Virtual spring damper method for nonholonomic robotic swarm self-organization and leader following

In this paper, we demonstrate a method for self-organization and leader following of nonholonomic robotic swarm based on spring damper mesh. By self-organization of swarm robots we mean the emergence of order in a swarm as the result of interactions among the single robots. In other words the self-organization of swarm robots mimics some natural behavior of social animals like ants among others. The dynamics of two-wheel robot is derived, and a relation between virtual forces and robot control inputs is defined in order to establish stable swarm formation. Two cases of swarm control are analyzed. In the first case the swarm cohesion is achieved by virtual spring damper mesh connecting nearest neighboring robots without designated leader. In the second case we introduce a swarm leader interacting with nearest and second neighbors allowing the swarm to follow the leader. The paper ends with numeric simulation for performance evaluation of the proposed control method

Ellipticity of gradient poroelasticity

We discuss the ellipticity properties of an enhanced model of poroelastic continua called dilatational strain gradient elasticity. Within the theory there exists a deformation energy density given as a function of strains and gradient of dilatation. We show that the equilibrium equations are elliptic in the sense of Douglis–Nirenberg. These conditions are more general than the ordinary and strong ellipticity but keep almost all necessary properties of equilibrium equations. In particular, the loss of the ellipticity could be considered as a criterion of a strain localization or material instability

Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions

The well-posedness of the boundary value problems for second gradient elasticity has been studied under the assumption of strong ellipticity of the dependence on the second placement gradients (see, e.g., Chambon and Moullet in Comput. Methods Appl. Mech. Eng. 193:2771–2796, 2004 and Mareno and Healey in SIAM J. Math. Anal. 38:103–115, 2006). The study of the equilibrium of planar pantographic lattices has been approached in two different ways: in dell’Isola et al. (Proc. R. Soc. Lond. Ser. A 472:20150, 2016) a discrete model was introduced involving extensional and rotational springs which is also valid in large deformations regimes while in Boutin et al. (Math. Mech. Complex Syst. 5:127–162, 2017) the lattice has been modelled as a set of beam elements interconnected by internal pivots, but the analysis was restricted to the linear case. In both papers a homogenized second gradient deformation energy, quadratic in the neighbourhood of non deformed configuration, is obtained via perturbative methods and the predictions obtained with the obtained continuum model are successfully compared with experiments. This energy is not strongly elliptic in its dependence on second gradients. We consider in this paper also the important particular case of pantographic lattices whose first gradient energy does not depend on shear deformation: this could be considered either a pathological case or an important exceptional case (see Stillwell et al. in Am. Math. Mon. 105:850–858, 1998 and Turro in Angew. Chem., Int. Ed. Engl. 39:2255–2259, 2000). In both cases we believe that such a particular case deserves some attention because of what we can understand by studying it (see Dyson in Science 200:677–678, 1978). This circumstance motivates the present paper, where we address the well-posedness of the planar linearized equilibrium problem for homogenized pantographic lattices. To do so: (i) we introduce a class of subsets of anisotropic Sobolev’s space as the most suitable energy space E relative to assigned boundary conditions; (ii) we prove that the considered strain energy density is coercive and positive definite in E; (iii) we prove that the set of placements for which the strain energy is vanishing (the so-called floppy modes) must strictly include rigid motions; (iv) we determine the restrictions on displacement boundary conditions which assure existence and uniqueness of linear static problems. The presented results represent one of the first mechanical applications of the concept of Anisotropic Sobolev space, initially introduced only on the basis of purely abstract mathematical considerations

