Order Of Stress Singularity Research Articles

Evaluation of the stress singularities of plane V-notches in bonded dissimilar materials

According to the linear theory of elasticity, there exists a combination of different orders of stress singularity at a V-notch tip of bonded dissimilar materials. The singularity reflects a strong stress concentration near the sharp V-notches. In this paper, a new way is proposed in order to determine the orders of singularity for two-dimensional V-notch problems. Firstly, on the basis of an asymptotic stress field in terms of radial coordinates at the V-notch tip, the governing equations of the elastic theory are transformed into an eigenvalue problem of ordinary differential equations (ODEs) with respect to the circumferential coordinate θ around the notch tip. Then the interpolating matrix method established by the first author is further developed to solve the general eigenvalue problem. Hence, the singularity orders of the V-notch problem are determined through solving the corresponding ODEs by means of the interpolating matrix method. Meanwhile, the associated eigenvectors of the displacement and stress fields near the V-notches are also obtained. These functions are essential in calculating the amplitude of the stress field described as generalized stress intensity factors of the V-notches. The present method is also available to deal with the plane V-notch problems in bonded orthotropic multi-material. Finally, numerical examples are presented to illustrate the accuracy and the effectiveness of the method.

Eigenspectra and orders of stress singularity at a mode I crack tip for a power-law medium

In this Note eigenspectra and orders of singularity of the stress field near a mode I crack tip in a power-law material are discussed. The perturbation theory technique is employed to pose the required asymptotic solution. The whole set of eigenvalues is obtained. It is shown that the eigenvalues of the nonlinear problem are fully determined by the corresponding eigenvalues of the linear problem and by the hardening exponent. To cite this article: L. Stepanova, C. R. Mecanique 336 (2008).

21209 異材接合体の応力解析(材料力学(3),一般講演,学術講演)

Recently, bonded structures have been noticed. Because, the improvement of the deformation, thermal and electrical properties and the reduction of the cost are expected. However, by the characteristic difference, the stress concentration is generated in the interface, and the reliability remarkably lowers in bonded structures. In this paper, by using the finite element method, stress singularity order and stress and displacement distribution at a vicinity of vertex in bonded structures are calculated. Concretely, the calculation was carried out by the change of jointing method, ratio of a Young's modulus and Poisson's ratio. In addition, the effect of the difference of deformation characteristic and joining method are examined.

An estimation of ceramic fracture at singularities by a statistical approach

A statistical analysis based on weakest link theory is employed to describe the brittle fracture induced at singularities in ceramic materials. Relationships are stated between the Weibull probability and the notch geometry. For low Weibull's modulus, survival probabilities only depend on the generalised stress intensity factor, whereas for high Weibull's modulus, probabilities also depend on the notch tip radius. The effect of the tip radius on the failure probabilities is equivalent to the exclusion of a small volume surrounding the notch tip. From these results, a numerical methodology based on the finite element analysis is proposed to state if singularity is harmful for a ceramic structure. Elsewhere, for a notch with high stress singularity order and symmetrically loaded, the Batdorf's theory gives the same results than the Weibull one. In the case of a low stress singularity order, the prediction of failure can strongly depend on the multi-axial criterion.

Standardization of strength evaluation methods using critical distance stress

The stress and displacement fields near the bonding edge, sharp notch, and contact edge show singularity behaviors, 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. The difficulty with this method was in obtaining the critical value of the intensity of the stress singularity parameter H c for each order of stress singularity λ. Then we have developed a method of formularizing critical value of stress singularity parameter H c for each order of stress singularity λ by utilizing critical distance stress theory. Firstly we apply this method to the delamination strength evaluation. The estimated delamination criteria H c( λ) agrees well with the experimental results. Then we apply this method to general sharp notch corner and contact edge fatigue problems. In these cases the fatigue-crack initiation criteria can be derived from two typical strength parameters, namely, fatigue limit σ w0 and threshold stress intensity factor range Δ K th. These estimated critical H th( λ) value agrees well with the experimentally measured value. Using this 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 can apply this new strength criterion to stress singularity fields for general stress-concentration structures.

