Transformation elasticity and anelasticity

Eigenstrains;Anelasticity;Residual stresses;Cloaking;Elastodynamics;Kirchhoff-Love plates;Transformation elasticity

Golgoon, Ashkan ; Yavari, Arash Civil and Environmental Engineering Garmestani, Hamid Paulino, Glaucio H. Rimoli, Julian J. Steigmann, David J. Suryanarayana, Phanish ; Yavari, Arash

University:Georgia Institute of Technology

Department:Civil and Environmental Engineering

关键词: Eigenstrains;    Anelasticity;    Residual stresses;    Cloaking;    Elastodynamics;    Kirchhoff-Love plates;    Transformation elasticity;   

Others  :  https://smartech.gatech.edu/bitstream/1853/62717/1/GOLGOON-DISSERTATION-2020.pdf

美国|英语

来源: SMARTech Repository

【 摘 要 】

We present a theoretical framework for studying a large class of elastic and anelastic problems in nonlinear solids. We specifically use the transformation properties of nonlinear and linearized elasticity in this theory. Given an anelastic deformation, a non-vanishing strain does not correspond to a non-vanishing stress. That part of strain that is related to the corresponding stress is called elastic strain. The remaining part of strain is called eigenstrain or pre-strain. Eigenstrains (or anelastic sources) such as inclusions, defects, growth, phase transformations, and nonuniform temperature changes can cause residual stresses. The relaxed (natural) configuration of a residually-stressed body is a non-Euclidean manifold that cannot be isometrically embedded in the Euclidean ambient space. Using transformation anelasticity, one can construct the Riemannian material manifold of the body. In particular, the material metric explicitly depends on the distribution of eigenstrains. In this PhD thesis we utilize transformation anelasticity to study the induced elastic fields of a circumferentially-symmetric distribution of finite eigenstrains in nonlinear elastic wedges; the stress field of a nonlinear elastic solid torus with a toroidal inclusion; nonlinear elastic inclusions in anisotropic solids as well as distributed line and point defects in nonlinear anisotropic solids. The goal in transformation elasticity is to transform the nonlinear or linearized boundary-value (or initial-boundary-value) problem of an elastic body to that of another elastic body using a diffeomorphism (or a smooth mapping). The diffeomorphism, in turn, explicitly determines how the different elastic fields (and elastic parameters) of the two bodies are related. In particular, it is noted that the two boundary-value problems are not related by push-forward or pull-back under the diffeomorphism. We apply this theory to formulate the nonlinear and linearized elastodynamic transformation cloaking problem in the context of the classical elasticity, the small-on-large theory of elasticity, i.e., linearized elasticity with respect to an initially stressed configuration, and in solids with microstructure, namely gradient and (generalized) Cosserat solids. In particular, we note that a cloaking transformation is neither a spatial nor a referential change of coordinates (frame). Rather, a cloaking map transforms the boundary-value problem of an isotropic and homogeneous elastic body (virtual problem) to that of an anisotropic and inhomogeneous elastic body with a finite hole covered by a cloak that is to be designed (physical problem). The virtual body has a desired mechanical (wave-guiding) response, whereas the physical body is designed such that the same response is mimicked outside the cloak using a cloaking transformation. Finally, starting from nonlinear shell theory, we utilize transformation elasticity to formulate the transformation cloaking problem for Kirchhoff-Love plates and for elastic plates with both the in-plane and the out-of-plane displacements.