A geometric formulation of Schaefer’s theory of Cosserat solids

The Cosserat solid is a theoretical model of a continuum whose elementary constituents are notional rigid bodies, having both positional and orientational degrees of freedom. Cosserat elasticity has had a revival of interest stemming from applications in soft robotics and active soft matter. Here we present a formulation of the mechanics of a Cosserat solid in the language of modern differential geometry and exterior calculus, motivated by Schaefer’s “motor field” theory. The solid is modelled as a principal fibre bundle, with the base labelling the positions of the constituents and the fibre accommodating their orientations. Configurations of the solid are related by translations and rotations of each constituent. This essential kinematic property is described in a coordinate-independent manner using a bundle map. Configurations are equivalent if this bundle map is a global isometry of Euclidean space. Inequivalent configurations, representing deformations of the solid, are characterised by the local structure of the bundle map. Using Cartan’s magic formula we show that the strain associated with infinitesimal deformations is the Lie derivative of a connection one-form on the bundle. The classical infinitesimal strain is thus revealed to be a Lie algebra-valued one-form. Extending Schaefer’s theory, we derive the strain associated with finite deformations by integrating the infinitesimal strain along a prescribed path. This is path independent when the curvature of the connection one-form is zero. Path dependence signals the presence of topological defects and the non-zero curvature is then recognised as the density of topological defects. Mechanical stresses are defined by a virtual work principle in which the Lie algebra-valued strain one-form is paired with a dual Lie algebra-valued stress two-form to yield a scalar work volume form. The d’Alembert principle for the work form provides the balance laws, which we obtain in the limit of vanishing inertia. The work form is shown to be integrable for a hyperelastic Cosserat solid and the breakdown of integrability, relevant to active oriented solids, is briefly examined. Our work elucidates the geometric structure of Schaefer’s theory of the Cosserat solid, aids in constitutive modelling of active oriented materials, and suggests structure-preserving integration schemes for numerical simulation.


I. INTRODUCTION
The Cosserat solid is a model of a continuum in which the material constituent has both positional and orientational degrees of freedom.It was presented by E. and F. Cosserat in their 1909 opus "Théorie des Corps deformables".The Cosserats' approach to elasticity lay neglected for half a century following publication before it was revisited in the middle of the twentieth century in the context of the microstructured materials Ericksen and Truesdell [1], Toupin [2], Mindlin [3], Schaefer [4] and Kröner [5].We refer the reader to the lucid historical reviews of Schaefer [6] and Eringen [7].
Cosserat elasticity makes an appearance in effective descriptions of rods and shells undergoing large deformations [8][9][10].Orientational degrees of freedom emerge when a dimensional reduction is employed to represent the deformation of the body as a combination of translations and rotations of its rigid cross-sections.Cosserat rod and shell theories have been greatly successful in modelling deformations of soft and slender structures including biological filaments [11], membranes or active metabeams [12].The underlying theoretical principles of the Cosserats has been generalised in many directions [13].Recent examples include extensions of general relativity [14], fracton gauge theories [15] and topological defects in complex media [16,17].Cosserat elasticity was an inspiration for Elie Cartan in his approach to differential geometry [18].
Despite its theoretical interest, the experimental validity of the Cosserat solid remained in question for many years.Recent advances in 3D printing techniques has made it possible to construct metamaterials whose material response is consistent with Cosserat elasticity [19][20][21][22].Active soft matter provides many experimental systems which contain both translational and rotational degrees of freedom and Cosserat elasticity, with suitably chosen constitutive assumptions [23,24] may be used to model them.The additional rotational degrees of freedom in Cosserat theory allow for the long-wavelength description of chiral materials, whether active or passive.Such materials cannot, a priori, be described by Cauchy elasticity in which only translational degrees of freedom are retained.
Given the theoretical and experimental importance of the Cosserat solid, it is surprising that the mathematical underpinnings of the theory have remained veiled.While the Cosserats used a most cumbersome notation that is hard to fathom for the modern reader, the majority of the following treatments relied on local coordinates and Cartesian tensors.This has led to a profusion of notation and confusion even over the fundamental strain measures [25].The pioneering work of Schaefer [4] stands out as a notable exception in being one of the earliest treatments that attempted to extract the geometric content of the Cosserat theory and express it in the language of differential forms.Focussing on infinitesimal deformations, Schaefer absorbed the infinitesimal displacement and rotation fields of a Cosserat solid into "motor" fields.The "motors" were taken to be elements of a six-dimensional vector space of infinitesimal translations and rotations, and had been introduced by von Mises in his treatment of rigid body mechanics [26].Strain was defined as a differential one-form measuring the deviation of the motor field from an infinitesimal rigid transformation, the latter defining a parallelism and a covariant derivative.Schaefer recognized the importance of duality between forces and velocities: he defined stress as a differential two-form taking values in the dual space of motors.This was motivated by the observation that forces should generally be thought of as covectors taking values in the vector space dual to velocities, with the duality pairing giving the rate of power expended by the force [27][28][29].For the Cosserat solid, the pairing of stress with a velocity field (represented by a motor field) yields a scalar-valued two-form that can be integrated on the boundary of the solid giving the rate of work done by stresses.Balance laws and and topological defects in Cosserat media were discussed within the framework provided by the motor calculus, i.e the combination of the algebra of motors and the calculus of differential forms.
Schaefer's approach, despite its originality, economy and clarity has not been widely adopted, probably owing to the fact that motor calculus is much less known and appreciated in the continuum mechanics and soft matter community than tensor calculus.Further, his derivations are often based on heuristic arguments and analogies, rather than established mathematical constructions.To remedy we present Schaefer's theory of the Cosserat solid in the language of modern differential geometry.The key mathematical concepts that appear are Lie groups and their algebras, principal bundles, and differential calculus that results from combining these structures.From this perspective, a nonlinear theory of the Cosserat solid, absent in Schaefer's theory, becomes available and linearisation in Schaefer's theory is elucidated.
