Order Of Stress Singularity Research Articles

Investigation of Order of Singularity in 3D Transversely Isotropic Piezoelectric Bimaterial Joints by FEM

The order of singularity near the vertex of bonded joints is one of the main factors responsible for debonding of electronic device under mechanical or electrical loading. The distribution of singularity field near the vertex of bonded joints is very important to maintain the reliability of smart electronic structure. Piezoelectric material, due to its characteristic direct-converse piezoelectric effect, has naturally received considerable attentions. Piezoelectric materials have been extensively used as transducers and sensors due to their piezoelectric effects that take place between electric fields and mechanical deformation. The order of singularity at a vertex and at a point on singularity line in 3D transversely isotropic piezoelectric joints is analyzed. Eigen analysis based on FEM is used for stress singularity field analysis of piezoelectric bimaterial joints. The eigen equation is used for calculating the order of stress singularity and the angular function of elastic displacement, electric potential, stress and electric displacement. The numerical result shows that the angular functions have large value near the interface edge than the inner portion of the joint. Therefore, there is a possibility to debond and delamination occurs at interface edge of the piezoelectric bimaterial joints, due to the higher stress and electric displacement concentration at the free edge.

The effects of different boundary conditions on three-dimensional cracked discs under anti-plane loading

Abstract Accordingly to the recent multi-scale model proposed by Sih and Tang, different orders of stress singularities are related to different material dependent boundary conditions associated with the interaction between the V-notch tip and the material under the remotely applied loading conditions. This induces complex three-dimensional stress and displacement fields in the proximity of the notch tip, which are worthy of investigation. Starting from Sih and Tang's model, in the present contribution the authors propose some analytical expressions for the calculation of the strain energy density (SED) averaged over a control volume embracing the V-notch tip. The expressions vary as a function of the different boundary conditions. Dealing with the specific crack case, the results from the analytical frame are compared with those determined numerically under linear-elastic hypotheses, by applying different constraints to the through-the-thickness crack edges in three dimensional discs subjected to Mode III loading. Free–free and free–clamped cases are considered. Due to three-dimensional effects, the application of a nominal Mode III loading condition automatically provokes coupled Modes (I and II). Not only the intensity of the induced modes but also their degree of singularity depend on the applied conditions on the crack flanks. The variability of local SED through the thickness of the disc is investigated by numerical analyses and compared with the theoretical trend. The capability of the SED to capture the combined three-dimensional effects is discussed in detail showing that this parameter is particularly useful when the definition of the Stress Intensity Factors (SIFs) is ambiguous or the direct comparison between SIFs with odd dimensionalities is not possible.

ANALYSIS OF ORDER OF STRESS SINGULARITY AT A VERTEX IN 3D TRANSVERSELY ISOTROPIC PIEZOELECTRIC DISSIMILAR BONDED JOINTS

The order of singularity near the vertex of bonded joints is one of the main factors responsible fordebonding under mechanical or thermal loading. The distribution of stress singularity field near the vertex ofbonded joints is very important to maintain the reliability of intelligent materials. In this paper, order of stresssingularity at vertex in 3D transversely isotropic piezoelectric dissimilar bonded joints is analyzed. Eigenanalysis based on FEM is used for stress singularity field analysis of piezoelectric bonded joints. The eigenequation is used for calculating the order of stress singularity, and the angular function. The numerical resultshows that the angular functions have large value near the interface edge than the inner portion of the joint.Therefore, there is a possibility to debond and delamination may occur at the interface edge of the piezoelectricbonded joints due to the higher stress and electric displacement concentration at the free edge.DOI: http://dx.doi.org/10.3329/jme.v44i1.19490

Asymptotic analysis of an adhered complete contact between elastically dissimilar materials

A complete contact problem between elastically dissimilar materials is studied using an asymptotic analysis. A quarter plane wedge on a half-plane represents the contact edge geometry. Two eigenvalues are obtained for pairs of contacting materials, and their characteristics are classified on the Dundurs parallelogram. Generalized stress intensity factors, KI and KII, are derived to use a two-term stress equation of dimensionless form with developing a mode separation angle. It is found that the order of stress singularity increases as the wedge becomes more rigid than the half-plane. Slipping characteristics on the contact interface are investigated in detail, especially for the case of KI < 0 < KII that represents a typical adhesive complete contact condition. An example case is given using a finite element model to provide calibration of the stress intensities for a specific material, geometry and load combination.

