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
SO(3)
References for the lectures
found on pages 5, 23, 27, 29
of these slides
Plan of lectures

Deformable thin films, macroscale to microscale

Bending

Nanoscale to microscale for films
Dynamic films

• Physical background
• Membrane theory: “tents”, “tunnels”, etc.
• Quantitative rigidity
• Bending theory
•
Piecewise rigid body mechanics
Two unifying threads in these
lectures
Think, first, in terms of energy wells
 Thinness and the geometry of SO(3)

Lecture 1

Deformable thin films, macroscale to microscale

Bending

Nanoscale to microscale for films
Dynamic films

• Physical background
• Membrane theory: “tents”, “tunnels”, etc.
• Quantitative rigidity
• Bending theory
•
Piecewise rigid body mechanics
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), 685689
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.
Lecture 2

Deformable thin films, macroscale to microscale

Bending (Joint work with G. Friesecke, S. Muller)

Nanoscale to microscale for films
Dynamic films

• Physical background
• Membrane theory: “tents”, “tunnels”, etc.
• Quantitative rigidity
• Bending theory
•
Piecewise rigid body mechanics
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.
The multiscale problem for atomic 
This slide courtesy of Karin Rabe,
continuum
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!
Lecture 3

Deformable thin films, macroscale to microscale

Bending

Nanoscale to microscale for films (Joint work with
G. Friesecke)
Dynamic films

• Physical background
• Membrane theory: “tents”, “tunnels”, etc.
• Quantitative rigidity
• Bending theory
•
Piecewise rigid body mechanics
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
Lecture 4

Deformable thin films, macroscale to microscale

Bending

Nanoscale to microscale for films
Dynamic films (Joint work with R. Rizzoni)

• Physical background
• Membrane theory: “tents”, “tunnels”, etc.
• Quantitative rigidity
• Bending theory
•
Piecewise rigid body mechanics
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.)
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