclusion located alongside a crack. The three-dimensional stress intensity factor for a center-crack, and stress singularity coefficients for/on a center-anticrack are then derived by incorporating an extension of the Stroh type approach in the present analysis. Through-thickness distribution of stress intensity factor and stress singularity coefficient for a crack and an anticrack, respectively, is also presented. Additionally, singular residual stress fields in the vicinity of the fronts of these cracks, anticracks and similar discontinuities are also discussed. Hitherto unavailable expressions for three-dimensional energy release and absorption rates for center-cracks and anticracks are derived by using Irwin’s crack closure and Eshelby’s eigenstrain approach, respectively. A heretofore unavailable expression for the energy release rate on the super-rigid inclusion is derived, using an approach which is analogous, in a reverse sense, to Irwin’s crack closure method. Finally, a new mode III crack deflection/bifurcation criterion is also derived. The crack deviation under antiplane shear loading is strongly correlated with the elastic stiffness constant, c45, of the monoclinic single crystal or off-axis composite lamina concerned.

Three-dimensional asymptotic mode I/II stress fields at the front of interfacial crack/anticrack discontinuities in trimaterial bonded plates

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.

Three-dimensional asymptotic stress fields at the front of a trimaterial junction

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.

Three-dimensional asymptotic antiplane shear stress fields at the front of interfacial crack/anticrack type discontinuities in trimaterial bonded plates

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 type discontinuity weakening/reinforcing an infinite pie-shaped trimaterial plate, of finite thickness, formed as a result of bimaterial (matrix/semiconductor/ARC plus reaction product/scatterer) deposit over a substrate (fiber/ARC/semiconductor). The wedge is of general (unsymmetric) geometrical configuration, and is subjected to antiplane shear (mode III) 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 III eigenvalues (or stress singularities) with various wedge aperture angles of the material 1 (reaction product/scatterer), are also presented. Hitherto generally unavailable results, pertaining to the through-thickness variations of stress intensity factors for symmetric parabolic load and its skew-symmetric counterpart that also satisfy the boundary conditions on the top and bottom surfaces of the trimaterial plates under investigation, form also an important part of the present investigation.

Three-dimensional singular stress field at the front of a crack weakening a unidirectional fiber reinforced composite plate

Abstract An eigenfunction expansion technique, based in part on separation of the thickness-variable and partly on the Eshelby–Stroh type affine transformation, is employed to derive hitherto unavailable three-dimensional asymptotic stress field in the vicinity of the front of a semi-infinite through-thickness crack weakening an infinite transversely isotropic unidirectional fiber reinforced composite 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, fixed or lubricated) surfaces of the lamina are exactly satisfied. The present investigation considers mainly two through-crack systems – (i) (0 1 0)[0 0 1] with the [1 0 0] propagation direction, and (ii) ( 1 ¯ 0 0 ) [ 0 0 1 ] with the [0 1 0] propagation direction. Explicit expressions for the singular stresses in the vicinity of the front of a through-thickness crack weakening a transversely isotropic unidirectional fiber reinforced composite plate, subjected to far-field mode I/II loadings, are presented. Hitherto unavailable numerical results, pertaining to the through-thickness variations of stress intensity factors for saw-tooth load and its skew-symmetric counterpart that also satisfy the boundary conditions on the top and bottom surfaces of the cracked unidirectional fiber reinforced composite (transversely isotropic) plates under investigation, bridge a longstanding gap in the fracture mechanics literature.

Three-dimensional singular stress field at the front of a crack and lattice crack deviation (LCD) in a cubic single crystal plate

A novel eigenfunction expansion technique, based in part on separation of the thickness variable, is developed to derive three-dimensional asymptotic stress fields in the vicinity of the front of a semi-infinite through-crack weakening an infinite plate made of a homogeneous cubic single crystal. Crack-side boundary conditions and those that are prescribed on the top and bottom (free) surfaces of the cubic crystal plate are exactly satisfied. Explicit expressions for singular stress fields in the vicinity of the front of through-thickness cracks, weakening cubic single crystal plates subjected to far-field extension/bending (mode I), sliding shear/twisting (mode II) and antiplane shear (mode III) loadings are presented. The present investigation considers three through-crack systems (crack plane)[crack front] × [propagation direction], (010)[001] × [100], and , weakening cubic crystals, and their relatively easier cleavage planes for propagation. It also introduces a new concept of lattice crack deviation (LCD) barrier, which can explain the reported discrepancy between simulations and experiments with regards to crack deviation from a “difficult” cleavage system to an easier one. Additionally, the relationships of the easier cleavage systems based on the present solutions with the structural chemistry aspects of various single crystals, such as bcc alkali metals, bcc transition alkali metals, fcc transition metals, group IVA (diamond cubic) elements, usually ionic compounds (rock salt and fluorite structures), covalent compounds (zinc blende structure), etc., are also discussed. Finally, the LCD parameter is strongly correlated with the anisotropic ratio for the cracked cubic crystal concerned.