S-FEM FOR FRACTURE PROBLEMS, THEORY, FORMULATION AND APPLICATION

This paper briefs some of recent works on applications of the smoothed finite element method (SFEM) to fracture problems. In the SFEM formulation for simulating the singular filed near the crack-tip, it is known that one can directly enrich the filed by adding in proper terms needed to simulate the singularity, which can be done in a number of ways (see the SFEM book). In this work a generalized technique called the "enriched linear PIM" for constructing shape functions is used to formulate a special "five-node (T5) singular crack-tip element" that can produce a proper order of stress singularity near the crack-tip. One layer of T5 crack-tip singular elements are used in a basic mesh of linear elements leading to a very simple model that can be crated automatically for complicated geometry. Because the SFEM uses weakened weak formulations, such a simple model does not require transition elements and the compatibility is ensured. In addition, the singular terms of functions as well as mapping procedures are no longer necessary to compute the stiffness matrix. Thus, the singular SFEM method is straightforward and can be easily implemented in the existing codes. The stress intensity factors of mix-modes can also be easily evaluated by an appropriate treatment during the domain form of the interaction integral, due to the use of simple triangular mesh. The effectiveness of the present singular T5 element is demonstrated via benchmark examples.

Analysis of a three-dimensional dissimilar material joint with one real singularity using a conservative integral

In the present study, a conservative integral based on the Betti reciprocal principle is formulated to determine the intensity of singularity at a vertex of the interface in three-dimensional dissimilar material joints with one real singularity. Eigenanalysis formulated using a three-dimensional finite element method (FEM) is used to calculate the order of stress singularity, angular functions of displacements and stresses. Models with various element sizes and various integral areas are used to investigate the effect of the integration area on the accuracy of the results. The results are compared with those obtained from the boundary element method (BEM) using a curve-fitting technique to calculate the intensity of singularity. In addition, models of various lengths and various material combinations are used to investigate the stress singularity characteristics in three-dimensional dissimilar material joints. The results of the present study indicate that the conservative integral can be used to determine the intensity of singularity in three-dimensional bi-material joints. The accuracy of the results can be improved by mesh refinement. Finally, the relationships among the intensity of singularity, the order of stress singularity and the model geometry are discussed.

Decoupling of Generalized Mode I and II Stress Intensity Factors in the Complete Contact Problem of Elastically Dissimilar Materials

Stress singularity of a complete contact problem is studied herein using an asymptotic analysis. It is considered that a half plane is indented by a semi-infinite sharp wedge for the contact geometry. It is also assumed that the contacting bodies have different elastic properties. It was found that the order of stress singularity is less than 0.5 and varies depending on the wedge angle and material mismatch. The variation in the stress singularity is illustrated in the Dundurs parallelogram. It provides an overall view of the material combination for reducing damage at the contact edge. An analysis is then carried out to evaluate the generalized stress intensity factors (GSIFs) of mode I and II (KI and KII) that calibrate the stress state in the vicinity of the contact edge incorporating the actual conditions. For decoupling the modes, it is developed to select an angle at which a shearing mode disappears for KI and an opening mode disappears for KII. It is found that the decoupling angle for mode I decreases and that for mode II increases as the second Dundurs parameter, β increases.

J0310206 パワーデバイスにおける封止樹脂のはく離評価([J031-02]電子情報機器・半導体デバイス・電子部品の冷却技術(2),材料力学部門)

Engine control units or power control unit are key technologies for new vehicles like hybrid cars, electric cars and hydrogen-fuel cell vehicles. These power devices must be protected by plastic packages from circumstances. However, severe circumstances in engine rooms cause delaminations between components in the plastic packages. In this study, we evaluated the stress intensity factors of a corner and a small crack along the interface between molding resin and a substrate in the plastic package of a power devise. Stress intensity factors of a corner were useless because the order of stress singularities of a corner change with temperature and the combination of components. However, evaluated stress intensity factors of a crack on the interface can evaluate the strength of the interface between molding resin and a substrate reasonably.