We provide a brief survey of our formulation before presenting the details below.Fibre bundles are manifolds which have a local product structure: any point of the bundle has a neighbourhood which looks like a product manifold of the form U × F , with U being an open subset of the base B of the bundle and another manifold F , called a typical fibre of the bundle [28].Two especially important classes of fibre bundles are vector bundles (when F is a vector space) and principal bundles (when F is a Lie group).In continuum mechanics, fibre bundles can naturally model media whose material particles possess a complex inner structure: the base manifold B represents the material particles and the fibre E is the collection of all possible configurations of the microstructure [30].They are also the underlying mathematical model in most field theories of soft condensed matter physics: the base B is the ambient space while the fibre F is an order parameter manifold corresponding to some broken symmetry [31].In the case of a Cosserat solid, the fibre bundle modelling the body is a principal bundle P [32][33][34][35] (as F can be identified with the orthogonal group) and therefore has a much richer structure than a general fibre bundle.We will show that Schaefer's space of motor fields is in fact a vector bundle associated to P , with the typical vector space being the Lie algebra of the Euclidean group, thus giving a precise meaning to motors.Invariance under rigid transformations will lead us to investigate the Maurer-Cartan form on the Euclidean group and view it as a Cartan connection ω on P [36].It will turn out that strain is the result of any changes in this connection form along a deformation -in particular, Schaefer's covariant derivative on the bundle of motor fields is induced by ω.Topological defects will be related to the curvature of the connection.Stress as a differential 2-form taking values in the dual bundle of motors will be introduced via a duality argument outlined above (for similar formulations of classical elasticity in terms of vector bundle valued differential forms, see [27,29]).Exploiting this duality, equations of motion will be derived from a virtual work principle.Since in the future we intend to apply our theory to active solids which are generally overdamped [37][38][39], we will not consider inertial forces.
A remarkable property of our formulation is that it only relies on the existence of a connection and does not directly use the metric of Euclidean space.This is a manifestation of the hierarchy of geometric structures: a Riemannian metric in differential geometry is a highlevel structure because it automatically gives rise to a unique connection through the Levi-Civita construction as well as providing an isomorphism between tangent and cotangent spaces and a distinguished volume form for orientable manifolds.It is interesting that a more low-level structure, namely a connection (which roughly speaking provides a way to parallel transport vectors along curves) is sufficient to describe deformations in these complex media.
The article is organized as follows.In Section II, the necessary preliminaries are given about the representation of Cosserat solids by principal bundles.In Section III the theory of strain is outlined while in Section IV compatibility conditions are discussed.In Section V stress is introduced and balance laws are derived from a virtual work principle.Finally, in Section VI constitutive laws are touched upon and conclusions are drawn in Section VII.We point the reader to the comprehensive textbook [28], which has an excellent section on geometric continuum mechanics in terms of bundle-valued differential Figure 1.Configuration of a simple body.In general, B is assumed to be just a smooth manifold that labels the material particles, but in many applications, B is identified with a submanifold of E 3 with the aid of a reference configuration.
forms; and to [40] which provides a brilliant introduction to exterior calculus.

II. PRELIMINARIES
In continuum mechanics, a body is modelled as a smooth 3-manifold B (often called the material manifold), and a configuration of the body is an embedding κ : B → E 3 into the ambient physical space E 3 , which is an affine space with underlying translational vector space E [41].Even though E is usually equipped with an inner product, we will not make direct use of it in the sequel, because, as we will demonstrate, strain and stress in a Cosserat solid are related to the action of the Euclidean group, not deformations in the metric structure.It is common practice to single out a reference configuration κ 0 : B → E 3 and label material points X ∈ B with their occupied position x = κ 0 (X) ∈ E 3 in the reference configuration, thereby identifying B with κ 0 (B) ⊂ E 3 [42].
However, this model only captures the translational degrees of freedom of the material particles.If the body has a more complex inner structure, an additional map ν : B → M on top of κ is introduced to describe the configuration of the microstructure, where M is the smooth manifold consisting of all possible microstructure configurations [13].Well-known examples are polar liquid crystals with M = S 2 (the two-sphere) or nematics with M = RP 2 (the real projective plane) [31].For Cosserat solids, M is generally taken to be SO(3) (the Lie group of rotations in E 3 ).This captures the property that the material points possess rotational degrees of freedom on top of translational ones.To describe a deformation with respect to a reference configuration κ 0 , ν 0 , one also needs to introduce a map which assigns to each element X ∈ B a transformation Ξ(X) of M that takes ν 0 (X) to ν(X).(It is usually assumed that a group K of transformations acts transitively on M, so Ξ(X) can be taken to be an element of K).
Figure 2. Configuration of a body with microstructure in the "traditional" formalism [13].
In this article we follow a slightly different approach to model Cosserat continua by enlarging the material manifold B to P = B × H, where H = O(3) is the proper orthogonal group [34].This way P becomes a (trivial) principal fibre bundle with structure group H over the base space B (also called the macromedium in this setting).It is instructive to think of elements p = (X, h) ∈ P = B×H as infinitesimal rigid bodies, where X labels their centre of mass, and h describes their orientation and chirality, e.g.via a body frame.The right action of H on P, given by p = (X, h) → p • k = (X, hk), can be thought of as a change of body frame and a change of chirality if det k = −1.The projection map π : P → B, (X, h) → X identifies the centre-of-mass label of a configuration of a material point, while a section s : B → P can be thought of as a specification of the orientation of the material particles.Since a configuration of the base is given by a map from B to E 3 , a configuration of P should be a map ψ . Configuration of a Cosserat continuum using principal fibre bundles.p = (X, S) ∈ B × H = P denotes an arbitrary element of the bundle, which gets mapped to from P to an H-bundle over E 3 .This bundle is usually taken to be the bundle of orthonormal (with respect to the standard inner product on E) frames F over E 3 .Nevertheless, it is convenient to single out an origin o ∈ E 3 and a Cartesian reference frame e i ∈ E, this way F can be identified with the group of isometries (generated by translations, rotations and reflections) The identity element of G will correspond to the reference frame which also defines the positive orientation.
