Proposal of FEM implemented with particle discretization for analysis of failure phenomena

A new method for numerical simulation of failure behavior, namely, FEM-β, is proposed. For a continuum model of a deformable body, FEM-β solves a boundary value problem by applying particle discretization to a displacement field; the domain is decomposed into a set of Voronoi blocks and the non-overlapping characteristic functions for the Voronoi blocks are used to discretize the displacement function. By computing average strain and average strain energy, FEM-β obtains a numerical solution of the variational problem that is transformed from the boundary value problem. In a rigorous form, FEM-β is formulated for a variational problem of displacement and stress with different particle discretization, i.e., the non-overlapping characteristic function of the Voronoi blocks and the conjugate Delaunay tessellations, respectively, are used to discretize the displacement and stress functions. While a displacement field is discretized with non-smooth functions, it is shown that a solution of FEM-β has the same accuracy as that of ordinary FEM with triangular elements. The key point of FEM-β is the ease of expressing failure as separation of two adjacent Voronoi blocks owing to the particle discretization that uses non-overlapping characteristic functions. This paper explains these features of FEM-β with results of numerical simulation of several example problems.     

Controlling parameter of the stress intensity factors for a planar interfacial crack in three-dimensional bimaterials

Numerical solutions of singular integral equations are discussed in the analysis of a planar rectangular interfacial crack in three-dimensional bimaterials subjected to tension. The problem is formulated as 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 density functions are chosen to express singular behavior along the crack front of the interface crack exactly. The calculation shows that the present method gives smooth variations of stress intensity factors along the crack front for various aspect ratios. The present method gives rapidly converging numerical results and highly satisfied boundary conditions throughout the crack boundary. The stress intensity factors are given with varying the material combination and aspect ratio of the crack. It is found that the stress intensity factors K_I and K_II are determined by the bimaterial constant ε alone, independent of elastic modulus ratio and Poisson’s ratio.     

Mechanical Quadrature Methods and Extrapolation Algorithms for Boundary Integral Equations with Linear Boundary Conditions in Elasticity

By potential theory, elastic problems with linear boundary conditions are converted into boundary integral equations (BIEs) with logarithmic and Cauchy singularity. In this paper, a mechanical quadrature method (MQMs) is presented to deal with the logarithmic and the Cauchy singularity simultaneously for solving the boundary integral equations. The convergence and stability are proved based on Anselone??s collective compact and asymptotical compact theory. Furthermore, an asymptotic expansion with odd powers of errors is presented, which possesses high accuracy order O(h ³). Using h ³?Richardson extrapolation algorithms (EAs), the accuracy order of the approximation can be greatly improved to O(h ⁵), and an a posteriori error estimate can be obtained for constructing a self-adaptive algorithm. The efficiency of the algorithm is illustrated by examples.     

Effects of cell shape and strut cross-sectional area variations on the elastic properties of three-dimensional open-cell foams

The Voronoi tessellation technique and the finite element (FE) method are utilized to investigate the microstructure-property relations of three-dimensional (3-D) cellular solids (foams) that have irregular cell shapes and non-uniform strut cross-sectional areas (SCSAs). Perturbations are introduced to a regular packing of seeds to generate a spatially periodic Voronoi diagram with different degrees of cell shape irregularity (amplitude a), and to the constant SCSA to generate a uniform distribution of SCSAs with different degrees of SCSA non-uniformity (amplitude b). Twenty FE models are constructed, based on the Voronoi diagrams for twenty foam samples (specimens) having the same pair of a and b, to obtain the mean values and standard deviations of the elastic properties. Spatially periodic boundary conditions are applied to each specimen. The simulation results indicate that for low-density imperfect foams, the elastic moduli increase as cell shapes become more irregular, but decrease as SCSAs get less uniform. When the relative density (R) increases, the elastic moduli of imperfect foams increase substantially, while the Poisson's ratios decrease moderately. The effect of the interaction between the two types of imperfections on foam elastic properties appears to be weak. In addition, it is found that the strut cross-sectional shape has a significant effect on the foam properties. Also, the elastic response of foams with the cell shape and SCSA imperfections appears to be isotropic regardless of changes in a, b and R and the strut cross-sectional shape.     

