Analytical and numerical studies on certain instabilities in slender structures based on gradient elasticity
Student thesis: Doctoral Thesis
Related Research Unit(s)
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 second-order 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 post-bifurcation solutions. To get the energy localization, we must refer to boundary-value 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 three-dimensional axisymmetric boundary-value problems of a slender cylinder composed of a nonlinear elastic material subjected to axial forces. In this model, whose energy-density depends not only on the gradient of the deformation, but also on its second-order 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 three-dimensional 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 Euler-Lagrange equation, we give an alterative derivation. Then, we discuss one boundary-value 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 traction-free boundary conditions up to the third order. We also study the uniaxial compression of a 2D rectangle, set-up the mathematical formulation of the problem from the field equations, traction-free boundary conditions and incompressible condition. By using the similar method which used in Chapter 2, we derive a leading-order axial strain and shear strain. With the sliding boundary conditions, not only we will obtain the numerical post-bifurcation solutions, but also we will obtain the approximate analytical post-bifurcation 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