American Institute of Mathematical Sciences

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May  2007, 7(3): 497-514. doi: 10.3934/dcdsb.2007.7.497

Analytical solutions for phase transitions in a slender elastic cylinder under non-deforming and other boundary conditions

Zong-Xi Cai 1, and  Hui-Hui Dai 2,

1. 

Department of Mechanics, Tianjin University, Tianjin, 300072, China
2. 

Department of Mathematics and Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, 83 Tat Chee Avenue, Hong Kong, China

Received  September 2006 Revised  January 2007 Published  February 2007

In this paper, we formulate the problem of phase transitions in a slender elastic cylinder induced by tension/extension as a boundary-value problem of a first-order dynamical system. One aim is to give analytical descriptions for some geometrical size effects observed in experiments. Three types of end boundary conditions corresponding to real physical situations are proposed. With the help of a phase-plane analysis analytical solutions for both a force-controlled problem and a displacement-controlled problem are obtained. It turns out that the value of the radius-length ratio has a great influence on the solutions. For a displacement-controlled problem it influences the number of all possible solutions. The engineering stress-strain curves plotted from the analytical solutions seem to capture the key features (e.g., stress peak, stress drop and stress plateau) of the curves measured in a few experiments in literature. Also, the analytical results reveal that smaller the radius is sharper the stress drop is and the width of the transformation front is of the order of the radius, which are in agreement with the experimental observations. We also compare the analytic solutions for the three types of boundary conditions, and a very interesting finding is that the engineering stress-strain curves are almost identical under these different boundary conditions.
Keywords: analytical solutions, nonlinear elasticity, Phase transitions, non-deforming condition, slender cylinder..
Mathematics Subject Classification: 74N10, 35B15, 35B32, 34C23, 41A6.
Citation: Zong-Xi Cai, Hui-Hui Dai. Analytical solutions for phase transitions in a slender elastic cylinder under non-deforming and other boundary conditions. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 497-514. doi: 10.3934/dcdsb.2007.7.497
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