Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 41, 3, pp. 593-621, Warsaw 2003 AFFINE TENSORS IN SHELL THEORY Géry De Saxcé Laboratoire de Mécanique de Lille, Université de Lille-I, France e-mail: gery.desaxce@univ-lille1.fr Claude Vallee Laboratoire de Modélisation Mécanique et de Mathématiques Appliquées Université de Poitiers, France e-mail: vallee@l3ma.univ-poitiers.fr Resultant force and moment are structured as a single object called the torsor. Excluding all metric notions, we define the torsors as skew- symmetric bilinear mappings operating on the linear space of the affine vector-valued functions. Torsors are a particular family of affine tensors. On this ground, we define an intrinsic differential operator called the af- fine covariant divergence.Next, we claim that the torsor field characteri- zing the behavior of a continuousmedium is affine covariant divergence free.Applying this generalprinciple to thedynamicsof three-dimensional media, Euler’s equations are recovered. Finally, we investigated more thoroughly the dynamics of shells. Using adapted coordinates, this ge- neral principle provides a consistent way to obtain new equations with non-expected terms involvingCoriolis’s effects and the time evolution of the surface. Key words: tensorial analysis, continuummechanics, dynamics of shells 1. Introduction Our starting point is closely related to a new setting developed in mecha- nics by Souriau (1992, 1997a) on the ground of two key ideas: a new definition of torsors and the crucial part played by the affine group of Rn. This group forwards on a manifold an intentionally poor geometrical structure. Indeed, this choice is guided by the fact that it contains both Galileo and Poincaré 594 G. De Saxcé, C. Vallee groups (Souriau, 1997b), that allows to involve the classical and relativistic mechanics at one go. This viewpoint implies that we do not use the trick of the Riemannian structure. In particular, the linear tangent space cannot be identified to its dual one and tensorial indices may be neither lowered nor raised. To each group corresponds a class of tensors. The components of these tensors are transformed according to the action of the considered group. The standard tensors discussed in the literature are those of the linear groupof Rn. We will call them linear tensors. A fruitful standpoint consists in considering the class of the affine tensors, corresponding to the affine group.To each group a family of connections allowing one to define covariant derivatives for the cor- responding classes of tensors is associated. The connections of the linear group are known throughChristoffel’s coefficients. They represent, as usual, infinite- simal motions of the local basis. From a physical viewpoint, these coefficients are force fields such as gravity or Coriolis’s force. To construct the connection of the affine group, we needChristoffel’s coefficients stemming from the linear group and additional ones describing infinitesimalmotions of the origin of the affine space associated with the linear tangent space. 2. Affine tensors Notations Let T be a linear space (or vector space) of the dimension n, and ( → eα) be a basis of T . The associated co-basis ( ← e α) is such that ← e α( → e β) = δ α β . A new basis → eα′ = P β α′ → eβ can be defined through the transformation matrix P =(P β α′).We denote the inverse matrix P −1. Tensor class In a previous paper (de Saxcé, 2002), we proposed a generalization of the usual concept of the tensor, relevant for themechanics: the tensors are objects the components of which are changed by a given group of transformations (more precisely, they are changed by a linear representation of the considered group). Considering the linear group GL(n), we recover the class of linear tensors. Nevertheless, other choices of transformation groups are possible. In many applications, people customarily handle the orthogonal group O(n), a subgroup of GL(n), that leads to the class of the Euclidean tensors. On Affine tensors in shell theory 595 the other hand, considering the affine group A(n), an extension of GL(n) obtained by adding the translations, we define the class of affine tensors. Affine space To define an origin Q of the affine space AT associated to T , we can use the column vector V0 collecting the components V α 0 , in the basis ( → eα), of the vector −→ Q0 joining Q to the zero of T considered as a point of AT . By the choice of this affine frame r =(V0,( → eα)), any point V of AT can be identified to the vector −−→ QV = Vα → eα. Now, let r ′ = (V ′0,( → eα′)) be a new affine frame of the origin Q′ and basis ( → eα′). Let C ′ be the column vector collecting the components Cα ′ of the translation −−→ Q′Q in the new basis. The set of all affine transformations a =(C′,P) is the affine group A(n). The transformation law for the affine components V β of the point V is Vα ′ =Cα ′ +(P−1)α ′ β V β (2.1) Affine functions Any affinemapping ψ from AT into R is called an affine function of AT . The affinemapping ψ is represented in an affine frame r by ψ(V )= χ+ΦαV α where χ = ψ(Q) and Φα are the components, in the co-basis ( ←−e α), of the unique covector ←− Φ associated to ψ.Wewill call (Φα,χ) the affine components of ψ. After a change of the affine frame, they are given by Φα′ = ΦβP β α′ χ ′ = χ−ΦβP β α′C α′ as it can be easily verified. The set A∗ T of such functions is a linear space of the dimension (n+1). Vector-valued torsors Let then R be a linear space of the dimension p ¬ n. We call the torsor any bilinear skew-symmetric mapping (ψ,ψ̂) 7→−→µ (ψ,ψ̂)∈R from A∗ T ×A∗ T into R. Following Souriau (1992) all the tensorial indices related to R will be located at the left hand of the tensor. With respect to an affine frame r =(V0,( → eα)) of AT and a basis (ρ −→η ) of R, the torsor is represented by −→ V =−→µ (ψ,ψ̂)= γµ(ψ,ψ̂) γ −→η = ( γJαβ Φα Φ̂β + γTα(χΦ̂α− χ̂Φα) ) γ −→η 596 G. De Saxcé, C. Vallee with γJαβ = −γJβα. Let r′ = (V ′0,( → eα′)) be a new frame of AT and γ′ −→η = ρ γ′Qρ −→η be a new basis of R. The corresponding transformation law is found to be γ′Tα ′ = γ ′ ρ(Q −1) (P−1)α ′ µ ρTµ (2.2) γ′Jα ′β′ = ( (P−1)α ′ µ (P −1)β ′ ν ρJµν +Cα ′( (P−1)β ′ µ ρTµ ) − − ( (P−1)α ′ µ ρTµ ) Cβ ′ ) γ′ ρ(Q −1) Proper frames and intrinsic torsors An affine frame will be called a proper frame if the zero vector of T is taken as the origin of the tangent affine space AT : V0 = 0. Any change of proper frames is a linear transformation (no translation C′ =0). Restricting the analysis to linear transformations, we define the class of intrinsic torsors. For any intrinsic torsor −→µ 0, transformation law (2.2) degenerates into γ′Tα ′ = γ ′ ρ(Q −1) (P−1)α ′ µ ρTµ γ′Jα ′β′ = γ ′ ρ(Q −1) (P−1)α ′ µ (P −1)β ′ ν ρJµν The components γTα are clearly components of a linear vector-valued tensor given by a linear mapping from T ∗ into R that we call the linear momen- tum. The components γJαβ of the intrinsic torsor can be interpreted as the components of a linear vector-valued tensor given by a bilinear mapping from T ∗×T ∗ into R that we call the intrinsic angular momentum or spin (Misner et al., 1973). The intrinsic tensor being given, the affine components of −→µ in any other affine frame r′ = (V ′0,( → eα′)) are deduced from general transforma- tion law (2.2). Thus, γ ′ Jα ′β′ is obtained as the sum of the component of the spin and of the additional term γ′J α′β′ C = ( Cα ′( (P−1)β ′ µ ρTµ ) − ( (P−1)α ′ µ ρTµ ) Cβ ′ )γ′ ρ (Q−1) called theorbital angularmomentum (Misner et al., 1973). In conclusion, there is a one-to-one correspondence −→µ0 7→ −→µ between the intrinsic torsors and torsors. Affine tensors in shell theory 597 3. Affine connection Affine tangent space While the difference of the components of two points of an affine space is definedwithout ambiguousness, the difference of the coordinates of two points in a manifold has no meaning in general. To get round this difficulty, the key idea is to consider that AT is the affine space associated to the tangent linear space TXM, denoted ATXM and called the affine tangent space at X. As observed by Cartan (1923): ”The affine space at point m could be seen as the manifold itself that would be perceived in an affine manner by an observer located at m”. Linear connection as sliding Let X be a point of a manifold M and X′ = X +dX be another point in the vicinity of X. Let us denote the zero vector at X as 0 and at X′ as 0′. For constructing a linear connection, we need to compare the tangent linear spaces at X and X′ by a suitable identification. A linear space is the corresponding affine space with the zero vector as the particular origin Q. Hence, a linear connection is obtained by a smooth sliding on themanifold of the origin Q =0of ATXM onto the origin Q ′ =0′ of ATX′M, as depicted in Figure1. Infinitesimalmotionsof thebasis are specifiedthroughtheconnection ∇ → eα = ω β α → eβ. Fig. 1. Linear connecting (sliding) Affine connection as rolling On the other hand, let us consider a rolling of the affine tangent space ATXM and let uswork on proper frames (Q =0, Q ′ =0′). The identification 598 G. De Saxcé, C. Vallee Fig. 2. Affine connection (rolling with initial origin at 0) Fig. 3. Affine connection (rolling with arbitrary origin) Fig. 4. Calculation of the affine connection by identifying two neighboring affine tangent spaces Affine tensors in shell theory 599 with theneighboring affine tangent space ATX′M shifts the origin Q =0onto a distinct point from Q′ =0′, as shown inFigure 2.Workingmore generally in arbitraryaffine frames, anaffineconnection is constructedbyrollingof ATXM with the origin Q shifted onto a distinct point from the origin Q′ of ATX′M, according to Figure 3.We define the affine connection ωαC as the components of the infinitesimal displacement −→ dQ = −−→ QQ′ of the origin when identifying both neighboring affine tangent spaces (Figure 4). Because of rolling, it holds −→ 00′ = dXα → eα Hence, the displacement is decomposed as follows ωαC → eα = −−→ QQ′ = −→ Q0+ −→ 00′+ −−→ 0′Q′ =(Vα0 +dX α) → eα−V α′ 0 → eα′ = = dXα → eα−∇(V α 0 → eα) that gives ωαC = dX α −∇Vα0 (3.1) In short, the affine connection provides a smoothvariation of themoving affine frames X 7→ r(X)= ( V0(X),( → eα(X)) ) Theusual connectionmatrix ωαβ gives a smoothvariationof thebasis,while the ωαC specify the motion of the affine space origin. The affine connections are due to Cartan (1923). In the present section, the key ideas are explained for readers interested in the mechanical science but who are not necessarily aware of advanced concepts of the differential geometry. A presentation using the geometry of principal bundles can be found in (de Saxcé, 2002) but, in any case, the final result is the same, whether it is obtained by the principal bundle theory or as before. Affine covariant derivative On this ground, we are able to calculate the intrinsic covariant derivative of the affine tensors of any type, that we call the affine covariant derivative and denote ∇̃ (in opposition to the usual covariant derivative ∇which should be called the linear covariant derivative). First of all, let us consider a field of the tangent vector. As a member of the linear space TXM, it has a linear covariant derivative, but as amember of the associated affine space ATXM, it has also an affine covariant derivative denoted ∇̃V . By the choice of an affine 600 G. De Saxcé, C. Vallee frame r = (V0,( → eα)), any point V of ATXM can be identified to the vector −−→ QV = Vα → eα. Thus, its linear covariant derivative is ∇ −→ V = −−→ Q′V ′− −−→ QV = −−→ Q′Q+ −−→ QV ′− −−→ QV The difference between the two last terms represents the infinitesimal va- riation of the field V as