# Science at the triple point between mathematics, mechanics and materials science

## Publication 53

### Study of the Cofactor Conditions: Conditions of Supercompatibility between Phases

##### Authors:

Xian Chen
University of Minnesota
110 Union St. SE, Minneapolis, MN 55455, USA

Vijay Srivastava
University of Minnesota
110 Union St. SE, Minneapolis, MN 55455, USA
E Global Research Center, 1 Research Circle, MB 274A Niskayuna, NY 12309, USA

University of Minnesota
110 Union St. SE, Minneapolis, MN 55455, USA

Richard D. James
University of Minnesota
110 Union St. SE, Minneapolis, MN 55455, USA

##### Abstract:
The cofactor conditions, introduced in James and Zhang(2005), are conditions of compatibility between phases in martensitic materials. They consist of three subconditions: (i) the condition that the middle principal stretch of the transformation stretch tensor $U$ is unity $(\lambda_2=1)$, (ii) the condition $a\cdot Ucof(U^2-I)n=0$, where the vectors a and n are certain vectors arising in the specification of the twin system, and (iii) the inequality View the MathML source. Together, these conditions are necessary and sufficient for the equations of the crystallographic theory of martensite to be satisfied for the given twin system but for any volume fraction $f$ of the twins, $0\leq f\leq1$. This contrasts sharply with the generic solutions of the crystallographic theory which have at most two such volume fractions for a given twin system of the form $f^*$ and $1-f^*$. In this paper we simplify the form of the cofactor conditions, we give their specific forms for various symmetries and twin types, we clarify the extent to which the satisfaction of the cofactor conditions for one twin system implies its satisfaction for other twin systems. In particular, we prove that the satisfaction of the cofactor conditions for either Type I or Type II twins implies that there are solutions of the crystallographic theory using these twins that have no elastic transition layer. We show that the latter further implies macroscopically curved, transition-layer-free austenite/martensite interfaces for Type I twins, and planar transition-layer-free interfaces for Type II twins which nevertheless permit significant flexibility (many deformations) of the martensite. We identify some real material systems nearly satisfying the cofactor conditions. Overall, the cofactor conditions are shown to dramatically increase the number of deformations possible in austenite/martensite mixtures without the presence of elastic energy needed for coexistence. In the context of earlier work that links the special case $\lambda_2=1$ to reversibility ( Cui et al., 2006, Zhang et al., 2009 and Zarnetta et al., 2010), it is expected that satisfaction of the cofactor conditions for Type I or Type II twins will lead to further lowered hysteresis and improved resistance to transformational fatigue in alloys whose composition has been tuned to satisfy these conditions.
##### Get the paper in its entirety
StudyCofactorJames.pdf

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