Order Of Stress Singularity Research Articles

Semi‐analytical computation of 3D stress singularities in linear elasticity

AbstractThe boundary finite element method (BFEM (Finite‐Element Modelling of Unbounded Structures. Wiley: Chichester, England, 1996, The Scaled Boundary Finite Element Method. Wiley: Chichester, England, 2003)) is employed for the investigation of the order of stress singularities for several classes of three‐dimensional singular stress concentration problems, namely notch, crack and laminate free‐edge situations. In all cases, the BFEM results agree excellently with reference results. The required computational effort is considerably lower compared to e.g. standard finite element computations and thus establishes the BFEM as a powerful tool for the semi‐analytical modelling of linear elastic solids. Copyright © 2005 John Wiley & Sons, Ltd.

Nonlinear mechanics of delamination in fiber-composite laminates: asymptotic solutions and computational results

A delamination in a composite laminate under mechanical loading involves an interface crack between dissimilar, anisotropic composite plies undergoing coupled I, II and III deformation modes. With the presence of high crack-tip local stresses, the effect of inherently nonlinear composite material constitutive properties on the delamination becomes an interest and important issue. In this paper, a nonlinear anisotropic laminate elasticity formulation is introduced to establish governing field equations for a delamination between dissimilar fiber-composite plies. In the context of nonlinear fracture mechanics, the dominant stress singularity and asymptotic solution structure are determined for a delamination in a composite laminate with nonlinear ply constitutive properties. A computational method is used to analyze the delamination field in a composite with complex lamination variables. Delamination energy release rates associated with individual modes are obtained for symmetric angle–ply composite laminates for illustration. Several case studies are conducted to compare the solutions for nonlinear delamination problems with those of corresponding linear interface-crack problems. Unique features of nonlinear composite delamination mechanics are presented, including the order of stress singularity, coupled tearing-mode contributions, and large delamination closure due to strong ply material anisotropy, discontinuity and Poisson’s effect.

916 Evaluation of Interface Strength at Free Edge between Thin Films

In this study, delamination tests using a sandwich type specimen are conducted for eight combinations of materials, thin films on silicon substrate, in order to develop a method for quantitative evaluation and comparison of crack initiation strength from the free edge. Stress distribution along interface at the crack initiation is analyzed by boundary element method. Since the order of stress singularities, λ, in the materials are less than 0.07, the stress field near the interface edge is almost constant in atomic level. Then, the critical strength for the interface cracking is quantitatively represented by the concentrated stress near the edge, σ_<ya>・Based on this critical stress, the effects of the several factors such as existence of oxidized interlayer, species of thin film materials and deposition process of thin film on the interface strength are examined as well.

Analysis of Stress Singularity Field in Three-Dimensional Joints Using Three-Dimensional Boundary Element Method

Stress singularity at a vertex in three-dimensional joints is closely related to the strength of joints. It is very important to perform an analysis for evaluating the reliability of a real three-dimensional joint. However, the stress singularity fields near the vertex and along stress singularity lines are still unclear. In the present study, we investigated the characteristics of stress singularity field at a vertex and a point located on the stress singularity line in three-dimensional joints using a three-dimensional BEM and an eigen analysis based on FEM. It was shown that the order of stress singularity at the vertex is larger than that at the point on the stress singularity line, and the order of logarithmic singularity along the singularity line is larger than that at vertex. In the present analysis, a triple root and a quintple root of eigen value, p=1, occur at the vertex and a point on the singularity line, respectively. The order of stress singularity determined from the eigen analysis was agreed with that from BEM. The value of coefficient in a power-law singularity increased as approaching to the vertex along the singularity line. The stress field along the singularity line is more influenced on the logarithmic singularity than that at the vertex. In particular, coefficients of terms in logarithmic singularity vary in the same way with the angle from an interface.

Stress singularity analysis around the singular point on the stress singularity line in three-dimensional joints

The characteristics of the stress fields around a singular point on the stress singularity line of dissimilar materials in three-dimensional joints are investigated using BEM. Contour for the order of stress singularity around the point is mapped on Dundurs’ parameters plane using eigen value analysis by FEM. The results in 3D joints are compared with those in 2D joints having the same cross section and material combination. The order of stress singularity around the singular point on the stress singularity line in 3D joints is almost identical with that in 2D joints in the singularity region. However, the zero boundary of singularity in 3D joints is slightly different from that in 2D joints. Furthermore, the multiple root of p = 1 exists in the eigen value analysis by FEM. Therefore, logarithmic singularity possibly occurs around the singular point on the stress singularity line. Then, the stress distributions around this point are expressed by the combination of the r λ term and logarithmic singularity terms. Finally, the characteristics of the stress intensity factors of the r λ term and logarithmic singularity terms around the singular points are investigated.

Modeling of the Crack Tip Plastic Zone by Two Slip Lines and the Order of Stress Singularity

Modeling of the Crack Tip Plastic Zone by Two Slip Lines and the Order of Stress Singularity

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.