The bundle structure on G is given by the quotient map q : G → G/H.The subgroup of orientation-preserving isometries of E 3 will be denoted by G + = SE(3).The configuration map ψ : P → G should induce a configuration κ : B → G/H of the macromedium and respect the rigidity of the microstructure.The first condition can be stated simply as: The second condition means that for any p ∈ P, h ∈ H, a rotated (and perhaps reflected) body frame p • h should get mapped to the rotated (and perhaps reflected) frame ψ(p) • h.Therefore: Equation ( 2) expresses that ψ is an H-equivariant map: in a certain sense, it is the mathematical manifestation of the rigidity of the material particles.Properties (1) and (2) imply that ψ is an H-bundle map between P and G [33,43].Let us again choose a reference configuration ψ 0 : P → G, then by identifying P with ψ 0 (P) ⊂ G so that ψ 0 ≡ Id, we assume from now on without loss of generality that P is a subbundle of G.In what follows, we will often use the following representation of elements of G by 4 × 4-matrices: where x ∈ G/H ∼ = R 3 and S ∈ O(3).The group element p in (3) corresponds to the orthonormal frame based at the point o + x a e a with basis vectors S ab e a .Now let ψ : P → G be an H-bundle map, then by the above stated properties it must be of the form for some functions y : B → R 3 and Q : B → H representing the deformation of the macromedium and the microstructure respectively, recovering the usual setting oulined in the second paragraph of this section.Since we are concerned with deformations that are continuously attainable from the identity, we will restrict Q(x) to have determinant +1 so that Q : B → SO(3) for the remainder of this article.We will denote the Lie algebra of G (which is the same as the Lie algebra of G + ) by g = se(3), and the Lie algebra of H by h = so(3).The matrix representation (3) induces a matrix representation of g given by: where u ∈ R 3 corresponds to an infinitesimal translation while Φ ∈ h is a 3 × 3 antisymmetric matrix corresponding to an infinitesimal rotation, which is usually identified with an axial vector φ ∈ R 3 .The Lie bracket on g is given by the usual matrix commutator: for w, z ∈ g, we have [w, z] = wz −zw, and the adjoint action of G on g is given by matrix conjugation: for g ∈ G, w ∈ g, Ad g w = gwg −1 .Note also that the Lie algebra splits as g = m ⊕ h, where m is the AdH-invariant subspace of infinitesimal translations (which can be identified with every tangent space of E 3 ), meaning that E 3 as a homogeneous space G/H is reductive [44].This slightly technical property of G and H will allow us to separate deformation and incompatibility measures into translational m and rotational h parts; otherwise, we could only treat them together as g-valued objects [36,45] (see below).

III. THEORY OF STRAIN
In continuum mechanics, one is interested in configurations up to rigid transformations.The building block of any continuum theory is a strain measure which captures the deviation of a mapping of a continuum body from a rigid transformation.It should be nonzero if and only if ψ differs from a global rigid body transformation.In the current framework, rigid motions are implemented by left multiplication L g : P → G, p → g • p by a constant group element g ∈ G + .Note that such a transformation acts simultaneously on the base (macromedium) and fibres (micromedium), therefore under a superimposed rigid transformation y → R • y + a for R ∈ SO(3), a ∈ R 3 the microstructure directors rotate together with the basepoint by R [38].This assumption, one of the most important in Cosserat theory, is often stated in the literature as the objectivity of microstructure directors [46].
In view of the above considerations, we consider the quantity: where ω is the Maurer-Cartan form on the group G and ψ * ω denotes the pullback of ω along the map ψ.The Maurer-Cartan form satisfies the well-known Maurer-Cartan structure equations: If we write ψ(p) = k(p) • p for a function k : P → G, then substituting into equation ( 6) yields:  [31], rotation of material particles of a Cosserat continuum independent of spatial rotations induces a deformation [38].This can be inferred by looking at the change in the angle between microstructure directors and vectors separating the centres of mass of material particles.
Equation (8) shows that E is identically zero if and only if k : P → G is a constant function, which is equivalent to ψ being a global rigid transformation.The deformation measure introduced in ( 6) is analogous to the classical Green-Lagrangian strain which is defined as the difference between the pullback of the spatial metric and the reference metric [42].In this case, the role of the metric as the indicator of deformation is replaced by the Maurer-Cartan form, which in turn can be viewed as a certain kind of connection (a Cartan connection) on the bundle P (for an in-depth presentation of Cartan connections, see [36]).In the appendix, we provide an informal motivation as to why the strain measure is a difference of connections.While the appearance of the Maurer-Cartan form in (6) might initially seem a bit surprising and perhaps difficult to grasp, it has been used implicitly and unconsciously in numerous contexts and applications related to elasticity and soft matter.For instance, strain measures in beam or plate theories [8][9][10]47] (also introduced based on left-invariance under rigid transformations) are in fact closely related to (6).
Let us now compute E explicitly.Using (3) and ( 4) we have: = 0 0 S T dx S T dS Similarly: 9) and ( 10) we find: Thus we recover the usual finite translational Q T dy − dx and rotational Q T dQ strain measures of the Cosserat solid, where Q(x) is the usual notation for the proper orthogonal rotation tensor describing the deformation of microstructure directors [25,48,49].The factors of S illustrate that E is tensorial: its components change in representations of O(3) under a change of section S, with the rotational strain measure unchanged if S = −I is an inversion, indicating that it is pseudovector valued.As we will see shortly, E is in fact a vector valued 1-form over B.
