Cosserat Rotations Research Articles

Quasiconvexity in a model of fiber-reinforced solids based on Cosserat elasticity theory

The quasiconvexity inequality associated with energy minimizers is derived in the context of a nonlinear Cosserat elasticity theory for fiber-reinforced elastic solids in which the intrinsic flexural and torsional elasticities of the fibers are taken into account explicitly. The derivation accounts for non-standard kinematic constraints, associated with the materiality of the embedded fibers, connecting the deformation gradient and the Cosserat rotation field.

Cosserat shape effects in the bending of foams

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.

A Cosserat Breakage Mechanics model for brittle granular media

This paper introduces a Breakage Mechanics model formulated in the framework of Cosserat continuum theory. This formulation allows us to account for the interaction of grain size evolution with shear band formation in a physically motivated and thermodynamically consistent way. We provide an upscaling procedure that introduces material parameters accounting for the contribution of the Cosserat rotations, as well as the micro-inertia of the entire grain size distribution. A particular highlight of the new formulation is the capacity to describe an evolving internal Cosserat length that takes account of the complete grain size distribution and its evolution. In addition, we specify a particular model that requires no additional calibration relative to the same model in the classical continuum. We apply this model to the study of a layer sheared under constant volume which simulates the fast undrained deformation observed in seismogenic faults. Through linear stability analysis we use the model to examine the effect of grain size polydispersity on the thickness of shear bands. By implementing the model using the finite element method we provide an explanation for the geological formation of double cataclastic shear bands.

Optimality of the relaxed polar factors by a characterization of the set of real square roots of real symmetric matrices

AbstractWe consider the problem to determine the optimal rotations which minimize urn:x-wiley:00442267:media:zamm201800120:zamm201800120-math-0002for a given diagonal matrix with positive entries . The objective function W is the reduced form of the Cosserat shear‐stretch energy, which, in its general form, is a contribution in any geometrically nonlinear, isotropic, and quadratic Cosserat micropolar (extended) continuum model. We characterize the critical points of the energy , determine the global minimizers and compute the global minimum. This proves the correctness of previously obtained formulae for the optimal Cosserat rotations in dimensions two and three. The key to the proof is the result that every real matrix whose square is symmetric can be written in some orthonormal basis as a block‐diagonal matrix with blocks of size at most two.

Non-classical continuum theories for solid and fluent continua and some applications

ABSTRACTThis paper presents two specific thermodynamically consistent non-classical continuum theories for solid and fluent continua. The first non-classical continuum theory for solid continua incorporates Jacobian of deformation in its entirety in the conservation and the balance laws and the derivation of the constitutive theories. The second non-classical continuum theory for solid continua considers Jacobian of deformation in its entirety as well as the Cosserat rotations in the conservation and balance laws as well as the constitutive theories. The first non-classical continuum theory for fluent continua presented here considers velocity gradient tensor in its entirety. The second non-classical continuum theory for fluent continua considers velocity gradient tensor in its entirety as well as Cosserat rotation rates in the derivation of the conservation and balance laws and the constitutive theories. Since the non-classical continuum theories for solid and fluent continua considered here incorporate additional physics of deformation due to rotations and rotation rates compared to classical continuum mechanics, the conservation and balance laws of classical continuum mechanics are shown to require modification as well as a new balance law balance of moment of moments is required to accommodate the new physics due to rotations and rotation rates. Eringen’s micropolar, micromorphic and microstretch theories, couple stress theories and nonlocal theories are also discussed within the context of the non-classical theories presented here for solid and fluent continua. Some applications of these theories are also discussed.

