Multi-material Wedges Research Articles

Stress singularity analysis of multi-material wedges under antiplane deformation

With the help of the coordinate transformation technique, the symplectic dual solving system is developed for multi-material wedges under antiplane deformation. A virtue of present method is that the compatibility conditions at interfaces of a multi-material wedge are expressed directly by the dual variables, therefore the governing equation of eigenvalue can be derived easily even with the increase of the material number. Then, stress singularity on multi-material wedges under antiplane deformation is investigated, and some solutions can be presented to show the validity of the method. Simultaneously, an interesting phenomenon is found and proved strictly that one of the singularities of a special five-material wedge is independent of the crack direction.

Evaluation of dynamic generalised stress intensity factors at cracks and multi-material wedges using the scaled boundary finite element method

Evaluation of dynamic generalised stress intensity factors at cracks and multi-material wedges using the scaled boundary finite element method

Singular harmonic problems at a wedge vertex: mathematical analogies between elasticity, diffusion, electromagnetism, and fluid dynamics

Multimaterial wedges are frequently observed in composite materials. They consist of two or more sectors of dissimilar materials joined together, whose interfaces converge at the same vertex. Due to the mismatch in material properties such as Young’s modulus, thermal conductivity, dielectric permittivity, or magnetic permeability, these geometrical configurations can lead to singular fields at the junction vertex. This paper discusses mathematical analogies, focused on singular harmonic problems, between antiplane shear problem in elasticity due to mode III loading or torsion, the steady-state heat transfer problem, and the diffraction of waves in electromagnetism. In the case of a single material wedge, a mathematical analogy between elasticity and fluid dynamics is also outlined. The proposed unified mathematical formulation is particularly convenient for the identification of common types of singularities (power-law or logarithmic type), the definition of a standardized method to solve nonlinear eigenvalue problems, and the determination of common geometrical and material configurations allowing the relief or removal of different singularities.

Evaluation of Dynamic Generalised Stress Intensity Factors at Cracks and Multi-Material Wedges Using the Scaled Boundary Finite Element Method

A unified definition of generalised stress intensity factors is proposed for multimaterial wedges based on the solution of the singular stress field obtained from the scaled boundary finite element method. This definition is consistent with the classical definitions for cracked problems. It is valid for real, complex and power-logarithmic singularities and avoids the breakdown of existing definitions for V-notched problems. The generalised stress intensity factors are evaluated by following standard stress recovery techniques used in the finite element method. This approach is also extended to the transient dynamic analysis of generalised stress intensity factors. The dynamic effect at high frequency is modelled by a high-order stiffness and mass matrix without internal meshing. Numerical examples are presented to verify the definition and the evaluation procedure. The simplicity and accuracy of this approach are demonstrated.

Stress singularity analysis of anisotropic multi-material wedges under antiplane shear deformation using the symplectic approach

Symplectic approach has emerged a popular tool in dealing with elasticity problems especially for those with stress singularities. However, anisotropic material problem under polar coordinate system is still a bottleneck. This paper presents a subfield method coupled with the symplectic approach to study the anisotropic material under antiplane shear deformation. Anisotropic material around wedge tip is considered to be consisted of many subfields with constant material properties which can be handled by the symplectic approach individually. In this way, approximate solutions of the stress and displacement can be obtained. Numerical examples show that the present method is very accurate and efficient for such wedge problems. Besides, this paper has extended the application of the symplectic approach and provides a new idea for wedge problems of anisotropic material.

A definition and evaluation procedure of generalized stress intensity factors at cracks and multi-material wedges

A definition of generalized stress intensity factors is proposed. It is based on a matrix function solution for singular stress fields obtained from the scaled boundary finite-element method. The dimensions of the matrix are equal to the number of singular terms. Not only real and complex power singularities but also power-logarithmic singularities are represented in a unified expression without explicitly determining the type of singularity. The generalized stress intensity factors are evaluated directly from the definition by following standard stress recovery procedures in the finite element method. Numerical examples are presented to valid the definition and evaluation procedure.

A definition and evaluation procedure of generalized stress intensity factors at cracks and multi-material wedges

This paper proposes a definition of generalized stress intensity factors that includes classical definitions for crack problems as special cases. Based on the semi-analytical solution obtained from the scaled boundary finite-element method, the singular stress field is expressed as a matrix power function with its dimension equal to the number of singular terms. Not only real and complex power singularities but also power-logarithmic singularities are represented in a unified expression without explicitly determining the type of singularity. The generalized stress intensity factors are evaluated directly from the scaled boundary finite-element solution for the singular stress field by following standard stress recovery procedures in the finite element method. The definition and evaluation procedure are valid to multi-material wedges composed of any number of isotropic and anisotropic materials. Numerical examples, including a cracked homogeneous plate, a bimaterial plate with an interfacial crack, a V-notched bimaterial plate and a crack terminating at a material interface, are analyzed. Features of this unified definition are discussed.

