Bimaterial Wedge Research Articles

A Nonlinear Finite Element Eigenanalysis of Singular Plane Stress Fields in Bimaterial Wedges Including Complex Eigenvalues

A finite element formulation is developed for the analysis of variable-separable singular stress fields in power law hardening materials under conditions of plane stress. The displacement field within a sectorial element is assumed to be quadratic in the angular coordinate and of the power type in the radial direction as measured from the singular point. An iteration scheme that combines the Newton method and matrix singular value decomposition is used to solve the nonlinear homogeneous eigenvalue problem, where the eigenvalues and eigenfunctions are obtained simultaneously. The formulation and iteration scheme apply when the eigenvalue is complex. The examples considered include the single material crack and wedge to demonstrate convergence, and the bimaterial interface crack and the bimaterial wedge to demonstrate geometric versatility and the ability to handle complex eigenvalues. It is found that the real part of the complex eigenvalue for the interface crack agrees with the HRR value. In this case the associated complex eigenfunction is converted into an approximate real-valued eigenfunction that is valid for any mode-mix. In addition, the behavior of separable solutions near certain 'wedge paradox' geometries where non-separable solutions occur is investigated.

ENRICHMENT OF FINITE ELEMENTS WITH NUMERICAL SOLUTIONS FOR SINGULAR STRESS FIELDS

Enriched 2-D and 3-D finite elements are formulated for analysis of solids having multi-material junction and wedge configurations that create singular stress fields due to the material and/or geometric discontinuities. The order and angular variation of the displacements associated with the singular fields are determined from a separate special finite element eigenanalysis and used in the enrichment process. The use of these numerically determined singular fields allows enriched elements to be developed for complex configurations for which analytical fields are not available. In addition to this added flexibility of application, the current formulation applies to elements that may or may not be in direct contact with the singular point. This allows multiple layers of enriched elements to be used around the singular point and traditional mesh refinement studies to be carried out in the enriched element region. Previous enriched formulations have not provided this important capability. For cases where analytical fields are available, such as cracked solids, the performance of elements developed with the current approach is shown to be equivalent to that obtained using analytically enriched elements. Mesh refinement techniques using enriched elements are described that allow accurate stress distributions and generalized stress intensity factors to be directly determined. The importance of high-order numerical integration, use of multiple layers of enriched elements, and proper choice of the size of the enriched region are demonstrated by comparison to existing solutions for solids with cracks. Application of enriched element modelling to a 2-D bi-material wedge and a 3-D stepped-thickness anisotropic composite laminate is demonstrated. © 1997 John Wiley & Sons, Ltd.

Finite deformations at the vertex of a bi-material wedge

The paper gives an asymptotic analysis of finite plane-strain deformations near the vertex of a bi-material wedge of arbitrary angle and subjected to the traction-free surface conditions. Each of two edge-bonded dissimilar wedges is assumed to be hyperelastic with the harmonic-type strain energy density introduced by John (1960). In contrast to the results of linearized elastostatics, the deformations and stresses around the vertex of an arbitrary bi-material wedge are found to be free of oscillatory singularities within the context of finite elastostatics. A simple relation is found between the material moduli and two wedge angles that classifies the higher-order singular field into two different types. The explicit conditions for the strict positivity of the Jacobian determinant in the vicinity of the vertex are given in terms of the known material and geometric parameters independently of the loading conditions. In particular, these conditions can be easily verified for several typical kinds of bi-material wedge. Finally, the expressions for the singular field near the wedge vertex are given and the main features of deformations and stresses are summarized.

Analysis of Bimaterial Wedges Using a New Singular Finite Element

This paper is concerned with the singular stress field at the vertex of a bimaterial wedge under in-plane loading. The boundary value problem is initially formulated in terms of the complex function method. The eigenequations are obtained using the continuity conditions along the interface and the traction-free conditions along the free edges, leading to the development of explicit expressions for the singular stress and displacement fields for a general bimaterial wedge. These expressions are then used to develop a new singular finite element. This element enables the determination of the singular stress field and the associated stress intensity factors reliably and efficiently. To establish the validity of the method, test cases are examined and compared with existing solutions. The method is then applied to evaluate the effect of the wedge geometry and the elastic mismatch upon the resulting stress intensity factors.

Interfacial stress singularities in a bimaterial wedge

In this paper methods of computing stress singularity factors for strain-hardening materials are reviewed. The methods are applied to calculation of these factors for a bimaterial wedge comprising two power-law hardening materials of arbitrary external and internal angles.