Ordered rate constitutive theories for non-classical thermoviscoelastic solids with memory incorporating internal and Cosserat rotations

This paper considers conservation and balance laws for non-classical solid continua in the presence of internal rotations ( $${}_i \pmb {\varvec{\varTheta }}$$ ) due to the Jacobian of deformation and Cosserat rotations ( $${}_e \pmb {\varvec{\varTheta }}$$ ) at each material point. In these balance laws, internal rotations are completely defined as functions of the displacement gradient tensor, but Cosserat rotations are additional three degrees of freedom at each material point. When these rotations are resisted by the deforming matter, conjugate moments are created. For thermoviscoelastic solids with memory, these result in additional energy storage, dissipation mechanism, and rheology. This paper presents a thermodynamically consistent derivation of constitutive theories for such solids based on the entropy inequality in conjunction with representation theorem. Material coefficients are derived and discussed. The constitutive theories are presented in the absence as well as in the presence of the balance of moment of moments as additional balance law for non-classical continuum theories, and the resulting theories are compared with classical continuum theories in which this balance law is not needed. Retardation moduli corresponding to the Cauchy stress tensor as well as the Cauchy moment tensor are derived. In this paper we only consider small strain, small deformation physics.

Ordered rate constitutive theories for thermoviscoelastic solids without memory incorporating internal and Cosserat rotations

The work in this paper is based on a non-classical continuum theory in the Lagrangian description for thermoviscoelastic solids without memory in which the conservation and balance laws are derived by incorporating internal rotations ( $${}_i \pmb {\varvec{{\varTheta } }}$$ ) arising from the Jacobian of deformation ( $$ \pmb {\varvec{J }}$$ ), as well as Cosserat rotations ( $${}_e \pmb {\varvec{{\varTheta } }}$$ ) at a material point. Such non-classical solids have additional energy storage due to rotations and additional dissipation due to rotation rates compared to classical continuum theories. Rotations $${}_i \pmb {\varvec{{\varTheta } }}$$ are completely defined by $$ \pmb {\varvec{J }}$$ , whereas displacements $$ \pmb {\varvec{u }}$$ and Cosserat rotations $${}_e \pmb {\varvec{{\varTheta } }}$$ are degrees of freedom at each material point. When $${}_i \pmb {\varvec{{\varTheta } }}$$ and $${}_e \pmb {\varvec{{\varTheta } }}$$ are resisted by the deforming matter, conjugate moments arise, which together with ( $${}_i \pmb {\varvec{{\varTheta } }},{}_e \pmb {\varvec{{\varTheta } }}$$ ) and ( $${}_i\overset{\,\text{. }}{ \pmb {\varvec{{\varTheta } }}},{}_e\overset{\,\text{. }}{ \pmb {\varvec{{\varTheta } }}}$$ ) result in additional work and rate of work. This paper utilizes thermodynamic framework for non-classical solids derived based on internal as well as Cosserat rotations and presents a thermodynamically consistent derivation of the constitutive theories that incorporate the aforementioned deformation physics. The constitutive theories are derived using the conditions resulting from the entropy inequality in conjunction with the representation theorem.

Necessity of law of balance of moment of moments in non-classical continuum theories for solid continua

In the non-classical continuum theories for solid continua the presence of internal rotations and their gradients arising due to Jacobian of deformation and/or consideration of Cosserat rotations as additional unknown degrees of freedom at a material point necessitate existence of moment tensor. For small deformation, small strains theories, in Lagrangian description the Cauchy moment tensor and the rates of rotation gradients are rate of work conjugate pair in addition to the rate of work conjugate Cauchy stress tensor and the strain rate tensor. It is well established that in such non-classical theories the Cauchy stress tensor is non-symmetric and the antisymmetric components of the Cauchy stress tensor are balanced by gradients of the Cauchy moment tensor, the balance of angular momenta balance law. In the non-classical continuum theories incorporating internal rotations and conjugate moment tensor that are absent in the classical continuum theories, the fundamental question is “are the conservation and balance laws used in classical continuum mechanics sufficient to ensure dynamic equilibrium of the deforming volume of matter”. At this stage the Cauchy moment tensor remains non-symmetric if we only consider standard balance laws that are used in classical continuum theories. Thus, requiring constitutive theories for the symmetric as well as anti-symmetric Cauchy moment tensors. The work presented in this paper shows that when the thermodynamically consistent constitutive theories are used for symmetric as well as antisymmetric Cauchy moment tensor non physical and spurious solutions result even in simple model problems. This suggests that perhaps the additional conjugate tensors resulting due to presence of internal rotations, namely the Cauchy moment tensor and the antisymmetric part of the Cauchy stress stress tensor must obey some additional law or restriction so that the spurious behavior is precluded. This paper demonstrates that in the non-classical theory with internal rotations considered here the law of balance of moment of moments and the consideration of the equilibrium of moment of moments are in fact identical. When this balance law is considered the Cauchy moment tensor becomes symmetric, hence eliminating the constitutive theory for the antisymmetric Cauchy moment tensor and thereby eliminating spurious and non physical solutions. The necessity of this balance law is established theoretically and is also demonstrated through model problems using thermoelastic solids with small strain small deformation as an example. The findings reported in this paper hold for thermoviscoelastic solids with and without memory as well as when deformation and strains are small. Extensions of the concepts presented here for finite deformation and finite strain will be presented in a follow up paper.