Modeling of the Crack Tip Plastic Zone by Two Slip Lines and the Order of Stress Singularity

Modeling of initial plastic zone by two slip-lines is considered. It is shown that such a model does not remove stress singularity at the crack tip.

The stress singularities at the apex of composite piezoelectric junctions

The stress singularities at the apex of a piezoelectric-conductor, -composite, or -piezoelectric junction are investigated under generalized plane deformation in this paper. The influencing parameters on the stress singularity order are the wedge angles, material properties, graphite fiber orientation, and piezoelectric poling direction. The singularities of decoupled inplane and antiplane electromechanical field s are investigated when piezoelectric materials are polarized in the (x-y)-plane or along the z-axis. From the numerical results, the conditions of strongest and weakest singularity orders can be determined.

Scattering of SH wave by a crack terminating at the interface of a bimaterial

A general method for solving the scattering of plane SH wave by a crack terminating at the interface of a bimaterial is presented. The crack can terminate at the interface in an arbitrary angle. In order to solve the proposed problem, the Green’s function for a point harmonic force applied at an arbitrary point of the bimaterial is established by the Fourier transformation method. Using the obtained Green’s function and the Betti-Rayleigh reciprocal theorem, the total scattered field of the crack is constructed. The total scattered field of the crack is divided into a regular part and a singular part. The hypersingular integral equation of the crack is obtained in terms of the regular and singular scattered field as well as the free wave field. The stress singularity order and singular stress at the terminating point are analyzed by the hypersingular integral equation and the singular scattered field of the crack. The dynamic stress intensity factor (DSIF) at the terminating point is defined in terms of the singular stresses at the terminating point. Numerical solution of the hypersingular integral equation gives the DSIFs at the crack tips. Comparison of our results with known results confirms the proposed method. Some numerical results and corresponding analysis are given in the paper.

Stress Intensity Factors for Woven Glass/Epoxy Laminates with Cracks at Cryogenic Temperatures

ABSTRACT We investigate the stress intensity factors for several crack configurations in G-11 woven glass/epoxy laminates with temperature-dependent properties under tension at cryogenic temperatures. A state of generalized plane strain is assumed. Cracks are assumed to have occurred in the transverse fiber bundles. Cases of a single stack and doubly-periodic arrays of cracks are considered. The order of stress singularities at the tip of a crack is obtained for the case where the crack is normal to and ends at an interface between two fiber bundles. A finite element method is used to determine the stress intensity factors at the crack tip of two-layered woven laminates. Special singular elements containing the exact stress singularity are placed around the crack tip. Numerical results for the stress intensity factor and the order of stress singularities at cryogenic temperatures are obtained and are presented in a graphical form.

THERMAL EFFECT ON THE SINGULAR BEHAVIOR OF MULTIBONDED ANISOTROPIC WEDGES

By employing the Stroh formalism for plane anisotropic thermoelasticity, closed-form solutions for the orders of stress and heat flux singularity of multibonded anisotropic wedges have been obtained. Moreover, the solutions for the temperature, heat flux, displacement, and stress in the field near the wedge apex have also been obtained analytically. Through the proper use of the key matrix introduced in our previous work, the general solutions for the present problems are presented in a simple and compact form. The generality of the present solutions includes the following. (1) Both of the mechanical and thermal properties are considered. (2) No restriction is required on the wedge numbers. (3) Each wedge can be composed of any kind of anisotropic materials such as isotropic, orthotropic, transversely isotropic, monoclinic, and so forth. (4) No restriction is required on the angle of each wedge: for example, the wedge angle can be set to 2π or π to simulate a crack or interfacial crack. (5) Several different boundary conditions are considered such as insulated or isothermal as well as free–free, fixed–fixed, free–fixed, or fixed–free boundary wedges. (6) The solution for the case of multibonded wedge space is also obtained.

Stress singularity analysis in the restorations of premolar class ii cavities

This paper presents the singular stress analysis near the apex of a structure formed during dental restoration of a premolar class II cavity. Based on the elasticity theory, the stresses may go to infinity at the junctions of different materials (e.g. dentine, enamel, restoration materials). Tensions will cause material separation and then material fracture. In order to reduce the failure probability, the degree of stress concentration has to be reduced. The stress singularity order and the stress intensity factor are two parameters, which are often used in fracture analysis. The objective of this paper is to find conditions such that non‐singular stress fields are possible. Three critical positions in the restoration structure are discussed. They are the tips of interface between (1) enamel and restoration; (2) dentine and restoration; and (3) enamel, dentine and restoration. In the last two cases, the restoration may be bonded or debonded to enamel or dentine. After employing Kolosov‐Muskhelishvili complex functions together with the eigenfunction expansion method, the singularity orders are computed theoretically. Weak stress singularity conditions can be sought by properly selecting cutting angles or restoration materials.

