Order Of Stress Singularity Research Articles

直交異方性異種接合材の界面端に生じる対数関数型応力特異性

When two materials are bonded, free-edge stress singularity usually develops at the intersection of free-surfaces and interface. The order of the free-edge stress singularity p can be obtained as a root of the characteristic equation deduced in terms of Airy's stress function. It is theoretically predicted that logarithmic free-edge singularity develops when the order of the singularity p has double root, but no numerical examination is available in the literature. In this study, stress distribution on the interface of the bonded dissimilar orthotropic materials is calculated by using the boundary element method (B.E.M.) under the condition where the order of the singularity has the double root of the characteristic equation. It was found that the logarithmic free-edge stress singularity developed under mechanical loading. The order of the stress singularity calculated by the nonlinear least squares method from the stress distribution obtained by B.E.M. analyses agreed well with one obtained by using the characteristic equation. The logarithmic free-edge stress singularity developed under thermal stress loading also.

Analysis for Three-Dimensional Singular Stress Field at a Vertex of Bonded Interface Edge in Single Lap Joint under Tensile-Shear Load

Dissimilar material joints are used in industrial products for improving functionality and reducing their weight. However, mechanism of delamination or crack for the dissimilar material joints is not still clarified. It is well known that singular stress occurs at the edge of the interface in the joints due to the mismatch of material properties. The stress on the singular stress field can be expressed by two parameters, i.e. the order of stress singularity and the intensity of stress singularity. In the present paper, an influence of the material properties of the adhesive in the shear lap joint on the intensity of stress singularity was investigated using a three-dimensional boundary element method analysis. Material property of adhesive in the shear lap joint is varied and that of adherend is fixed at Young's modulus 100GPa and Poisson's ratio 0.3. The intensities of stress singularity arranged by the different values of the order of stress singularity are expressed as a power function of the ratio of Young's modulus of adherend to adhesive.

エレメントフリーガラーキン法による三次元異材接合体の応力解析に基づく接合界面端角部における特異応力場の強さに関する評価

We present evaluation of intensity of stress singularity at vertex on interface of three-dimensional bonded structure based on mesh free analysis. A bonded structure consists of mild steel and aluminum is employed as computational model. Order of stress singularity is obtained by eigen analysis based on the finite element method. In this study, relationship between intensity of stress singularity, interface width of bonded structure and the height of aluminum is investigated. It is found that intensity of stress singularity for radius r direction increases with increasing interface width, for the aluminum height h and the interface width b change. In addition, it is found that if least square approximation is carried out for intensity of stress singularity K1ij and interface width b by K1θθ=ξbη, parameter η is close to the order of stress singularity λ at vertex.

OS1206 三次元エレメントフリーガラーキン法による自由表面の介在物と交点近傍における特異応力場の解析 : 角部の曲率半径が特異応力場に与える影響

Intensity of stress singularity near interface edge for bonded structures is investigated by a lot of researchers [1], [2]. Variation of intensity of stress singularity with respect to thickness of adhesive layer, interface width, slanted side surface have already investigated. As for the computational methods, the boundary element method (BEM) and the finite element method (FEM) are frequently used to compute stress distribution. In addition, the element free Galerkin method (EFGM) is also recently employed for stress analysis[3]. In this study, the EFGM is employed in stress analysis for bonded structures. In addition, eigen analysis based on the FEM is carried out to obtain order of stress singularity[4]. As the computational model, a structure made by resin with silicon plate inclusion is employed. We especially focus on influence of radius of curvature at vertex for stress singularity field in this study(See Fig.1).

A least squares procedure for the evaluation of multiple generalized stress intensity factors at 2D multimaterial corners by BEM

Detailed characterization of linear elastic stress states at corners and crack tips requires knowledge of the stress singularity orders, the characteristic angular functions and the generalized stress intensity factors (GSIF). Typically a high accuracy is found in the literature for the evaluation of the stress singularity orders and characteristic angular functions (numerically computed from analytical expressions in most cases). Nevertheless, GSIF values, evaluated by means of a numerical model using FEM or BEM and usually by postprocessing the results, are often reported with a lower level of confidence. A robust procedure is presented in this work for the evaluation of the GSIF at multimaterial corners. The procedure is based on a simple least squares technique involving stresses and/or displacements, computed by BEM, at the neighborhood of the corner tip. A careful verification of the robustness and accuracy of the procedure using a few benchmark problems in the literature has been carried out. Applications of the procedure developed to the evaluation of GSIFs appearing at corners in metal-composite adhesive lap joints are presented.