Ordered Rate Constitutive Theories for Non-Classical Thermoviscoelastic Fluids Incorporating Internal and Cosserat Rotation Rates

This paper considers conservation and balance laws for non-classical fluent continua in the presence of internal rotation rates due to the velocity gradient tensor and the rotation rates due to Cosserat rotations. In these balance laws, the internal rotation rates are completely defined as functions of the velocity gradient tensor, but the Cosserat rotation rates are additional three degrees of freedom at each material point. When these rotation rates are resisted by the deforming continua, conjugate moments are created. For thermoviscoelastic fluent continua, these result in additional dissipation mechanism as well as rheology. This paper presents a thermodynamically consistent derivation of constitutive theories for such fluids based on the entropy inequality in conjunction with the representation theorem using integrity, i.e., complete basis. Material coefficients are derived and discussed. The constitutive theories are presented in the absence as well as presence of the balance of moments of moments as a balance law and are compared with the classical continuum theories. Retardation moduli corresponding to the Cauchy stress tensor, both symmetric and antisymmetric, as well as the symmetric and antisymmetric Cauchy moment tensors are derived. The constitutive theories presented in this paper are ordered rate theories. These incorporate convected time derivatives of the strain tensor up to order [Formula: see text] and the convected time derivatives of the stress and moment tensors up to certain orders. Simplified forms of the constitutive theories are also derived. It is shown and concluded that the constitutive theories for thermoviscoelastic fluids based on classical continuum theory such as Maxwell model, Oldroyd-B model, Giesekus model, etc., are all a subset of the constitutive theories presented in this paper for non-classical physics. Both compressible and incompressible thermoviscoelastic fluids are considered.

Grioli’s Theorem with weights and the relaxed-polar mechanism of optimal Cosserat rotations

Let F \in \mathrm {GL}^+(3) and consider the right polar decomposition F = \mathrm {R_p}(F) \cdot U into an orthogonal factor \mathrm {R_p}(F) \in \mathrm {SO}(3) and a symmetric, positive definite factor U = \sqrt{F^TF} \in \mathrm {Psym}(3) . In 1940 Giuseppe Grioli proved that \argmin\limits_{R\,\in\,\mathrm{SO}(3)} \| R^T F - 1\!\! 1 \|^2= \{\, \mathrm{R}_p(F)\,\} = \argmin\limits_{R\,\in\,\mathrm{SO}(3)}\|F - R\|^2\; . This variational characterization of the orthogonal factor \mathrm {R_p}(F) \in \mathrm {SO}(n) holds in any dimension n \geq 2 (a result due to Martins and Podio-Guidugli). In a similar spirit, we characterize the optimal rotations \mathrm{rpolar}_{\mu,\mu_c}(F) \coloneqq \argmin\limits_{R\in \mathrm{SO}(n)} \{\mu \|\mathrm{sym}(R^TF - 1\!\! 1)\|^2 + \mu_c\|\mathrm{skew}(R^TF - 1\!\! 1)\|^2\} for given weights \mu > 0 and \mu_c \geq 0 . We identify a classical parameter range \mu_c \geq \mu > 0 for which Grioli’s Theorem is recovered and a non-classical parameter range \mu > \mu_c \geq 0 giving rise to a new type of globally energy-minimizing rotations which can substantially deviate from \mathrm {R_p}(F) . In mechanics, the weighted energy subject to minimization appears as the shear-stretch contribution in any geometrically nonlinear, quadratic, and isotropic Cosserat theory.