On three-dimensional singular stress/residual stress fields at the front of a crack/anticrack in an orthotropic/orthorhombic plate under anti-plane shear loading

Abstract A novel eigenfunction expansion technique, based in part on separation of the thickness-variable, is developed to derive three-dimensional asymptotic stress field in the vicinity of the front of a semi-infinite through-thickness crack/anticrack weakening/reinforcing an infinite orthotropic/orthorhombic plate, of finite thickness and subjected to far-field anti-plane shear loading. Crack/anticrack-face boundary conditions and those that are prescribed on the top and bottom (free, fixed and lubricated) surfaces of the orthotropic plate are exactly satisfied. Five different through-thickness crack/anticrack-face boundary conditions are considered: (i) slit crack, (ii) anticrack or perfectly bonded rigid inclusion, (iii) transversely rigid inclusion (longitudinal slip permitted), (iv) rigid inclusion in part perfectly bonded, the remainder with slip, and (v) rigid inclusion located alongside a crack. Explicit expressions for the singular stress fields in the vicinity of the fronts of the through-thickness cracks, anticracks or mixed crack–anticrack type discontinuities, weakening/reinforcing orthotropic/orthorhombic plates, subjected to far-field anti-plane shear (mode III) loadings, are presented. In addition, singular residual stress fields in the vicinity of the fronts of these cracks, anticracks and similar discontinuities are also discussed.

Three-dimensional asymptotic stress field in the vicinity of an adhesively bonded scarf joint interface

A recently developed three-dimensional eigenfunction expansion approach for prediction of the singular stress field in the neighborhood of the interfacial front of an adhesively bonded scarf joint is presented. The plate is subjected to extension/bending (mode I) and in-plane shear/twisting (mode II) far field loading. Each material is assumed to be isotropic and elastic, but with different material properties. Numerical results include the dependence of the lowest eigenvalue (or stress singularity) on the wedge aperture angle of the plate material. Variation of the same with respect to the shear moduli ratio of the component plate and adhesive layer materials is also an important part of the present investigation. Hitherto unobserved interesting and physically meaningful conclusions are also presented.

Three-dimensional asymptotic stress field in the vicinity of the circumference of a bimaterial penny-shaped interfacial discontinuity

An eigenfunction expansion method is presented to obtain three-dimensional asymptotic stress fields in the vicinity of the circumference of a bimaterial penny-shaped interfacial discontinuity, e.g., crack, anticrack (infinitely rigid lamella), etc., located at the center, edge or corner, and subjected to the far-field torsion (mode III), extension/bending (mode I), and sliding shear/twisting (mode II) loadings. Five different discontinuity-surface boundary conditions are considered: (1) bimaterial penny-shaped interface anticrack or perfectly bonded thin rigid inclusion, (2) bimaterial penny-shaped interfacial jammed contact, (3) bimaterial penny-shaped interface crack, (4) bimaterial penny-shaped interface crack with partial axisymmetric frictionless slip, and (5) bimaterial penny-shaped interface thin rigid inclusion alongside penny-shaped crack. Solutions to these cases except (3) are hitherto unavailable in the literature. Closed-form expressions for stress intensity factors subjected to various far-field loadings are also presented. Numerical results presented include the effect of the ratio of the shear moduli of the layer materials, and also Poisson’s ratios on the computed lowest real parts of eigenvalues for the case (5). Interesting and physically meaningful conclusions are also presented, especially with regard to cases (1) and (2).

