On the geometric character of stress in continuum mechanics

On the geometric character of stress in continuum mechanics Eva Kanso, Marino Arroyo, Yiying Tong, Arash Yavari, Jerrold E. Marsden1and Mathieu Desbrun Abstract. This paper shows that the stress field in the classical theory of continuum mechanics may be taken to be a covector-valued differential two-form. The balance laws and other fundamental laws of continuum mechanics may be neatly rewritten in terms of this geometric stress. A geometrically attractive and covariant derivation of the balance laws from the principle of energy balance in terms of this stress is presented. Mathematics Subject Classification (2000). Keywords. Continuum mechanics, elasticity, stress tensor, differential forms. 1. Motivation This paper proposes a reformulation of classical continuum mechanics in terms of bundle-valued exterior forms. Our motivation is to provide a geometric description of force in continuum mechanics, which leads to an elegant geometric theory and, at the same time, may enable the development of space-time integration algorithms that respect the underlying geometric structure at the discrete level. In classical mechanics the traditional approach is to define all the kinematic and kinetic quantities using vector and tensor fields. For example, velocity and traction are both viewed as vector fields and power is defined as their inner product, which is induced from an appropriately defined Riemannian metric. On the other hand, it has long been appreciated in geometric mechanics that force should not be viewed as a vector, but rather a one-form. This fits naturally with one of the main properties of a force, namely that when paired with a displacement (a vector), one gets work. No metric is needed for this operation of course when force is thought of as a one form. One also sees the same thing when one looks at the tensorial nature of the Euler–Lagrange equations: the equations themselves are natually one-form equations, not vector equations. Despite this, the notion of force as a one-form has not properly been put into the foundations of continuum mechanics. In the 01 Research partially supported by the California Institute of Technology and NSF-ITR Grant ACI-0204932. Kanso E., Arroyo M., Tong Y., Yavari A., Marsden J.E. and Desbrun M, On the geometric character of stress in continuum mechanics, Zeitschrift für Angewandte Mathematik und Physik (ZAMP), Vol. 58, Issue 5, pp. 843-856, 2007 2E. Kanso et al. geometric approach to continuum mechanics proposed in this paper, traction is defined as an exterior one-form. Consequently, one also has a metric-independent notion of power as the natural pairing between the velocity vector field and the traction one-form. Although the importance of the geometric character of these fields is already known in mechanics (see, for example, [17] and [3]), the classical derivation of the balance laws as presented in most works does not reflect this geometric understanding. One of the purposes of the present work is to fill this gap. An outcome of this approach is that the stress field is naturally described as a bundle-valued two-form. The balance laws are then rewritten in terms of the new geometric stress by appealing to tools from differential calculus on bundlevalued forms; that is, in terms of Cartan’s calculus [2]. It is worth noting that the notion of stress as a covector-valued two-form appears in the recent literature, e.g., [18, 9, 13, 6], but a reformulation of the balance laws in terms of this stress in arbitrary Riemannian ambient spaces has remained open. This paper fills that gap and provides a complete treatment of continuum mechanics, including balance laws and constitutive equations, in terms of this geometric notion of the stress. The reformulation of elasticity in terms of (bundle-valued) exterior forms brings the theory closer to Discrete Exterior Calculus (DEC), see, e.g., [14] and [5], and, therefore, may