To obtain a consistent linearization of E, we consider infinitesimal deformations of the continuum, which are vector fields V on P such that their flows preserve the bundle structure [50] (i.e. they are H-bundle maps).This puts the following constraint on the vector field: As V is a vector field on a Lie group G, we can represent it by a Lie algebra valued function ξ : P → g such that V (p) = p • ξ(p), or equivalently ξ(p) = ι V ω| p , where ι denotes the interior product.Note also that ξ(p) satisfies: Hence ξ is a section of the associated vector bundle W = P × H g transforming in the adjoint representation of H on g.This vector bundle is what Schaefer [4] calls the space of motors.The Lie derivative, by definition, is the infinitesimal version or first order approximation of the difference ψ * − Id along the flow map of a vector field.Therefore, the linearization e of E is obtained by taking the Lie derivative of ω along the vector field V , which measures the infinitesimal change of ω along the vector field V [18,42].This is further elaborated in the appendix.Using Cartan's magic formula [28] and the Maurer-Cartan structure equations (7) we get: = dξ + ad ω ξ If V is not an infinitesimal displacement but a velocity vector field corresponding to a motion of the Cosserat continuum, then (14) defines the strain rate.One may also argue that ( 14) is in fact a more fundamental strain measure than (11) because it does not assume the existence of an arbitrary reference configuration.In addition, it is closer in spirit to differential geometry, where every meaningful operation or comparison can only be done locally, and global results can be obtained by integration.In the next section, an explicit demonstration of this viewpoint will be shown by taking e to be a general Lie algebra valued 1-form not necessarily coming from a vector field V , leading to the appearance of topological defects.Moreover, every reasonable elastic deformation can be decomposed into a sequence of infinitesimal deformations, consequently a finite strain measure may be obtained by integrating (14) along the motion [51,52].In the autonomous case when the vector field is independent of time parametrizing the motion, this integration gives the exponential of the Lie derivative operator, which results in the pullback operation along the flow of the vector field V , consistent with (11).
Let us now compute e in Cartesian coordinates on B. Formally, this is obtained by pulling back e to B via the section corresponding to S = I.Locally ξ is represented by a g-valued function on B, which we can specify by a pair of infinitesimal displacement and microrotation fields u : B → R 3 and Φ : B → so(3) [53].In Cartesian coordinates, since ω pulls back to the R 3 -valued 1-form dx (corresponding to the identity tensor on B) we get: Under the usual identification of elements so(3) with R 3 , Φ is identified with the axial vector field φ : B → R 3 and the matrix product Φdx becomes a vector cross product φ×dx.This way we recover the well-known infinitesimal strain measures of the Cosserat continuum du − φ × dx and dφ [54] [7].In components (ϵ ijk denotes the Levi-Civita permutation symbol, coming from the commutation relations of g): Both E and e are one-forms with values in the bundle W : this is due to the fact that both of them are essentially differences of connections on P, and as such are tensor-valued (in this case W -valued) one-forms on B. Therefore equation ( 14) can also be interpreted as the covariant derivative Dξ of ξ with respect to the connection on W induced by ω, the connection coefficients in Cartesian coordinates can be read off from ( 16) and ( 17).This result was first obtained by Schaefer [4].Hence the infinitesimal strain measure can be equally viewed as a covariant derivative of a section of W or the Lie derivative of the Maurer-Cartan form along a vector field on G.This also has an analogue in classical elasticity, where infinitesimal strain is the Lie derivative L U g of the metric g along a vector field U , but can also be expressed as 1  2 ∇U + (∇U ) T with the aid of the Levi-Civita connection ∇ corresponding to g [55].It is also interesting to remark that the infinitesimal strain measures ( 16)-( 17) have been recently derived in the context of symmetry breaking and high energy physics [56] via a coset construction starting from the Maurer-Cartan form on the Galilei group.Furthermore, Cartan connections play an important role in the extended theories of general relativity [14,45] (which have, in turn, also been influenced by the theory of continua with microstructure).
It should be apparent from the development above that the Maurer-Cartan form is the basic building block of almost any classical continuum theory respecting locality and invariance under rigid body transformations because it furnishes a set of differential invariants that can subsequently be used to construct Lagrangians or free energies for conservative systems in equilibrium.The most well-known example of this procedure is the Frenet-Serret framing [57]: for a curve γ : (a, b) → E 3 ∼ = G/H embedded in Euclidean space, the set of tangent, normal and binormal vectors give an adapted frame that provides a lift γ : (a, b) → G of the curve from the Euclidean (homogeneous) space to the Euclidean group .The curvature and torsion (not to be confused with curvature and torsion of a connection [58] in later sections) of the curve are nothing else but the components of the pullback of the Maurer-Cartan form on G via the lift γ.The simplest resulting elasticity theory of curves in two dimensions is the famous theory of the elastica dating back to Euler [59].The classification of submanifolds of general homogeneous spaces has led to the deep and beautiful method of moving frames [60,61], pioneered by Élie Cartan, where the Maurer-Cartan form plays a fundamental role.

IV. COMPATIBILITY CONDITIONS
If e is an W -valued one-form, a natural question to ask is that when it is compatible with a displacement field, i.e. whether there exist u, Φ satisfying (15).Necessary conditions can be elegantly obtained by means of exterior calculus, utilizing the exterior covariant derivative induced on W -valued differential forms by the connection ω [4].This is done as follows: any W -valued p-form α can be equivalently characterized as a g-valued p-form on P that is H-equivariant and horizontal [62].Now the exterior covariant derivative Dα is the g-valued p+1-form on P defined as: It can be verified that ( 18) is again an equivariant and horizontal form, hence descends to a W -valued p+1-form on B. The compatibility conditions follow from the fact that the connection is flat.One can show that: Therefore a necessary [63] condition for a W -valued 1form e to be integrable (i.e to be of the form Dξ for some section ξ of W ) is In coordinates (with a slight abuse of notation as τ here corresponds to an so(3)-valued 1-form): Using the definition of the exterior derivative and the correspondence between the action of so(3) on vectors and the vector cross product we obtain [7]: Any nonzero quantities on the right-hand side of ( 22) and ( 23) can be interpreted as measures of incompatibility.On the right hand side of (22) there can be a displacement-valued two-form T , usually associated with torsion, while on the right-hand side of (23) a microrotation-valued two-form Ω, associated with curvature [64].Therefore, in components, in general we have: They can be viewed together as a more general W -valued 2-form, J, satisfying the Bianchi identity (27) [65].
For the finite strain measure E, a similar compatibility condition can be derived based on the Maurer-Cartan structure equations: since the integrability condition reads: In general, one can model a Cosserat solid with defects by an abstract principal H-bundle P equipped with a Cartan connection η, that is, a g-valued 1-form on P satisfying the following properties [36]: (p • exp(tς)).