Investigation of Order of Singularity in 3D Two-phase Transversely Isotropic Piezoelectric Dissimilar Joints by Eigen Analysis Method

The distribution of stress singularity field near the vertex of two-phase material components is very important to maintain the reliability of intelligent materials. Piezoelectric material, due to its characteristic direct-converse piezoelectric effect, has naturally received considerable attentions. In this paper, stress singularity at a vertex in transversely isotropic piezoelectric dissimilar material joints is analyzed. Eigen analysis based on FEM is used for stress singularity field analysis of piezoelectric dissimilar material joints. The Eigen equation is used for calculating the order of stress singularity, and the angular function of elastic displacement, electric potential, stress and electric displacement. Stress and electric displacement fields demonstrate that the value near the free edge of the joint is larger than that at the inner portion. From the numerical result, it is suggested that delamination of the interface may occur at the interface edge of the dissimilar piezoelectric material joints due to the higher stress concentration at the free edge.

平面V形切口塑性应力奇异性分析

A higher-order analysis of the stress singularity of the plane V-notches and cracks in power law hardening materials is studied. First the asymptotic displacement field in terms of radial coordinates at the notch tip is adopted. When the material near the notch tips arises in plastic deformation, the Von Mises yield criterion and the plastic‘total strain’theory are used. By introducing the displacement expressions into the governing differential equations of the plastic theory, it results in a set of the eigenvalue problem of nonlinear ordinary differential equations with the stress singularity orders and the associated eigenfunctions. Then the interpolating matrix method established by the first author is used to solve the eigenvalue problem by an iteration process. The several leading plastic stress singularity orders of plane V-notches and cracks have been obtained. Simultaneously the associated eigenvectors of the displacement and stress fields in the notch tip region have been determined with the same degree of accuracy. Finally two numerical examples have been given to illustrate the accuracy and the effectiveness of the present method to determine the singularity orders of the plastic V-notch and crack. Based on the computed results, some characteristics related to the stress singularity of the plastic notches and cracks are discussed.

A three-dimensional ES-FEM for fracture mechanics problems in elastic solids

The edge-based smoothed finite element method (ES-FEM) using triangular mesh was recently proposed to model the fracture problems in 2D solids. This paper contains the following ingredients: (1) the ES-FEM is extended to three-dimensional (3D) ES-FEM using tetrahedral elements to compute the stress intensity factors and simulate crack propagation in 3D elastic solids; (2) to model the singular fields of arbitrary order near the crack front, a layer of specially designed seven-noded crack-tip element is constructed; (3) the displacement is then enriched with ease to reproduce the necessary order of stress singularity; and (4) the enrichment is done without losing the essential properties of partition-of-unity and the linear function reproduction. Because the singular ES-FEM uses the strain smoothing technique and it is a typical weakened weak (W2) formulation, the system stiffness matrix is computed employing only the shape function values on the surface of the smoothing domains created based on the edges of elements. No derivatives of the shape functions are needed, and thus no mapping and integration for the W2 form is required. Several numerical examples are presented to validate the effectiveness of the proposed method.

Energetic-statistical size effect in fracture of bimaterial hybrid structures

This paper investigates the size effect on the strength of bimaterial hybrid structures, which consist of a weak bimaterial interface. A general scaling relation was recently derived by combining the energetic scaling of fracture of the bimaterial notch and the finite weakest link model. This scaling relation is now studied through a stochastic cohesive crack model, which is able to capture the fracture process at the bimaterial notch tip as well as the random fracture properties of the bimaterial interface. The numerical example includes a series of metal-composite hybrid beams with a centered V-notch under three-point bending. A wide range of notch angles is considered, which represents various orders of stress singularities. The simulation shows that, as the stress singularities get weaker, there exists a transition from the energetic scaling to the statistical scaling. Within this transition range, the size effect on the structural strength can be explained by a combined energetic-statistical mechanism.

Plastic stress singularity near interface edge of elasto-plastic/elastic bi-material

Stress field near the interface edge of Cu/Si bi-material is numerically analyzed under the small and large scale yielding conditions. Fifteen two-dimensional models with the different surface–interface angles and hardening exponents are used to investigate. The obtained results reveal that the elastic stress singularity order λel increases with increasing of the plastic surface–interface angle θ1 and the elastic surface–interface angle θ2, while the plastic stress singularity order λpl only increases in the range of 45°⩽θ1<90° and approaches to a constant value in the remain range (90°⩽θ1⩽180°). At a specific plastic surface–interface angle θ1, the plastic stress singularity order λpl has the same magnitude regardless of the change of the elastic surface–interface angle θ2. Especially, the elastic stress singularity order λel revealed is not always larger than the plastic stress singularity order λpl. In addition, the plastic stress singular investigated increases with decreasing of the hardening exponent n of the elasto-plastic material.