contribute to the development of discrete mechanics and structurepreserving integration schemes. The systematic design of algorithms that preserve exactly the conservation laws of momentum and energy or exhibit dissipation consistent with the continuous systems (no spurious numerical dissipation) for any step-size is an active area of research; see, for example, the work on symplectic and variational integrators in [15], [19], [20], [22] and references therein. While such time integrators for finite-dimensional mechanical systems are well-understood, space-time integration algorithms that respect the geometric character of the physical quantities (such as stress and strain) and the symmetries of the equations (such as conservation of momentum and energy) remain a challenge. Discrete Exterior Calculus alone might not, in itself, be sufficient for the design of such conserving algorithms but may provide some useful tools for this undertaking. We view the present study, that is, the geometric reformulation of elasticity, as a first step in our research project on developing a consistent theory of discrete elasticity that will lead to the design of geometric space-time integration algorithms. The organization of this paper is as follows: In §2 we introduce the stress as a bundle-valued form and rewrite the classical balance laws and constitutive relations in terms of this geometric stress. In §3, we assume the existence of a stress form, with no reference to the stress tensor, and present a covariant derivation of the balance laws and constitutive equations. The results are summarized in §4. 4E. Kanso et al. mappings TR × T∗R → Ror T∗R × TR → R. We denote the space of covariant 2-tensor fields on Rby T0 2(R), the space of contravariant 2-tensors by T2 0(R) and mixed tensors by T1 1(R) or T1 1(R). This notation extends in the obvious way to k-tensors. Further, let Ωk(R) denote the space of k-forms, or alternating k-tensors, on R. In particular, Ω0(R) is the space of smooth functions on Rand Ω1(R) is the space of smooth sections of T∗R. Similarly, we introduce a basis Eion TR0 and its dual Eion T∗R0and adopt analogous notation for tensors on R0as well as for two-point tensors, that is, tensors that can have “legs” in both Rand R0 connected through the diffeormorphism ϕ:R0→ R. For example, T0,0 1,1(R,R0) denotes the space of two-point 2-tensors or bilinear maps TR × TR0→R. Finally, the flat (·)[and sharp (·)]operations refer to lowering and raising tensor indices. On Rthis would mean using the metric g. For example, [:TR → T∗Ris defined by v[, w®=g(v, w) and its inverse is ]:T∗R → TR. Similar operations are defined on the reference configuration R0with respect to a metric G. Here, the symbol h·,·i is used to denote the natural pairing of a contravariant field with a covariant field, such as the pairing of a vector field and a one-form or covector. We shall sometimes use the notation hh·,·ii to denote the inner product between two covariant or contravariant fields with respect to the corresponding metric. Continuum Mechanics. The motion (1) is assumed to occur due to the action of body forces per unit mass and surface traction forces per unit area of the boundary ∂R. Continuum mechanics aims at providing the dynamical equations governing the motion under these conditions. In this paper, we adopt the standpoint that forces are one-forms as explained in §1, hence, the surface traction tand the body force bare naturally defined as one-forms and represented as vector fields through the ]operator. This view is important to the development of Elasticity in terms of bundle-valued forms. 