3. η| p : T p P → g is a linear isomorphism for all p ∈ P.
Albeit rather formally (for an intuitive interpretation of the above properties of η, see e.g.[45]), these three conditions capture and generalize the main properties of the Maurer-Cartan form on the principal H-bundle G → G/H.The crucial difference is that the Maurer-Cartan structure equations do not necessarily hold: the two-form Θ = dη + η ∧ η is in general nonzero.This quantity is called the curvature of the Cartan connection, representing the incompatibility of the underlying Cosserat solid.Note that this curvature is more general than the curvature of a linear connection because it takes values in the Lie algebra of the Euclidean group, not a subgroup of the general linear group.In some sense, one can view Θ as the "unification" of torsion and curvature into a single object, whose significance is highlighted by the fact that rotational and translational defects are intimately coupled in Cosserat solid as expressed by (24) and (25).
Modelling bodies with continuous distributions of topological defects (dislocations, disclinations, etc.) as abstract manifolds equipped with an extra structure incompatible with Euclidean space has a long and distinguished history, starting from the work of Nye [66], Bilby et al. [67], Kröner [68], Kondo [69] and others in the 1950s and 1960s.These authors have independently discovered that the Burgers vector density of continuous distributions of dislocations can be mapped to the torsion of a certain affine connection on the material manifold.It has also been realized that continuous distributions of disclinations and point defects are associated with the curvature (in the usual Riemannian sense) and the nonmetricity of an affine connection on the material manifold, see e.g.[70,71] for a modern exposition.
While the compatibility conditions and stress fields around defects for Cosserat solids have been obtained by various authors, the geometric origin of incompatibilities remained unclear.Based on the above discussion and inspired by [70], we propose a simple model of an incompatible Cosserat solid as a principal H-bundle P equipped with a non-flat Cartan connection η.The finite strain measure E becomes ψ * ω − η and (28) changes to: where D η is the exterior covariant derivative with respect to the connection η.This definition of strain as a difference of connections also sheds light on the appearance of the contorsion tensor in previous works on topological defects in general relativity and complex media [16,17].In addition, with the aid of the theory of stress and and constitutive relations given in the following sections, one can calculate the stress fields around defects in Cosserat solids [72].

V. THEORY OF STRESS AND BALANCE LAWS
Most treatments of classical elasticity derive the governing equations of elasticity via the following train of thought [73]: interactions between material particles are assumed to be short-ranged, which, together with the action-reaction principle, implies the existence of a second-rank stress tensor through Cauchy's tetrahedron argument.Equations of motion are then obtained by postulating the balance of linear and angular momentum on each subbody of a macroscopic body, with the former yielding Cauchy's momentum equations while the latter the symmetry of the stress tensor upon localization.However, there are a number of difficulties encountered when trying to extend this method to complex materials [74].
First, the above assumptions restrict the nature of boundary interactions between subbodies to be only force-like depending linearly on the normal of the boundary surface.However, one can envisage cases when "higher-order" forces corresponding to media sensitive to higher gradients in displacements or the curvature of boundary surfaces [74,75].In the case of complex materials, torques or even torque dipoles could be transmitted through boundaries.In general, it is unclear what kind of balance laws one should postulate for these quantities.
Second, the relationship between the balance of angular momentum and the symmetry of stress tensor is quite mysterious: in Cauchy elasticity [73], this balance law is only used to argue that the stress tensor is symmetric, but subsequently neglected in the solution of any practical problem [76].Furthermore, if material points in a Cauchy continuum are assumed to be structureless point masses only possessing translational degrees of freedom, why does the balance of angular momentum constitute an independent equation?It is in stark contrast with the Newtonian mechanics of point masses, where Newton's second law is the only equation needed to work out the motion of a point mass.
Finally, from a differential geometric standpoint, the vector-valued integrals in balance laws are ill-defined because in non-flat spaces one cannot identify distant tangent spaces unambiguously.
To address these problems, we construct the theory of stress and balance laws from another perspective, based on the principle of virtual work [77].Even though this approach is much less popular and appreciated than Cauchy's, it also has a long and distinguished history, essentially originating from Piola who extended the ideas of d'Alembert to the mechanics of continua [75].The fundamental assumption is that applying a rigid transformation to a body requires no work, which motivates the introduction of strain as a measure of deviation from a rigid transformation (in classical elasticity, it is the change in the metric along the deformation, while in the present case it is the change in the connection as argued in Section III).Stress is defined as a dual quantity to strain: the duality pairing gives the virtual work done by stress during an infinitesimal variation of configuration.This variation or virtual displacement is represented by a vector field on the current configuration respecting the imposed boundary conditions (in classical elasticity, it is a vector field on the ambient space, while for Cosserat solids it is a vector field exactly analogous to V in ( 12)).Equations of motion are obtained by a version of the d'Alembert's principle which states that the total virtual work is zero for any admissible virtual displacement field.This method answers all the above concerns as follows.
First, in the variational formalism, stress has exactly as many degrees of freedom as the model of the continuum has: for rotational degrees of freedom, one has moment stresses, and for higher-gradient forces additional "hyperstresses" [75,78,79].Balance laws are obtained from a single application of the principle of virtual work.
Second, the duality between stress and strain clearly highlights the number of degrees of freedom in a model and the underlying symmetries.For example, in classical elasticity the infinitesimal strain 1  2 (∂ i u j + ∂ j u i ) is an element of the vector space of symmetric second-rank tensors, therefore the stress tensor, being an element of the dual vector space, is also automatically symmetric.The symmetry of the stress tensor has thus been identified as a consequence of the number of degrees of freedom in a Cauchy continuum (because the strain only depends on the displacement vector) and the axiom that rigid transformations require no work [77].This way in Cauchy elasticity the only balance law following from the principle of virtual work is Cauchy's momentum equation.
Finally, the principle of virtual work involve scalarvalued integrals only which are well-defined on any smooth manifold.
