Application of hypersingular integral equation method to three-dimensional crack in electromagnetothermoelastic multiphase composites

A three-dimensional crack problem in electromagnetothermoelastic multiphase composites (EMTE-MCs) under extended loads is investigated in this paper. Using Green’s functions, the extended general displacement solutions are obtained by the boundary element method. This crack problem is reduced to solving a set of hypersingular integral equations coupled with boundary integral equations, in which the unknown functions are the extended displacement discontinuities. Then, the behavior of the extended displacement discontinuities around the crack front terminating at the interface is analyzed by the main-part analysis method of hypersingular integral equations. Analytical solutions for the extended singular stresses, the extended stress intensity factors (SIFs) and the extended energy release rate near the crack front in EMTE-MCs are provided. Also, a numerical method of the hypersingular integral equations for a rectangular crack subjected to extended loads is put forward with the extended displacement discontinuities approximated by the product of basic density functions and polynomials. In addition, distributions of extended SIFs varying with the shape of the crack are presented. The results show that the present method accurately yields smooth variations of extended SIFs along the crack front.     

Mixed-mode stress intensity factors for graded materials

In this paper, we present a general method for the calculation of the various stress intensity factors in a material whose constitutive law is elastic, linear and varies continuously in space. The approach used to predict the stress intensity factors is an extension of the interaction integral method. For this type of material, we also develop a systematic method to derive the asymptotic displacement fields and use it to achieve better-quality results. A new analytical asymptotic field is given for two special cases of graded materials. Numerical examples focus on materials with space-dependent Young modulus.     

Analysis of stress intensity factors of a ring-shaped interface crack

In this paper, numerical solutions of singular integral equations are discussed in the analysis of axi-symmetric interface cracks under torsion and tension. The problems of a ring-shaped interface crack are formulated in terms of a system of singular integral equations on the basis of the body force method. In the numerical analysis, unknown body force densities are approximated by the products of the fundamental density functions and power series, where the fundamental densities are chosen to express a two-dimensional interface crack exactly. The accuracy of the present analysis is verified by comparing the present results with the results obtained by other researchers for the limiting cases of the geometries. The calculation shows that the present method gives rapidly converging numerical results for those problems as well as for ordinary crack problems in homogeneous material. The stress intensity factors of a ring-shaped interface crack are shown in tables and charts with varying the material combinations and also geometrical conditions.     

A study on stress intensity factors and singular stress fields of three-dimensional interface crack

By using the finite-part integral concepts and limit technique, the hypersingular integrodifferential equations of three-dimensional (3D) planar interface crack were obtained; then the dominant-part analysis of 2D hypersingular integral was further used to investigate the stress fields near the crack front theoretically, and the accurate formulae were obtained for the singular stress fields and the complex stress intensity factors. After that, a numerical method is proposed to solve the hypersingular integrodifferential equations of 3D planar interface crack, and the problem of elliptical planar crack is then considered to show the application of the method. The numerical results obtained are satisfactory. Project supported by the Foundation of Solid Mechanics Open Research Laboratory of State Education Commission at Tongji University and the National Natural Science Foundation.     

Thermal stress intensity factors for an interface crack in a functionally graded layered structures

The fracture behavior of a functionally graded layered structure (FGLS) with an interface crack under thermal loading is investigated. Considering new boundary conditions, it is assumed that interface crack is partly insulated, and the temperature drop across the crack surfaces is the result of the thermal resistance due to the heat conduction through the crack region. The problem is formulated in terms of a system of singular integral equations. Numerical results are presented to show the influence of the material nonhomogeneity parameters and the dimensionless thermal resistance on the thermal stress intensity factors (TSIFs).     

Multiple 3D flaws in coupled electro-magneto-thermo-elastic multiphase composites by extended hypersingular integral equation method

This work presents extended hypersingular integral equation (E-HIE) method to analyze the multiple 3D mixed-mode flaws problem in fully coupled electro-magneto-thermo-elastic multiphase composites under extended electro-magneto-thermo-elastic coupled loads through intricate theoretical analysis and numerical simulations. First, the problem is reduced to solving a set of E-HIEs. Analytical solutions for the extended singular stresses, the extended stress intensity factors (E-SIFs), the extended energy release rate and the extended strain energy density factors (E-SEDFs) near the flaws front are obtained. Then, the numerical method for the E-HIEs for two 3D flaws subjected to extended coupled loads is proposed. Finally, numerical solutions of E-SIFs and E-SEDFs of some examples are given, and the effect of flaws orientation, interaction and shielding is discussed.     