point of the affine space, i.e. its affine covariant derivative. Hence, we obtain the relation ∇ −→ V =(−ωαC + ∇̃V α) → eα fromwhich one we deduce ∇̃Vα =∇Vα+ωαC (3.2) Next, we calculate the affine covariant derivative of the affine functions. According to the rule of differentiating a product, one has ∇̃(ψ(V ))= ∇̃(χ+ΦαV α)= ∇̃χ+(∇̃Φα)V α+Φα(∇̃V α) As the components Φα represent a linear object, the covector ←− Φ associated to ψ, its affine derivative is just its linear one. Owing to equation (3.2), it holds ∇̃(ψ(V ))= ∇̃χ+(∇Φα)V α+Φα(∇V α+ωαC) Hence, one has ∇̃(ψ(V ))= ∇̃χ+∇(ΦαV α)+Φαω α C = ∇̃χ+∇(ψ(V ))−∇χ+Φαω α C On the other hand, for any field V , we have to satisfy ∇̃(ψ(V ))= d(ψ(V ))=∇(ψ(V )) Finally, the affine covariant derivatives of the components of ψ are given by ∇̃Φα =∇Φα ∇̃χ =∇χ−Φαω α C (3.3) 4. Affine covariant divergence of vector-valued torsors Thin body Let M be a manifold of the dimension n representing the physical space in statics (n = 3) and the space-time in dynamics (n = 4). The mapping Affine tensors in shell theory 601 N → M : ξ 7→ X = f(ξ) defines a sub-manifold of the dimension p enable one to represent three-dimensional bodies (p = n) or thin ones (p < n). In the sequel, Rwill be the tangent space TξN at ξ, while AT will be the affine tangent space ATXM, that is the tangent space TXM at X = f(ξ) endowed with the structure of the affine space. By a choice of the coordinate systems (Xα) on M and (ξβ) on N , the tangent mapping to f is given by βU α = ∂Xα ∂ξβ (4.1) Linear covariant divergence It is assumed that N is equipped with a symmetric connection α βω = α ρβγdξ ρ, where αρβγ = α βργ are Christoffel’s connection coefficients. The covariant derivative of the tangent vector field −→ V = γV γ −→η on N is given by ∇ γV = dξβ β∇ γV β∇ γV = ∂γV ∂ξβ + γ βργ ρV (4.2) Themanifold M is equipped with a symmetric connection ωαβ = Γ α ρβ dX β (4.3) usingChristoffel’s connection coefficients Γαρβ = Γ α βρ. The covariant derivative of any covector field ←− Φ = Φα ← e α on M is ∇Φα = dX β ∇βΦα ∇βΦα = ∂Φα ∂Xβ −Γ ρ βα Φρ Considering the restriction to the sub-manifold N , it holds γ∇Φα = ∂Xβ ∂ξγ ∇βΦα = ∂Φα ∂ξγ − γU β Γ ρ βαΦρ (4.4) Now,weconsider a tensorfield ξ 7→ T(ξ) of the components γTα asdefined before. We hope to calculate its linear covariant derivative, namely γ∇ γTα. According to the rule of differentiating a product, one has for any covector field on M of the components Φα γ∇( γTαΦα)= (γ∇ γTα)Φα+ γTα(γ∇Φα) The left handmember, representing the divergence of a vector field on N , can be developed using (4.2), while, in the right hand member, the last term is transformed owing to (4.4). After simplification, it remains ∂γTα ∂ξγ Φα+ γ γργ ρTα Φα =(γ∇ γTα)Φα− γTα γU β Γ ρ βα Φρ 602 G. De Saxcé, C. Vallee In the last term, replacing α, β, ρ in turn by β, ρ, α, we obtain (γ∇ γTα)Φα = (∂γTα ∂ξγ + γγργ ρTα+ γTβ γU ρ Γαρβ ) Φα With the covector field Φα being arbitrary, the previous relation is satisfied if and only if γ∇ γTα = ∂γTα ∂ξγ + γγργ ρTα+ γTβ γU ρ Γαρβ (4.5) This formula allows one to calculate the linear divergence of this class of tensors. Affine covariant divergence Now, we are able to calculate the affine covariant derivative of a vector- valued torsor. For any covector field ←− F = γF γ←−η on N , we have ∇̃ (←− F (−→µ (ψ,ψ̂) )) = ∇̃ ( γF γµ(ψ,ψ̂) ) = ∇̃ (( γJαβ Φα Φ̂β+ γTα(χΦ̂α−χ̂Φα) ) γF ) As the affine derivatives of the components γF, Φα, Φ̂β and γTα representing linear objects are equal to their linear derivatives, it holds, according to the rule of differentiating products ∇̃ (←− F (−→µ (ψ,ψ̂) )) = [ (∇̃ γJαβ)ΦαΦ̂β + γJαβ∇(ΦαΦ̂β)+(∇ γTα)(χΦ̂α− χ̂Φα)+ +γTα ( (∇̃χ)Φ̂α− (∇̃χ̂)Φα ) + γTα ( χ(∇Φ̂α)− χ̂(∇Φα) )] γF + γµ(ψ,ψ̂)∇ γF Taking into account expression (3.3) of the derivative of the affine functions, we obtain after some rearrangements ∇̃ (←− F (−→µ (ψ,ψ̂) )) = [ (∇̃ γJαβ)ΦαΦ̂β + γJαβ ∇(ΦαΦ̂β)+ +∇ ( γTα(χΦ̂α− χ̂Φα) ) +(γTα ω β C −ω α C γTβ) ] γF + γµ(ψ,ψ̂)∇ γF which can be simplified as follows ∇̃ (←− F (−→µ (ψ,ψ̂) )) = [ (∇̃ γJαβ)ΦαΦ̂β +∇ ( γµ(ψ,ψ̂) ) − (∇ γJαβ)ΦαΦ̂β + +(γTα ω β C −ω α C γTβ)ΦαΦ̂β ] γF + γµ(ψ,ψ̂)∇ γF Affine tensors in shell theory 603 On the other hand, because the value of ←− F for −→µ is a scalar field, we have ∇̃ (←− F (−→µ (ψ,ψ̂) )) = d (←− F (−→µ (ψ,ψ̂) )) =∇ (←− F (−→µ(ψ,ψ̂) )) = =∇ ( γµ(ψ,ψ̂) ) γF + γµ(ψ,ψ̂)∇ γF Hence, for any ψ, ψ̂ and ←− F , it holds ( ∇̃ γJαβ −∇ γJαβ −ωαC γTβ + γTα ω β C ) γFΦα Φ̂β =0 With the affine functions and covectors being arbitrary, we obtain an expres- sion of the affine covariant derivative of the vector-valued torsors ∇̃ γTα =∇ γTα ∇̃ γJαβ =∇ γJαβ +ωαC γTβ − γTα ω β C (4.6) By analogy with (4.3), we introduce the affine connection coefficients ΓαρC such that ωαC = Γ α ρC dX ρ = ΓαρC γU ρ dξγ Owing to (3.1), one has ΓαρC = δ α ρ − ∂V α0 ∂Xρ −ΓαρβV β 0 (4.7) As a particular case of (4.6), we obtain the affine covariant divergence of a vector-valued torsor γ∇̃ γTα = γ∇ γTα (4.8) γ∇̃ γJαβ = γ∇ γJαβ + γU ρ ΓαρC γTβ − γTα γU ρ Γ β ρC where the divergence of γTα is given by (4.5) and γ∇ γJαβ = ∂γJαβ ∂ξγ + γJρβ γU µ Γαµρ+ γJαρ γU µ Γβµρ+ γ γργ ρJαβ (4.9) which can be easily obtained by reasoning as for the proof of (4.5). 