Micropolar Elasticity Research Articles

Topology optimization of link mechanisms for comprehensive synthesis of component arrangement and structure using micropolar elasticity model

This paper proposes a method for topology optimization of link mechanisms with multiple outputs, using a multi-material micropolar elasticity model. This approach allows for comprehensive optimization of both the arrangement and structure of the link mechanism components. By utilizing a continuum model that incorporates micropolar elasticity, we can specify bending stiffness independently from tensile stiffness, resulting in deformation characteristics that approximate a link mechanism. The optimization problem of designing link mechanisms for multiple outputs is reformulated as a boundary value problem within this model framework. The design goal is to synthesize a link mechanism that not only follows a desired path but also possesses the required degrees of freedom. To achieve this, the objective function is defined by the displacement error under external force and the strain energy in the links. The multi-material micropolar elasticity model is then optimized through a gradient-based optimization method, focusing on this objective function. The effectiveness and applicability of our methodology are demonstrated through several numerical case studies.

Hard interfaces with microstructure: the cases of strain gradient elasticity and micropolar elasticity.

As the size of a layered structure scales down, the adhesive layer thickness correspondingly decreases from macro- to micro-scale. The influence of the material microstructure of the adhesive becomes more pronounced, and possible size effect phenomena can appear. This paper describes the mechanical behaviour of composites made of two solids, bonded together by a thin layer, in the framework of strain gradient and micropolar elasticity. The adhesive layer is assumed to have the same stiffness properties as the adherents. By means of the asymptotic methods, the contact laws are derived at order 0 and order 1. These conditions represent a formal generalization of the hard elastic interface conditions. A simple benchmark equilibrium problem (a three-layer composite micro-bar subjected to an axial load) is developed to numerically assess the asymptotic model. Size effects and non-local phenomena, owing to high strain concentrations at the edges, are highlighted. The example proves the efficiency of the proposed approach in designing micro-scale-layered devices.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

The Nye Cells and Figures for Athermic Hemitropic, Isotropic, and Ultraisotropic Micropolar Elastic Solids

The Nye Cells and Figures for Athermic Hemitropic, Isotropic, and Ultraisotropic Micropolar Elastic Solids

Finite deformation micropolar peridynamic theory: Variational consistency of wryness measure

This paper concerns developments in both classical and peridynamic (PD) finite deformation micropolar elasticity theory. The first part presents an alternative proof of a theorem that demonstrates variational consistency of the wryness measure in classical finite deformation micropolar elasticity which is one of the novel contributions of this work. The linearized version of the proposed wryness measure is compared with the wryness measure of small deformation micropolar elasticity. In the second part of this work, a finite deformation micropolar PD theory is presented. After proposing an additional integro-differential equation along with the standard PD equation of motion, the global balance of angular momentum is proved to be satisfied. The balance of virtual work is derived for the micropolar PD theory. Next, constitutive correspondence approach is used to relate PD force and moment states with their classical counterparts. The nonlocal versions of the strain and the proposed wryness measure are used in the constitutive correspondence. The constitutive equations of the classical micropolar theory are presented and how they can be incorporated in PD framework is discussed. We introduce a new bond breaking criterion for PD micropolar materials based on critical stretch and critical relative rotation. Quasi-static numerical simulations on deformations of plate with a hole and fracture of a double notched sheet are presented. Dynamic simulations concern wave propagation through a solid specimen and plate with a hole. Validation of the simulation results with the finite element solutions, boundary element solutions, and experimental observations demonstrate the potential of our approach.