2.1. The stress field as a covector-valued two-form Surface traction and Cauchy’s stress. In the classical non-relativistic theory, the basic postulate for formulating the dynamical equations of motion is the existence of a stress field t](x, t;n) defined everywhere in R. Physically, t](x, t;n) represents the force per unit area exerted on a surface element dain Roriented with unit normal n. It is also convenient to introduce the stress field p](X, t;N) acting on surface elements in Rbut measured per unit area of the corresponding surface elements in R0, that is, p](X, t;N) dA=t]¡x(X, t), t;n) da, (2) where dAis an oriented surface element in R0with unit normal Nand NdAis related to ndaby the Piola formula: nda=JF−TNdA. Here, the deformation gradient2is denoted F=∂ϕ/∂X( = Fa Aea⊗EA) is a mixed 2-tensor ∈T1,0 0,1(R,R0), 2We cannot resist making the standard remark that despite its misleading name, Fis not a On the geometric character of stress in continuum mechanics 5 J= det(F)pdet(g)/det(G)(det is the determinant), and (·)Tdenotes the transpose. Cauchy’s stress theorem states that there are second-order stress tensors called, respectively, the Cauchy stress tensor σand the two-point Piola-Kirchhoff stress tensor P, such that t(x, t;n) = hσ(x, t),ni,p(X, t;N) = hP(X, t),Ni.(3) This means that t(x, t;n) and p(X, t;N) depend linearly on nand N, respectively. Rewriting the stress as a covector-valued two-form. Although the physical interpretation of the notion of stress is geometric, their vectorial and tensorial representations fail to exploit, or even reveal, their geometric character. One of the main goals of the present work is to clarify this geometric nature by rewriting the stress fields as covector-valued two-forms. We will take an approach to stress that considers them to be covector valued two-forms and regards them as fundamental quantities in a manner similar to the way one postulates the existence of t(x, t;n) in the standard approach. However, before taking this point of view, we show how they will end up being related to the standard notions. Namely, if we imagine the standard quantities being given, we define the “new” stresses Tand Pin terms of them by applying the Hodge star operation ∗2to the second ‘leg’ of σand Prespectively as follows: T=∗2σ,P=∗2P.(4) That is, in coordinate notation, one has: T=σab ea⊗(∗eb), and P=PaA ea⊗ (∗EA). By definition, one obtains T∈Ω1(R)⊗Ω2(R) and P∈Ω1(R)⊗Ω2(R0).3 Physically, Tand Pcan be interpreted as follows: the stress, upon pairing with a velocity field, provides an area-form that is ready to be integrated over a surface to give the rate of work done by the stress on that surface — this point is elaborated further in §2.2. Another point is worth mentioning. That is, part of the linearity of tis that it switches sign under a change of sign of n. This property is nicely built into the new tensors Tand Psimply because they are two forms—changing the arguments as two-forms switches their signs and this may be regarded as a reflection of the change of orientation of the surface to which nis normal. However, note that when we say Tand Pare covector valued two forms, there is no need to mention nor a surface element daas such—unless one wants to reconstitute the classical stresses from them using oriented surface elements. The Piola transformation. Recall that the standard stress tensors σand P are related through the Piola transformation: Jσ=PFT.(5) gradient at all, but simply is the derivative of the map ϕ. 3Or, T∈TR ⊗ Ω2(R) and P∈TR ⊗ Ω2(R0), depending on the representation of σand P. 6E. Kanso et al. Clearly, Pis not the pull-back of σby the motion ϕ. But this equation, when written in terms of the stress-forms Tand P, reads as: P=ϕ∗2T,(6) where ϕ∗2is defined as the pull back by the mapping ϕof the area-form of a covector-valued two-form, e.g., T∈T∗(R)×Ω2(R), or of the second ‘leg’ of a two-tensor, e.g., σ∈T∗(R)×T∗(R). That is, the Piola transformation in (6) has a clear geometric interpretation: Pis the pull-back of the area-form part of Tthat does nothing to the covector-valued part. 2.2. Physical interpretation of the stress form Rate of work done by the stress. The rate of work Rtdone by the traction forces ton an oriented surface Sof the continuum can be written as Rt=ZS hv,tida=ZS