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Rigorous derivation of a homogenized bending-torsion theory for inextensible rods from 3d elasticity
| Content Provider | Semantic Scholar |
|---|---|
| Author | Neukamm, Stefan |
| Copyright Year | 2011 |
| Abstract | We present a rigorous derivation of a homogenized, bending-torsion theory for inex- tensible rods from three-dimensional nonlinear elasticity in the spirit of -convergence. We start with the elastic energy functional associated to a nonlinear composite materia l. In a stress-free ref- erence configuration it occupies a thin cylindrical domain with thickness h - 1. We consider com- posite materials that feature a periodic microstructure with period " - 1. We study the behavior as " and h simultaneously converge to zero and prove that the energy (scaled by h −4 ) -converges towards a non-convex, singular energy functional. The energy is only finite for configurations that correspond to pure bending and twisting of the rod. In this case, the energy is quadratic in curvature and torsion. Our derivation leads to a new relaxation formula that uniquely determines the homogenized coefficients. It turns out that their precise structure additionally depend s on the ratio h/" and, in particular, different relaxation formulas arise for h - ", " - h and " - h. Although, the initial elastic energy functional and the limiting functional are non-convex, our analysis leads to a relaxation formula that is quadratic and involves only relaxation over a sing le cell. Moreover, we derive an explicit formula for isotropic materials in the cases h - " and h - ", and prove that the -limits associated to homogenization and dimension reduction in general do not commute. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://www.mis.mpg.de/preprints/2011/preprint2011_74.pdf |
| Alternate Webpage(s) | http://www.mis.mpg.de/preprints/2011/preprint2011_74.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
