Finite element recovery techniques for local quantities of linear problems using fundamental solutions

We show that local quantities of interest such as displacements or stresses of a FE–solution can be calculated with improved accuracy if fundamental solutions are employed. The approach is based on Bettis theorem and an integral representation of the local quantities via Greens function. The unknown Greens function is split into a regular part and a fundamental solution so that only the regular part must be approximated on the finite element ansatz space. Some numerical studies for linear elasticity will illustrate our approach.     

Electro-elastic stress analysis for a Zener-Stroh crack interacting with a coated inclusion in a piezoelectric solid

Summary. The problem of a Zener-Stroh crack initiated near a coated circular inclusion in a piezoelectric medium is investigated in this paper. By using the solution of a single piezoelectric screw dislocation near a coating inclusion as the Greens function, a Mode III displacement loaded crack is investigated. The proposed problem is formulated as a set of singular integral equations which are solved by numerical techniques. The influence of various parameters, such as the material constants of the inclusion, the coating, the matrix, the coating layer thickness, etc., on the crack behavior is studied. The stress and electric displacement intensity factors of the crack are derived. Several numerical examples are given and the results obtained are discussed in detail.     

Numerical solutions of the singular integral equations in the crack analysis using the body force method

In this paper, numerical solutions of the singular integral equations of the body force method in the crack problems are discussed. The stress fields induced by two kinds of displacement discontinuity are used as fundamental solutions. Then, the problem is formulated as a hypersingular integral equation with the singularity of the form r ². In the numerical calculation, two kinds of unknown functions are approximated by the products of the fundamental density function and the Chebyshev polynomials. As examples, the stress intensity factors of the oblique edge crack, kinked crack, branched crack and zig-zag crack are analyzed. The calculation shows that the present method gives accurate results even for the extremely oblique edge crack and kinked crack with extremely short bend which has been difficult to analyze by the previous method using the approximation by the products of the fundamental density function and the stepped functions etc.     

The dependence of stress intensity factors and weight functions on elastic constants

We investigate the dependence of stress intensity factors (SIFs) on elastic constants in two-dimensional elastic isotropic bodies. Bueckners weight function theory is used for the derivation of the dependence of SIFs on elastic constants. The dependence of SIFs on Poissons ratio shows up when the resultant tractions on each of the contours L _k separately is not zero in multiply connected bodies. As an example we calculate K ₁ for Griffith crack under concentrated loading applied on the upper crack face.     

Stress intensity formulas for three-dimensional cracks in homogeneous and bonded dissimilar materials

This paper is concerned with maximum stress intensity factors of arbitrary shaped defects or cracks under mixed mode loading and also cracks terminating at an interface. A convenient formula is proposed in terms of parameter, where “area” is the projected area of the defects or cracks. First, a rectangular crack under mixed mode loading is considered with varying the aspect ratio and compared with the results of elliptical cracks. Then, parameter is found to be useful under mixed mode loading. Second, a rectangular crack, which is perpendicular to and terminating at a bimaterial interface, is investigated with varying the combinations of materials constants. At the crack front the maximum stress intensity factors are expressed as a function of the elastic ratio of the materials. On the other hand, the generalized stress intensity factors at the interface are expressed as a function of Dundurs parameters α and β. Proposed formulas are usefully evaluating defects with any aspect ratio under any combinations of the materials.     

Stress intensity factors for cracks of arbitrary shape near an interfacial boundary

Modes I, II and III stress intensity factors for a crack of arbitrary planar shape near a bimaterial interface are calculated. The solution utilizes the body-force method and requires Green's functions for perfectly bonded elastic half-spaces. The formulation leads to a system of two-dimensional singular integral equations whose solutions represent the three modes of crack opening displacement. Numerical examples of a semicircular or semielliptical crack terminating at the interface and circular or elliptical cracks contained in one material are given for both internal pressure and farfield tension.     

A new boundary integral equation method for cracked 2-D anisotropic bodies

By using integration by parts to the traditional boundary integral formulation, a traction boundary integral equation for cracked 2-D anisotropic bodies is derived. The new traction integral equation involves only singularity of order 1/r and no hypersingular term appears. The dislocation densities on the crack surface are introduced and the relations between stress intensity factors and dislocation densities near the crack tip are induced to calculate the stress intensity factors. The boundary element method based on the new equation is established and the singular interpolation functions are introduced to model the singularity of the dislocation density (in the order of ) for crack tip elements. The proposed method can be directly used for the 2-D anisotropic body containing cracks of arbitrary geometric shapes. Several numerical examples demonstrate the validity and accuracy of BEM based on the new boundary integral equation.     

