nwhm/Related experimental work

Elastic waves in materials which undergo martensitic phase transformations may be related. A simple one dimensional model is:

\(\begin{eqnarray*} \rho X_{tt} - \beta X_{xxt} = (X_x^3 -X_x)_x - \epsilon X_{xxxx} \end{eqnarray*}\)

The displacement \(X_x\) has stable equilibrium equilibrium values at \(X_x=\pm1\) and in a simple situation, one can assume that interfaces are uniformly space with period one so that

\(\begin{eqnarray*} X_x=1\quad x\in[0,1) \\ X_x=-1\quad x\in[1,2) \end{eqnarray*}\)

which is then extended periodically. If we consider small perturbations about this,

\(\begin{eqnarray*} X=X^0+\tilde{X} \end{eqnarray*}\)

where \(X^0\) is the equilibrium configuration and \(\tilde{X}\) is the small perturbation, one might expect a non-constant coefficient wave equation

\(\begin{eqnarray*} \rho \tilde{X}_{tt} - \beta \tilde{X}_{xxt} = 3\tilde{X}_{xx} - \epsilon \tilde{X}_{xxxx} \end{eqnarray*}\) For this choice of nonlinear strain energy function, the elastic terms that are linear in \(\tilde{X}_x\) are the same at first order due to symmetry conditions which would not occur if the equilibria had different elastic constants. For example if the strain energy function \(((X_x-1)(X_x+2))^2\) was used instead of the current \((X_x^2-1)^2\), the resulting equation for propagation of small perturbations about equilibrium is

\(\begin{eqnarray*} \rho \tilde{X}_{tt} - \beta \tilde{X}_{xxt} = b(x)\tilde{X}_{xx} - \epsilon \tilde{X}_{xxxx} \end{eqnarray*}\)

where

\(\begin{eqnarray*} b(x)=9/4\quad x\in[0,2/3) \\ b(x)=9/2 \quad x\in[2/3,1) \end{eqnarray*}\)

and is extended periodically. We note that there may be mathematical difficulties in finding \(\tilde{X}_{xxxx}\) at points where \(b(x)\) is discontinuous.

In the three dimensional setting people have tried to measure elastic wave constants by phononic techniques. An example paper in this area is

Potashnik S., Tulenko A., Stenger T.E., and Trivisonno J. `` Phonon focusing and the martensitic phase transformation in NiAl’’ Ultrasonics Symposium, 1999. Proceedings. 1999 IEEE (Volume:1 ) pg. 543 - 547 link

A good overview of phononic imaging techniques is in (this book also contains a chapter on lattice models):

Wolfe, J.P. ``Imaging Phonons: Acoustic Wave Propagation in Solids’’ Cambridge University Press (1998) link

An introduction to martensitic materials can be found in

Bhattacharya, K. ``Microstructure of Martensite: why it forms and how it gives rise to the shape-memory effect’’ Oxford University Press (2003)

The partial differential equation used here has been examined in numerous studies. Some references are:

Truskinovsky, L. ``Kinks versus shocks’’, in: J.E. Dunn, R. Fosdick, M. Slemrod (Eds.), Shock induced transitions and phase structures in general media, in: IMA Vol. Math. Appl., volume 52, pg. 185-229 (1993). Mathscinet review

Andrews, G.; Ball, J. M. ``Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity.’’ J. Differential Equations 44(2), 306–341 (1982) Mathscinet review

Andrews, G. ``On the existence of solutions to the equation utt=uxxt+σ(ux)x’’. J. Differential Equations 35(2), 200–231 (1980) Mathscinet review