NECESSARY CONDITIONS FOR ENERGY MINIMIZERS IN THE NONLINEAR THEORY OF SIX-PARAMETER ELASTIC SHELLS

A general nonlinear theory of elastic shells

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.

This chapter will discuss the nonlinear theories of webs, membranes and shells. The webs and membranes are extensively used to model bio-tissues and membranes experiencing arbitrary initial configurations. Since any webs cannot resist any compressive forces, it is very difficult to determine the corresponding deformations and stresses in the webs. Traditionally, one used the membrane theory with certain constraints to determine the final configuration of webs. However, the webs can be of non-continuum and continuum. Thus, the network non-continuum web will be presented first from the cable theory, and the continuum web will be developed. Further, the nonlinear theory of membranes will be developed in an analogy way. The nonlinear theory of shells will be developed from the general theory of the 3-dimensional deformable body, and such a theory of shells can be easily reduced to the established theories.

The Nonlinear Theory of Elastic Shells

Elastic shells are pervasive in everyday life. Examples of these thin-walled structures range from automobile hoods to basketballs, veins, arteries and soft drink cans. This book explains shell theory, with numerous examples and applications. As a second edition, it not only brings all the material of the first edition entirely up to date, it also adds two entirely new chapters on general shell theory and general membrane theory. Aerospace, mechanical and civil engineers, as well as applied mathematicians, will find this book a clearly written and thorough information source on shell theory.

We develop the general nonlinear theory of elastic shells with an account of phase transitions in the shell material. Our formulation is based on the dynamically and kinematically exact through-the-thickness reduction of three-dimensional description of the phenomenon to the two-dimensional form written on the shell base surface. In this model shell displacements are expressed by work-averaged translations and rotations of the shell cross-sections. All shell relations are then found from the variational principle of the stationary total potential energy. In particular, we derive the new global dynamic continuity condition at the singular surface curve modelling the phase interface. We also discuss particular forms of the local dynamic continuity conditions at coherent and incoherent interface curves. The results are illustrated by an example of a phase transition in an infinite plate with a circular hole.

By using spline functions, a unified expression to describe various continuous or discontinuous variables in sandwich shells and laminated shells is derived. Then a general nonlinear theory of anisotropic sandwich shells faced with laminated composites is developed using the assumption of a smooth layer-wise curvilinear coordinate θ after deformation. The theory combines the global theory and the discrete-layer theory of laminated shells in view of the structural characteristics of anisotropic sandwich shells faced with laminated composites. A series of refined theories for sandwich and laminated shells can be obtained directly by simplifying the general theory.

A version of a refined non-linear theory of thin elastic sandwich shells of iteration type

Drilling couples and refined constitutive equations in the resultant geometrically non-linear theory of elastic shells

The set of equations for the geometrically non-linear theory of thin elastic shells is usually expressed in terms of displacements as basic independent variables of the shell deformation. Various general and reduced displacemental forms of bending shell equations are summarized, for example, by MUSHTARI and GALIMOV [1], KOITER [2], PIETRASZKIEWICZ [3, 4], SCHMIDT [5] and BA§AR and KRATZIG [6], where further references may be found. When displacement field is determined from the shell equations, strains, rotations and stresses may be obtained by prescribed algebraic or differential procedures.

A comparison of shell theories for large-amplitude vibrations of circular cylindrical shells: Lagrangian approach

A general nonlinear theory of isothermal shells is presented in which the only approximations occur in the conservation of energy and in the consequent constitutive relations, which include expressions for the shell velocity and spin. No thickness expansions or kinematic hypotheses are made. The introduction of a dynamic mixed-energy density avoids ill-conditioning associated with near inextensional bending or negligible rotational momentum. It is shown that a variable scalar rotary inertia coefficient exists that minimizes the difference between the exact kinetic-energy density and that delivered by shell theory. Finally, it is shown how specialization of the dynamic mixed-energy density provides a simple and logical way to introduce a constitutive form of the Kirchhoff hypothesis, thus avoiding certain unnecessary constraints (such as no thickness changes) imposed by the classical kinematic Kirchhoff hypothesis.

On a version of the nonlinear dynamical theory of thin multilayered shells: PMM vol. 40, n≗1, 1976, pp. 180–185

Non-linear theory of shells

We consider the general theory of 6-parameter shells, in which material points on the midsurface are endowed with 3 translational and 3 rotational degrees of freedom. In this framework, we derive quasiconvexity conditions and rank-one convexity conditions. These inequalities represent necessary conditions for energy minimizers; they are the two-dimensional counterparts of the well-known relaxed convexity conditions in three-dimensional finite elasticity. As a specific feature, the quasiconvexity inequality for shells contains the gradients in the tangent plane of the variation fields associated to deformation and microrotation. Finally, we also deduce the Legendre-Hadamard condition for shells, as a consequence of the rank-one convexity inequality.