Material symmetry group of the non-linear polar-elastic continuum

By Victor A. Eremeyev, W. Pietraszkiewicz. 2012

134 citations recorded by the source corpus.

We extend the material symmetry group of the non-linear polar-elastic continuum by taking into account microstructure curvature tensors as well as different transformation properties of polar and axial tensors. The group consists of an ordered triple of tensors which makes the strain energy density of polar-elastic continuum invariant under change of reference placement. An analog of the Noll rule is established. Four simple specific cases of the group with corresponding reduced forms of the strain energy density are discussed. Definitions of polar-elastic fluids, solids, liquid crystals and subfluids are given in terms of members of the symmetry group. Within polar-elastic solids we discuss in more detail isotropic, hemitropic, cubic-symmetric, transversely isotropic, and orthotropic materials and give explicitly corresponding reduced representations of the strain energy density. For physically linear polar-elastic solids, when the density becomes a quadratic function of strain measures, reduced representations of the density are established for monoclinic, orthotropic, cubic-symmetric, hemitropic and isotropic materials in terms of appropriate joint scalar invariants of stretch, wryness and undeformed structure curvature tensors.

Subfields: Biomedical Engineering, Materials Chemistry, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/d99396b0f6246abaebb5e352e8ffd941def95fd3

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In Biomedical Engineering

In Materials Chemistry

In Mechanics of Materials

Citations in the corpus

Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence (2015, methodology): “For other types of fiber configurations, the symmetry analysis performed in [22,23,25] may be useful for construction of the 2D or 3D strain energy densities within the framework of the generalized models of continua with directors.”
Material symmetry group and constitutive equations of micropolar anisotropic elastic solids (2016, methodology): “Extending the results of [28, 30] we present the lists of additional joint invariants appearing inW describing the monoclinic, orthotropic and transversely isotropic micropolar solids.”
A unifying perspective: the relaxed linear micromorphic continuum (2013, background): “Recently, the theory of material symmetry for the Cosserat continuum was extended in [148].”
Continuum and discrete models for structures including (quasi-) inextensible elasticae with a view to the design and modeling of composite reinforcements (2015, methodology): “Moreover, the most general constitutive modeling of 2D anisotropic structures made of flexible nets of fibers should rely on the methods based on the material symmetry groups (see e.g. Bouchet and Brechet, 2009; Eremeyev and Pietraszkiewicz, 2006; Eremeyev and Pietraszkiewicz, 2012).”
Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models (2016, background): “Some future developments concern contact problem between fibers, see, for instance, [82–84]; although stability problem is not explicitly considered in this paper, the extension to these cases could be based on the following papers [85–90] while pantographic structures with initially curved beams ca”