Micropolar Continuum Research Articles

Double‐surface plasticity for micropolar continuum

Double‐surface plasticity for micropolar continuum

Fracture and size effect in mechanical metamaterials

We resort to variational methods to evaluate the asymptotic behavior of fine metamaterials as a function of cell size. To zeroth order, the metamaterial behaves as a micropolar continuum with both displacement and rotation degrees of freedom, but exhibits linear-elastic fracture mechanics scaling and therefore no size effect. To higher order, the overall energetics of the metastructure can be characterized explicitly in terms of the solution of the zeroth-order continuum problem by the method of Γ-expansion. We present explicit expressions of the second-order correction for octet frames. As an application, we evaluate the compliance of double-cantilever octet specimens to second order and use the result to elucidate the dependence of the apparent toughness of the specimen on cell size. The analysis predicts the discreteness of the metamaterial lattice to effectively shield the crack-tip, a mechanism that we term lattice shielding. The theory specifically predicts anti-shielding, i. e., coarser is weaker, in agreement with recent experimental observations.

A B-spline based gradient-enhanced micropolar implicit material point method for large localized inelastic deformations

The quasi-brittle response of cohesive-frictional materials in numerical simulations is commonly represented by softening plasticity or continuum damage models, either individually or in combination. However, classical models, particularly when coupled with non-associated plasticity, often suffer from ill-posedness and a lack of objectivity in numerical simulations. Moreover, the performance of the finite element method significantly degrades in simulations involving finite strains when mesh distortion reaches excessive levels. This represents a challenge for modeling cohesive-frictional materials, given their tendency to experience strongly localized deformations, such as those occurring during shear band dominated failure. Hence, accurate modeling of the response of cohesive-frictional solids is a demanding task. To address these challenges, we present an extension of the material point method (MPM) for the unified gradient-enhanced micropolar continuum, aiming at the analysis of finite localized inelastic deformations in cohesive-frictional materials. The generalized gradient-enhanced micropolar continuum formulation is employed to tackle challenges related to localization and softening material behavior, while the MPM addresses issues arising from excessive deformations. The method utilizes a B-spline formulation for the rigid background mesh to mitigate the well-known cell crossing errors of the MPM. To demonstrate the performance of the method, 2D and 3D numerical studies on localized failure in sandstone in plane strain compression and triaxial extension tests are presented. A comparison with finite element results confirms the suitability of the formulation. Moreover, an efficient numerical implementation of the formulation is presented, and it is demonstrated that the additional MPM specific overhead is negligible.

Overall constitutive properties of stratified lattices with alternating chirality.

The present research focuses on a continuum description of stratified metamaterials achieved through the superposition of layers with alternating chirality. Each layer is constructed as a periodic assembly of centre-symmetric periodic cells, formed by a recurring arrangement of rigid circular discs connected by elastic ligaments. The layers are interconnected through elastic pins passing through the centres of aligned discs, allowing for either restrained or free relative rotation. A micropolar continuum model is used to describe each individual layer. The overall response of the metamaterials to in-plane forces is derived using a multi-field non-local model, expressed in terms of the average and difference of displacement and rotational fields. The overall micropolar and standard (Cauchy) constitutive tensors have been determined in closed form. The validity of the equivalent generalized micropolar model has been confirmed through comparison with discrete Lagrangian solutions of representative examples. In addition, a detailed analysis of a pseudo-indentation test has been carried out. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 2)'.

Numerical Analysis of Strain Localization in Granular Soils by Modified Cam-Clay Model Based on Micropolar Continuum Theory

Numerical Analysis of Strain Localization in Granular Soils by Modified Cam-Clay Model Based on Micropolar Continuum Theory

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.

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.

Geotechnical analysis involving strain localization of overconsolidated soils based on unified hardening model with hardening variable updated by a composite scheme

AbstractStrain localization simulation of overconsolidated soils with high overconsolidation ratio (OCR) has been a long‐standing challenge. Some critical state soil models, including the modified Cam‐clay (MCC) model, have been widely applied, but they may not predict the shear dilatancy of overconsolidated soils well in some cases. Hence, the unified hardening (UH) model, which may be viewed as a generalized version of the MCC model, is implemented. It has been recognized, nonetheless, that without resorting to the regularization mechanism, the standard finite element method (FEM) or the second‐order cone programming optimized finite element method (FEM‐SOCP) often experiences instability or interruption of calculating the hardening‐softening responses of overconsolidated soils. To resolve the aforementioned difficulty, the UH model is developed and implemented in the framework of FEM‐SOCP based on the micropolar continuum (mpcFEM‐SOCP) to predict strain localizations of overconsolidated soils. Furthermore, to obviate non‐convexity of mpcFEM‐SOCP induced by material softening, an effective composite update scheme of hardening variable pc, which refers to the implicit variable (IV) scheme for the hardening stage and then refers to the explicit variable (EV) scheme for the softening stage, is proposed. Based on one biaxial compression problem and one rigid strip footing problem, numerical analyses disclose that by applying mpcFEM‐SOCP in conjunction with the composite update scheme of pc, the UH model of micropolar continuum can effectively predict the strain localization behavior of overconsolidated soil during its failure stage, and the stable hardening‐softening responses of overconsolidated soils can be readily attained.