5. Dynamics of three-dimensional bodies Three-dimensional bodies Let a continuous medium (a solid or a fluid) occupying an open domain Ω ⊂ R3 that we call also a body. In order to model its evolution betwe- en the instants t0 and t1, we consider the sub-manifold of the space-time 604 G. De Saxcé, C. Vallee f : ]t0, t1[×Ω → M. For convenience, we choose the same coordinate sys- tem on N and M. Hence, the local expression of f is the identity mapping Xα = ξα. The distinction between the left and right hand indices becomes ir- relevant and, in the present section, we put all the indices at the right hand as usual.Thus,we have βU α = δαβ and γ∇̃= ∇̃γ.Moreover, wewrite γTα = Tαγ and γJαβ = Jαβγ. The behavior of the continuous medium is described by a torsor field X 7→ −→µ (X). We claim that the balance (or conservation law) of momentum of the continuous medium says that the torsor field is affine covariant divergence free ∇̃γ T αγ =0 ∇̃γ J αβγ =0 (5.1) Following Souriau (1992, 1997a), the first equation traduces the balance of linear momentum. The second one is a full covariant version of the balance of angular momentum as presented byMisner et al. (1973, p.156). Balance of the angular momentum Three-dimensional continuous media are mainly considered in the litera- ture as non polarized media, according to Cauchy’s famous theory. Let (Tαγ,Jαβγ) be the affine components of the unique intrinsic torsor field X 7→ −→µ0(X), associated to X 7→ −→µ (X), in a moving proper frame X 7→ r(X) = (0,( → eα(X)). We claim that the intrinsic torsor is spin free Jαβγ =0.Thus, accounting for (4.7), (4.8)2, thebalance of angularmomentum (5.1)2 leads to ∇̃γ J αβγ = Tβα−Tαβ =0 In this symmetry condition of the linearmomentum, the reader can clearly recognize the classical hypothesis of Cauchy’s media. Galilean tensors All what has been said so far may be applied as much for the general relativity theoryas for the classicalmechanics.Henceforth,we shall restrict the analysis to the latter theory. In the sequel,Greek indices are 0 to 3while Latin ones run from 1 to 3 (associated to the space coordinates only). Any point X of the space-time M represents an event occurring at position r and time t. With an appropriate coordinate system, it is represented by Xi = ri, X0 = t. Let us considerGalileo’s group, a subgroupof the affine group A(4), collecting Affine tensors in shell theory 605 the Galilean transformations, that is the affine transformations a = (C′,P) such that (Souriau, 1997b) C′ = [ τ k ] P = [ 1 0 u R ] where u ∈R3 is aGalilean boost, R ∈SO(3) is a rotation, k ∈R3 is a spatial translation and τ ∈R is a clock change. Any coordinate change representing a rigid bodymotion and a clock change r′ =(R(t))⊤ ( r−r0(t) ) t′ = t+ τ0 where t 7→ R(t) ∈ SO(3) and t 7→ r0(t) ∈ R 3 are smooth mappings and τ0 ∈ R is a constant, is called a Galilean coordinate change. Indeed, the cor- responding Jacobeanmatrix is a linear Galilean transformation P = ∂X′ ∂X = [ 1 0 u R ] where u = ̟(t)× (r − r0(t))+ ṙ0(t), is the well-known velocity of transport. It involves Poisson’s vector ̟ such that Ṙ = j(̟)R, where j(̟) (sometimes also denoted ad(̟)) is the skew-symmetric matrix representing the cross- product by ̟ : ∀v ∈R3, j(̟)v = ̟ ×v. Galilean connections At each group of the transformation G a family of connections and the corresponding geometry (called the G-structure by Dieudonné (1971)) is as- sociated. We call Galilean connections the symmetric connections associated to Galileo’s group. In a Galilean coordinate system, they are given by ω = [ 0 0 j(Ω)dr −gdt j(Ω)dt ] (5.2) where g is a column-vector collecting the gj = −Γ j 00 and identified to the gravity (Cartan, 1923), while Ω is a column-vector associated by themapping j−1 to the skew-symmetric matrix the elements of which are Ωij = Γ i j0 and interpreted as Coriolis’s effects (Souriau, 1997a). 606 G. De Saxcé, C. Vallee Balance of the linear momentum In the present sub-section, we shall follow the reasoning proposed by Souriau (1992, 1997a). Let be an Eulerian representation of the continuous medium in which any event is represented in a Galilean coordinate system by Euler’s coordinates Xi = ri and X0 = t. On the other hand, let be a La- grangean representation in which the same event is represented byLagrange’s coordinates of the material particle Xi ′ = si ′ and X′0 = t′, which are not in general Galilean coordinates. They are related to the previous representation by a smooth coordinate change like ri = ϕi(sj ′ , t′) t = t′ We introduce the deformation gradient F ij′ = ∂r i/∂sj ′ , and the velocity ui = ∂ri/∂t. The particle of the coordinates (sj ′ ) being at rest in the Lagrangean re- presentation has the trajectory such that dsj ′ =0. Differentiating, we obtain dX = [ dt dr ] = [ 1 0 u F ][ dt′ ds ] = P dX′ (5.3) To adjust to usual convention in the continuummechanics (compressive stres- ses are negative), we put Si ′j′ = −T i ′j′. The components Si ′j′ are generally recognized as representing the internal forces or stresses in the Lagrangean representation, and are customarily called symmetric Piola-Kirchhoff stresses. The component ρ = T ′00 can be interpreted as themass density. The particles being at rest in this particular representation, the components T0i ′ = T i ′0, interpreted as the linear momentum, are supposed to vanish. In Euler’s coor- dinates, it holds, owing to (2.2)1 T = PT ′P⊤ = [ 1 0 u F ][ ρ 0 0 −S ][ 1 u⊤ 0 F⊤ ] = [ ρ ρu⊤ ρu −σ+ρuu⊤ ] (5.4) where σk = ρuu ⊤ collects kinetic stresses and, according to Simo (1988) σ = FSF⊤ collects Cauchy’s stresses. In Euler’s representation, the events being given by a Galilean coordinate system, the connection matrix is given by (5.2). Thus, accounting for (4.8)1, equation (5.1)1 can be interpreted as Euler’s equations of the continuous medium ∂ ∂rj (ρuj)+ ∂ρ ∂t =0 ρ (∂ui ∂t +uj ∂ui ∂rj ) = ∂σij ∂rj +ρ(gi −2Ωiju j) Affine tensors in shell theory 607 6. Adapted coordinates Moving surface We want to model problems of the dynamics of shells and plates within the frame of the classical mechanics.We have to represent the time evolution of a smooth material surface. Hence, we are in the case n =4 and p =3 < n. In the sequel, the indices a,b,c,e associated to the surface parameters takes only values 1 or 2,while the other Latin indices such that i,j,k are running from 1 to 3 as before. In the Eulerian representation, let us consider aGalilean coordinate system (Xα) on the space-time M. Interpreting the last coordinate ξ0 on the submanifold N