PPseudomomentum: origins and consequences
H. Singh ∗ Institute of Mathematics, Laboratory for Computation and Visualisation in Mathematics and Mechanics,´Ecole polytechnique f´ed´erale de Lausanne, MA C1 612 (Bˆatiment MA), Station 8, CH-1015 Lausanne, Switzerland.
J. A. Hanna † Department of Mechanical Engineering, University of Nevada,1664 N. Virginia St. (0312), Reno, NV 89557-0312, U.S.A. (Dated: July 14, 2020)The balance of pseudomomentum is discussed and applied to first gradient elasticity, ideal fluids,and the mechanics of inextensible rods and sheets. A general framework is presented in which thesimultaneous variation of an action with respect to position, time, and material labels yields bulkbalance laws and jump conditions for momentum, energy, and pseudomomentum. The example offirst gradient elasticity of space-filling continua is treated at length. Then the pseudomomentumbalance is employed to derive several results, beginning with the conservation of vorticity, circu-lation, and helicity in ideal fluids. A mathematical similarity is noted between the evaluation ofcirculation along a material loop and the J-integral of fracture mechanics. Integration of the pseu-domomentum balance, making use of a prescription for singular sources derived by analogy withthe continuous form of the balance, directly provides the propulsive force driving passive reconfig-uration or locomotion of confined, inhomogeneous elastic rods. The conserved angular momentumand pseudomomentum are identified in the classification of conical sheets with rotational inertia orbending energy.
I. INTRODUCTION
Pseudomomentum is a property of material continua. It is variously referred to as material momentum, configu-rational momentum, Eshelbian momentum, quasimomentum, and impulse. Our impression is that its balance law isunder-utilized and under-recognized in many branches of continuum and structural mechanics. The current effort isour attempt to synthesize and expand a body of prior work on variational approaches and symmetry in continua, aswell as revisit and reinterpret several results in the mechanics of thin structures.While a material element acquires momentum by virtue of its motion in space, pseudomomentum is a propertyassociated with the motion of an element or defect through the material medium itself. As Peierls [1] commented, thedistinction between the momentum and the pseudomomentum of an electromagnetic or other field is only meaningfulin the presence of a material medium. An important early paper in this area is that of Rogula [2], who defineda Lagrangian density of fields interacting with a homogeneous body, and constructed unnamed conservation lawsassociated with shifts in material labels. He remarked on the difference between fields defined per unit mass of materialand those associated with the embedding space. The influential work of Eshelby defined the “force” on an elasticsingularity as a dynamic quantity conjugate to a kinematic parameter characterizing the position of the defect withinthe material medium [3]. This material or configurational “force” and the pseudomomentum have the same relationshipto each other as the standard force and momentum. Later, Eshelby introduced a four-dimensional elastic “energy-momentum tensor” [4, 5] by analogy with the energy-momentum tensors of classical field theories [6]. The three-dimensional non-temporal part of this tensor has come to be known as the Eshelby tensor; its elements are materialcomponents corresponding to pseudomomentum rather than spatial components corresponding to momentum as inthe classical construction . The Eshelby tensor is related to the path independent integrals of Rice and Cherepanovin fracture mechanics [8, 9]. Rogula later revisited material forces and corresponding balance laws in the presenceof localized inhomogeneities [10]. In a series of papers, A. Golebiewska Herrmann presented a unified Lagrangiantreatment of continuum mechanics for systems that admit an action principle, deriving a complete set of dynamicbalance laws [11–13]. She showed that while conservation of momentum follows from homogeneity of the ambient space,conservation of material momentum implies a homogeneous material “space”. The inference was that the balanceof material momentum follows from variation of the material coordinates, although this process was not explicitly ∗ harmeet.singh@epfl.ch † [email protected] Whereas some fields such as gravity are associated with space-time, other fields such as displacements in elasticity are attached to a set ofmaterial elements. The state of a body is described using both material coordinates describing locations within the medium and spatialcoordinates associated with the embedding of the body in the ambient space. The non-temporal part of an “energy-momentum tensor”can be obtained when the action is varied with respect to either material or spatial coordinates, yielding either the Eshelbian or classicalresult. These are different physical quantities. As the flux associated with momentum is known as “stress” in continuum mechanics,that associated with pseudomomentum (material momentum) could be called pseudostress (material stress). Although, as Ericksen [7]points out, Eshelby’s exposition was in a linear-elastic context, where the distinction between spatial and material coordinates is blurred. a r X i v : . [ phy s i c s . c l a ss - ph ] J u l performed. She constructed several balance laws corresponding to symmetries of the material space and related themto path independent integrals. Further applications of the pseudomomentum concept have been extensively exploredby Kienzler and G. Herrmann [14]. Maugin was a prolific promoter of these ideas whose viewpoint can be found inseveral books and review articles [15–18]. A recent book by O’Reilly [19] makes a case for the utility of materialmomentum balance in the context of rod mechanics.In the broader context of dynamical phase transitions, where the second law of thermodynamics needs to be con-sidered, some researchers including Fried, Gurtin, and Podio-Guidugli [20–22] consider the balance of configurationalmomentum to be a basic law of continuum mechanics, on par with the balance of momentum and independent ofany constitutive considerations. O’Reilly also agrees with this interpretation [19]. This viewpoint contrasts with thatof Rogula, Maugin, Kienzler and G. Herrmann, Rajagopal and Srinivasa [23], and Yavari and co-workers [24], whoconsider the material momentum balance law to be merely the pullback of the spatial momentum balance onto thematerial’s reference configuration. Rajagopal and Srinivasa [23] attribute the existence of configurational forces toevolving reference configurations of the body. In the present work, which restricts itself to conservative mechanicalsystems governed by an action principle, both the energy and pseudomomentum balances could be derived from themomentum balance without invoking additional assumptions. However, instead we employ a variational formulationthat is a non-standard extension of the usual action principle of classical mechanics, although it is one not withoutprecedent [25–27].Other notable early works include those of Sturrock on the “pseudo-energy-pseudo-momentum tensor” of waves inplasmas [28], Gilbert and Mollow on “tensor momentum” associated with elastic vibrations [29], Broer’s presentationof the pseudomomentum balance for a moving string [30], Knowles and Sternberg’s derivation of material conserva-tion laws for finite elastostatics by application of Noether’s theorem to “translations” and “rotations” of materialcoordinates [31], and Fletcher’s extension of their results to elastodynamics [32]. Edelen presented a comprehensivetreatment of variational elastostatics and its associated conservation laws [25]. R. Hill derived the Eshelby tensor forelastostatics using the principle of virtual work [33]. Variational derivations of the balance of pseudomomentum maybe found in Nelson [34] and Thellung [35], in the context of interaction between an electromagnetic field and an elasticdielectric. The material relabeling symmetry of fluids was exploited by Eckart [36], Newcomb [37], Bretherton [38],and Salmon [39], and later by M¨uller [40] and Padhye and Morrison [41], to derive a variety of known conservationlaws. Benjamin provided a broad commentary on variational principles and conservation laws in fluids [42], using theterm “impulse” for Noether’s charge in a conservation law arising from variations of independent coordinates. He didnot distinguish between material and spatial coordinates, but remarked that the impulse is in many cases distinctfrom the momentum. The term “impulse” was inherited by Maddocks and Dichmann in the context of conservationlaws in rod dynamics, although they did not invoke a variational principle [43]. McIntyre emphasized the distinctionbetween momentum and pseudomomentum of fluid waves as respectively arising from translational invariance of theentire system and of the medium [44].In this paper, we examine the concept of pseudomomentum in a variational setting, deriving bulk and singularbalances of momentum, energy, and pseudomomentum from an action principle. The derivation systematically dis-plays the conjugate relationships between the variations and the associated physical quantities. Just as the balance ofmomentum can be associated with position, the balances of energy and pseudomomentum can be respectively asso-ciated with the independent