A geometric formulation of Schaefer’s theory of Cosserat solids Open Access

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 the Cosserat brothers in their 1909 opus Théorie des Corps Déformables.¹ 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,² Toupin,³ Mindlin,⁴ Schaefer⁵ and Kröner.⁶ We refer the reader to the lucid historical reviews of Schaefer⁷ and Eringen.⁸ 

Cosserat elasticity makes an appearance in effective descriptions of rods and shells undergoing large deformations.^9–11 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 modeling deformations of soft and slender structures including biological filaments,¹² membranes or active metabeams.¹³ The underlying theoretical principles of the Cosserats has been generalised in many directions.¹⁴ Recent examples include extensions of general relativity,¹⁵ fracton gauge theories¹⁶ and topological defects in complex media.^17,18 Cosserat elasticity was an inspiration for Élie Cartan in his approach to differential geometry.¹⁹ 

Despite its theoretical interest, the experimental validity of the Cosserat solid remained in question for many years. Recent advances in 3D printing techniques have made it possible to construct metamaterials whose material response is consistent with Cosserat elasticity.^20–23 Active soft matter provides many experimental systems which contain both translational and rotational degrees of freedom and Cosserat elasticity, with suitably chosen constitutive assumptions^24,25 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.²⁶ The pioneering work of Schaefer⁵ 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. Focusing 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.²⁷ 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.^28–30 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. Fiber bundles are manifolds which have a local product structure: any point of the bundle has a neighborhood 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 fiber of the bundle.²⁹ Two especially important classes of fiber bundles are vector bundles (when F is a vector space) and principal bundles (when F is a Lie group). In continuum mechanics, fiber bundles can naturally model media whose material particles possess a complex inner structure: the base manifold B represents the material particles and the fiber E is the collection of all possible configurations of the microstructure.³¹ 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 fiber F is an order parameter manifold corresponding to some broken symmetry.³² In the case of a Cosserat solid, the fiber bundle modeling the body is a principal bundle P^33–36 (as F can be identified with the orthogonal group) and therefore has a much richer structure than a general fiber 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, which is a different type of connection on a principal bundle from the more widely used principal connections.³⁷ 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 Refs. 28 and 30). 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,^38–40 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 high-level structure because it automatically gives rise to a unique connection through the Levi–Cività 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 Sec. II, the necessary preliminaries are given about the representation of Cosserat solids by principal bundles. In Sec. III the theory of strain is outlined while in Sec. IV compatibility conditions are discussed. In Sec. V stress is introduced and balance laws are derived from a virtual work principle. Finally, in Sec. VI constitutive laws are touched upon and conclusions are drawn in Sec. VII. We point the reader to the comprehensive textbook,²⁹ which has an excellent section on geometric continuum mechanics in terms of bundle-valued differential forms; and to Ref. 41 which provides a brilliant introduction to exterior calculus.

In continuum mechanics, a body is modeled as a smooth 3-manifold $B$ (often called the material manifold), and a configuration of the body is an embedding $κ:B→E3$ into the ambient physical space $E3$ (Fig. 1), which is an affine space with underlying translational vector space $E$⁠.⁴² 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 $κ0:B→E3$ and label material points $X∈B$ with their occupied position $x=κ0(X)∈E3$ in the reference configuration,⁴³ thereby identifying $B$ with $κ0(B)⊂E3$⁠.

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.¹⁴ Well-known examples are polar liquid crystals with $M=S2$ (the two-sphere) or nematics with $M=RP2$ (the real projective plane).³² For Cosserat solids, $M$ is generally taken to be SO(3) (the Lie group of rotations in $E3$⁠). 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 κ₀, ν₀, one also needs to introduce a map which assigns to each element $X∈B$ a transformation Ξ(X) of $M$ that takes ν₀(X) to ν(X) (Fig. 2). [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].

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 (Fig. 3).³⁵ This way $P$ becomes a (trivial) principal fiber 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 center 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 $E3$⁠, a configuration of $P$ should be a map ψ from $P$ to an H-bundle over $E3$⁠. This bundle is usually taken to be the bundle of orthonormal (with respect to the standard inner product on $E$⁠) frames $F$ over $E3$⁠. Nevertheless, it is convenient to single out an origin $o∈E3$ and a Cartesian reference frame $ei∈E$⁠, this way $F$ can be identified with the group of isometries (generated by translations, rotations and reflections) G = E(3) of $E3$⁠. 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 $E3$ will be denoted by G⁺ = SE(3).