Transient dynamic response of a coated piezoelectric strip with a vertical crack

Abstract In this study, the dynamic response of a coated piezoelectric strip containing a crack vertical to the interfaces under normal impact load is considered. Based on the superposition principle and the integral transform techniques, the solution in the Laplace transformed plane is obtained in terms of a singular integral equation. The order of stress singularity around the tip of the terminated crack is also obtained. The singular integral equation is solved by using the Gauss–Jacobi integration formula, and the numerical Laplace inversion is then carried out to obtain the resulting dynamic stress and electric displacement intensities. The effects of the material properties and the geometric parameters on the dynamic stress intensity factor and the dynamic energy density factors are shown graphically.

Singular stresses near apex of wedge by finite element analysis

Based on the theoretical solution of stress singularity order, the corresponding stress intensity factors for a one or two‐bonded wedge, under mechanical or thermal load, is determined using the finite element approach. The r-λ (λ is real or complex) and ln(r) stress singularity types are the major concern. For a wedge with two singularity orders, a modified least square method is proposed. Good agreement was observed when analytical results compared to experimental results. In the case that λ is complex, the numerical errors in calculating the angular functions expressed in the stress terms should be carefully examined.

Enhancement of surface layer to a cracked piezoelectric half-plane underimpact

In this study, the impact response of a coated piezoelectric half-plane containing acrack perpendicular to the boundary is considered. The Laplace and Fouriertransform techniques are used to formulate the problem in terms of a singularintegral equation. The order of stress singularity around the tip of the terminatedcrack is also obtained. The singular integral equation is solved by using theGauss–Jacobi integration formula. Numerical calculations are carried out, andthe effects of the geometric parameters on the dynamic stress intensityfactor and the dynamic energy density factors are shown graphically.

Singularity analysis of anisotropic multimaterial corners

Singular stress states induced at the tip of linear elastic multimaterial corners are characterized in terms of the order of stress singularities and angular variation of stresses and displacements. Linear elastic materials of an arbitrary nature are considered, namely anisotropic, orthotropic, transversely isotropic, isotropic, etc. Thus, in terms of Stroh formalism of anisotropic elasticity, the scope of the present work includes mathematically non-degenerate and degenerate materials. Multimaterial corners composed of materials of different nature are typically present at any metal-composite, or composite-composite adhesive joint. Several works are available in the literature dealing with a singularity analysis of multimaterial corners but involving (in the vast majority) only materials of the same nature (e.g., either isotropic or orthotropic). Although many different corner configurations have been studied in literature, with almost any kind of boundary conditions, there is an obvious lack of a general procedure for the singularity characterization of multimaterial corners without any limitation in the nature of the materials. With the procedure developed here, and implemented in a computer code, multimaterial corners, with no limitation in the nature of the materials and any homogeneous orthogonal boundary conditions, could be analyzed. As a particular case, stress singularity orders in corners involving extraordinary degenerate materials are, to the authors’ knowledge, presented for the first time. The present work is based on an original idea by Ting (1997) in which an efficient procedure for a singularity analysis of anisotropic non-degenerate multimaterial corners is introduced by means of the use of a transfer matrix.

A Key Matrix N for the Stress Singularity of the Anisotropic Elastic Composite Wedges.

By employing the Stroh formalism, a general solution satisfying the basic laws of two-dimensional linear anisotropic elasticity has been written in a complex variable formulation. To study the stress singularity, suitable stress functions have been assumed in the exponential form. The singular order near the anisotropic elastic composite wedge apex can then be found by satisfying the boundary conditions. Since there are many material constants and boundary conditions involved, the characteristic equation for the singular order usually becomes cumbersome or leaves in the form of a system of simultaneous algebraic equations. It is therefore difficult to get any important parameters to study the failure initiation of the composite wedges. Through a careful mathematical manipulation, a key matrix N^ that contains the information of material properties and wedge geometries has been found to be a dominant matrix for the determination of the singular order. A closed-form solution for the order of stress singularity is thus written in a simple form. Special cases such as the wedge corners, cracks, interfacial joints or cracks, a crack terminating at the interface, etc. can all be studied in a unified manner.

Stress singularities of two special geometries of wedges with free–mixed boundary conditions

The order of stress singularities at the apex of a wedge where the boundaries of which are allowed to be in a mixed type and the materials of which are considered to be general anisotropic is investigated. Selecting the complex function f( z) to be in the form of z δ+1 in the Stroh formalism where δ (−1< Re[δ]<0) is the magnitude of the stress singularities, combined with appropriate boundary conditions under consideration, the characteristic equations for the δ are then derived from which the values of δ are investigated. Numerical results for the order of singularities with the extended wedge angles equal to π/2 or π and with mixed types of boundary conditions, which are discussed in the paper, are given. Furthermore, the effect of the alignment of the principal axes of the material on the order of singularities is also investigated.

Impact response of a piezoelectric layered composite plate with a crack

The dynamic response of a central crack in a piezoelectric layered composite plate under normal impact is analyzed. The crack is oriented normally to the interfaces. The Laplace and Fourier transform techniques are used to formulate the problem in terms of a singular integral equation. The order of stress singularity around the tip of the terminated crack is also obtained. Numerical calculations are carried out, and the main results presented are the variations of the dynamic stress intensity factor and the dynamic energy density factor versus time as functions of the geometric parameters and the piezoelectric material properties of the layered composite plate.