Interlaminar stresses and delamination of composite laminates under extension and bending

The metis element method (Hung 1978) has been applied to analyse free edge interlaminar stresses and delamination in composite laminates, which are subjected to extension and bending. The paper recalls Lekhnitskii?s solution for generalized plane strain state of composite laminate and Wang?s singular solution for determination of stress singularity order and of eigen coefficients Cm for delamination problem. Then the formulae of metis displacement finite element in two-dimensional problem are established. Computation of the stress intensity factors and the energy release rates are presented in details. The energy release rate, G, is computed by Irwin?s virtual crack technique using metis elements. Finally, results of interlaminar stresses, the three stress intensity factors and the energy release rates for delamination crack in composite laminates under extension and bending are illustrated and compared with the literature to demonstrate the efficiency of the present method.

P003 三次元ピエゾ接合体の特異性のオーダーに対する接合端部角度の影響(フェロー賞表彰対象ポスターセッション)

The 3DFEM of piezoelectric joints is presented. The influence of the angle of corner on the order of stress singularity in the case of Three-dimensional corner shape of having various angles, various piezoelectric joints angle and various piezoelectricity poled directions, is analyzed in this paper. The eigen equation determining the stress singular order and, the stress, displacement and electric displacement fields near the interface are deduced in details.

Effect of Thickness in Intermediate Layer on Intensity of Stress Singularity at a Vertex in Three-Dimensional Bonded Joints

In the present paper, singular stress fields at the vertex on the interface in three-dimensional bonded joints are analyzed using BEM. 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 simple relationship between the thickness of layer and a value of stress is introduced. Then, stress distributions for various thicknesses of layer are unified using the relationship, and the unified intensity of stress singularity is determined.

Initiation of interface crack at free edge between thin films with weak stress singularity

Delamination tests using sandwich type specimens are conducted for eight combinations of materials: thin films formed on silicon substrates which are relatively popular in micro-electronic industry, to develop a method for quantitative evaluation and comparison of crack initiation strength at the free edge. The difficulty stems from the difference of stress singularity, K ij / r λ ( K ij : stress intensity, r: distance from free edge and λ: order of stress singularity), where λ is depending on the combination of materials. Thus, the critical K ij has different dimensions, MPa m λ , in each interface. Using the experimentally observed delamination load, the stress distribution along the interface is analyzed by boundary element method. Since the orders of stress singularity, λ, in the materials are less than 0.07 (weak singularity), the stress field near the interface edge is almost constant in atomic (nanometer) level. Then, the critical strength for the interface cracking is quantitatively represented by the concentrated stress near the edge. The effects of the several factors such as species of thin films, oxidized interlayers and deposition processes of thin films on the interface strength are evaluated on the basis of this critical stress as well.

Corner stress singularities in an FGM thin plate

Describing the behaviors of stress singularities correctly is essential for obtaining accurate numerical solutions of complicated problems with stress singularities. This analysis derives asymptotic solutions for functionally graded material (FGM) thin plates with geometrically induced stress singularities. The classical thin plate theory is used to establish the equilibrium equations for FGM thin plates. It is assumed that the Young’s modulus varies along the thickness and Poisson’s ratio is constant. The eigenfunction expansion method is employed to the equilibrium equations in terms of displacement components for an asymptotic analysis in the vicinity of a sharp corner. The characteristic equations for determining the stress singularity order at the corner vertex and the corresponding corner functions are explicitly given for different combinations of boundary conditions along the radial edges forming the sharp corner. The non-homogeneous elasticity properties are present only in the characteristic equations corresponding to boundary conditions involving simple support. Finally, the effects of material non-homogeneity following a power law on the stress singularity orders are thoroughly examined by showing the minimum real values of the roots of the characteristic equations varying with the material properties and vertex angle.

Analytical study of the singularities arising at multi-material interfaces in 2D linear elastic problems

The problem of stress singularities due to multi-material junctions in the material microstructure is addressed in the framework of the eigenfunction expansion method, provided that the eigenvalues are explicitly assumed complex. A novel efficient numerical procedure for computing the order of the stress singularity is presented. The numerical method can also be used to solve analogous singularity problems in anisotropic elasticity. According to this procedure, new problems of bi- and tri-material junctions occurring in the material microstructure are examined, paying special attention to the role played by Mode-I and Mode-II deformation. The effects of both the properties of the matrix and of the presence of transgranular cracks on the order of stress singularities are carefully investigated.

