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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 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]

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]