Richard D. James

Fine Phase Mixtures as Minimizers of Energy

Solid-solid phase transformations often lead to certain characteristic microstructural features involving fine mixtures of the phases. In martensitic transformations one such feature is a plane interface which separates one homogeneous phase, austenite, from a very fine mixture of twins of the other phase, martensite. In quartz crystals held in a temperature gradient near the α-β transformation temperature, the α-phase breaks up into triangular domains called Dauphine twins which become finer and finer in the direction of increasing temperature. In this paper we explore a theoretical approach to these fine phase mixtures based on the minimization of free energy.

Magnetostriction of martensite

Abstract A general strategy is described for inducing magnetostriction in ferromagnetic martensitic materials. An analysis of domain redistribution caused by a magnetic field is given, and certain relations between material constants that promote this effect are described. These relations suggest a ‘constrained theory of magnetostriction’ which is used to predict strain against field in a tetragonal martensite subject to an orthogonal biaxial magnetic field and uniaxial stress. These predictions are compared with the corresponding experiments in Fe70Pd30. Reversible field-induced strains of 0.6% are exhibited in this system. Microstructural observations confirm that these strains are caused by a field-induced redistribution of martensitic twins.

Proposed Experimental Tests of a Theory of Fine Microstructure and the Two-Well Problem

Predictions are made based on an analysis of a new nonlinear theory of martensitic transformations introduced by the authors. The crystal is modelled as a nonlinear elastic material, with a free-energy function that is invariant with respect to both rigid-body rotations and the appropriate crystallographic symmetries. The predictions concern primarily the two-well problem, that of determining all possible energy-minimizing deformations that can be obtained with two coherent and macroscopically unstressed variants of martensite. The set of possible macroscopic deformations obtained is completely determined by the lattice parameters of the material. For certain boundary conditions the total free energy does not attain a minimum , and the finer and finer oscillations of minimizing sequences are interpreted as corresponding to microstructure. The predictions are am enable to experimental tests. The proposed tests involve the comparison of the theoretical predictions with the mechanical response of properly oriented plates subject to simple shear.

Magnetic and magnetomechanical properties of Ni2MnGa

Abstract Magnetocrystalline anisotropy and magnetostrictive properties are reported for the austenitic and martensitic phases of ferromagnetic shape memory Heusler alloy Ni 2 MnGa. In the low-temperature martensitic phase, the phenomenon of field-induced variant rearrangement provides a mechanism which can produce large strains, while at the same time causing anomalous effects in the apparent anisotropy. These anomalies and their effects on measured M – H and torque curves are clarified. Magnetomechanical experiments were performed in the martensitic phase to characterize the work output of a suitably oriented specimen at various stresses, and with proper stress biasing of the initial microstructure produced the largest magnetostrictive strains to date of 4.3%.

Martensitic transformations and shape-memory materials ☆

The authors review theoretical research on martensitic phase transformations in shape-memory materials, with emphasis on recently derived theory and predictions of interest for alloy development. Research on special lattice parameters corresponding to certain microstructures, complex crystal structures and 6M martensite, the relation of micro-scale to macro-scale deformations, ferromagnetic and ferroelectric martensites, and martensite at small scales is covered.

A theory of thin films of martensitic materials with applications to microactuators

A direct derivation is given of a theory for single crystal thin films, starting from three dimensional nonlinear elasticity theory augmented by a term for interfacial energy. The derivation involves no a priori choice of asymptotic expansion or ansatz. It yields a frame-indifferent Cosserat membrane theory with one Cosserat vector field. The theory is applied to multi-well energy functions appropriate to martensitic materials. It is found that, unlike in bulk materials, which generally only support finely twinned austenite/martensite interfaces as energy minimizing states, the thin film theory predicts the existence of exact, untwined austenite/martensite interfaces. These are used to construct some simple energy minimizing deformations—”tents” and “tunnels”—that could possibly be the basis of simple large-deformation microactuators. Explicit results are given for martensitic materials in the systems NiMnGa, NiTi,NiTiCu, and NiAl. A certain alloy of precise composition Ni_(30.5) Ti_(49.5) Cu_(20.0) is predicted to support a four-sided “tent” on an (001) film, which furthermore is predicted to collapse to the substrate upon heating. A formal derivation is given of higher order theories, which yields two additional Cosserat vectors and an explicit form of the bending energy. The derivation indicates an approach to plate-shell-thin film theories that is rather different from the ones usually followed.

Ferromagnetic shape memory in the NiMnGa system

Strain versus field measurements for a ferromagnetic shape memory alloy in the NiMnGa system demonstrate the largest magnetostrictive strains to date of nearly 1.3%. These strains are achieved in the martensitic state through field-induced variant rearrangement. An experimental apparatus is described that provides biaxial magnetic fields and uniaxial compressive prestress with temperature control while recording microstructural changes with optical microscopy. The magnetostrictive response is found to be sensitive to the initial state induced by stress-biasing the martensitic variant structure, and exhibits rate effects related to twin boundary mobility. Experiments performed with constant stress demonstrate work output capacity. Experimental results are interpreted by using a theory based on minimization of a micromagnetic energy functional that includes applied field, stress, and demagnetization energies. It is found that the theory provides a good qualitative description of material behavior, but significantly overpredicts the amount of strain produced. Issues concerning the martensitic magnetic anisotropy and variant nucleation are discussed with regard to this discrepancy.

A constrained theory of magnetoelasticity

Abstract A simple variational theory for the macroscopic behavior of materials with high anisotropy is derived rigorously from micromagnetics. The derivation leads to a constrained theory in which the state of strain and magnetization lies very near the ‘energy wells’ on most of the body. When specialized to ellipsoidal specimens and constant applied field and stress, the theory becomes a finite dimensional quadratic programming problem. Streamlined methods for solving this problem are given. The theory is illustrated by a prediction of the magnetoelastic behavior of the giant magnetostrictive material Tb0.3Dy0.7Fe2. The theory embodies precisely the assumptions that have been postulated for ideal ferromagnetic shape memory, in which the magnetization stays rigidly attached to the easy axes of a martensitic material in the martensitic phase. More generally, the framework can be viewed as a prototype for the derivation of constrained theories for materials that change phase, and whose free-energy density grows steeply away from its minima.

The propagation of phase boundaries in elastic bars

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 1. Dynamic Elastic Bar Theory . . . . . . . . . . . . . . . . . . . . . . . 126 2. Existence of Solutions with Phase Boundaries . . . . . . . . . . . . . . . 129 3. Extensions of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . 133 A. Data prescribed on both sides of the phase boundary . . . . . . . . . . . 133 B. Global solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 C. Invariance groups . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4. Interaction of Sound Waves with a Phase Boundary . . . . . . . . . . . . . 136 5. The Riemann Problem and Admissibility . . . . . . . . . . . . . . . . . 143 A. The Riemann problem. Single phase boundary . . . . . . . . . . . . . 143 B. The Riemann problem. Double phase boundary . . . . . . . . . . . . . 147 C. Admissibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 1. Consistency with static stability theory . . . . . . . . . . . . . . . . 150 2. Consistency with viscoelastic bar theory . . . . . . . . . . . . . . . 153 3. Consistency with the maximal rate of decay of entropy . . . . . . . . . 156 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Frustration in ferromagnetic materials

We examine the theory of micromagnetics developed by W. F. Brown. We show that in the case often considered, with exchange energy omitted, the minimum of the free energy is not attained for uniaxial materials but is attained for cubic materials. A study of the minimizing sequences reveals that these accurately model many features of observed domain structure. Finally, we reexamine the so-called “coercivity paradox” from the viewpoint of nonlinear stability theory.