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 $Lg:P→G,p↦g⋅p$ by a constant group element g ∈ G⁺. Note that such a transformation acts simultaneously on the base (macromedium) and fibers (micromedium), therefore under a superimposed rigid transformation y ↦ R · y + a for $R∈SO(3),a∈R3$ the microstructure directors rotate together with the basepoint by R (Figs. 5 and 6)).³⁹ This assumption, one of the most important in Cosserat theory, is often stated in the literature as the objectivity of microstructure directors.⁴⁷ 

It is important to stress that a Cartan connection (a precise definition is given in the next section) is different from a principal connection, despite having very similar properties. From a technical viewpoint, just as a principal connection, a Cartan connection is given by a Lie algebra valued one-form on a principal H-bundle $P$⁠, but unlike a principal connection, it takes values not in the Lie algebra $h$ of H but in the Lie algebra $g$ of a larger Lie group G which contains H as a subgroup. Furthermore, while principal connections always have kernels furnishing a horizontal vector subbundle of the tangent bundle $TP$ of $P$ complementary to the vertical subbundle, the Cartan connection form, just like the Maurer-Cartan form, has to provide a linear isomorphism between $TP$ and $P×g$⁠. Conceptually, a Cartan connection identifies the base of the bundle with the homogeneous space G/H and the bundle itself with G on an infinitesimal level while allowing for curvature (in the sense that the Maurer-Cartan structure equations might no longer hold for the connection), hence generalizing Klein’s Erlangen program.³⁷ A Cartan connection is thus more restrictive than a principal connection, which can be defined on an arbitrary principal H-bundle irrespective of the base.

Thus we recover the usual finite translational Q^Tdy − dx and rotational Q^TdQ 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.^26,48,49 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 $B$⁠.

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. Strain measures in beam or plate theories^9–11,50 (also introduced based on left-invariance under rigid transformations) are in fact closely related to (6). For instance, in Ref. 50, the time-dependent configurations of a Cosserat rod are described using a pair of maps $Λ(s,t),r(s,t)$⁠, where t denotes time, s a centerline coordinate, Λ(s, t) ∈ SO(3) the orientation of each cross-section with respect to a fixed spatial frame and $r(s,t)∈R3$ the position of the centerline, measured from a fixed origin and in the fixed spatial frame. This pair of maps is then interpreted as a section of a trivial fiber bundle $Q≜X×SE(3)→X$⁠, where the base X denotes “spacetime,” i.e., the collection of labels (s, t). The group of rotations about the origin — as a subgroup of isometries of $E3$ — acts on the fibers of $Q$ from the left, thus turning $Q$ to a left principal bundle with structure group SO(3). The rotational part of the right Maurer-Cartan form on the fibers gives rise to a natural principal connection on the bundle $Q→Q/SO(3)$⁠, from which the strain measures Λ^TdΛ and Λ^Tdr are eventually obtained, of which the first is the rotational part of our strain measure E, while the second corresponds to the translational part of the pullback ψ*ω (it is not exactly a strain measure because it can never vanish).

The key conceptual difference between our approach and Ref. 50 is that in our case, the principal bundle models the Cosserat solid, while in Ref. 50, it plays the role of the configuration space for a Lagrangian field theory. We do not interpret the bundle map ψ as a section of any bundle. The physical requirement of left invariance under rigid transformations (both rotations and translations) directly leads to the study of the left Maurer-Cartan form as a Cartan connection on the bundle, rather than introducing another principal fiber bundle and a principal connection via the left action of a symmetry group.

If V is not an infinitesimal displacement but a velocity vector field corresponding to a motion of the Cosserat continuum, then (17) defines the strain rate. One may also argue that (17) is in fact a more fundamental strain measure than (14) 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 (17) along the motion.^52,53 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 (14).

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 Eq. (17) 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 (19) and (20). This result was first obtained by Schaefer.⁵ 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 $LUg$ of the metric g along a vector field U, but can also be expressed as $12∇U+∇UT$ with the aid of the Levi–Cività connection ∇ corresponding to g.⁵⁵ It is also interesting to remark that the infinitesimal strain measures (19) and (20) have been recently derived in the context of symmetry breaking and high energy physics⁵⁶ 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^15,46 (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 $γ:(a,b)→E3≅G/H$ embedded in Euclidean space, the set of tangent, normal and binormal vectors give an adapted frame that provides a lift $γ̃:(a,b)→G$ 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 connection⁵⁸ 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 $γ̃$⁠. 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.

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:³⁷ 

1. $Rh*η=Adh−1η∀h∈H$⁠, i.e., η is H-equivariant.

2. $ηp⋅ς=ς∀ς∈h$⁠, where $p⋅ς=ddtt=0p⋅exp(tς)$⁠.

3. $η|p:TpP→g$ is a linear isomorphism for all $p∈P$⁠.

Albeit rather formally (for an intuitive interpretation of the above properties of η, see e.g., Ref. 46), these three conditions capture and generalize the main properties of the Maurer-Cartan form on the principal H-bundle G → G/H. It is also interesting to note that the first two conditions are identical to those defining a principal H-connection on the bundle $P$⁠, but η takes values in $g$ rather than $h$⁠. The crucial difference between a general Cartan connection η and the Maurer-Cartan form ω on G 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 principal 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 (27) and (28).

