Singular Elastic Fields Research Articles

Stress discontinuity and singularity around the vertex of a triangular inhomogeneity

The stress field around a vertex of angular inhomogeneity has been investigated by Eshelby’s equivalent inclusion method (EIM). Different from Eshelby’s problem for ellipsoidal inhomogeneity with a uniform or polynomial eigenstrain, a singular eigenstrain field is derived by Airy’s stress function and asymptotic analysis, in which the singular elastic fields can be expressed in terms of the distance to the vertex. The discontinuity of Eshelby’s tensor along the boundary has been analytically derived and are used in the stress equivalence condition at the vertex, which can be converted into an eigenvalue problem with the integral of the Green’s function and singular eigenstrain over the inhomogeneity. To verify the solution, when the opening angle 2β at the vertex approaches zero, the triangular void reduces to a slit-like crack, and the paper reproduces the classic solution of 1/r singularity. When β increases from 0 to π/2, the singularity parameter λ around the vertex of the triangular void reduces at different pace under the symmetric and antisymmetric loading conditions. When a triangular inhomogeneity exhibits nonzero stiffness and different angles, λ changes with the stiffness ratio, β, and loading conditions, and the dominant stress singularity around the vertex and stress discontinuity across the boundary are analytically provided. Particularly, the stress singularity for an adhesive interface with varying stiffness provides insight for structure repair and integration.

Oscillatory singularity for bending of a partially clamped nanoplate with consideration of surface effect

This article studies a partially clamped flexible ultrathin elastic nanoplate with surface effect. An emphasis is to analyze the singular elastic behavior near the tips of partially clamped boundary. A mixed boundary value problem associated with a partially clamped nanoplate having parabolic deflection is presented and solved. Using the Kirchhoff plate theory incorporating the Gurtin–Murdoch surface elasticity, some basic equations for nanoplates with surface elasticity are established. With the aid of the Fourier transform and the superposition principle, two coupled singular integral equations with Cauchy kernel are derived during bending of the nanoplate and then converted to a singular integral equation with two Cauchy kernels. The exact solution to the singular integral equation with two Cauchy kernels is determined and the oscillatory singular elastic fields near the tips of the clamped boundary are given in elementary functions. The bending moments and the normal stress components in the vicinity of the tips simultaneously have the oscillator singular behavior like r−3/2−iɛ, r being the distance from the tip and ɛ relying on the material properties. Besides, the effective shear force possesses an r−5/2−iɛ singular behavior. The influences of the geometrical parameters and material properties of nanoplates on the oscillatory singular behavior are presented graphically. When surface effect is neglected, the solution is applicable to classical thin plates with partially clamped boundary.

Analytical Solution for a 1D Hexagonal Quasicrystal Strip with Two Collinear Mode-III Cracks Perpendicular to the Strip Boundaries

We considered the problem of determining the singular elastic fields in a one-dimensional (1D) hexagonal quasicrystal strip containing two collinear cracks perpendicular to the strip boundaries under antiplane shear loading. The Fourier series method was used to reduce the boundary value problem to triple series equations, then to singular integral equations with Cauchy kernel. The analytical solutions are in a closed form for the stress field, and the stress intensity factors and the energy release rates of the phonon and phason fields near the crack tip are expressed using the first and third complete elliptic integrals. The effects of the geometrical parameters of the crack configuration on the dimensionless stress intensity factors are presented graphically. The studied crack model can be used to solve the problems of a periodic array of two collinear cracks of equal length in a 1D hexagonal quasicrystal strip and an eccentric crack in a 1D hexagonal quasicrystal strip. The propagation of cracks produced during their manufacturing process may result in the premature failure of quasicrystalline materials. Therefore, it is very important to study the crack problem of quasicrystalline materials with defects as mentioned above. It can provide a theoretical basis for the application of quasicrystalline materials containing the above defects.

Does elastic stress modify the equilibrium corner angle?

We consider the influence of elasticity and anisotropic surface energy on the energy-minimizing shape of a two-dimensional void under biaxial loading. In particular, we consider void shapes with corners for which the strain energy density is singular at the corner. The elasticity problem is formulated as a boundary integral equation using complex potentials. By incorporating the asymptotic behavior of the singular elastic fields at corners of the void, we develop a numerical spectral method for determining the stress for a class of arbitrary void shapes and corner angles. We minimize the total energy of surface energy and elastic potential energy using calculus of variations to obtain an Euler-Lagrange equation on the boundary that is coupled to the elastic field. The shape of the void boundary is determined using a numerical spectral method that simultaneously determines the equilibrium void shape and singular elastic fields. Our results show that the precise corner angles that minimize the total energy are not affected by the presence of the singular elastic fields. However, the stress singularity on the void surface at the corner must be balanced by a singularity in the curvature at the corner that effectively changes the macroscopic geometry of the shape and effectively changes the apparent corner angle. These results reconcile the apparent contradiction regarding the effect of elasticity on equilibrium corner angles in the results of Srolovitz and Davis (2001) and Siegel, Miksis and Voorhees (2004), and identify an important nontrivial singular behavior associated with corners on free-boundary elasticity problems.