О механизме разрушения ледяных пленок метастабильных газогидратов и его возможном приложении к процессу эмиссии метана в Арктике

The release of methane deposits in the Arctic, which is bound in the form of gas hydrates, can accelerate global warming. The threat of increasing methane content in the atmosphere is associated with the decomposition of gas hydrates located in the sedimentary layer in a metastable state. The authors formulate destruction mechanisms that can affect the dissociation of natural gas hydrates and assess the risks of possible gas emissions during the advancement of slow tectonic waves. They have studied the instability of moving granules of the sedimentary strata and obtained the condition for the bifurcation of the micropolar continuum, which may be associated with the emission of methane from zones of accumulation of metastable relict gas hydrates.

Block preconditioning strategies for generalized continuum models with micropolar and nonlocal damage formulations

AbstractIn this work, preconditioning strategies are developed in the context of generalized continuum formulations used to regularize multifield models for simulating localized failure of quasi‐brittle materials. Specifically, a micropolar continuum extended by a nonlocal damage formulation is considered for regularizing both, shear dominated failure and tensile cracking. For such models, additional microrotation and nonlocal damage fields, and their interactions, increase the complexity and size of the arising linear systems. This increases the demand for specialized preconditioning strategies when iterative solvers are adopted. Herein, a block preconditioning strategy, employing algebraic multigrid methods (AMG) for approximating the application of sub‐block inverses, is developed and tested in three steps. First, a block preconditioner is introduced for linear systems resulting from micropolar models. For this case, a simple sparse Schur complement approximation, which is practical to compute, is proposed and analyzed. It is tested for three different discretizations. Second, the developed preconditioner is extended to reflect the additional nonlocal damage field. This extended three‐field preconditioner is tested on the simulation of a compression test on a sandstone sample. All numerical tests show an improved performance of the block preconditioning approach in comparison to a black‐box monolithic AMG approach. Finally, a problem‐adapted preconditioner setup strategy is proposed, which involves a reuse of the multigrid hierarchy during nonlinear iterations, and additionally accounts for the different stages occurring in the simulation of localized failure. The problem‐adapted preconditioning strategy has the potential to further reduce the total computation time.

Hexahedral finite elements with enhanced fixed‐pole interpolation for linear static and vibration analysis of 3D micropolar continuum

AbstractThe spotlight of this research is on the application of the fixed‐pole interpolation, sometimes used in the analysis of three‐dimensional (3D) geometrically non‐linear beams, but for which no attempts have been made to apply it to linear analysis so far. Particular attention is given to the correlation between the linearised forms of the fixed‐pole and helicoidal interpolation with the linked interpolation. Knowledge of this interdependence is crucial for identifying paths for possible enhancement and extension from the Timoshenko beam of arbitrary order to the new hexahedral finite element of arbitrary order for linear analysis of 3D micropolar continuum. After ensuring the convergence of the newly developed micropolar element through a set of patch tests, three numerical examples of a 3D micropolar continuum in static equilibrium and free vibration of 3D micropolar plates with different geometric properties and boundary conditions have been analysed. Based on these results, the newly proposed finite elements have been critically assessed against the conventional elements.

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.

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.

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.

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.

Workshop “Micropolar continua and beyond”

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

Geometric structures of micropolar continuum with elastic and plastic deformations based on generalized Finsler space

Elasto-plastic deformations of micropolar continuum are discussed by a non-Riemannian geometry. The non-locality of micropolar continuum is described in a second-order vector bundle of displacements and microrotations. With a decomposition of total elasto-plastic field, geometric quantities are divided into the elastic and plastic components independently. Especially, when an intrinsic parallelism of displacements and microrotations holds, integrability conditions of the elasto-plastic field are represented by a torsion tensor or the curvature of nonlinear connection. Then, Burgers and Frank vectors and an energy release rate around crack tips are related to the torsion tensor or the curvature of nonlinear connection. Moreover, the non-locality of microrotation is discussed based on a kink band as a disclination. It is found a generalized expression of Burgers vector which can describe the kink interface including the disclination.

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.