Konstantin A. Lurie

On the effective conductivity of polycrystals and a three‐dimensional phase‐interchange inequality

We derive optimal bounds on the effective conductivity tensor of polycrystalline aggregates by introducing an appropriate null‐Lagrangian that is rotationally invariant. For isotropic aggregates of uniaxial crystals an outstanding conjecture of Schulgasser is proven, namely that the lowest possible effective conductivity of isotropic aggregates of uniaxial crystals is attained by a composite sphere assemblage, in which the crystal axis is directed radially outwards in each sphere. By laminating this sphere assemblage with the original crystal we obtain anisotropic composites that are extremal, i.e., attaining our bounds. These, together with other results established here, give a partial characterization of the set of all possible effective tensors of polycrystalline aggregates. The same general method is used to prove a conjectured phase interchange inequality for isotropic composites of two isotropic phases. This inequality correlates the effective conductivity of the composite with the effective tensor...

Invariant Properties of the Stress in Plane Elasticity and Equivalence Classes of Composites

Attention is drawn to the invariance of the stress field in a two-dimensional body loaded at the boundary by fixed forces when the compliance tensor S(x) is shifted uniformly by S1 (λ, - λ), where λ is an arbitrary constant and S1 (k,u)is the compliance tensor of a isotropic material with two-dimensional bulk and shear moduli k and μ. This invariance is explained from two simple observations: first, That in two dimensions the tensor S(1/2, -1/2) acts to locally rotate the stress by 90° and the second that this rotated field is the symmetrized gradient of a vector field and therefore can be treated as a strain. For composite materials the invariance of the stress field implies that the effective compliance tensor S * also gets shifted by S1l( (λ, - λ) when the constituent moduli are each shifted by S(λ, - λ). This imposes constraints on the functional dependence of S* on the material moduli of the components. Applied to an isotropic composite of two isotropic components it implies that when the inverse bulk modulus is shifted by the constant 1/λ and the inverse shear modulus is shifted by — 1/λ, then the inverse effective bulk and shear moduli undergo precisely the same shifts. In particular it explains why the effective Young’s modulus of a two-dimensional media with holes does not depend on the Poisson’s ratio of the matrix material.

Regularization of optimal design problems for bars and plates, part 1

In this paper, we consider a number of optimal design problems for elastic bars and plates. The material characteristics of rigidity of an elastic nonhomogeneous medium are taken as the control variables. A linear functional of the solutions to the equilibrium boundary-value problem is minimized under additional restrictions upon the control variables.Special variations of the control within a narrow strip provide a necessary condition for a strong local minimum (Weierstrass test). This necessary condition can be used for the detailed analysis of the following problems: bar of extremal torsional rigidity; optimal distribution of isotropic material with variable shear modulus within a plate; and optimal orientation of principal axes of elasticity in an orthotropic plate. For all of these cases, the stationary solutions violate the Weierstrass test and therefore are not optimal. This is because, in the course of the approximation of the optimal solution, the curve dividing zones occupied by materials with different rigidities displays rapid oscillations sweeping over a two-dimensional region. Within this region, the material behaves as a composite medium assembled of materials of the initial class.

Effective Characteristics of Composite Materials and the Optimal Design of Structural Elements

This paper is concerned with structural optimization problems related to a design of inhomogeneous continuous media. Many natural formations, such as the trunk of a tree, a leaf, or bone tissue, have sharply defined internal inhomogeneity leading to the anisotropy of their physical properties varying from one point to another. Since these formations are highly expedient from the point of view of structural mechanics, it would be reasonable to expect that the requirement of optimality for a construction that is artificially designed from a given set of materials would by itself bring into existence the composite media having the best microstructure.

Sliding regimes and anisotropy in optimal design of vibrating axisymmetric plates

Abstract This paper deals with optimal design of solid, elastic, axisymmetric plates performing free, transverse vibrations. It is the objective to determine the plate thickness distribution from the condition that the plate volume is minimized for a given value of the fundamental natural frequency, or for a given higher order natural frequency that corresponds to a vibration mode with a prescribed number of nodal diameters. It is found that the Weierstrass necessary condition for optimality is generally not satisfied for a traditional formulation of this problem, and that the optimal design is characterized by a sliding regime of control where the plate thickness exhibits an infinite number of discontinuities, as a system of infinitely thin, circumferential stiffeners are formed on the optimal axisymmetric plate. This inherent anisotropy of the optimal design is taken into account in a regularization of the initial optimization problem by establishing the tensorial character of the plate bending rigidity and using the concentration of thin, circumferential stiffeners as a new design variable (control). It is shown that the new formulation of the problem can be solved numerically, and examples of optimal designs are presented in the paper.

EFFECTIVE PROPERTIES OF SMART ELASTIC LAMINATES AND THE SCREENING PHENOMENON

Non-stationary phenomena arising in structures can be effectively controlled via smart (intelligent) materials. The corresponding mathematical problems contain controls in the coefficients of hyperbolic equations, these controls depending both on position and time. When appropriately activated, such controls may provide 100% screening of some extended parts of the structure from the invasion of dynamic disturbances. For bars exposed to longitudinal vibrations, the material pattern producing such a screening effect can be the combination of rank-one laminar composites in space-time. Such a composite is described as an array of alternating segments with different pairs (p,, k,) and (pz, kJ of density and stiffness of smart material, this whole array traveling along the bar with a suitable constant speed V. 0 1997 Elsevier Science Ltd. All rights reserved. The appearance of smart materials has made it realistic to claim for the structures properties that may vary both in space and time, thus making these structures highly responsive to the non-stationary environment. We shall place emphasis on the design of material patterns able to block the propagation of undesirable disturbances caused by impulsive loads, impacts, etc., into some extended parts of the structure vulnerable to a material damage. Such screening may be achieved through the deployment of special assemblages of smart materials, e.g., smart laminar composites, activated at appropriate location at appropriate time. Mathematically, the problem reduces to that of the control of coefficients in hyperbolic equations both in space and time. This control is assumed to be materialized through the sensors and actuators distributed throughout a smart structure.

An introduction to the mathematical theory of dynamic materials

A General Concept of Dynamic Materials.- An Activated Elastic Bar: Effective Properties.- Dynamic Materials in Electrodynamics of Moving Dielectrics.- G-closures of a Set of Isotropic Dielectrics with Respect to One-Dimensional Wave Propagation.- Rectangular Microstructures in Space-Time.- Some Applications of Dynamic Materials in Electrical Engineering and Optimal Design.

On the existence of solutions to some problems of optimal design for bars and plates

AbstractThe problem of the optimal control of the material characteristics of continuous media necessitates an extension of the initial class of materials to the set of composites assembled from elements belonging to the initial class. Such an extension guarantees the existence of an optimal control and is equivalent to the construction of theG-closureGU of the initial setU. In this paper, we consider some problems of constructingG-closures for the operators ▽·2D·▽ and ▽·▽·4D··▽▽, where2D and4D denote self-adjoint tensors of 2nd and 4th rank, respectively, their components belonging to bounded sets ofL∞. These operators arise in the theory of the torsion of bars and in the theory of bending of thin plates. A procedure is suggested that provides estimates of some sets Σ containingGU. These estimates are expressed through weak limits of certain functions of the elements of theU-set. The estimates are based on the weak convergence of the elastic energy and, for operators of 4th order, also on the weak convergence of the second invariant of deformation,

Method of control and optimization of microwave heating in waveguide systems

I_2 (e) = w_{xx} w_{yy} - w_{xy}^2 - I_2 (e^0 ) = w_{xx}^0 w_{yy}^0 - (w_{xy}^0 )^2 .