The relaxed-polar mechanism of locally optimal Cosserat rotations for an idealized nanoindentation and comparison with 3D-EBSD experiments

The rotation ${\rm polar}(F) \in {\rm SO}(3)$ arises as the unique orthogonal factor of the right polar decomposition $F = {\rm polar}(F) \cdot U$ of a given invertible matrix $F \in {\rm GL}^+(3)$. In the context of nonlinear elasticity Grioli (1940) discovered a geometric variational characterization of ${\rm polar}(F)$ 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 $\mu_c \geq 0$ (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 $F := \nabla\varphi: \Omega \to {\rm GL}^+(3)$ and the microrotation field $R: \Omega \to {\rm SO}(3)$. The corresponding set of non-classical energy-minimizing rotations $$ {\rm rpolar}^\pm_{\mu,\mu_c}(F) := \substack{{\rm argmin}\\ R\,\in\,{\rm SO(3)}} \Big\{ W_{\mu, \mu_c}(R\,;F) := \mu\, || {\rm sym}(R^TF - 1)||^2 + \mu_c\, ||{\rm skew}(R^TF - 1)||^2 \Big\} $$ 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 $\varphi_{\rm nano}$ which models an idealized nanoindentation and compare the corresponding optimal rotation patterns ${\rm rpolar}^\pm_{1,0}(F_{\rm nano})$ 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.

The geometrically nonlinear Cosserat micropolar shear–stretch energy. Part II: Non‐classical energy‐minimizing microrotations in 3D and their computational validation***

In any geometrically nonlinear, isotropic and quadratic Cosserat micropolar extended continuum model formulated in the deformation gradient field and the microrotation field , the shear–stretch energy is necessarily of the form urn:x-wiley:00442267:media:zamm201600030:zamm201600030-math-0003We aim at the derivation of closed form expressions for the minimizers of in SO(3), i.e., for the set of optimal Cosserat microrotations in dimension , as a function of . In a previous contribution (Part I), we have first shown that, for all , the full range of weights and can be reduced to either a classical or a non‐classical limit case. We have then derived the associated closed form expressions for the optimal planar rotations in SO(2) and proved their global optimality. In the present contribution (Part II), we characterize the non‐classical optimal rotations in dimension . After a lift of the minimization problem to the unit quaternions, the Euler–Lagrange equations can be symbolically solved by the computer algebra system Mathematica. Among the symbolic expressions for the critical points, we single out two candidates which we analyze and for which we can computationally validate their global optimality by Monte Carlo statistical sampling of SO(3). Geometrically, our proposed optimal Cosserat rotations act in the plane of maximal stretch. Our previously obtained explicit formulae for planar optimal Cosserat rotations in SO(2) reveal themselves as a simple special case. Further, we derive the associated reduced energy levels of the Cosserat shear–stretch energy and criteria for the existence of non‐classical optimal rotations.

Non-classical continuum theory for solids incorporating internal rotations and rotations of Cosserat theories