Analysis of stress intensity factor in orthotropic bi-material mixed interface crack

Adopting the complex function approach, the paper studies the stress intensity factor in orthotropic bi-material interface cracks under mixed loads. With consideration of the boundary conditions, a new stress function is introduced to transform the problem of bi-material interface crack into a boundary value problem of partial differential equations. Two sets of non-homogeneous linear equations with 16 unknowns are constructed. By solving the equations, the expressions for the real bi-material elastic constant εt and the real stress singularity exponents λt are obtained with the bi-material engineering parameters satisfying certain conditions. By the uniqueness theorem of limit,undetermined coefficients are determined, and thus the bi-material stress intensity factor in mixed cracks is obtained. The bi-material stress intensity factor characterizes features of mixed cracks. When orthotropic bi-materials are of the same material, the degenerate solution to the stress intensity factor in mixed bi-material interface cracks is in complete agreement with the present classic conclusion. The relationship between the bi-material stress intensity factor and the ratio of bi-material shear modulus and the relationship between the bi-material stress intensity factor and the ratio of bi-material Young's modulus are given in the numerical analysis.     

Mixed-mode thermal stress intensity factors from the finite element discretized symplectic method

A finite element discretized symplectic method is introduced to find the thermal stress intensity factors (TSIFs) under steady-state thermal loading by symplectic expansion. The cracked body is modeled by the conventional finite elements and divided into two regions: near and far fields. In the near field, Hamiltonian systems are established for the heat conduction and thermoelasticity problems respectively. Closed form temperature and displacement functions are expressed by symplectic eigen-solutions in polar coordinates. Combined with the analytic symplectic series and the classical finite elements for arbitrary boundary conditions, the main unknowns are no longer the nodal temperature and displacements but are the coefficients of the symplectic series after matrix transformation. The TSIFs, temperatures, displacements and stresses at the singular region are obtained simultaneously without any post-processing. A number of numerical examples as well as convergence studies are given and are found to be in good agreement with the existing solutions.     

多层材料结构的界面裂纹尖端复应力强度因子

本文建立了一种多层材料复合结构的界面裂纹问题分析模型。当两种材料之间插入第三种薄层弹性材料,裂纹位于第三种材料与第一或第二种弹性材料的界面上,且插页材料3~#的厚度相对于裂纹尺寸或平面内其他尺寸很小时,可以得到该问题裂纹尖端的复应力强度因子通式。本文用有限元法对结果进行了数值验证,并进行了有关问题的讨论。     

Effect of crack position on stress intensity factor in particle-reinforced metal-matrix composites

In this study, a numerical model was developed to study the effects of mechanical properties of the particle and matrix materials, the crack position (in particle/in matrix) and loading conditions (mode 1 and mixed-mode) in particle-reinforced metal-matrix composites. The finite element technique was used to calculate the stress intensity factors for crack at and near-interface. The Displacement Correlation Method was used to calculate the stress intensity factors K₁ and K₂. In the present model, the particle and matrix materials were modeled in linear elastic conditions. The interface crack was considered between the particle and matrix, without the presence of the interface. For near-interface crack problem, two different crack positions (in particle/in matrix) were selected. The obtained results show the key role on the stress intensity factors played by the relative elastic properties of the particle and matrix. The results also show that loading condition has an important effect on the K₂ stress intensity factor and the crack deflection angle.     

Stress intensity factors of a rectangular crack meeting a bimaterial interface

Using the hypersingular integral equation method based on body force method, a planar crack meeting the interface in a three-dimensional dissimilar materials is analyzed. The singularity of the singular stress field around the crack front terminating at the interface is analyzed by the main-part analytical method of hypersingular integral equations. Then, the numerical method of the hypersingular integral equation for a rectangular crack subjected to normal load is proposed by the body force method, which the crack opening dislocation is approximated by the product of basic density functions and polynomials. Numerical solutions of the stress intensity factors of some examples are given.     

估算裂纹应力强度因子的新方法

根据裂纹形状与裂纹尖端应力强度因子分布之间的固有关系,在线弹性断 裂力学条件下,提出了一种按已知I型裂纹应力强度因子分布规律求裂纹形状及相应应力强 度因子的无梯度迭代法. 通过有限厚度、有限宽度板穿透裂纹和表面裂纹的数值模拟实例验 证了所提出方法的有效性和实用性,并对不同应力强度因子分布规律对裂纹形状以及相应的 应力强度因子大小的影响进行了分析和讨论. 所提出的方法有助于提高实际扩展裂纹应 力强度因子的估算精度以及更合理地预测疲劳裂纹形状演化.     

Mixed-mode stress intensity factors and critical angles of cracks in bolted joints by weight function method

Summary   Mechanical joints, such as bolted or riveted joints, are widely used in structural components. Reliable determination of stress intensity factors for cracks in bolted joints is required to evaluate their safety and fatigue life. The weight function method is an efficient technique to calculate stress intensity factors for various loading conditions by the stress analysis of an uncracked model. In this paper, the mixed-mode stress intensity factors for cracks in bolted joints are analyzed by the weight function method, and coefficients included in the weight function are determined by finite element analysis for reference loadings. The critical angle at which mode I stress intensity factor becomes maximum is determined, and the effects of the amount of clearance and crack length on the critical angle are investigated. Received 28 February 2001; accepted for publication 22 June 2001 RID=" ID=" The authors are grateful for the support provided by a grant from the Korea Science & Engineering Foundation (KOSEF) and Safety and Structural Integrity Research Center at the Sungkyunkwan University.     

