An analytical study on the evolution of localization and the onset of failure in a slender elastic cylinder


Student thesis: Master's Thesis

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  • Yanhong HAO

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Awarding Institution
Award date2 Oct 2007


In this thesis, we study several three-dimensional axisymmetric boundary-value problems of a slender cylinder composed of a nonlinear elastic material subjected to axial forces. 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, in a three-dimensional setting there is not any analytical solution for localization available in literature. One main purpose of this thesis is to present some analytical solutions to capture some interesting phenomena of the deformations observed in experiments. Another main purpose of this thesis is to give the methods judging the onset of failure of a slender elastic cylinder subjected to tension/ extention through the analytical description of the localization of the strain energy. To get the energy localization, we must refer to boundary-value problems. Here, the material is incompressible and the strain energy function Φ has a general form, so the results can be used in any special case. We formulate the field equations by treating the slender cylinder as a three-dimensional object. Through novel series and asymptotic expansion, we derive nonlinear ordinary differential equation which governs the axial strain. By using the Euler-Lagrange equation, we give an alterative derivation. Then, in chapter 3 and 5 we discuss two boundary-value problems, and give two or one nontrivial types of solutions respectively for a 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 get the analytical forms of the total elongation, the potential energy, the strain energy per unit volume, the radial displacement, the axial displacement, especially the energy localization and concentration. We examine the pre-peak portion of the stress-strain curves which show no significant effect due to the radius; however, the post-peak curves are highly dependent on the specimen radius. Different material coefficients have different localization sizes, but for the same material the localization width is uniform with different external forces. In the curves of the total elongation, the thinner the specimen is, the steeper the post-peak curves is, which is in agreement with the experimental results. Then, according to the theory of fracture mechanics, through the strain energy per unit volume, we get a new analytical criterion for the onset of failure in a slender elastic cylinder subject to tension/extention.

    Research areas

  • Elastic plates and shells, Localization theory, Cylinders, Fracture mechanics