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Irreducible representations for constitutive equations of anisotropic solids III: crystal and quasicrystal classes D[2m+1h] and D[2md]

Xiao H., Bruhns O., Meyers A.

Archives of Mechanics

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2000

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Vol. 52, nr 3

347-395

EN

A simple, unified procedure is applied to derive irreducible nonpolynomial representations for scalar-, vector-, skewsymmetric and symmetric second order tensor-valued anisotropic constitutive equations involving any finite number of vector variables and second order tensor variables. In this part, our concern is for the crystal classes and quasicrystal classes D[2m+1h] and D[2md] for all integers m>=1.

2

Irreducible representations for constitutive equations of anisotropic solids II: crystal and quasicrystal classes D[2m+1d], D[2m+1] and C[2m+1v]

Xiao H., Bruhns O., Meyers A.

Archives of Mechanics

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2000

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Vol. 52, nr 1

55-88

EN

A simple, unified procedure is applied to derive irreducible nonpolynomial representations for scalar-, vector-, skewsymmetric and symmetric second order tensor-valued anisotropic constitutive equations involving any finite number of vector variables and second order tensor variables. In this part, our concern is for all crystal classes and quasicrystal classes D[2m+1d], D[2m+1] and C[2m+1v] for all integers m >= l.

3

Hencky's elasticity model with the logarithmic strain measure : a study on Poynting effect and stress response in torsion of tubes and rods

Bruhns O., Xiao H., Meyers A.

Archives of Mechanics

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2000

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Vol. 52, nr 4-5

489-509

EN

Hencky's elasticity model is a finite strain elastic constitutive equation derived by replacing the infinitesimal strain measure in the classical strain-energy function of infinitesimal isotropic elasticity with Hencky's logarithmic strain measure. ANAND [1, 2] has demonstrated that, with only the two classical Lame elastic constants measurable at infinitesimal strains, predictions of the just-mentioned simple model for a wide class of materials for moderately large deformations may be in better agreement with experimental data than other known finite elasticity models. The deformation modes considered in Anand's work are simple tension and compression, simple shear, and simple torsion and combined extension-torsion of solid cylinders, etc. Here, we indicate some remarkable properties of this Hencky model and, mainly, we investigate the large deformation responses of this model in torsion of cylindrical tubes and rods with free ends. It is noticeable that, with only the material properties measurable at infinitesimal strains, the Hencky model can predict the just-mentioned second order effects, in particular the Poynting effect, and its predictions are in good accord with experiments reported in the literature.

4

Irreducible representations for constitutive equations of anisotropic solids I: crystal and quasicrystal classes D[2mh], D[2m] and G[2mv]

Xiao H., Bruhns O., Meyers A.

Archives of Mechanics

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1999

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Vol. 51, nr 5

559-603

EN

A simple, unified procedure is applied to derive irreducible nonpolynomial representations for scalar-, vector-, skewsymmetric and symmetric second order tensor-valued anisotropic constitutive equations involving any finite number of vector variables and second order tensor variables. In the paper consisting of three parts, we consider all kinds of material symmetry groups as subgroups of the cylindrical group D[infinity]h. This paper, together with a previous work, covers all kinds of material symmetric groups of solids, except for the five cubic crystal classes and the two icosahedral quasicrystal classes. In this part, our concern is with all crystal classes and quasicrystal classes D[2mh], D[2m] and G[2mv] for all integers m [is greater than or equal to] 2.

5

On symmetric tensor-valued isotropic functions of a symmetric tensor and a skewsymmetric tensor and related transversely isotropic functions

Xiao H., Bruhns O., Meyers A.

Archives of Mechanics

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1998

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Vol. 50, nr 4

731-741

EN

A new generating set consisting of seven polynomial tensor generators is presented for symmetric second order tensor-valued isotropic functions of a symmetric second order tensor and a skewsymmetric second order tensor. It is smaller than the existing corresponding generating set consisting of eight tensor generators and shown to be minimal in all possible generating sets consisting of homogeneous polynomial tensor generators. This result indicates that the well-known results for isotropic functions may be sharpened. In addition, from the presented results a minimal generating set consisting of six tensor generators is derived for the symmetric second order tensor-valued transversely isotropic functions of a symmetric second order tensor relative to the transverse isotropy group C [...] consisting of all rotations about a fixed axis.

6

Objective corotational rates and unified work-conjugacy relation between Eulerian and Lagrangean strain and stress measures

Xiao H., Bruhns O., Meyers A.

Archives of Mechanics

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1998

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Vol. 50, nr 6

1015-1045

EN

By virtue of objective corotational rates and related corotating frames, a unified work-conjugacy relation between Eulerian and Lagrangean strain and stress measures is established, which is a natural extension of Hill's work-conjugacy relation between Lagrangean strain and stress measures. It turns out that the latter is the particular case of the former one when a corotating frame with the well-known spin ------ is concerned, where R is the rotation tensor defined by the polar decomposition of the deformation gradient. The work-conjugate stress measure of an arbitrary Hill's strain measure (either Eulerian or Lagrangean) with regard to any kind of objective corotational rate is determined in the seanse of the introduced unified work-conjugacy relation. The result is presented both in the principal component form and explicit basis-free form valid for all cases of the principal stretches. In particular, the intrinsic, unique relationship between Hencky's logarithmic strain measures --------------- and the fundamental mechanical quantities, i.e. the Eulerian and Lagrangean stretching tensors D and ------- and Eulerian and Lagrangean Kirchhoff stress measures -- and --------- are disclosed. It is shown that there are objective corotational rates of --------- that are identical with the Eulerian and Lagrangean stretching tensors D and ------- respectively, and further that only --------- enjoy the just-stated favourable properties. As a result, the two pairs of strain and stress measures, (-------) and (----------), form a work-conjugate Eulerian strain-stress pair and a work-conjugate Lagrangean strain-stress pair, respectively, in the sense of the introduced work conjugacy relation. Finally, application of the unified work-conjugacy notion in formulating the rate - type constitutive relations is indicated.