The penny-shaped crack at a bonded plane with localized elastic non-homogeneity

This paper examines the axisymmetric problem pertaining to a penny-shaped crack which is located at the bonded plane of two similar elastic halfspace regions which exhibit localized axial variations in the linear elastic shear modulus, which has the form G(z)=G₁+G₂e^±ζz. The equations of elasticity governing this type of non-homogeneity are solved by employing a Hankel transform technique. The resulting mixed boundary value problem associated with the penny-shaped crack is reduced to a Fredholm integral equation of the second kind which is solved in a numerical fashion to generate the crack opening mode stress intensity factor at the tip.     

Mechanical quadrature methods and extrapolation for solving nonlinear boundary Helmholtz integral equations

This paper presents mechanical quadrature methods (MQMs) for solving nonlinear boundary Helmholtz integral equations. The methods have high accuracy of order O(h ³) and low computation complexity. Moreover, the mechanical quadrature methods are simple without computing any singular integration. A nonlinear system is constructed by discretizing the nonlinear boundary integral equations. The stability and convergence of the system are proved based on an asymptotical compact theory and the Stepleman theorem. Using the h ³-Richardson extrapolation algorithms (EAs), the accuracy to the order of O(h ⁵) is improved. To slove the nonlinear system, the Newton iteration is discussed extensively by using the Ostrowski fixed point theorem. The efficiency of the algorithms is illustrated by numerical examples.     

Solution of a mixed dynamic problem on the displacements given on the boundary of a half-plane lying on the surface of an isotropic homogeneous elastic half-space

We consider the problem on the motion of an isotropic elastic body occupying the half-space z ≥ 0 on whose boundary, along the half-plane x ≥ 0, the horizontal components of displacement are given, while the remaining part of the boundary is stress-free. We seek the solution by the method of integral Laplace transforms with respect to time t and Fourier transforms with respect to the coordinates x, y; the problem is reduced to a system of Wiener-Hopf equations, which can be solved by the methods of singular-integral equations and circulants. We invert the integral transforms and reduce the solution to the Smirnov-Sobolev form. We calculate the tangential stress intensity coefficients near the boundary z = 0, x = 0, |y| < ∞ of the half-plane. The circulant method for solving the Wiener-Hopf system was proposed in [1]. A static problem similar to that considered in the present paper was solved earlier. The Hilbert problem was reduced to a system of Fredholm integral equations in [2]. In the present paper, we solve the above problem by reducing the solution to quadratures and a quasiregular system of Fredholm integral equations. We give a numerical solution of the Fredholm equations and calculate the integrals for the tangential stress intensity coefficients.     

Elasticity solutions for a transversely isotropic functionally graded circular plate subject to an axisymmetric transverse load qrk

This paper considers the bending of transversely isotropic circular plates with elastic compliance coefficients being arbitrary functions of the thickness coordinate, subject to a transverse load in the form of qr^k (k is zero or a finite even number). The differential equations satisfied by stress functions for the particular problem are derived. An elaborate analysis procedure is then presented to derive these stress functions, from which the analytical expressions for the axial force, bending moment and displacements are obtained through integration. The method is then applied to the problem of transversely isotropic functionally graded circular plate subject to a uniform load, illustrating the procedure to determine the integral constants from the boundary conditions. Analytical elasticity solutions are presented for simply-supported and clamped plates, and, when degenerated, they coincide with the available solutions for an isotropic homogenous plate. Two numerical examples are finally presented to show the effect of material inhomogeneity on the elastic field in FGM plates.     

