Search

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]

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]

Spectral derivatives of scalar-valued, vector-valued and tensor-valued tensor functions are derived. The method used here is able to obtain spectral derivatives of tensor functions that cannot be explicitly expressed in terms of deformation gradient tensor but can be written explicitly in terms of the eigenvalues and eigenvectors of the right and left stretch tensors. [ABSTRACT FROM AUTHOR]