To relate continuum theories of elastic phenomena to typical atomic or molecular theories, one needs some way of correlating changes in positions of the entities in the latter with descriptions of deformation used in the former. For crystals, the most commonly used bridge used for this is the Cauchy—Born rule. In recent years, there has been a flurry of activity aimed at better understanding it and in trying to generalize it. I will describe and comment on important ideas that have been generated, point out some successes and failures of the rule and indicate kinds of applications that have motivated such work.
Ericksen, J.L.The Cauchy and Born hypotheses for crystals , in Phase Transformations and Material Instabilities in Solids , pp. 61-77, ed. M. E. Gurtin, Academic , New York, 1984.
2.
Zanzotto, G.Twinning in minerals and metals: remarks on the comparison of a thermoelastic theory with some experimental results, Nota 1 and Nota 2, Atti Accademia Nazionale dei Lincei, 82, 725-741 and 743-756 (1988).
3.
Goetz, G. and Hergenrother, R.C.X-ray studies of the thermal expansion of bismuth single crystals, Physical Review, 40, 643-661 (1932).
4.
Balzer, R. and Sigvaldson, H.Equilibrium vacancy concentration measurements on zinc single crystals, Journal of Physics F: Metal Physics , 9, 171-178 (1979).
5.
Jay, A.H.The thermal expansion of quartz, Proceedings of the Royal Society of London, A142, 237-247 (1933).
6.
Pitteri, M. and Zanzotto, G.Continuum Models for Phase Transitions and Twinning in Crystals, CRC/Chapman and Hall, London , 2000.
7.
Ericksen, J.L.On nonessential descriptions of crystal multilattices, Mathematics and Mechanics of Solids, 4, 363-392 (1998).
8.
Ericksen, J.L.On the symmetry of crystals (in Russian), in Problems of Mechanics of Deformable Solid Bodies, pp. 493-496, ed. T. N. Robotnov and L. I. Sedov, Academy of Science of the USSR, 1970.
9.
Ericksen, J.L.Nonlinear elasticity of diatomic crystals , International Journal of Solids and Structures, 6, 951-957 (1970).
10.
Ericksen, J.L.Mechanics and Mathematics of Crystals, ed. M. F. Beatty and M. A. Hayes, World Scientific, Singapore, 2005.
11.
Pitteri. M.Reconciliation of local and global symmetries of crystals, Journal of Elasticity, 14, 175-190 (1984).
12.
Pitteri, M.On (ν+1) lattices, Journal of Elasticity, 15, 3-25 (1985).
13.
Duistermaat, J.J. and Kolk, J.A.C.Lie Groups, Springer , Berlin, 1999.
14.
Parry, G.P.Low-dimensional lattice groups, Archive for Rational Mechanics and Analysis, 145, 1-22 (1998).
15.
Friesecke, G. and Theil, F.Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice, Journal of Nonlinear Science, 12, 445-478 (2002).
16.
Ericksen, J.L.A modified theory of magnetic fields in elastic materials, Mathematics and Mechanics of Solids, 11, 23-47 (2006).
17.
Charlotte, M. and Truskinovsky , L.Linear elastic chain with a hyper-stress, Journal of Mechanics and Physics of Solids , 50, 217-251 (2002).
18.
Conti, S. , Dolzmann, G.Kirchheim, B. and Múller, S.Sufficient conditions for the validity of the Cauchy-Born rule close to S O (n), Journal of the European Mathematical Society, 8, 515-530 (2006).
19.
Blanc, X. , LeBris, C. and Lions, P.-L.From molecular models to continuum mechanics , Archive for Rational Mechanics and Analysis, 164, 341-381 (2002).
20.
Dunn, J.E. and Serrin, J.On the thermomechanics of interstitial working , Archive for Rational Mechanics and Analysis, 88, 95-133 (1985).
21.
Sunyk, R. and Steinmann, P.On higher gradients in continuum atomistic modeling, International Journal of Solids and Structures, 40, 6877-6896 (2003 ).
22.
Weinan, E. and Ming, P.Cauchy-Born rule and the stability of crystalline solids: static problems , Archive for Rational Mechanics and Analysis, 183, 241-297 (2007).
23.
Wuttig, M., Lurebrink, D., Wamwangi, D., Welnick, W., Gilleben, M. and Dronskowski, R.The role of vacancies and local distortion in the design of new phase-change materials, Nature Materials, 6, 168-174 (2007).
24.
Conti, S. and Zanzotto, G.A variational model for reconstructive phase transformations in crystals and their relation to dislocations and plasticity, Archive for Rational Mechanics and Analysis, 173, 69-88 (2004).
25.
Balandraud, X. and Zanzotto, G.Stressed microstructures in thermally induced M9R-M18R Martensite, Journal of Mechanics and Physics of Solids , 55, 194-224 (2007).
