Micropolar Theory Research Articles

A micropolar damage model for size-dependent concrete fracture problems and crack propagation simulated by PDDO method

Lots of experiments observe the size-dependent phenomena of concrete mechanical behaviors. In this paper, the micropolar theory is adopted to describe the size effects by introducing two size-related parameters into their constitutive equations, named coupling number N and characteristic length l. A micropolar damage model is built for size-dependent concrete fracture problems utilizing the bond-based peridynamic (PD) idea. That is, the material damage and fracture behaviors are determined by the local damage factors of the PD bonds. Then, the tensile strength of a concrete specimen is numerically estimated by the PD differential operator (PDDO) method. The influences of the size parameters N and l on the tensile strength are studied. By comparing with the transformed Bažant size-effect law and the fitting error analysis, the reasonable values of N and l are determined for a certain reinforced concrete composite. Finally, by using the determined micropolar damage model, the crack propagation paths in concrete members are numerically simulated.

A Hierarchical Nano to Micro Scale Modelling of 3D Printed Nano-Reinforced Polylactic Acid: Micropolar Modelling and Molecular Dynamics Simulation.

Fused deposition modelling (FDM) is an additive manufacturing technique widely used for rapid prototyping. This method facilitates the creation of parts with intricate geometries, making it suitable for advanced applications in fields such as tissue engineering, aerospace, and electronics. Despite its advantages, FDM often results in the formation of voids between the deposited filaments, which can compromise mechanical properties. However, in some cases, such as the design of scaffolds for bone regeneration, increased porosity can be advantageous as it allows for better permeability. On the other hand, the introduction of nano-additives into the FDM material enhances design flexibility and can significantly improve the mechanical properties. Therefore, modelling FDM-produced components involves complexities at two different scales: nanoscales and microscales. Material deformation is primarily influenced by atomic-scale phenomena, especially with nanoscopic constituents, whereas the distribution of nano-reinforcements and FDM-induced heterogeneities lies at the microscale. This work presents multiscale modelling that bridges the nano and microscales to predict the mechanical properties of FDM-manufactured components. At the nanoscale, molecular dynamic simulations unravel the atomistic intricacies that dictate the behaviour of the base material containing nanoscopic reinforcements. Simulations are conducted on polylactic acid (PLA) and PLA reinforced with silver nanoparticles, with the properties derived from MD simulations transferred to the microscale model. At the microscale, non-classical micropolar theory is utilised, which can account for materials' heterogeneity through internal scale parameters while avoiding direct discretization. The developed mechanical model offers a comprehensive framework for designing 3D-printed PLA nanocomposites with tailored mechanical properties.

Contact formulations for analysis of micropolar media with finite continuum beam elements

Microscale beam-like structures are standard components of micromechanical systems in many devices. However, the small dimensionality affects their deformation characteristics and leads to misinterpretations of the results. Presenting such size dependency is possible with advanced continuum descriptions via introducing additional variables compared to the classical one. Hence, the problems where contact occurs require the readjustment of boundary conditions considering these extra parameters. That burdens the already challenging task of contact problem calculations and restricts most demonstration examples to two-dimensional problems and geometrical linearity. The resolution of the imposed restrictions within finite element modeling further emphasizes the usage of advanced media in design and facilitates its application to various problems, which is the aim of this paper. The delivery of that task is as follows. Firstly, we present the kinematics and material descriptions for the micropolar media. The authors propose to use a newly developed continuum-based micropolar beam formulation to avoid an overwhelming computational burden and, at the same time, deal with nonlinear stress–strain relations. Secondly, the work develops a contact approach within the micropolar theory from two-dimensional to three-dimensional elasticity, although the contact is considered frictionless. Finally, it compares two existing contact formulations, including the developed one, for the contact beam problems, using the examples of two collinear sliding beams’ bending.

Inconsistency between the micropolar theory and non-equilibrium thermodynamics in the case of polar fluids

Abstract The balance equation of angular momentum in anisotropic fluids includes a couple stress contribution, also responsible for an antisymmetric contribution to the force stress tensor. We herein derive all balance equations for the simplest anisotropic fluid, i.e., a polar fluid, using the GENERIC formalism of non-equilibrium thermodynamics. In doing so, we find that there is an inconsistency between the internal energy density evolution equation derived using non-equilibrium thermodynamics and the one usually considered in micropolar theory.

