Equilibrium stability of nonlinear elastic sphere with distributed dislocations
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O R I G I NA L A RT I C L E
Evgeniya V. Goloveshkina · Leonid M. Zubov
Equilibrium stability of nonlinear elastic sphere with distributed dislocations
Received: 24 January 2020 / Accepted: 3 March 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract An exact formulation and a solution of the stability problem for a three-dimensional elastic body containing distributed dislocations are given. The buckling phenomenon for a hollow nonlinear elastic sphere of the semi-linear (harmonic) material with edge dislocations is studied. The study is carried out within the framework of the continuum theory of continuously distributed dislocations using the bifurcation method of buckling analysis. The bifurcation method is to find the equilibrium positions of an elastic body, which differ little from the subcritical (unperturbed) state. The perturbed equilibrium state is described by a linearized boundary value problem. By solving a homogeneous linear boundary value problem, the minimum critical value of the external pressure at which the sphere loses stability is found. The influence of dislocations on the buckling of both thin and thick spherical shells is analyzed. Keywords Nonlinear elasticity · Dislocation density · Large deformations · Equilibrium stability · Nanostructured materials · Semi-linear material 1 Introduction The study of stability loss for elastic bodies is a fairly developed field of mechanics of deformable solids. Most studies concern the stability of thin rods, plates, and shells [4,16,20,33,36,39–41,43]. At the same time, it is advisable to study a phenomenon of stability loss for three-dimensional elastic bodies that allow large deformations. The stability problems for elastic bodies in the framework of the three-dimensional nonlinear theory of elasticity were considered, for example, in works [3,13,15,17,30,32,34,45,51,53,54]. The most common method for studying the equilibrium stability of spatial nonlinear elastic bodies is the bifurcation method also called the static Euler method [9,15,29,30,32,45,53]. Within the framework of this method, a stability analysis is reduced to solving a homogeneous boundary value problem linearized near the ground state, that is, to determining the eigenvalues (critical loads) and eigenfunctions (instability forms). A necessary condition for the correctness of the bifurcation method is a conservatism of external loads. Instability under tensile stresses [10,11,18,26,37,55] is an example of problems in which an elastic body in a subcritical state undergoes large deformations. In [56], on the basis of the three-dimensional nonlinear elasticity, the stability of the hollow circular infinite cylinder under three-parameter loading is studied: axial tension, torsion, and inflation. The instability of the stretched hollow cylinder inflated by internal pressure was investigated in [5,18,19]. The effect of torsion on the stability of the continuous circular cylinder under tension was analyzed in [57]. Communicated by Andreas Öchsner. E. V. Goloveshkina (
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