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The paper develops a systematic procedure for formulating finite elements on manifolds. The theoretical developments give rise to a modular computational framework for composing coordinate transformations and manifold parameterizations. The procedure is demonstrated with the Cosserat rod model furnishing a novel finite element formulation that rectifies the lack of objectivity of existing finite elements without violating the director constraints or compromising the symmetry of the tangent stiffness at equilibrium. The framework is element‐independent, allowing its implementation as a wrapper to existing element libraries without modification of the element state determination procedures. [ABSTRACT FROM AUTHOR]

A thermomechanical large deformation Eulerian formulation of a multi-phase soft tissue model with homeostasis and phase transformation has been developed. The model has been simplified by considering a single velocity field and a single temperature field with evolution equations for volume fractions of the phases which are not driven by diffusion. Specific constitutive equations are developed for an exponential form of the Helmholtz free energy. An example of spherical expansion of a three-phase material is considered in the absence of thermal effects. This example considers phase transformations from healthy tissue to a composite of healthy, diseased and necrotic tissues. Although much work is needed to model the mechanobiological aspects of the disease progress, this simplified model indicates that the state of the tissue at the onset of its non-zero volume fraction (i.e. when it begins to influence stress) can significantly effect the resulting stress distribution. [ABSTRACT FROM AUTHOR]

A thermomechanical generalized Cosserat membrane has no resistance to bending but it allows for deformations through the membrane's thickness. This theory is generalized to allow for growth due to mass supply and is constrained to eliminate shear deformations but retain thickness changes. An Eulerian formulation of isothermal elastic-viscoplastic constitutive equations is developed using evolution equations for elastic dilatation and elastic distortional deformations. These evolution equations include time-dependent inelastic effects of homeostasis which cause a tendency for the elastic deformation measures to approach their homeostatic values. An important feature of the Eulerian evolution equations is that they depend only on the current state of the membrane and therefore are appropriate for modeling growth. Examples of zero-stress growth and constrained growth are considered. [ABSTRACT FROM AUTHOR]

In classical pure bending of a bar of square cross sections, two lateral surfaces undergo anticlastic curvature and the other two lateral surfaces undergo tilt. If the bar is made of a Cosserat solid, the tilted lateral surfaces deform in a sigmoid shape in addition to tilt. Parametric studies reveal the bulge depends on the bending characteristic length, the coupling number, and the ratio of two Cosserat rotation gradient constants. This sigmoid bulge deformation is observed in two foams and is found to be consistent with predictions using elastic constants obtained from size effect studies. [ABSTRACT FROM AUTHOR]

Composite material properties are dependent on their microstructure. To adequately model these microstructural effects, a formulation of elasticity that accounts for microstructural effects must be considered. Size-dependent behaviour is an inherent property of such materials, resulting in a need for non-classical continuum theories for adequate characterization. A three-dimensional model for an exponentially graded composite with microstructural effects is developed under the framework of Cosserat elasticity. The mixed boundary value problem is formulated and solved through the finite element method, and implemented in the commercial software Abaqus through two user developed elements. Recovery of the classical limit for the Dirichlet problem is discussed as an initial model validation until complete experimental measurements can be conducted. [ABSTRACT FROM AUTHOR]

The design of many pneumatic soft actuators is based on multiple chambers in parallel alignment. The Cosserat beam theory is an established technique for modeling this kind of actuator, where existing approaches mainly differ in the parameters being required for simulation. The modeling approach presented in this study particularly aims at finding the beam parameters necessary for a simulation even with limited experimental methods. Importantly, it provides a straightforward relationship between the bending stiffness, the extensional stiffness and the axial stretch of the actuator. If the actuator to be modeled has an elementary design, axial measurements are sufficient to identify the parameters to perform three-dimensional simulations, which is of interest to adopters with limited testing equipment. The experimentally parameterized model of such an actuator of elementary design shows high accuracy. Both without load and with a weight of 1 N applied to the tip, the mean error of the tip position in vertical orientation is less than 3.4% for a constant extensional stiffness and less than 2.7% for a pressure-dependent extensional stiffness. Further reduction of the error could be achieved by more refined identification techniques that decompose the complex interrelationship of pressurization, forces and material stiffness. [ABSTRACT FROM AUTHOR]