Micropolar theory of solid subjected to initial stress

The basic of the initial stress in an elastic continuum is revisited, then this investigation studies micropolar elastic solid subjected to initial stress using the classical method. The strain energy density function was developed to derive the constitutive relations and field equations when the influence of initial stress is taken into account. The case of linear-homogeneous isotropic and anisotropic condition of field and constitutive relations of the solid is obtained.

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.

Equivalent micropolar model for porous guided bone regeneration mesh: Optimum design for desired mechanical properties

In the present work, a micropolar continuum model is adopted to homogenise a heterogeneous porous model of Guided Bone Regeneration (GBR) meshes. GBR meshes are used in dentistry as mechanical barriers to isolate and protect the area of bone loss from the surrounding tissue while allowing new bone growth. The mechanical constants of the continuum are derived based on the strain energy equivalence of a periodic porous plate with its equivalent ortho-tetragonal micropolar model under prescribed boundary conditions. The effects of various architectural features such as pore shapes, patterns and sizes on the material parameters are investigated. The results show that the micropolar theory provides a better prediction of the response of the 2D porous geometries considered for the GBR mesh, compared to the classical Cauchy theory. The collected equivalent material parameters are further used for GBR mesh design, considering both mechanical and biomedical requirements. As an example, different materials and arrangements are analysed to find micropolar constitutive parameters that are comparable to bone parameters reported in the literature. This allows the GBR mesh to possess the mechanical performance that matches the adjacent bones.

Photo-thermo-elastic interactions in micropolar generalized thermoelasticity theory in the framework of Green-Naghdi theory

The purpose of the current study is to establish a novel mathematical model in the micropolar theory of generalized thermoelasticity under the framework of photothermal theory. For the solution of the required problem, the integral transforms along with the potential displacement approach has been utilized. For graphical representation of different physical quantities such as displacement components, force and couple stress components, carrier density as well as the temperature distribution, Matlab software has been used.

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.

An efficient numerical method for the quasi-static behaviour of micropolar viscoelastic Timoshenko beams for couple stress problems

A viscoelastic Timoshenko micro-beam is investigated using micropolar viscoelasticity theory. The governing equations are derived by applying the principle of virtual displacements, and can be used in a variety of problems. The equations are then reduced to the special case of a homogeneous uniform beam and solved for quasi-static elastic and viscoelastic cases. The viscoelastic material is modeled via standard linear solid model. For the elastic case, a closed-form solution is reached, but for the viscoelastic case, the inverse Laplace transform is calculated numerically by utilizing De Hoog's algorithm. It is shown that the proposed model can capture the size effects, and converges to the classical Timoshenko beam if the beam is thick enough. The model also converges to the micropolar Euler-Bernoulli model if the thickness of the beam is small. The role of distributed couple stress is also investigated in this paper. The results indicate that in a cantilever beam, the distributed couple stress has a high influence on the deflection of the beam, micro-rotation, stress components and couple stress. However, in a beam resting on three supports, it only affects the micro-rotation and the shear stress component in transverse direction (σxz). Further investigation shows an extremum in some of the results over time, illustrating the importance of taking viscoelastic behaviour into account.

A three-dimensional micropolar beam model with application to the finite deformation analysis of hard-magnetic soft beams

The main purpose of this contribution is to develop a three-dimensional (3D) nonlinear beam model based on the micropolar continuum theory. To do so, a kinematic model based on the deformation of three directors and accounting for the micro-rotation tensor of the micropolar theory is introduced. One of the main characteristics of the present beam model is that 3D constitutive equations without any modification can be directly used in the formulation. Furthermore, it is known that a body couple field is induced in hard-magnetic soft materials (HMSMs) when subjected to external magnetic fluxes. Therefore, the stress tensor in HMSMs is asymmetric, in general. Since the asymmetry of stress is one of the main features of the micropolar theory, the present formulation can be used for analyzing the deformation of beams made of HMSMs. Accordingly, the virtual external work of the present model is formulated so that it accounts for the contribution from uniform or constant-gradient external magnetic fluxes on the beam. Moreover, a Total Lagrangian (TL) nonlinear finite element (FE) formulation to provide numerical solutions of the related problems is developed. Several numerical examples are solved to investigate the capability of the developed formulation. It is shown that the present formulation can model the size-dependent behavior of beam-like structures if the material length-scale parameter of the micropolar constitutive model is comparable to the thickness of the beam. Moreover, the proposed model can successfully predict the finite deformation of 3D beams made of HMSMs subjected to magnetic loading.

