Search

We discuss ellipticity property within the linear couple-stress elasticity. In this theory, there exists a deformation energy density introduced as a function of strains and gradient of macrorotations, where the latter are expressed through displacements. So the couple-stress theory could be treated as a particular class of strain gradient elasticity. Within the micropolar elasticity, the model is called Cosserat pseudocontinuum or medium with constrained rotations. Applying the classic definitions of ordinary ellipticity and strong ellipticity to static equations of the couple-stress theory, we conclude that these equations are neither elliptic nor strongly elliptic. As a result, one should be aware of extending properties of full strain gradient models such as Toupin–Mindlin strain gradient elasticity to models with incomplete set of second derivatives. [ABSTRACT FROM AUTHOR]

In this paper, we discuss well-posedness of the boundary-value problems arising in some "gradient-incomplete" strain-gradient elasticity models, which appear in the study of homogenized models for a large class of metamaterials whose microstructures can be regarded as beam lattices constrained with internal pivots. We use the attribute "gradient-incomplete" strain-gradient elasticity for a model in which the considered strain energy density depends on displacements and only on some specific partial derivatives among those constituting displacements first and second gradients. So, unlike to the models of strain-gradient elasticity considered up-to-now, the strain energy density which we consider here is in a sense degenerated, since it does not contain the full set of second derivatives of the displacement field. Such mathematical problem was motivated by a recently introduced new class of metamaterials (whose microstructure is constituted by the so-called pantographic beam lattices) and by woven fabrics. Indeed, as from the physical point of view such materials are strongly anisotropic, it is not surprising that the mathematical models to be introduced must reflect such property also by considering an expression for deformation energy involving only some among the higher partial derivatives of displacement fields. As a consequence, the differential operators considered here, in the framework of introduced models, are neither elliptic nor strong elliptic as, in general, they belong to the class so-called hypoelliptic operators. Following (Eremeyev et al. in J Elast 132:175–196, 2018. 10.1007/s10659-017-9660-3) we present well-posedness results in the case of the boundary-value problems for small (linearized) spatial deformations of pantographic sheets, i.e., 2D continua, when deforming in 3D space. In order to prove the existence and uniqueness of weak solutions, we introduce a class of subsets of anisotropic Sobolev's space defined as the energy space E relative to specifically assigned boundary conditions. As introduced by Sergey M. Nikolskii, an anisotropic Sobolev space consists of functions having different differential properties in different coordinate directions. [ABSTRACT FROM AUTHOR]

We discuss the material symmetry group of the micropolar continuum and related consistently simplified constitutive equations. Following Eremeyev and Pietraszkiewicz (Int J Solid Struct 2012; 49: 1993–2005; Generalized continua as models for materials, Heidelberg: Springer, 2013, 77–90) we extend the definition of the group proposed by Eringen and Kafadar (Continuum physics, vol. 4, New York, NY: Academic Press, 1976, 1–75) by taking into account the microstructure curvature tensor as well as different transformation properties of polar (true) and axial (pseudo) tensors. Our material symmetry group consists of ordered triples of tensors which make the strain energy density of the micropolar continuum invariant under change of the reference placement. Within micropolar solids we discuss the isotropic, hemitropic, orthotropic, transversely isotropic and clinotropic materials and give explicitly the consistently reduced representations of the strain energy density. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]