Three-dimensional asymptotic stress field at the front of an unsymmetric bimaterial wedge associated with matrix cracking or fiber break

A combined approach for prediction of the singular stress fields at the interfacial fronts of fiber breaks and matrix cracks is presented. A recently developed eigenfunction expansion method is employed for obtaining three-dimensional asymptotic displacement and stress fields in the vicinity of a point located at the front of a bimaterial wedge of general (unsymmetric) geometrical configuration (with respect to bimaterial interface), and subjected to extension/bending (mode I) and in-plane shear/twisting (mode II) far field loading and free–free wedge-side boundary condition. Each material is isotropic and elastic, but with different material properties. The material 2 (fiber in the case of matrix cracking or matrix in the case of fiber break) is always taken to be a half-space, while the wedge aperture angle of the material 1 is varied to represent varying matrix cracking or fiber break incidence angles at the interfaces. Numerical results pertaining to the variation of the lowest eigenvalues (or stress singularities) for various wedge aperture angles of the material 1, subjected to the afore-mentioned wedge-side boundary condition, are also presented. Variation of the same with the shear moduli ratio of the component material phases is also an important part of the present investigation. The conclusion drawn from the present asymptotic analysis of the matrix cracking problem is in agreement with that of an earlier investigation based on the energy based criterion. Similar conclusion drawn from the present asymptotic analysis of fiber break problem is in agreement with that of an energy based criterion derived here in analogy to the corresponding matrix crack problem by another investigator.

Three-dimensional asymptotic stress field in the vicinity of the circumferential tip of a fiber–matrix interfacial debond

A heretofore unavailable asymptotic solution pertaining to the stress field in the neighborhood of the circumferential tip of an interfacial debond, between the fiber or inclusion and the unreinforced plate made of the matrix material, and subjected to far-field extension-bending (mode I), inplane shear-twisting (mode II) and torsional (mode III) loadings, is presented. A local orthogonal curvilinear coordinate system ( ρ, φ, θ), is selected to describe the local deformation behavior of the afore-mentioned plate in the vicinity of the afore-mentioned circumferential line of interfacial debond. One of the components of the Euclidean metric tensor, namely g 33, is approximated ( ρ/ a≪1) in the derivation of the kinematic relations and the ensuing governing system of three partial differential equations. A recently developed eigenfunction approach coupled with a novel local curvilinear coordinate system, is utilized to compute the asymptotic displacement and stress fields. The oscillatory behavior at the debond tip may be considered to be a first-order approximation of the interdiffusion of the component phases followed by molecular entanglement and other similar microscopic phenomena studied by materials scientists.

Three-dimensional asymptotic stress field in the vicinity of the line of intersection of a circular cylindrical through/part–through open/rigidly plugged hole and a plate

Heretofore unavailable asymptotic solutions to a class of problems pertaining to the stress fields in the neighborhood of the circumferential corner line or line of intersection of a circular cylindrical through or part-through (embedded) open or rigidly plugged hole and a bounding or interior surface of an isotropic plate, subjected to far-field extension-bending (mode I), inplane shear-twisting (mode II) and torsional (mode III) loadings, are presented. A local orthogonal curvilinear coordinate system (ρ,φ,θ), is selected to describe the local deformation behavior of the afore-mentioned plate in the vicinity of the afore-mentioned circumferential corner line. One of the components of the Euclidean metric tensor, namely g33, is approximated (ρ/a⋘1) in the derivation of the kinematic relations and the ensuing governing system of three partial differential equations. Nine different combinations of boundary conditions are considered, five of which relate to a through hole or infinitely rigid inclusion, while the remaining four pertain to a part–through (embedded) hole or infinitely rigid inclusion. Numerical results presented include the effect of Poisson's ratio, wherever applicable, on the computed lowest eigenvalue(s).

Eigenfunction expansion solutions for three-dimensional rigid planar inclusion problems

An eigenfunction expansion method is presented to obtain three-dimensional asymptotic stress fields in the vicinity of the front of a semi-infinite infinitely rigid inclusion, subjected to the far-field antiplane shear and extension/bending loadings. Four different rigid inclusion side-surface boundary conditions are considered: (i) anticrack or perfectly bonded rigid inclusion, (ii) transversely rigid inclusion (longitudinal slip permitted), (iii) rigid inclusion in part perfectly bonded, the remainder with slip, (iv) rigid inclusion located alongside a crack. The computed stress singularity for an anticrack is the same as its plane strain counterpart, while the expression for the stress intensity factor varies in the thickness direction.