Study on the Crack Initiation Criterion in Fretting Fatigue of LZ50 Steel

Fretting-fatigue often occurs at the contact area between two materials. There exist strong stress singularity exist at the contact edge, crack usually initiation at here. In this paper, the singular stress fields near the contact edge is obtained by using commercial code ABAQUS, then, the curve fitting method is used to determine the stress-singularity order and stress-strength parameter. Finally, the criterion of fretting-fatigue crack initiation in LZ50 steel is determined by using stress-strength parameters.

Elastic-Plastic Stress Singularities of Plane V-Notches in Power-Hardening Materials

Extensive studies have been carried out to deal with the stress singularity of V-notch problems in linear elasticity theory. In fact, the plastic deformation consequentially arises in the notch tip region because of the high stress concentration. The solution of linear elasticity is not adequate to explain the fracture failure of V-notch structures. Because of the difficulties of the nonlinear analysis and the singularity behavior, few results are given for the plastic stress singularities of general V-notch structures. In this paper, the plane V-notch structures in a power law hardening materials are considered. The Von Mises yield criterion and the plasticity total theory are adopted when the materials arise in plastic status. Similar to methods used in the elastic analysis, the plastic stress field near V-notch tips is assumed as an asymptotic expansion with respect to the radial coordinate originating from the notch tip. The governing equations of plastic behavior of plane V-notch are transformed to eigenvalue problems of nonlinear ordinary differential equations (ODEs) contained by the stress singularity order and the associated eigenfunctions. Consequently all of the stress singularities who are less than zero and the associated eigenvectors are accurately determined for the plane V-notches with arbitrary opening angle.

Asymptotic Analysis for the Singular Stress Behaviour around an Interface Edge of Dissimilar Power-Law Hardening Materials Joint

An asymptotic analysis for singular stress fields around an interface-edge of dissimilar power-law hardening materials joint has been presented under plane strain condition and J2 deformation plasticity theory. Both the balance of force and the continuity of displacement are satisfied on the interface. In the higher order approximation, the nonlinear effective stress term was expanded by Taylor series. An iteration method is proposed for the determination of singular fields around interface edge. Multiple stress singular terms exist for in the higher order approximation. The order of stress singularity has a dependency with the combination of hardening exponents, .

Analysis of the stress singularity field at a vertex in 3D-bonded structures having a slanted side surface

The stress singularity that occurs at a vertex in a joint with a slanted side surface is investigated. The orders of stress singularity at a vertex and at a point on stress singularity lines for various material properties are determined using eigenanalysis. The stress distribution on an interface and the intensity of stress singularity at the vertex are investigated using BEM. It is shown that the order of stress singularity at the vertex in the joints can be reduced by slanting a side surface so as to decrease the angle between the interface and the side surface. The results of BEM analysis reveal that the distribution of stress on the interface is influenced by the slanted side surface. Finally, the 3D intensities of the singularity for stress components which are continuous at the interface are newly defined and determined for various material combinations.

Influence of angle between side surface on singular stress field in 3D joints under a tensile loading

Characteristics of singular stress field at the vertexes of two interfaces in three-dimensional joints with three layers are investigated using eigen analysis formulated by FEM and BEM using fundamental solutions for two-phase materials. Angle between two side surfaces at the vertex is varied from 30° to 150°. An expression for stress field near the vertex on the interface is firstly derived from the result of eigen analysis. The order of stress singularity decreases with the increase of the angle. Angular functions corresponding to stress components are deduced from eigen analysis. A coefficient in the power law term of distance from a side surface in the angular function is a function of an angle between two side surfaces, and the value of coefficient increases with the increase of the angle. A coefficient of power law term in a stress distribution for the distance from the vertex has an extremum at 90° of the angle between side surfaces. It was found that stress singularity field at the vertex with a weaker singularity is affected largely by that at the vertex with a stronger singularity. There are two factors, which define the singular stress filed near the vertex; 1) the intensity of singularity defined by the distance from the vertex, 2) an angular function defined by the distance from stress singularity lines. The intensity of singularity at the vertex of interface was defined considering those factors. The value of the intensity of singularity at the vertex increases with the increase of the angle between side surfaces.

