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This work proposes a generalized Lagrangian strain function f α (that depends on modified stretches) and a volumetric strain function g α (that depends on the determinant of the deformation tensor) to characterize isotropic/anisotropic strain energy functions. With the aid of a spectral approach, the single-variable strain functions enable the development of strain energy functions that are consistent with their infinitesimal counterparts, including the development of a strain energy function for the general anisotropic material that contains the general 4th order classical stiffness tensor. The generality of the single-variable strain functions sets a platform for future development of adequate specific forms of the isotropic/anisotropic strain energy function; future modellers only require to construct specific forms of the functions f α and g α to model their strain energy functions. The spectral invariants used in the constitutive equation have a clear physical interpretation, which is attractive, in aiding experiment design and the construction of specific forms of the strain energy. Some previous strain energy functions that appeared in the literature can be considered as special cases of the proposed generalized strain energy function. The resulting constitutive equations can be easily converted, to allow the mechanical influence of compressed fibres to be excluded or partial excluded and to model fibre dispersion in collagenous soft tissues. Implementation of the constitutive equations in Finite Element software is discussed. The suggested crude specific strain function forms are able to fit the theory well with experimental data and managed to predict several sets of experimental data. [ABSTRACT FROM AUTHOR]

This article presents important constitutive refinements and simplifications in the theory of polar elasticity of materials reinforced by a single family of fibers resistant in bending. One of these simplifications is achieved by paying attention to forms of the strain energy which are symmetric with respect to the symmetric and the antisymmetric parts of the fiber gradient tensor. This leads to the identification of a restricted version of the theory that is predominantly influenced by the fiber-splay mode of deformation. The lack of ellipticity of the governing equations of polar elasticity and the anticipation of existence of weak discontinuity surfaces even in the small deformation regime are also investigated. The manner in which potential activation of such surfaces is related with the action of either the fiber-bending or the fiber-splay deformation mode, as well as with their conjoined combination and coupling with their fiber-twist counterpart, is examined. The proposed constitutive equations can be simplified via the use of a new set of fourteen independent spectral invariants of the deformation. This set serves as an irreducible functional basis of relevant invariants or as an irreducible integrity basis of polynomial invariants. For instance, its use here enables identification of fourteen classical invariants that emerge as mutually independent from the known set of thirty-three in total classical invariants. In the special case of polynomial invariants, this result paves the way for identification of a corresponding minimal integrity basis. [ABSTRACT FROM AUTHOR]