Nonlinearly Elastic Shells Research Articles

Axisymmetric equilibrium states of non-linear elastic cylindrical shells

The problem of nonuniqueness of static axisymmetric solutions for a constrained cylindrical shell under a compressive thrust is studied. Both in the elastic and hyperelastic case, we prove the existence of buckled states. For a simple choice of the elastic potential, a Hamiltonian formulation is provided.

A justification of the Marguerre-von K�rm�n equations

The method of asymptotic expansions, with the thickness as the small parameter, is applied to the general three-dimensional equations for the equilibrium of nonlinearly elastic shells with specific geometries, subjected to suitable loadings and boundary conditions. Then it is shown that the leading term of the expansion is the solution of a system of equations equivalent to those of Marguerre-von Karman (the case of a clamped shell is also considered). In addition, without making any a priori assumption regarding the variation of the unknowns across the thickness of the shell, it is found that the displacement field is of Kirchhoff-Love type, and that the stresses have polynomial variations with respect to the thickness.

Qualitative properties of large buckled states of spherical shells

In this paper we study the global qualitative behavior of axisymmetric buckled states of homogeneous isotropic nonlinearly elastic shells that can suffer flexure, compression, and shear. Our model is geometrically exact in the sense that a geometric quantity, such as sin θ, is not replaced by an approximation, such as θ or θ — θ 3/6. (The usual justification for such a replacement is that θ is known to be small for the physical situation under study. But it is quite possible that a mathematical model with exact geometry permits only small solutions, while those with approximate geometry have large solutions.) We allow the material properties to be described by a very general class of nonlinear constitutive relations. Consequently our governing equations form a quasilinear sixth-order system of ordinary differential equations.

Dynamic loss of stability of flexible, nonlinearly elastic shallow shells

Dynamic loss of stability of flexible, nonlinearly elastic shallow shells

Asymptotic shapes of inflated spheroidal nonlinearly elastic shells

In this paper we study the asymptotic behaviour of large axisymmetric deformations of closed axisymmetric nonlinearly elastic shells under internal hydrostatic pressure. These shells can suffer flexure, extension, and shear. Since there are spherical shells that can enclose an arbitrarily large volume at a finite pressure (cf. [1]), we take the volume rather than the pressure as the large parameter.

Stress and strain determination in a nonlinearly elastic conical shell subjected to aerodynamic loading and heating

Stress and strain determination in a nonlinearly elastic conical shell subjected to aerodynamic loading and heating

Investigation of the contact interaction between nonlinear elastic shells of finite shear rigidity and punches

Investigation of the contact interaction between nonlinear elastic shells of finite shear rigidity and punches

The dynamical behaviour of nonlinear elastic spherical shells

The dynamical behaviour of nonlinear elastic spherical shells

Ordinary differential equations of non-linear elasticity I: Foundations of the theories of non-linearly elastic rods and shells

Ordinary differential equations of non-linear elasticity I: Foundations of the theories of non-linearly elastic rods and shells

Free vibrations of nonlinearly elastic shells

Free vibrations of nonlinearly elastic shells

Existence and nonuniqueness of axisymmetric equilibrium states of nonlinearly elastic shells

Existence and nonuniqueness of axisymmetric equilibrium states of nonlinearly elastic shells

Addendum: Existence and Nonuniqueness of Axisymmetric Equilibrium States of Nonlinearly Elastic Shells

Addendum: Existence and Nonuniqueness of Axisymmetric Equilibrium States of Nonlinearly Elastic Shells

Lagrangian Formulation of Statics of Shells

A Lagrangian formulation of the equations of equilibrium of nonlinear thin elastic shell theory referred to nonorthogonal midsurface coordinates is presented. In analogy with the definition of the Lagrangian stress tensor of nonlinear continuum mechanics, stress and moment resultants are introduced and the equations of equilibrium with reference to the undeformed state derived. All results are “exact” within the Kirchhoff-Love hypothesis and are stated both in tensorial as well as in physical component form.

On the theory of nonlinearly elastic nonuniform flexible shells

On the theory of nonlinearly elastic nonuniform flexible shells

Physical Strains in Nonlinear Thin Shell Theory

Expressions for the physical tangential and bending strains of nonlinear thin elastic shell theory in nonorthogonal coordinates are derived in a new, relatively compact form. The geometric and physical significance of the results is interpreted, where possible. In addition, the quantities describing the differential geometry of a deformed surface are also given. All results obtained are exact in the sense that no restrictions are imposed on the magnitude of strains in the derivations. The expressions for the nonlinear case are compared with the corresponding formulas of the linear theory. The results obtained together with further results to be published on the strains of the parallel surface, the statics of shells, and constitutive relations find direct application in the analysis of the nonlinear behavior of thin shells.

Axisymmetric static and dynamic buckling of spherical caps due to centrally distributed pressures

Spherical caps axisymmetric static and dynamic buckling under load, using axisymmetric nonlinear elastic shell theory approximation and finite difference equations