The aim of this thesis is to cover several aspects of the theory of nonlinear elasticity. We discuss the role of physically motivated constitutive assumptions (to ensure realistic material behavior) as well as mathematically motivated conditions (to allow for the application of strong mathematical results).

To this end, we reconsider fundamental constitutive concepts of elasticity theory like polyconvexity, quasiconvexity and rank-one convexity as well as more specific questions of the occurrence of anti-plane shear deformations, difference between shear stress and simple shear deformations, and the request for Truesdell's empirical inequalities. In particular, we discuss the crucial difference between additional a priori and a posteriori constraints on the minimization problem

\[ I(\varphi)=\int_\Omega W(\nabla\varphi(x))\,\dx\to\min\,,\qquad\varphi\in M\subset W^{1,p}(\Omega)\,. \]

All obtained constitutive requirements are evaluated for different energy functions from the literature. One of the major open problems in the calculus of variations is Morrey's problem in the planar case:

Does rank-one convexity imply quasiconvexity?

For some specific families of energies, there are precise conditions known under which rank-one convexity even implies polyconvexity. We will extend these findings to the more general family of energies $W\col\GLp(2)\to\R$ with an additive volumetric-isochoric split, i.e.

\[ W(F)=W_{\rm iso}(F)+W_{\rm vol}(\det F)=\widetilde W_{\rm iso}\bigg(\frac{F}{\sqrt{\det F}}\bigg)+W_{\rm vol}(\det F)\,, \]

which is the natural finite extension of isotropic linear elasticity. We give a precise analysis of rank-one convexity criteria for this case by showing that the Legendre-Hadamard ellipticity condition separates and simplifies in a suitable sense. Starting from the classical two-dimensional criterion by Knowles and Sternberg, we can reduce the conditions for rank-one convexity to a family of one-dimensional coupled differential inequalities and also introduce weaker regularity assumptions. This new criterion is used to find candidates in the class of volumetric-isochoric split energies which are barely rank-one convex and therefore are interesting energies to test for quasiconvexity. We focus on

\[ W_{\rm magic}^+(F)=\frac{\lambdamax}{\lambdamin}-\log\frac{\lambdamax}{\lambdamin}+\log\det F=\frac{\lambdamax}{\lambdamin}-2\.\log\lambdamin \]

and show a surprising connection between $W_{\rm magic}^+(F)$ and the work of Burkholder and Iwaniec in the field of complex analysis. Besides, we reveal different non-trivial settings such that $W_{\rm magic}^+(F)$ exhibits non-homogeneous deformations which have the same energy value $I(\varphi)$ as the homogeneous solution.