RIGOROUS DERIVATION OF A HOMOGENIZED BENDING-TORSION THEORY FOR INEXTENSIBLE RODS FROM 3D ELASTICITY
RIGOROUS DERIVATION OF A HOMOGENIZED BENDING-TORSION THEORY FOR INEXTENSIBLE RODS FROM 3D ELASTICITY
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从 3D 弹性严格推导不可伸展杆的均质弯扭理论
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发表时间：
2011
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通讯作者：
S. Neukamm
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作者：
S. Neukamm
本文从三维非线性弹性理论出发，严格推导了不可拉伸杆的均匀化弯扭理论。我们从非线性复合材料的弹性能量泛函出发。在无应力参考配置中，它占据厚度为h1的薄圆柱形域。我们考虑复合材料的特点是周期性的微观结构与周期“-1。我们研究了as“和h同时收敛到零的行为，并证明了能量（以h −4为尺度）-收敛到一个非凸的奇异能量泛函。能量仅对对应于杆的纯弯曲和扭转的构型是有限的。在这种情况下，能量是曲率和挠率的平方。我们的推导导致一个新的松弛公式，唯一确定的均匀化系数。事实证明，它们的精确结构还取决于h/”的比值，特别是，对于h“，”h“和“h”，会出现不同的松弛公式。虽然，初始弹性能量泛函和极限泛函是非凸的，但我们的分析导致松弛公式是二次的，并且仅涉及单个细胞的松弛。此外，我们推导了各向同性材料在h ′“和h ′“情况下的显式公式，并证明了与均匀化和降维有关的-极限一般不交换。
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.