Regularized BE formulations for the analysis of fracture in thin plates

A boundary integral formulation for the analysis of cracks in thin Kirchhoff plates is presented. The numerical solution of the relevant equations is addressed following three different approaches: two single integration methodologies initially introduced for 2D elastic solids are here reformulated, compared with a third (Galerkin) double integration approach and extended to the analysis of cracks in thin plates. exploiting an analogy with 2D elastic fracture mechanics. Comparative numerical testing, in terms of stress intensity factors, is performed with reference to straight and curved cracks in unbounded domains. This revised version was published online in August 2006 with corrections to the Cover Date.     

Evaluation of stress intensity factor and fatigue growth of surface cracks in tension plates

A closed form solution for the stress intensity factor (K) of semielliptical surface cracks in tension plates, obtained by Newman and Raju using 3D finite elements, is critically evaluated. This solution is compared to a recommended solution by the Society of Experimental Stress Analysis (SESA). It is also compared to photoelastic K-measurements. In addition, the aspect ratio variability of surface cracks under fatigue loading is predicted and compared with experimental data and also with the predictions of other empirical equations proposed by Portch, Kawahara et al. and Iida. Finally, the usefulness of this equation for the prediction of fatigue life of surface cracks is evaluated by a comparison with experimental results.     

Assessment of stress intensity factor and aspect ratio variability of surface cracks in bending plates

In this paper, a closed-form solution for the stress intensity factor (K) of semielliptical surface cracks in plates subjected to pure bending, obtained by Newman and Raju using 3D finite elements, is critically evaluated. This solution is compared to a recommended solution by the Society of Experimental Stress Analysis (SESA). It is also compared to photoelastic K measurements. In addition, in this study the aspect-ratio variability of surface cracks under fatigue loading is predicted and compared with experimental data and also with the predictions of other empirical equations proposed by Porten, Kawahara and Kurihara, and lida. Finally, the usefulness of this equation for the prediction of fatigue life of surface cracks is evaluated by a comparison with experimental results.     

基于变分渐近法的功能梯度板高保真简化模型

为有效分析非均质功能梯度板在载荷下的力学性能, 基于变分渐近方法(VAM)建立高保真简化模型。根据Hamilton扩展原则建立功能梯度板的三维能量方程;利用板固有小参数将三维能量渐近扩展为系列二维近似能量方程, 从而将三维各向异性弹性问题简化为沿板厚向的一维分析和二维板分析;提供重构关系以准确预测沿厚度方向的三维场分布。通过SiC-Al功能梯度板的柱形弯曲算例验证:基于该理论和模型重构的位移和应力分量与三维精确解相一致;在应变很小时, 可考虑任意大位移和全局旋转, 并可准确捕捉板所有的几何非线性。     

Stress intensity factor analysis of an interface crack between dissimilar anisotropic materials under thermal stress using the finite element analysis

New numerical methods were presented for stress intensity factor analyses of two-dimensional interfacial crack between dissimilar anisotropic materials subjected to thermal stress. The virtual crack extension method and the thermal M-integral method for a crack along the interface between two different materials were applied to the thermoelastic interfacial crack in anisotropic bimaterials. The moving least-squares approximation was used to calculate the value of the thermal M-integral. The thermal M-integral in conjunction with the moving least-squares approximation can calculate the stress intensity factors from only nodal displacements obtained by the finite element analysis. The stress intensity factors analyses of double edge cracks in jointed dissimilar isotropic semi-infinite plates subjected to thermal load were demonstrated. Excellent agreement was achieved between the numerical results obtained by the present methods and the exact solution. In addition, the stress intensity factors of double edge cracks in jointed dissimilar anisotropic semi-infinite plates subjected to thermal loads were analyzed. Their results appear reasonable.     

A comprehensive study on the 2D boundary element method for anisotropic thermoelectroelastic solids with cracks and thin inhomogeneities

This paper develops Somigliana type boundary integral equations for 2D thermoelectroelasticity of anisotropic solids with cracks and thin inclusions. Two approaches for obtaining of these equations are proposed, which validate each other. Derived boundary integral equations contain domain integrals only if the body forces or distributed heat sources are present, which is advantageous comparing to the existing ones. Closed-form expressions are obtained for all kernels. A model of a thin pyroelectric inclusion is obtained, which can be also used for the analysis of solids with impermeable, permeable and semi-permeable cracks, and cracks with an imperfect thermal contact of their faces. The paper considers both finite and infinite solids. In the latter case it is proved, that in contrast with the anisotropic thermoelasticity, the uniform heat flux can produce nonzero stress and electric displacement in the unnotched pyroelectric medium due to the tertiary pyroelectric effect. Obtained boundary integral equations and inclusion models are introduced into the computational algorithm of the boundary element method. The numerical analysis of sample and new problems proved the validity of the developed approach, and allowed to obtain some new results.     

