TY - JOUR

T1 - Two-Phase Solutions for One-Dimensional Non-convex Elastodynamics

AU - Kim, Seonghak

AU - Koh, Youngwoo

N1 - Publisher Copyright:
© 2018, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2019/4/1

Y1 - 2019/4/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85055566781&partnerID=8YFLogxK

U2 - 10.1007/s00205-018-1326-1

DO - 10.1007/s00205-018-1326-1

M3 - Article

AN - SCOPUS:85055566781

SN - 0003-9527

VL - 232

SP - 489

EP - 529

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

IS - 1

ER -