variables of time and material labels. After a general discussion, we apply the conceptsto various examples taken from first gradient elasticity, ideal fluids, and thin structures.We begin in Section II with a procedure for varying a material action simultaneously with respect to materialcoordinates and time (the independent fields) and the present configuration (the dependent field). We conduct ourcalculations in the present configuration, and identify the changes in the Lagrangian density induced by the shifts inthe fields. A direct consequence of the variational treatment is that the bulk balance equations for material momentumand energy are projections of the momentum balance onto the material tangents and material velocity, respectively.In Section III, the balance laws for a first gradient mechanical field theory are presented, where the Lagrangian densityis assumed to be a function of the material velocity and the deformation gradient and to possess explicit dependencieson the independent field variables. Through a rearrangement of the pseudomomentum and energy equations, wesee that the source terms for these balance laws are the explicit partial derivatives of the Lagrangian density withrespect to the reference configuration and time, respectively. An equivalent derivation in the referential frame ispresented in Appendix A. We invoke Noether’s theorem to identify conservation laws arising due to invariance undershifts in the present and reference configurations, and relate the latter to the path independent J-integral. In SectionIV, we consider material symmetry and the balance laws of an ideal fluid, show that several results pertaining tovorticity, circulation, and helicity can be derived from the general form of the balance of pseudomomentum, and notea connection between circulation and the J-integral. In Section V, we consider the bulk and singular balance laws forquasistatic elastica with inhomogeneous bending stiffness, and apply these to understand the propulsive and reactionforces observed in recent studies of confined rods [45, 46]. This is perhaps the best illustration of the potential powerof the pseudomomentum balance, which provides a simple, almost effortless, derivation of the propulsive “force” onthe body after a reasonable prescription for singular sources is provided by analogy with the bulk balance law. Weconclude in Section VI with two examples of conical surface mechanics, for which conserved quantities associatedwith spatial and material rotational symmetry can be used to classify equilibrium configurations: rotating inertialmembranes [47], and plates with bending energy [48]. II. BALANCE LAWS
We begin by deriving balance laws for momentum, energy, and pseudomomentum from variation of an actionsimultaneously with respect to dependent and independent variables. Other such treatments exist in the literature[2, 25–27], and the required variational machinery can be found in several places [49–52]. Our approach differs fromthese others in several ways. We derive the laws in the current configuration, although all of our fields are definedper unit volume in the reference configuration. We present the derivation so as to delineate the Noether charges andcurrents associated with each type of variation. We also allow for a propagating non-material singular interface inthe material, and so obtain both field equations and jump conditions.We will consider a body B as a differentiable, orientable manifold with boundary ∂ B . Physically, this manifoldcomprises a collection of material elements labeled by attached material coordinates η i , where the index i can run overone, two, or three dimensions. The configuration of the body at time t is an embedding x ( η i , t ) in three-dimensionalEuclidean space E . This embedding induces a metric on B . Many quantities will be defined in a static referenceconfiguration ¯ x ( η i ). In some situations, it may be necessary or convenient to think of the manifold as Riemannian,carrying its own metric rather than obtaining it from a reference configuration. Incompatible-elastic bodies have noreference configuration, and low-dimensional bodies have many possible reference configurations.We will require explicit partial derivatives ∂ t and ∂ i ≡ ∂∂η i , as well as a material time derivative d t , material(noncovariant) derivatives d i , and covariant derivatives ∇ i and ¯ ∇ i constructed with the present and reference metricswhose components in the present and reference coordinate bases are, respectively, g ij = d i x · d j x = ∇ i x · ∇ j x and¯ g ij = d i ¯ x · d j ¯ x = ¯ ∇ i ¯ x · ¯ ∇ j ¯ x . We also define reciprocal bases such that ∇ i x · ∇ j x = δ ij and ¯ ∇ i ¯ x · ¯ ∇ j ¯ x = δ ij .We construct an action using a Lagrangian density ¯ L (cid:0) η i , t ; x (cid:1) , where dependence on temporal and material deriva-tives of x is implied, but not explicitly shown for brevity of notation. The arguments are written such that thedependent fields appearing after the semicolon are considered to be functions of the independent fields before it. Thedensity is defined per unit reference volume of the body, which in the examples considered in this paper is equivalentto per unit mass. However, with the exception of Appendix A, we will work in the present configuration, and thuswrite the action in the second of the two ways below, A = (cid:90) t t dt (cid:90) B d ¯ V ¯ L ( η i , t ; x ) = (cid:90) t t dt (cid:90) B dV J − ¯ L ( η i , t ; x ) . (1)The present and reference volume forms dV = √ g dη dη dη and d ¯ V = √ ¯ g dη dη dη use the metric determinants g ≡ det g ij and ¯ g ≡ det ¯ g ij , respectively, and are related by the Jacobian (determinant) J = (cid:112) g/ ¯ g such that dV = Jd ¯ V . The description (1) contrasts with that of geometric energies, such as those describing soap films, inwhich case it is more natural to work with Lagrangian densities per unit present volume (area).In the spirit of several prior investigators [2, 11–15, 25–27], we subject the action (1) to a set of transformations ofboth independent and dependent fields: η i → η (cid:48) i , t → t (cid:48) , x ( η i , t ) → x (cid:48) ( η (cid:48) i , t (cid:48) ). The transformed action is A (cid:48) = (cid:90) t (cid:48) t (cid:48) dt (cid:48) (cid:90) B (cid:48) dV (cid:48) J (cid:48)− ¯ L ( η (cid:48) i , t (cid:48) ; x (cid:48) ) . (2)Shifting the independent variables also transforms the domains of integration in time and space. We assume that thetransformation involves small shifts of the form η (cid:48) i = η i + δη i ( η j , t ) , t (cid:48) = t + δt , x (cid:48) ( η (cid:48) i , t (cid:48) ) = x ( η i , t ) + δ x ( η i , t ) , (3)where in keeping with a classical treatment, the time shift δt is just a uniform constant. For simplicity, we expressthe small variations as functions of the original un-transformed independent fields, but these could instead be written Throughout the text the symbol δ will also be used to denote a small variation in a quantity; no confusion should arise with theindex-bearing Kronecker δ ij . as functions of the transformed fields; for example, δ x ( η (cid:48) i , t (cid:48) ) = δ x ( η i , t ) + ∂δ x ∂η i δη i + ∂δ x ∂t δt + . . . , with all the terms onthe right except the first being of higher order [49].The δ operator in (3) measures both changes in x due to changes in the independent fields, as well as changes in x through physical deformation at a fixed material point . Because the two sides of equation (3) are functions oftwo different labels η i and η (cid:48) i , the δ operator does not commute with the material derivative. We thus define [49]an operator ˜ δ which measures changes in the field variable at a fixed label , that is, fixed values of η i , and thereforecommutes with the material derivative, x (cid:48) ( η i , t ) = x ( η i , t ) + ˜ δ x ( η i , t ) . (4)Using (3) and (4), the two variational operators can be related by δ x = ˜ δ x + ∇ j x δη j + d t x δt . (5)The second term on the right is the shift in x due to shifts in the parameterization alone. Note that for a staticlow-dimensional body such as an elastic surface, any normal variation of the position vector is contained in ˜ δ x .The change in the action due to the transformation (3) is the difference between (1) and (2),∆ A = (cid:90) t (cid:48) t (cid:48) dt (cid:48) (cid:90) B (cid:48) dV (cid:48) J (cid:48)− ¯ L ( η (cid:48) i , t (cid:48) ; x (cid:48) ) − (cid:90) t t dt (cid:90) B dV J − ¯ L ( η i , t ; x ) . (6)To evaluate this difference to first order, we manipulate the shifted integral so that it corresponds to the originaldomain to obtain [53, 54] δA = (cid:90) t t dt d t (cid:18) δt (cid:90) B dV J − ¯ L (cid:19) + (cid:90) t t dt (cid:90) B dV ∇ i (cid:0) J − ¯ L δη i (cid:1) + (cid:90) t t dt (cid:90) B dV J − ˜ δ ¯ L . (7)In writing the bulk term on the far right, we note that ˜ δ ( J − dV ) = ˜ δd ¯ V = 0. This bulk term contains both Euler-Lagrange content as well as pieces that will contribute to the charge and current on the boundaries after integrationby parts.At this point, several choices are available to us when manipulating the integral. For the present discussion, wechoose to express everything in terms of an integral over the present volume. Recalling that δt is uniform in space,and noting that the material integration limits and the reference volume form d ¯ V = J − dV are independent of time,we may rewrite (7) simply as δA = (cid:90) t t dt (cid:90) B dV (cid:104) J − d t (cid:0) ¯ L δt (cid:1) + ∇ i (cid:0) J − ¯ L δη i (cid:1) + J − ˜ δ ¯ L (cid:105) . (8)This expression separates changes in the action due to shifts in the independent and dependent variables. The firsttwo terms account for the shift in the domain of integration in the material coordinates and time, whereas the thirdterm represents the change due to shifts ˜ δ x in the dependent variable at a fixed label . In terms of formal calculation,the computation of this term involves nothing but the familiar process of variation in which one shifts the dependentfields alone. This final term, involving the variation of the Lagrangian density, will generate both bulk and boundaryterms through integration by parts, which we write schematically as J − ˜ δ ¯ L = E · ˜ δ x + J − d t E ( Q ) + ∇ i E i ( J ) . (9)Here E ( ¯ L ) is the Euler-Lagrange operator, and E ( Q ) and E ( J ) the temporal and material boundary terms, associatedwith the Lagrangian density ¯ L . The variation of the action (8) may be arranged as δA = (cid:90) t t dt (cid:90) B dV (cid:104) J − d t Q + ∇ · J + E · ˜ δ x (cid:105) , (10) Q = ¯ L δt + E ( Q ) , (11) J i = J − ¯ L δη i + E i ( J ) . (12)The form of the Euler-Lagrange term E is in general the same as the boundary terms, but with the possibility ofadditional source terms. The charge density Q and current density J are functions of the variations of all the fields. While the appearance of J − may look a bit strange outside of the time derivative, consider that inertial terms are generally proportionalto a reference density ¯ ρ that is independent of time. Due to conservation of mass, one has J − = (cid:112) ¯ g/g = ρ/ ¯ ρ , and moving the referencedensity through the time derivative, one is left with the present density as coefficient. The third, Euler-Lagrange term in the integral (10) delivers the bulk balance laws corresponding to shifts in thedependent and independent variables, in an extension [25–27] of the usual Hamilton-Lagrange-d’Alembert principle ofstationary action. When the field variations are symmetries of the system, δA = 0 independently of any considerationsregarding the stationarity of A , and the essence of Noether’s theorem is that the first two terms inside the integral(10) provide conservation laws associated with these symmetries when the Euler-Lagrange term vanishes.In the following section, we will consider in detail the specific case of a space-filling body with a Lagrangian densitydependent on, at most, the first derivative of the position vector. The classic example of such a “first gradient” theoryis the elasticity of simple materials [15]. Because of its prominence in modern solid mechanics, we will also present areferential version of the above derivation, along with that of Section III below, in Appendix A.Some energies are more naturally formulated with respect to a reference configuration, and others with respectto the present. An example of the latter would be a geometric energy such as that of a soap film, dependent onthe current area of the film and independent of any reference density distribution. The above framework, discussedwith respect to a Lagrangian density ¯ L defined with respect to a reference volume, can be augmented with geometricterms L = J − ¯ L = ( ρ/ ¯ ρ ) ¯ L . For example, an “inertial soap film” with surface energy γ could be represented either by L = ρd t x · d t x − γ or ¯ L = ¯ ρd t x · d t x − Jγ . III. FIRST GRADIENT THEORY WITH A REFERENCE CONFIGURATION
In this section, we consider field theories with the form ¯ L (¯ x , t ; x , d t x , F ), where the independent material coordinatesare represented by a time-independent reference configuration ¯ x , and first derivatives of position are represented bya “deformation gradient” [55] F ≡ ¯ ∇ x ≡ d x d ¯ x = ∇ i x ¯ ∇ i ¯ x that applies the gradient ¯ ∇ of the reference space to theposition vector x of the present configuration (note that F also serves to transform between bases; for example d i x δη i = F · d j ¯ x δη j , and ¯ ∇ () = ∇ () · F if () has no free indices). This description is appropriate, and indeed quitetraditional in solid mechanics, for the description of simple elastic bodies that fill some portion of three-dimensionalspace (that is, i ∈ { , , } ), but is not suitable for the description of incompatible elastic systems or lower-dimensionalbodies such as the elastic surfaces we will consider later in Section VI.We will identify the balance and conservation laws that arise from variation of the dependent and independentvariables. The temporal and material boundary terms from (9) can be computed as E ( Q ) = ∂ ¯ L ∂d t x · ˜ δ x and E ( J ) = J − F · (cid:104) ∂ ¯ L ∂ F (cid:105) T · ˜ δ x . The Euler-Lagrange, charge, and current terms corresponding to the variation (10) are E = J − ∂ ¯ L ∂ x − J − d t (cid:18) ∂ ¯ L ∂d t x (cid:19) − ∇ · (cid:32) J − · F · (cid:20) ∂ ¯ L ∂ F (cid:21) T (cid:33) , (13) Q = (cid:18) ¯ L δt + ∂ ¯ L ∂d t x · ˜ δ x (cid:19) , (14) J = J − (cid:32) ¯ L F · δ ¯ x + F · (cid:20) ∂ ¯ L ∂ F (cid:21) T · ˜ δ x (cid:33) . (15)However, to clearly identify the terms power-conjugate to each of the different variational quantities, we rewrite thevariations at a fixed material label appearing in (14-15), to express the charge and current in terms of the totalvariation δ x at a fixed material point as well as variations with respect to the independent material and temporalvariables. Using the relation (5) to substitute for ˜ δ x , we rearrange equations (14-15) to obtain Q = (cid:20) ∂ ¯ L ∂d t x · δ x + (cid:18) ∂ ¯ L ∂d t x · F (cid:19) · ( − δ ¯ x ) + (cid:18) ∂ ¯ L ∂d t x · d t x − ¯ L (cid:19) ( − δt ) (cid:21) , (16) J = (cid:34)(cid:32) J − F · (cid:20) ∂ ¯ L ∂ F (cid:21) T (cid:33) · δ x + (cid:32) J − F · (cid:20) ∂ ¯ L ∂ F (cid:21) T · F − J − ¯ L F (cid:33) ( − δ ¯ x ) + (cid:32) J − F · (cid:20) ∂ ¯ L ∂ F (cid:21) T · d t x (cid:33) ( − δt ) (cid:35) . (17)In this form, we may identify several familiar quantities. In (16), the charges conjugate to δ x , − δ ¯ x , and − δt are, respectively, the (spatial) momentum, pseudomomentum , and Hamiltonian density [54]. In (17), the currents A strict conservation law form can be seen in a derivation in the referential frame; see for example Appendix A. A. Golebiewska Herrmann’s material momentum [12], or the negative of Peierls’s pseudomomentum in equation (2.10) of [1]. conjugate to δ x , − δ ¯ x , and − δt are, respectively, the (spatial) stress, pseudostress , and power expended by the stress.As we are working in the present configuration, the stress in question is that of Cauchy [55], J − F · (cid:104) ∂ ¯ L ∂ F (cid:105) T . Thisis the Piola transform of the first Piola-Kirchhoff stress ∂ ¯ L ∂ F whose transpose will appear naturally in the referentialframe in Appendix A. Similarly, the Eshelby tensor conjugate to − δ ¯ x in (17) is a transformed version of the usualreferential form of this tensor.We now proceed to integrate (10) by parts to obtain bulk balance laws as well as boundary and jump conditions.The latter are singular balance laws that hold at an internal non-material surface of discontinuity S ( t ). This surfaceis assumed to move with some “velocity” through the coordinates, whose normal component is denoted U . Let ˆn and ˆN be the unit normals to the external boundary and internal surface of discontinuity. The relevant forms of thedivergence and transport theorems for a piecewise continuous tensorial quantity A are [55, 56] (cid:90) B dV ∇ · A = (cid:90) ∂ B dA ˆn · A − (cid:90) S ( t ) dA (cid:114) ˆN · A (cid:122) , (18) d t (cid:90) B dV J − P = (cid:90) B dV J − d t A − (cid:90) S ( t ) dA (cid:113) J − U A (cid:121) , (19)where (cid:74) (cid:75) denotes the jump in the enclosed quantity across the discontinuity. In general U should be continuous, andthus can be moved outside the brackets. In the present consideration of space-filling bodies, ˆN will be continuous aswell. Using (18-19) and (5), we obtain δA = (cid:90) t t dt (cid:90) ∂ B dA ˆn · J + (cid:90) t t dt (cid:90) S ( t ) dA (cid:114) − ˆN · J + J − U Q (cid:122) + (cid:90) t t dt (cid:90) B dV [ E · ( δ x − F · δ ¯ x − d t x δt )] , (20)where we have used δ x = F · δ ¯ x . These three integrals provide the (free) boundary conditions, jump conditions,and bulk field equations. We now consider separately the balance laws conjugate to the variation in the currentconfiguration, material coordinates, and time.Pure variations δ x of the current configuration, with δ ¯ x = and δt = 0, provide the bulk equation, the boundarycondition, and the jump condition for momentum, E ( ¯ L ) = J − ∂ ¯ L ∂ x − J − d t (cid:18) ∂ ¯ L ∂d t x (cid:19) − ∇ · (cid:32) J − F · (cid:20) ∂ ¯ L ∂ F (cid:21) T (cid:33) = on B , (21) ˆn · (cid:32) J − F · (cid:20) ∂ ¯ L ∂ F (cid:21) T (cid:33) = on ∂ B , (22) (cid:116) − ˆN · (cid:32) J − F · (cid:20) ∂ ¯ L ∂ F (cid:21) T (cid:33) + J − U ∂ L ∂d t x (cid:124) = on S ( t ) . (23)Pure variations δ ¯ x of the reference configuration, with δ x = and δt = 0, provide balance laws for pseudomomentum, E ( ¯ L ) · F = (cid:34) J − ∂ ¯ L ∂ x − J − d t (cid:18) ∂ ¯ L ∂d t x (cid:19) − ∇ · (cid:32) J − F · (cid:20) ∂ ¯ L ∂ F (cid:21) T (cid:33)(cid:35) · F = on B , (24) ˆn · (cid:32) J − F · (cid:20) ∂ ¯ L ∂ F (cid:21) T · F − J − ¯ L F (cid:33) = on ∂ B , (25) (cid:116) − ˆN · (cid:32) J − F · (cid:20) ∂ ¯ L ∂ F (cid:21) T · F − J − ¯ L F (cid:33) + J − U (cid:18) ∂ ¯ L ∂d t x · F (cid:19) (cid:124) = on S ( t ) . (26) Material stress, or Eshelby’s tensor [4].