hv,σ(·,n)ida=ZS σ(v,n) da=ZS hσ(v,·),ndai(7) where v=v(x, t) is the spatial velocity field. Surface integrals (over oriented surfaces) are more naturally expressed in terms of two-forms (that replace nda). To this end, one can readily check that Rt=ZS hσ(v,·),ndai=ZS ∗2σ(v,·) = ZS hv,∗2σi=ZS hv,Ti(8) The above equation reads naturally as follows: the rate of work done by the stress on an oriented hypersurface Sis obtained by pairing the stress Twith the velocity field and integrating the resulting area-form over S.Notice that if the orientation of Sswitches, then the sign of the integral automatically switches and this corresponds to the change of sign of nin the traditional approach. Similar relations hold for Pand p, namely, Rt=RS0 hV,Pi,where Vis the material velocity field defined by V¡X, t¢=∂ϕ(X, t)/∂t and v(x, t) = V¡X(x, t), t¢. The resultant force in Euclidean space. Let the body deform in a Euclidean space. The notion of a resultant force acting on a surface Swith unit normal ndepends on the Euclidean structure of the ambient space and is given by f= RStda=RShσ,ndai.This force can be rewritten as f=ZS T=ZS ∗2σ=ZS ea⊗ ∗(σabeb) = ZS ea⊗ ∗ta=ea⊗ZS ∗ta,(9) where the Euclidean structure allows us to “factor out” the basis vectors of the traction field and integrate the area-form component-wise. The force in the latter expression does not explicitly depend on the normal nda. Also, it is clear that Tautomatically obeys Cauchy’s lemma in the sense that the resultant force changes sign if we change the orientation of S, as we have mentioned previously. On the geometric character of stress in continuum mechanics 7 2.3. Differentation of bundle-valued forms In this section, we define an operation that will be of importance for rewriting the balance laws in terms of Tand Pin §2.4, namely, a differentiation operation d of vector- and covector-valued forms. The differentiation dcombines the exterior derivative d, that has a topological character, with the covariant derivative ∇with respect to the Riemannian connection, that has a metric character, see, e.g., [1] and [8]. To this end, recall that, in component notation, the covariant derivative ∇vof a vector field v=vieion TRis given by ∇jvi=vi |j=∂vi/∂xj+γi jkvk, where γi jk are the Christoffel symbols, also called the connection coefficients. This suggests that ∇vcan be expressed as a mixed 2-tensor, that is, a vector-valued one-form ∇v=vi |jei⊗ej. In particular, one has ∇ej=ei⊗γi jkek=ei⊗ωi j, where ωi j=γi jkekare called the connection one-forms. The derivative d. Let Tdenote either TRor T∗R, and let kbe any integer ≤3. We define the differential operator d:T⊗Ωk−1(R)−→ T⊗Ωk(R); T7−→ dT by hu,dTi=d(hu,Ti)− ∇u˙ ∧T,(10) for all u∈T∗, where ˙ ∧is, by definition, an inner product or a pairing on the first ‘leg’ and a wedge product on the second ‘leg’. For example, if one considers T=a⊗band S=c⊗dboth in T(R)⊗Ω1(R), one gets T˙ ∧S=hha,ciib∧din Ω2(R).4Note that for k= 0, dreduces to the regular covariant derivative, while for k= 3, dis identically zero. Now, in order for (10) to provide a valid definition of d, one needs to show that its right hand side depends only on the point values of uand, hence, uniquely defines the differential dT. To this end, note that for any function f∈Ω0(R), one has d(hfu,Ti) = d(f∧ hu,Ti) = (df)∧ hu,Ti+fd(hu,Ti).(11) On the other hand, one can readily verify that ∇(fu)˙ ∧T= (u⊗df)˙ ∧T+f∇u˙ ∧T= (df)∧ hu,Ti+f∇u˙ ∧T,(12) which proves our claim. Note that the differential operator dis closely related to Cartan’s exterior covariant differential (reviewed in [9, Chapter 9], see also [21] and [16]), which was originally developed to express connections and curvatures in terms of forms. Motivated by the fact that the usual pull-back of forms commutes with the exterior derivative, we are able to use dto define a derivative Don elements of 4Clearly, the ˙ ∧operation can be extended to all tensors by linearity. 