Modeling on the dynamic mechanical response of single-crystalline Ni–Mn–Ga alloys based on Hamilton’s principle

In this paper, a variational approach is proposed to study the dynamic mechanical response of a single-crystalline Ni–Mn–Ga sample. First, some constitutive assumptions are adopted to describe the material properties of single-crystalline Ni–Mn–Ga alloys. Hamilton’s action integral is then formulated for the mechanical system being studied, which depends on the position vector field and the variant state distribution in the sample. By calculating the variation of the action integral with respect to the position vector field, the equation of motion, as well as the boundary condition and the twin interface connection condition, can be obtained. By further calculating the variation of the action integral with respect to the variant state distribution (through twin interface movements), the expression of the driving force on the twin interfaces is derived, based on which the twin interface movement criterion is established. Combining the equation of motion and the twin interface movement criterion, the governing system for modeling the dynamic response of the single-crystalline Ni–Mn–Ga sample can be formulated. To show the validity of the governing system, a simple example is studied and some analytical results are constructed. Especially, the relation between the external mechanical load and the twin interface velocity is revealed, which is consistent with the experimental observations.