Finally, purely temporal variations δt , with δ x = δ ¯ x = ), provide balance laws for energy, E ( ¯ L ) · d t x = (cid:34) J − ∂ ¯ L ∂ x − J − d t (cid:18) ∂ ¯ L ∂d t x (cid:19) − ∇ · (cid:32) J − F · (cid:20) ∂ ¯ L ∂ F (cid:21) T (cid:33)(cid:35) · d t x = 0 on B , (27) ˆn · (cid:32) J − F · (cid:20) ∂ ¯ L ∂ F (cid:21) T · d t x (cid:33) = 0 on ∂ B , (28) (cid:116) − ˆN · (cid:32) J − F · (cid:20) ∂ ¯ L ∂ F (cid:21) T · d t x (cid:33) + J − U (cid:18) ∂ ¯ L ∂d t x · d t x − ¯ L (cid:19) (cid:124) = 0 on S ( t ) . (29)While the bulk balances for pseudomomentum and energy are simply projections of the momentum balance ontothe deformation gradient and velocity, respectively, the boundary and jump conditions are distinct. Note also thatthe two sets of vector equations correspond to different “legs” of the two-point tensorial quantities they contain; theextant leg in the momentum equations corresponds to the present configuration while that in the pseudomomentumequations corresponds to the reference configuration. Referential forms of these balance laws are presented in AppendixA. Because equations (24-26) arise from a continuous shift in material coordinates made possible by a continuumdescription of a body, they have no analogue in a discrete set of particles [11, 13].Although the simple relationship between the bulk balance laws in our system implies that satisfaction of thebalance of momentum (21) means that the other balances (24) and (27) hold as well, this obscures a crucial point,namely that the conserved quantities associated with the corresponding symmetries are not identical. In the followingsection, we will rearrange these equations to illustrate that the source terms arising from broken spatial, temporal,and material symmetries are mutually independent quantities. A. Forces and material forces
Following A. Golebiewska Herrmann [11] and Maugin [15], we recast the balances of energy (27) and pseudomo-mentum (24) into a standard form that clearly reveals the form of the source terms. Referential expressions thatfollow a strict conservation law form are presented in Appendix A.First note that the balance of momentum (21) can be easily written with a source term on the right hand side, J − d t (cid:18) ∂ ¯ L ∂d t x (cid:19) + ∇ · (cid:32) J − F · (cid:20) ∂ ¯ L ∂ F (cid:21) T (cid:33) = J − ∂ ¯ L ∂ x . (30)Any explicit dependence of the Lagrangian density on the position x , such as the presence of a gravitational potential,breaks the symmetry of the embedding space and provides a source of momentum.The balance of energy (27) can be rearranged by employing the chain rule J − d t ¯ L = J − (cid:32) ∂ ¯ L ∂t + ∂ ¯ L ∂ x · d t x + ∂ ¯ L ∂d t x · d t d t x + (cid:20) ∂ ¯ L ∂ F (cid:21) T : ( ∇ d t x · F ) (cid:33) , (31)where the final term involves the material time derivative of F ; the notation means that the ∇ leg is contracted withthe present leg of F , and double contraction associates present and referential legs with their respective counterparts.After some integration by parts, (27) becomes J − d t (cid:18) ∂ ¯ L ∂d t x · d t x − ¯ L (cid:19) + ∇ · (cid:32) J − F · (cid:20) ∂ ¯ L ∂ F (cid:21) T · d t x (cid:33) = − J − ∂ ¯ L ∂t . (32)As one might expect, any explicit dependence of the Lagrangian density on the time t manifests as a source term inthe energy balance.Similarly, the balance of pseudomomentum (24) can be rearranged with the help of the chain rule J − ∇ ¯ L · F = J − ∂ ¯ L ∂ ¯ x + J − ∂ ¯ L ∂ x · F + J − ∂ ¯ L ∂d t x · d t F + J − F · (cid:20) ∂ ¯ L ∂ F (cid:21) T : ∇ F , (33)where the double contraction involves the two present legs of the gradient of F . After some integration by parts anduse of the Piola identity ∇ · (cid:0) J − F (cid:1) = , (24) becomes d t (cid:18) J − ∂ ¯ L ∂d t x · F (cid:19) + ∇ · (cid:32) J − F · (cid:20) ∂ ¯ L ∂ F (cid:21) T · F − J − ¯ L F (cid:33) = − J − ∂ ¯ L ∂ ¯ x . (34)Any explicit dependence of the Lagrangian density on the reference configuration ¯ x breaks the symmetry of thematerial continuum and provides a source of pseudomomentum. Forms of the balance law (34) in present or referentialform, with or without the source term, can be found in [2, 11–16, 31, 32, 35].The source terms in the three balances (30), (32), and (34) are entirely independent. In particular, the balance ofpseudomomentum is related to the symmetry of the material continuum, a feature independent of any properties ofthe embedding space. Just as the source term in (21) is often interpreted as a body force, we may interpret the sourceterm in (24) as a “material body force”. However, it is important to note that these forces are not of the same type.While spatial (Newtonian) forces are vectors associated with the embedding space, material (Eshelbian) forces areassociated with a material space. Although in the example under consideration, the material space can be thought ofas a reference configuration embedded in the same space as the present configuration, with material forces associatedwith vectors in the reference configuration, and although the present configuration of a space-filling body is oftenassociated with the embedding space itself, this does not mean that spatial and material forces can be conflated oradded together in any meaningful way. They pertain, respectively, to the motion of material bodies in space and themotion of non-material objects within a material. B. A few symmetries and conservation laws
Here we apply Noether’s theorem to obtain conservation laws for momentum and pseudomomentum. We insertvariations corresponding to spatial and material symmetries of the action into the general expression J − d t Q + ∇ · J = 0 , (35)where Q and J are given by (16) and (17). A static version of this general statement in elasticity can be found inEdelen [25].The embedding space is symmetric under translations δ x = D and rotations δ x = D × x , where D is a (small)constant vector. With δ ¯ x = and δt = 0, we obtain linear and angular momentum conservation laws, J − d t (cid:18) ∂ ¯ L ∂d t x (cid:19) + ∇ · (cid:32) J − F · (cid:20) ∂ ¯ L ∂ F (cid:21) T (cid:33) = , (36) J − d t (cid:18) x × ∂ ¯ L ∂d t x (cid:19) + ∇ · (cid:32) x × J − F · (cid:20) ∂ ¯ L ∂ F (cid:21) T (cid:33) = . (37)By comparing [24] the linear momentum conservation law (36) with the balance law (30), we can see that conservationimplies that the Lagrangian density cannot depend explicitly on position x .If the material is uniform, “translations” in material coordinates δ ¯ x produce the linear pseudomomentum conser-vation law J − d t (cid:18) ∂ ¯ L ∂d t x · F (cid:19) + ∇ · (cid:34)(cid:32) J − F · (cid:20) ∂ ¯ L ∂ F (cid:21) T · F − J − ¯ L F (cid:33)(cid:35) = . (38)In the present context, this means [24] that the Lagrangian density cannot depend explicitly on the reference configu-ration ¯ x , as can be seen by comparing (38) with (34). Material “rotational” symmetry and angular pseudomomentumconservation will be exploited in Section VI. Pseudomomentum conservation laws can be found in [2, 11–16, 31, 32, 35]. We use the term “conservation law” loosely.