Effects of local fiber discontinuity on the fatigue strength parameter at the fiber inclusion corner in fiber-reinforced composites

Abstract To comprehend the fatigue failure mechanism at the fiber discontinuity in fiber-reinforced composites, it is necessary to evaluate the local mechanical behaviors. The fatigue strength depends on the stress distribution at the fiber inclusion corner. An improved advanced finite element method (IAFEM) is proposed for the stress intensity factor (SIF) analysis at the fiber inclusion corner. In the IAFEM, the element stiffness matrix of singular inclusion corner element (SICE) is obtained, and the singular elastic field at the tip of the fiber inclusion is determined. The effects of load direction, fiber distribution, fiber geometry, and material properties on SIFs are analyzed numerically using the IAFEM. The difference in stress field distribution between two-dimensional and three-dimensional fiber inclusions is discussed. The IAFEIM and calculation results can provide reference for fatigue strength analysis and preparation of composite materials.

Singular elastic field induced by a rigid line adhering to a micro/nanoscale plate during bending

In this work, the elastic analysis of a micro/nanoscale plate with a stiffer ribbon adhering to it is studied. The stiffer ribbon is understood as a rigid line. The problem is solved by superposition and the singular elastic field induced by a rigid line is concerned when the plate is during bending. The singular elastic field of a micro/nanoscale elastic thin plate with surface elasticity is determined for a parabolic rigid line. The theoretical analysis is based on a mixed boundary value problem that is solved through the Fourier integral transform technique. According to the classical Kirchhoff thin plate theory incorporating surface elasticity, some basic equations are established. The mixed boundary value problem is reduced to a singular integral equation of the first kind with Cauchy kernel. Explicit expressions for the moments and effective shear forces as well as the surface and bulk stress components in any position of the whole nanoplate are obtained. The singularity coefficients of the moment and effective shear force at the tip of the rigid line are determined in closed form. From the obtained results, we find that the normal stress components for bulk and surface phases and bending moments at the rigid line tips have a usual inverse square-root singularity, but the effective shear forces behave like r−3/2, r being the distance from the tip of the rigid line or admit an r−3/2 singularity. The numerical results are displayed graphically and show that the singularity coefficients are heavily dependent on the surface and bulk material constants.

Singular elastic field induced by a rigid line inclusion in a thin nanoplate with surface elasticity

This paper analyzes the mechanical behavior of a Kirchhoff thin nanoplate with surface stresses containing a thickness-through rigid line inclusion. Based on the bulk and surface elasticity, a rigid line inclusion problem of a prescribed displacement is reduced to an associated mixed boundary value problem. By using the Fourier transform, the problem is converted to dual integral equations, then to a weakly singular integral equation with the logarithmic kernel. An exact solution for a rigid line inclusion with an off-nanoplate rigid rotation is obtained and full elastic fields including the deflection, bending moments, and effective shear forces at any position of the nanoplate are determined explicitly in terms of the elementary functions. The singularity coefficients of the bending moments and effective shear forces near the tip of the rigid line inclusion are evaluated. The obtained results show that the bending moments and the stress components exhibit an r−3/2 singularity near the rigid line inclusion tip, where r is the distance from the rigid line inclusion tip. The effective shear forces have an r−5/2 singularity. The influence of surface elasticity on the singularity coefficients is examined.

Smooth interface crack between two bonded dissimilar orthotropic elastic media under shear loading

An interface crack in an orthotropic bi-material is studied under applied loading. Crack faces are assumed to contact smoothly and only slide relatively due to exertion of applied loading. A singular elastic field disturbed by the interface crack is concerned, and the Fourier transform technique is used to solve the singular elastic field. Dual integral equations involving trigonometric functions are derived and solved analytically. An analytic solution is given and explicit expressions for the full elastic fields at any position are obtained for constant and linear shear loading at the crack faces. Stress intensity factors and energy release rate are given explicitly. The obtained results show that no oscillation behavior occurs for smooth interface cracks, but only the usual square-root singularity near the crack tip. The influence of material properties on the crack initiation angle is presented graphically.

