Micropolar Model Research Articles

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.

A micropolar model accounting for asymmetric behavior of cold‐box sand in relation to tensile and compression tests

AbstractCold‐box sand (CBS) belongs to the granular materials and consists of sand and a binder. The behavior of CBS is simulated with a micropolar model, whereby the additional degree of freedom of the model describes the rotation of the sand grains. The model is used to generate a shear band under pressure for three different meshes, where the force‐displacement curves of the three meshes converge so that no mesh dependence occurs. Another requirement of the model is the consideration of asymmetric behavior for compression and tension. Due to the additional degree of freedom the implicit implementation of the micropolar continuum is very time‐consuming. Therefore, an explicit implementation is considered as an alternative possibility. This paper compares the advantages and disadvantages of both methods and the results for both calculations.

Simulation of micropolar fluid flow with ternary nanoparticles over a permeable stretching surface involving nonlinear thermal radiation

ABSTRACT The primary goal of this research is to look into the rheological aspects of blood flow that contains ternary nanoparticles ( C u O + T i O 2 + A l 2 O 3 ). For this, the micropolar fluid model is adopted to simulate blood flow transport. Inclined magnetization and porosity characteristics are used to reveal the momentum attributes of flowing fluid, and nonlinearized radiating thermal flux is used to predict the thermal characteristics of the concerned flow. Further, the aspects of suction and injection of fluid over a permeable stretching sheet are also acknowledged. Governing system of partial differential equations are transformed into nonlinear ordinary differential equations by using similarity transformation. An effective built-in routine known as bvp4c, which is based on finite differencing, is used to acquire the solution. The results showed that the trihybrid nanofluid significantly boost fluid motion and temperature profiles more in the mass injection effect in comparison to suction phenomena. The increase in heat transfer rate is also noticed for augmented values of non-linear radiation parameter and micropolar parameter. There is a detailed discussion of the effects of the contributing parameters on the resulting dimensionless profiles. This research has significant implications for the delivery of magnetized blood, hyperthermia therapy, understanding of the bloodstream, transfer of complex bio-waste liquids, and capillary heat exchange.

Two different micro-polar phase-field models for brittle fracture and their open-source finite element implementation

Realizing the limitations of classical phase-field fracture models and the need for the development of non-local phase-field models that can capture the experimentally observed size effects in brittle and quasi-brittle materials, we propose thermodynamically consistent volumetric-deviatoric and spectral decomposition-based phase-field models (PFMs) for brittle fracture in micro-polar continua. For developing the proposed models, we have considered the micro-rotation as an additional kinematic descriptor, and have derived the balance equations by invoking the virtual power principle. On satisfying the thermodynamic laws, we have determined the constitutive relations for the thermodynamic fluxes and demonstrated that any dissipative effect can easily be incorporated into the proposed models. We have provided an open-source finite element (FE) implementation of the micro-polar PFMs using Gridap in Julia. The implementation includes various benchmark problems to illustrate how the responses in a specimen change with the variations in micro-polar material parameters. Through a series of numerical examples with different values of micro-polar material parameters, we have showcased both the capabilities and limitations of the proposed volumetric-deviatoric and spectral decomposition-based micro-polar PFMs.

Computational Cosserat periporomechanics for strain localization and cracking in deformable porous media

Strain localization and cracking in porous media are significant issues in engineering and science. Periporomechanics is a strong nonlocal framework for modeling the mechanics and physics of porous media with evolving discontinuities. In periporomechanics, the horizon that usually lacks a physical meaning serves as a nonlocal parameter. In this article, as a new contribution, we formulate a Cosserat periporomechanics paradigm incorporating a micro-structure related length scale for modeling shear banding and cracking in dry porous media. In this new Cosserat-periporomechanics framework, each material point is endowed with both translational and rotational degrees of freedom following the Cosserat continuum theory. We formulate a stabilized Cosserat constitutive correspondence principle through which classical micro-polar constitutive models for porous media can be used in Cosserat periporomechanics. We have numerically implemented the Cosserat periporomechanics paradigm through an explicit Lagrangian meshfree algorithm. We first present numerical examples to validate the implemented computational Cosserat periporomechanics paradigm for modeling shear bands and cracks. We then present numerical examples to demonstrate the efficacy and robustness of the Cosserat periporomechanics for modeling the shear banding bifurcation and crack branching in dry porous media.