C. J-integral
As demonstrated by A. Golebiewska Herrmann [12, 13], conservation laws arising from invariance of material spaceare intimately related to well-known path independent integrals of hyperelastic fracture mechanics [8, 9]. Considerthe conservation of “translational” material momentum (38) integrated over an arbitrary volume V with boundary ∂V and unit normal ˆ ν , (cid:90) V dV (cid:34) J − d t (cid:18) ∂ ¯ L ∂d t x · F (cid:19) + ∇ · (cid:32) J − F · (cid:20) ∂ ¯ L ∂ F (cid:21) T · F − J − ¯ L F (cid:33)(cid:35) = . (39)If V encapsulates a defect such as an inclusion or crack tip– a point where the conservation law fails to hold– the righthand side of the above equation need not be zero. Such a source term would represent the total material force on thedefect that seeks to drive it through the material rather than through space. Equation (39) is known as the dynamicgeneralization of the J-integral [17, 57, 58]. Markenscoff [57] also discusses the corresponding “rotational” L-integral.For the static case, the time derivative vanishes and the divergence term may be written as a surface integral (cid:90) ∂V dA ˆ ν · (cid:32) J − F · (cid:20) ∂ ¯ L ∂ F (cid:21) T · F − J − ¯ L F (cid:33) = , (40)the original J-integral of Rice [8] and Cherepanov [9]. IV. IDEAL FLUID
In this section, we derive the balances of momentum, pseudomomentum, and energy for an inviscid, incompressiblefluid in the framework of Section II. We also demonstrate that the conservation of several important quantities, namelyvorticity, circulation, and helicity, can be seen as a consequence of pseudomomentum balance and material symmetry.Variational derivations of both Lagrangian and Eulerian inviscid fluid equations exist [36, 38, 39, 59]. As fluidmechanics is often considered from an Eulerian point of view, the utility of material symmetry may not be immediatelyobvious. However, many classical results in fluid mechanics are of a material character, such as Kelvin’s circulationtheorem describing the conservation of ideal fluid impulse evaluated over a material loop.Rather than a reference configuration in the sense of an elastic solid, the fluid will be given an arbitrarily chosen setof material labels that convect with the flow. We still define a Lagrangian density in terms of the density ¯ ρ at somereference state (presumed uniform for simplicity), although as the fluid is incompressible the distinction betweenthis and the present density ρ affects only formal manipulations. In terms of the present configuration of a fluid x ≡ x ( η i , t ), A = (cid:90) t t dt (cid:90) B dV J − (cid:2) ¯ ρd t x · d t x + p ( J − (cid:3) , (41)where the pressure p is a Lagrange multiplier enforcing the incompressibility constraint J = 1. We employ ˜ δJ = J ∇ i x · ∇ i ˜ δ x and subsequently invoke J − ¯ ρ = ρ and J = 1 to obtain δA = (cid:90) t t dt (cid:90) B dV (cid:18) d t (cid:20) ρd t x · δ x + ρd t x · ∇ j x (cid:0) − δη j (cid:1) + ρd t x · d t x ( − δt ) (cid:21) + ∇ i (cid:20) p ∇ i x · δ x + (cid:0) p − ρd t x · d t x (cid:1) (cid:0) − δη i (cid:1) + p ∇ i x · d t x ( − δt ) (cid:21) + (cid:20) − d t ( ρd t x ) − ∇ i (cid:0) p ∇ i x (cid:1) (cid:21) · (cid:0) δ x − ∇ j x δη j − d t x δt (cid:1) (cid:19) . (42)The bulk equations for momentum, pseudomomentum, and energy are thus ρd t x + ∇ i (cid:0) p ∇ i x (cid:1) = , (43) d t ( ρd t x · ∇ i x ) + ∇ i (cid:0) p − ρd t x · d t x (cid:1) = 0 , (44) d t (cid:0) ρd t x · d t x (cid:1) + ∇ i (cid:0) p ∇ i x · d t x (cid:1) = 0 . (45)0Note that because of the way we have written the dependencies of the action, the pseudomomentum balance (44) isobtained in component form. The pseudomomentum is the quantity whose components are inside the time derivative,also known as the impulse [42] or “vortex momentum” [60].Terms such as ∇ x appear in the momentum (43) and energy (45) equations. If x were a surface, these wouldrepresent normal vectors, but if x is merely a space-filling blob of fluid in flat space, these terms vanish, giving forexample ρd t x + ∇ p = for the momentum equation. For the energy equation, it is more useful to note that the flowis incompressible ( ∇ i v i = 0, where v ≡ d t x ), giving d t (cid:0) ρ v · v (cid:1) + v i ∇ i p = 0. Noting that d t = ∂∂t + v i ∇ i , we canwrite d t (cid:18) ρ v · v + p (cid:19) = ∂p∂t , (46)a form of Bernoulli’s equation equivalent to Eckart’s (3.14) [36]. Alternately, keeping in mind that ρ is uniform andconstant, we can write a more familiar expression, v · (cid:20) ∂ v ∂t + ∇ (cid:18) v · v + pρ (cid:19)(cid:21) = 0 , (47)involving the streamline derivative v · ∇ .The pseudomomentum and energy equations are projections of the momentum equation onto the tangents d i x = ∇ i x and velocity d t x , respectively. One consequence is that when the flow can be expressed as a steady velocity field, thestreamline component of the pseudomomentum equation expresses the same content as the energy equation. Thisis partly why one of the present authors misleadingly identified the conserved quantity associated with the materialsymmetry of a flowing string with Bernoulli’s constant in [61]. A. Vorticity
The vorticity is defined as the curl of the velocity, ω i ≡ (cid:15) ijk ∇ j v k . Applying the curl to the pseudomomentumequation (44) and noting that for constant ρ , the time derivative commutes with the metric determinant implicit inthe alternating tensor, we obtain ρd t ω i = 0 . (48)The vorticity equation (48) takes this simple form because the contravariant components, in material coordinates,of any field convecting with (“frozen in”) the flow are such that their material time derivative vanishes [62]. Thesecomponents of vorticity may be identified with Cauchy’s invariants [63, 64].Integrating the pseudomomentum equation (44) with respect to time, and noting that ∇ i is equivalent to d i whenacting on a scalar, so can be interchanged with time derivatives and integrals, we obtain the Cauchy-Weber integralrelation [65, 66], ρ ( v i − v i | t =0 ) + d i (cid:20)(cid:90) t dt (cid:0) p − ρ v · v (cid:1)(cid:21) = 0 . (49) B. Circulation
Integrating the component of the pseudomomentum equation (44) directed along a closed material loop ( i = l ), (cid:72) dη l , and noting the equivalence of ∇ l and d l , the time-independence of dη l and ρ , and the uniformity of ρ , we obtainKelvin’s circulation theorem, ρd t (cid:20)(cid:73) d t x · d l (cid:21) = 0 . (50)The connection between this theorem and material symmetry, also known as “relabeling symmetry” or “exchangeinvariance”, is discussed in several works [36–41].In two dimensions, the integration performed here is analogous to the derivation of the J-integral in Section III C.This analogy between solids and fluids has been noted by Cherepanov [67], A. Golebiewska Herrmann [13], Atilgan[68], and Maugin [16]. Were material symmetry to be broken, for example by the presence of a body inside the loop,the integral could be nonzero. It would be interesting to consider the connection between the resulting material forcethat seeks to drive the body through the fluid, and the conventional concepts of lift and drag.1 C. Helicity
To apply Noether’s theorem, we use the pseudomomentum conservation law associated with a general symmetry inmaterial coordinates, d t (cid:0) ρv i δη i (cid:1) + ∇ i (cid:2)(cid:0) p − ρ v · v (cid:1) δη i (cid:3) = 0 , (51)and consider a coordinate shift that follows the vorticity field, δη i = (cid:15)ω i , where (cid:15) is a small constant. This shift hasthe property that ∇ j (cid:0) (cid:15)ω j (cid:1) = 0 [40, 41]. Integrating over a material volume, noting that ρdV is time-independent,and applying the divergence theorem, d t (cid:90) V dV ρv i ω i + (cid:90) ∂V dA n i (cid:0) p − ρ v · v (cid:1) ω i = 0 . (52)If the surface ∂V is such that n i ω i = 0, the helicity within the volume V is conserved [69, 70], ρd t (cid:20)(cid:90) V dV v i ω i (cid:21) = 0 , (53)where the final manipulation uses the uniformity of ρ and the individual time-independence of ρ and dV . V. NON-UNIFORM
ELASTICA
Elastic beams or rods whose properties vary along their length provide excellent one-dimensional demonstrationsof concepts related to pseudomomentum. One such property is a variable bending stiffness arising from changes incross section. An early theoretical investigation is that of Kienzler and G. Herrmann, who considered discontinuitiesin stiffness in a beam [71]. More recently, Bigoni and co-workers have performed a very interesting series of experi-ments involving both continuous and discontinuous variation in the bending and torsional stiffness of rods as well asconfinement conditions imposed on these rods [46, 72, 73]. This group has also offered a theoretical analysis that, webelieve, incorrectly conflates forces and material forces, lumping them both under a general heading of configurationalor “Eshelby-like” effects. In this section, we present our perspective, which is heavily influenced by the analyses ofKienzler and G. Herrmann [71] and O’Reilly [19, 74, 75], and partially laid out in prior publications [76, 77]. Afterpresenting the bulk and singular balance laws for momentum and pseudomomentum for a non-uniform planar Eu-ler elastica , we apply our approach specifically to the problem of planar serpentine locomotion of a rod through acurved channel [45, 46], and attempt to delineate which forces appearing in the problem are actual forces and whichare configurational forces. We demonstrate that the propulsive material force on the confined rod can be obtaineddirectly by integrating the pseudomomentum balance, and obtain reaction forces at points of geometric and materialdiscontinuity from the singular pseudomomentum balance without any appeal to micromechanical arguments [46].As in O’Reilly’s analysis [75, 76] of Bigoni’s sleeve constraint [72], this approach requires a prescription for singularsources of pseudomomentum, but once this leap has been taken the results follow immediately.To facilitate comparison we adopt, to the extent possible, notation from prior works referred to in this section.