Topological boundary states in micropolar gyroelastic continua

The study of topology in elastic media has been primarily focused on achieving non-trivial topological states in discrete elastic lattices through active or chiral microscopic interactions. Realization of such topological states in continuous elastic media remains largely unexplored. In this study, a new continuum theory of micropolar gyroelasticity is developed and applied to attain non-trivial topological boundary states in elastic continua. According to the new theory, an elastic continuum is composed of elastically interacting micro-volume elements that can translate and rotate and are connected at their mass centers to gyroscopes, which contribute to the linear and orbital angular momenta but not to the spin angular momentum of the continuum. By applying this micropolar gyroelasticity theory to elastic media with both periodic and finite domains, the emergence of topological boundary states in 2D micropolar gyroelastic continua is demonstrated. Through using the Floquet–Bloch method for periodic domains, the bulk-boundary correspondence is analytically established, and the emergence of non-trivial topological bulk states characterized by Mexican-hat band structures is observed. In addition, by employing an asymptotic analytical model based on the extended Bloch theorem and performing numerical analyses of micropolar gyroelastic continua with finite domains of different geometries, it is shown that the non-trivial Mexican-hat band structure is associated with and provides protection for topological boundary states confined at the boundaries. Finally, the application of the newly developed micropolar gyroelasticity theory to Zinc-blende structured materials (including ZnTe, GaP, InP and ZnS) reveals that the emergence of the topological boundary states in an elastic continuum is not triggered solely by the gyroscopic effect but also depends on the material properties of the micropolar continuum. This study provides new insights into extending notions and methods of topology to analyze elastic continua, paving the way for the practical implementation of topological mechanical systems in various engineering applications.

An implicit Material Point Method for micropolar solids undergoing large deformations

Modelling the mechanical behaviour of structural systems where the system size approaches that of the material microstructure (such as in MEMS) presents challenges to the standard continuum assumption and classical models can fail to predict important phenomena. Of the various non-conventional continuum frameworks developed to tackle this issue, the micropolar (Cosserat) continuum is widely acknowledged as a suitable and rigorous alternative for its ability to naturally predict size effects by introducing characteristic length scales. This work proposes an implementation of geometrically non-linear micropolar theory using an implicit Material Point method, for the purpose of simulating nanoscale large-deformation problems involving Hookean materials. The framework employs an analytically-derived consistent tangent, and is verified with a novel benchmark problem derived using the Method of Manufactured Solutions. Due to similarities between the methods, many aspects of the formulation could be used to construct an Updated Lagrangian Finite Element Method.

Modeling of blood flow in the framework of micropolar theory

In this paper, we study the blood flow through blood vessels of various radii (including the case of variable cross section as well as modeling the blood flow through venae and arteries). Two approaches are discussed in order to mimic the dependence of blood viscosity on red blood cells aggregation, which changes with the shear rate and position inside the vessel: Two microstructural parameters together with empirical constitutive equations as a characteristic of aggregation are proposed, namely the microinertia as well as the volume fraction of blood particles (erythrocytes, platelets and leukocytes). Consequently, the Navier–Stokes system of equations for an incompressible fluid is supplemented by a constitutive equation for the moment of inertia in one case and for the volume fraction in another. The problems are solved numerically by the finite volume method for vessels of various geometries in spatial description. A comparison with experimental data for a narrow capillary shows the efficiency of the proposed constitutive equations for describing blood flow. Also, velocity profiles are obtained on the basis of compiled empirical formula for various sections of a blood vessel of variable radius. In addition, the flow through vessels of the human circulatory system, such as the inferior vena cava and the carotid artery, are studied.