Dynamic continualization of masonry-like structured materials

Block-lattice materials featuring periodic planar running-bond tessellation of regular rigid blocks and linear elastic homogeneous isotropic interfaces are considered. The governing equations of the discrete masonry-like Lagrangian model are properly manipulated via the novel enhanced continualization scheme, in such a way as to obtain equivalent integral type non-local continua, whose band structure turns out to be coincident with that of the corresponding discrete models. The formal Taylor series expansion of the integral kernels allows deriving homogeneous generalized micropolar higher-order continuum models, characterized by non-local constitutive and inertial terms. The enhanced continualization exhibits thermodynamic consistency in the definition of the overall non-local constitutive tensors, as well as qualitative agreement and quantitative convergent matching of the complex frequency band structure in the regime of both homogeneous and non-homogeneous Bloch waves. The theoretical findings are effectively validated by studying the dispersion relations and the spatial attenuation properties, as referred to realistic representative cases of masonry-like block-lattice micro-structures.

On the optimized energy transport rate of magnetized micropolar fluid via ternary hybrid ferro-nanosolids: A numerical report

In the current era, a chemical, industrial, or production process may not be devoid of heat transfer processes through fluids. This is seen in evaporators, distillation units, dryers, reactors, refrigeration and air conditioning systems, and others. On the other hand, the micropolar model effectively simulates microstructured fluids like animal blood, polymeric suspensions, and crystal fluid, paving the way for new potential applications based mainly on complex fluids. This investigation attempts to figure out and predict the thermal behavior of a polar fluid in motion across a solid sphere while considering the Lorentz force and mixed convection. To support the original fluid's thermophysical characteristics, two types of ternary hybrid ferro-nanomaterials are used. The problem is modelled using a single-phase model. Then, using the Keller box approximation, a numerical finding is obtained. The study reveals that Increasing the volume fraction of the ternary hybrid nonsolid results in optimized values of Nusselt number, velocity, and temperature. The presence of Lorentz forces effectively mitigates flow strength, skin friction, and energy transfer rate. The mixed convection factor contributes significantly to enhanced energy transfer and improved flow scenarios. For maximum heat transfer efficiency, employing Fe3O4–Cu–SiO2 is recommended over Fe3O4–Al2O3–TiO2.

Thermal enhancement, thermophoretic diffusion, and Brownian motion impacts on MHD micropolar nanofluid over an inclined surface: Numerical simulation

The majority of liquids are undesirable for engineering due to their low heat conductivity. Scientists and researchers are developing nanofluids, which are made of nanoparticles (NPs) scattered in a base fluid, to boost the capacity for heat transfer in a variety of industries, including molten metals, pharmaceuticals, and computers. In this study, the magneto-hydrodynamics (MHD) flow of a micropolar nanofluid model with particle microstructure and inertial properties is investigated. A micropolar nanofluid’s flow is examined concerning the effects of a magnetic field. Aspects of heat and mass transfer are investigated using thermal radiation, thermophoresis, Brownian motion, and double diffusion theory toward an inclined surface. With the right level of similarity, the process converts partial differential equations emerging in nanofluidic systems into nonlinear differential equation systems. The FDM finite difference approach (FDM) (Lobatto IIIA) is used for the nonlinear nanofluid issue with the precision of order 4–5 and is implemented using a variety of collocation locations. The strength of Lobatto IIIA is its effectiveness in handling coupled differential equations that are extremely nonlinear. The higher-order differential equations are converted into a first-order method utilizing the boundary value dilemma (bvp4c) solver, which is part of the MATLAB software package, to computationally analyze the simplified mathematical model. The data obtained demonstrated a high degree of accuracy and symmetry when measured against previously published studies. Numerous variables’ effects on fluid flow are examined and graphically displayed with obtained numerical data. The results show that the temperature profile behaves in a way that is consistent with increasing thermophoresis and Brownian motion forces. This work provides insights into practical applications such as nanofluidic, energy conservation, friction reduction, and power generation. However, the work makes a significant point that the flow of a micropolar nanofluid including NPs can be regulated by appropriately modifying the thermophoresis parameter and Brownian motion parameter.

Extension of moving particle simulation by introducing rotational degrees of freedom for dilute fiber suspensions

AbstractWe develop a novel Moving Particle Simulation (MPS) method to reproduce the motion of fibers floating in sheared liquids accurately. In conventional MPS schemes, if a fiber suspended in a liquid is represented by a one‐dimensional array of MPS particles, it is entirely aligned to the flow direction due to the lack of shear stress difference between fiber–liquid interfaces. To address this problem, we employ the micropolar fluid model to introduce rotational degrees of freedom into the MPS particles. The translational motion of liquid and solid particles and the rotation of solid particles are calculated with the explicit MPS algorithm. The fiber is modeled as an array of micropolar fluid particles bonded with stretching, bending and torsional potentials. The motion of a single rigid fiber is simulated in a three‐dimensional shear flow generated between two moving solid walls. We show that the proposed method is capable of reproducing the fiber motion predicted by Jeffery's theory which is different from the conventional MPS simulations.