A. Balance laws for planar elastica
We consider a static planar configuration of an inextensible elastic curve x ( s ), where s is both arc length and amaterial coordinate. A relevant action for an elastica with position-dependent stiffness B ( s ) is A = (cid:90) s s ds (cid:0) − (cid:1) [ B ( s ) Ω · Ω + σ ( d s x · d s x − , (54)where Ω ≡ d s x × d s x is the Darboux vector, pointing out of the plane, whose squared magnitude is the square of therod curvature κ . We write B ( s ) to emphasize that the bending stiffness varies along the length of the rod; all otherquantities appearing in the brackets, including the Lagrange multiplier σ , are also functions of s . Since the rod isinextensible, comparison can be made to the general approach by noting that the volume form is 1 ds and J = 1.The variation of the action may be written [77] in terms of the contact force n , the contact moment m , and theonly component of the material stress c , δA = (cid:90) s s ds [ d s ( − n · δ x − m · δ Ω + c δs ) + d s n · ( δ x − d s x δs )] , (55) n = σd s x − d s (cid:0) B ( s ) d s x (cid:1) , m = B ( s ) Ω , c = n · d s x + m · Ω − B ( s ) Ω · Ω . (56)2Furthermore, assuming a single point of discontinuity at s = s and applying the divergence theorem for piecewisecontinuous fields, δA = ( − n · δ x − m · δ Ω + c δs ) | s s + (cid:74) n · δ x + m · δ Ω − c δs (cid:75) | s = s + (cid:90) s s ds [ d s n · ( δ x − d s x δs )] , (57)from which expression we may directly obtain boundary conditions, jump conditions, and bulk field equations. Mo-mentum balance is given by those terms conjugate to δ x with δs = 0, and pseudomomentum by those conjugate to δs with δ x = .The bulk momentum and pseudomomentum balances are d s n = , (58) d s c = − ∂ s B ( s ) κ . (59)The pseudomomentum balance d s n · d s x = 0 is rearranged into the form (59) by a chain rule procedure akin to thatin Section III A. The explicit dependence of the action on the coordinate s breaks the material symmetry of the rod,giving rise to the source term on the right hand side.The corresponding singular balances at the point of discontinuity s are R + (cid:74) n (cid:75) = , (60) Y + (cid:74) c (cid:75) = 0 , (61)with allowance for singular supplies of momentum R and pseudomomentum Y that do not explicitly appear in theaction [76]. The quantity R represents such things as reaction forces from external constraints at the discontinuity,as might arise at the edge of a sleeve [72]. The interpretation of Y is still a matter of discussion in the literature[19, 74, 76] and will be considered for our specific problem in the following section. B. Spatial and material sources
While the identification of the momentum source R is straightforward, we lack a general prescription for thepseudomomentum source Y . There is currently no generally agreed upon conceptual framework that provides aphysical understanding of what a source of pseudomomentum means. The prescription of the pseudomomentumsource term at a point of discontinuity was inferred from the bulk balance law for a simple transversely loaded elasticbeam with continuous curvature by Kienzler and G. Herrmann [71]. O’Reilly [74] has argued that, in general, thesource term Y is a constitutive parameter related to the power input ˜ E across a moving discontinuity s ( t ) through Y ˙ s = ˜ E − R · v − M · ω , where ˙ s is the “velocity” of the discontinuity through the body, R and M are a pointforce and moment acting at the discontinuity, and v and ω are the velocity and angular velocity of the spatial pointassociated with the discontinuity. The particular class of problems we are considering consists of the motion of anelastic rod through a space-fixed, frictionless channel. There are discontinuities in the constraint— for example, thecurvature of the channel— as well as in the properties of the rod itself. The former type of discontinuity is fixed inspace ( v = ω = ), and as no net power is input or dissipated, ˜ E = 0 and thus Y also vanishes. For the latter type,we might seek insight from the balance of energy across the discontinuity. Instead, however, we propose to infer theprescription of Y for any discontinuity from the source term in the continuous balance law for material momentum(59), leading to Y = 0 for discontinuities in the channel properties , (62) Y = (cid:74) B ( s ) (cid:75) κ for discontinuities in the material properties , (63)where clearly (62) is just a specific case of (63). The prescription (62) is consistent with O’Reilly [74] and with ourprior work [76], while the prescription (63) is consistent with Kienzler and G. Herrmann [71].Thus we make a distinction between broken symmetries in the environment and in the material itself, with onlythe latter giving rise to a material force. In the original problem considered by Bigoni and co-workers [72], a uniformrod slides in and out of a frictionless sleeve constraint. The relevant discontinuity is the edge of the sleeve, and theprescription Y = 0 directly provides an interesting relationship between reaction forces and moments at this point For a different perspective that would lead to the same conclusions, see [76].
C. Serpentine locomotion
The passive motion of a variable-property rod through a variable-curvature channel, as presented by previousauthors [45, 46], is portrayed in Figure 1. A planar rod of length l moves in a frictionless channel of length L . Thecoordinates s and S are the arc length values measured from the left ends of the rod and the channel, respectively.In terms of the channel coordinate, the left end of the rod is located at a time-dependent location S = ξ ( t ). Thereis a discontinuity in channel curvature χ ( S ) at the point S = L , and a discontinuity in rod stiffness B ( s ) at thepoint s = l . For simplicity, we consider only these single discontinuities, the curvature and stiffness being uniformelsewhere. Continuous variation in these properties can be treated straightforwardly using bulk balance laws. Therod is fully constrained so that its curvature κ ( s, t ) = χ ( ξ ( t ) + s ) matches that of the channel. The rest curvature ofthe rod is zero, in contrast to the variable rest curvature considered in [45]. FIG. 1:
After [46]. A planar rod of length l moving in a frictionless channel of length L . Here s and S denote the arc length values measured from the left end of the rod and the channel,respectively, with the left end of the rod located at S = ξ ( t ). The thin lines indicate adiscontinuity in channel curvature at S = L , and the two colors in the rod indicate adiscontinuity in stiffness s = l . If we allow for motion of the rod, the static pseudomomentum balance (59) will be augmented by an inertiaterm, namely the tangential projection which, in the present constrained case, is the only component of the inertia.Integrating the static equation (59) over the rod provides the unbalanced “propulsive force” [45, 46] that tends tomove the rod through the channel: P ( t ) = (cid:90) l ds (cid:2) d s c + ∂ s B ( s ) κ (cid:3) , = − c (0 , t ) + c ( l, t ) − (cid:74) c ( l , t ) (cid:75) + (cid:90) l − ds ∂ s B ( s ) κ + (cid:90) ll +1 ds ∂ s B ( s ) κ . (64)In this expression, we have left out a jump term at S = L , because we anticipate that c will be continuous there byway of the prescription (62). There will be no contribution to the propulsive material force from discontinuities inthe imposed constraints. Using the definition (56), singular balance (61), and prescription (63), we can evaluate (64)to be P ( t ) = B (0) κ (0 , t ) − B ( l ) κ ( l, t ) + (cid:74) B ( l ) (cid:75) κ ( l , t ) + (cid:90) l − ds ∂ s B ( s ) κ ( s, t ) + (cid:90) ll +1 ds ∂ s B ( s ) κ ( s, t ) , (65)in agreement with equation (2.14) of [46].The first term in the pseudomomentum (56) is the tangential component of the contact force. Thus, we may alsoobtain the tangential reaction forces at the rod ends and at the points of discontinuity in the rod stiffness and the4channel curvature directly from this definition, the jump condition (61), and the prescription (62-63): n · d s x | = − B (0) κ (0 , t ) , (66) n · d s x | l = − B ( l ) κ ( l, t ) , (67) (cid:74) n · d s x (cid:75) | l = − (cid:74) B ( l ) (cid:75) κ ( l , t ) , (68) (cid:74) n · d s x (cid:75) | L = − B ( L − ξ ) (cid:113) χ ( L ) (cid:121) . (69)These relations are equations (2.27-2.30) of [46], where they were computed by micromechanical arguments. VI. ISOMETRIC DEFORMATIONS OF CONICAL SHEETS