Propagation of Elastic Waves along a Cylindrical Rod of Micropolar Elastic Solid with Stretch under Impedance Boundary Conditions

Objective: The objective of this article is to derive the dispersion relations of elastic waves along a cylindrical rod of homogeneous, isotropic micropolar elastic solid with stretch under impedance boundary conditions. Method: The basic equations and constitutive relations are converted into cylindrical coordinates and solved by adopting the method of plane harmonic solution. Findings: Displacements, micro rotations, and micro stretch relations are derived analytically. The dispersion relations pertaining to elastic waves with using impedance boundary conditions also derived analytically and these relations are compared with the results of Tomar. Novelty: With the use of the MATLAB program, the effects of impedance parameters, micropolarity and stretch are discussed for a particular model. The speed of the elastic wave is proportional to the ratio of the impedance parameter in microstretch elastic solids, but there is no significant effect of the impedance ratio in micropolar elastic solids. Keywords: Elastic Waves, Cylindrical rod, Micropolarity, Micro stretch, Impedance Boundary Conditions

Continuum-molecular modeling of planar micropolar media: Anisotropy, chiral properties and length-scale effects

This paper presents a continuum-molecular formulation for bi-dimensional micropolar media within the mathematical formalism of a revised peridynamic theory with oriented material points. A variational procedure is considered to derive the fundamental governing equations of the model, which postulates that material points interact through pair potentials allowing generalized pairwise actions to be derived as energy conjugates to properly defined pairwise deformation measures. While continuity of mass is assumed, constitutive laws are produced for long-range micro-interactions to reproduce the effective behavior of the material at the macro-scale. The definition of proper micromoduli functions respecting material symmetries and invariance properties allows then to obtain a non-local micropolar continuum based on pseudo-discrete kinematics, while providing a mechanism-based description of material anisotropy and coupling behaviors uncovered by classical elasticity. An analytic micro-macro moduli correspondence procedure is also established, based on the formal analogy with the constitutive tensor of centrosymmetric planar micropolar continua. As an important result of this research, we show that distinctive chiral effects of micropolar elasticity can be reproduced by introducing a directional independent pseudo-scalar pair potential, which turns out to be analytically vanishing when the rotationally invariant part of the corresponding continuum elastic tensor is invariant to mirror reflections as well. Moreover, the proposed formulation demonstrates sensitivity to elastic bending size-effect as specific property of structured materials homogenized as micropolar continua, and related to the characteristic length of their underlying microstructure. The resulting constitutive non-locality, which plays an important role in fracture problems, is conceptually separated with respect to the intrinsic non-local character of the model related to the integral nature of its governing equations, and then does not vanish when the horizon reduces to zero. The theoretical findings and the effectiveness of the model are successfully verified through illustrative examples referred to representative cases of structured materials homogenized as micropolar continua, including length-scale dependent quasi-static crack propagation as well as the mechanism-based description of the coupling between bulk strain and micro-rotation in elastic bi-dimensional homogenized chiral lattices.

On micropolar elastic foundations

The modelling of heterogeneous and architected materials poses a significant challenge, demanding advanced homogenisation techniques. However, the complexity of this task can be considerably simplified through the application of micropolar elasticity. Conversely, elastic foundation theory is widely employed in fracture mechanics and the analysis of delamination propagation in composite materials. This study aims to amalgamate these two frameworks, enhancing the elastic foundation theory to accommodate materials exhibiting micropolar behaviour. Specifically, we present a novel theory of elastic foundation for micropolar materials, employing stress potentials formulation and a unique normalisation approach. Closed-form solutions are derived for stress and couple stress reactions inherent in such materials, along with the associated restoring stiffness. The validity of the proposed theory is established through verification using the double cantilever beam configuration. Concluding our study, we elucidate the benefits and limitations of the developed theory by quantifying the derived parameters for materials known to exhibit micropolar behaviour. This integration of micropolar elasticity into the elastic foundation theory not only enhances our understanding of material responses but also provides a versatile framework for the analysis of heterogeneous materials in various engineering applications.

Symplectic solutions of the plane annular sectors in micropolar elasticity

The symplectic approach is utilized to derive the solutions of the plane annular sector in micropolar elasticity. According to the Hellinger-Reissner variational principle, the Hamiltonian canonical equations are obtained for the plane problem in an annular sectorial region. Through the method of separation of variables, we obtain an eigenvalue equation of the Hamiltonian operator matrix which is a system of linear differential equations with variable coefficients. By the method of series solutions, its general solutions can be expressed as the linear combination of Bessel functions and power functions. The corresponding eigensolutions for three types of homogeneous boundary conditions are derived. According to the adjoint symplectic orthogonality of the eigensolutions, the solution of the micropolar plane annular sector is expressed as a series expansion of eigensolutions. Numerical analyses are presented to demonstrate the validity of the proposed method and confirm its ability to accurately capture the size effect.