as the time, let us suppose that the mapping N → M : ξ 7→ X = f(ξ) is represented in local coordinates by given equations Xi = ri = pi(ξγ)= pi(ξa,ξ0)= pi(ξa, t) X0 = t = ξ0 Adapted coordinates Aclassical tool of the theory of surfaces is the tangent plane to the current point. In order to separate in-plane and off-plane components of the torsor and the balance of momentum, we introduce another coordinates (Xβ ′ ) of the space-time. The new spatial coordinates are X′a, denoted θa, and X′3, denoted θ3. The time coordinate X′0, denoted t′, is unchanged. The key- idea is to choose the new coordinates in such a way that the equation of the material surface at a given time is merely θ3 =0 In these adapted coordinates, the local representation ξ 7→ X′ of themap- ping f defining the sub-manifold N is θa = ξa θ3 =0 t′ = ξ0 (6.1) The adapted coordinates are related to the previous ones through equations like ri = pi(θa, t)+θ3ni(θa, t) t = t′ (6.2) For convenience, following for instance (Naghdi, 1972), we choice ni such that 3∑ i=1 πian i =0 3∑ i=1 nini =1 (6.3) 608 G. De Saxcé, C. Vallee with πia = ∂pi ∂θa vi = ∂pi ∂t βia = ∂ni ∂θa wi = ∂ni ∂t Let n be a column vector of the components ni and π (resp. β) be amatrix the element ofwhich at the ath rowand ith column is πia (resp. β i a).Asusual, n is interpreted as the unit vector normal to the material surface and π as the projector onto the tangent plane (to the material surface at the current time) (Valid, 1995). The uniqueness of n is ensured by conditions (6.3). The column vector v, collecting the components vi, represents the velocity and the column vector w, collecting the components wi, represents the time rate of the unit normal vector. Hence equations (6.3) read πn =0 n⊤n =1 (6.4) Differentiating (6.4)2 leads to n ⊤dn =0. Thus βn =0 n⊤w =0 (6.5) Calculation of the connection matrix Notice that the new coordinates (Xβ ′ ) are not generally Galilean. Hence the connectionmatrix ω′ in this coordinate systemdoes not have the standard form of (5.2). Differentiating (6.2) gives successively P = [ 1 0 0 v+θ3w π⊤+θ3β⊤ n ] (6.6) dP = [ 1 0 0 dv+wdθ3+θ3dw π⊤+β⊤dθ3+θ3dβ⊤ dn ] Putting θ3 =0 on thematerial surface, leads to P = [ 1 0 0 v π⊤ n ] dP = [ 1 0 0 dv+wdθ3 π⊤+β⊤dθ3 dn ] (6.7) Owing to (6.4), the inverse transformation matrix is P−1 =   1 0 −vt a −1π −v3 n⊤   (6.8) Affine tensors in shell theory 609 where, the symmetric matrix a = ππ⊤, represents the first fundamental form of the material surface, vt = a −1πv is the in-plane velocity and v3 = n⊤v is theoff-plane component of thevelocity. Because of the classical transformation law ω′ = P−1ωP +P−1dP and taking into account (5.2) and (6.7-8), the connectionmatrix in theadapted coordinate system is ω′ =   0 0 0 a−1πA1 a −1π(A2+β ⊤dθ3) a−1π(dn+ j(Ω)ndt) n⊤A1 n ⊤A2 0   (6.9) where A1 = j(Ω)(dr+vdt)−gdt+dv+wdθ 3 A2 = dπ ⊤+ j(Ω)π⊤dt Its elements have to be expressed with respect to the differential of the adapted coordinates dθi and dt′. For convenience, we introduce the column vector dθt collecting dθ a, and we put dt′ = dt. By introducing the linear operators dθt = dθ a ∂ ∂θa d dt = I3 ∂ ∂t + j(Ω) where I3 is the 3×3 identity matrix, we have dπ⊤+ j(Ω)π⊤dt = dθtπ ⊤+ dπ⊤ dt dt dn+ j(Ω)ndt = dθtn+ dn dt dt Let gp be a column vector representing the acceleration of transport gp = ∂v ∂t = ∂2p ∂t2 and ∂v ∂θt = ∂2p ∂θt∂t = ∂π⊤ ∂t Accounting for (6.5), it holds j(Ω)(dr+vdt)−gdt+dv+wdθ3 = j(Ω)(π⊤dθt+ndθ 3+2vdt)−gdt+ +gpdt+ ∂π⊤ ∂t dθt+ ∂n ∂t dθ3 = dπ⊤ dt dθt+ dn dt dθ3−g∗dt 610 G. De Saxcé, C. Vallee where g∗ = g−2Ω ×v −gp, is the gravity in the adapted coordinate system, obtained by subtracting Coriolis’s acceleration gc = 2Ω × v and the acce- leration of transport gp from the gravity in the Galilean coordinate system. Because of (6.5)2, one has n⊤ dn dt = n⊤w+n⊤j(Ω)n =0 The connection matrix (6.9) becomes ω′ =   0 0 0 a−1π ( A3+ dn dt dθ3 ) a−1π ( A4+β ⊤dθ3 ) a−1π ( dθtn+ dn dt dt ) n⊤A3 n ⊤A4 0   (6.10) where A3 = dπ⊤ dt dθt−g ∗dt A4 = dθtπ ⊤+ dπ⊤ dt dt As we shall be working latter on in the adapted coordinate system, for sake of easiness, we cancel the prime symbol. It clearly results from (6.1) that aU b = δba aU 3 =0 0U i =0 aU 0 =0 0U 0 =1 (6.11) 7. Shell variables Balance of momentum Let a continuous medium of an arbitrary dimension p ¬ n, the behavior of which is described by a torsor field ξ 7→ −→µ (ξ) on N . Generalizing the approach of Section 5, we claim that the balance of momentum says that this torsor field is affine covariant divergence free γ∇̃ γTα =0 γ∇̃ γJαβ =0 (7.1) Now, we want to particularize this general principle to the shells. Affine tensors in shell theory 611 Discussion Anytorsor has pn components γTα and, owing to the skew-symmetrywith respect to the right hand side indices, pn(n−1)/2 independent components γJαβ. Then, it has pn(n+1)/2 affine components. On the other hand, a torsor field is subjected to n(n +1)/2 independent scalar equations (7.1). In the statics of shells (p = 2,n = 3), only 6 equations are available to determine 12 variables. In the dynamics (p = 3,n = 4), the difficulty is higher with 10 equations for 30 variables. Fortunately, it is possible in theGalilean setting to reduce the redundancy of the shell by introducing additional hypothesis related to the modeling. A resistingmaterial surface can be seen as an approximation of a three dimensio- nalmedium, as a consequence of the fact that it is thin in the normal direction to themiddle surface. There is a broad variety of situations such as a smooth curved sheet or a composite laminate but also a smooth surface approximating a corrugated sheet, a lattice or a fluidmoving between two close sheets and so on. Although the general modeling proposed as before is relevant to represent this wide range of situations, it would be a heavy task to examine every one of them.Hence, we only wish