Modeling 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 Kondo⁶⁹ 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 non-metricity of an affine connection on the material manifold, see e.g., Refs. 70 and 71 for a modern exposition.

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 Sec. 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 (15)]. 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 $12∂iuj+∂jui$ 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 scalar-valued integrals only which are well-defined on any smooth manifold.

Motivated by the above discussion and inspired by other geometric treatments of elasticity,^28–30 for Cosserat solids we therefore define stress Σ as an $W̃*$-valued 2-form⁸⁰ on the current configuration $ψ(P)$⁠, where $W̃*$ is the dual vector bundle of $W̃=ψ(P)×Hg$ (this is the space of dual motors in the language of Ref. 5). The intuitive reason behind this definition is that (Eulerian) velocities are sections of $W̃$ [analogously as in (15)], 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 − 1-submanifold to give the rate of power on a hypersurface, see e.g., Refs. 29 and 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 − 1-forms and vectors afforded by the interior product which only requires a volume form, so the existence of a metric structure is not required.

Suppose that external volume forces and torques act on the Cosserat solid, given by an $W̃*$-valued 3-form F on the current configuration $B̃$⁠, as well as traction forces and couples on the boundary $∂B̃$⁠, represented by a $W̃*$-valued 2-form T. Let us impart a virtual deformation ξ on the current configuration $B̃$⁠, given by a section of $W̃$⁠, then the principle of virtual work states that the total virtual work done by all forces and stresses vanishes.^77–79 The three contributions (neglecting inertia) to the virtual work are:

• The virtual work of external volume forces and torques: $δWext=∫B̃⟨ξ,F⟩$⁠.

• The virtual work of traction forces and torques: $δWtrac=∫∂B̃⟨ξ,T⟩$⁠.

• The virtual work of stress in the bulk (the minus sign is conventional): $δWint=−∫B̃⟨Dξ,Σ⟩$⁠.

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.⁸⁹ 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.⁴³ 

In components, E is $QTdy−dx,QTdQ=Ydx,Γdx$ from (14), where Γ corresponds to the axial vector-valued wryness tensor, transforms to $ER=RTYR,detRRTΓR$⁠, taking into account the axial vector nature⁹⁰ of Γ. 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 reflections⁹¹ 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 matter⁹² while entirely absent in Cauchy elasticity, and experimentally engineered mechanical metamaterials have recently been demonstrated to exhibit fascinating chiral effects such as acoustical activity,^21–23 confirming predictions of micropolar elasticity.

In this paper a geometric theory of Cosserat solids has been outlined, modeling the body as a principal fiber bundle $P$ over a three-dimensional base and O(3) as the typical fiber. 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 modeled 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 modeled 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^4,8 can also be cast in this language by taking G to be the general affine group GA(3) of line-preserving transformations of $E3$ 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 S² with G = O(3) and H = O(2). It is also possible to study other complex materials using fiber bundles,^14,31 for example configurations of polar liquid crystals can be viewed as bundle maps between principal bundles over $E3$ and typical fiber S². However, 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 time-dependent, 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 setting⁵³). Analogously to (15) and (16), the velocity vector field V can again be interpreted as a section of a vector bundle over $B$ with typical fiber $g$⁠. For the dynamical equations of motion one has to add an inertial contribution to the virtual work principle (39), usually deriving from the variation of a kinetic energy, modeled as a bundle metric on the vector bundle of velocities. However, the choice of this kinetic energy metric is far from obvious:^94,95 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 velocites⁹⁶ or a more complicated expression for the kinetic energy.⁹⁷ 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 Eringen⁸ is more controversial⁹⁸ (in particular, it has been suggested in the literature⁹⁹ 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.¹⁰⁰ 

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.^86–88 We shall address this in forthcoming work where the geometrisation will be implemented in the alternative setting of a field theory.¹⁰¹ 

We thank Professor M. E. Cates and Dr. Lukas Kikuchi for many helpful discussions and a critical reading of the manuscript. This work was supported by the Engineering and Physical Sciences Research Council (B.N., Award No. EP/W524141/1).

The authors have no conflicts to disclose.

Balázs Németh: Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal). Ronojoy Adhikari: Formal analysis (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

Equations (A1)–(A4) 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.^29,102 The deformation measures are essentially differences of the material connection and solder forms $Ωij$ and θⁱ and the pullback connection and solder forms $Ω̃ij$ and $θ̃i$⁠. Equations (A12) and (A13) also highlight the fact that Cartan connections “encapsulate” the solder forms and connection forms into a larger Lie algebra-valued form.