The gyroid lattice is prepared in chiral and non‐chiral forms. The chiral gyroid lattice is observed to exhibit temperature‐induced twist with the direction of twist corresponding to the sense of chirality. This effect is a nonclassical effect that cannot occur in classical elasticity or classical thermo‐elasticity but is allowable in Cosserat solids. Poisson's ratio of the gyroid is known to be about 0.3 with minimal dependence on size. In contrast to squeeze–twist coupling in which substantial size effects occur with slender specimens twisting much more than thicker ones, thermal–twist coupling exhibits opposite and less consistent size effects of much smaller magnitude. [ABSTRACT FROM AUTHOR]

Poisson's ratio in chiral Cosserat elastic solids is considered. Chirality allows the Poisson's ratio to exceed classical bounds, even if the material is directionally isotropic and all elastic moduli are within thermodynamic limits based on strain energy density. Poisson's ratio in chiral rods depends on the chiral elastic constants as well as on the shear and bulk moduli, assumed positive. Poisson's ratio can be greater than 0.5 or smaller than −1 for slender chiral specimens. [ABSTRACT FROM AUTHOR]

The mechanical response of materials such as fiber and particle composites, rocks, concrete, and granular materials, can be profoundly influenced by the existence of voids. The aim of the present work is to study the dynamic behavior of hexagonal microstructured composites with voids by using a discrete model and homogenizing materials, such as micropolar and classical Cauchy continua. Three kinds of hexagonal microstructures, named regular, hourglass, and skew, are considered with different length scales. The analysis of free vibration of a panel described as a discrete system, as a classical and as a micropolar continuum, and the comparison of results in terms of natural frequencies and modes show the advantage of the micropolar continuum in describing dynamic characteristics of orthotropic composites (i.e., regular and hourglass microstructures) with respect to the Cauchy continuum, which gives a higher error in frequency evaluations for all three hexagonal microstructured materials. Moreover, the micropolar model also satisfactorily predicts the behavior of skewed microstructured composites. Another advantage shown here by the micropolar continuum is that, like the discrete model, this continuum is able to present the scale effect of microstructures, while maintaining all the advantages of the field description. The effect of void size is also investigated and the results show that the first six frequencies of the current problem decrease by increasing in void size. [ABSTRACT FROM AUTHOR]

The main objective of this work is to extend finite element-based topology optimization problem to the two-dimensional, size-dependent structures described using weakly non-local Cosserat (micropolar) and strongly non-local Eringen’s theories, the latter of which finds an application for the first time, to the best of Authors’ knowledge. The optimum material layouts that minimize the structural compliance are attained by means of Solid Isotropic Material with Penalization approach, while the desired smooth, mesh-independent, binary solutions are obtained using density filter accompanied by volume preserving Heaviside projection method. The algorithms are enhanced by including an element removal and reintroduction strategy to reduce the computational cost, and to prevent spurious excessive distortion of elements with very low density. Example problems of practical importance are investigated under the assumption of linear elasticity to validate the code and to clearly demonstrate the influence of internal length scales and different non-locality mechanisms on final configurations. Obtained macro-scale optimum topologies admit the characteristics of corresponding continuum theories, and appear to be in agreement with the mechanical response governed by particle interactions in micro/nanoscale. [ABSTRACT FROM AUTHOR]

This paper describes the development of an exponentially graded constitutive model with micropolar considerations for a homogenized composite material. The existence and uniqueness of the solution to the associated boundary value problems are demonstrated, with a sample numerical solution included to demonstrate applicability of the model. [ABSTRACT FROM AUTHOR]

• Indentation of a rigid cylinder on an microstructural elastic half-plane. • The micropolar theory of elasticity is incorporated to describe the microstructural characteristics of the material. • Dependence of the pressure shape and the width of the contact site upon the material microstructural characteristics. • The material response becomes stiffer, thus suggesting a decreased measured contact width and increased average pressure. • An upper bound for the stiffness is provided by couple stress elasticity and a lower bound by classical elasticity. Experiments existing in the literature present a scale dependence of the mechanical properties of certain materials. Such scale dependence is observable also in indentation experiments where the attained contact area and average pressure among other experimental outputs show a strong dependence upon the microstructure of the indented material. In this work we will examine the indentation of a rigid cylinder on an elastic half-plane within the context of the micropolar elasticity (also known as Cosserat elasticity). The micropolar theory of elasticity incorporates in addition to the displacement field, an independent kinematical microrotation field. Our results show the dependence of the shape of the pressure distribution, the average pressure and the width of the contact site below the indenter upon the microstructure of the material described through the material length as introduced through the micropolar theory. A comparison of the results with those obtained by the classical theory and the couple stress theory as well as a full field solutions regarding equivalent stresses are presented. [ABSTRACT FROM AUTHOR]