Interfacial Stress Singularities in a Bimaterial Wedge

In this paper methods of computing stress singularity factors for strain-hardening materials are reviewed. The methods are applied to calculation of these factors for a bimaterial wedge comprising two power-law hardening materials of arbitrary external and internal angles.

Analysis of stress singular fields at a bimaterial wedge corner

The corner stress singularities at an adhesive interface have been investigated analytically for various butt joint geometries with different adhesive material behavior. Singularities for a rigid interface with elastic and elasto-plastic adhesives have been determined using finite element and finite element iterative methods. In addition, the singular zone geometries, the stress intensity factors and the plastic zone dimensions have been evaluated for the joints investigated. The results obtained in this study have shown excellent agreement with previously published analytical and numerical data. It has been found that the corner singularity becomes weaker as the plastic zone develops.

Singular stress behavior at an adhesive interface corner

The singular stress behavior at a bimaterial wedge corner was studied analytically using the Williams' Airy stress function approach. In addition, the Finite Element Method (FEM) and the Finite Element Iterative Method (FEIM) were used to investigate the asymptotic field in adhesive butt joints of various geometries. In particular, the singular power λ and the geometry of the singular stress zones in the joints were analyzed as a function of the elastic properties of the bonded materials. It was found that the singular power λ and the singular zone geometry at the interface corners in the joints are strongly dependent on the elastic properties of the adhesive and adherend. Moreover, the numerical results of the singular powers obtained from the FEIM analysis showed excellent agreement with previously published analytical data.

The Effects of Inclined Free Edges on the Thermal Stresses in a Layered Beam

A stress-function-based variational method is used to determine the thermal stresses in a layered beam with inclined free edges at the two ends. The stress functions are expressed in terms of oblique cartesian coordinates, and polynomial expansions of the stress functions with respect to the thickness coordinate are used to obtain approximate solutions. Severe interlaminar stresses act across end segments of the layer interfaces. Local concentration of such stresses may be significantly affected by the inclination angle of the end planes. Variational solutions for a two-layer beam show generally beneficial effects of free-edge inclination in dispersing the concentration of interlaminar stresses. The significance of these effects is generally not indicated by the power of the stress singularity as computed from an elasticity analysis of a bimaterial wedge.

Stress Singularities for a Full Plane Consisting of Trimaterial Wedges.

In this study, the stress singularity at the jointed corner tip of a full plane consisting of trimaterial wedges is considered. The eigenequation for the stress singularity is given in an explicit closed form. The eigenequation is reduced from a 4-dimensional determinant. The eigenequations for a single wedge or for a bimaterial wedge can be seen as special cases of the present eigenequation for a trimaterial wedge.

Antiplane Problems of Monoclinic Material

The antiplane problem of anisotropic materials is studied by the Mellin transform technique. The material considered in this study possesses a symmetry plane at z=0. The general solutions of the anisotropic case are obtained in the Mellin transform domain, which are found to have a similar form as for the isotropic case. The exact full‐field solutions for stresses and displacement of the boundary‐value problem for anisotropic bimaterial wedge subjected to prescribed traction on the wedge surfaces are investigated in detail. The asymptotic behavior of the stress in the vicinity of the apex for an anisotropic bimaterial wedge is presented. The field solutions of interfacial crack along two anisotropic materials and a single anisotropic wedge subjected to a pair of concentrated point loads are expressed explicitly by infinite series. This solution is the Green's function for more general problems. The results show that the stress and displacement fields have reduced dependence on the elastic constants, signi...

On the singular behavior at the vertex of a bi-material wedge

Information on the singular behavior at the vertex of a bi-material wedge is the objective of this paper. A summary of the necessary conditions, which depend heavily on the associated eigenvalue equation, for stress singularities of O(r -λ 1n r) as r→0 or O(r -λ) as r→0 is stated. The eigenvalue equations arising from a wide range of boundary and interface conditions are then provided. Bi-material wedge problems that have been subjected to singularity analyses of some generality in the literature are briefly reviewed.

The order of singularity at a multi-wedge corner of a composite plate

The plane problem of a composite body consisting of many dissimilar isotropic, homogeneous and elastic wedges, perfectly bonded along their common interfaces, so that the plate forms a composite full plane with a number of corners, where a number of interfaces coalesce, is considered. The loading of the plate is due to prescribed surface tractions at the outside boundary of the plate. The particular behavior of the stress and displacement fields at the close vicinity of each interface corner is studied and the dependence of the order of singularity there was established in relation with the mechanical properties of the wedges coalescing at the particular corner considered. Applications of this general theory to simple cases of a single wedge and a bimaterial wedge gave results coinciding with already existing simple relations.