Quasiconvex functions incorporating volumetric constraints are rank-one convex
Quasiconvex functions incorporating volumetric constraints are rank-one convex
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结合体积约束的拟凸函数是一阶凸函数
DOI：
10.1016/j.matpur.2008.04.009
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发表时间：
2008
期刊：
影响因子：
--
通讯作者：
S. Conti
中科院分区：
文献类型：
--
作者：
S. Conti
我们证明了在集合F ={F：detF=1}上有限的拟凸函数W：Mn×n→[0，∞]是秩一凸的，因此在集合F上是连续的;对子式的约束也是如此.这意味着秩一凸包络给出了对不可压缩材料建模的任何能量密度的拟凸包络的上界。我们的结果是基于一个适当的分段仿射函数u的建设，使<$u∈ <$几乎无处不在。
We prove that a quasiconvex function W:Mn×n→[0,∞] which is finite on the set Σ={F:detF=1} is rank-one convex, and hence continuous, on Σ; and the same for constraints on minors. This implies that the rank-one convex envelope gives an upper bound on the quasiconvex envelope of any energy density modeling an incompressible material. Our result is based on the construction of an appropriate piecewise affine function u such that ∇u∈Σ almost everywhere.