Modeling of Branched Needle Microstructures at the Edge of a Martensite Laminate

We present models of the microstructure of martensite that occurs when twinned layers of martensite meet a pure variant. This microstructure often occurs below the A, temperature, with loading that favors the passage from one pure variant of martensite to another. This microstructure gives a mechanism for variant rearrangement. In this microstructure, the layers are observed to pinch down to sharply pointed needles. If the laminate is sufficiently long, branching of the needles is observed. In materials with a reasonably large transformation strain, the needles exhibit a characteristic bent shape; the bending is in the opposite sense as might be expected from the overall shear. Our analytical and numerical results imply that this microstructure forms as a result of energy minimization in the transition layer between the martensite laminate and the pure variant. The bending of the needles plays a key role in lowering the energy of this transition layer. 1. GEOMETRICALLY LINEAR AND NONLINEAR THEORIES OF MARTENSITE A picture of the microstructure of interest here is shown in Figure 1, taken from the biaxial loading experiments Figure 1. Branched needle microstructure. When a twinned laminate of martensite meets a pure variant as in Figure 1, there are certain matching conditions that must hold which are exactly analogous to those of the crystallographic theory of martensite. We prefer to think of them as a consequence of energy minimization (see [3], [4] and [5] for the explanation of how the crystallographic theory follows from energy minimization). In this viewpoint the bulk free energy density of the specimen is a function cp of the deformation gradient, F = V y , (a 3 x 3 matrix) and the temperature 8. Throughout this paper we hold 8 fixed below A, and so we omit it from the notation. The deformation y ( x ) maps a suitable reference configuration into 3-dimensional space. The total bulk free energy is given by the integral ] P ( V Y ( X ) ) dx . (1)