Deformable Thin Films : from Macroscale to Microscale and from Nanoscale to Microscale

1. 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 james@umn.edu References for the lectures found on pages 5, 23, 27, 29 of these slides

2. Plan of lectures • Deformable thin films, macroscale to microscale • Physical background • Membrane theory: “tents”, “tunnels”, etc. • Bending • Quantitative rigidity • Bending theory • Nanoscale to microscale for films • Dynamic films • Piecewise rigid body mechanics

3. Two unifying threads in these lectures • Think, first, in terms of energy wells • Thinness and the geometry of SO(3)

4. Lecture 1 • Deformable thin films, macroscale to microscale • Physical background • Membrane theory: “tents”, “tunnels”, etc. • Bending • Quantitative rigidity • Bending theory • Nanoscale to microscale for films • Dynamic films • Piecewise rigid body mechanics

5. References: Lecture 1 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.

6. Lecture 2 • Deformable thin films, macroscale to microscale • Physical background • Membrane theory: “tents”, “tunnels”, etc. • Bending (Joint work with G. Friesecke, S. Muller) • Quantitative rigidity • Bending theory • Nanoscale to microscale for films • Dynamic films • Piecewise rigid body mechanics

7. References: Lecture 2 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 james@umn.edu) 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.

8. The multiscale problem for atomic  continuum 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!

9. Lecture 3 • Deformable thin films, macroscale to microscale • Physical background • Membrane theory: “tents”, “tunnels”, etc. • Bending • Quantitative rigidity • Bending theory • Nanoscale to microscale for films (Joint work with G. Friesecke) • Dynamic films • Piecewise rigid body mechanics

10. Reference: Lecture 3 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

11. Lecture 4 • Deformable thin films, macroscale to microscale • Physical background • Membrane theory: “tents”, “tunnels”, etc. • Bending • Quantitative rigidity • Bending theory • Nanoscale to microscale for films • Dynamic films (Joint work with R. Rizzoni) • Piecewise rigid body mechanics

12. References: Lecture 4 R. D. James and Raffaella Rizzoni, Piecewise rigid body mechanics (Preprint available from james@umn.edu) 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.)

13. References: Lecture 4, cont’. 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