Abstract
We provide existence theorems in nonlinear elasticity for minimum problems modeling the deformations of a crystal with a given dislocation. A key technical difficulty is that due to the presence of a the dislocation the elastic deformation gradient cannot be in L 2. Thus one needs to consider elastic energies with slow growth, to which the original results of Ball cannot be applied directly.
References
Ambrosio L., Fusco N., Pallara D. (2000). Functions of bounded variation and free discontinuity problems. Oxford, UP
Ball J.M. (1976). Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63: 337–403
Conti S., De Lellis C. (2003). Some remarks on the theory of elasticity for compressible Neohhookean materials. Ann. Sci. Norm. Super. Pisa Cl. Sci. 3: 521–549
Federer H. (1969). Geometric Measure Theory. Springer, New York
Federer H., Fleming W. (1960). Normal and integral currents. Ann. Math. 72: 458–520
Giaquinta, M., Modica, G., Souček, J.: Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 106, 97–159 (1989). Erratum and addendum. Arch. Ration. Mech. Anal. 109, 385–392 (1990)
Giaquinta M., Modica G., Souček J. (1998). Cartesian Currents in the Calculus of Variations I and II. Springer, Berlin
Hirth J.P., Lothe J. (1968). Theory of Dislocations. McGraw-Hill, New York
Müller S. (1990). Det=det. A remark on the distributional determinant. C. R. Acad. Sci. Paris 311: 13–17
Ortiz, M.: Lectures at the Vienna summer school on microstructures. Vienna, 25–29 September, (2000)
Phillips R. (2001). Crystals, Defects and Microstructures: Modeling Across Scales. Cambridge University Press, New York
Šverák V. (1988). Regularity properties of deformations with finite energy. Arch. Ration. Mech. Anal. 100: 105–127
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Müller, S., Palombaro, M. Existence of minimizers for a polyconvex energy in a crystal with dislocations. Calc. Var. 31, 473–482 (2008). https://doi.org/10.1007/s00526-007-0120-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-007-0120-y
Mathematics Subject Classification (2000)
- 74N05
- 58A25
- 74B20
- 49J45
- 46E40