Motivated by the above discussion and inspired by other geometric treatments of elasticity [27][28][29], for Cosserat solids we therefore define stress Σ as an W *valued 2-form [80] on the current configuration ψ(P), where W * is the dual vector bundle of W = ψ(P) × H g (this is the space of dual motors in the language of [4]).The intuitive reason behind this definition is that (Eulerian) velocities are sections of W (analogously as in ( 12)), and the natural duality pairing between stress and velocity gives a scalar-valued 2-form that can be integrated on the current configuration to obtain the power of stresses exerted on surfaces inside the body.(More generally, stress can be defined in n dimensions as an n − 1 form taking values in a vector bundle dual to velocities/virtual displacements, and thus can be integrated on any n − 1submanifold to give the rate of power on a hypersurface, see e.g.[28,81]).While this would initially suggest that Σ is described by three indices (one bundle index and two form indices), one usually reduces this to two indices overall to obtain the second rank Cauchy stress tensor in the presence of a metric and a volume form by exploiting the isomorphism provided by the Hodge duality between 2-forms and 1-forms, i.e. by associating area elements with corresponding normal vectors.It turns out that there is a more general correspondence between n − 1forms and vectors afforded by the interior product which only requires a volume form, so the existence of a metric structure is not required.
The duality pairing between stress and infinitesimal strain or strain rate gives a scalar-valued 3-form on B = φ(B), which can be integrated over any 3-dimensional submanifold of B to give the virtual work or the rate of work done by stresses, respectively [27].This pairing can be defined for general p-and q-forms taking values in W * and W , outputting a scalar-valued p + q-form on B. For forms µ ⊗ α and s ⊗ β, where µ and s are sections of W * and W , α and β are p and q-forms on B, it is given by ⟨µ ⊗ α, s ⊗ β⟩ := ⟨µ, s⟩α ∧ β, then extended linearly to all forms (here ⟨µ, s⟩ is the natural pairing of sections of W and W * ).
Let us compute this pairing explicitly for the stress 2form Σ and the infinitesimal strain 1-form e.We choose a basis {v i , r i } of g, where the {v i } generate infinitesimal translations (spanning the subspace m) and the {r i } generate infinitesimal rotations (spanning the subspace h).Suppose that in local Cartesian coordinates x i , i = 1, 2, 3, e is represented by the pair of mand h-valued 1-forms ε ij v j dx i and τ ij r j dx i as in ( 16) and ( 17).(We do not distinguish upper and lower indices in this section.)Let vol be the standard volume form in Euclidean space, then the 2-forms A i = ι ∂i vol form a basis of 2-forms satisfying dx i ∧ A j = δ ij vol, moreover, dA i = 0 for Cartesian coordinates.Let {v * i , r * i } be the dual basis of g * to {v i , r i }, and expand Σ as: where σ ij is the generalization of the Cauchy stress and χ ij represents moment stresses, the first index corre-sponds to the area element while the second to the direction of force/moment respectively.We have: Another important mathematical tool is the exterior covariant derivative operator D * , induced on W * -valued forms on B by ω, which will allow us to integrate by parts and write down local forms of balance laws.There are multiple ways to define it, one of them is through the Leibniz rule: for a W * -valued p-form Π, D * Π is the unique W * -valued p + 1-form such that for any section ξ of W we have: [27] d⟨ξ, Π⟩ = ⟨ξ, D * Π⟩ + ⟨Dξ, Π⟩ This is very similar to how one induces the covariant derivative on the cotangent bundle of a manifold from a linear connection on its tangent bundle.Another possible definition comes from the observation that W * is also an associated bundle to ψ(P) by the coadjoint representation of H on g * .We can again view any W * -valued p-form Π as a g * -valued equivariant horizontal p-form on ψ(P), then its exterior covariant derivative D * Π is nothing else but where ad * denotes the coadjoint representation of g on g * , defined via: The differential operator in (33) is similar to expressions which appear in the theory of the Euler-Poincarï¿oe equations arising from the variation of group-invariant Lagrangians on Lie group configuration spaces [82,83].Suppose that external volume forces and torques act on the Cosserat solid, given by an W * -valued 3-form F on the current configuration B, as well as traction forces and couples on the boundary ∂ B, represented by a W *valued 2-form T .Let us impart a virtual deformation ξ on the current configuration B, given by a section of W , then the principle of virtual work states that the total virtual work done by all forces and stresses vanishes [77][78][79].The three contributions (neglecting inertia) to the virtual work are: a): The virtual work of external volume forces and torques: δW ext = B⟨ξ, F ⟩.

b):
The virtual work of traction forces and torques: The virtual work of stress in the bulk (the minus sign is conventional): δW int = − B⟨Dξ, Σ⟩.
The principle of virtual work asserts that: for all virtual displacements ξ.Substituting the integral expressions into (35), using Stokes' theorem [28] and the exterior covariant derivative (32) to integrate by parts yields: As (36) holds for any virtual displacement field ξ, we deduce the following equilibrium equations and boundary conditions [4]: Equation ( 37) provides yet another interpretation of the operator D * : it is the generalization of the divergence operator appearing in the usual Cauchy momentum equation.(37) is written in the Eulerian picture: to obtain the analougous Lagrangian equilibrium equations, one can pull back (37) to the reference configuration via ψ, the W * -valued 2-form S = ψ * Σ is going to be the analogue of the second Piola-Kirchhoff stress tensor in finite elasticity [42].The first Piola-Kirchhoff stress can be found from Σ by pulling back the form "part" from B to the reference configuration B but doing nothing to the vector part [27] (it is useful because it helps performing the integral (36) in Lagrangian coordinates on B).