Analysis method of planar interface cracks of arbitrary shape in three-dimensional transversely isotropic magnetoelectroelastic bimaterials

Using the fundamental solutions for three-dimensional transversely isotropic magnetoelectroelastic bimaterials, the extended displacements at any point for an internal crack parallel to the interface in a magnetoelectroelastic bimaterial are expressed in terms of the extended displacement discontinuities across the crack surfaces. The hyper-singular boundary integral–differential equations of the extended displacement discontinuities are obtained for planar interface cracks of arbitrary shape under impermeable and permeable boundary conditions in three-dimensional transversely isotropic magnetoelectroelastic bimaterials. An analysis method is proposed based on the analogy between the obtained boundary integral–differential equations and those for interface cracks in purely elastic media. The singular indexes and the singular behaviors of near crack-tip fields are studied. Three new extended stress intensity factors at crack tip related to the extended stresses are defined for interface cracks in three-dimensional transversely isotropic magnetoelectroelastic bimaterials. A penny-shaped interface crack in magnetoelectroelastic bimaterials is studied by using the proposed method.The results show that the extended stresses near the border of an impermeable interface crack possess the well-known oscillating singularity r^?1/2±iε or the non-oscillating singularity r^?1/2±κ. Three-dimensional transversely isotropic magnetoelectroelastic bimaterials are categorized into two groups, i.e., ε-group with non-zero value of ε and κ-group with non-zero value of κ. The two indexes ε and κ do not coexist for one bimaterial. However, the extended stresses near the border of a permeable interface crack have only oscillating singularity and depend only on the mechanical loadings.     

The degenerate scale problem for the Laplace equation and plane elasticity in a multiply connected region with an outer circular boundary

This paper investigates the degenerate scale problem for the Laplace equation and plane elasticity in a multiply connected region with an outer circular boundary. Inside the boundary, there are many voids with arbitrary configurations. The problem is analyzed with a relevant homogenous BIE (boundary integral equation). It is assumed that all the inner void boundary tractions are equal to zero, and tractions on the outer circular boundary are constant. Therefore, all the integrations in BIE are performed on the outer circular boundary only. By using the relation z * conjg(z) = a * a, or conjg(z) = a * a/z on the circular boundary with radius a, all integrals can be reduced to an integral for complex variable and they can be integrated in closed form. The degenerate scale a = 1 is found in the Laplace equation and in plane elasticity regardless of the void configuration.     

Variation of the stress intensity factor along the front of a 3-D rectangular crack subjected to mixed-mode load

Summary  The singular integral equation method is applied to the calculation of the stress intensity factor at the front of a rectangular crack subjected to mixed-mode load. The stress field induced by a body force doublet is used as a fundamental solution. The problem is formulated as a system of integral equations with r ⁻³-singularities. In solving the integral equations, unknown functions of body-force densities are approximated by the product of polynomial and fundamental densities. The fundamental densities are chosen to express two-dimensional cracks in an infinite body for the limiting cases of the aspect ratio of the rectangle. The present method yields rapidly converging numerical results and satisfies boundary conditions all over the crack boundary. A smooth distribution of the stress intensity factor along the crack front is presented for various crack shapes and different Poisson's ratio. Received 5 March 2002; accepted for publication 2 July 2002     

INITIAL PRESSURE DISTRIBUTION DUE TO JET IMPACT ON A RIGID BODY

The analysis in this paper shows that, after an impulse due to a two-dimensional jet having velocityU and density ρ hitting a rigid body, the initial pressure distribution over the wall has the constant valueρU² relative to the ambient pressure. It also reveals that a discontinuity exists in the pressure at the intersection of the surface of the body and the surface of the jet. These results have been confirmed by a numerical solution based on a boundary element method.     