The effect of contact stress intensity factors on fatigue crack propagation

Using the technique of Dimensional Analysis the phenomenon of crack closure is modelled using the concept of a contact stress intensity factor Kc. For constant amplitude loading, a simple expression, Kc_max = g(R) ΔK, is obtained without making idealized assumptions concerning crack tip behaviour. Further, by assuming that crack closure arises from the interaction of residual plasticity in the wake of the crack and crack tip compressive stresses, the function g(R) is shown to be constant for non-workhardening materials. This implies that any dependency of Kc_max on R must be attributed to the workhardening characteristic of the material. With Kc known, an “effective” stress intensity factor Ke may be calculated and incorporated into a crack growth law of the form da/dn = f(ΔKe). From analysis, it can be deduced that for a workhardening material, Kc_max will decrease as R increases and the effective stress intensity factor will increase. This means that the fatigue crack propagation rate will increase with R, in accordance with experimental observations.     

求解混合型裂纹应力强度因子的围线积分法

本文用复变函数理论推导出裂纹的辅助场，并用Betti功互等定理给出求解混合型裂纹应力强度因子的远场围绕积分法．此方法与积分路径的选择无关，用有限元法计算出远离裂纹尖端的位移场和应力场，就可通过计算绕裂端的围线积分，精确地给出混合型裂纹的应力强度因子KⅠ和KⅡ的数值解．     

Inclined semi-elliptical crack for predicting crack growth direction based on apparent stress intensity factors

Linear elastic criterion of the inclined semi-elliptical crack growth direction is elaborated on the basis of the strain energy density theory. Stress and displacement fields are presented for higher order terms asymptotic expansion. Solutions for elastic stress intensity factors are accounting for the function describing of the crack tip fields near the free surface of plate. The mixed mode behavior of crack growth direction angle along the semi-elliptical crack front for different combination of biaxial loading, inclination crack angle and surface flaw geometry is determined.     

Determining stress intensity factors in orthotropic composites from far-field measured temperatures

We demonstrate the ability to determine stress intensity factors in orthotropic materials directly from measured temperatures away from the crack and using far-field expressions for the stresses. This is advantageous, recognizing that recorded thermoelastic data can be very unreliable near the tip of a crack. In addition to singular terms that govern in the immediate vicinity of the crack tip, the present series expressions for the stresses contain higher-order finite terms. Little measured input information is needed and data acquisition positions can be selected largely at the user's discretion.     

Thermal stress intensity factors for a crack in an anisotropic half plane

On the basis of the two-dimensional theory of anisotropic thermoelasticity, a solution is given for the thermal stress intensity factors due to the obstruction of a uniform heat flux by an insulated line crack in a generally anisotropic half plane. The crack is replaced by continuous distributions of sources of temperature discontinuity and dislocations. First, the particular thermoelastic dislocation solutions for the half plane are obtained; then the corresponding isothermal solutions are superposed to satisfy the traction-free conditions on the crack surfaces. The dislocation solutions are applied to calculate the thermal stress intensity factors, which are validated by the exact solutions. The effects of the uniform heat flux, the ply angle and the crack length are investigated.     

Analysis of dynamic stress intensity factors of three-point bend specimen containing crack

A new formula is obtained to calculate dynamic stress intensity factors of the three-point bending specimen containing a single edge crack in this study. Firstly, the weight function for three-point bending specimen containing a single edge crack is derived from a general weight function form and two reference stress intensity factors, the coefficients of the weight function are given. Secondly, the history and distribution of dynamic stresses in uncracked three-point bending specimen are derived based on the vibration theory. Finally, the dynamic stress intensity factors equations for three-pointing specimen with a single edge crack subjected to impact loadings are obtained by the weight function method. The obtained formula is verified by the comparison with the numerical results of the finite element method (FEM). Good agreements have been achieved. The law of dynamic stress intensity factors of the three-point bending specimen under impact loadings varing with crack depths and loading rates is studied.     

Analysis of dynamic stress intensity factors of three-point bend specimen containing crack

A new formula is obtained to calculate dynamic stress intensity factors of the three-point bending specimen containing a single edge crack in this study. Firstly, the weight function for three-point bending specimen containing a single edge crack is derived from a general weight function form and two reference stress intensity factors, the coefficients of the weight function are given. Secondly, the history and distribution of dynamic stresses in uncracked three-point bending specimen are derived based on the vibration theory. Finally, the dynamic stress intensity factors equations for three-pointing specimen with a single edge crack subjected to impact loadings are obtained by the weight function method. The obtained formula is verified by the comparison with the numerical results of the finite element method (FEM). Good agreements have been achieved. The law of dynamic stress intensity factors of the three-point bending specimen under impact loadings varing with crack depths and loading rates is studied.