We conclude our set of examples with two problems in the mechanics of thin sheets, modeled as flexible, inextensiblesurfaces, with conical metrics. As was already implicit in our treatment of elastic rods in Section V, these bodies arebest considered as Riemannian manifolds endowed with a low-dimensional reference metric, as there is no meaningfuldistinction between a continuum of possible reference configurations that might correspond to a strain-free embeddingof this metric. The action is written in terms of an embedding x ( η α , t ), α ∈ { , } , in E , and a reference metric¯ a αβ ( η γ ). In these particular examples, the actual metric adopted a αβ = ∇ α x · ∇ β x will be identical to the referencemetric, as enforced by a Lagrange multiplier σ αβ .These systems are characterized by a rotational symmetry of the embedding space, as well as a “rotational”material symmetry of the surface, each corresponding to a conserved quantity. However, the embedding itself willbe a generalized cone, not spatially rotationally symmetric except in special cases. While our analysis in terms ofpseudomomentum is new, much of the structure and results of this section are a result of prior work, in particular theformalism of Guven and co-workers [47, 48]. One difference is that our actions are written with respect to a referencearea, but as all deformations are isometric this has only a formal significance in redefining the multiplier associatedwith the metric.To facilitate comparison we adopt, to the extent possible, notation from prior works referred to in this section. Toavoid confusion with other quantities denoted by a letter J , the areal inverse Jacobian is written using the metricdeterminants explicitly.The surfaces we consider are conical embeddings x ( r, s, t ) = r ˆu ( s, t ) parameterized by radial and circumferentialmaterial coordinates r and s and the time t . The two tangents to the surface are d r x = ˆu and d s x = r ˆt , and theonly nonzero components of the metric are a rr = 1 and a ss = r . The normal is ˆu × ˆt = ˆn . The following relationsdescribe the rotation of a surface-adapted orthonormal frame [48] along the circumferential coordinate s , d s ˆu = ˆt , d s ˆt = − ˆu − k ˆn , d s ˆn = k ˆt , (70)where k ( s, t ) is a measure of curvature that does not depend on the radial coordinate. The coordinates r and s areattached to the sheet, and as such are entirely distinct from spatial cylindrical coordinates that might be also usedto describe any embedding of the sheet. The conical singularity at r = 0 breaks “translational” material symmetryof the sheet. The remaining symmetries of the system are the “rotational” material symmetry of the sheet aroundthe singularity, and the translational and rotational symmetries of space. Thus, linear and angular momentum,and angular pseudomomentum, are conserved. We will discuss and exploit both angular quantities in the followingsections. A. Inertia: rotation and circumferential flow
An action for a perfectly flexible, inextensible sheet involves inertia and a constraint on the metric [47], A = (cid:90) t t dt (cid:90) B dA (cid:112) ¯ a/a (cid:2) ¯ ρd t x · d t x − σ αβ ( a αβ − ¯ a αβ ) (cid:3) , (71)5where dA = √ a dη dη . Performing a variation in position and material coordinates, and subsequently invoking (cid:112) ¯ a/a ¯ ρ = ρ and (cid:112) ¯ a/a = 1, we obtain the variation in terms of the stress f α and the (symmetric) material stress T αγ , δA = (cid:90) t t dt (cid:90) B dA (cid:18) d t (cid:20) ρd t x · ( δ x − ∇ γ x δη γ ) (cid:21) + ∇ α (cid:18) − f α · δ x + T αγ δη γ (cid:19) + (cid:20) − d t ( ρd t x ) + ∇ α f α (cid:21) · ( δ x − ∇ γ x δη γ ) (cid:19) , (72) f α = σ αβ ∇ β x , T αγ = f α · ∇ γ x + δ αγ (cid:0) ρd t x · d t x (cid:1) . (73)We will be concerned with the general conservation laws associated with spatial and material symmetries δ x and δη γ , d t ( ρd t x · δ x ) = ∇ α ( f α · δ x ) , (74) d t ( ρd t x · ∇ γ x δη γ ) = ∇ α (cid:0) T αγ δη γ (cid:1) . (75)These we will apply to a prescribed equilibrium motion, consisting of rotation of a steady (generalized-)conical shapeabout a spatial axis ˆZ with an additional s -tangential flow superposed along the shape. Defining spatial cylindricalcoordinates using the axis of rotation, we can express the direction of the position vector as ˆu = R ˆR + Z ˆZ . Thevelocity is thus a combination of a rotation with angular velocity ω in the direction ˆΘ = ˆZ × ˆR and a tangential flowwith “angular velocity” τ in the circumferential s direction, d t x = ωrR ˆΘ + τ d s x , where d s x = r ˆt . Figure 2 illustratesthe two sets of coordinates and the adapted frame. FIG. 2:
A conical surface with material coordinates r, s and adapted frame ˆu , ˆt , ˆn . Materialproperties are symmetric in s . Also shown are cylindrical coordinates R, Θ , Z . For such equilibria, there is a symmetry about the rotational axis, and the Z component of the conserved angularmomentum vector is itself conserved. We start by inserting a small spatial rotational shift δ x = (cid:15) ˆZ × ˆx = (cid:15)rR ˆΘ into(74). If, as in prior work [47], boundary conditions allow us to set off-diagonal terms σ rs = 0, we can simplify theright hand side to an s -derivative, because ˆu · ˆΘ = 0 and the Christoffel symbol Γ sss = 0. Additionally, because theconfiguration is steady, the quantities inside the outer time derivative on the left hand side change only through thetangential motion of material along the steady shape. Thus, we may substitute τ d s for the outer d t . Finally, as τ isuniform, it may be moved inside the s -derivative, and thus all terms can be grouped into a single conserved quantity,0 = (cid:15)r d s (cid:104) ˆt · ˆΘ R (cid:0) σ ss − ρτ (cid:1) − ρωR τ (cid:105) . (76)A material “rotation” in two dimensions, or rather circumferential shift δr = 0, δs = (cid:15) , around the singularity r = 0, will provide us with the only component of the angular material momentum. Inserting this shift into (75) andfollowing similar lines, noting that T rs = 0 and σ ss = r σ ss , we obtain0 = (cid:15)r d s (cid:2) σ ss + ρ (cid:0) ω R − τ (cid:1)(cid:3) . (77)The two conserved quantities bracketed in (76) and (77), along with the flow parameter τ , are sufficient to classifyall such rotating, flowing equilibria of perfectly flexible conical sheets [47]. Eliminating the σ ss term, using the relation6 R + Z = 1 (and its s -derivative), and noting that ˆt · ˆR = d s R , ˆt · ˆZ = d s Z , and (cid:16) ˆt · ˆR (cid:17) + (cid:16) ˆt · ˆΘ (cid:17) + (cid:16) ˆt · ˆZ (cid:17) = 1,it is possible to construct an equation in terms of R and three constants that is first order in s -derivatives of R . Weleave this as an exercise for the enthusiastic reader. Alternatively, one could similarly construct a quadrature for Z,as in the prior work [47]. A related work on rotating, flowing strings [61] used an alternate method in which theconserved Z -component of linear momentum was used to classify the equilibria. This quantity can be captured in thepresent example by inserting a spatial translational shift δ x = (cid:15) ˆZ into (74). B. Bending elasticity
Our next example is a static one in which inertia is neglected, but isometric flexure of the sheet is penalized witha bending energy quadratic in mean curvature [48] with associated uniform stiffness B , A = (cid:90) B dA (cid:112) ¯ a/a (cid:0) − (cid:1) (cid:2) B H + σ αβ ( a αβ − ¯ a αβ ) (cid:3) , (78)where 2 H = b αα , and these components of the extrinsic curvature tensor may be defined either by b αβ = d β ∇ α x · ˆn orthrough the Gauss-Weingarten relations ∇ β ∇ α x = b αβ ˆn and ∇ α ˆn = − b βα ∇ β x . The variation of the action may bewritten [48, 78] in terms of the stress f α , the moment µ , and the material stress T αγ , after invoking (cid:112) ¯ a/a = 1, δA = (cid:90) B dA (cid:20) ∇ α (cid:0) − f α · δ x − µ · ∇ α δ x + T αγ δη γ (cid:1) + ∇ α f α · ( δ x − ∇ γ x δη γ ) (cid:21) , (79) f α = σ αβ ∇ β x + 2 B ( H ∇ α ˆn − ∇ α H ˆn ) , µ = 2 B H ˆn , T αγ = f α · ∇ γ x + µ · ∇ α ∇ γ x − B H δ αγ . (80)Note that σ αγ = f α · ∇ γ x + µ · ∇ α ∇ γ x . For the conical parametrization x = r ˆu ( s, t ), the only nonzero curvature is b ss = − rk , and thus H = − k/ r .For this system with bending and without inertia, the general conservation laws associated with spatial and materialsymmetries δ x and δη γ take the form ∇ α ( f α · δ x + µ · ∇ α δ x ) = 0 , (81) ∇ α (cid:0) T αγ δη γ (cid:1) = 0 . (82)The expression (82) is a rearrangement of ∇ α f α · ∇ γ x δη γ = 0, making use of the Codazzi relation ∇ α b γβ = ∇ γ b αβ toidentify ∇ γ H = H ∇ α b αγ .The rotational symmetry of the embedding space around any arbitrary constant axis ˆD gives rise to the conservationof angular momentum. We insert a small shift of the form δ x = (cid:15) ˆD × x in (81) and express the result in terms of theconserved torque m α [48], ∇ α ( x × f α + ∇ α x × µ ) ≡ ∇ α m α = . (83)The boundary quantity corresponding to the surface divergence ∇ α is m α ν α , where ν r = ± ν s = ± r are thecomponents of the unit tangents normal ± ˆu and ± ˆt . However, if σ rs = 0 as before, the m r term does not contributeto this quantity and we may, following [48], consider the quantity J ≡ r m s which is conserved along the s coordinatelines.The material circumferential symmetry corresponding to δr = 0, δs = (cid:15) gives rise to the conservation of angularmaterial momentum. Using (82) as before, we find d s T ss = 0 and, anticipating our result, define a constant C , T ss = σ ss − B H = − B Cr . (84)Making use of (84), we can express J as J = − B (cid:2) k ˆu + ∂ s k ˆt + (cid:0) k + C (cid:1) ˆn (cid:3) . (85)Its squared modulus is a quadrature for k that can be arranged into the same form as that of planar Euler elastica [48], thus classifying all static equilibria of inextensible conical sheets with quadratic mean curvature energy in termsof the ratio of magnitudes of the angular momentum and pseudomomentum [77].7 VII. OTHER AVENUES
Pseudomomentum is an under-utilized concept in many corners of continuum and structural mechanics. Beyondthe examples in this paper and in the recent monograph by O’Reilly [19], we suggest that the most fruitful andinteresting targets for the application of these approaches may be problems involving evolving or active matter [79]or fluid-structure interactions, including general locomotion or the motion of triple points at capillary contact lines[80] or during water entry of structures [81].