Nanoscale mode-III interface crack in a bimaterial with surface elasticity

This paper studies a mode-III nanocrack at the interface between two bonded dissimilar materials under antiplane shear loading. The classical elasticity incorporating surface elasticity is applied to solve a mixed boundary value problem associated with an anti-plane shear interface crack. The influence of surface elasticity on the crack-tip field for a nanoscale mode-III crack is analyzed. By use of the Fourier transform, the problem is reduced to a set of hypersingular integro-differential equations. The displacement and bulk stress jumps are expanded as the Chebyshev orthogonal polynomials and the Galerkin method is used to approximately determine the singular elastic field near the interface crack tips. Consideration of surface elasticity does not cause the disappearance of crack-tip singularity. A usual inverse square-root singularity is derived near the crack tips. The influences of surface elasticity on the stress intensity factor are examined and displayed graphically. The surface residual stress does not alter the stress field for a mode-III interface crack.

A brief note on the self-balanced singular elastic field caused by an edge dislocation

Four physical quantities describing the self-balanced singular elastic field caused by an edge dislocation before or after loading are obtained by using SW-integral. These quantities are directly related to mechanisms of dislocation plasticity and are compared with that of a concentrated force. An edge dislocation and a concentrated force interacting with the first stress invariant and the maximum shear stress at infinity, respectively, are also presented.

VISCO-ELASTIC AND PLASTIC SINGULAR BEHAVIORS OF AN INTERFACE CORNER IN ELECTRONIC PACKAGES

Failure of an electronic package generally initiates from the interface corners, due to the stress and strain singular behaviors near the corners. Since the solder materials are viscous elastic and even maybe viscous plastic, the singular behavior near the solder joint corner is thereby time-dependent, and is quite different from that of an elastic one. It is necessary to clarify such a singular behavior for the strength and life evaluation of an electronic package. This paper firstly has deduced the approximated viscous elastic singular fields theoretically based on the principle of correspondence and the Laplace transformation techniques, and secondly has investigated the viscous plastic singular behavior of an elastic-plastic interface corner by the analogue method; finally the numerical analysis of an electronic package has been carried out to show the details of the singular behaviors.

Evaluation of fracture mechanics parameters for free edges in multi-layered structures with weak singularities

This study focuses on the stress intensity factors for free edges in multi-layered structural components. The effects of elastic constants of various material combinations on the weak singularity at free edges are analyzed. Using the H-integral approach, the effects of elastic mismatch parameters, the bond area and the thickness of the thin metal layer on the stress intensity factor are quantified and the results are compared with detailed finite element solutions. A good agreement between numerical predictions obtained from the H-integral and the detailed FE results is achieved, showing the applicability of this approach. Similar to a crack problem, the singular elastic field dominates in an annular region adjacent to the notch tip. The relationship between the valid range of the K-dominated field and the thin-film thickness is then demonstrated. Furthermore, the competition of crack initiation between the free edge interface (180° opening angle) and a 90° notch interface in a generic specimen is investigated, in order to find out which is the prevailing failure mode. Comparison between isotropic Si and anisotropic Si substrate is also illustrated. Anisotropy of the Si substrate has a significant influence on the stress intensity factor when combined with an Au or Al metal layer but not with a Cu layer. Additionally, standardized numerical formulae of the dimensionless stress intensity factor have been derived to guide the engineering application.

A novel hybrid finite element analysis of inplane singular elastic field around inclusion corners in elastic media

This paper deals with the inplane singular elastic field problems of inclusion corners in elastic media by an ad hoc hybrid-stress finite element method. A one-dimensional finite element method-based eigenanalysis is first applied to determine the order of singularity and the angular dependence of the stress and displacement field, which reflects elastic behavior around an inclusion corner. These numerical eigensolutions are subsequently used to develop a super element that simulates the elastic behavior around the inclusion corner. The super element is finally incorporated with standard four-node hybrid-stress elements to constitute an ad hoc hybrid-stress finite element method for the analysis of local singular stress fields arising from inclusion corners. The singular stress field is expressed by generalized stress intensity factors defined at the inclusion corner. The ad hoc finite element method is used to investigate the problem of a single rectangular or diamond inclusion in isotropic materials under longitudinal tension. Comparison with available numerical results shows the present method is an efficient mesh reducer and yields accurate stress distribution in the near-field region. As applications, the present ad hoc finite element method is extended to discuss the inplane singular elastic field problems of a single rectangular or diamond inclusion in anisotropic materials and of two interacting rectangular inclusions in isotropic materials. In the numerical analysis, the generalized stress intensity factors at the inclusion corner are systematically calculated for various material type, stiffness ratio, shape and spacing position of one or two inclusions in a plate subjected to tension and shear loadings.

Adhesive Fracture of an Epoxy Joint Under Thermal and Mechanical Loadings

The failure of an assembly made of an epoxy joint between aluminium substrates under thermal and mechanical loadings is addressed. The structure undergoes stress concentrations characterized by singular elastic fields at the end of the interface between the epoxy joint and the aluminium substrates. An already existing onset criterion is extended to thermo-elastic problems and used to predict the failure loading. Numerical results are compared with experimental measures taken from the literature. They exhibit a satisfying agreement but it is clear that a better knowledge of the curing process and then of the residual stresses would allow a better fitting in all cases.