Estimating terminal velocity of rough cracks in the framework of discrete fractal fracture mechanics

In this paper we first obtain the order of stress singularity for a dynamically propagating self-affine fractal crack. We then show that there is always an upper bound to roughness, i.e. a propagating fractal crack reaches a terminal roughness. We then study the phenomenon of reaching a terminal velocity. Assuming that propagation of a fractal crack is discrete, we predict its terminal velocity using an asymptotic energy balance argument. In particular, we show that the limiting crack speed is a material-dependent fraction of the corresponding Rayleigh wave speed.

Fracture Toughness Test of Epoxy Adhesive Dissimilar Joint with Various Adhesive Thicknesses

In this study, effect of bond thickness upon the strength and fracture toughness of epoxy adhesive dissimilar joint is investigated. Tensile and three-point-bending (abbreviated as 3PB hereafter) fracture tests are conducted. Finite element method (abbreviated as FEM hereafter) analysis is also executed to analyze the stress distribution at an interface corner of dissimilar joint. From FEM analysis results, it is found that the stress singularity in the dissimilar joint exists pronouncedly at the SUS304/adhesive interface corner and the order of stress singularity in the tensile model is higher than that in the 3PB model. Moreover, the order of stress singularity in the dissimilar joint having bond thickness of 1.0 mm is quite identical to the value obtained from analytical solution under the plane stress condition. From 3PB test and tensile test, it has been confirmed that the failure stress of dissimilar joint slightly increases with the decreasing bond thickness and can be well predicted by using the interface corner toughness, Hc parameter. The failure of dissimilar joints always originates from the SUS304/adhesive interface corner and the failure stress for dissimilar joint of 3PB test is higher than that of tensile test. For the specimens failed at the ALU/adhesive interface corner, the poor wettability of ALU adherend’s surface plays an important role. For the dissimilar joint with an interfacial crack, the fracture toughness, Jc is calculated by J integral method in FEM analysis. Fracture toughness, Jc for cohesively fractured specimens is more or less constant but shows some dependency on bond thickness for interfacially fractured specimens. Locus of fracture can be best interpreted in terms of stress singularity order at the interfacial crack tip.

609 エポキシ系異材継手のせん断強度及び破断特性の評価(OS10-(1)オーガナイズドセッション《材料損傷の機構・実験評価》)

In this paper, the effect of bond thickness upon the shear strength of epoxy adhesively bonded joints with dissimilar adherends is addressed. The bond thickness, t between the adherends is controlled to be ranged from 0.1 mm to 1.2 mm. Finite element analyses are also executed by ANSYS 11 code to investigate the stress distributions in adhesive layer of adhesive joints. The order of stress singularity is obtained from the analytical solution of Bogy's characteristics equation. As a result, shear strength of adhesive joint reduces with increasing bond thickness. Moreover, the failure of dissimilar adherends bonded shear joint has been initiated at a location with critical stress singularity order which is the interface corner of aluminium/adhesive. It is found that the shear strength prediction can be satisfactorily done by interface corner toughness, Hc parameter.

Characteristics of Singular Stress Distribution at a Vertex in Transversely Isotropic Piezoelectric Dissimilar Material Joints

Stress singularity frequently occurs at a vertex in an interface due to a discontinuity of materials. The stress singularity fields are one of the main factors responsible for debonding under mechanical or thermal loadings. The stress distribution near the vertex in the interface is very important to maintain the reliability of joints. In this paper, stress singularity at a vertex in transversely isotropic piezoelectric dissimilar material joints are 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 values near stress singularity line of the joint are 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 angular function at the free edge.

