Yury Grabovsky

Microstructures minimizing the energy of a two phase elastic composite in two space dimensions. II: The vigdergauz microstructure

Abstract For modeling coherent phase transformations, and for applications to structural optimization, it is of interest to identify microstructures with minimal energy or maximal stiffness. The existence of a particularly simple microstructure with extremal elastic behavior, in the context of two-phase composites made from isotropic components in two space dimensions, has previously been shown. This “Vigdergauz microstructure” consists of a periodic array of appropriately shaped inclusions. We provide an alternative discussion of this microstructure and its properties. Our treatment includes an explicit formula for the shape of the inclusion, and an analysis of various limits. We also discuss the significance of this microstructure (i) for minimizing the maximum stress in a composite, and (ii) as a large volume fraction analog of Michell trusses in the theory of structural optimization.

The cavity of the optimal shape under the shear stresses

The problem of optimal shape of a single cavity in an infinite 2-D elastic domain is analyzed. An elastic plane is subjected to a uniform load at infinity. The cavity of the fixed area is said to be optimal if it provides the minimal energy change between the homogeneous plane and the plane with the cavity. We show that for the case of shear loading the contour of the optimal cavity is not smooth but is shaped as a curved quadrilateral. The shape is specified in terms of conformal mapping coefficients, and explicit analytical representations for components of the dipole tensor associated with the cavity are employed. We also find the exact values of angles at the corners of the optimal contour. The applications include the problems of optimal design for dilute composites.

Bounds and Extremal Microstructures for Two-Component Composites: A Unified Treatment Based on the Translation Method

The overall energy or stiffness of an elastic composite depends on the microgeometry. Recently, there has been a lot of work on ‘extremal microstructures’ for elastic composites, for example microstructures which minimize the elastic energy at a given macroscopic strain. However, most attention has been focused on composites made of the elastically isotropic component materials. Breaking with this tradition, we consider composites made of two fully anisotropic phases. Our approach, based on the well-known translation method, provides not only the energy bound but also necessary and sufficient conditions for optimality in terms of the local strain field. These optimality conditions enable us to look for optimal microstructures in a more systematic way than before. They also provide clarification of the relations between different problems, for example bounding effective conductivity of a conducting composite versus minimizing strain energy of an elastic composite. Our analysis shows that anisotropy of the constituent materials is very important in determining optimal microgeometries. Some constructions of extremal matrixinclusion composites made from isotropic components cease to be available when the matrix material is anisotropic, even when the degree of anisotropy is small. Most of our analysis is restricted to two space dimensions.

Exact relations for effective tensors of composites: Necessary conditions and sufficient conditions

Typically the elastic and electrical properties of composite materials are strongly microstructure dependent. So it comes as a nice surprise to come across exact formulae for effective moduli that are universally valid no matter what the microstructure. Such exact formulae provide useful benchmarks for testing numerical and actual experimental data and for evaluating the merit of various approximation schemes. They can also be regarded as fundamental invariances existing in a given physical context. Classic examples include Hills formulae for the effective bulk modulus of a two-phase mixture when the phases have equal shear moduli, Levins formulae linking the effective thermal expansion coefficient and effective bulk modulus of two-phase mixtures, and Dykhnes result for the effective conductivity of an isotropic two-dimensional polycrystalline material. Here we present a systematic theory of exact relations embracing the known exact relations and establishing new ones. The search for exact relations is reduced to a search for matrix subspaces having a structure of special Jordan algebras. One of many new exact relations is for the effective shear modulus of a class of three-dimensional polycrystalline materials. We present complete lists of exact relations for three-dimensional thermoelectricity and for three-dimensional thermopiezoelectric composites that include all exact relations for elasticity, thermoelasticity, and piezoelectricity as particular cases.

The G-closure of two well-ordered, anisotropic conductors.

We give a complete solution to the G-closure problem for mixtures of two well ordered possibly anisotropic conductors. Both the G-closure with fixed volume fractions and the full G-closure are computed. The conductivity tensors are considered in a fixed frame and no rotations are allowed.

Exact Scaling Exponents in Korn and Korn-Type Inequalities for Cylindrical Shells

Understanding asymptotics of gradient components in relation to the symmetrized gradient is important for the analysis of buckling of slender structures. For circular cylindrical shells we obtain the exact scaling exponent of the Korn constant as a function of shells thickness. Equally sharp results are obtained for individual components of the gradient in cylindrical coordinates. We also derive an analogue of the Kirchhoff ansatz, whose most prominent feature is its singular dependence on the slenderness parameter, in marked contrast with the classical case of plates and rods.

Scaling Instability in Buckling of Axially Compressed Cylindrical Shells

In this paper, we continue the development of mathematically rigorous theory of “near-flip” buckling of slender bodies of arbitrary geometry, based on hyperelasticity. In order to showcase the capabilities of this theory, we apply it to buckling of axially compressed circular cylindrical shells. The theory confirms the classical formula for the buckling load, whereby the perfect structure buckles at the stress that scales as the first power of shell’s thickness. However, in the case of imperfections of load, the theory predicts scaling instability of the buckling stress. Depending on the type of load imperfections, buckling may occur at stresses that scale as thickness to the power 1.5 or 1.25, corresponding to the lower and upper ends, respectively, of the historically accumulated experimental data.

Exact Relations for Effective Conductivity of Fiber-Reinforced Conducting Composites with the Hall Effect via a General Theory

In this paper we apply the general theory of exact relations to derive all microstructureindependent relations for effective conductivity of fiber-reinforced composites with Hall effect. We also derive all possible links between effective conductivities of two composites that have the same microstructure but are built using different materials. Our results hold for any number of constituents with any anisotropy. This paper is a record of the work of 14 undergraduate students in the NSF-sponsored REU (Research Experience for Undergraduates) program led by the author.

Algebra, Geometry, and Computations of Exact Relations for Effective Moduli of Composites

In this paper we will review and extend the results of [21], which covered the case of 3D thermopiezoelectric polycrystals. In that context the settings of conductivity, elasticity, pyroelectricity, piezoelectricity, thermoelectricity, and thermoelasticity can be viewed as particular cases. We will consider a class of composites more general than polycrystals, where the set of allowable materials is not constrained in any way. In addition, the tensors of material properties are not assumed to be symmetric—an assumption we made in [21]. For example, the Hall effect for conduction in a weak magnetic field is described by a nonsymmetric conductivity tensor. We explain the step-by-step process of finding all exact relations for the simple example of the 2D Hall effect. The paper concludes with a discussion of new algebraic and geometric questions posed by the theory of exact relations.

Marginal Material Stability

Marginal stability plays an important role in nonlinear elasticity because the associated minimally stable states usually delineate failure thresholds. In this paper we study the local (material) aspect of marginal stability. The weak notion of marginal stability at a point, associated with the loss of strong ellipticity, is classical. States that are marginally stable in the strong sense are located at the boundary of the quasi-convexity domain and their characterization is the main goal of this paper. We formulate a set of bounds for such states in terms of solvability conditions for an auxiliary nucleation problem formulated in the whole space and present nontrivial examples where the obtained bounds are tight.