Bimaterial Wedge Research Articles

The anisotropic and angularly inhomogeneous elastic wedge under a monomial load distribution

In this study, the generally anisotropic and angularly inhomogeneous wedge under a monomial type of distributed loading of order n of, the radial coordinate r at its external faces is considered. At first, using variable separable relations in the equilibrium equations, the strain–stress relations and the strain compatibility equation, a differential system of equations, is constructed. Decoupling this system, an ordinary differential equation is derived and the stress and displacement fields may be determined. The proposed procedure is also applied to the elastostatic problem of an isotropic and angularly inhomogeneous wedge. The special cases of loading of order n=−1 and n=−2, where the self-similarity approach is not valid, are examined and the stress and displacements fields are derived. Applications are presented for the cases of an angularly inhomogeneous wedge and in the case of a bi-material isotropic wedge.

A boundary value method for the singular behavior of bimaterial systems under inplane loading

Based on the governing equations of linear elasticity, this paper develops a novel boundary value method to study the singular behavior of elastic stress fields at the corners of bimaterial wedges and junctions by using the eigenfunction expansion technique. The resulted one-dimensional differential system, which consists of the reduced equilibrium equations and boundary conditions, just relates to an angular coordinate in the polar coordinate system. Implementing discretization of this differential system by the finite cloud method, we readily derive the so-called generalized eigenproblem in the singular eigenvalue. The performance of the methodology is subsequently verified through the well-known crack and interface crack problems, demonstrating high accuracy and quick convergence characteristics. In addition, a selected set of practically useful models is numerically analyzed to examine the angular variations of the displacement and stress fields, and the influences of wedge-side boundary conditions to singular behavior are also studied.

Disappearance Conditions of Stress Singularities for Anisotropic Bimaterial Half-Plane Wedges Under Antiplane Shear

Based on the anisotropic elasticity theory and Lekhnitskii’s complex potential functions, the analytical eigenequations of anisotropic bimaterial half-plane wedges under antiplane shear are derived in brief forms. The boundary surfaces of half-plane wedges can be combinations of free and/or clamped edges. The disappearance conditions of stress singularities can be obtained directly from the derived eigenequations, which can be applied to improve the safety of the structures. The interesting phenomenon on the periodic appearance of the singularity orders is proposed and discussed as well.

Some problems in the antiplane shear deformation of bi-material wedges

The antiplane shear deformation of a bi-material wedge with finite radius is studied in this paper. Depending upon the boundary condition prescribed on the circular segment of the wedge, traction or displacement, two problems are analyzed. In each problem two different cases of boundary conditions on the radial edges of the composite wedge are considered. The radial boundary data are: traction–displacement and traction–traction. The solution of governing differential equations is accomplished by means of finite Mellin transforms. The closed form solutions are obtained for displacement and stress fields in the entire domain. The geometric singularities of stress fields are observed to be dependent on material property, in general. However, in the special case of equal apex angles in the traction–traction problem, this dependency ceases to exist and the geometric singularity shows dependency only upon the apex angle. A result which is in agreement with that cited in the literature for bi-material wedges with infinite radii. In part II of the paper, Antiplane shear deformation of bi-material circular media containing an interfacial edge crack is considered. As a special case of bi-material wedges studied in part I of the paper, explicit expressions are derived for the stress intensity factor at the tip of an edge crack lying at the interface of the bi-material media. It is seen that in general, the stress intensity factor is a function of material property. However, in special cases of traction–traction problem, i.e., similar materials and also equal apex angles, the stress intensity factor becomes independent of material property and the result coincides with the results in the literature.