Reissner-Sagoci problem for a homogeneous coating on a functionally graded half-space

This paper is concerned with the axisymmetric elastostatic problem related to the rotation of a rigid punch which is bonded to the surface of a nonhomogeneous half-space. The half-space is composed of an isotropic homogeneous coating in the form of layer, which is attached to the functionally graded half-space. The shear modulus of the FGM is assumed to vary in the direction of axis Oz normal to the boundary as μ₁(z) = μ₀(1 + αz)^β, where μ₀, α, β are positive constants. The punch undergoes rotation due to the action of the internal loads. By using Hankel's integral transforms, the mixed boundary value problem is reduced to dual integral equations, and next, to a Fredholm's integral equation of the second kind, which is solved numerically for the case of β = 2. The final results show the effect of non-homogeneity on the shear stresses and an unknown moment of punch rotation.     

Crack reinforcement by distributed springs

An analysis is presented for a centre crack which is reinforced by linear springs over a distance l from both tips. The reinforcement introduces a characteristic length k⁻¹ where k denotes a suitably defined spring constant. Two functions F, V are defined corresponding respectively to the stress intensity factor and the maximum spring-stretch, normalized relative to their values in the absence of reinforcement, for modes I, II or III. Asymptotic expansions for these functions are derived for the limiting cases of soft springs (kl ⪡ 1) and hard springs (kl ⪢ 1). Interpolating functions constructed from these asymptotic expansions are shown to agree with the numerical solution of the governing integral equation over the intermediate range of kl, to within an error of less than 1%. It is noted that the leading term of the expansions for hard springs can be derived by physical reasoning, requiring relatively simple calculations, which can also be used for nonlinear springs.     

Poisson–Nernst–Planck Systems for Ion Flow with Density Functional Theory for Hard-Sphere Potential: I–V Relations and Critical Potentials. Part II: Numerics

We consider a one-dimensional steady-state Poisson–Nernst–Planck type model for ionic flow through membrane channels. Improving the classical Poisson–Nernst–Planck models where ion species are treated as point charges, this model includes ionic interaction due to finite sizes of ion species modeled by hard sphere potential from the Density Functional Theory. The resulting problem is a singularly perturbed boundary value problem of an integro-differential system. We examine the problem and investigate the ion size effect on the current–voltage (I–V) relations numerically, focusing on the case where two oppositely charged ion species are involved and only the hard sphere components of the excess chemical potentials are included. Two numerical tasks are conducted. The first one is a numerical approach of solving the boundary value problem and obtaining I–V curves. This is accomplished through a numerical implementation of the analytical strategy introduced by Ji and Liu in [Poisson–Nernst–Planck systems for ion flow with density functional theory for hard-sphere potential: I–V relations and critical potentials. Part I: Analysis, J. Dyn. Differ. Equ. (to appear)]. The second task is to numerically detect two critical potential values V _c and V ^c .The existence of these two critical values is first realized for a relatively simple setting and analytical approximations of V _c and V ^c are obtained in the above mentioned reference. We propose an algorithm for numerical detection of V _c and V ^c without using any analytical formulas but based on the defining properties and numerical I–V curves directly. For the setting in the above mentioned reference, our numerical values for V _c and V ^c agree well with the analytical predictions. For a setting including a nonzero permanent charge in which case no analytic formula for the I–V relation is available now, our algorithms can still be applied to find V _c and V ^c numerically.     

Asymmetric problem of a row of revolutional ellipsoidal cavities using singular integral equations

This paper deals with numerical solution of singular integral equations of the body force method in an interaction problem of revolutional ellipsoidal cavities under asymmetric uniaxial tension. The problem is solved on the superposition of two auxiliary loads; (i) biaxial tension and (ii) plane state of pure shear. These problems are formulated as a system of singular integral equations with Cauchy-type singularities, where the unknowns are densities of body forces distributed in the r, θ, z directions. In order to satisfy the boundary conditions along the ellipsoidal boundaries, eight kinds of fundamental density functions proposed in our previous papers are applied. In the analysis, the number, shape, and spacing of cavities are varied systematically; then the magnitude and position of the maximum stress are examined. For any fixed shape and size of cavities, the maximum stress is shown to be linear with the reciprocal of squared number of cavities. The present method is found to yield rapidly converging numerical results for various geometrical conditions of cavities.     

