SO(3). Deformable Thin Films : from Macroscale to Microscale and from Nanoscale to Microscale. CNA Summer School 2001. Richard D. James University of Minnesota USA [email protected] References for the lectures found on pages 5, 23, 27, 29 of these slides. Plan of lectures.
K. Bhattacharya and R. D. James, A theory of thin films of martensitic materials with applications to microactuators, J. Mech. Phys. Solids 47 (1999), 531-576
K. Bhattacharya, A. DeSimone, K. F. Hane, R. D. James, C. J. Palmstrom, Tents and tunnels on martensitic films, Materials Science and Engineering A273-275 (1999), 685-689
Y. C. Shu, Heterogeneous thin films of martensitic materials, Arch. Rational Mech. Anal. 153 (2000), 39-90
For some interesting recent numerical analysis/computation of “tents” see: Pavel Belik and Mitchell Luskin, On the numerical modelling of the indentation and shape memory of a martensitic thin film, preprint. (Among other things they show what happens when the conditions for the tent are approximately, but not exactly, satisfied)
For general background on epitaxial growth of films see C. Palmstrom, Epitaxy of dissimilar materials, Ann, Rev. Mater. Sci. 25 (1995), 389.
G. Friesecke, R. D. James and S. Muller, A quantitative Reshetnyak-Liouville theorem and its application to the derivation of nonlinear plate theory from three dimensional elasticity (Preprint and brief announcement available from [email protected])
G. Kirchhoff, Uber das Gleichgewicht und die Bewegung einer elastischen Scheibe, J. reine angew. Math. 40 (1850), 51-88; see also pp. of A. E. H. Love, The Mathematical Theory of Elasticity (4th ed) Dover Press, 515-613
The simple argument concerning the form of Young measures supported on SO(3) is given in Section 4 of, R. D. James and D. Kinderlehrer, Theory of diffusionless phase transitions, Lecture notes in Physics 344 (Springer), 51-84
For a complete treatment of the analysis of the wafer curvature measurement, and lots of information on the behavior of multilayer films (food for thought re: various possible gamma limits), see Chapters 2 and 3 of, L. B. Freund and S. Suresh, Mechanical Behavior of Thin Film Materials,to appear early 2002.
This slide courtesy of Karin Rabe, Physics, Rutgers University
Density Functional Theory (DFT):
high-accuracy calculations for structural energetics of solids
map to an independent-electron problem V(r)
compute wavefunctions in a periodic unit cell
scaling with cell volume worse than N2
example: total energy, forces, stress for NiTi (no symmetry)
N = 2: 25000 secs on Alpha
N = 4: 150000 secs
nanoscale actuators: 1000 nm3 (N=8000) 2 million years
for just one atomic arrangement!
G. Friesecke and R. D. James, A scheme for the passage from atomic to continuum theory for thin films, nanotubes and nanorods, Journal of the Mechanics and Physics of Solids 48 (2000), 1519-1540
R. D. James and Raffaella Rizzoni, Piecewise rigid body mechanics (Preprint available from [email protected])
R. D. James, Part I. Real and configurational forces in magnetism; Part II. Analysis of a microscale cantilever, preprint available. A version of these notes, together with simulations of the cantilever in a magnetic field, to be submitted by R. D. James and H. Tang to a volume of Continuum Mechanics and Thermodynamics in honor of I. Muller, with the anticipated publication date of February, 2002.
(See the thesis of Anja Schlömerkemper MPI Leipzig (adv. S.Müller), for a rigorous derivation of the formula for the force on a subregion of a ferromagnetic body from an atomic model (a lattice of dipoles); her formula disagrees with the accepted one. This formula plays an important role in the magnetic version of the theory, PRMM.)
General background on theory for ferromagnetic shape memory materials can be found in, R. D. James and M. Wuttig, Magnetostriction of martensite, Phil. Mag. A77 (1998), 1273. The structure of this theory and the prescription for the energy wells follows largely from, R. D. James and D. Kinderlehrer, A theory of magnetostriction with applications to TbxDy1-xFe2, Phil. Mag. 68 (1993), 237-274