Investigation of non-smooth solutions in finite elasticity using the phase-plane method
DOI:
https://doi.org/10.5540/03.2023.010.01.0064Keywords:
Phase-Plane Method, Phase Portrait, Finite Elasticity, AnisotropyAbstract
We use the phase-plane method to investigate a class of problems in finite elasticity for which the spatial derivative of a solution may have a finite jump at an interior point of the domain. In particular, we consider the equilibrium of a nonlinearly elastic annular disk fixed on its inner surface and subjected to a constant uniform pressure on its outer surface. We show that the solution of this problem is non-diferentiable at an interior point when the applied pressure exceeds a certain value. This value serves as an upper bound for the pressure that can be applied without violating the range of validity of the infinitesimal theory. On the other hand, non-smooth deformation fields are of interest in the study of crystalline materials that can exist in more than one crystal structure, such as the shape-memory alloys.
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References
R. Abeyaratne and J. K. Knowles, Evolution of Phase Transitions. Cambridge University Press, May 2006. ISBN: 9780521661478. DOI: 10.1017/CBO9780511547133.
S. S. Antman and P. V. Negrón-Marrero, The remarkable nature of radially symmetric equilibrium states of aeolotropic nonlinearly elastic bodies. Journal of Elasticity 18:2 (1987), 131-164. ISSN: 03743535. DOI: 10.1007/BF00127554
I. M. Daniel and O. Ishai, Engineering Mechanics of Composite Materials. 2nd ed. Engineering mechanics of composite materials v. 13. Oxford University Press, 2006 ISBN: 9780195150971.
S. G. Lekhnitskii, Anisotropic plates. 2nd ed. New York: Gordon & Breach, 1968. ISBN: 2881242006, 9782881242007.