On time-dependent nonlinear dynamic response of micro-elastic solids

A new approach to the mechanical response of micro-mechanic problems is presented using the modified couple stress theory. This model captured micro-turns due to micro-particles' rotations which could be essential for microstructural materials and/or at small scales. In a micro media based on the small rotations, sub-particles can also turn except the whole domain rotation. However, this framework is competent for a static medium. In terms of dynamic investigations of micro materials, it is required to involve micro-rotations' mass inertias. This fact persuades us to pay particular attention to the micro mechanics' samples and directed us to re-derive the modified couple stress model to propose and represent a new micro-mechanic approach which is well-deserved, especially for dynamic studies of microstructures. In carrying out this job, the classical beam has provided the basic form of formulation procedure. The continuum medium has been limited to a square flat non-porous beam deducing a homogeneous isotropic micromaterial. As long as the time-dependent results are concerned due to studying micro-mass inertia in time history, there would be two solution steps. The Galerkin decomposition technique is imposed in accord with an analytical postulate to issue the algebraic problem distributing time-dependent equations. The latter, the Homotopy perturbation method delivers time-dependent outcomes. The solution methods have been validated by building numerical models in Abaqus software. On the new achievements of this study, one can declare that both static and dynamic length scale parameters are very effective in order to study vibrations of microstructures. If the values of these characteristic lengths are considerable, the nonlinear frequency analysis will be essential. Furthermore, the stiffness of the structure will be higher if the values of both length scale parameters increase

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

AbstractWe 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

On the well posedness of static boundary value problem within the linear dilatational strain gradient elasticity

AbstractIn this paper, it is proven an existence and uniqueness theorem for weak solutions of the equilibrium problem for linear isotropic dilatational strain gradient elasticity. Considered elastic bodies have as deformation energy the classical one due to Lamé but augmented with an additive term that depends on the norm of the gradient of dilatation: only one extra second gradient elastic coefficient is introduced. The studied class of solids is therefore related to Korteweg or Cahn–Hilliard fluids. The postulated energy naturally induces the space in which the aforementioned well-posedness result can be formulated. In this energy space, the introduced norm does involve the linear combination of some specific higher-order derivatives only: it is, in fact, a particular example of anisotropic Sobolev space. It is also proven that aforementioned weak solutions belongs to the space $$H^1(div,V)$$ H 1 ( d i v , V ) , i.e. the space of $$H^1$$ H 1 functions whose divergence belongs to $$H^1$$ H 1 . The proposed mathematical frame is essential to conceptually base, on solid grounds, the numerical integration schemes required to investigate the properties of dilatational strain gradient elastic bodies. Their energy, as studied in the present paper, has manifold interests. Mathematically speaking, its singularity causes interesting mathematical difficulties whose overcoming leads to an increased understanding of the theory of second gradient continua. On the other hand, from the mechanical point of view, it gives an example of energy for a second gradient continuum which can sustain externally applied surface forces and double forces but cannot sustain externally applied surface couples. In this way, it is proven that couple stress continua, introduced by Toupin, represent only a particular case of the more general class of second gradient continua. Moreover, it is easily checked that for dilatational strain gradient continua, balance of force and balance of torques (or couples) are not enough to characterise equilibrium: to this aim, externally applied surface double forces must also be specified. As a consequence, the postulation scheme based on variational principles seems more suitable to study second gradient continua. It has to be remarked finally that dilatational strain gradient seems suitable to model the experimentally observed behaviour of some material used in 3D printing process

Workshop “Micropolar continua and beyond”

We present a review of the recent workshop “Micropolar Continua and beyond” which held in March 28–31, 2023, at Technische University of Berlin, Germany

Anti-plane shear waves in an elastic strip rigidly attached to an elastic half-space

We consider the anti-plane shear waves in a domain consisting of an infinite layer with a thin coating lying on an elastic half-space. The elastic properties of the coating, layer, and half-space are assumed to be different. On the free upper surface we assume the compatibility condition within the Gurtin–Murdoch surface elasticity, whereas at the plane interface we consider perfect contact. For this problem there exist two possible regimes related to waves exponentially decaying in the half-space. The first one, called transversely exponential–transversely exponential (TE–TE) regime, is related to waves described by exponential in transverse direction functions; the second, transversely harmonic–transversely exponential (TH–TE) regime, corresponds to waves in the upper layer which have the harmonic behaviour in the transverse direction. Detailed analysis of the derived dispersion equations for both regimes is provided. In particular, the effects of surface stresses, the layer thickness as well as of the ratio of shear moduli of the upper layer and half-space on the dispersion curves is analysed