Three-dimensional Singular Stress Research Articles

Three-Dimensional Singular Stress Fields and Interfacial Crack Path Instability in Bicrystalline Superlattices of Orthorhombic/Tetragonal Symmetries

First, a recently developed eigenfunction expansion technique, based in part on the separation of the thickness variable and partly utilizing a modified Frobenius-type series expansion technique in conjunction with the Eshelby–Stroh formalism, is employed to derive three-dimensional singular stress fields in the vicinity of the front of an interfacial crack weakening an infinite bicrystalline superlattice plate, made of orthorhombic (cubic, hexagonal, and tetragonal serving as special cases) phases of finite thickness and subjected to the far-field extension/bending, in-plane shear/twisting, and anti-plane shear loadings, distributed through the thickness. Crack-face boundary and interface contact conditions as well as those that are prescribed on the top and bottom surfaces of the bicrystalline superlattice plate are exactly satisfied. It also extends a recently developed concept of the lattice crack deflection (LCD) barrier to a superlattice, christened superlattice crack deflection (SCD) energy barrier for studying interfacial crack path instability, which can explain crack deflection from a difficult interface to an easier neighboring cleavage system. Additionally, the relationships of the nature (easy/easy, easy/difficult, or difficult/difficult) interfacial cleavage systems based on the present solutions with the structural chemistry aspects of the component phases (such as orthorhombic, tetragonal, hexagonal, as well as FCC (face-centered cubic) transition metals and perovskites) of the superlattice are also investigated. Finally, results pertaining to the through-thickness variations in mode I/II/III stress intensity factors and energy release rates for symmetric hyperbolic sine-distributed loads and their skew-symmetric counterparts that also satisfy the boundary conditions on the top and bottom surfaces of the bicrystalline superlattice plate under investigation also form an important part of the present investigation.

On the three-dimensional singular stress field near the corner front of revolution-shaped inclusions

On the three-dimensional singular stress field near the corner front of revolution-shaped inclusions

A new computational approach for three-dimensional singular stress analysis of interface voids

Defects in terms of three-dimensional voids are commonly encountered at bi-material interfaces. In the current study, the singular stress field near the circumferential corner line of a three-dimensional axisymmetric interfacial void is analyzed using our newly established singular interface edge elements. Under the premise that $$\rho {\ll }R$$ , the proposed singular element method does not depend on the size of the element; thereby, it is not necessary to use refined elements at the interface corner line. The numerical results reveal the intensity of the stress singularity at the interface line of the three-dimensional axisymmetric voids. The obtained stress intensity parameters can be used to judge the local fatigue crack initiation. The geometry effect of the void on the singular stress field at the circumferential interface corner line is studied and discussed in detail.

CFRP単一重ね継手における三次元特異応力場の保存積分による評価

CFRP単一重ね継手における三次元特異応力場の保存積分による評価

Three-dimensional singular stress fields near the circumferential junction corner line of an island/substrate system either free-standing or fully/partially bonded to a rigid block

The primary motivation of the present investigation stems from the need for understanding the three-dimensional singular stress fields in the vicinity of the re-entrant circumferential bimaterial junction corner line between a thin film island and the underlying substrate, either free-standing or fully/partially bonded to a super-rigid block. A three-dimensional eigenfunction expansion approach is employed for determination of the singular stress field in the vicinity of the re-entrant circumferential bimaterial junction corner line. The island-substrate bimaterial system is subjected to extension/bending (mode I), shear/twisting (mode II) and torsional (mode III) far field loadings. Both the island and substrate materials are assumed to be isotropic and elastic. The boundary conditions, prescribed on either side of the interfacial bond line between the island and substrate materials (free–free, fixed–fixed, free–fixed and fixed–free), are exactly satisfied. Numerical results include the dependence of the lowest eigenvalue(s) (or most severe order(s) of stress singularity) on the wedge aperture angle of the island material (looking at a axisymmetric cross-section). Variation of the same with respect to the shear moduli ratio of the constituent island and substrate materials is also an important part of the present investigation. Hitherto unobserved interesting and physically meaningful conclusions in regards to the mixed free–fixed type singularity and delamination type flaw sensitivity of an island-substrate bimaterial system are also presented.