Analysis of Stress Singularities at Bi-Material Corners in Reddy's Theory of Plate Bending

AbstractThe order of stress singularity at a sharp corner of a plate needs to be known before a numerical approach can be taken to determine accurately the stress distribution of a plate with irregular geometry (such as a V-notch) under loading. This work analyzes the order of the stress singularity at a bi-material corner of a thick plate under bending, based on Reddy's third-order shear deformation plate theory. An eigenfunction expansion technique is used to derive the asymptotic displacement field in the vicinity of the sharp corner by solving the equilibrium equations in terms of displacement functions. This paper explicitly shows the first known characteristic equations for determining the order of the stress singularity at the interface corner of a bonded dissimilar isotropic plate. Moreover, the numerical results are given in graphic form for the order of stress singularity at the interface corner in bonded dissimilar isotropic plates and at the vertex of a bi-material wedge with free radial edges. The results presented herein fill some of the gaps in the literature

Efficient computation of order and mode of three-dimensional stress singularities in linear elasticity by the boundary finite element method

The Boundary Finite Element Method (BFEM), a novel semi-analytical boundary element procedure solely relying on standard finite element formulations, is employed for the investigation of the orders and modes of three-dimensional stress singularities which occur at notches and cracks in isotropic halfspaces as well as at free edges and free corners of layered plates. After a comprehensive literature review and a concise introduction to the standard three-dimensional BFEM formulation for the static analysis of general unbounded structures, we demonstrate the application of the BFEM for the computation of the orders and modes of two-dimensional and three-dimensional stress singularities for several classes of problems within the framework of linear elasticity. Special emphasis is placed upon the investigation of stress concentration phenomena as they occur at straight free edges and at free corners of arbitrary opening angles in composite laminates. In all cases, the BFEM computations agree excellently with available reference results. The required computational effort is found to be considerably lower compared to e.g. standard Finite Element Method (FEM) computations. In the case of free laminate corners, numerous new results on the occurring stress singularities are presented. It is found that free-corner problems generally seem to involve a more pronounced criticality than the corresponding free-edge situations.

Intensity of Stress Singularity Fields at a Vertex in Three-dimensional Bonded Joints with an Interlayer (1st Report, In Case of a Large Difference of Young's Moduli in Interlayer and Adherend)

In the present paper, a singular stress field at a vertex in three-dimensional bonded structures is investigated. The order of stress singularity is determined through an eigen analysis using FEM, and then the intensity of singularity is determined using BEM as a function of the thickness of interlayer. It was found that stress values at the vertex reduce with decreasing the thickness of interlayer and stress, σθθ which is related to delamination of interface, reduces to the minimum at γ/h= 0.4 on the interface in a softer interlayer than the adherend, where γ represents a distance from the vertex and h the thickness of interlayer. Furthermore, a ratio of the displacement in loading direction near the stress singularity field to that interior the bonded structure is introduced, and it is demenstrated to be a useful parameter for arranging the singular stress field. Finally, a unified intensity of singularity is determined using a least square method. Once the unified intensity of singularity is known, the intensity of singularity for an arbitrary thickness of interlayer will be estimated using the parameter

Multi-scale and multi-order singularity approach to non-equilibrium mechanics: Coupling of atomic-micro-macro damage

Offered in this work is the development of a macro/meso/micro model that covers the lineal scale of 10−11 to 100 by application of the volume energy density function. Boundary constraints and defect geometries are shown to play a role at the smaller scale in the same way as those at the macroscopic scale. Different orders of stress (or energy density) singularities are used to describe the defect geometry and prevailing constraint via the boundary conditions in a way similar to singularity adopted in classical fracture mechanics. Two classes of singularities have been identified in addition to classical one without violating the finiteness conditions of the local displacement and energy density. Still the connection of results from the different scales is no small task and is made possible by application of a scale multiplier. It is determined by considering the interactive effects of the parameters at the different scales from the atomic to the macroscopic. Unlike the classical boundary value problem approach, application of the scale multiplier has led to closed-form asymptotic multiscale solutions that otherwise would not have been made possible. The procedure is demonstrated for the anti-plane shear of a macro-micro-atomic model that accounts for imperfection at the different scales