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.

Computation of micropolar nanofluid from a wedge with heterogeneous carbon/metallic nanoparticles, viscous dissipation and heat sink/source: rheological nanocoating flow simulation

ABSTRACT Motivated by nanotechnological coating applications, a theoretical study is presented for the laminar, steady-state, incompressible nonlinear boundary layer flow of a non-Newtonian nanofluid external to a wedge-shaped configuration. The wedge surface is assumed to be isothermal. The Eringen micropolar model is deployed for rheological properties of the nanofluid. The dimensionless thermal perimeter layer equations are solved with the efficient MATLAB bvp4c numerical scheme. Validation with earlier studies is conducted. Aqueous-based nano-polymers are examined with either metallic/metallic oxide or carbon-based nanoparticles. The influence of Hartree pressure gradient parameter, Eringen vortex viscosity parameter, nanoparticle volume fraction, heat absorption parameter, Prandtl number and nanoparticle type on velocity, angular velocity, temperature, skin friction function and Nusselt number function are visualized graphically and in tables. Temperature is strongly elevated with increasing micropolar parameter and nanoparticle volume fraction. Velocity is boosted with increasing nanoparticle volume fraction. Temperatures are elevated with heat source but suppressed with heat sink. Temperatures are a maximum for silver and progressively lower for copper, diamond and with a minimum for titania. Skin friction is boosted with pressure gradient parameter whereas Nusselt number is depleted. Nusselt number is observed to be a maximum for diamond whereas it is a minimum for silver.

A thermodynamics-based micro-macro elastoplastic micropolar continuum model for granular materials

A micro–macro elastoplastic micropolar continuum model for granular materials in the dry and pendular states is proposed on the basis of thermodynamics. The independent rotational degrees of freedom are considered besides the translational degrees of freedom for particles among the discrete particle system, which is described by the equivalent micropolar continuum. In the framework of the micropolar continuum, the elastic part and yield locus at macroscale are firstly rigorously deduced from the general elastic constitutive relation and Coulomb-type yield criterion at inter-particle contacts, respectively. Furthermore, the micro–macro elastoplastic micropolar continuum model is proposed, in which the model parameters have clear physical meanings. Subsequently, a return mapping algorithm for the integration of the micropolar elastoplastic model is presented. Finally, the proposed model, coded into ABAQUS software, is validated by comparisons of test and discrete element method results in terms of the stress–strain relation and strain localization phenomenon. The stress–strain relation and shear band modes are well reproduced, and the effects of the confining pressure and the partially saturated state on granular materials can also be correctly predicted by the proposed model.

FreeFEM++ based heat transfer analysis of an electrically induced magnetic flow within the framework of micropolar continuum

In this article, we aim to analyze electrically induced magnetic incompressible flow and heat transfer through a channel within the framework of micropolar continuum. Finite element method is employed to compute the numerical solution of the governing dynamics formulated in the form of PDEs along with associated boundary conditions on the channel. The weak form is presented and variational problem is implemented in FreeFem++ language. To validate, the correct implementation of the problem and obtained numerical results, a reduced micropolar magnetohydrodynamic model is taken into consideration. The analytical solution is presented. Computed numerical results are compared with the exact solution in case of reduced micropolar model and a good agreement is achieved. An excellent convergence of the numerical solution to the exact solution is shown through Tables where L 2 -norm and H 1 -errors are computed at refined meshes. Effect of different physical parameters such as magnetic Reynolds number ( R m ), Hartmann number (Ha), micorpolar constants (m and l 1 2 ), micropolar coupling number (N), Prandtl number (Pr) and Brinkmann number (Br) are studied and discussed in detail. The development of thermal, translational and micro-rotational velocity profiles over the cross-sectional domain of the channel is shown for varying values of these material parameters. Some interesting and new findings in this investigation are presented and discussed. It is observed that the maximum temperature in the medium shifts from the center of the domain toward the boundaries if the micropolar coupling constant N is increased. Maximum micro-rotations are always found near the channel’s boundary where it is evidenced that the particle’s micro-rotations have a direct proportionality with the micropolar constant m. Moreover, the counter rotations of the continuum particles are evidenced at the channel’s boundary. Furthermore, the micorpolar coupling constant l 1 2 is found to resist the micro motions of the particles in the channel. The successful implementation of one and higher dimensional magnetohydrodynamic model in FreeFEM++ in general and within the context of micorpolar continuum in particular is delineated. This shows that it provides an efficient platform to simulate magnetohydrodynamic problems in higher order continuum which has applications in designing magneto-rheological airbags, aircraft take-off gear, mechanical heart valves and cooling systems.