8E. Kanso et al. the space T⊗Ωk−1(R0) (k≤3) such that the following diagram commutes T⊗Ωk−1(R0)ϕ∗2 ←−−−− T⊗Ωk−1(R) D  y  y d T⊗Ωk(R0)←−−−− ϕ∗2 T⊗Ωk(R) where ϕ∗2denotes a partial pullback of the “(k−1)−form” part of the tensors but does nothing to the vector or covector values. This diagram simply means that for ν∈T, ω ∈Ωk−1(R), we have d(ν⊗ω) (u, ϕ∗V1, ..., ϕ∗Vk) = D(ν⊗ϕ∗ω) (u,V1, ..., Vk),(13) for all u∈T∗and V1, ..., Vk∈TR0. Equation (13) provides a definition for D. Useful identities. The following identities hold:5 (div σ]2)⊗µ=d(∗2σ),(Div P]2)⊗µ0=D(∗2P).(14) where div and Div denote the divergence operator on Rand R0, respectively, while σ]2=σb aea⊗eband P]2=PB AEA⊗EB, that is, (.)]2acts on the second ‘leg’ only. Here, µand µ0are volume forms in Ω3(R) and Ω3(R0), respectively. 2.4. Balance laws and constitutive relations Balance of linear momemtum. The pointwise equations of balance of linear momentum can be written either with respect to the current configuration R (Eulerian form) or with respect to the reference configuration R0(Lagrangian form) as follows ρ˙ v[= div σ]2+ρb, ρ0˙ V[= Div P]2+ρ0B,(15) where the overdot denotes the material time derivative. Here, ρis the mass density in R, while ρ0is the referential mass density. Also, one has B(X, t) = b(x(X, t), t). It is worth emphasizing here that V(X, t), B(X, t) and p(X, t) are expressed in terms of a base point Xin R0but take their values in the tangent (or cotangent) fiber TxR(or T∗ xR) above the corresponding point x=ϕ(X, t) in R. Take the tensor product of the point-wise balance of linear momentum (15) with the volume forms µand µ0, respectively, and use the identities in (14) to get ˙ v[⊗ρµ =dT+b⊗ρµ, ˙ V[⊗ρ0µ0=DP+B⊗ρ0µ0.(16) Balance of angular momemtum. Balance of angular momentum states that σT=σ,(17) 5One can prove analogous results for any 2-tensor. On the geometric character of stress in continuum mechanics 9 Consequently, the tensor PFT=Jσis also symmetric. This symmetry translates in terms of T, viewed as a vector-valued two-form, to the following equality: (ν⊗β)˙ ∧T= (β⊗ν)˙ ∧T,(18) for all ν, β ∈Ω1(R). The symmetry of Tcan be interpreted physically as follows. Consider a surface Snwith unit normal nmoving at a velocity ve=ve, where eis a unit vector. From (18), one gets that (ve[⊗n[)˙ ∧T=n[∧hve,Ti=e[∧hvn,Ti, where vn=vn. Combining this with (8), we see that the power per unit area expended by the stress as a surface Snwith unit normal nmoves at a velocity ve=veis equal to the power per unit area expended by the stress as a surface Se with unit normal e moves at a velocity vn=vn. Finally, note that the statement of balance of angular momentum for P(as a vector-valued two-form) can be equivalently expressed as (ν⊗ϕ∗β)˙ ∧P= (β⊗ϕ∗ν)˙ ∧P.(19) Conservation of mass. Recall that the pointwise equation of conservation of mass is usually written as ρ0=ρJ, (20) which can be expressed as ϕ∗(ρµ) = ρ0µ0. Balance of energy. For an elastic material one assumes that there exists a strain energy function eper unit mass whose change represents the change in the internal energy due to mechanical deformations. Balance of energy may be written as ZV ρhv,biµ+ZS hv,Ti=d dt³1 2ZV ρh hv,vii µ+ZV eρµ´,(21) or, equivalently, on R0as ZV0 ρ0hV,Biµ0+ZS0 hV,Pi=d dt³1 2ZV0 ρ0h hV,Vii µ0+ZV0 eρ0µ0´.(22) Here, the hh,ii denotes the inner products both on TRand TR0, respectively. The volume integrals are taken over an arbitrary subset V⊆ R while V0=ϕ−1(V)⊆ R0and the area integrals are taken over the bounding surfaces S=∂V and S0=∂V0. One can readily check, using Stokes’ theorem, the definition of din (10) and the balance of linear momentum (16), that the rate of change of internal energy is equal to d dtZV eρµ =ZV ∇v˙ ∧T,d dtZV0 eρ0µ0=ZV0 ∇V˙ ∧P.(23) That is, the stress power can be expressed in terms of the stress form as in ∇v˙ ∧T in Rand ∇V˙ ∧Pin R0. In the classical theory, the stress power is defined as the inner product hh∇v,σii in Ror hh∇V,Pii in R0, which can also be written as hh ˙ F,Pii.