Free-Edge Stress Singularity in Bonded Dissimilar Orthotropic Materials Under Thermal Stress Leading

When two dissimilar materials are bonded, thermal stresses develop due to the difference in coefficient of thermal expansion of the materials. In recent years, not only isotropic materials but also anisotropic materials are used for making bonded dissimilar materials. In this study, free-edge stress singularity of bonded dissimilar orthotropic materials under thermal stress loading is investigated theoretically. The thermal stresses show the free-edge stress singularity similar to that under mechanical loading, when the thermoelastic constant stress term is subtracted. Stress distributions on the interface are calculated by using the boundary element method. The value of the order of thermal stress singularity determined by using boundary element results agrees well with the theoretical value obtained by using the characteristic equation. It is shown that the thermal stress singularity disappears for certain ranges of wedge angles of a pair of materials.

A generalized stress singularity approach for material failure prediction and its application to adhesive joint strength analysis

The concept of stress is very useful to describe the effect of external loads on structures. However, as a basis for the prediction of failure the concept of stress becomes meaningless when the structure encompasses singularities as a result of discrete stiffness steps or geometric anomalies such as cracks. In this article it is argued that the concept of failure stress is incorrect and should be replaced by a generalized concept based on stress intensity factors and singularity orders. It appears that material failure stress is the critical stress intensity factor for a zero-order singularity stress field. By plotting the critical stress intensity factor as a function of singularity order, the strength of a material can be characterized in a general fashion that integrates tensile strength, fracture toughness and critical singularities in adhesive joints. It is also shown that plasticity does not eliminate the stress singularity in an adhesive joint but changes the order of the singularity due to the induced change in interface corner angle between the dissimilar materials in the joint.

A method for eliminating the effect of 3-D bi-material interface corner geometries on stress singularity

The stress singularities at three-dimensional (3-D) interface corners and edges have been investigated numerically using common finite element methods. The effects of variation of edge angle and vertex angle between the two free side surfaces on the order of stress singularity at bi-material interface corners are determined. It is found that the order of stress singularity at interface corner depends not only on the combination of materials and edge angles but also on the vertex angle between the two free side surfaces. The effect of the vertex angle on the order of stress singularity at interface corner can be eliminated by smoothing the intersection of interface edges, which can be achieved simply by generating a circular-arc fillet at the intersection of the two free side surfaces. The numerical results show that, after smoothing the intersection of interface edges, the order of stress singularity along the interface edges become continuous, i.e. the order of the corner singularity can be reduced to the level of the edge singularity.

The Boundary Finite Element Method for the Efficient Computation of Orders and Modes of Three‐Dimensional Stress Singularities

AbstractIn this contribution the Boundary Finite Element Method (BFEM) is employed for the computation of the orders of stress singularities for several three‐dimensional stress concentration problems in linear elastic fracture mechanics. The BFEM combines the advantages of both the FEM and the BEM: while only a discretization on a structural boundary is required, the actual surface mesh consists of standard displacement based finite elements. In contrast to the BEM, no fundamental solution is required. The BFEM is an ef.cient analysis tool which leads to highly accurate results with significantly lesser computational effort when compared to e.g. standard FEM procedures. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the stress singularity of dissimilar anisotropic wedges and junctions in antiplane shear

In this paper, r λ−1 type antiplane stress singularities in dissimilar anisotropic wedges and junctions are studied. For some special wedge angles, the eigenequations derived are a function of fiber orientations and the elastic constants in the principal material coordinates. The boundary surfaces of the wedges can be subjected to combinations of free and clamped boundary conditions. The repeated occurrence of stress singularity orders is examined. By properly selecting the fiber orientations, the degree of the stress singularities can be diminished or even disappeared. The singularities of the materials reduced to isotropic are discussed as well.