Hinged lattices that attain extremal values of Poisson's ratio do not in general obey the theory of elasticity. Hinged structures that attain a Poisson's ratio of − 1 are easy to stretch or deform volumetrically but they resist bending, in contrast to the predictions of elasticity theory. Their behavior corresponds to that of a Cosserat solid with divergent characteristic length. Hinged structures that attain a maximum Poisson's ratio are rigid with respect to hydrostatic tension, are easy to shear or to stretch but are unstable with respect to hydrostatic compression. [ABSTRACT FROM AUTHOR]

The design of many pneumatic soft actuators is based on multiple chambers in parallel alignment. The Cosserat beam theory is an established technique for modeling this kind of actuator, where existing approaches mainly differ in the parameters being required for simulation. The modeling approach presented in this study particularly aims at finding the beam parameters necessary for a simulation even with limited experimental methods. Importantly, it provides a straightforward relationship between the bending stiffness, the extensional stiffness and the axial stretch of the actuator. If the actuator to be modeled has an elementary design, axial measurements are sufficient to identify the parameters to perform three-dimensional simulations, which is of interest to adopters with limited testing equipment. The experimentally parameterized model of such an actuator of elementary design shows high accuracy. Both without load and with a weight of 1N applied to the tip, the mean error of the tip position in vertical orientation is less than 3.4% for a constant extensional stiffness and less than 2.7% for a pressure-dependent extensional stiffness. Further reduction of the error could be achieved by more refined identification techniques that decompose the complex interrelationship of pressurization, forces and material stiffness.

The variational discrete element method developed in Marazzato et al. (Int J Numer Methods Eng, 121(23):5295–5319, 2020) for dynamic elasto-plastic computations is adapted to compute the deformation of elastic Cosserat materials. In addition to cellwise displacement degrees of freedom (dofs), cellwise rotational dofs are added. A reconstruction is devised to obtain P 1 non-conforming polynomials in each cell and thus constant strains and stresses in each cell. The method requires only the usual macroscopic parameters of a Cosserat material and no microscopic parameter. Numerical examples show the robustness of the method for both static and dynamic computations in two and three dimensions. [ABSTRACT FROM AUTHOR]

The mechanical response of materials such as fiber and particle composites, rocks, concrete, and granular materials, can be profoundly influenced by the existence of voids. The aim of the present work is to study the dynamic behavior of hexagonal microstructured composites with voids by using a discrete model and homogenizing materials, such as micropolar and classical Cauchy continua. Three kinds of hexagonal microstructures, named regular, hourglass, and skew, are considered with different length scales. The analysis of free vibration of a panel described as a discrete system, as a classical and as a micropolar continuum, and the comparison of results in terms of natural frequencies and modes show the advantage of the micropolar continuum in describing dynamic characteristics of orthotropic composites (i.e., regular and hourglass microstructures) with respect to the Cauchy continuum, which gives a higher error in frequency evaluations for all three hexagonal microstructured materials. Moreover, the micropolar model also satisfactorily predicts the behavior of skewed microstructured composites. Another advantage shown here by the micropolar continuum is that, like the discrete model, this continuum is able to present the scale effect of microstructures, while maintaining all the advantages of the field description. The effect of void size is also investigated and the results show that the first six frequencies of the current problem decrease by increasing in void size.

Experimental tests of rotation sensitivity in elastic materials and in empty space are surveyed. Pioneers in the theory for each field of study were aware of the other field. Sensitivity to rotation has been demonstrated experimentally in elastic solids but tests for rotation gradient sensitivity in gravity have found no such effects. Insight from comparison of experimental approaches may lead to new experimental protocols. [ABSTRACT FROM AUTHOR]

We derive the Green's functions (concentrated force and couple in an infinite space) for the isotropic planar relaxed micromorphic model. Since the relaxed micromorphic model particularizes into the micro-stretch, Cosserat (micropolar), couple-stress, and linear elasticity model for certain choices of material parameters, we recover the fundamental solutions in all these cases. • Analytical derivation of Green's functions for planar isotropic relaxed micromorphic medium. • Other generalized continua (micro-stretch, Cosserat-micropolar, couple stress) appear as limits of the relaxed micromorphic continua. • Unlike other generalized continua, the stiffness of the relaxed micromorphic model remains bounded as the sample size becomes smaller. [ABSTRACT FROM AUTHOR]