Two‐dimensional stress distributions in tensioned orthotropic plates weakened by blunt V‐shaped notches

In this work, a comprehensive analytical and numerical study on the two‐dimensional stress fields in orthotropic plates with blunt notches is carried out. Initially, a quick review of the available solutions for lateral notches is presented. Later on, using Lekhnitskii's approach, a new analytical solution is derived which explicitly accounts for the local geometry of the notch, such as the notch root radius and the notch opening angle, as well as the elastic properties of the material. Theoretical predictions based on the new solution are compared with the results from a bulk of 2D finite element analyses carried out on tensioned plates weakened by lateral hyperbolic and blunt V‐shaped notches, showing a very satisfactory agreement.     

Static stresses in a periodically layered anisotropic elastic composite containing a periodic array of planar cracks

Summary The problem of calculating the static elastic stresses in a periodically layered anisotropic composite containing a periodic array of planar cracks is considered. We formulate the problem in terms of a system of simultaneous finite-part singular integral equations which can be solved numerically using collocation techniques. The solution of the integral equations enables relevant quantities such as the stress intensity factors to be computed. Numerical results are obtained for specific cases of the problem.     

求解含裂纹各向异性板与单向复合材料板应力强度因子的变分解法

本文推导了含边缘裂纹各向异性板与单向复合材料板的应力与位移的函数项级数表达式。该级数每一项均满足平衡方程、协调方程与裂纹表面静力边界条件。通过最小势能原理的变分方程来满足其余静力边界条件以确定级数中的待定系数,从而求得应力强度因子。最后,本文给出了长条板拉伸试件、紧凑拉伸试件与三点弯曲试件的计算结果。收敛是迅速的,几乎所有算例都能满足两位收敛。     

An extended layerwise/solid-element method of stiffened composite plates with delaminations and transverse crack

The stiffened composite plates with the transverse crack and delamination were studied in this paper, and an extended layerwise/solid-element (XLW/SE) method was developed. In the proposed method, the governing equations of composite plates and stiffeners were established based on the extended layerwise method and 3D solid elements, respectively. The final governing equation of stiffened composite plates is assembled by using the compatibility conditions and internal force equilibrium conditions at the joint interface between the plates and stiffeners. For the stiffened composite plates with damages, the XLW/SE method can obtain the local stress and displacement fields accurately and simulate the in-plane transverse cracks and delaminations simultaneously, considering complicated stiffeners without any assumptions. In the numerical examples, the results obtained by the proposed method are compared with those obtained by the 3D elastic models developed in the general finite element code, and the good agreements were achieved for the stiffened composite plates with/without delaminations and/or transverse crack.     

Bending analysis of thick exponentially graded plates using a new trigonometric higher order shear deformation theory

An analytical solution of the static governing equations of exponentially graded plates obtained by using a recently developed higher order shear deformation theory (HSDT) is presented. The mechanical properties of the plates are assumed to vary exponentially in the thickness direction. The governing equations of exponentially graded plates and boundary conditions are derived by employing the principle of virtual work. A Navier-type analytical solution is obtained for such plates subjected to transverse bi-sinusoidal loads for simply supported boundary conditions. Results are provided for thick to thin plates and for different values of the parameter n, which dictates the material variation profile through the plate thickness. The accuracy of the present code is verified by comparing it with 3D elasticity solution and with other well-known trigonometric shear deformation theory. From the obtained results, it can be concluded that the present HSDT theory predict with good accuracy inplane displacements, normal and shear stresses for thick exponentially graded plates.     

Regularized boundary integral equations for two-dimensional crack problems in multi-field media

A systematic procedure is followed to develop a set of regularized boundary integral equations for modeling cracks in 2D linear multi-field media. The class of media treated is quite general and includes, as special cases, anisotropic elasticity, piezoelectricity and magnetoelectroelasticity. Of particular interest is the development of a pair of weakly-singular, weak-form integral equations for ‘generalized displacement’ and ‘generalized stress’; these serve as the basis for a weakly-singular symmetric Galerkin boundary element method.     

Stroh formalism based boundary integral equations for 2D magnetoelectroelasticity

This paper presents a novel approach for obtaining boundary integral equations of 2D anisotropic magnetoelectroelasticity. This approach is based on the holomorphy theorems and the Stroh formalism and allows developing of the integral equations for the aperiodic, singly and doubly periodic problems of magnetoelectroelasticity. Obtained equations contain the unknown discontinuities of displacement, electric and magnetic potentials and also traction, electric displacement and magnetic induction that allow adopting the existing boundary element procedures for their solution. Analytical solutions for systems of collinear permeable or impermeable cracks are obtained. Numerical boundary element solutions are obtained for the singly and doubly periodic sets of permeable and impermeable cracks in the magnetoelectroelastic medium and a half-plane. Comparison with analytical solutions and other available results validate the present formulations and numerical computation.     