to illustrate ourmethod by focussing the attention on the most simple case of an homogeneous curved thin sheet of the current thickness h. The behavior of a three dimensional body is characterized by a spin free torsor field with components Tαγ, as discussed in Section 5.We hope to build a shell torsor field with the affine components γTα and γJαβ by a suitable integration over the thickness. The shell variables γTα, γJαβ are related to an infinitesimal surface element modeling a piece of the three dimensional body occupying the volume over and above the surface element in the thickness direction, and called the shell element. Three dimensional continuum torsor In a first draft, we adopt two usual hypotheses. On the shell element scale, the surface curvature is neglected and the strain is small (but not necessarily the displacements and rotations). Amore sophisticatedmethod using the con- cept of the affine transport is proposed in (de Saxcé, 2002a), but we shall not follow this way in the present work. According to the approach of Section 5, let Xi ′ = si ′ be Lagrange’s coordinates of the material particles of the three dimensional continuum and X′0 = t′. Neglecting the strains, the motion of the shell element can be locally approximated by a rigid motion r = ϕ(s,t) = p(t)+R(t)s t = t′ 612 G. De Saxcé, C. Vallee where p(t)∈R3 represents the position of the point on themiddle surface (in short, themiddle point) and the timedependent rotationmatrix R(t)∈SO(3) describes the rigidmotion of the shell element around this point.Relation (5.3) degenerates into dX = [ dt dr ] = [ 1 0 u R ][ dt′ ds ] = PdX′ involving the velocity of transport of themiddle point v = ṗ and of any point of the shell element u = v+̟ × (r −p) where ̟ is Poisson’s vector Ṙ = j(̟)R. Taking into account (6.2), it holds u = v+θ3̟ ×n (7.2) On the other hand, (6.6) says that u = v+θ3w (7.3) Identifying (7.2) to (7.3), leads to w = ∂n ∂t = ̟ ×n (7.4) In the Eulerian representation, the torsor components Tαγ are given by (5.4). On the considered scale, the surface curvature effects can be neglected. The inverse transformation matrix is approximated by its value (6.8) on the middle surface. Thus, in the adapted coordinates, the new components Tα ′γ′ are given by T ′ = P−1TP−⊤ =   1 0 −vt a −1π −v3 n⊤   [ ρ ρu⊤ ρu −σ+ρuu⊤ ][ 1 v⊤t −v 3 0 (a−1π)⊤ n ] Accounting for (6.5) and (6.2)-(6.3), it holds T ′ =   ρ ρθ3(a−1πw)⊤ 0 ρθ3a−1πw a−1π ( −σ+ρ(θ3)2ww⊤ ) (a−1π)⊤ −a−1πσn 0 −n⊤σ(a−1π)⊤ −n⊤σn   Affine tensors in shell theory 613 Therefore, in the adapted coordinates, the stress components are σ′ab = 3∑ i,j=1 aae πie σ ij πjc a cb σ′33 = 3∑ i,j=1 ni σij nj σ′b3 = σ′3b = 3∑ i,j=1 aae πie σ ij nj while the new velocity components are w′a = 3∑ i=1 aae πie w i Canceling the prime symbol for sake of easiness, we obtain the torsor compo- nents of the three dimensional medium in the adapted coordinates Tab =−σab+ρ(θ3)2wawb Ta0 = T0a = ρθ3wa Ta3 = T3a =−σa3 T30 = T03 =0 T00 = ρ (7.5) This result has been obtained according to the two classical previous hypothe- ses. They could be eliminated by amore pervasive analysis using for instance tools developed in (Hamdouni et al., 1999) by considering a shell as a stacking- up of curve sheets. Integration over the thickness If the torsor components are calculated with respect to the current point r of the three dimensional medium in a proper frame, the spin components vanish. It is recalled that everypoint X of themanifold M corresponds toanevent occurring at a given position and time. If we neglect once again the curvature effects on the shell element scale, the manifold M can be approximated by the affine tangent space ATXM at the current point X. Naturally, we are working with the basis ( → eα) associated to the considered adapted coordinate system. The current point X of coordinates (X0 = t,Xa = θa,X3 = θ3) is identified to the origin of the proper frame, that is zero of the tangent linear space TXM. Concerning the position of the middle point at the same time, the point X′ of the coordinates (X0 = t,Xa = θa,X3 =0) is represented by the point of the affine tangent space ATXM with the affine coordinates V =   t θ1 θ2 0   −   t θ1 θ2 θ3   =   0 0 0 −θ3   614 G. De Saxcé, C. Vallee Let us take this point as a new origin. In the non-proper frame r′ = (V,( → eα)), the considered point has vanishing coordinates V ′ = 0. Ac- cording to transformation law (2.1), we consider the affine transformation a =(C′,P) with P being the identity of R4, and C′ = V ′−V =   0 0 0 θ3   (7.6) Owing to transformation law (2.2), the components of the linear momentum are unchangedwhile those of the angularmomentumdo not vanish due to the existence of the orbital angular momentum any longer Jα ′β′γ′ = Cα ′ Tβ ′γ′ −Tα ′γ′Cβ ′ For easy notations, we cancel the prime symbol. Owing to (7.6), the non- vanishing components of the angular momentum in the new frame are J3bγ =−Jb3γ = θ3Tbγ J30γ =−J03γ = θ3T0γ (7.7) Under the previous approximations, the torsor at every point of the shell and at the considered time is given by its components with respect to the affine frame associated to themiddle point. By integrating them over the thickness, we obtain the shell variables γTα = h/2∫ −h/2 Tαγ dθ3 γJαβ = h/2∫ −h/2 Jαβγ dθ3 (7.8) Combining (7.5), (7.7) and (7.8) leads to aTb =− h/2∫ −h/2 σab dθ3+ ρh3 12 wawb aT3 =− h/2∫ −h/2 σa3 dθ3 0T0 = ρh (7.9) aJb3 =−aJ3b = h/2∫ −h/2 θ3σab dθ3 0J3b =−0Jb3 = ρh3 12 wb the other variables being zero. The previous result is neither general nor exact but it illustrates a method to construct the shell variables and provides their physical interpretation.Accounting for the symmetryofCauchy’s stress tensor, it remains 11 independent non-zero shell variables in the Galilean setting, instead of 30 in the general affine geometry. Affine tensors in shell theory 615 8. Usual theory