In the present study, the effect of material microstructure on the mechanical response of a two-dimensional elastic layer perfectly bonded to a substrate is examined under surface loadings. In the current model, the substrate is treated as an elastic half plane as opposed to a rigid base, and this enables its applications in practical cases when the modulus of the layer (e.g., the coating material) and substrate (e.g., the coated surface) are comparable. The material microstructure is modeled using the generalized continuum theory of couple stress elasticity. The boundary value problems are formulated in terms of the displacement field and solved in an analytical manner via the Fourier transform and stiffness matrix method. The results demonstrate the capability of the present continuum theory to efficiently model the size-dependency of the response of the material when the external and internal length scales are comparable. Furthermore, the results indicated that the material mismatch and substrate stiffness play a crucial role in the predicted elastic field. Specifically, the study also addresses significant discrepancy of the response for the case of a layer resting on a rigid substrate. [ABSTRACT FROM AUTHOR]

Increasing application of composite structures in engineering field inherently speed up the studies focusing on the investigation of non-homogeneous bodies. Due to their capability on capturing the size effects, and offering solutions independent of spatial discretization, enriched non-classical continuum theories are often more preferable with respect to the classical ones. In the present study, the sample problem of a plate with a circular inclusion subjected to a uniform tensile stress is investigated in terms of both 'implicit'/'weak' and 'explicit'/'strong' non-local descriptions: Cosserat (micropolar) and Eringen theories, by employing the finite element method. The material parameters of 'implicit' model is assumed to be known, while the nonlocality of 'explicit' model is optimized according to stress concentration factors reported for infinite Cosserat plates. The advantages/disadvantages, and correspondence/non-correspondence between both non-local models are highlighted and discussed apparently for the first time, by comparing the stress field provided for reference benchmark problem under various scale ratios, and material parameter combinations for matrix-inclusion pair. The results reveal the analogous character of both non-local models in case of geometric singularities, which may pave the way for further studies considering problems with noticeable scale effects and load singularities. [ABSTRACT FROM AUTHOR]

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. [Display omitted] • A new method for simulating large deformation problems with microstructural effects. • Combines nonlinear micropolar/Cosserat theory with an implicit MPM. • Analytically derives a consistent tangent for use in a Newton–Raphson scheme. • Presents a novel verification scheme using the Method of Manufactured Solutions. [ABSTRACT FROM AUTHOR]

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. • Formulate a Cosserat periporomechanics paradigm for modeling shear bands and cracks in dry porous media. • Formulate a stabilization scheme for the proposed Cosserat constitutive correspondence principle. • Numerically implement the computational framework through an explicit Newmark scheme. • Validate the efficacy of the computational paradigm for modeling shear banding and crack branching in dry porous media. [ABSTRACT FROM AUTHOR]

Cubic 3D lattices were designed, made by 3D printing, and studied experimentally. One lattice was simple cubic with ribs of diameter 1/5 the cell size. The second lattice was chiral with spiral ribs. The chiral lattice, but not the achiral lattice, exhibited squeeze-twist coupling with size effects. The squeeze-twist coupling cannot occur in a classically elastic solid but is anticipated by theory in a Cosserat solid. Both lattices exhibited size effects in bending and torsion. [ABSTRACT FROM AUTHOR]

This paper develops a micropolar constitutive model for a transversely isotropic composite material comprised of a polymer matrix and unidirectional fibers. The constitutive law follows Eringen's model for a generally anisotropic micropolar medium and reduces the model to the transversely isotropic case. The model is then used to develop a boundary value problem with a homogenization procedure, with the goal of obtaining an analytical solution for the homogenized elastic moduli once solved. A variational principle is used to develop the boundary conditions. [ABSTRACT FROM AUTHOR]

