Theory of elasticity for a semilinear material

Lur'e, A. I.

doi:10.1016/0021-8928(68)90034-8

Cited by 27 publications

(36 citation statements)

References 1 publication

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“…Here we find that ν = 1 2 implies the kinematic constraint on the deformation that i 1 = λ 1 + λ 2 + λ 3 = 3. The latter is associated with the notion of incompressibility in linear elasticity in the form tr E = 0 and in the present context can be viewed as a "semilinear" feature, in keeping with the descriptor [23] for the strain energy function (19). The kinematic condition, Div x = 3 or equivalently…”

Section: The Limit Ofmentioning

confidence: 95%

Hyperelastic cloaking theory: transformation elasticity with pre-stressed solids

Norris

Parnell

2012

Proc. R. Soc. A.

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Transformation elasticity, by analogy with transformation acoustics and optics, converts material domains without altering wave properties, thereby enabling cloaking and related effects. By noting the similarity between transformation elasticity and the theory of incremental motion superimposed on finite pre-strain, it is shown that the constitutive parameters of transformation elasticity correspond to the density and moduli of smallon-large theory. The formal equivalence indicates that transformation elasticity can be achieved by selecting a particular finite (hyperelastic) strain energy function, which for isotropic elasticity is semilinear strain energy. The associated elastic transformation is restricted by the requirement of statically equilibrated pre-stress. This constraint can be cast as trF = constant, where F is the deformation gradient, subject to symmetry constraints, and its consequences are explored both analytically and through numerical examples of cloaking of anti-plane and in-plane wave motion.

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Section: The Limit Ofmentioning

confidence: 95%

Hyperelastic cloaking theory: transformation elasticity with pre-stressed solids

Norris

Parnell

2012

Proc. R. Soc. A.

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“…To avoid confusion in terminology, it should be pointed out that ten years later potential (2.46) was given a different name: a material with potential (2.46) was called a semilinear material in [43,44] (this term was somehow adopted in the Russian scientific literature). This situation was mentioned in the monographs [26,27,69].…”

Section: Determination Of Follower Loadmentioning

confidence: 99%

Stability of elastic bodies under uniform compression (review)

Guz

2012

Int Appl Mech

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The main results on the three-dimensional theory of stability of compressible and incompressible hyperelastic simply connected bodies under uniform compression are analyzed. The problems are classified according to the type of loading (dead or follower loads, acting on the whole or a part of the surface) and the type of boundary conditions (the same conditions on the whole surface or different conditions on different parts of the surface). Approaches based on the three-dimensional linearized theory of stability (theory of finite subcritical deformations, first and second theories of small subcritical deformations, incremental-deformation theory, theory of small average rotations) and an approximate approach (linear equations; the loading parameter is approximately included in the boundary conditions) to solving these problems are discussed. Related problems (rock pressure manifestations, folding in the Earth's crust, wave-like formations on the surface of structural members, stability of laminated composite materials) are briefly commented. Regarding isotropic compressible and incompressible hyperelastic materials, the three-dimensional theory of elastic stability, theory of finite subcritical deformations, and first and second theories of small subcritical deformations are considered. The sufficient conditions for the applicability of the static (Euler's) method, the sufficient conditions for the stability of equilibrium state, and the general solutions to anti-plane, plane, and spatial problems for homogeneous subcritical states are formulated. The exact solutions, obtained with the above-mentioned general approaches, for compressible and incompressible isotropic bodies (strip, rectangular and circular plates, circular cylinder, sphere, and body of arbitrary geometry) under dead or follower loading are presented. These solutions were obtained for isotropic hyperelastic materials with an arbitrary elastic potential under uniform (hydrostatic, biaxial, or triaxial) pressure. The reviewed results were originally reported in the author's monograph (A. N. Guz, Stability of Elastic Bodies under Uniform Compression [in Russian], Naukova Dumka, Kyiv (1979)) and articles listed in the ReferencesKeywords: stability of simply connected isotropic hyperelastic bodies, three-dimensional linearized theory, uniform loading, dead follower loads Introduction.Here we provide an introductory discussion of the problem of interest and related problems, including a brief historical sketch, classification of problems, brief description of approaches, relation to phenomena observed in related fundamental and scientific-and-technological fields, and exemplary analysis of recent publications.1.1. Brief Historical Sketch. The present paper reviews results related to hyperelastic isotropic compressible or incompressible simply connected bodies with an arbitrary elastic potential. In this case (simply connected isotropic elastic bodies), uniform hydrostatic compression (of arbitrarily shaped bodies) and uniform uniaxial, biaxial, or triaxia...

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“…We employ convected coordinates and a mixture of referential and present bases, revealing interesting relationships between components of various tensors. Our Biot-quadratic energy is surprisingly rare in the nonlinear elasticity literature, although its roots go back to early work by Lurie [22] and John [23]. Inspired by the works of Atluri and Murakawa [24], Wiśniewski [25], and Merlini [26], we propose a variational principle in terms of position, Biot strain, and rotation fields, and show that constitutive stress-strain relations and balance equations can be written in terms of positions and stretches alone.…”

Section: Introductionmentioning

confidence: 97%

Quadratic-stretch elasticity

Vitral

Hanna

2021

Mathematics and Mechanics of Solids

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A nonlinear small-strain elastic theory is constructed from a systematic expansion in Biot strains, truncated at quadratic order. The primary motivation is the desire for a clean separation between stretching and bending energies for shells, which appears to arise only from reduction of a bulk energy of this type. An approximation of isotropic invariants, bypassing the solution of a quartic equation or computation of tensor square roots, allows stretches, rotations, stresses, and balance laws to be written in terms of derivatives of position. Two-field formulations are also presented. Extensions to anisotropic theories are briefly discussed.

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Theory of elasticity for a semilinear material

Cited by 27 publications

References 1 publication

Hyperelastic cloaking theory: transformation elasticity with pre-stressed solids

Hyperelastic cloaking theory: transformation elasticity with pre-stressed solids

Stability of elastic bodies under uniform compression (review)

Quadratic-stretch elasticity

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