Preface* Chapter 1. Background* Chapter 2. The Equations of Motion for Extensible Strings* Chapter 3. Elementary Problems for Elastic Strings* Chapter 4. Planar Steady-State Problems for Elastic Rods* Chapter 5. Introduction to Bifurcation Theory and it's Applications to Elasticity* Chapter 6. Global Bifurcation Problems for Strings and Rods* Chapter 7. Variational Methods* Chapter 8. Theory of Rods Deforming in Space* Chapter 9. Spatial Problems for Rods* Chapter 10. Axisymmetric Equilibria of Shells* Chapter 11. Tensors* Chapter 12. 3-Dimensional Continuum* Chapter 13. 3-Dimensional Theory of Nonlinear Elasticity* Chapter 14. Problems in Nonlinear Elasticity* Chapter 15. Large-Strain Plasticity* Chapter 16. General Theories of Rods* Chapter 17. General Theories of Shells* Chapter 18. Dynamical Problems* Chapter 19. Appendix: Topics in Linear Analysis* Chapter 20. Appendix: Local Nonlinear Analysis* Chapter 21. Appendix: Degree Theory and it's Applications* References* Index

Nonlinear problems of elasticity

One of the research direction of Horst Lippmann during his whole scientific career was devoted to the possibilities to explain complex material behavior by generalized continua models. A representative of such models is the Cosserat continuum. The basic idea of this model is the independence of translations and rotations (and by analogy, the independence of forces and moments). With the help of this model some additional effects in solid and fluid mechanics can be explained in a more satisfying manner. They are established in experiments, but not presented by the classical equations. In this paper the Cosserat-type theories of plates and shells are debated as a special application of the Cosserat theory.

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On generalized Cosserat-type theories of plates and shells: a short review and bibliography

The present literature review focuses on geometrically non-linear free and forced vibrations of shells made of traditional and advanced materials. Flat and imperfect plates and membranes are excluded. Closed shells and curved panels made of isotropic, laminated composite, piezoelectric, functionally graded and hyperelastic materials are reviewed and great attention is given to non-linear vibrations of shells subjected to normal and in-plane excitations. Theoretical, numerical and experimental studies dealing with particular dynamical problems involving parametric vibrations, stability, dynamic buckling, non-stationary vibrations and chaotic vibrations are also addressed. Moreover, several original aspects of non-linear vibrations of shells and panels, including (i) fluid–structure interactions, (ii) geometric imperfections, (iii) effect of geometry and boundary conditions, (iv) thermal loads, (v) electrical loads and (vi) reduced-order models and their accuracy including perturbation techniques, proper orthogonal decomposition, non-linear normal modes and meshless methods are reviewed in depth.

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Non-linear vibrations of shells: A literature review from 2003 to 2013

https://hal.archives-ouvertes.fr/hal-00835661/document

On natural strain measures of the non-linear micropolar continuum

Dynamic problems for metamaterials: Review of existing models and ideas for further research

The stored energy functional of a homogeneous isotropic elastic body is invariant with respect to translation and rotation of a reference configuration. One can use Noether's Theorem to derive the conservation laws corresponding to these invariant transformations. These conservation laws provide an alternative way of formulating the system of equations governing equilibrium of a homogeneous isotropic body. The resulting system is mathematically identical to the system of equilibrium equations and constitutive relations, generally, of another material. This implies that each solution of the system of equilibrium equations gives rise to another solution, which describes the reciprocal deformation and solves the system of equilibrium equations of another material. In this paper we derive conservation laws and prove the theorem on conjugate solutions for two models of elastic homogeneous isotropic bodies – the model of a simple material and the model of a material with couple stress (Cosserat continuum).

Conservation laws and conjugate solutions in the elasticity of simple materials and materials with couple stress

Constitutive inequalities in general static and dynamic theory of elastic shells undergoing finite deformation are discussed. Constitutive inequalities are well known in continuum mechanics. They express physical or mathematical restrictions for constitutive equations of 3D elastic materials. In this paper we discuss the analogs of the strong ellipticity, Hadamard and Coleman-Noll (GCN-condition) inequalities for nonlinear elastic shells. It is shown that the GCN-condition implies the strong elipticity for shell theory whereas the strong ellipticity is equivalent to the existence conditions of acceleration waves in shell.

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On constitutive inequalities in nonlinear theory of elastic shells

The theory of line defects (dislocations and disclinations) in elastic bodies has been revisited. A consistent application of the formal limiting passage from isolated defects to the continuous distribution of these allows one to obtain a complete system of equations describing internal stresses in a body with distributed defects. Special emphasis is placed on disclinations in planar systems like graphene which very often demonstrate nonlinear elastic response. As a specific example an exact solution has been found for the problem of the eigenstresses in a disc of nonlinear elastic material due to a given density of continuously distributed disclinations.

Disclinations in nonlinear elasticity

Within the framework of linear elasticity of anisotropic solids, the strong ellipticity (SE) condition is discussing. In this paper, the SE-condition is deriving for 19 classes of 32 known ones of anisotropy. For each class SE-condition reduces to a finite set of elementary inequalities. As a special case, incompressible materials are also considered.

Leonid M. Zubov

Papers

Conservation laws and conjugate solutions in the elasticity of simple materials and materials with couple stress

On constitutive inequalities in nonlinear theory of elastic shells

Disclinations in nonlinear elasticity

On necessary and sufficient conditions of strong ellipticity of equilibrium equations for certain classes of anisotropic linearly elastic materials

The torsion of a composite, nonlinear-elastic cylinder with an inclusion having initial large strains