On isotropic extension of anisotropic constitutive functions via structural tensors

Xiao, Heng; Bruhns, Otto T.; Meyers, A.

doi:10.1002/zamm.200410226

Cited by 7 publications

(6 citation statements)

References 22 publications

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“…Besides transverse isotropy and orthotropy, Liu’s list includes groups pertaining to crystal classes in the triclinic, monoclinic, and rhombic systems. More recently, Xiao et al [9] demonstrated that “any number of vectors and second order tensors can merely characterize and represent one of the cylindrical groups and the triclinic, monoclinic, rhombic crystal classes.” They proceed to assert what follows:This suggests that, for anisotropic functions relative to any anisotropic material symmetry group other than those just mentioned, the widely used isotropic extension method via structural tensors has to result in isotropic extension functions involving at least one structural tensor variable of order higher than two.…”

Section: Methods Of Isotropic Extension As Formulated By Boehler Anmentioning

confidence: 99%

“…Besides transverse isotropy and orthotropy, Liu's list includes groups pertaining to crystal classes in the triclinic, monoclinic, and rhombic systems. More recently, Xiao et al [9] demonstrated that "any number of vectors and second order tensors can merely characterize and represent one of the cylindrical groups and the triclinic, monoclinic, rhombic crystal classes." They proceed to assert what follows:…”

Section: Qs( V A)qmentioning

confidence: 99%

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Remarks on isotropic extension of anisotropic constitutive functions via structural tensors

Man

Goddard

2017

Mathematics and Mechanics of Solids

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The following is an elaboration on the linear non-local model of viscoelastic fluids proposed in a previous work (Int. J. Eng. Sci. 48 (2010), 1279-1288). As a recapitulation of that work, the basic theory is presented in terms of the temporal frequency and spatial wave number in the Laplace-Fourier domain. Taylor-series expansions in these variables provides a weakly non-local theory in spatio-temporal gradients that is more comprehensive than the "bi-velocity" model of Brenner. The linearized Chapman-Enskog kinetic theory is shown to provide a confirmation of the more general theory, from which one can reconstruct a fully non-local integral model. Following the work of Davis and Brenner (J. Acoust. Soc. Am. 132 (2012), 2963-2969), the general theory is employed to derive dispersion relations for acoustic, thermal and shear-wave propagation in compressible viscoelastic fluids. At Burnett order the Chapman-Enskog theory gives a cubic polynomial in wave number squared which reduces in the dissipative quasi-static limit to a quadratic like that given by the classical Navier-Stokes-Fourier model and the bi-velocity modification of that model. With minor modification, the present analysis applies to viscoelastic shear and dilatational wave propagation in solids with higher-gradient and Cosserat effects, where it may, for example, find application to the field of rotational seismology.

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Section: Methods Of Isotropic Extension As Formulated By Boehler Anmentioning

confidence: 99%

Section: Qs( V A)qmentioning

confidence: 99%

Remarks on isotropic extension of anisotropic constitutive functions via structural tensors

Man

Goddard

2017

Mathematics and Mechanics of Solids

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“…This is E-mail address: marc.francois@u-psud.fr 1 consistent with the Curie principle: the symmetry group of the consequences (the elasticity tensor) includes the symmetry group of the causes (the microstructure). Although informations on microstructure can be described by the use of structure tensors (Xiao et al, 2006), we suppose in this introductory article that the symmetry group of the material is also the symmetry group of the elasticity tensor.…”

Section: Damage Mechanicsmentioning

confidence: 99%

A damage model based on Kelvin eigentensors and Curie principle

François

2012

Mechanics of Materials

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A general anisotropic damage model is developed that accounts for the thermodynamics of irreversible processes in the framework of generalized standard materials and Kelvin tensor decomposition. Damage is described by fourth-rank tensors, one per eigenspace of the initial stiffness tensor. Their number thus ranges from two for an initially isotropic material to six for an initially triclinic material. The yield criteria are expressed in terms of a limiting energy for each eigenspace. The second-rank eigentensors (at most six) of the fourth-rank damage tensors define the direction of influence of the damage, while the associated eigenvalues characterize its intensity. These eigentensors evolve during loading, inducing an evolution of the symmetry group of the elastic tensor subject to the constraints of the Curie principle.

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“…It appears possible for many crystalline symmetry groups, although perhaps not all [80], to obtain an isotropic extension w E of Eq. (12) represented by an isotropic function of C and a set of vectors and second-rank tensors designated as structural tensors, which, to be definite, are assumed to be independent of C. To make the point, we focus attention on the special case of a single vector b…”

Section: Frommentioning

confidence: 99%

“…, according to Xiao et al [80]. Degeneracy (confluence of two eigenvalues) in one invariant subspace of A leads once again to transverse isotropy.…”

Section: Frommentioning

confidence: 99%

Continuum Modeling of Granular Media

Goddard

2014

Applied Mechanics Reviews

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This is a survey of the interesting phenomenology and the prominent regimes of granular flow, followed by a unified mathematical synthesis of continuum modeling. The unification is achieved by means of "parametric" viscoelasticity and hypoplasticity based on elastic and inelastic potentials. Fully nonlinear, anisotropic viscoelastoplastic models are achieved by expressing potentials as functions of the joint isotropic invariants of kinematic and structural tensors. These take on the role of evolutionary parameters or "internal variables," whose evolution equations are derived from the internal balance of generalized forces. The resulting continuum models encompass most of the mechanical constitutive equations currently employed for granular media. Moreover, these models are readily modified to include Cosserat and other multipolar effects. Several outstanding questions are identified as to the contribution of parameter evolution to dissipation; the distinction between quasielastic and inelastic models of material instability; and the role of multipolar effects in material instability, dense rapid flow, and particle migration phenomena.

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On isotropic extension of anisotropic constitutive functions via structural tensors

Cited by 7 publications

References 22 publications

Remarks on isotropic extension of anisotropic constitutive functions via structural tensors

Remarks on isotropic extension of anisotropic constitutive functions via structural tensors

A damage model based on Kelvin eigentensors and Curie principle

Continuum Modeling of Granular Media

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