Mixed 4-node shell elements with the drilling rotation and Cosserat-type strain measures based on the three-field Hu–Washizu principle are proposed. In the formulation, apart from displacement and rotation fields, both strain and stress resultant fields are treated as independent. The elements are derived in the framework of a general nonlinear 6-parameter shell theory dedicated to the analysis of multifold irregular shells. The novelty of the developed elements stems from the fact that the measures of assumed strains and stress resultants are asymmetric. The original interpolation of drilling and bending components of strains and stress resultants is proposed. In the formulation of new mixed elements, two variants of the interpolation of membrane components are used and interpolation of the independent fields is defined in the natural or skew coordinates. Accuracy and efficiency of the developed elements are tested in several linear and nonlinear numerical examples. It is shown that smaller number of independent parameters in the interpolation of membrane components gives more accurate results. The proposed mixed 4-node elements enable the use of large load steps in nonlinear computations. Moreover, they require significantly less equilibrium iterations than other shell elements formulated in the 6-parameter shell theory. [ABSTRACT FROM AUTHOR]

A dense closed cell foam is studied to determine which continuum theory of elasticity is applicable. Size effects inconsistent with classical elasticity are observed. The material exhibits a characteristic length scale considerably larger, by more than a factor 6, than the largest observable structure size. The Cosserat coupling number N is shown to be small, via measurements of size effects in square section bars, comparison with size effects in round section bars, and determination of warp of a square section bar in torsion. For this material, the couple stress theory is excluded and the modified couple stress theory is excluded. Theories that force constants to their thermodynamic limits do not apply to this foam. The role of other generalized continuum theories is considered. [ABSTRACT FROM AUTHOR]

Lattices composed of cubic and triangular prismatic unit cells with polymeric Sarrus linkage rib elements are designed, fabricated via 3D printing and studied experimentally. Size effects in these lattices are observed experimentally; slender specimens appear more rigid in torsion and in bending than expected via classical elasticity. Effects are analyzed via Cosserat elasticity. The magnitude of size effects is sensitive to geometry of the lattices; triangular cells with short ribs revealed the most extreme effects, also the largest characteristic length in relation to cell size. The torsion coupling number is 1, its upper bound, for all lattices. A path to the attainment of arbitrarily large nonclassical effects is delineated. [ABSTRACT FROM AUTHOR]

A 3D lattice structure consisting of pivoting cubes with flexible links at the corners is presented. It exhibits anisotropic negative Poisson's ratio -0.54 and -0.75. Size effects occur in torsion and bending; these effects are consistent with Cosserat elasticity but not with classical elasticity. Cosserat elastic solids exhibit sensitivity to strain gradients; size effects occur in torsion and bending; also reduction in stress concentrations. The observed Cosserat effects also reveal anisotropy. [ABSTRACT FROM AUTHOR]

This note proposes the use of horospherical coordinates to overcome the problems associated with multiple eigenvalues occurring in the Cosserat microrotation problem. The advantages of this coordinate system are argued. [ABSTRACT FROM AUTHOR]

Cellular solids usually possess random microstructures that may contain a characteristic length scale, such as the cell size. This gives rise to size dependent mechanical properties where large systems behave differently from small systems. Furthermore, these structures are often irregular, which not only affects the size dependent behavior but also leads to significant property variations among different microstructure realizations. The computational model for cellular microstructures is based on networks of Timoshenko beams. It is a computationally efficient approach allowing to obtain statistically representative averages from computing large numbers of realizations. For detailed analysis of the underlying deformation mechanisms an energetically consistent continuization method was developed which links the forces and displacements of discrete beam networks to equivalent spatially continuous stress and strain fields. This method is not only useful for evaluation and visualization purposes but also allows to perform ensemble averages of, e.g., continuous stress patterns – an analysis approach which is highly beneficial for comparisons and statistical analysis of microstructures with respect to different degrees of structural disorder. [ABSTRACT FROM AUTHOR]