On the Variational Principle of Lagrange in the Micropolar Theory of Elasticity at Nonisothermal Processes

On the Variational Principle of Lagrange in the Micropolar Theory of Elasticity at Nonisothermal Processes

Composite material identification as micropolar continua via an optimization approach

A strategy based on material homogenization and heuristic optimization for the structural identification of composite materials is proposed. The objective is the identification of the constitutive properties of a micropolar continuum model employed to describe the mechanical behaviour of a composite material made of rigid blocks and thin elastic interfaces. The micropolar theory (Cosserat) has been proved to be capable of properly accounting for the particles arrangements as well as their size and orientation. The constitutive parameters of the composite materials, characterized by different textures and dimensions of the rigid blocks, are identified through a homogenization procedure. Thus, the identification is repeated exploiting the static or modal response of the composite materials and using the Differential Evolution algorithm. The benchmark structures assumed as target are represented by discrete models implemented in ABAQUS where the blocks and the elastic interfaces are modelled by rigid bodies and elastic interfaces, respectively. The obtained results show that proposed strategies provide accurate results paving the way to the experimental validation and in field applications.

Strain localization of Mohr-Coulomb soils with non-associated plasticity based on micropolar continuum theory

To address the problems of strain localization, the exact Mohr-Coulomb (MC) model is used based on second-order cone programming (mpcFEM-SOCP) in the framework of micropolar continuum finite element method. Using the uniaxial compression test, we focused on the earth pressure problem of rigid wall segment involving non-associated plasticity. The numerical results reveal that when mpcFEM-SOCP is applied, the problems of mesh dependency can be effectively addressed. For geotechnical strain localization analysis involving non-associated MC plasticity, mpcFEM-SOCP in conjunction with the pseudo-time discrete scheme can improve the numerical stability and avoid the unreasonable softening issue in the pressure-displacement curves, which may be encountered in the conventional FEM. It also shows that the pressure-displacement responses calculated by mpcFEM-SOCP with the pseudo-time discrete scheme are higher than those calculated by mpcFEM-SOCP with the Davis scheme. The inclination angle of shear band predicted by mpcFEM-SOCP with the pseudo-time discrete scheme agrees well with the theoretical solution of non-associated MC plasticity.

Buckling analysis of thick cylindrical shells using micropolar theory

Modeling complex materials with internal structure such as porous solids is challenging as in some cases the classical theory cannot provide precise responses. By considering the scale effects through additional kinematic descriptors and six constants for isotropic materials, the micropolar theory can accurately model complex materials like bone. This paper studies the buckling of a thick cylindrical shell using classical and non-classical theories. The cylinder’s material is considered bone and isotropic, and the critical load value for different geometries and boundary conditions has been obtained. Finally, the size-effect and importance of micropolar theory in micro dimensions are investigated. The micropolar equations are solved by a numerical solution using the 3D finite element method. The results show that decreasing the macroscopic length increases the stiffness of the cylinder more than that predicted by classical theory; In addition, by increasing the thickness of the cylinder and the importance of shear stresses, the micropolar theory predicts a more critical load than the classical one, and the result differences become more significant between micropolar and classical theories. Also, the characteristic length of the micropolar is investigated. The results show that the change of the critical load increased by moving toward the micro dimensions.

Creeping Micropolar Flow past a Porous Oblate Spheroid

A mathematical model for the creeping micropolar flow past a porous oblate spheroidal body is presented. The oblate spheroidal body possesses permeability and uniform stream velocity is assumed far away from the body and along its axis of symmetry. The flow inside the particle obeys the Darcy- Brinkman laws of porosity and the fluid outside is governed by the A.C. Eringen micropolar flow theory. Drag is determined with varying values of the permeability parameter and the coupling number . Also, the micropolar parameter is studied numerically.

A micropolar shell model for hard‐magnetic soft materials

AbstractHard‐magnetic soft materials (HMSMs) are particulate composites that particles with high coercivity are dispersed in a soft matrix. Since applying the magnetic loading induces a body couple in HMSMs, the resulting Cauchy stress is predicted to be asymmetric. Therefore, the micropolar continuum theory can be employed to capture the deformation of these materials. On the other hand, the geometries and structures made of HMSMs often possess small thickness compared to the overall dimensions of the body. Accordingly, in the present contribution, a 10‐parameter micropolar shell formulation to model the finite elastic deformation of thin hard‐magnetic soft structures under magnetic stimuli is developed. The proposed shell formulation allows for using three‐dimensional constitutive laws without any need for modification to apply the plane stress assumption in thin structures. A nonlinear finite element formulation is also presented for the numerical solution of the governing equations. To alleviate the locking phenomenon, the enhanced assumed strain method is employed. Several examples are presented that demonstrate the performance and effectiveness of the proposed formulation.