Band gap of shear horizontal waves for one-dimensional phononic crystals with chiral materials.

Extensive applications of chiral lattice structures in the field of acoustic wave manipulation and vibration modulation show the effectiveness of chirality route to the design of phononic crystals. However, how and to what extent the material chirality affects the band gap properties of phononic crystals remains unclear. In this study, one-dimensional phononic crystals made of chiral materials is proposed, and a theoretical model of shear horizontal (SH) wave propagation in the chiral phononic crystals is developed based on the noncentrosymmetric micropolar elasticity theory. Through the transfer matrix method, the dispersion relationship of SH wave propagation is obtained and the effects of material chirality on the band-gap properties are investigated. Our work demonstrates that the change of material chirality can significantly affect the dispersion relationship of phononic crystals, leading to the wide band gap and low frequency. In a unit cell, when the chiral coefficients of the two parts have opposite signs but the same magnitude and the chiral directions are consistent with the vibrational direction, it is the most favorable for the phononic crystals to achieve the lowest frequency and widest band gap. This study suggests that the material chirality can be harnessed to effectively tune the band-gap properties of phononic crystals. The present study provides insight for the chirality route to the design of phononic crystals.

Application of the Reduced and Selected Integration Method in Problems of Micropolar Elasticity Theory

Application of the Reduced and Selected Integration Method in Problems of Micropolar Elasticity Theory

Применение метода редуцированного и селективного интегрирования в задачах микрополярной теории упругости

В рамках теории микрополярного континуума с использованием вариационного принципа Лагранжа, метода Ритца и кусочно-полиномиальных базисных функций трилинейного типа составлена схема редуцированного и селективного интегрирования в тензорно-блочной форме для исключения эффекта запирания в почти несжимаемых изотропных средах с центром симметрии.

Об одном способе построения фигурная в асимметричных теориях демитропной микрополярной упругости

В настоящей работе процедура построения двумерных фигур Ная модифицируется для асимметричных матриц и применяется для представления определяющих псевдотензоров демитропных микрополярных упругих континуумов. Указанные матричные представления используются для упрощения тензорной записи уравнений анизотропных тел. Метод матричного представления Ная позволяет изобразить тензоры и псевдотензоры четвертого и второго рангов в виде своеобразных двумерных фигур. Получена матричная форма асимметричных определяющих уравнений демитропного микрополярного упругого тела. Основное изложение статьи проводится в декартовой прямоугольной системе координат в терминах инвариантного элемента объема. In this work, the process of constructing two-dimensional figures is modified for asymmetric matrices and is used to represent the defining pseudotensors of demitropic micropolar elastic continua. The specified matrix representations are used to simplify the tensor representation of the material of anisotropic bodies. This method allows us to represent tensors and pseudotensors of the fourth and second ranks in the form of peculiar two-dimensional figures. A matrix form is obtained that defines the basis of a demitropic micropolar elastic body. The main presentation of the article is carried out in a Cartesian rectangular coordinate system.

Odd Cosserat elasticity in active materials.

Stress-strain constitutive relations in solids with an internal angular degree of freedom can be modeled using Cosserat (also called micropolar) elasticity. In this paper, we explore Cosserat materials that include chiral active components and hence odd elasticity. We calculate static elastic properties and show that the static response to rotational stresses leads to strains that depend on both Cosserat and odd elasticity. We compute the dispersion relations in odd Cosserat materials in the overdamped regime and find the presence of exceptional points. These exceptional points create a sharp boundary between a Cosserat-dominated regime of complete wave attenuation and an odd-elasticity-dominated regime of propagating waves. We conclude by showing the effect of Cosserat and odd-elasticity terms on the polarization of Rayleigh surface waves.

