TY - JOUR

T1 - Energy minimizing twinning with variable volume fraction, for two nonlinear elastic phases with a single rank-one connection

AU - Conti, Sergio

AU - Kohn, Robert V.

AU - Misiats, Oleksandr

N1 - Publisher Copyright:
© 2022 World Scientific Publishing Company.

PY - 2022

Y1 - 2022

N2 - In materials that undergo martensitic phase transformation, distinct elastic phases often form layered microstructures-a phenomenon known as twinning. In some settings the volume fractions of the phases vary macroscopically; this has been seen, in particular, in experiments involving the bending of a bar. We study a two-dimensional (2D) model problem of this type, involving two geometrically nonlinear phases with a single rank-one connection. We adopt a variational perspective, focusing on the minimization of elastic plus surface energy. To get started, we show that twinning with variable volume fraction must occur when bending is imposed by a Dirichlet-type boundary condition. We then turn to paper's main goal, which is to determine how the minimum energy scales with respect to the surface energy density and the transformation strain. Our analysis combines ansatz-based upper bounds with ansatz-free lower bounds. For the upper bounds we consider two very different candidates for the microstructure: one that involves self-similar refinement of its length scale near the boundary, and another based on piecewise-linear approximation with a single length scale. Our lower bounds adapt methods previously introduced by Chan and Conti to address a problem involving twinning with constant volume fraction. The energy minimization problem considered in this paper is not intended to model twinning with variable volume fraction involving two martensite variants; rather, it provides a convenient starting point for the development of a mathematical toolkit for the study of twinning with variable volume fraction.

AB - In materials that undergo martensitic phase transformation, distinct elastic phases often form layered microstructures-a phenomenon known as twinning. In some settings the volume fractions of the phases vary macroscopically; this has been seen, in particular, in experiments involving the bending of a bar. We study a two-dimensional (2D) model problem of this type, involving two geometrically nonlinear phases with a single rank-one connection. We adopt a variational perspective, focusing on the minimization of elastic plus surface energy. To get started, we show that twinning with variable volume fraction must occur when bending is imposed by a Dirichlet-type boundary condition. We then turn to paper's main goal, which is to determine how the minimum energy scales with respect to the surface energy density and the transformation strain. Our analysis combines ansatz-based upper bounds with ansatz-free lower bounds. For the upper bounds we consider two very different candidates for the microstructure: one that involves self-similar refinement of its length scale near the boundary, and another based on piecewise-linear approximation with a single length scale. Our lower bounds adapt methods previously introduced by Chan and Conti to address a problem involving twinning with constant volume fraction. The energy minimization problem considered in this paper is not intended to model twinning with variable volume fraction involving two martensite variants; rather, it provides a convenient starting point for the development of a mathematical toolkit for the study of twinning with variable volume fraction.

KW - Nonlinear elasticity

KW - Solid-solid phase transitions

KW - Twinning

UR - http://www.scopus.com/inward/record.url?scp=85134372842&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85134372842&partnerID=8YFLogxK

U2 - 10.1142/S0218202522500397

DO - 10.1142/S0218202522500397

M3 - Article

AN - SCOPUS:85134372842

JO - Mathematical Models and Methods in Applied Sciences

JF - Mathematical Models and Methods in Applied Sciences

SN - 0218-2025

ER -