TY - JOUR
T1 - On a three-dimensional axisymmetric boundary-value problem of nonlinear elastic deformation
T2 - Asymptotic solution and exponentially small error
AU - Dai, Hui-Hui
AU - Wang, Fan-Fan
PY - 2007/12
Y1 - 2007/12
N2 - In this paper, we study a three-dimensional axisymmetric boundary-value problem of a slender cylinder composed of a nonlinearly elastic material subjected to an axial force. Starting from the field equations, after a transformation and proper scalings, we identify a small variable and two small parameters, which characterize the present problem. Then, by an approach involving compound series-asymptotic expansions, a nonlinear ODE is derived, which governs the axial strain (the first-term in the series expansion). By imposing the zero radial displacement conditions at two ends, we manage to get the analytical solution of the axial strain, from which all other physical quantities can be deduced and thus the three-dimensional displacement field can be determined. Graphical results are presented, which show that there are two boundary layers near the two ends while the middle part is in a state of almost uniform extension. The asymptotic structure of the analytical solution is derived, which offers clear explanations to the structure of the deformed configuration and shows that the thickness of both boundary layers is of the order of the radius. We also point out the relevance of the present results to the St. Venant's problem. In particular, we obtain the explicit uniformly-valid exponentially small error term, when the obtained deformed configuration is compared to the configuration of a uniform extension. © 2007 Elsevier Ltd. All rights reserved.
AB - In this paper, we study a three-dimensional axisymmetric boundary-value problem of a slender cylinder composed of a nonlinearly elastic material subjected to an axial force. Starting from the field equations, after a transformation and proper scalings, we identify a small variable and two small parameters, which characterize the present problem. Then, by an approach involving compound series-asymptotic expansions, a nonlinear ODE is derived, which governs the axial strain (the first-term in the series expansion). By imposing the zero radial displacement conditions at two ends, we manage to get the analytical solution of the axial strain, from which all other physical quantities can be deduced and thus the three-dimensional displacement field can be determined. Graphical results are presented, which show that there are two boundary layers near the two ends while the middle part is in a state of almost uniform extension. The asymptotic structure of the analytical solution is derived, which offers clear explanations to the structure of the deformed configuration and shows that the thickness of both boundary layers is of the order of the radius. We also point out the relevance of the present results to the St. Venant's problem. In particular, we obtain the explicit uniformly-valid exponentially small error term, when the obtained deformed configuration is compared to the configuration of a uniform extension. © 2007 Elsevier Ltd. All rights reserved.
KW - Analytical solution
KW - Asymptotic analysis
KW - Finite elasticity
KW - Nonlinear boundary-value problem
KW - St. Venant's problem
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U2 - 10.1016/j.ijengsci.2007.08.003
DO - 10.1016/j.ijengsci.2007.08.003
M3 - RGC 21 - Publication in refereed journal
VL - 45
SP - 951
EP - 967
JO - International Journal of Engineering Science
JF - International Journal of Engineering Science
SN - 0020-7225
IS - 12
ER -