Maximizing pseudoconvex transportation problem: a special type

The paper discusses a non-concave fractional programming problem aiming at maximization of a pseudoconvex function under standard transportation conditions. The pseudoconvex function considered here is the product of two linear functions contrasted with a positive valued linear function. It has been established that optimal solution of the problem is attainable at an extreme point of the convex feasible region. The problem is shown to be related to indefinite quadratic programming which deals with maximization of a convex function over the given feasible region. It has been further established that the local maximum point of this quadratic programming problem is the global maximum point under certain conditions, and its optimal solution provides an upper bound on the optimal value of the main problem. The extreme point solutions of the indefinite quadratic program are ranked to tighten the bounds on the optimal value of the main problem and a convergent algorithm is developed to obtain the optimal solution.     

MODELLING OF 3-D MIXED-MODE FRACTURES NEAR PLANAR BIMATERIAL INTERFACES USING SURFACE INTEGRALS

Mixed-mode fractures of arbitrary orientation with respect to a planar bimaterial interface have been effectively modelled using a surface integral approach. By requiring only that the surface of the fracture be discretized, the surface integral method circumvents the practical difficulties associated with having to mesh the interacting dual singularities in stress along the three-dimensional (3-D) crack front and at the interface. The key elements of this numerical capability are discussed in detail. These include: the derivation of the fundamental solutions for a generalized fracture event near a planar bimaterial interface, formulation of the governing integral equation including its decomposition into singular and non-singular terms, development of analytical and numerical techniques for performing the singular integrations, and efficient numerical integration of the non-singular terms using non-dimensionalized surface approximations of the dipole solutions. The problem of a pressurized planar crack near a bimaterial interface was used to assess convergence. The effect of material contrast and crack shape on tendencies for crack growth were also examined.     

Green’s functions of a half-infinite piezoelectric body: Exact solutions

Summary. Greens functions of a half-infinite piezoelectric space play an important role in electroelastic analyses of piezoelectric media. However, almost all works available on the topic are based on the assumption that the normal component of the electric displacement is zero on the surface of the piezoelectric solid, neglecting the effect of polarized surface charge. In the present work, we develop an exact solution for the Greens functions of a half-infinite piezoelectric solid by means of the Stroh formalism. The solution is based on using the exact electric boundary conditions at the interface between the solid and the air medium. First, Greens function for an arbitrary line load in the solid is derived taking into account the effect of polarized charge at the interface, and then the surface Greens function for a surface load is obtained as a special example. Finally, by using the superposition principle, a general expression for the polarized charge distribution on the surface of the piezoelectric solid is presented when an arbitrarily distributed force is exerted on the boundary. It is shown that the normal component of the electric displacement on the solid surface is not zero and it is dependent on the applied loads and the electro-elastic constants of the piezoelectric material and air.     

Systematic derivation of contour integration formulae for laplace and elastostatic gradient BIE's

A complete set of contour integrands is derived for the primary BIE's of elastostatics and potential flow. Because of surface-independent properties of vector potentials, these apply to nonplanar surfaces and can be differentiated at the fixed point, producing contour integrands for both the so-called hypersingular and Cauchy singular parts of the gradient BIE. The results are applicable to far field, near field and on surface cases. Numerical examples demonstrate exact agreement with surface quadrature, and contour plots are given showing variation of the hypersingular integrands in on surface cases.     

Two bonded half planes with a crack going through the interface

The plane problem of two bonded elastic half planes containing a finite crack perpendicular to and going through the interface is considered. The problem is formulated as a system of singular integral equations with generalized Cauchy kernels. Even though the system has three irregular points, it is shown that the unknown functions are algebraically related at the irregular point on the interface and the integral equations can be solved by a method developed previously. The system of integral equations is shown to yield the same characteristic equation as that for two bonded quarter planes in the general case of the through crack, and the characteristic equation for a crack tip terminating at the interface in the special case. The numerical results given in the paper include the stress intensity factors at the crack tips, the normal and shear components of the stress intensity factors at the singular point on the interface, and the crack surface displacements.     

Numerical determination of a class of generalized stress intensity factors

A modification of the collocation method for the numerical solution of Cauchy-type singular integral equations appearing in plane elasticity and, especially, crack problems is proposed. This modification, based on a variable transformation, applies to the case when the unknown function of the singular integral equation behaves like A(x ? c)^α + B(x ? c)^β, where α < 0, 0 < β ? α < 1, near an endpoint c of the integration interval. In plane elasticity such a point is either a crack tip or a corner point of the boundary of the elastic medium. Thus the method seems to be quite efficient for the numerical evaluation of generalized stress intensity factors near such points. A successful application of the method to the classical plane elasticity problem of an antiplane shear crack terminating at a bimaterial interface was also made.     

Stress intensity factors of an inclined elliptical crack near a bimaterial interface

In this paper the stress intensity factors are discussed for an inclined elliptical crack near a bimaterial interface. The solution utilizes the body force method and requires Green’s functions for perfectly bonded semi-infinite bodies. The formulation leads to a system of hypersingular integral equation whose unknowns are three modes of crack opening displacements. In the numerical calculation, unknown body force densities are approximated by using fundamental density functions and polynomials. The results show that the present method yields smooth variations of stress intensity factors along the crack front accurately. Distributions of stress intensity factors are presented in tables and figures with varying the shape of crack, distance from the interface, and elastic modulus ratio. It is found that the inclined crack can be evaluated by the models of vertical and parallel cracks within the error of 24% even for the cracks very close to the interface.     

