New integral estimates for deformations in terms of their nonlinear strains
New integral estimates for deformations in terms of their nonlinear strains
复制标题
根据非线性应变对变形进行新的积分估计
DOI：
10.1007/bf00250837
复制
发表时间：
1982
影响因子：
2.5
通讯作者：
R. Kohn
中科院分区：
文献类型：
--
作者：
R. Kohn
如果u是有界Lipschitz域Ω在ℓn(n≧2)中的双Lipschitzian变形，我们证明了与u相关的某个“非线性应变函数”e(U)的Lp范数(p≧1，p≠n)控制了Lq(q=np/(n−p)if pn)中从u到ℝn的适当选择的刚体运动的距离.本文推广了F.John的工作，他在u具有“一致小应变”的假设下证明了p}>；1的相应估计.我们还得到了Du在L2中振荡的一个界。这些估计显然是第一次在没有先验的逐点假设的情况下应用于u的应变。在ℝ3中，积分
$$\int\Limits_\Omega{}$$
E(U)2dℋ3由文献中提出的用于模拟橡胶行为的典型超弹性能量泛函所支配；因此，当n=3，p=2时，根据相关的非线性弹性功给出了这类材料变形的第一个一般界限。
AbstractIf u is a bi-Lipschitzian deformation of a bounded Lipschitz domain Ω in ℓn (n≧2), we show that the LP norm (p≧1, p≠n) of a certain “nonlinear strain function” e(u) associated with u dominates the distance in Lq (q= np/(n−p) if pn) from u to a suitably chosen rigid motion of ℝn. This work extends that of F. John, who proved corresponding estimates for p}>1 under the hypothesis that u has “uniformly small strain”. We also obtain a bound for the oscillation of Du in L2. These estimates are apparently the first to apply with no a priori pointwise hypotheses upon the strain of u. In ℝ3 the integral
$$\int\limits_\Omega {}$$
e(u)2dℋ3 is dominated by typical hyperelastic energy functionals proposed in the literature for modeling the behavior of rubber; thus the case n=3, p=2 gives the first general bound for the deformations of such materials in terms of the associated nonlinear elastic work.