Two-Dimensional Nye Figures for Some Micropolar Elastic Solids

The paper is devoted to problems related to two-dimensional Nye figures for micropolar continua. The representation technique known from studies on crystallography for 4th, 3rd and 2nd rank tensors is employed by two-dimensional matrices, supplied by relationships between their elements. Such representations are commonly used to simplify a script of the equations of athermic micropolar elastic solids. This method allows us to graphically represent micropolar constitutive tensors and pseudotensors in the form of specific two-dimensional blocks. The Nye figures for the micropolar elastic solids are obtained. The Nye figures enables us to discriminate anisotropic solids and figure out general anisotropic, hemitropic, isotropic, and ultraisotropic continua.

Micropolar Elasticity in Physically-Based Animation

We explore micropolar materials for the simulation of volumetric deformable solids. In graphics, micropolar models have only been used in the form of one-dimensional Cosserat rods, where a rotating frame is attached to each material point on the one-dimensional centerline. By carrying this idea over to volumetric solids, every material point is associated with a microrotation, an independent degree of freedom that can be coupled to the displacement through a material's strain energy density. The additional degrees of freedom give us more control over bending and torsion modes of a material. We propose a new orthotropic micropolar curvature energy that allows us to make materials stiff to bending in specific directions. For the simulation of dynamic micropolar deformables we propose a novel incremental potential formulation with a consistent FEM discretization that is well suited for the use in physically-based animation. This allows us to easily couple micropolar deformables with dynamic collisions through a contact model inspired from the Incremental Potential Contact (IPC) approach. For the spatial discretization with FEM we discuss the challenges related to the rotational degrees of freedom and propose a scheme based on the interpolation of angular velocities followed by quaternion time integration at the quadrature points. In our evaluation we validate the consistency and accuracy of our discretization approach and demonstrate several compelling use cases for micropolar materials. This includes explicit control over bending and torsion stiffness, deformation through prescription of a volumetric curvature field and robust interaction of micropolar deformables with dynamic collisions.

Observation of Chirality‐Induced Roton‐Like Dispersion in a 3D Micropolar Elastic Metamaterial

AbstractA theoretical paper based on chiral micropolar effective‐medium theory suggested the possibility of unusual roton‐like acoustical‐phonon dispersion relations in 3D elastic materials. Here, as a first novelty, the corresponding inverse problem is solved, that is, a specific 3D chiral elastic metamaterial structure is designed, the behavior of which follows this effective‐medium description. The metamaterial structure is based on a simple‐cubic lattice of cubes, each of which not only has three translational but also three rotational degrees of freedom. The additional rotational degrees of freedom are crucial within micropolar elasticity. The cubes and their degrees of freedom are coupled by a chiral network of slender rods. As a second novelty, this complex metamaterial is manufactured in polymer form by 3D laser printing and its behavior is characterized experimentally by phonon‐band‐structure measurements. The results of these measurements, microstructure finite‐element calculations, and solutions of micropolar effective‐medium theory are in good agreement. The roton‐like dispersion behavior of the lowest phonon branch results from two aspects. First, chirality splits the transverse acoustical branches as well as the transverse optical branches. Second, chirality leads to an ultrastrong coupling and hybridization of chiral acoustical and optical phonons at finite wavevectors.

Anisotropy can imply exponential decay in micropolar elasticity

In this note, two problems arisen in micropolar elasticity are considered from the analytical point of view. Following the Kelvin–Voigt theory of micropolar viscoelasticity, two dissipative mechanisms are imposed: in the first problem, it is defined on the microscopic structure and, for the second problem, on the macroscopic structure. Then, an existence and uniqueness result, as well as an exponential energy decay, are proved for the first problem. Since similar arguments can be used for the second problem, only the main key points are commented.

Cosserat micropolar elasticity: classical Eringen vs. dislocation form

Cosserat micropolar elasticity: classical Eringen vs. dislocation form

A Study of Thermo-Mechanical Interactions in the Rotating Micropolar Elastic Solid with Two Temperatures Using Memory-Dependent Derivative

A Study of Thermo-Mechanical Interactions in the Rotating Micropolar Elastic Solid with Two Temperatures Using Memory-Dependent Derivative

On the Variational Principle of Lagrange of the Micropolar Elasticity Theory in the Case of Orthotropic Medium

On the Variational Principle of Lagrange of the Micropolar Elasticity Theory in the Case of Orthotropic Medium