Earthquake faults are the result of severe strain localization in rocks deep down in the earth's crust. This localized deformation is controlled by the size of the microstructure and various Thermo-Hydro-Mechanical (THM) couplings, whose modeling is central for understanding earthquake nucleation and seismic energy release. We model this challenging system using the Cosserat theory and by considering large shear deformations during seismic slip. Our numerical results justify the rate and state friction law (Dieterich, 1992; Ruina, 1983a), which describes empirically the fault tribology. This shows the importance of the underlying physics of our model for fault friction. In our analyses traveling shear bands along the thickness of the fault are present, leading to oscillations in the fault's frictional response. Existing numerical analyses presented in Rattez et al. (2018a, 2018b, 2018c) do not capture this behavior, which goes beyond the established models of uniform shear (Lachenbruch, 1980) and shear on a mathematical plane (Rice, 2006a). Recent experimental results, that insulate thermal pressurization from other weakening mechanisms (Badt et al., 2020), corroborate our numerical results. Our results motivate us to extend the classical model of thermal pressurization in Mase & Smith (1987) and Rice (2006b) to incorporate different strain localization modes, temperature and pore fluid pressure boundary conditions. In particular, we start our analysis by using the normalized coupled system of partial differential equations that include the THM couplings for the case of a Cosserat continuum. We then perform a bifurcation analysis, which indicates that traveling shear bands are possible inside the fault gouge. Next, we derive our non linear mesh independent numerical results accounting for the influence of large displacements by using and Adaptive Lagrangian Eulerian (ALE) procedure. We introduce viscosity in our numerical analyses for the rate and state phenomenology to emerge. We corroborate our numerical results comparing them to similar laboratory experiments. Furthermore, we modify the Volterra integral equation of the classical model of thermal pressurization in Mase & Smith (1987) and Rice (2006b), which we solve by a semi analytical procedure, in order to capture the effect of isothermal drained boundary conditions and traveling shear bands. Our results reappraise and extend the established models of frictional weakening due to thermal pressurization during coseismic slip. [Display omitted] • We study fault friction under thermo-hydro-mechanical couplings during large coseismic slip. • Our numerical analyses present frictional regain and frictional oscillations. • Our numerical analyses present a traveling principal slip zone inside the fault gouge. • Our physics-based model exhibits rate and state phenomenology. • We extend the established model of fault friction under thermal pressurization. [ABSTRACT FROM AUTHOR]

The rotation $${{\mathrm{polar}}}(F) \in {{\mathrm{SO}}}(3)$$ arises as the unique orthogonal factor of the right polar decomposition $$F = {{\mathrm{polar}}}(F)\,U$$ of a given invertible matrix $$F \in {{\mathrm{GL}}}^+(3)$$ . In the context of nonlinear elasticity Grioli (Boll Un Math Ital 2:252-255, 1940) discovered a geometric variational characterization of $${{\mathrm{polar}}}(F)$$ as a unique energy-minimizing rotation. In preceding works, we have analyzed a generalization of Grioli's variational approach with weights (material parameters) $$\mu > 0$$ and $$\mu _c \ge 0$$ (Grioli: $$\mu = \mu _c$$ ). The energy subject to minimization coincides with the Cosserat shear-stretch contribution arising in any geometrically nonlinear, isotropic and quadratic Cosserat continuum model formulated in the deformation gradient field $$F :=\nabla \varphi : \Omega \rightarrow {{\mathrm{GL}}}^+(3)$$ and the microrotation field $$R: \Omega \rightarrow {{\mathrm{SO}}}(3)$$ . The corresponding set of non-classical energy-minimizing rotations represents a new relaxed-polar mechanism. Our goal is to motivate this mechanism by presenting it in a relevant setting. To this end, we explicitly construct a deformation mapping $$\varphi _\mathrm{nano}$$ which models an idealized nanoindentation and compare the corresponding optimal rotation patterns $${{\mathrm{rpolar}}}^\pm _{1,0}(F_\mathrm{nano})$$ with experimentally obtained 3D-EBSD measurements of the disorientation angle of lattice rotations due to a nanoindentation in solid copper. We observe that the non-classical relaxed-polar mechanism can produce interesting counter-rotations. A possible link between Cosserat theory and finite multiplicative plasticity theory on small scales is also explored. [ABSTRACT FROM AUTHOR]

In any geometrically nonlinear, isotropic and quadratic Cosserat micropolar extended continuum model formulated in the deformation gradient field [ABSTRACT FROM AUTHOR]

In any geometrically nonlinear quadratic Cosserat-micropolar extended continuum model formulated in the deformation gradient field [ABSTRACT FROM AUTHOR]

Strong Cosserat elastic effects are observed in a designed composite consisting of unidirectional corrugated tubes in a hexagonal array. The torsional characteristic length is much larger than the tube diameter. The effective coupling number N approaches its upper bound of 1. Extremely large size effects are observed, about a factor of 128 in torsion. [ABSTRACT FROM AUTHOR]

Chiral dielectrics are considered from the perspective of continuum representations of spatial heterogeneity. Static effects in isotropic chiral dielectrics are predicted, provided the electric field has nonzero third spatial derivatives. The effects are compared with static chiral phenomena in Cosserat elastic materials which obey generalized continuum constitutive equations. Dynamic monopole-like magnetic induction is predicted in chiral dielectric media. [ABSTRACT FROM AUTHOR]