Three-dimensional singular antiplane shear stress fields at the fronts of interfacial crack/anticrack/contact type discontinuities in tricrystal anisotropic plates

A recently developed eigenfunction expansion technique, based in part on separation of the thickness-variable and partly utilizing a modified Frobenius type series expansion in conjunction with the Eshelby–Stroh formalism, is employed to derive three-dimensional singular stress fields in the vicinity of crack/anticrack/junction front of an infinite tricrystal plate, made of monoclinic/hexagonal/orthorhombic/cubic phases, of finite thickness and subjected to far-field antiplane shear loading. Such tricrystals are used in solar cells and superconductivity application among others. Crack/anticrack-face boundary and/or interface contact conditions and those that are prescribed on the top and bottom (free or fixed) surfaces of the tri-crystal plate are exactly satisfied. Five different through-thickness boundary conditions are considered: (i) slit crack, (ii) anticrack or perfectly bonded rigid inclusion, (iii) tri-crystal junction, and (iv and v) rigid inclusion located alongside a crack. Numerical results pertaining to the variation of the mode III order of stress singularity with various wedge aperture angles of the material 1 (e.g., scatterer in a solar cell), are also presented. Finally, results, pertaining to the through-thickness variations of mode III stress intensity factors or stress singularity coefficients for symmetric exponentially growing distributed load and their skew-symmetric counterparts that also satisfy the boundary conditions on the top and bottom surfaces of the tricrystal plate under investigation, also form an important part of the present investigation.

Three-dimensional singular stress field near the interfacial bond line of a tapered jointed plate either free-standing (notch) or (fully/partially) attached to a super-rigid inclusion (antinotch)

A three-dimensional eigenfunction expansion approach is employed for determining singular stress fields in the vicinity of interfacial bond line of a tapered jointed plate either free-standing (notch) or (fully/partially) attached to a super-rigid inclusion (antinotch). The tapered jointed bi-material plate is subjected to extension/bending (mode I) and shear/twisting (mode II) far field loading (extension and shear being in the plane of the cross-section and bending/twisting about mid-width). Hitherto unavailable results, pertaining to the width-wise variations of stress intensity factors corresponding to symmetric and skew-symmetric (about mid-width) cosine function loads, in the vicinity of the interfacial bond line, are presented.

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.

Analysis for Three-Dimensional Singular Stress Field at a Vertex of Bonded Interface Edge in Single Lap Joint under Tensile-Shear Load

Dissimilar material joints are used in industrial products for improving functionality and reducing their weight. However, mechanism of delamination or crack for the dissimilar material joints is not still clarified. It is well known that singular stress occurs at the edge of the interface in the joints due to the mismatch of material properties. The stress on the singular stress field can be expressed by two parameters, i.e. the order of stress singularity and the intensity of stress singularity. In the present paper, an influence of the material properties of the adhesive in the shear lap joint on the intensity of stress singularity was investigated using a three-dimensional boundary element method analysis. Material property of adhesive in the shear lap joint is varied and that of adherend is fixed at Young's modulus 100GPa and Poisson's ratio 0.3. The intensities of stress singularity arranged by the different values of the order of stress singularity are expressed as a power function of the ratio of Young's modulus of adherend to adhesive.

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.

Do Notched Thick Plates Have Strength in Shear?

This paper summarizes recent theoretical studies on the three-dimensional singular stress state at sharp notches in infinite (or large) plates of finite thickness subjected to in-plane loading. In general, such loading produces a number of singular states: in-plane singularities (normally described as KI and KII fracture modes and known as Williams’ solution), singular states associated with corners and the out-of-plane singularity (KO mode), which is generated due to the Poisson’s effect. The latter mode has an interesting behavior and its intensity increases as a power function with the increase of the plate thickness when the notch is stresses in shear mode. From finite fracture mechanics considerations it is clear that at some certain thickness the out-of-plane singular mode will dominate over the fracture zone and with the further increase of the plate thickness will affect the strength of the notched plate, virtually reducing it to zero.