Singular, hypersingular and singular free electromagnetic fields at wedge tips in metamaterials

The engineering response of metamaterials has a dramatic impact on the physics, optics and engineering communities, because they offer electromagnetic properties that are difficult or impossible to achieve with conventional materials. In this paper, an asymptotic analysis of the electromagnetic fields at multi-material wedges composed of metamaterials is proposed. This is made possible by removing the assumption of positive electric permittivities and magnetic permeabilities, an hypothesis which usually applies to conventional materials. Exploring the whole range of variability of these electromagnetic properties, it is shown that, in addition to the classical real eigenvalues 0 ⩽ λ < 1 leading to power-law singularities of the type O( r λ−1 ) as r → 0, it is also possible to find imaginary eigenvalues leading to hypersingular solutions, as well as nonsingular configurations for a suitable choice of the negative electric permittivities and magnetic permeabilities of the media. Moreover, to fully characterize the asymptotic fields, the analysis is not only restricted to the determination of the lowest real and complex eigenvalues, but is also extended to the evaluation of the higher-order nonsingular ones. The obtained analytical results collected in synthetic diagrams are expected to have impact on the design of micro- and nano-electro-mechanical systems.

A Mathematical Analogy and a Unified Asymptotic Formulation for Singular Elastic and Electromagnetic Fields at Multimaterial Wedges

In the present contribution, the mathematical analogy existing between the singular stress field in elasticity due to antiplane loading and the singular electromagnetic fields in electromagnetism is derived with reference to the problem of isotropic multimaterial wedges. These configurations, where dissimilar sectors converge to the same vertex, are commonly observed in composite materials and may lead to singularities. The proposed analogy permits to extend several elastic solutions for the power of the stress-singularity already available in the elasticity literature to the analogous electromagnetic problems and viceversa. Finally, electromagnetic structures that cannot be treated according to the proposed analogy, such as those related to bi-isotropic multimaterial wedges, are discussed.

Singular stress analyses of V-notched anisotropic plates based on a novel finite element method

A super singular wedge tip element whose stiffness matrix is based on numerical eigensolutions is incorporated into standard hybrid-stress finite elements to study singular stress fields around the vertex of anisotropic multi-material wedge. The numerical eigensolutions are obtained by an ad hoc finite element eigenanalysis method. To demonstrate the validity of the method, singular stresses for some typical anisotropic single-material/bimaterial wedges are investigated. All numerical results show present finite element method converges rapidly to available solutions with few elements. The present method is applicable to dealing with the problems with more complex geometries.

Solution Methods for General Stress Concentrators in Anisotropic Heterogeneous Media

The increasing use of fibre-reinforced composites in high performance structures has brought a renewed interest in the analysis of cracks, wedges, and multi-material wedges in anisotropic materials. This paper will address three crucial stages of the general stress concentrator analysis: i) numerical procedures for the determination of eigenvalues and eigenvectors in Williams-like asymptotic expansion for multi-material wedge; ii) approaches to an accurate calculation of the near crack tip fields – the application of so-called two-state (or mutual) conservation integrals; iii) application of fracture criteria for the assessment of fracture inception at the general stress concentrators - concept of the so called finite fracture mechanics.

A study of multiple singularities in multi-material wedges and their use in analysis of microelectronic interconnect structures

The nature of the singular stress field at bi-material and multi-material wedges with de-bonded or bonded interfaces is investigated in this paper with the aim of relating singular stress field to failures in microelectronic interconnect structures. The dependence of the strength and the number of the singularities on the relative material behavior as well as the wedge angle is extensively analyzed using planar isotropic elasticity. The procedure is used to analyze modern copper interconnect structures; the orders of singularities and angular functions are asymptotically determined, based on which the stress intensity factors are calculated.

FINITE ELEMENT ANALYSIS OF PIEZOELECTRIC ELASTICITY WITH SINGULAR INPLANE ELECTROELASTIC FIELDS

An ad hoc one-dimensional finite element formulation is developed for the eigenanalysis of inplane singular electroelastic fields at material and geometric discontinuities in piezoelectric elastic materials by using the eigenfunction expansion procedure and the weak form of the governing equations for prismatic sectorial domains composed of piezoelectrics, composites or air. The order of the electroelastic singularities and the angular variation of the stress and electric displacement fields are obtained with the formulation. The influence of wedge angle, polarization orientation, material types, and boundary and interface conditions on the singular electroelastic fields and the order of their singularity are also examined. The simplicity and accuracy of the formulation are demonstrated by comparison to several analytical solutions for piezoelectric and composite multi-material wedges. The nature and speed of convergence suggests that the present eigensolution could be used in developing hybrid elements for use along with standard elements to yield accurate and computationally efficient solutions to problems having complex global geometries leading to singular electroelastic states.