穿过界面的刚性线夹杂对SH波的散射

利用两个联结半平面中简谐集中力的格林函数,得出了穿过界面刚性线的散射场。刚性线的散射场可分解为有界部分和奇异部分。利用散射场的有界部分和奇异部分得出了刚性线的在SH波作用下的Cauchy型奇异积分方程。根据所得奇异积分方程和Cauchy型积分的端点性质,得出了确定刚性线和界面交点处奇异性阶数的特征方程。根据刚性线和界面交点处的奇性应力定义了交点处的应力奇异因子。对所得Cauchy型奇异积分方程的数值求解,可得刚性线端点和交点处的应力奇异因子。     

Integral equation approach for 3D multiple-crack problems

In this paper, a traction integral equation containing no hypersingular integrals is presented to study the interaction of multiple cracks in an infinite elastic medium. 8-node quadratic quadrilateral elements are used to discretize general crack surfaces, and special crack tip elements are employed along surface boundaries to model the variation of displacements near the crack fronts. Thus, the method possesses the merits of the traction integral equation without hypersingular integrals and those of the special crack tip elements for modeling variation of displacements near the crack tips. The stress intensity factors at the crack front are evaluated using one point formulation and the results are compared with available solutions.     

A nonlinear finite elment eigenanalysis of singular stress fields in bimaterial wedges for plane strain

A displacement-based finite element formulation for the analysis of singular stress fields in power law hardening materials under conditions of plane strain is presented. The displacement field within a sectorial element is quadratic in the angular coordinate and of the power type in the radial direction as measured from the singular point. A hydrostatic pressure variable, which is linear in the angular coordinate, is introduced to account for the incompressibility of the material. The Newton method is combined with matrix singular value decomposition to iteratively solve the resulting nonlinear homogeneous eigenvalue problem where the eigenvalues and eigenfunctions are obtained simultaneously. The examples considered include the single material wedge, the bimaterial interface crack and the bimaterial wedge. In particular, the case of a single material wedge bonded to a rigid material along one edge is examined to study the possibility of the existence of mixed mode solutions for arbitrary wedge angles, including the important case of an interface crack when the wedge angle is 180 . This behavior is distinctly different from that of plane stress where a complex singularity is obtained. The possibility of the existence of nonseparable solutions is also discussed.     

Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity

Summary A modification of the form of the singular integral equation for the problem of a plane crack of arbitrary shape in a three-dimensional isotropic elastic medium is proposed. This modification consists in the incorporation of the Laplace operator into the integrand. The integral must now be interpreted as a finite-part integral. The new singular integral equation is equivalent to the original one, but simpler in form. Moreovet, its form suggests a new approach for its numerical solution, based on quadrature rules for one-dimensional finite part integrals with a singularity of order two. A very simple application to the problem of a penny-shaped crack under constant pressure is also made. Moreover, the case of straight crack problems in plane isotropic elasticity is also considered in detail and the corresponding results for this special case are also derived.With 2 Figures     

A definition and evaluation procedure of generalized stress intensity factors at cracks and multi-material wedges

This paper proposes a definition of generalized stress intensity factors that includes classical definitions for crack problems as special cases. Based on the semi-analytical solution obtained from the scaled boundary finite-element method, the singular stress field is expressed as a matrix power function with its dimension equal to the number of singular terms. Not only real and complex power singularities but also power-logarithmic singularities are represented in a unified expression without explicitly determining the type of singularity. The generalized stress intensity factors are evaluated directly from the scaled boundary finite-element solution for the singular stress field by following standard stress recovery procedures in the finite element method. The definition and evaluation procedure are valid to multi-material wedges composed of any number of isotropic and anisotropic materials. Numerical examples, including a cracked homogeneous plate, a bimaterial plate with an interfacial crack, a V-notched bimaterial plate and a crack terminating at a material interface, are analyzed. Features of this unified definition are discussed.     

Quasistatic temperature stresses in a multilayer thermally sensitive cylinder

An approach to the evaluation of nonstationary temperature fields and the thermoelastic state of multilayer hollow long thermally sensitive cylinders based on the use of the Kirchhoff substitution, generalized functions, and Greens functions of the corresponding problems for a hollow cylinder with piecewise constant physicomechanical characteristics is illustrated. The problem of heat conduction is reduced to the solution of a Fredholm-Volterra nonlinear integral equation of the second kind for the Kirchhoff variable. In the problem of thermoelasticity, the coefficients of the equation (continuous inside each layer) are approximated by piecewise constant functions. The boundary conditions at the ends of the cylinder are satisfied in the integral form. Numerical analysis is performed for the case of a three-layer cylinder.Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 40, No. 4, pp. 7–16, July–August, 2004.