Three-dimensional singular stress field near a partially debonded cylindrical rigid fiber

Asymptotic solutions to a class of problems pertaining to the stress fields in the neighborhood of a circular cylindrical partially debonded infinitely rigid fiber (inclusion) in a homogeneous isotropic linearly elastic matrix (plate), subjected to far-field extension-bending (mode I), inplane shear-twisting (mode II) and torsional (mode III) loadings, are presented. Two different combinations of boundary conditions are considered: (i) infinitely rigid fiber with partial axisymmetric debond, and (ii) infinitely rigid fiber with partial axisymmetric frictionless slip (Comninou type contact zone). Introduction of the Comninou type contact zone at the circumferential tip of an infinitely rigid fiber–matrix axisymmetric interfacial debond eliminates the oscillatory character of the deformation in the case of mode I/II loading, and the stress singularity altogether in the case of mode III loading. Numerical results presented include, whenever applicable, the effect of Poisson’s ratio on the computed eigenvalue(s).

Semi‐analytical computation of 3D stress singularities in linear elasticity

AbstractThe boundary finite element method (BFEM (Finite‐Element Modelling of Unbounded Structures. Wiley: Chichester, England, 1996, The Scaled Boundary Finite Element Method. Wiley: Chichester, England, 2003)) is employed for the investigation of the order of stress singularities for several classes of three‐dimensional singular stress concentration problems, namely notch, crack and laminate free‐edge situations. In all cases, the BFEM results agree excellently with reference results. The required computational effort is considerably lower compared to e.g. standard finite element computations and thus establishes the BFEM as a powerful tool for the semi‐analytical modelling of linear elastic solids. Copyright © 2005 John Wiley & Sons, Ltd.

A finite element approach to three-dimensional singular stress states in anisotropic multi-material wedges and junctions

A finite element formulation is developed to determine the order and angular variation of singular stress states due to material and geometric discontinuities in anisotropic materials. The formulation applies to any two-dimensional geometry that is prismatic in the third direction and has three-dimensional displacement fields. In some special cases the three-dimensional fields become uncoupled antiplane and inplane fields and this formulation yields the uncoupled results. The formulation provides for the determination of the asymptotic stress and displacement fields present at interior singular points of three-dimensional structures. The displacement field of the sectorial finite element is quadratic in the angular coordinate direction and asymptotic in the radial direction measured from the singular point. The formulation of Yamada and Okumura [(1983) Hybrid and Mixed Finite Element Methods, pp. 325–343. Wiley, Chichester] for inplane problems is adapted for this purpose. The simplicity and accuracy of the formulation are demonstrated by comparison with several analytical solutions for both isotropic and anisotropic multi-material wedges and junctions. The nature and speed of convergence associated with the element suggests that it could be employed in developing two-dimensional and three-dimensional enriched elements for use along with standard elements to yield accurate and computationally efficient solutions to problems having complex global geometries leading to singular stress states.

Surface singularity and crack propagation

The three-dimensional singular stress field near the terminal point 0 of the crack front edge at the surface of an elastic body is investigated, using spherical coordinates r, θ, φ and assuming all three displacements to be of the form rλppF(θ, λ) where p = distancefromthesingularityline (crack front edge or notch edge) and p = givenconstant. The variational principle governing the displacement distribution on a unit sphere about point 0, which has previously been obtained from the differential equations of equilibrium, is now derived more directly from potential energy. The previously developed finite element method on the unit sphere is used to reduce the problem to the form k(λ)X = 0 where X = columnmatrixofthenodalvaluesofdisplacementsontheunitsphere and k(λ) = squarematrix, all coefficients of which are quadratic polynomials in λ. It is proven that the variational principle as well as matrix k must be nonsymmetric, which means that complex eigenvalues A are possible. The dependence of A upon Poisson's ratio v for Mode I cracks whose front edge is normal to the surface is solved numerically and it closely agrees with the analytical solution of Benthem. Previously unavailable solutions for Modes II and HI and for cracks (of all modes) with inclined front edge and inclined crack plane are also obtained. By energy flux argument, it is found that the front edge of a propagating crack must terminate at the surface obliquely, at a certain angle whose dependence upon the inclination of the crack plane is also solved. The angle is the same for Modes II and III, but different for Mode I. For this mode, the surface point trails behind the interior of the propagating crack, while for Modes II and III it moves ahead. Consequently, a combination of Mode I with Modes II and III is impossible at the surface terminal point of a propagating crack whose plane is orthogonal. When the plane is inclined, the three stress intensity factors can combine only in certain fixed ratios. The angle of crack edge is a function of the angle of crack plane. Some results with complex λ for two-material interfaces are also given.