On the assessment of cracks within the interface of thin layers on a substrate

AbstractAlthough bimaterial wedge or notch configurations are identified as potential weak locations, the assessment of the degree of criticality of cracks in such regions is still a demanding problem. The singular character of the stress field at cracks or at bimaterial notches can be calculated analytically or numerically. The angle of the direction of potential crack initiation may also be determined, but the decisive question is whether a hypothetical crack will be initiated or not.An essential question in the context of crack assessment is to find a criterion for crack nucleation. For that aim, the hypothesis of Leguillon is modified. Herein, the crack is assumed to be critical when and only when both the released energy and the local stress reach critical values along a hypothetical crack of finite length. This concept can be transferred to a bimaterial interface configuration of a thin layer on a substrate. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The Effect of an Imperfect Interface on the Singular Stress Field of an Orthotropic Bimaterial Wedge

In this analysis, the effect of an imperfect interface on the stress singularity of an orthotropic elastic bimaterial wedge subjected to traction free boundary conditions, is investigated, where the planes of symmetry are aligned, and one symmetry plane is along the interface and another symmetry plane coincides with the cross-sectional plane. The Stroh formalism with the method of separation of variables are used to obtain the relevant expressions for displacements and stresses. At the interface, only the interfacial tractions and the displacement normal to the interface are assumed to be continuous. The imperfect bond is modeled using a local coordinate system, where each tangential traction component in this local coordinate system is directly proportional to the corresponding displacement discontinuity and inversely proportional to the distance from the wedge apex. The order of singularity is computed numerically for graphite/epoxy wedges and presented for various imperfect interface conditions. The numerical results agree with the available results for the fully bonded case. It is expected that when the axes of local and global coordinate systems are aligned, that the in-plane and antiplane problems are uncoupled, and this feature also can be seen in the numerical results.

Singular Stress Fields of Anisotropic Bimaterial Wedges with Imperfect Interfaces

ABSTRACTThe effect of an imperfect interface on the stress singularity of anisotropic bimaterial wedges subjected to traction free boundary conditions are investigated. The interfacial tractions are assumed to be continuous, directly proportional to the displacement jumps and inversely proportional to the radial coordinate. The characteristic equation for the order of singularity is obtained and numerical results are given for the angle-ply bimaterial composite wedge.

Antiplane stress singularities in a bonded bimaterial piezoelectric wedge

In this paper, the eigen-equations governing antiplane stress singularities in a bonded piezoelectric wedge are derived analytically. Boundary conditions are set as various combinations of traction-free, clamped, electrically open and electrically closed ones. Application of the Mellin transform to the stress/electric displacement function or displacement/electric potential function and particular boundary and continuity conditions yields identical eigen-equations. All of the analytical results are tabulated. It is found that the singularity orders of a bonded bimaterial piezoelectric wedge may be complex, as opposed to those of the antiplane elastic bonded wedge, which are always real. For a single piezoelectric wedge, the eigen-equations are independent of material constants, and the eigenvalues are all real, except in the case of the combination C–D. In this special case, C–D, the real part of the complex eigenvalues is not dependent on material constants, while the imaginary part is.

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.

Corner singularities in bi-material Mindlin plates

An eigenfunction expansion solution is first developed to find stress singualrities for bi-material wedges by directly solving the governing equations of the Mindlin plate theory in terms of displacement components. The singularity orders of moments and shear forces at corners are determined from the corresponding asymptotic solutions having the lowest order in r and satisfying the radial boundary conditions and continuity conditions. The present solution is applied to thoroughly examine the singularities occurring at the interface joint of bonded dissimilar isotropic plates and at the vertex of a bi-material wedge with two simply supported radial edges. The corresponding characteristic equations for determining the singularity orders of moments and shear forces are explicitly given. The singularity orders of moment are shown in graphic form as functions of the flexural rigidity ratio and corner angle, while the shear force singularity orders are given as functions of the corner angle and the shear modulus ratio multiplied by the thickness ratio. The order of moment singularity obtained here for bonded dissimilar plates is also compared with that based on the classical plate theory.

Screw Dislocation Interacting with an Interfacial Edge Crack between Two Bonded Dissimilar Piezoelectric Wedges

This letter is concerned with an interfacial edge crack in a piezoelectric bimaterial wedge interacting with a screw dislocation under antiplane mechanical and in-plane electric loading. In addition to a discontinuous electric potential across the slip plane, the dislocation is subjected to a line-force and a line-charge at the tip. The out-of-plane displacement and electric potentials are obtained in closed-form based on conformal mapping technique and the solution for the screw dislocation in an infinite piezoelectric bimaterial with a semi-infinite interfacial crack. The intensity factors (IFs) and energy release rate (EER) are obtained explicitly. These solutions can be used as a base for constructing solutions for arbitrary coupled antiplane mechanical and in-plane electrical loadings.