For an explicit coordinate representation of (37), let us work again in Cartesian coordinates, writing F = (f i v * i + m i r * i ) vol for the volume forces and torques.As in (15), ω is locally given by the 1-form v i dx i , thus using (30) and ( 33) Using the commutation relations of g we get ad * vi v * j = ϵ ijk r * k and ad * vi r * j = 0, therefore: Separating the coefficients of v * i and r * j and substituting into (37) we obtain the familiar local balance of linear and angular momentum: When F = 0, (37) takes the simple form D * Σ = 0. Since ω is flat, D * D * = 0 (just like in (19)), so the solution is of the form Σ = D * Y for a stress potential Y , which is a W * -valued 1-form that is only defined up to a gauge transformation Y → Y + D * α for some section α of W * .While stress potentials for Cosserat solids were derived in e.g.[84], it is important to note that similar potentials have been extensively utilized in many areas of classical physics, such as scalar and vector potentials in electromagnetism, Airy and Maxwell stress functions for certain problems in Cauchy elasticity and the Papkovich-Neuber representiation of Stokes flows.By formulating the theory of Cosserat solids in terms of these stress potentials (and with the inclusion of inertia too), one obtains a gauge theory similar to fracton theory in condensed matter physics [15].Equations ( 37) and ( 38) also suggest geometric structure-preserving numerical schemes using discrete exterior calculus [85][86][87].

VI. CONSTITUTIVE RELATIONS
In order to be able to close the above system of equations, one needs to specify a constitutive law between stress and strain, characteristic of the material in question [88].In general, these can be quite complicated nonlinear relations which may depend on time rates of tensors too.In this paper, we restrict our attention to the simple case when the constitutive law gives the second Piola-Kirchhoff stress S at a point X ∈ B as a function S(X, E(X)) of X and the strain measure E at the point X.An important class of materials is the set of hyperelastic materials, for which the stress derives from a potential or stored energy function per unit mass U (X, E(X)) as S = ∂U ∂E .These bodies are conservative: they do not perform any work along a closed cycle in deformation space [42].
The number of independent parameters in the constitutive law is restricted by the symmetries of the material.In our setting, a material symmetry is an H-bundle automorphism of P that leaves the functional form of the constitutive relation invariant.More specificially, the material symmetry group for a hyperelastic Cosserat solid at a point X ∈ B is a subgroup K X ≤ H of the orthogonal group such that: for any possible strain E(X), where L R : P → G is a rigid rotation (and perhaps reflection) of the reference configuration P by R about X, i.e.: The motivation behind definition (43) is that the action of a material symmetry is given by replacing ψ with ψ R = ψ • L R and the connection ω on P by L * R ω (c.f.rotating or reflecting the reference configuration); this way (11), where Γ corresponds to the axial vector-valued wryness tensor, transforms to E R = R T Y R, (det R) R T ΓR , taking into account the axial vector nature of Γ [89].
Hence if the Cosserat solid is centrosymmetric (equivalent to achirality in three dimensions), meaning −I ∈ K X , then U (X, E) = U (X, Y, Γ) = U (X, Y, −Γ), so there cannot be any translation-rotation coupling in the constitutive relation.If K X is a subgroup of SO(3), then the material is not symmetric under reflections [90] and that the microstructural constitutents are chiral.Other important cases are when the material is isotropic so that K X = O(3) or hemitropic when K X = SO(3), in these cases the constitutive relation can only involve certain invariants of the strain E(X).Chirality is one of the most exciting features of the theory of Cosserat solids because it is ubiquitous in nature and soft matter [91] while entirely absent in Cauchy elasticity, and experimentally engineered mechanical metamaterials have recently been demonstrated to exhibit fascinating chiral effects such as acoustical activity [20][21][22], confirming predictions of micropolar elasticity.
Linear constitutive relations are often a good approximation to constitutive behaviour, especially when deformations are small.Suppose that in local coordinates and with respect to a basis {w A } of g, we expand E as a g-valued 1-form E = E Ai w A dx i .Using the dual basis {w * A } of g * and a dual basis of 2-forms A i as in the previous section, we expand S as the g * -valued 2-form S = S Bj w * B A j .A linear constitutive law is then given by: for a general stiffness tensor C ABij .Compared with the two Lamï¿oe constants of classical elasticity, a linear isotropic or hemitropic constitutive relation contains six or nine material constants respectively [90].If the material is hyperelastic, then S Ai = ∂U/∂E Ai for a stored energy function U (X, E), and the symmetry of second partial derivatives implies the major symmetry: However, there are recent so-called odd elastic models of active solids [92] with rich phenomenology where the major symmetry ( 46) is broken and the material is able to perform work along a closed cycle in deformation space via using energy produced from an internal mechanism.Odd elasticity has also been extended to and experimentally realized in micropolar beams with piezoelectric activity [12].It is quite likely that most oriented solids in nature [91] also violate the symmetry (46), therefore it is an interesting future direction of research to study further consequences of such an "odd" constitutive relation [23].Finally, for a more comprehensive discussion of constitutive relations involving possible material symmetry groups and questions of material uniformity and homogeneity in Cosserat media, we refer to e.g.[34,89].

VII. DISCUSSION
In this paper a geometric theory of Cosserat solids has been outlined, modelling the body as a principal fibre bundle P over a three-dimensional base and O(3) as the typical fibre.Configurations of the continuum were defined as bundle maps ψ from P to an ambient O(3)-bundle G, while the strain measure as the difference between two Cartan connections: a material one and the pullback of the Maurer-Cartan form on G along ψ.Compatibility conditions were expressed using exterior calculus, and incompatibilities were identified as the curvature of the material connection on the Cartan geometry P. Stress was introduced as a dual quantity of strain, and balance laws were obtained using the principle of virtual work.Constitutive laws relating stress and strain were also briefly discussed, along with issues of chirality and hyperelasticity.
Our work can be extended in a multitude of different directions.One can take any Klein geometry G/H and consider a generalized Cosserat solid as a Cartan geometry modelled on G/H (i.e. a principal H-bundle P with a Cartan connection η) and bundle maps ψ : P → G as configurations.(The ambient space can in principle be another arbitrary Cartan geometry modelled on G/H).The notions of strain, stress, compatibility conditions and constitutive relations can be defined exactly analogously as it was done in the main text.For example, micromorphic elasticity [3,7] can also be cast in this language by taking G to be the general affine group GA(3) of line-preserving transformations of E 3 and H = GL(3) the general linear group.However, the physical relevance of such models is unclear as the introduction of the strain measure was postulated based on invariance under rigid body transformations, not under e.g.general affine transformations, not to mention the potentially extremely large number of material parameters involved in the constitutive relations.Nevertheless, one can certainly consider two-dimensional Cosserat continua on the plane with G = E(2) and H = O(2) or on the 2-sphere S 2 with G = O(3) and H = O(2).It is also possible to study other complex materials using fibre bundles [13,30], for example configurations of polar liquid crystals can be viewed as bundle maps between principal bundles over E 3 and typical fibre S 2 .Nevertheless, these bundles are no longer principal, therefore strain and stress measures become more involved for these media.
