Analytical and numerical studies on certain instabilities in slender structures based on gradient elasticity
利用彈性梯度理論研究細長結構一定的不穩定性
Student thesis: Doctoral Thesis
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Award date  15 Jul 2014 
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Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(46377dcf38e7441aaf224ac95fc733a1).html 

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Abstract
One main purpose of this thesis is to derive the normal form of the equilibrium
equation, where the component of the strain energy function depends on the second
gradient quadratic in the secondorder gradient showed by the small parameter
B[i, j, k]. The equation is given by a fourth order semilinear system of ODE.
Another main purpose of this thesis is to present some numerical solution
to capture instabilities of the deformations observed in experiments, such as the
description of the localization of the solution and the postbifurcation solutions.
To get the energy localization, we must refer to boundaryvalue problems. Here,
the classical part of the strain energy function Φ has a general form, so the results
can be used in any special case.
In this thesis, we first study a threedimensional axisymmetric boundaryvalue
problems of a slender cylinder composed of a nonlinear elastic material subjected
to axial forces.
In this model, whose energydensity depends not only on the gradient of the
deformation, but also on its secondorder gradient. So much phenomena of phase
transitions and localization in a variety of materials, including shape memory alloys
have been studied by using such models. Due to the importance of localization
phenomena in structural safety assessment, much research has been conducted to
resolve experimental, theoretical and computational issues associated with localization
problems. However, as far as we know, for a threedimensional setting
there is not any analytical solution for localization available in literature.
We formulate the field equations by treating the slender cylinder as a threedimensional
object. Through novel series and asymptotic expansion, we derive nonlinear ordinary differential equation which governs the axial strain, of course,
this equilibrium equation includes the small parameter B[i, j, k]. By using the
EulerLagrange equation, we give an alterative derivation. Then, we discuss one
boundaryvalue problem, but this equation is a fourth order ordinary differential
equation, we can’t give the analytical solution only numerical solution for this
boundary value problem of the asymptotic model which takes into account the
influences of the radial deformation as well as the tractionfree boundary conditions
up to the third order.
We also study the uniaxial compression of a 2D rectangle, setup the mathematical
formulation of the problem from the field equations, tractionfree boundary
conditions and incompressible condition. By using the similar method which used
in Chapter 2, we derive a leadingorder axial strain and shear strain. With the sliding
boundary conditions, not only we will obtain the numerical postbifurcation
solutions, but also we will obtain the approximate analytical postbifurcation solutions.
A bifurcation analysis is carried out in order to find at which point(s) the
material failure occurs first.
 Elasticity, Mathematical models, Structural stability