26.
Ball, J. and James, R.D.Proposed experimental tests of a theory of fine microstructures and the two-well problem, Philosophical Transactions of the Royal Society of London, A333, 389-450 (1992).
27.
James, R.D. and Hane, K.F.Martensitic transformations and shape-memory materials, Acta Materiali, 48, 197-222 (2000).
28.
Christian, J.W. and Mahajan, S.Deformation twinning, Progress in Materials Science, 39, 1-157 (1995).
29.
Wooster, W.A. and Wooster, N.Control of electrical twinning in quartz, Nature, 159, 405-406 (1946).
30.
Wooster, W.A. and Wooster, N.Piezocrcesence-the growth of Dauphiné twinning in quartz under stress, Proceedings of the Royal Society of London, A208, 43-63 (1951).
31.
Bhattacharya, K.Comparison of geometrically nonlinear and linear theories of Martensitic transformations, Continuum Mechanics and Thermodynamics, 5, 205-242 (1993).
32.
Ericksen, J.L.On the theory of cyclic growth twins, Mathematics and Mechanics of Solids, 11, 3-22 (2006).
33.
Zanzotto, G.Geobarothermometric properties of growth twins and analysis of quartz data for a broad range of temperatures and pressures , Physics and Chemistry of Minerals.169, 783-789 (1989).
34.
Zanzotto, G.Thermoelastic stability of multiple growth twins in quartz and general barothermometric implications, Journal of Elasticity, 23, 253-287 (1990).
35.
Zanzotto, G.On the material symmetry group of elastic crystals and the Born rule, Archive for Rational Mechanics and Analysis,121, 1-36 (1992).
36.
Tolédano, P. and Dmitriev, V.Reconstructive Phase Transitions, World Scientific Publishing Company, Singapore, 1996.
37.
Bhattacharya, K., Conti, G., Zanzotto, G. and Zimmer, J.Symmetry and reversibility of Martensitic transformations, Nature, 428, 55-59 (2004).
38.
Engel, P.Geometric Crystallography, D. Reidel Publishing Co., Dordrecht, 1986.
39.
Parry, G.P.On the elasticity of monatomic crystals, Mathematical Proceedings of the Cambridge Philosophical Society, 80, 189-211 (1976).
40.
Yang, J.Z. and Weinan, E.Generalized Cauchy-Born rules for elastic deformation of sheets, plates and rods, Physical ReviewB74, 184110-1-11 (2006).
41.
Friesecke, G. and James, R.D.A scheme for the passage from atomic to continuum theory for thin films, Journal of Mechanics and Physics of Solids, 48, 1519-1540 (2000 ).
42.
Schmidt, B.On the passage from atomic theory to continuum theory for thin films, to appear in Archive for Rational Mechanics and Analysis.
43.
Schmidt, B.Qualitative properties of a continuum theory for thin films, to appear in Annales de l' Institut Henri Poincaré.
44.
Arroyo, M. and Belytschko, T.An atomic based finite deformation membrane for single layer crystalline films, Journal of Mechanics and Physics of Solids, 50, 1941-1977 (2002 ).
45.
Arroyo, M. and Belytschko, T.Finite crystal elasticity of carbon nanotubes based on the exponential Cauchy-Born rule, Physical Review , B69, 115415-1-11 (2004).
46.
Guo, X., Wang, J.B. and Zhang, H.W.Mechanical properties of single-walled nanotubes based on higher order Cauchy-Born rule. International Journal of Solids and Structures, 43, 1276-1290 (2006 ).
47.
Wang, J.B. , Guo, X., Zhang, H.W., Wang, L. and Liao, J.B.Energy and mechanical properties of single-walled carbon nanotubes predicted using the higher order Cauchy-Born rule, Physical Review, B73, 115428-1-14 (2006).
48.
Chandreseker, K., Mukherjee, S. and Mukherjeem, Y.X.Modifications to the Cauchy-Born rule: applications in the deformation of single-walled carbon nanotubes, International Journal of Solids and Structures, 41, 7128-7144 (2006).
49.
James, R.D.Objective structures, Journal of Mechanics and Physics of Solids, 54, 2354-2390 (2006 ).
50.
Falk, W. and James, R.D.An elasticity theory for self-assembled protein lattices with application to the Martensitic phase transition in bacteriophage T4 tail sheath, Physical Review, E73, 011917-1-23 (2006).
51.
Ericksen, J.L.Equilibrium theory for x-ray observations of crystals, Archive for Rational Mechanics and Analysis, 139, 181-200 (1997).
52.
Ericksen, J.L.Notes on the x-ray theory, Journal of Elasticity, 55, 201-218 (1999).
53.
Ericksen, J.L.Twinning analyses in the x-ray theory, International Journal of Solids and Structures, 38, 967-995 (2001).
