Shmuel Vigdergauz

Complete Elasticity Solution to the Stress Problem in a Planar Grained Structure

This paper considers a planar elastic composite fonned by identical inclusions of a smooth shape embedded periodically in a background. Both materials are taken to be homogeneous and linearly isotropic. The paper addresses two major topics related to such media: the problem of solving their stress states and the attendant problem of computing the composite effective parameters. At given averages, the local stresses/strains are calculated to high accuracy using the well-known Kolosov-Muskhelishvili potentials, which perform well in many situations. This papers contribution to this approach is twofold. First, the author introduces a new representation of the potentials that incorporates the stress periodicity and given average fields in a simple manner. Second, the author formulates the initial boundary value problem in a nonintegral form that saves computational effort when solving the resultant system of algebraic equations.

Artificial boundary conditions for 2D problems in geophysics

Abstract The numerical solution of a class of two-dimensional problems in geophysics is considered. The earth is modeled as a semi-infinite acoustic or elastic medium. In order to make the computational domain finite, a semi-circular artificial boundary oB is introduced, thus truncating the original semi-infinite domain. An exact nonlocal relation is then derived on oB. This relation is used as a boundary condition on oB, which is combined with the finite element scheme employed in the computational domain. The approach is applied to the Helmholtz equation, modeling time-harmonic acoustic waves, and to problems in elastostatics. Numerical examples validate the method and demonstrate its performance.

Finite element analysis of wave scattering from singularities

Abstract A combined analytic-finite element method is used for the efficient solution of the Helmholtz equation in the presence of geometrical singularities. In particular, time-harmonic waves in a membrane which contains one or more fixed-edge cracks (stringers) are investigated. The Dirichlet-to-Neumann (DtN) map is used in the procedure, to enable the replacement of the original singular problem by an equivalent regular problem, which is then solved by a finite element scheme. The method yields the solution in the entire membrane, as well as the dynamic “stress intensity factor.” Numerical results are presented for a circular membrane containing an edge stringer, two edge stringers and an internal stringer. The first few critical wave numbers of the membrane are also found.

Three-dimensional grained composites of extreme thermal properties

Abstract A new class of two-phase composites possessing optimal macroscopic tensor of heat conductance is found, in which non-identical inclusions are spaced periodically in a matrix making a piecewise homogeneous medium with a microstructure. Geometrical forms of these inclusions minimizing the total thermal energy can be determined numerically after the analytical investigation of a specific inverse boundary problem by the usage of the Newton volume potential. Grained materials of extreme coefficients of heat expansion are also described in the case of cubic symmetry. These results are obtained by the application of the equi-strength concept used before in the theory of elasticity.

Genetic algorithm perspective to identify energy optimizing inclusions in an elastic plate

Abstract Analytical methods in optimization of elastic structures are often problematic due to the high order of the governing equations. It is therefore useful to try various numerical schemes in parallel to better understand the optimized object behavior. The paper describes the theoretical grounds and technical features that had to be introduced for effective implementation of a regular genetic algorithm (GA) to the shape optimization problems in planar elasticity. These features concern the fitness calculation and imposing specific geometric constraints as a way to drastically reduce the computational efforts. Finally we demonstrate a successful GA application for numerical shape optimization of a hole or rigid inclusion in a plate, arbitrarily loaded at infinity. Not only the known results have been reliably reproduced for the energy minimizing holes, but this approach has allowed us to extend the findings for the “worst” (energy maximizing) inclusions as well.

Two-dimensional grained composites of minimum stress concentration

In this paper, we consider a regular two-phase microstructure which is formed by identical foreign inclusions spaced periodically in an elastic matrix. The optimization problem of finding the inclusion shape that minimizes the stress concentration over the whole region is solved explicitly (without recourse to approximate or numerical methods) when both materials have the same shear modulus differing only in Poissons ratios. It is rigorously shown that the stress concentration reduces to the absolute minimum (dictated only by a global part of a given average stress tensor) if the inclusions have equi-stress boundaries identified before in the closely-related context of energy-wise optimization. The stress concentration factors at non-optimal circular inclusions are also found in a closed form to estimate the gain from applying optimal contours.

Optimal stiffening of holes under equibiaxial tension

Abstract In this paper, the optimal configuration—in a remote uniform tension field—is investigated for stiffening rings (made of a different material) in a perforated elastic plate. The complex variable approach and the Kolosov-Muskhelishvili potentials are used to determine the unknown shape of the multiconnected region in which we pose the equations of equilibrium. It turns out that, for any number and relative spacing of the holes, the ring boundaries should represent equalstrength contours. This finding is extended to the case of multilayered and composite materials with elastic moduli varying continuously in a given direction.

Thermoelastic stresses in a cylinder or disk with cubic anisotropy

The thermoelastic stresses in a crystal in the shape of a circular cylinder or disk are considered. The crystal is a cubically-orthotropic linear elastic solid, with three independent elastic properties. The cubic anisotropy renders the problem asymmetric, despite the axisymmetry of the geometry and thermal loading. This problem is motivated by a thermoelastic model used for certain crystal growth processes. Two simplifying assumptions are made here: (a) the problem is two-dimensional with plane strain or plain stress conditions, and (b) the elastic properties do not depend on the temperature. A new Fourier-type perturbation method is devised and an analytic asymptotic solution of a closed form is obtained, based on the weak cubic anisotropy of the crystal as a perturbation parameter. A general solution technique is described which yields the asymptotic solution up to a desired order. Numerical results are presented for typical parameter values.

An effective method for computing the elastic field in a finite cracked disk

The method of analytic continuation is used for analyzing the stress state of a circular disk, with rectilinear cracks subjected to an arbitrary self-balanced loading at the circumference. The use of this method together with the Fourier-series expansion leads to a relatively simple computational procedure, with the stress intensity factor being obtained as a by-product of the solution. Some numerical examples are given for a disk with a single edge crack under uniform normal stresses or two opposite concentrated loads.

A generalization of the equi-stress principle in optimizing the mechanical performance of two-dimensional grained composites

A new optimization criterion of minimizing the variations of the boundary tractions in 2D bi-material elastostatic problems is proposed as a relaxation of the well-known equi-stress principle far beyond its primary application. This integral-type assessment of the local stresses offers significant numerical advantages over their direct minimization. In particular, it allows us to obtain reliable results at moderate computational cost through the same flexible scheme as in the author’s previous research on optimizing perforated plates. The scheme combines a genetic algorithm optimization with an efficient direct solver and with an economic shape parametrization, both formulated in the complex variable terms. After an extended analysis of the criterion, its effectiveness is illustrated, as before, by applying to a checkerboard grained plate where the equi-stress inclusions cease to exist at not-too-small volume fractions. The results obtained permit us to make some empirical conclusions which may stimulate further studies in both theoretical and practical directions.