A local PUFEM modeling of stress singularity in sliding contact with minimal enrichment for direct evaluation of generalized stress intensity factors

The order of stress singularity around sharp corners is studied by solving the characteristic equation numerically. The corresponding displacement and stress fields around the sharp corners, which accurately satisfy the compatibility of deformation and stress states on the two sides of the slave corner, are derived for various contact configurations. The dominant mode of infinite asymptotic stress field for contact problems is then implemented with minimum enrichments (2 functions for each enriched node), for the first time, within the framework of partition of unity finite element. An increased rate of convergence is achieved and the generalized stress intensity factor can be obtained directly from the additional unknowns. Numerical examples demonstrate the superior accuracy of the present approach to capture the sliding contact stress singularities near sharp corners.

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.

Evaluation of Intensity of Singularity for Three-Materials Joints with Power-Logarithmic Singularities using an Enriched Finite Element Method

In this study, the enriched finite element method (enriched FEM) is applied to evaluate the intensity of singularity for three-material joints with power-logarithmic singularities. Using this method, the intensity of singularities can be directly evaluated and very refined meshes around the singular point are unnecessary. Analyses for eigenvalue and eigenvector are conducted to calculate the order of stress singularities and the asymptotic displacement fields on the enriched elements. Accuracy of the results for different mesh types (4-node and 8-node elements) and different sizes of the enriched region are compared with the value obtained from extrapolation for stress distribution based on FEM with very fine mesh. Finally, the models with various lengths and thicknesses are investigated to study an influence of geometry on the singular stress field.

OS0108 第ゼロ節点法による単純重ね合わせ継ぎ手の特異応力場の解析

In this paper, the crack tip stress method is extended so that the intensity of the singular stress field of the SLJ with two real stress singularity orders can be analyzed. Two types of the reference models are proposed; one is the tensile force model; the other one is the shear force model. The intensities of the singular stress fields of the reference models are calculated by the reciprocal work contour integral method (RWCIM). The intensities of the singular stress field of the SLJ are calculated by superposing those of the reference models. Then the accuracy of the present method is discussed by comparing the present results with the solutions which are calculated by the RWCIM.

Analysis of a wedge in an anisotropic bimaterial under antiplane loads

An anisotropic bimaterial wedge under antiplane shear is investigated. The problem is formulated by using a linear transformation method. Employing complex function representations of elastic fields, a complete form of elastic fields near the apex of an anisotropic composite wedge is derived. An anisotropic composite wedge subjected to concentrated loads is also analyzed. The transformed isotropic bimaterial wedge is first solved, and the solution for the anisotropic wedge is then obtained based on the linear transformation method. Expressions of the stress intensity factor for the composite wedge are obtained. Numerical computations are carried out to obtain the order of stress singularity and the stress intensity factors for three types of wedge shapes.

Stress singularities of contact problems with a frictional interface in anisotropic bimaterials

ABSTRACTThis paper develops an effective approach to analyse the asymptotic displacement and stress fields near the singular points in laminated anisotropic composite joints with a fictional contact along the interface. The Ting's theory, extended from the Stroh's formulation for anisotropic elasticity problems, is employed to represent the asymptotic fields near the singular points of the contact problems. The characteristic equations of the contact problems are constructed from the boundary conditions, where the continuity conditions at the frictional interface governed by the Coulomb's law of friction are considered. The order of stress singularities and the asymptotic fields near the singular points are determined by solving the developed non‐linear coupled characteristic equations for the general contact problems of friction in anisotropic bimaterials from an iterative procedure. Only real values of root representing the order of stress singularities are found for the general frictional contact problems. In the special case when the frictional contact exists in two half plane composite laminates, stress singularities may appear if the material constants satisfy certain requirements. Explicit solutions for the root of the characteristic equations and the asymptotic fields for the special case can be directly obtained without requiring the iterative procedure. The theoretical and numerical results show that both the friction coefficient within the slip interface and the wedge angle of anisotropic materials have significant effects on the stress singularities for the frictional contact problems. Further investigations are undertaken for several special cases, including studies for isotropic materials that are confirmed by existing theoretical results and experimental data available.