scholarly journals Existence and convergence of solutions of the boundary value problem in atomistic and continuum nonlinear elasticity theory

2016 ◽  
Vol 55 (5) ◽  
Author(s):  
Julian Braun ◽  
Bernd Schmidt
Keyword(s):  
Boundary Value Problem ◽  
Elasticity Theory ◽  
Nonlinear Elasticity ◽  
Boundary Value ◽  
Nonlinear Elasticity Theory ◽  
Convergence Of Solutions

Influence of Distributed Dislocations on Stability of Hollow Nonlinearly Elastic Sphere

2020 ◽  
pp. 17-21
Author(s):  
Evgeniya V. Goloveshkina
Keyword(s):  
Boundary Value Problem ◽  
Elasticity Theory ◽  
Elastic Body ◽  
Nonlinear Elasticity ◽  
External Pressure ◽  
Elastic Sphere ◽  
Unperturbed State ◽  
Nonlinear Elasticity Theory ◽  
Edge Dislocations ◽  
Distributed Dislocations

The phenomenon of stability loss of a hollow elastic sphere containing distributed dislocations and loaded with external hydrostatic pressure is studied. The study was carried out in the framework of the nonlinear elasticity theory and the continuum theory of continuously distributed dislocations. An exact statement and solution of the stability problem for a three-dimensional elastic body with distributed dislocations are given. The static problem of nonlinear elasticity theory for a body with distributed dislocations is reduced to a system of equations consisting of equilibrium equations, incompatibility equations with a given dislocation density tensor, and constitutive equations of the material. The unperturbed state is caused by external pressure and a spherically symmet-ric distribution of dislocations. For distributed edge dislocations in the framework of a harmonic (semi-linear) mate-rial model, the unperturbed state is defined as an exact spherically symmetric solution to a nonlinear boundary value problem. This solution is valid for any function that characterizes the density of edge dislocations. The perturbed equilibrium state is described by a boundary value problem linearized in the neighborhood of the equilibrium. The analysis of the axisymmetric buckling of the sphere was performed using the bifurcation method. It consists in determining the equilibrium positions of an elastic body, which differ little from the unperturbed state. By solving the linearized problem, the value of the external pressure at which the sphere first loses stability is found. The effect of dislocations on the buckling of thin and thick spherical shells is analyzed.

scholarly journals Stability of a composite thick plate with an inhomogeneous field of initial stresses

2018 ◽  
Vol 226 ◽  
pp. 01013
Author(s):  
Vadim V. Eremeev ◽  
Natalia Y. Shubchinskaya ◽  
Denis V. Ivashchenko ◽  
Vsevolod A. Minko
Keyword(s):  
Stability Analysis ◽  
Boundary Value Problem ◽  
Linear Stability ◽  
Nonlinear Elasticity ◽  
Elastic Plate ◽  
Linear Stability Analysis ◽  
Thick Plate ◽  
Boundary Value ◽  
Initial Stresses ◽  
Inhomogeneous Field

We discuss the linear stability analysis for a rectangular twolayered elastic plate. Each layer of the plate was obtained due to inflation of an annular wedge of a cylinder. As a result, in the plate there are initial stresses. The main aim of the paper is to analyse the influence of the initial stresses on the buckling. Here we use the 3D nonlinear elasticity technique. The linearized boundary-value problem is derived and its non-trivial solutions were obtained.

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