A damage model based on singular elastic fields

At re-entrant corners in elastic structures, the stress field is known to increase to infinity following a power law. From the material viewpoint it is paradoxical that it can locally sustain such an overburden. To avoid this paradox we propose a damage model where the Young's modulus of the material decreases (damage) also following a power law and such that the resulting stress field remains bounded. To cite this article: D. Leguillon, C. R. Mecanique 336 (2008).

Elastic and elastic–plastic singular fields around a crack-tip in particulate-reinforced composites with progressive debonding damage

This paper deals with elastic and elastic–plastic singular fields around a crack-tip in particulate-reinforced composites with debonding damage of particle-matrix interface. Numerical analyses are carried out on a crack-tip field in elastic-matrix and elastic–plastic-matrix composites reinforced with elastic particles, using a finite element method developed based on an incremental damage theory of particulate-reinforced composites. A particle volume fraction and interfacial strength between particles and matrix of the composites are parametrically changed. In the elastic-matrix composites, a unique elastic singular field is created on the complete damage zone in the vicinity of a crack-tip in addition to the conventional elastic singular field on the no damage zone. The macroscopic stress level around a crack-tip is reduced by the debonding damage while the microscopic stress level of the matrix remains unchanged. In the elastic–plastic-matrix composites, the damage zone develops in addition to the plastic zone due to matrix plasticity, and both the macroscopic and microscopic stress revels around a crack-tip are reduced by the debonding damage. It is concluded from the numerical results that the toughening due to damage could be expected in the elastic–plastic-matrix composites, while it is questionable in the elastic-matrix composites.

The minimum plastic zone radius criterion for crack initiation direction applied to surface cracks and through-cracks under mixed mode loading

The theories for crack initiation orientation under biaxial loading based on the singular elastic field may become inaccurate as the crack tip plasticity increases. In this study, a criterion based on the relative minimum plastic zone radius (MPZR) is investigated for prediction of fatigue crack initiation angles. The MPZR criterion is applied to inclined surface crack specimens and through-crack specimens under mixed mode loading, and its performance is compared with those of the maximum tensile stress theory (MTS) and the minimum strain energy density theory (MSED). It is found that the MPZR criterion provides somewhat better overall prediction of the crack initiation angle than the MTS and MSED theories. The approach is also applied to predict crack initiation under mixed monotonic loading for through-crack specimens. The MPZR criterion is found to give conservative results, while the MTS criterion yields best correlation.

粒子分散複合材料のき裂先端近傍の弾性・弾塑性特異場に及ぼすはく離損傷の影響

This paper deals with the influence of debonding damage between particles and matrix on elastic and elastic-plastic singular fields around a crack-tip in particle-reinforced-composites. Numerical analyses are carried out on a cracktip field in elastic-matrix and elastic-plastic-matrix composites reinforced with elastic particles using a finite element method developed based on an incremental damage theory of particle-reinforced composites. A particle volume fraction and interfacial strength between particles and matrix of the composites are parametrically changed. In the elastic-matrix composites, the complete damage zone, in which all particles are debonded, and the surrounding progressive damage zone develop around a crack-tip. A unique elastic singular field is created on the complete damage zone in addition to the conventional elastic singular field of the non-damaged zone. The macroscopic stress level around a crack-tip is reduced by the debonding damage while the microscopic stress level of the matrix remains no change. In the elastic-plastic-matrix composites, the damage zone develops in addition to the plastic zone due to matrix plasticity, and both the macroscopic and microscopic stress levels around a crack-tip are reduced by the debonding damage. It is concluded from the numerical results that the toughening due to damage could be expected in the elastic-plastic-matrix composites, while it is suspicious in the elastic-matrix composites.

Computation of 3D Singular Elastic Fields for the Prediction of Failure at Corners

Computation of 3D Singular Elastic Fields for the Prediction of Failure at Corners

Failure initiation in an epoxy joint between two steel plates

An Irwin-like criterion, initially developed to predict crack initiation in monolithic v-notched specimens, is used to examine the failure of two steel plates bonded together by an epoxy joint. It is based on an estimate of the critical value of the intensity factor of the elastic singular fields appearing in some regions of the structure, due to the geometry and to the elastic mismatch of the components. The analysis is worked out at two scales: a macroscopic one ignoring the thin epoxy layer and considering the two plates as perfectly bonded along an idealized interface and a microscopic one taking into account the elastic fields within the joint. It is shown that these two approaches are equivalent and that a macroscopic failure criterion can be developed. The characteristic failure parameter is a macroscopic critical intensity factor. It can be derived from the knowledge of microscopic data such as the strength and the toughness of the epoxy.