VIII. CONCLUSIONS
We have presented a general variational framework, within which we have applied the balance of pseudomomentumto derive and interpret results on a variety of continua including elastic solids, ideal fluids, and thin structures.
ACKNOWLEDGMENTS
This work was supported by U.S. National Science Foundation grant CMMI-1462501. H.S. acknowledges partialsupport by Swiss National Science Foundation grant 200020 182184 to J. H. Maddocks. We thank A. Gupta and A.Yavari for helpful suggestions.
Appendix A: Referential description of first gradient theory
Here we rederive some of the results of Section III while working in the referential frame. Defining F ≡ ∇ i x ¯ ∇ i ¯ x and F − ≡ ¯ ∇ i ¯ x ∇ i x , the two sets of results can be translated into each other by use of the Piola transforms J ∇ · () =¯ ∇ · (cid:0) J F − · () (cid:1) and J − ¯ ∇ · () = ∇ · (cid:0) J − F · () (cid:1) , special cases of which are the Piola identities that hold when () isunity, one of which was used in Section III A.We begin by expressing the action using a reference volume integral, A = (cid:82) t t dt (cid:82) B d ¯ V ¯ L (¯ x , t ; x , d t x , F ). Undertransformations of the dependent and independent fields, the change in the action is δA = (cid:90) t t dt (cid:90) B d ¯ V (cid:104) d t (cid:0) ¯ L δt (cid:1) + ¯ ∇ i (cid:0) ¯ L δη i (cid:1) + ˜ δ ¯ L (cid:105) , (A1)which may be arranged as δA = (cid:90) t t dt (cid:90) B d ¯ V (cid:104) d t Q + ¯ ∇ · J ( R ) + E ( R ) · ˜ δ x (cid:105) , (A2)with the Euler-Lagrange term E ( R ) = ∂ ¯ L ∂ x − d t (cid:18) ∂ ¯ L ∂d t x (cid:19) − ¯ ∇ · (cid:32)(cid:20) ∂ ¯ L ∂ F (cid:21) T (cid:33) , (A3)and charge and current terms Q = ∂ ¯ L ∂d t x · δ x + (cid:18) ∂ ¯ L ∂d t x · F (cid:19) · ( − δ ¯ x ) + (cid:18) ∂ ¯ L ∂d t x · d t x − ¯ L (cid:19) ( − δt ) , (A4) J ( R ) = (cid:20) ∂ ¯ L ∂ F (cid:21) T · δ x + (cid:32)(cid:20) ∂ ¯ L ∂ F (cid:21) T · F − ¯ L I (cid:33) · ( − δ ¯ x ) + (cid:32)(cid:20) ∂ ¯ L ∂ F (cid:21) T · d t x (cid:33) ( − δt ) . (A5)8As in Section III, we obtain balance laws for momentum, E ( ¯ L ) ≡ ∂ ¯ L ∂ x − d t (cid:18) ∂ ¯ L ∂d t x (cid:19) − ¯ ∇ · (cid:32)(cid:20) ∂ ¯ L ∂ F (cid:21) T (cid:33) = on B , (A6) ˆ¯n · (cid:20) ∂ ¯ L ∂ F (cid:21) T = on ∂ B (A7) (cid:116) − ˆ¯N · (cid:20) ∂ ¯ L ∂ F (cid:21) T + U ∂ ¯ L ∂d t x (cid:124) = on S ( t ) (A8)pseudomomentum, E ( ¯ L ) · F ≡ (cid:34) ∂ ¯ L ∂ x − d t (cid:18) ∂ ¯ L ∂d t x (cid:19) − ¯ ∇ · (cid:32)(cid:20) ∂ ¯ L ∂ F (cid:21) T (cid:33)(cid:35) · F = on B , (A9) ˆ¯n · (cid:32)(cid:20) ∂ ¯ L ∂ F (cid:21) T · F − ¯ L I (cid:33) = on ∂ B , (A10) (cid:116) − ˆ¯N · (cid:32)(cid:20) ∂ ¯ L ∂ F (cid:21) T · F − ¯ L I (cid:33) + U (cid:18) ∂ ¯ L ∂d t x · F (cid:19) (cid:124) = on S ( t ) , (A11)and energy, E ( ¯ L ) · d t x ≡ (cid:34) ∂ ¯ L ∂ x − d t (cid:18) ∂ ¯ L ∂d t x (cid:19) − ¯ ∇ · (cid:32)(cid:20) ∂ ¯ L ∂ F (cid:21) T (cid:33)(cid:35) · d t x = 0 on B , (A12) ˆ¯n · (cid:32)(cid:20) ∂ ¯ L ∂ F (cid:21) T · d t x (cid:33) = 0 on ∂ B , (A13) (cid:116) − ˆ¯N · (cid:32)(cid:20) ∂ ¯ L ∂ F (cid:21) T · d t x (cid:33) + U (cid:18) ∂ ¯ L ∂d t x − ¯ L (cid:19) (cid:124) = 0 on S ( t ) , (A14)where ˆ¯n and ˆ¯N are unit normals to the external boundary and internal surface of discontinuity in the referenceconfiguration.The balance of momentum has the form d t (cid:18) ∂ ¯ L ∂d t x (cid:19) + ¯ ∇ · (cid:32)(cid:20) ∂ ¯ L ∂ F (cid:21) T (cid:33) = ∂ ¯ L ∂ x . (A15)The balance of energy can also, with the help of (31), be rearranged into d t (cid:18) ∂ ¯ L ∂d t x · d t x − ¯ L (cid:19) + ¯ ∇ · (cid:32)(cid:20) ∂ ¯ L ∂ F (cid:21) T · d t x (cid:33) = − ∂ ¯ L ∂t . (A16)Finally, with the chain rule ¯ ∇ ¯ L = ∂ ¯ L ∂ ¯ x + ∂ ¯ L ∂ x · F + ∂ ¯ L ∂d t x · d t F + (cid:20) ∂ ¯ L ∂ F (cid:21) T : ¯ ∇ F , (A17)where the double contraction involves the referential ¯ ∇ leg and the present leg of F , we can rearrange the balance ofpseudomomentum (A9) to obtain d t (cid:18) ∂ ¯ L ∂d t x (cid:19) + ¯ ∇ · (cid:32)(cid:20) ∂ ¯ L ∂ F (cid:21) T · F − ¯ L I (cid:33) = − ∂ ¯ L ∂ ¯ x . (A18)9The quantity appearing inside the referential divergence is the most commonly presented form of the Eshelby tensor[5, 15]. [1] R. Peierls. Momentum and pseudomomentum of light and sound. In F. Bassani, F. Fumi, and M. P. Tosi, editors, Proceedings of the International School of Physics “Enrico Fermi”: Highlights of Condensed-Matter Theory , pages 237–255. North-Holland, Amsterdam, 1985.[2] D. Rogula. Noether’s theorem for a continuous medium interacting with external fields.
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