An obvious limitation of our theory is that it only considers static deformations.A straightforward way to incorporate time is to make the configuration ψ timedependent, then material velocity can be defined as a vector field V obtained by taking the partial derivative of ψ with respect to time.(An alternative and conceptually cleaner approach would be to work in a covariant spacetime setting [52]).Analogously to ( 12) and ( 13), the velocity vector field V can again be interpreted as a section of a vector bundle over B with typical fibre g.For the dynamical equations of motion one has to add an inertial contribution to the virtual work principle (35), usually deriving from the variation of a kinetic energy, modelled as a bundle metric on the vector bundle of velocities.However, the choice of this kinetic energy metric is far from obvious [93,94]: it is usually assumed (by analogy with a standard rigid body) that it is given by a mass density times the spatial metric ρg ij on the translational part and ρI ij on the rotational part where I ij is a symmetric moment of inertia density tensor.In general, one cannot rule out a term coupling angular and translational velocites [95] or a more complicated expression for the kinetic energy [96].An additional issue is that the evolution of inertial quantites (mass and moment of inertia density) are usually governed by conservation laws, and while mass conservation is a natural assumption, moment of inertia conservation as proposed by Eringen [7] is more controversial [97] (in particular, it has been suggested in the literature [98] that one should consider moment of inertia production terms as well).In any case, most experimental systems of interest only undergo small deformations or are overdamped, in which case either a linearised treatment -where the inertial terms can be added in straightforwardly -is sufficient, or inertia terms are absent entirely and one may include viscous damping terms on a phenomenological basis [99].
We conclude by mentioning that the geometrisation of Schaefer's theory of the Cosserat solid immediately suggests methods for numerically integration that preserve geometric structures [85][86][87].We shall address this in forthcoming work where the geometrisation will be implemented in the alternative setting of a field theory [100].
In this appedix we give a heuristic derivation of the infinitesimal strain measures, based on Cartan's method of moving frames [101].This rationalizes as to why the strain measures can be considered as differences of connections.Here we view the Cosserat body as a continuum whose material points have an orthonormal set of directors attached to them, in line with the classical treatment of Ericksen and Truesdell [1] and others.Consider two The translational strain can be measured by looking at the change in the angle between the vectors ei and dx as ei goes to fi and dx goes to dy.Similarly, rotational strain is captured by the change in the relative angle of infinitesimally close directors ei and ei + dei as they go to fi and fi + dfi respectively.
infinitesimally close points x and x + dx of a Cosserat solid, with corresponding directors e i and e i + de i .Now suppose that the body undergoes an infinitesimal deformation, such that x gets mapped to y, x+dx gets mapped to y + dy, and the directors transform to f i and f i + df i .
In the micropolar theory, the directors are assumed to be rigid, which means that f i = Q(e i ) for some proper orthogonal tensor Q.If we view dx and dy as infinitesimal vectors based at x and y respectively, we can write: Similarly, for the infinitesimal relative orientations of the directors we get: The deformation is encoded in the relative changes ∆θ i = θ i − θ i and ∆Ω j i = Ω j i − Ω j i .We cannot directly subtract dy and dx or df i and de i because they live in different vector spaces.However, we can use Q −1 = Q T to bring back the directors f i to x.This way we find the following deformation measures: Q T (df i ) − de i = Q T f j Ω j i − e j Ω j i = e j ∆Ω j i (52) (52) can also be written as: Hence we recover the usual strain measures Q T dy − dx and Q T dQ [25,102].Now suppose that Q = I + tω + O(t 2 ) and y = x + tv + O(t 2 ), where ω ∈ so(3) corresponding to ω ∈ R 3 then we find: The infinitesimal strain measures are recovered if the derivative wrt.t is taken as t → 0. We can also phrase the above discussion as follows.Let the tuple E = (x, e i ) be a frame at x, viewed as a row vector formed from x and e i , similarly F = (y, f i ).Then we have: dE = (dx, de j ) = (x, e j ) • 0 0 dF = (dy, df j ) = (y, f j ) • 0 0 Then our strain measures can be viewed together as: Equations ( 47),( 48),( 49) and ( 50) are known as Cartan's structure equations, describing an affine connection with in a local trivialization (i.e. a choice of directors e i and f i ) using vector-valued differential forms [28,101].The deformation measures are essentially differences of the material connection and solder forms Ω j i and θ i and the pullback connection and solder forms Ω j i and θ i .Equations ( 53)-( 54) also highlight the fact that Cartan connections "encapsulate" the solder forms and connection forms into a larger Lie algebra-valued form.

Figure 4 .
Figure 4. Commutative diagram corresponding to the bundle map ψ : P → G.

Figure 5 .
Figure 5. Rigid transformation of a Cosserat continuum.Observe that the material particles rotate together with spatial points, therefore spatial and microstructural rotations are not independent.

Figure 6 .
Figure 6.Unlike magnetic systems or the O(n) model in statistical field theory[31], rotation of material particles of a Cosserat continuum independent of spatial rotations induces a deformation[38].This can be inferred by looking at the change in the angle between microstructure directors and vectors separating the centres of mass of material particles.

Figure 7 .
Figure 7. Illustration of deformation in Cosserat medium.The translational strain can be measured by looking at the change in the angle between the vectors ei and dx as ei goes to fi and dx goes to dy.Similarly, rotational strain is captured by the change in the relative angle of infinitesimally close directors ei and ei + dei as they go to fi and fi + dfi respectively.