Three-dimensional asymptotic stress field in the vicinity of the circumference of a penny shaped discontinuity

An eigenfunction expansion method is presented to obtain three-dimensional asymptotic stress fields in the vicinity of the front of a penny shaped discontinuity, e.g., crack, anticrack (infinitely rigid lamella), etc., subjected to the far-field torsion (mode III), extension/bending (mode I) and sliding shear/twisting (mode II) loadings. Five different discontinuity-surface boundary conditions are considered: (i) penny shaped crack, (ii) penny shaped anticrack or perfectly bonded thin rigid inclusion, (iii) penny shaped thin transversely rigid inclusion (frictionless planar slip permitted), (iv) penny shaped thin rigid inclusion in part perfectly bonded, the remainder with frictionless slip, and (v) penny shaped thin rigid inclusion alongside penny shaped crack. The computed stress singularity for a penny shaped anticrack is the same as that of the corresponding crack. The main difference is, however, that all the stress components at the circular tip of an anticrack depend on Poisson’s ratio under modes I and II.

Three-dimensional asymptotic stress field at the front of an unsymmetric bimaterial pie-shaped wedge under antiplane shear loading

Three-dimensional asymptotic stress field in the vicinity of the front of a bimaterial pie-shaped wedge of general (unsymmetric) geometry is obtained by a recently developed eigenfunction expansion method. The bimaterial pie-shaped wedge is subjected to four combinations of wedge-side boundary conditions – free–free, clamped–clamped, free–clamped and clamped–free. Each material is isotropic and elastic, but with different material properties. Additionally, numerical results pertaining to the variation of the lowest eigenvalues (or stress singularities) with respect to the wedge opening angle of the bimaterial wedge, subjected to the aforementioned wedge-side boundary conditions, are also presented. Dependence of the same on the material properties of the component material phases is also an important part of the present investigation.

Three-dimensional asymptotic stress field in the vicinity of the circumferential line of intersection of an inclusion and plate surface

A heretofore unavailable asymptotic solution pertaining to the stress field in the neighborhood of the circumferential line of intersection of an inclusion and one of the bounding (top or bottom) surfaces of a plate, subjected to far-field extension/bending (mode I), inplane shear-twisting (mode II) and torsional (mode III) loadings, is presented. A local orthogonal curvilinear coordinate system (ρ,φ,θ), is selected to describe the local deformation behavior of the afore-mentioned plate in the vicinity of the afore-mentioned circumferential line of intersection. One of the components of the Euclidean metric tensor, namely g33, is approximated (ρ/a<<1) in the derivation of the kinematic relations and the ensuing governing system of three partial differential equations. Four different combinations of boundary conditions are considered: (i) free bounding surface of the plate and uncapped inclusion (free-free), (ii) bounding surface of the plate clamped by an infinitely rigid washer and inclusion covered with an infinitely rigid cap (clamped-clamped), (iii) free bounding surface of the plate and inclusion covered with an infinitely rigid cap (free-clamped), and (iv) bounding surface of the plate clamped by an infinitely rigid washer and uncapped inclusion (clamped-free). For dissimilar inclusion and plate (matrix) materials, the asymptotic stress fields in the vicinity of the line of intersection of an inclusion and the bottom (or top) surface of a plate, subjected to far-field mode I and mode II loadings, are singular, in all cases of the above described boundary conditions. In the case of the mode III (torsional) loading, the asymptotic stress fields are singular only for the free-clamped (or clamped-free) boundary condition. Numerical results presented include the effect of the ratio of the shear moduli of plate and inclusion materials, and also Poisson's ratios on the computed lowest eigenvalues.

Stress singularities for three-dimensional corners using the boundary integral equation method

The boundary integral equation method is developed to study three-dimensional asymptotic singular stress fields at vertices of a pyramidal notch or inclusion in an isotropic elastic space. Two-dimensional boundary integral equations are used for the infinite body with pyramidal notches and inclusions when either stresses or displacements are specified on its surface. Applying the Mellin integral transformation reduces the problem to one-dimensional singular integral equations over a closed, piece-wise smooth line. Using quadrature formulas for regular and singular integrals with Hilbert and logarithmic kernels, these integral equations are reduced to a homogeneous system of linear algebraic equations. Setting its determinant to zero provides a characteristic equation for the determination of the stress singularity power. Numerical results are obtained and compared against known eigenvalues from the literature for an infinite region with a conical notch or inclusion, for a Fichera vertex, and for a half-space with a wedge-shaped notch or inclusion.