Vector and matrix monotonicity characterisation of the "load forces" in nonlinear elasticity: compressible versus incompressible response

In this paper, we study the relation between different notions of monotonicity for isotropic hyperelastic compressible and incompressible materials. We study how to define the monotonicity concept on the manifold ${\mathrm{S}\mathrm{y}\mathrm{m}}^{++}(n)\cap \mathrm{S}\mathrm{L}(n)$ of positive definite symmetric tensors with unit determinant and the corresponding spectral manifold $\mathcal{S}=\{({\lambda }_1,\dots ,{\lambda }_n)\in {\mathbb{R}}_{+}^n|{\lambda }_1{\lambda }_2\cdots {\lambda }_n=1\}$, and give a precise characterisation of monotonicity in the case of hyperelastic materials. In particular, we consider the monotonicity properties of the Biot stress tensor $T_{\mathrm{B}\mathrm{i}\mathrm{o}\mathrm{t}}(U)$, the engineering stress, or the load tensor.