Three-dimensional asymptotic stress field at the front of a bimaterial wedge of symmetric geometry under antiplane shear loading

Three-dimensional asymptotic stress field in the vicinity of the front of a bimaterial wedge is obtained by a new eigenfunction expansion method, subjected to three combinations of wedge-side boundary conditions – free–free, clamped–clamped and free–clamped. The bimaterial wedge has symmetric geometrical configuration with respect to the bimaterial interface. Each material is isotropic and elastic, but with different elastic properties. Additionally, numerical results pertaining to the variation of the lowest eigenvalues (or stress singularities) with respect to the half-aperture angle of the bimaterial wedge, subjected to the afore-mentioned wedge-side boundary conditions, are also presented. Dependence of the same on the material properties of the component material phases (if any) is also presented.

A novel hybrid finite element analysis of bimaterial wedge problems

An assumed hybrid-stress finite element model together with a new super singular wedge-tip element with numerical eigensolutions is developed to study the bimaterial wedge/notch problems. The establishment of the super wedge-tip element consists of (1) a finite element method-based eigenanalysis is developed and applied to determine the order of the stress singularity and the angular dependences of the stress and displacement fields, (2) these fields are subsequently used to develop a finite element surrounding the wedge tip. To demonstrate the validity of the method, three types of the bimaterial wedge problems are examined and compared with analytical/referenced solutions. The high accuracy and general applicability of the present technique are shown.

Analysis of singular stresses in bonded bimaterial wedges by computed eigen solutions and hybrid element method

AbstractThis paper concerns the determination of the singular stress fields in bonded bimaterial wedges under inplane loading. The mathematical complexity required for deriving the eigen solutions is avoided by an ad hoc developed one‐dimensional finite element formulation. The computed eigen solutions are here adopted in the assumed stress fields of hybrid stress elements which are employed to determine the stress intensities. To illustrate the efficacy of the suggested procedure, stress intensities for cracks in homogeneous, interfacial cracks, bimaterial free edges and other configurations are computed and compared with the existing analytical/reference solutions. Copyright © 2001 John Wiley & Sons, Ltd.

Investigation of the underfill delamination and cracking in flip-chip modules under temperature cyclic loading

In this paper, stress singularity in electronic packaging is described and three general cases are summarized. The characteristics of each stress singularity are briefed. In order to predict the likelihood of delamination at a bimaterial wedge, where two interfaces are involved, a criterion is proposed and the corresponding parameters are defined. The propagation of a crack inside a homogeneous material with the effects of delamination and stress singularity is predicted by the maximum hoop stress criterion. The proposed criteria are adopted in the analysis of a flip-chip with underfill under thermal cyclic loading. A finite element (FE) model for the package is built and the proper procedures in processing FE data are described. The proposed criterion can correctly predict the interface where delamination is more likely to occur. It can be seen that the opening stress intensity factor along the interface (or peeling stress) plays a very important role in causing interfacial failure. The analytical results are compared with experimental ones and good agreement is found. The effects of delamination and cracking inside the package on the solder balls are also mentioned. Further investigation into the fatigue model of the underfilled solder ball is discussed.

Singular stress fields of angle-ply and monoclinic bimaterial wedges

The characteristic equations for the order of stress singularity of anisotropic bimaterial wedges subjected to traction boundary conditions are investigated. For an angle-ply bimaterial wedge, both fully bonded and frictional interfaces are considered, whereas for a monoclinic bimaterial wedge, a frictional interface is considered. Here, the Stroh formalism and the separation of variables technique are used. In general, the order of stress singularity can be real or complex, but for the special geometry of a crack along the frictional interface of a monoclinic composite, it is always real. Explicit characteristic equations for the order of singularity are presented for an aligned orthotropic composite with a frictional interface. Numerical results are given for an angle-ply bimaterial wedge and a monoclinic bimaterial wedge consisting of a graphite/epoxy fiber-reinforced composite.

