Access Type

Open Access Dissertation

Date of Award

January 2023

Degree Type

Dissertation

Degree Name

Ph.D.

Department

Mechanical Engineering

First Advisor

Victor L. Berdichevsky

Second Advisor

Xin Wu

Abstract

Tensors that are invariant with respect to subgroups of 3D group of rotations were constructed first by Lokhin and Sedov (J. Appl. Math. Mech. (PMM), 1963, 27, 597-629). In this paper we do one more step and derive formulas for the elastic modulus tensors for all symmetry groups from the Lokhin-Sedov results. The tensor basis obtained in this way has an advantage of being built upon the same tensors that are used in construction of nonlinear tensor functions. One consequence of the formulas derived is that the number of types of anisotropic elastic behaviors is ten. There is a long- standing controversy concerning this number; many authors claim that the correct number is eight. We discuss this controversy in detail, and add several points to the Fedorov-Khatkevich argument that initiated this controversy. We formulate the tensor relationship which is the underlying cause of the Fedorov-Khatkevich argument. This relationship yields several generalizations (extension to cubic symmetries, the inversed argument). We argue that for crystals ten types of elastic response cannot be reduced to eight due to built-in intrinsic structure determining the crystal symmetry. For polycrystals with a priori unknown elastic symmetry, the problem of determining the symmetry by elastic moduli is shownto be ill-posed. This paper is a review of Hashin-Shtrikman type bounds for effective moduli of conductivity and elasticity of polycrystals and composites written from the perspective of the variational principle for probabilistic measure. The results for such bounds are rederived in probabilistic terms. Remarkably, in probabilistic terms the Hashin-Shtrikman approach gets especially simple form. Besides, a clear distinction arises between the basic assumption, the choice of the trial field, and the simplifying assumptions, like geometrical isotropy, physical isotropy, texture isotropy, etc. We filled out several gaps. First, we derive an integral equation to be solved to get the bounds when the simplifying assumptions do not hold. Second, we extend the bounds for polycrystals with the cubic symmetry of crystallites to all thermodynamically possible crystallites; previously such bounds were found for crystallites with special elastic properties . One practical outcome considered is the derivation of approximate formuli for the temperature dependence of effective elastic moduli. Third, for crystallites with noncubic symmetries, we formulated algebraic variational problems to be solved numerically to obtain the bounds, and solved these problems for several materials.

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