This paper presents a non-classical continuum theory in Lagrangian description for solids in which the conservation and the balance laws are derived by incorporating both the internal rotations arising from the Jacobian of deformation and the rotations of Cosserat theories at a material point. In particular, in this non-classical continuum theory, we have (i) the usual displacements \(( \pmb {\varvec{u }})\) and (ii) three internal rotations \(({}_i \pmb {\varvec{{\varTheta } }})\) about the axes of a triad whose axes are parallel to the x-frame arising from the Jacobian of deformation (which are completely defined by the skew-symmetric part of the Jacobian of deformation), and (iii) three additional rotations \(({}_e \pmb {\varvec{{\varTheta } }})\) about the axes of the same triad located at each material point as additional three degrees of freedom referred to as Cosserat rotations. This gives rise to \( \pmb {\varvec{u }}\) and \({}_e \pmb {\varvec{{\varTheta } }}\) as six degrees of freedom at a material point. The internal rotations \(({}_i \pmb {\varvec{{\varTheta } }})\), often neglected in classical continuum mechanics, exist in all deforming solid continua as these are due to Jacobian of deformation. When the internal rotations \({}_i \pmb {\varvec{{\varTheta } }}\) are resisted by the deforming matter, conjugate moment tensor arises that together with \({}_i \pmb {\varvec{{\varTheta } }}\) may result in energy storage and/or dissipation, which must be accounted for in the conservation and the balance laws. The Cosserat rotations \({}_e \pmb {\varvec{{\varTheta } }}\) also result in conjugate moment tensor which, together with \({}_e \pmb {\varvec{{\varTheta } }}\), may also result in energy storage and/or dissipation. The main focus of the paper is a consistent derivation of conservation and balance laws that incorporate aforementioned physics and associated constitutive theories for thermoelastic solids. The mathematical model derived here has closure, and the constitutive theories derived using two alternate approaches are in agreement with each other as well as with the condition resulting from the entropy inequality. Material coefficients introduced in the constitutive theories are clearly defined and discussed.

The geometrically nonlinear Cosserat micropolar shear–stretch energy. Part I: A general parameter reduction formula and energy‐minimizing microrotations in 2D

In any geometrically nonlinear quadratic Cosserat‐micropolar extended continuum model formulated in the deformation gradient field and the microrotation field , the shear–stretch energy is necessarily of the form urn:x-wiley:00442267:media:zamm201500194:zamm201500194-math-0003where is the Lamé shear modulus and is the Cosserat couple modulus. In the present contribution, we work towards explicit characterizations of the set of optimal Cosserat microrotations as a function of and weights and . For , we prove a parameter reduction lemma which reduces the optimality problem to two limit cases: and . In contrast to Grioli's theorem, we derive non‐classical minimizers for the parameter range in dimension . Currently, optimality results for are out of reach for us, but we contribute explicit representations for which we name and which arise for by fixing the rotation axis a priori. Further, we compute the associated reduced energy levels and study the non‐classical optimal Cosserat rotations for simple planar shear.

Effects of periodic fluctuations of micro‐polar boundary conditions on shear localizations in granular soil–structure interaction

SUMMARYIn this work, the interface behavior between an infinite extended narrow granular layer and a rough surface of rigid body is investigated numerically, using finite element method in the updated Lagrangian (UL) frame. In this regard, the elasto‐plastic micro‐polar (Cosserat) continuum approach is employed to remove the limitations caused by strain‐softening of materials in the classical continuum. The mechanical properties of cohesionless granular soil are described with Lade's model enhanced by polar terms, including Cosserat rotations, curvatures, and couple stresses. Furthermore, the mean grain diameter as the internal length is incorporated into the constitutive relations accordingly. Here, the evolution and location of shear band, within the granular layer in contact with the rigid body, are mainly focused. In this regard, particular attention is paid to the effects of homogeneous distribution and periodic fluctuation of micro‐polar boundary conditions, prescribed along the interface. Correspondingly, the effects of pressure level, mean grain diameter, and stratified soil are also considered. The finite element results demonstrate that the location and evolution of shear band in the granular soil layer are strongly affected by the non‐uniform micro‐polar boundary conditions, prescribed along the interface. It is found that the shear band is located closer to the boundary with less restriction of grain rotations. Furthermore, the predicted thickness of shear band is larger for higher rotation resistance of soil grains along the interface, larger mean grain diameter, and higher vertical pressure. Regarding the stratified soil, comprising a thin layer with slightly different initial void ratio, the shear band moves towards the layer with initially higher void ratio. Copyright © 2011 John Wiley & Sons, Ltd.