A tale of two saints: St. Venant and ‘St. Nick’ — does St. Venant's principle apply to bi-material straight-edge and wedge-singularity problems?

A rigorous approach is presented to investigate the applicability of St. Venant's principle to the class of bi-material straight edge stress singularity problems, first investigated by Pipes and Pagano in the special case of a free edge and anti-plane shear loading. A recently developed eigenfunction expansion method is utilized to obtain three-dimensional asymptotic stress field in the vicinity of a point located at the bimaterial wedge front, subjected to free-free, clamped-clamped and clamped-free wedge-side boundary conditions. The present study considers the straight edge as a special case of a bi-material wedge, consisting of two isotropic materials, with half the aperture angle θ 0= π 2 . The present study is devoted to establishment of rigorous conditions as to whether the presence of bi-material straight edge (free or otherwise) singularity can validate or invalidate the assumption implicit in St. Venant's principle that more or less guarantees the occurrence of tensile or compressive failure in the gauge section of a carefully prepared test specimen. This is accomplished by applying Cherepanov's general approach to the solution of canonically singular problems, based on the concept of proper boundary-value problem and the theorem of homogeneous solutions, and classification of boundary value problems (BVP) of three-dimensional elasticity theory into class S (St.Venant) or class N (non-St. Venant or ‘St. Nick'). It is concluded that the class of bimaterial straight edge (free-free, clamped-clamped and clamped-free) stress singularity problem belongs to class N, which explains the frequent occurrence of end tab failures in tensile (or compressive) laminate test specimens.

Photoelastic analysis of the singular stress field in a bimaterial wedge

This paper presents an experimental investigation of the singular stress field near the vertex of a bimaterial wedge using a digital photoelastic technique. Special attention is given to the casting of bimaterial wedge specimens and analysis technique for extracting stress intensity factors from photoelastic samples. Different bimaterial wedge specimens are made of two different photoelastic materials bonded through a special casting procedure and loaded in simple tension. A new multiple-parameter method is developed to obtain the stress intensity factor reliably from the isochromatic fringe patterns and the series representation of the stress field at the vertex of the wedge. Experimental results are compared with finite element predictions, and good agreement is observed.

ADHESIVELY BONDED COMPOSITE IOSIPESCU SPECIMENS WITHOUT SINGULAR STRESS FIELDS

The asymptotic singular stress fields at bi-maSerial wedge corners in adhesively bonded composite losipescu specimens have been studied using two-dimensional finite-element techniques. In particular, the singular powers of the fields have been analyzed using the finite-element iterative method. Different combinations of the elastic properties of the composite adherends have been considered, taking into account various volume fractions of fibers as well as different fiber orientations with respect to the adherend/adhesive interface. It has been shown that there are critical angles of the interfaces that separate the regions with positive and negative singular powers. This creates stress fields that are not singular in nature. It appears that the critical angles are dependent on the elastic properties of the composite adherends. In the second part of this work, the critical angle approach has been applied to the design of adhesively bonded composite losipescu specimens free from singular stress fields at their bi-material wedge corners. The samples will not develop singular stress fields either in shear or under biaxial shear-dominated loading conditions.

A nonlinear finite elment eigenanalysis of singular stress fields in bimaterial wedges for plane strain

A displacement-based finite element formulation for the analysis of singular stress fields in power law hardening materials under conditions of plane strain is presented. The displacement field within a sectorial element is quadratic in the angular coordinate and of the power type in the radial direction as measured from the singular point. A hydrostatic pressure variable, which is linear in the angular coordinate, is introduced to account for the incompressibility of the material. The Newton method is combined with matrix singular value decomposition to iteratively solve the resulting nonlinear homogeneous eigenvalue problem where the eigenvalues and eigenfunctions are obtained simultaneously. The examples considered include the single material wedge, the bimaterial interface crack and the bimaterial wedge. In particular, the case of a single material wedge bonded to a rigid material along one edge is examined to study the possibility of the existence of mixed mode solutions for arbitrary wedge angles, including the important case of an interface crack when the wedge angle is 180\(\circ \). This behavior is distinctly different from that of plane stress where a complex singularity is obtained. The possibility of the existence of nonseparable solutions is also discussed.