arXiv subjects

Christian Thiel

Publications and source records attributed to Christian Thiel.

5 recordsLinked to original sources

arXiv2018

Do we need Truesdell's empirical inequalities? On the coaxiality of stress and stretch

Truesdell's empirical inequalities are considered essential in various fields of nonlinear elasticity. However, they are often used merely as a sufficient criterion for semi-invertibility of the isotropic stress strain-relation, even though weaker and much less restricting constitutive requirements like the strict Baker-Ericksen inequalities are available for this purpose. We elaborate the relations between such constitutive conditions, including a weakened version of the empirical inequalities, and their connection to bi-coaxiality and related matrix properties. In particular, we discuss a number of issues arising from the seemingly ubiquitous use of the phrase "$X,Y$ have the same eigenvectors" when referring to commuting symmetric tensors $X,Y$.

math.AP↗

arXiv2018

Shear, pure and simple

In a 2012 article in the International Journal of Non-Linear Mechanics, Destrade et al. showed that for nonlinear elastic materials satisfying Truesdell's so-called empirical inequalities, the deformation corresponding to a Cauchy pure shear stress is not a simple shear. Similar results can be found in a 2011 article of L. A. Mihai and A. Goriely. We confirm their results under weakened assumptions and consider the case of a shear load, i.e. a Biot pure shear stress. In addition, conditions under which Cauchy pure shear stresses correspond to (idealized) pure shear stretch tensors are stated and a new notion of idealized finite simple shear is introduced, showing that for certain classes of nonlinear materials, the results by Destrade et al. can be simplified considerably.

math.AP↗

arXiv2015

The sum of squared logarithms inequality in arbitrary dimensions

We prove the \emph{sum of squared logarithms inequality} (SSLI) which states that for nonnegative vectors $x, y \in \mathbb{R}^n$ whose elementary symmetric polynomials satisfy $e_k(x)\le e_k(y)$ (for $1\le k < n$) and $e_n(x)=e_n(y)$, the inequality $\sum_i (\log x_i)^2 \le \sum_i (\log y_i)^2$ holds. Our proof of this inequality follows by a suitable extension to the complex plane. In particular, we show that the function $f\colon M\subseteq \mathbb{C}^n\to \mathbb{R}$ with $f(z)=\sum_i(\log z_i)^2$ has nonnegative partial derivatives with respect to the elementary symmetric polynomials of $z$. This property leads to our proof. We conclude by providing applications and wider connections of the SSLI.

math.CA↗

arXiv2015

On the convexity of nonlinear elastic energies in the right Cauchy-Green tensor

We present a sufficient condition under which a weak solution of the Euler-Lagrange equations in nonlinear elasticity is already a global minimizer of the corresponding elastic energy functional. This criterion is applicable to energies $W(F)=\widehat{W}(F^TF)=\widehat{W}(C)$ which are convex with respect to the right Cauchy-Green tensor $C=F^TF$, where $F$ denotes the gradient of deformation. Examples of such energies exhibiting a blow up for $\det F\to0$ are given.

math.AP↗

arXiv2015

On the sum of squared logarithms inequality and related inequalities

We consider the sum of squared logarithms inequality and investigate possible connections with the theory of majorization. We also discuss alternative sufficient conditions on two sets of vectors $a,b\in\mathbb{R}_+^n$ so that $\sum_{i=1}^n(\log a_i)^2\ \leq\ \sum_{i=1}^n(\log b_i)^2\,.\notag $ Generalizations of some inequalities from information theory are obtained, including a generalized information inequality and a generalized log sum inequality, which states for $a,b\in\mathbb{R}_+^n$ and $k_1,...,k_n\in [0,\infty)$: $ \sum_{i=1}^na_i\,\log\prod_{s=1}^m(\frac{a_i}{b_i} + k_s)\ \geq\ \log\prod_{s=1}^m(1+k_s)\,.\notag $

math.CA↗