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A new class of constitutive equation is proposed for isotropic thermoelastic solids, wherein the Hencky strain tensor is assumed to be a function of the Cauchy stress tensor, via a Gibbs potential. The solid is assumed to be incompressible in the referential state, but the volume can change due to differences in the temperature relative to a reference temperature. The change in volume only depends on temperature. Some restrictions are found for the Gibbs potential, resulting in a constitutive equation for isotropic solids, wherein the volume depends on temperature. Using the resulting constitutive equation, some boundary value problems are studied, considering some relatively simple distributions for the temperature, deformations and stresses. [ABSTRACT FROM AUTHOR]

In this article, we prove that for isotropic functions that depend on |$P$| vectors, |$N$| symmetric tensors and |$M$| non-symmetric tensors (a) the minimal number of irreducible invariants for a scalar-valued isotropic function is |$3P+9M+6N-3,$| (b) the minimal number of irreducible vectors for a vector-valued isotropic function is |$3$| and (c) the minimal number of irreducible tensors for a tensor-valued isotropic function is at most   |$9$|⁠. The minimal irreducible numbers given in (a), (b) and (c) are, in general, much lower than the irreducible numbers obtained in the literature. This significant reduction in the numbers of irreducible isotropic functions has the potential to substantially reduce modelling complexity. [ABSTRACT FROM AUTHOR]

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]

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]

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]

In this communication a spectral constitutive equation for nonlinear viscoelastic-electroactive bodies with short-term memory response is developed, using the total stress formulation and the electric field as the electric independent variable. Spectral invariants, each one with a clear physical meaning and hence attractive for use in experiment, are used in the constitutive equation. A specific form for constitutive equation containing single-variable functions is presented, which are easy to analyze compared to multivariable functions. The effects of viscosity and an electric field are studied via the results of boundary value problems for cases considering homogeneous distributions for the strains and the electric field, and some these results are compared with experimental data. [ABSTRACT FROM AUTHOR]

In the present article, a spectral model is developed for residually stressed electro-elastic bodies. The model uses a total energy function that depends on the right stretch tensor, residual stress tensor and one of the electric variables. Some boundary value results with cylindrical symmetry are given. Results for the inflation of a hollow sphere, where the residual stress is assumed to depend only on the radial position, are also given. The constitutive formulation contains spectral invariants that have an immediate physical interpretation which is useful in a rigorous construction of a specific form of the total energy function via an appropriate experiment [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]

In the literature, residual stress problems are generally formulated using classical invariants despite most of them having an unclear physical meaning and not having experimental advantages. In this article, we give an alternative formulation for residual stress problems using a set of spectral invariants. These invariants have a clear physical meaning which may facilitate the design of a residual stress experiment. For the case of an energy function that depends on the right Cauchy tensor and the residual stress tensor, we show that only nine of the classical invariants are independent, not 10 as commonly assumed. Details of the spectral formulation are given and several boundary value problems are illustrated. [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]

A strain energy function which depends on five independent variables that have immediate physical interpretation is proposed for finite strain deformations of transversely isotropic elastic solids. Three of the five variables (invariants) are the principal stretch ratios and the other two are squares of the dot product between the preferred direction and two principal directions of the right stretch tensor. The set of these five invariants is a minimal integrity basis. A strain energy function, expressed in terms of these invariants, has a symmetry property similar to that of an isotropic elastic solid written in terms of principal stretches. Ground state and stress–strain relations are given. The formulation is applied to several types of deformations, and in these applications, a mathematical simplicity is highlighted. The proposed model is attractive if principal axes techniques are used in solving boundary-value problems. Experimental advantage is demonstrated by showing that a simple triaxial test can vary a single invariant while keeping the remaining invariants fixed. A specific form of strain energy function can be easily obtained from the general form via a triaxial test. Using series expansions and symmetry, the proposed general strain energy function is refined to some particular forms. Since the principal stretches are the invariants of the strain energy function, the Valanis–Landel form can be easily incorporated into the constitutive equation. The sensitivity of response functions to Cauchy stress data is discussed for both isotropic and transversely isotropic materials. Explicit expressions for the weighted Cauchy response functions are easily obtained since the response function basis is almost mutually orthogonal. [ABSTRACT FROM PUBLISHER]