Assessment of plate theories for multilayered angle-ply plates

The accuracy of laminated plate theories for the analysis of angle-ply multilayered plates is investigated. Classical Lamination Theory, First Order and Reddy's High Order Shear Deformation Theories are considered, together with a new two-dimensional theory recently proposed by the second author (2D-theory). Deflections and through-the-thickness stress distributions are compared with the exact 3D elasticity solution. Energy errors of stress fields obtained from the various theories through both constitutive and equilibrium equations are also evaluated. It is shown that equilibrium equations are required to obtain sufficiently accurate stress distributions from CLT, FSDT and HSDT, whereas stresses can be obtained directly from constitutive equations for the 2D-theory.     

An assumed displacement hybrid finite element model for linear fracture mechanics

This paper deals with a procedure to calculate the elastic stress intensity factors for arbitrary-shaped cracks in plane stress and plane strain problems. An assumed displacement hybrid finite element model is employed wherein the unknowns in the final algebraic system of equations are the nodal displacements and the elastic stress intensity factors. Special elements, which contain proper singular displacement and stress fields, are used in a fixed region near the crack tip; and the interelement displacement compatibility is satisfied through the use of a Lagrangean multiplier technique. Numerical examples presented include: central as well as edge cracks in tension plates and a quarter-circular crack in a tension plate. Excellent correlations were obtained with available solutions in all the cases. A discussion on the convergence of the present solution is also included.     

Generalized approach to estimation of strains and stresses at blunt V‐notches under non‐localized creep

Geometrical discontinuities such as notches need to be carefully analysed by engineers because of the stress concentration generated by them. Notches become even more important when the component is subjected, in service, to very severe conditions, such as high‐temperature fatigue and imposed viscoplastic behaviour such as creep. The knowledge of strains and stresses in such stress concentration zones is essential for an efficient and safe design process. The aim of the paper is to present an improvement and extension of the existing notch‐tip creep stress–strain analysis method developed by Nuñez and Glinka, validated for U‐notches only, to a wide variety of blunt V‐notches. The key in obtaining the extension to blunt V‐notches is the substitution of the Creager–Paris equations with the more generalized Lazzarin–Tovo solution, allowing a unified approach to the evaluation of linear elastic stress fields in the neighbourhood of both cracks and notches. Numerous examples have been analysed to date, and the stress fields obtained according to the proposed method were compared with appropriate finite element data, resulting in a very good agreement. In view of the promising results discussed in the paper, authors are considering possible further extension to sharp V‐notches and cracks introducing the concept of the strain energy density.     

Hypersingular integral equations for a thermoelastic problem of multiple planar cracks in an anisotropic medium

The problem of calculating the thermoelastic stress around an arbitrary number of arbitrarily located planar cracks in an infinite anisotropic medium is considered. The cracks open up under the action of suitably prescribed heat flux and traction. With the aid of suitable integral solutions, we reduce the problem to solving a system of Hadamard finite-part (hypersingular) integral equations. The hypersingular integral equations are solved for specific cases of the problem.     

Stress intensity factors and energy release rates of delaminations in composite laminates

Due to the oscillatory characteristics of stresses near interface crack tips, the stress intensity factor K_i, i = I, II, III, should be modified and the energy release rate G_i, i = 1, 2, 3, of each fracture mode calculated by the virtual crack closure method may not exist. Based upon a near-tip solution for interface cracks between dissimilar anisotropic media, a proper definition for the stress intensity factors and energy release rates for general anisotropic bimaterial interface cracks is provided in this paper, which is applicable for the delaminated composites. Moreover, this definition can be reduced to the classical definition for a crack tip in homogeneous media when the two materials become the same. A simple quadratic relation between K_i and _Gi is derived, which is further reduced explicitly for orthotropic bimaterials. The influence of fiber orientation and the coupling among opening, shearing and tearing mode fracture are studied numerically. The results show that the classical stress intensity factors and energy release rates are still the dominant stress intensity and energy release rate of the mixed mode condition induced by the interface.     

Frobenius’ method for curved cracks

The distribution of stresses produced by an undulated crack in a plane elastic solid, and in particular, at its tips where stresses approach infinity, requires the solution of two coupled singular integral equations. Except for simple crack geometries such as rectilinear and circular arcs in infinite plates, for which explicit analytic solutions have been obtained, the integral equations require numerical solutions. We propose a treatment of the integral equations by Frobenius’ method, which is particularly suitable for evaluating the stress intensity factors of slightly curved cracks.