Numerical Investigation on Polar Effects in Localized Shear Zone

Numerical simulations related to polar effects in an infinite extended granular layer under shearing movement and constant vertical pressure are presented. The mechanical behavior of cohesionless granular soil is described within the framework of micro-polar (Cosserat) continuum and using an elasto-plastic constitutive relation. The influence of Cosserat rotations and couple stresses are taken into account using the mean grain size as characteristic length. Finite element method in Updated Lagrangian (UL) frame is used to consider large deformations during calculations. The numerical results demonstrate that for large shearing movement, the shear deformations within the granular layer are localized into a narrow zone. The FE-calculations indicate that the polar effects manifest by the appearance of noticeable grain rotations, high void ratios, pronounced volume changes within the localized shear zone.

Effect of periodic fluctuation of soil particle rotation resistance on interface shear behaviour

The interface behaviour between infinite extended narrow granular layer and bounding structure is numerically investigated using finite element method. The micro-polar (Cosserat) continuum approach within the framework of elasto-plasticity is employed to remove the numerical difficulties caused by strain-softening of materials in classical continuum mechanics. Mechanical properties of cohesionless granular soil are described with Lade's model enhanced with polar terms including Cosserat rotations, curvatures and couple stresses via mean grain diameter as the internal length. The main attention of paper is laid on the influence of spatial periodic fluctuation of rotation resistance of soil particles interlocked with the surface of bounding structure on evolution and location of shear band developed inside granular body. The finite element results demonstrate that the location and evolution of shear localization in granular body is strongly affected by prescribed non-uniform micro-polar kinematic boundary conditions along the interface.

Strain localization and failure load predictions of geosynthetic reinforced soil structures

This study illustrates the differences between the elasto-plastic cap model and Lade\'s model with Cosserat rotation through the analyses of two large-scale geosynthetic-reinforced soil (GRS) retaining wall tests that were brought to failure using a monotonically increasing surcharge pressure. The finite element analyses with Lade\'s model were able to reasonably simulate the large-scale plane strain laboratory tests. On average, the finite element analyses gave reasonably good agreement with the experimental results in terms of global performances and shear band occurrences. In contrast, the cap model was not able to simulate the development of shear banding in the tests. In both test simulations the cap model predicted failure loads that were substantially less than the measured ones.

Capturing strain localization in reinforced soils

Lade’s single hardening soil model with Cosserat rotation embodied in the finite element method is employed to investigate the behavior of geosynthetic reinforced soils with special attention to the development of shear banding. The ability of the finite element model to detect shear banding in a reinforced soil is examined against three high quality small-scale laboratory plane strain tests on Toyoura sand with and without reinforcement. These three tests were chosen because of the clear failure surfaces that developed in the soil during loading. The FEM analyses were able to reasonably simulate the plane strain laboratory tests including both unreinforced and reinforced cases. The FEM analyses gave reasonably good agreement with the experimental results in terms of global stress–strain relationships and shear band occurrences. Furthermore, and based on FE analyses of a hypothetical geosynthetic reinforced soil (GRS) retaining wall, it is shown that the geosynthetic reinforcements are very effective in hindering the formation of shear bands in GRS retaining walls when small spacing between the reinforcement layers was used. When used properly, the geosynthetic reinforcements made the soil behave as a truly reinforced mass of considerable stiffness and strength.

Effect of rolling on dissipation in fault gouges

Sliding and rolling are two outstanding deformation modes in granular media. The first one induces frictional dissipation whereas the latter one involves deformation with negligible resistance. Using numerical simulations on two-dimensional shear cells, we investigate the effect of the grain rotation on the energy dissipation and the strength of granular materials under quasistatic shear deformation. Rolling and sliding are quantified in terms of the so-called Cosserat rotations. The observed spontaneous formation of vorticity cells and clusters of rotating bearings may provide an explanation for the long standing heat flow paradox of earthquake dynamics.