Influence of Interlayer Thickness on the Intensity of Singular Stress Field in 3D Three-Layered Joints under an External Load

In the present paper, an influence of interlayer thickness on the singular stress field in 3D adhesive joints with three layers are investigated using eigen analysis formulated by finite element method (FEM) and boundary element method (BEM) using fundamental solutions for two-phase materials. A model for analysis is a three-layered block joint composed of Si, resin and FR-4.5 (Flame Retardant type 4.5, a kind of glass epoxy resin), which constitutes CSP (Chip Size Package) in the electric device. All stress components are expressed as spherical coordinate systems where their origins are located at the vertex of each interface. Firstly, the orders of stress singularity for vertex and lines characterizing singular stress fields are calculated from eigen analysis. Secondly, distributions of stress for the model subjected to an external load are calculated using BEM. Then, the intensity of singularity is derived from these results. Furthermore, 3D intensity of singularity is determined for evaluating a bonding strength in the 3D three-layered bonded joints. A relationship between the 3D intensity of singularity and the thickness of interlayer is finally examined. It was found that the 3D intensity of singularity becomes a small value when the interlayer thickness reduces, and the 3D intensity of singularity attains an upper limit when the interlayer thickness is larger than the width of the model.

Fatigue Strength Evaluation Methods Using Stress Distributions

The stress and displacement fields near the bonding edge, sharp notch, and contact edge show singularity behaviours, so methods of evaluating the strength of these points using maximum stresses calculated by a numerical stress analysis, such as the finite element method, are generally not valid. We have previously presented a new method of evaluating the strength of these singular points using two stress singularity parameters H and λ. In this paper we have developed a method of formularizing critical stress-singularity parameter Hth for each order of stress singularity λ by utilizing critical distance stress theories (point method and line method), which can be derived from two typical strength parameters, namely, fatigue limit σw0 and threshold stress-intensity factor range ΔKth. These estimated critical Hth (λ) value agreed well with the experimentally measured value. Using these simple critical distance stress approach we estimated the fretting-fatigue-crack initiation criteria for any contact edge angle and optimized the contact-edge geometry. Moreover, we apply this new strength criteria to general stress concentration structures.

Evaluation of multiple stress singularity orders of a V-notch by the boundary element method

A new technique concerning the evaluation of multiple stress singularity orders of a V-notch with the boundary element method is proposed. For linear elastic material, the asymptotic displacement field in a V-notch tip region is expressed as a series expansion with respect to the radial coordinate from the V-notch tip. The series expansion of the asymptotic field is then substituted into the boundary integral equation established on the V-notch. By boundary element discretization, the boundary integral equation is transformed into an eigen-equation with respect to the stress singularity orders. Hence the eigen-values, which are the singularity orders, can be obtained simultaneously by solving the eigen-equation. An evident feature of the present method is that the real and complex singularity orders together with the non-singularity ones can be obtained at the same time.

Cohesive Zone Model Applied to Creep Crack Initiation at an Interface Edge between Submicron Thick Films

A crack initiates at an interface edge between submicron thick films and leads to the malfunction of microelectronic devices. In this study, the cohesive zone model method with a cohesive law based on the damage mechanics concept is developed to simulate the creep crack initiation at an interface edge between tin and silicon films. Experiments on delamination at the Sn/Si interface using a micro-cantilever bend specimen were conducted. The cohesive law is applied to the elements in the UEL user subroutine in the finite element code ABAQUS. The parameters characterizing the cohesive law are calibrated by fitting displacement-time curves obtained by experiments and FEM simulations. It is revealed that the order of stress singularity increases with time and has a significant jump in its value at the crack initiation.

Geometrically Induced Stress Singularities of a Thick FGM Plate Based on the Third-Order Shear Deformation Theory

Asymptotic solutions for a functionally graded material (FGM) plate are developed to elucidate stress singularities at a plate corner, using a third-order shear deformation theory. The characteristic equations are given explicitly for determining the order of stress singularity at the vertex of a corner with two radial edges having various boundary conditions. The non-homogeneous elasticity properties are present only in the characteristic equations for a corner with one of its two edges simply supported. The effects of material non-homogeneity on the stress singularities are extensively examined. The present results are very useful for developing accurate numerical solutions for an FGM plate under static or dynamic loading when the plate involves stress singularities, such as a V-notch or crack.

Stress Singularity of Thermal Stresses in Three-Dimensional Bonded Structures with an Interlayer

In the present paper, singular stress field at the vertex on the interface in three-dimensional bonded joints is analyzed using BEM and eigen analysis. The order of stress singularity is determined solving an eigen equation based on FEM formulation and the stress distribution is expressed using the result of the eigen-value analysis. A relationship between the thickness of interlayer and residual thermal stresses is presented. Then, a three-dimensional intensity of singularity is determined.