of plates and shells in static equilibrium Shell variables Let us assume that all the points of the three dimensional body are at rest in the Galilean coordinate system (Xα) at any time. The function ni does not explicitly depend on t = ξ0 and, consequently, the velocity w va- nishes. The components Nab = −aTb are line densities of membrane forces, and the components Qa =−aT3 are line densities of transverse shear forces. The component ρs = 0T0 is interpreted as a surface density of mass. The components Mab = aJb3 can be interpreted as line densities of bending and twisting couples. Shell variables (7.9) are reduced to Nab =−aTb = h/2∫ −h/2 σab dθ3 Qa =−aT3 = h/2∫ −h/2 σa3 dθ3 (8.1) ρs = 0T0 = ρh Mab = aJb3 =−aJ3b = h/2∫ −h/2 θ3σab dθ3 the other variables are zero. Taking into account the symmetry of the stress tensor, it remains 9 independentvariables in statics, instead of 11 in dynamics. Static equilibrium Let us assume that all the points of the material surface are at rest in the Galilean coordinate system (Xα) at any time.The functions ri does not expli- citly depend on t = ξ0 and, consequently, the velocity v and the acceleration of transport gp vanish. Besides, we suppose that Coriolis’s effects are absent Ω =0. Therefore, it holds dπ⊤ dt =0 dn dt =0 g∗ = g Connection matrix (6.10) becomes ω′ =   0 0 0 −gtdt a −1πdθtπ ⊤+β⊤dθ3 a−1πdθtn −gndt n⊤dθtπ ⊤ 0   with: gt = a −1πg, g3 = n⊤g. Moreover, we assume that the connection on the submanifold N is the connection induced by the one of M. The element 616 G. De Saxcé, C. Vallee of the matrix a−1 at the bth row and eth column will be denoted abe. For convenience, we put cai = a aeπie The following Christoffel’s coefficients are generated Γabc = a bcγ = c a i ∂πib ∂θc Γa3b = c a i ∂ni ∂θb =−bab Γ a 00 =−c a ig i =−gat (8.2) Γ3ab = 3∑ i=1 ni ∂πib ∂θa = bab Γ 3 00 =−g 3 with theother onesbeingzero.Asusual, the coefficients bab define the2nd fun- damental form of the surface. Let us examine the particular form of the balance of linear momentum (7.1)1 in the adapted coordinates.Accounting for (6.11), (8.1) and (8.2),many terms disappear. For the in-plane translation equilibrium, it remains −γ∇ γTa = Nba ∣∣∣ b − babQ b+ρsg a t =0 (8.3) where Nba ∣∣∣ b = ∂Nba ∂θb +ΓabcN bc+ΓccbN ba is the linear covariant divergence with respect the connection on thematerial surface. Similarly, for the off-plane translation equilibrium equation, it holds −γ∇ γT3 = babN ab+Qb ∣∣∣ b +ρsg 3 =0 (8.4) where Qb ∣∣∣ b = ∂Qb ∂θb +ΓcbcQ b The last equation provides the balance of mass in the Lagrangean repre- sentation γ∇ γT0 = ∂ρs ∂t =0 (8.5) For the balance of angular momentum (7.1)2, we need to evaluate first the linear covariant divergence by (4.9), next the affine one by (4.8). For the in- plane rotation equilibrium, we have γ∇ γJb3 = ∂Mab ∂θa +ΓbacM ac+ΓcacM ab = Mab ∣∣∣ a Affine tensors in shell theory 617 Thus, it holds γ∇̃ γJb3 =Mab ∣∣∣ a −Qb =0 (8.6) For the off-plane rotation equilibrium, one has successively γ∇ γJ21 =−b1aM a2+ b2aM a1 (8.7) γ∇̃ γJ21 = γ∇ γJ21−N21+N12 =(N12− b1aM a2)− (N21− b2aM a1)= 0 Let us define the symbol εcb such that ε12 =−ε21 =1 ε11 = ε22 =0 Equation (8.7) reads γ∇̃ γJ21 = εcb(N cb − bcaM ab)= 0 (8.8) which can be recognized as the classical symmetry relation of the usual shell theory (Valid, 1995).We recover the standard system of equilibriumequations of shells (8.3)-(8.6), (8.8) (Green and Zerna, 1968) but as the expression in an adapted coordinate system of the free affine covariante torsor principle (7.1). Nevertheless, these 7 scalar equations are not sufficient to determine the 9 independent shell variables (8.1), andwe need additional conditions, the constitutive laws, but this topic will not be treated here. 9. Consistent formulation of the dynamics of plates and shells Dynamic equilibrium We start again with general expression (6.10) of the connection matrix in the adapted coordinates. In addition to (8.2), new Christoffel’s coefficients arise, due to both Coriolis’s effects and the time evolution of the surface Γab0 = c a i (∂πib ∂t +Ωijπ j b ) = Φab Γ a 30 = c a i (∂ni ∂t +Ωijn j ) = Φa (9.1) Γ3b0 = 3∑ i=1 ni (∂πib ∂t +Ωijπ j b ) = Φb 618 G. De Saxcé, C. Vallee According to definitions (8.1), non-zero shell variables (7.9) are aTb =−Nab+ ρh3 12 wawb aT3 =−Qa 0T0 = ρh aJb3 =−aJ3b = Mab 0J3b =−0Jb3 = ρh3 12 wb (9.2) In the adapted coordinates, the balance of linear momentum (7.1)1 gives the following equations. For the in-plane translation equation, it holds −γ∇ γTa = ( Nba− ρh3 12 wawb )∣∣∣ b −babQ b+ρsc a i ( gi−2Ωijv j ) −ρs ∂vi ∂t =0 (9.3) The off-plane translation equation is −γ∇ γT3 = bab ( Nab− ρh3 12 wawb ) +Qb ∣∣∣ b + 3∑ i=1 ni ( ρs(g i −2Ωijv j)−ρs ∂vi ∂t ) =0 (9.4) Bycomparisonwithcorrespondingstatic equilibriumequations (8.3) and(8.4), we obtain additional terms reflecting expected Coriolis’s and inertia effects. Less classical arekinetic terms similar to those arising in theequationsof three- dimensional continuousmediumgiven inSection 5. Inmechanics of solids, they are generally neglected, but we have to notice that their magnitude could become significant at the high velocity, for instance in the case of impact. Finally, the last equation provides the balance of mass γ∇ γT0 = ∂ρs ∂t +Φaaρs =0 (9.5) Next, we calculate the linear covariant divergence of the angular momentum by (4.9). The non-zero components are γ∇ γJ21 =−b1aM a2+ b2aM a1+Φ10J23−Φ20J13 (9.6) γ∇ γJb3 = ∂γJb3 ∂ξγ +Γbac cJa3+Γcρc ρJb3+Φba 0Ja3 The affine covariant divergence of the angular momentum is given by (4.8). Let us determine the particular form of the balance of angular momentum (7.1)2 in the adapted coordinates. The off-plane rotation equation is γ∇̃ γJb3 = Mab ∣∣∣ a −Qb− ρh3 12 (∂wb ∂t +Φbaw a+Φccw b ) =0 (9.7) Affine tensors in shell theory 619 The first two terms are the same as in static equilibrium