On universal relations in orthotropic material subclasses

Universal relations that hold true for specific classes or subclasses of elastic materials form an integral part of the characterisation process for any material that can deform elastically. On the other hand, it has become understood that any class of hyper-elastic materials possessing anisotropic properties more complicated to those of transverse isotropy obeys no universal relations. However, there are met in practice special cases (subclasses) of anisotropic hyper-elastic materials that, under certain circumstances, may still obey a certain number of universal relations. Material characterisation purposes suggest therefore that there is a practical purpose in identifying and classifying different subclasses of anisotropic hyper-elastic materials that may obey any kind of universal relations and, hence, to derive and gradually build up catalogues or tables of different universal relations that such material subclasses obey when subjected to different deformation patterns. This paper heads towards this research direction by initially considering the orthotropic material class which is alternatively identified as the hyper-elastic crystal class of rhombic symmetry [A.E. Green, J.E. Adkins, Large Elastic Deformations, Clarendon Press, Oxford, 1960]. Then, for a certain set of appropriately chosen orthotropic and transverse isotropic hyper-elastic material subclasses, it develops and presents a considerable number of relevant universal relations, most of which are new in the literature.