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.

Hiemenz flow with heat transfer in a slip condition micropolar fluid model: Exact solutions

The impingement of Hiemenz stagnation-point flow and a flexible surface which has linear stretching/shrinking is studied. The flow is governed by the wall's velocity, gyration velocity, transpiration and velocity slip parameters. The stretching sheet problem offers single solution, whereas for the shrinking sheet dual solutions are obtained in the region s∗ ≤ s < ∞ , and λ ≤ λ∗. The stretching/shrinking strength parameter (c) under different values produces numerous exact solutions. The existence of critical numbers explains the increasing or decreasing nature of solutions. Dual solutions are found for skin friction, couple stress friction and temperature gradient. In the process of boundary layer, increasing suction or injection decreases velocity. The effect of velocity slip indicates a decrease in linear wall velocity. It was found that the number and formation of solutions are dependent on a parameter measuring correlation of the external flow strength to the surface stretching/shrinking parameter which is a key player in determining exact solutions. Exact solutions for the uniform surface temperature and linearly growing wall temperature are derived for different special cases.

Equivalent Beam Model and Improved Structure Design of Large Space Antenna Truss With Geometric Nonlinearity

AbstractThis paper proposes a extend linear equivalent method that can extend the linear equivalent micropolar beam model to the nonlinear equivalent micropolar beam model to analyze nonlinear vibration for large space truss structure, and also proposes a lattice enhancement method to improve the cantilever truss for the buckling problem that exists in space cantilever truss structures. The nonlinear equivalent model is obtained by introducing a corotating coordinate system into the linear equivalent beam model. Since the instability of the fixed root end of the cantilever truss is the main reason for buckling, the method of strengthening the longeron of the truss lattice by lattice is proposed. The accuracy of the equivalent geometric nonlinear model and the effectiveness of the improved cantilever truss structure are verified by four numerical simulation examples. The methods proposed in this paper provide some reference for studying the dynamics analysis of large space trusses and the design of structures.

A novel micropolar peridynamic model for rock masses with arbitrary joints

A novel micropolar peridynamic model is proposed to simulate fracture behaviors of rock masses with arbitrary joints. The rock masses are discretized into a group of material points linked through two kinds of bonds, i.e. ordinary and joint bonds, which represent the mechanical behaviors of rock block and joint, respectively. The peridynamic parameters of ordinary and joint bonds are introduced to keep the consistence of the strain energy obtained from the proposed peridynamic model and from the continuum mechanics. The stress condition of the bond in micropolar peridynamic model is defined in terms of the displacement gradients of material points. Then a new stress-based failure criterion is proposed to describe the dynamic fracture for jointed rocks. The proposed model is verified by comparing its predictions with those from experimental observations. Numerical examples demonstrate that the wave transmission and crack development process of rock masses with different joint sets are well captured by the proposed model. Moreover, the influence of the joint geometry on failure characteristics of jointed rocks with an opening is further investigated by the proposed model.

Random micropolar beams: response and identification

The objective of this paper is to provide a simplified procedure for the random analysis of axially functionally graded Timoshenko–Ehrenfest beams in the field of the linearized micropolar continuum. New approximated deterministic closed form solutions are proposed, which are also useful for the study of classic 1D structural elements characterized by defects distributed along their axes, bleeding phenomena and step geometric properties. Hence, the proposed results meet several technical applications. The Maximum Entropy (MaxEnt) principle and Monte Carlo (MC) simulations are used to solve the direct and inverse stochastic problems. In order to simplify the presentation, simply-supported and clamped macro- and micro-beams are considered. It is shown that the analysis of 1D micropolar models allows us to identify one of the material parameters related to 3D micropolar models.

Thermal analysis of a rotating micropolar medium using a two-temperature micropolar thermoelastic model with higher-order time derivatives

Thermal analysis of a rotating micropolar medium using a two-temperature micropolar thermoelastic model with higher-order time derivatives