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Two-Phase Solutions for One-Dimensional Non-convex Elastodynamics

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Abstract

We explore the local existence and properties of classical weak solutions to the initial-boundary value problem for a class of quasilinear equations of elastodynamics in one space dimension with a non-convex stored-energy function, a model of phase transitions in elastic bars proposed by Ericksen (J Elast 5(3–4):191–201,1975). The instantaneous phase separation and formation of microstructures of such solutions are observed for all smooth initial data with initial strain having its range that overlaps with the phase transition zone of the Piola–Kirchhoff stress. Moreover, we can select those solutions in a way that their phase gauges are close to a certain number inherited from a modified hyperbolic problem and thus give rise to an internal strain–stress hysteresis loop. As a byproduct, we prove the existence of a measure-valued solution to the problem that is generated by a sequence of weak solutions but not a weak solution itself. It is also shown that the problem admits a local weak solution for all smooth initial data and local weak solutions that are smooth for a short period of time and exhibit microstructures thereafter for certain smooth initial data.

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Acknowledgements

The authors would like to thank the anonymous referees for many helpful suggestions and corrections that greatly improved the presentation of the paper. S. Kim was supported by the National Research Foundation of Korea under the Grant Number NRF-2018R1C1B6001308. Y. Koh was supported by the National Research Foundation of Korea under the Grant Number NRF-2016R1D1A1B03932049.

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Correspondence to Seonghak Kim.

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Communicated by C. De Lellis

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Kim, S., Koh, Y. Two-Phase Solutions for One-Dimensional Non-convex Elastodynamics. Arch Rational Mech Anal 232, 489–529 (2019). https://doi.org/10.1007/s00205-018-1326-1

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