A boundary element formulation using the simple method

A new boundary element method is described for calculation of the steady incompressible laminar flows. The method is based on the well-known SIMPLE algorithm. The new boundary element method allows one to find the fields of the pressure and velocity corrections without inner iterations, thus reducing the computational time drastically. This makes it different from the method developed by Patankar and Spalding.³² However, the new method demands a much larger computer strorage. The boundary integral equations are discretized with the help of constant boundary elements and constant cells. The values of the integrals along the boundary elements and the cells for the two-dimensional domain are found analytically. To preserve the stability in the iteration process, under-relaxation for the convection terms is used. This paper gives the results of calculations of the flows between two plane parallel plates at Re = 20 and Re = 200, the flows in a square cavity with a moving upper lid at Re = 1 and Re = 100 and the flow in a plane channel with sudden symmetric expansion at Re =46·6.     

Elastic properties of model random three-dimensional open-cell solids

Most cellular solids are random materials, while practically all theoretical structure-property relations are for periodic models. To generate theoretical results for random models the finite element method (FEM) was used to study the elastic properties of open-cell solids. We have computed the density (ρ) and microstructure dependence of the Young's modulus (E) and Poisson's ratio (ν) for four different isotropic random models. The models were based on Voronoi tessellations, level-cut Gaussian random fields, and nearest neighbour node-bond rules. These models were chosen to broadly represent the structure of foamed solids and other (non-foamed) cellular materials. At low densities, the Young's modulus can be described by the relation E∝ρⁿ. The exponent n and constant of proportionality depend on microstructure. We find 1.3<n<3, indicating a more complex dependence than indicated by periodic cell theories, which predict n=1 or 2. The observed variance in the exponent was found to be consistent with experimental data. At low densities we found that ν≈0.25 for three of the four models studied. In contrast, the Voronoi tessellation, which is a common model of foams, became approximately incompressible (ν≈0.5). This behaviour is not commonly observed experimentally. Our studies showed the result was robust to polydispersity and that a relatively large number (15%) of the bonds must be broken to significantly reduce the low-density Poission's ratio to ν≈0.33.     

A Nodal Method for Phase Change Moving Boundary Problems

A semi-analytic numerical scheme has been developed to solve the one-dimensional, moving boundary phase change problem with time-dependent boundary conditions. Locally analytic, approximate solutions are developed for the position of the moving boundary, and for temperature distribution. Set of discrete equations are obtained by applying these solutions over space-time nodes, and by imposing continuity of temperature and heat flux. Application of this so-called nodal integral approach to the nonlinear Stefan problem shows that the scheme is O(Δx ²), and that it predicts the position of the moving boundary and the temperature distribution within the domain very accurately. For example, with as little as two nodes in the spatial domain, the location of the moving boundary for the case of an exponentially increasing surface temperature on the boundary, after one dimensionless time unit, is found with an error of less than 1%. In addition to large size nodes in space, this scheme also allows the use of very large size time steps. Comparison of numerical results with reference solutions is presented.     

Variational approach to non-linear boundary value problems for elasto-plastic incompressible bending plate

Non-linear boundary value problems for inelastic isotropic homogeneous incompressible bending plate, within the range of J₂-deformation theory, are considered. An existence of the weak solution of the non-linear problem with clamped boundary condition is obtained in H²(Ω) by using monotone operator theory and Browder-Minty theorem. For linearization of the non-linear problem a monotone iteration scheme is constructed. It is shown that the sequence of potentials obtained from the sequence of approximate solutions (i.e. iterations), is a monotone decreasing one. Convergence of the iteration process in H²-norm is proved by using the convexity argument. Numerical solutions, based on finite-difference scheme, are given for linear bending problems with rigid clamped as well as simply supported boundary conditions. Further numerical examples are presented to illustrate the convergence of approximate solutions and monotonicity of the potentials as applied to the non-linear problems.