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In the past, to model fibre stiffness of finite-radius fibres, previous finite-strain (nonlinear) models were mainly based on the theory of non-linear strain-gradient (second-gradient) theory or Kirchhoff rod theory. We note that these models characterize the mechanical behaviour of polar transversely isotropic solids with infinitely many purely flexible fibres with zero radius. To introduce the effect of fibre bending stiffness on purely flexible fibres with zero radius, these models assumed the existence of couple stresses (contact torques) and non-symmetric Cauchy stresses. However, these stresses are not present on deformations of actual non-polar elastic solids reinforced by finite-radius fibres. In addition to this, the implementation of boundary conditions for second gradient models is not straightforward and discussion on the effectiveness of strain gradient elasticity models to mechanically describe continuum solids is still ongoing. In this paper, we develop a constitutive equation for a non-linear non-polar elastic solid, reinforced by embedded fibers, in which elastic resistance of the fibers to bending is modelled via the classical branches of continuum mechanics, where the development of the theory of stresses is based on non-polar materials; that is, without using the second gradient theory, which is associated with couple stresses and non-symmetric Cauchy stresses. In view of this, the proposed model is simple and somewhat more realistic compared to previous second gradient models. [ABSTRACT FROM AUTHOR]

Mechanical models of residually stressed fibre-reinforced solids, which do not resist bending, have been developed in the literature. However, in some residually stressed fibre-reinforced elastic solids, resistance to fibre bending is significant, and the mechanical behavior of such solids should be investigated. Hence, in this paper, we model the mechanical aspect of residually stressed elastic solids with bending stiffness due to fibre curvature, which up to the authors' knowledge has not been mechanically modeled in the past. The proposed constitutive equation involves a nonsymmetric stress and a couple-stress tensor. Spectral invariants are used in the constitutive equation, where each spectral invariant has an intelligible physical meaning, and hence they are useful in experiment and analysis. A prototype strain energy function is proposed. Moreover, we use this prototype to give results for some cylindrical boundary value problems. [ABSTRACT FROM AUTHOR]

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]

Invariants play an important role in continuum mechanics. Knowing the number of independent invariants is crucial in modelling and in a rigorous construction of a constitutive equation for a particular material, where it is determined by doing tests that hold all, except one, of the independent invariants constant so that the dependence in the one invariant can be identified. Hence, the aim of this paper is to prove that the number of independent invariants for a set of n symmetric tensors and m unit vectors is at most 2m+6n-3. The prove requires the construction of spectral invariants. All classical invariants can be explicitly expressed in terms spectral invariants. We show that the number of spectral invariants in an irreducible functional basis is reduced to 2m+6n-3; a significant reduction to that obtained in the literature if the value of m or n is large. Relations between classical invariants in a classical-minimal integrity basis are given. [ABSTRACT FROM AUTHOR]

A constitutive equation is provided for nonlinear transversely isotropic elastic solids, wherein the linearized strain tensor is assumed to be a function of the Cauchy stress tensor; this elastic constitutive equation belongs to a subclass of a more general set of implicit constitutive relations proposed in the recent years. The proposed constitutive equation is valid for both compressible and incompressible bodies and can be simply modified, to exclude the mechanical influence of compressed fibres and to model inextensible fibres. A crude specific constitutive model is proposed to compare with a uniaxial experimental data on Marcellus shale. Some simple boundary value problems are analyzed. [ABSTRACT FROM AUTHOR]

Classical invariants, despite most of them having unclear physical interpretation and not having experimental advantages, have been extensively used in modeling nonlinear magneto-elastic materials. In this paper, a new set of spectral invariants, which have some advantages over classical invariants, is proposed to model the behavior of transversely isotropic nonlinear magneto-elastic bodies. The novel spectral invariant formulation, which is shown to be more general, is used to analytically solve some simple magneto-mechanical boundary value problems. With the aid of the proposed spectral invariants it is possible to study, in a much simpler manner, the effect of different types of deformations on the response of the magneto-elastic material. [ABSTRACT FROM AUTHOR]

The motivation for this work was the absence of closed form solutions that can reasonably describe the axial deformation behaviour of stress-softening polymer bearings. In this paper, new closed form solutions that exhibit Mullins phenomenon are developed. We show that the apparent Young modulus depends on the shape factor and the minimal infinitesimal strain. We furthermore found that, in a nonlinear deformation, the shape factor plays an important role in stress softening. The solutions are design friendly and are consistent with expected results. [ABSTRACT FROM AUTHOR]