Abstract
After introducing the topics that will be covered in this work we review important concepts from the calculus of variations in elasticity theory. Subsequently the following three topics are discussed: The first originates from the work of Post and Sivaloganathan [____emph{Proceedings of the Royal Society of Edinburgh, Section: A Mathematics}, 127(03):595--614, 1997] in the form of two scenarios involving the twisting of the outer boundary of an annulus $A$ around the inner. It seeks minimisers of $∫_A ____frac{1}{2}|∇u|^2 ____d x$ among deformations $u$ with the constraint $____det ∇u ≥0$ a.e.~as well as of $∫_A ____frac{1}{2}|∇u|^2 + h(____det ∇u) ____d x$ in which $h$ penalises volume compression so that $____det ∇u > 0$ a.e.~is imposed on minimisers. In the former case we find infinitely many explicit solutions for which $____det ∇u = 0$ holds on a region around the inner boundary of $A$. In the latter we expand on known results by showing similar growth properties of the solutions compared to the previous case while contrasting that $____det ∇u>0$ holds everywhere. In the second we introduce a new semiconvexity called $n$-polyconvexity that unifies poly- and rank-one convexity in the sense that for $f:ℝ^{d×D}→____bar{ℝ}$ we have that $n$-polyconvexity is equivalent to polyconvexity for $n=____min____{d,D____}=:d∧D$ and equivalent to rank-one convexity for $n=1$. For $d,D≥3$ we gain previously unknown semiconvexities in hierarchical order ($2$-polyconvexity, ____dots, $(d∧D-1)$-polyconvexity, weakest to strongest). We further define functions which are `$n$-polyaffine at $F$' and find that they are not necessarily polyaffine for $n<d∧D$ (unlike rank-one affine functions). As one of the main results we obtain that $1$-polyconvex (i.e.~rank-one convex) functions $f:ℝ^{d×D} →ℝ$ are the pointwise supremum of $1$-polyaffine functions at $F$ for every $F∈ℝ^{d×D}$. In addition and among other things, we discuss envelopes, generalised $T_k$ configurations and relations to quasiconvexity. The third involves a generalisation of the theory of abstract convexity which allows one to include cases like $1$-polyconvex functions as the pointwise supremum of $1$-polyconvex functions at $F$ for every $F$, while this is not possible within the classical theory. We review the most important results of the classical theory and present results on generalised hull operators, subgradients, conjugations and Legendre-Fenchel transforms for our new theory. In particular we obtain an operator that is reminiscent of the rank-one convexification process via lamination steps for a function. Moreover, we show that directional convexity is a special case of the generalised abstract convexity theory. Finally, we conclude each topic, pointing out possible directions of further research.