Abstract
For micropolar media we present a new definition of the local material symmetry group considering invariant properties of the both kinetic energy and strain energy density under changes of a reference placement. Unlike simple (Cauchy) materials, micropolar media can be characterized through two kinematically independent fields, that are translation vector and orthogonal microrotation tensor. In other words, in micropolar continua we have six degrees of freedom (DOF) that are three DOFs for translations and three DOFs for rotations. So the corresponding kinetic energy density nontrivially depends on linear and angular velocity. Here we define the local material symmetry group as a set of ordered triples of tensors which keep both kinetic energy density and strain energy density unchanged during the related change of a reference placement. The triples were obtained using transformation rules of strain measures and microinertia tensors under replacement of a reference placement. From the physical point of view, the local material symmetry group consists of such density-preserving transformations of a reference placement, that cannot be experimentally detected. So the constitutive relations become invariant under such transformations. Knowing a priori a material’s symmetry, one can establish a simplified form of constitutive relations. In particular, the number of independent arguments in constitutive relations could be significantly reduced.
Highlights
The model of micropolar medium was originally introduced by Cosserat brothers [1]
The definition of the material symmetry group for micropolar materials was proposed by Eringen and Kafadar [20] who included a single microinertia tensor as an argument of constitutive relations
As in case of rigid body dynamics, for micropolar media a kinetic energy density has more general form than for simple materials, its symmetry should be included in a symmetry characterization of a micropolar material
Summary
The model of micropolar medium was originally introduced by Cosserat brothers [1]. Recently, it founds various applications for modelling of media with complex inner microstructure such as granular [2,3], porous [4,5], composite [6,7] materials, masonries [8,9], bones [10,11], textiles and beam-lattices [12,13,14,15,16], suspensions [17,18,19], see [20,21,22,23,24]. The definition of the material symmetry group for micropolar materials was proposed by Eringen and Kafadar [20] who included a single microinertia tensor as an argument of constitutive relations. In micropolar dynamics the form and possible symmetries of a kinetic energy density may play an important role. As in case of rigid body dynamics, for micropolar media a kinetic energy density has more general form than for simple materials, its symmetry should be included in a symmetry characterization of a micropolar material. The aim of this paper is to consider a general form of kinetic constitutive equation, i.e., a dependence of a kinetic energy density, and include its symmetries into the definition of the material local symmetry group. We discuss the further applications of the proposed approach to other generalized media such as strain-gradient elasticity and micromorphic media
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