equation (8.6). The third one represents inertia effects and is expected. On the other hand, the last two terms are non-standard in the literature as subtly resulting from the time evolution of the surface geometry through new Christoffel’s coefficients (9.1). Besides, owing to (9.6), the in-plane rotation equation γ∇̃ γJ21 =−εcb( cTb+ bca aJb3−Φc0Jb3)= 0 generalizes the symmetry relation of the usual shell theory to the dynamics. Owing to (9.2), one has equivalently γ∇̃ γJ21 = εcb ( Ncb− bcaM ab − ρh3 12 (Φc+wc)wb ) =0 (9.8) Once again, new terms appear, taking into account kinetic terms and the time evolution of the surface geometry. Finally, it can be seen that the remai- ning equations are automatically satisfied γ∇̃ γJ03 =0 γ∇̃ γJb0 =0 In short, the behavior of the torsor field is governed by a system of 7 scalar equations, namely (9.3)-(9-5) and (9.7)-(9.8). On the other hand, we have 11 independent unknowns, Qb, Nab, Mab, h, wb linked to shell variables (9.2) and 3 additional variables vi. Initially equal to 30− 10 = 20 in the most general case, the redundancy degree is now reduced to 14−7=7. To get rid of this indeterminacy, we must introduce additional assumptions concerning the constitutive laws. What about the 30−11=19 other shell variables?Under various assump- tions introduced in Sections 6 to 9, they are in awayasleep.Nevertheless, their existence is predictedby thegeneral theorywithin the frameof theaffinegroup geometry. They could be waken up by considering other idealization than the one of a smooth curved thin sheet undergoing small deformation, and above all in a relativistic context. 10. Conclusions Firstly, although the affine geometry could appear as a poverty-stricken mathematical frame, we think it is sufficient to describe the fundamental tools of the continuum mechanics. It leads to a definition of the torsors which is 620 G. De Saxcé, C. Vallee completely relieved of all metric features. Next, we developed the correspon- ding Affine Tensor Analysis enabling one to propose a general principle of an affine divergence free torsor. We showed that this principle allows one to recover the balance equations of the statics of three dimensional bodies. For the dynamics of shells, we revealed the existence of new terms, depending on velocities and involving time variation of the surface geometry. Of course, so- me open problems deserve more pervasive investigations. We have to develop at first applications of the new shell theory in order to assess the magnitu- de of the predicted terms. Other subjects of interest would be the dynamics of beams. Finally, we would like to point out related topics. In the previous work, the first author proved that for the dynamics of particles and rigid bo- dies, the well-known theorem of angular momentum is also a consequence of our principle of balance (de Saxcé, 2002). Besides, there is a subtle link with the symplectic mechanics which leads to a nice extension of Kirillov-Kostant- Souriau theorem. References 1. Cartan É., 1923,Sur les variétés à connexionaffineet la théoriede la relativité généralisée (première partie),Ann. Ecole Normal. Sup., 40, 325-412 2. De Saxcé G., 2002,Affine tensors inmechanics and covariance of the angular momentum law, Submitted for publication inEur. J. Mech. A/Solids 3. Dieudonné J., 1971, Eléments d’analyse, Tome IV, cahiers scientifiques, Gauthiers-Villars, Paris 4. Green A.E., Zerna W., 1968, Theoretical Elasticity, 2nd edition, At the Clarendon Press, Oxford 5. HamdouniAb., Vallée C., FortunéD., 1999,Correspondence between 2D and 3D compatibility conditions, Submitted for publication 6. Naghdi, 1972, The Theory of Shells and Plates, in: Flügge S. (Chief Ed.), Encyclopedia of Physics, Truesdell C. (Ed.), Vol. VIa/2, Mechanics of Solids II, 425-640 7. Misner W.M., Thorne K.S., Wheeler J.A., 1973, Gravitation, W.H.Freeman nd co., San Francisco 8. Simo J.C., 1988, A framework for finite strain elastoplasticity based on ma- ximum plastic dissipation and the multiplicative decomposition: part I. Conti- nuum formulation,Computer Methods in Applied mechanics and Engineering, 66, 199-219 Affine tensors in shell theory 621 9. Souriau J.-M., 1992,Mécanique des états condensés de la matière,Proc. 1er Séminaire International de la Fédération Mécanique de Grenoble, France, may 19-21 10. Souriau J.-M., 1997a, Milieux continus de dimension 1, 2 ou 3: statique et dynamique,Proc. 13èmeCongrès Français deMécanique, Poitiers-Futuroscope, Sept. 1-5, 41-53 11. Souriau J.-M., 1997b, Structure of Dynamical Systems, a Symplectic View of Physics, Progress inMathematics, Birkhäuser Verlag, NewYork 12. Valid R., 1995, The Nonlinear Theory of Shells Through Variational Princi- ples, JohnWiley and Sons, Chichester Tensory afiniczne w teorii powłok Streszczenie Strukturę formalna wypadkowej siły i momentu można ująć w postaci pojedyn- czego obiektu zwanego torsorem. Wyłączając wszystkie pojęcia metryczne, torsory definiujemy jako skośno-symetryczne dwuliniowe odwzorowania w przestrzeni linio- wej w dziedzinie funkcji wektorowych. Torsory stanowią szczególną rodzinę afinicz- nych tensorów. Na tej podstawie zdefiniowano wewnętrzny operator różniczkowania zwanyafiniczną kowariantnądywergencją.Następniewysunięto postulat, że zachowa- nie się ośrodka ciągłego opisane polem torsorowymposiada zerową taką dywergencję. Zastosowawszy tę ogólną zasadę, użyto równańEuleraw opisie dynamiki ciał trójwy- miarowych.W dalszej części pracy skoncentrowano się na dynamice powłok. Poprzez użycie odpowiednio zaadaptowanych współrzędnych wykazano, że zastosowanie tej ogólnej zasady stanowi spójną metodę otrzymywania równań z nieoczekiwanie po- jawiającymi się członami odpowiedzialnymi za efekty przyspieszenia Coriolisa oraz zmian powierzchni powłoki w czasie. Manuscript received December 2, 2002; accepted for print March 5, 2003