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A non-ellipticity result, or the impossible taming of the logarithmic strain measure. (arXiv:1712.04846v1 [math.CA])
来源于:arXiv
The logarithmic strain measures $\lVert\log U\rVert^2$, where $\log U$ is the
principal matrix logarithm of the stretch tensor $U=\sqrt{F^TF}$ corresponding
to the deformation gradient $F$ and $\lVert\,.\,\rVert$ denotes the Frobenius
matrix norm, arises naturally via the geodesic distance of $F$ to the special
orthogonal group $\operatorname{SO}(n)$. This purely geometric characterization
of this strain measure suggests that a viable constitutive law of nonlinear
elasticity may be derived from an elastic energy potential which depends solely
on this intrinsic property of the deformation, i.e. that an energy function
$W\colon\operatorname{GL^+}(n)\to\mathbb{R}$ of the form \begin{equation}
W(F)=\Psi(\lVert\log U\rVert^2) \tag{1} \end{equation} with a suitable
function $\Psi\colon[0,\infty)\to\mathbb{R}$ should be used to describe finite
elastic deformations.
However, while such energy functions enjoy a number of favorable properties,
we show that it is not possible to find a strictly m 查看全文>>