Green’s function of bimaterials comprising all cases of material degeneracy

For bimaterials with planar interfaces subjected to a line force and dislocation, Green’s functions are determined for all types of anisotropic materials including the nondegenerate, degenerate and extra-degenerate cases. The changes in Green’s function caused by material degeneracy are twofold: (i) implicit changes, attributable to material effects only and characterized by high-order eigenvectors and their intrinsic coupling in the higher-order eigensolutions; (ii) explicit changes, influenced by boundary and interface conditions, that cause additional terms in Green’s function. Material degeneracy affects the angular variation of the singular stress field, which may have significant implication on the failure prediction of strongly anisotropic materials. For all material types, Green’s functions are obtained for bimaterials with a planar interface, and for multi-material wedges subjected to a line force and dislocation at the vertex. The results are expressed in a concise notation in terms of the complete set of eigenvectors and kernel matrices of analytic functions.

The Plane Wedge Problem Loaded at Its Apex ? the Self-similarity Property and the Characteristic Vector

In the present paper the elastostatic problem of a generally anisotropic and angularly inhomogeneous plane wedge loaded at its apex by a concentrated force, is studied in linear elasticity. At first the self-similarity property is formulated and the stress field of the inhomogeneous anisotropic self-similar wedge problem, is deduced. The wedge is radially separated and the plane wedge problem is reformulated by the introduction of a characteristic vector. Furthermore, the angular distribution of the load is determined. The multi-material wedge problem in terms of a formulation based on the isotropic angularly inhomogeneous wedge, is confronted, and necessary conditions that ensure the self-similarity property, are found. Finally, the “similar” elastostatic wedge problems and the involution between stresses, are studied.

The Singularity in Multi-Material Wedges under Thermal Loading

By employing the Stroh formalism for plane anisotropic thermoelasticity, closed-form solutions for the orders of stress and heat flux singularities of multi-material wedges have been obtained. Several different boundary conditions are considered in this paper such as insulated or isothermal as well as free-free or fixed-fixed or free-fixed or fixed-free wedge boundaries. The solutions show that the singular orders are influenced by the wedge configurations (n wedge angles), boundary conditions, elastic constants and heat conduction coefficients, but are independent of the thermal moduli.

The plane elasticity problem of an isotropic wedge under normal and shear distributed loading––application in the case of a multi-material problem

The problem of a multi-material composite wedge under a normal and shear loading at its external faces is considered with a variable separable solution. The stress and displacement fields are determined using the equilibrium conditions for forces and moments and the appropriate Airy stress function. The infinite isotropic wedge under shear and normal distributed loading along its external faces is examined for different values of the order n of the radial coordinate r. The proposed solution is applied to the elastostatic problem of a composite isotropic k-materials infinite wedge under distributed loading along its external faces. Applications are made in the case of the two-materials composite wedge under linearly distributed loading along its external faces and in the case of a three-materials composite wedge under a parabolically distributed loading along its external faces.

General transfer matrix method for the multi-material junction and wedge problems

A general transfer matrix method (GTMM) is developed to derive the characteristic equation for the multi-material junction and wedge problems. Using complex variable techniques, the boundary conditions can be represented by a matrix equation. The matrix itself can be simplified to be a square matrix of order 4, for both the junction and wedge problems. In the case of a uniform temperature field, non-homogeneous terms occur on the interface of different kinds of materials. Including the logarithmic terms of potential for temperature, the GTMM can also be applied in this case. Since the derivation of the method involves only a matrix of order 4 or less, the GTMM is much more convenient in the implementation of computer programs.

Stress singularities in anisotropic multi-material wedges and junctions

The general problem of stress singularities in anisotropic multi-material wedges and junctions due to geometric and material discontinuities is investigated. The anisotropic wedges and junctions discussed here comprise two or three materials with different geometric configurations and different material properties. The interfaces in multi-material junctions are either all perfectly bonded or a disbond is introduced along one interface. The order of stress singularities is obtained by solving the characteristic equations that are developed from the boundary conditions of the problem. Numerical examples for the order of stress singularities are given for the general problem including two and three-material wedges, a crack terminating at an interface between two materials, bonded and disbonded two and three-material junctions as well. An “extra” root of the characteristic equations in addition to three roots corresponding to higher singular stresses is observed for the cases of disbonded anisotropic two and three-material junctions. The stress singularities affected by geometric configurations and material properties of anisotropic materials are also investigated.

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.