The d'Alembert-lagrange principle for gradient theories and boundary conditions
aa r X i v : . [ phy s i c s . f l u - dyn ] J a n November 12, 2018 11:38 WSPC - Proceedings Trim Size: 9in x 6in 2007˙AMNLWP THE D’ALEMBERT-LAGRANGE PRINCIPLE FORGRADIENT THEORIES AND BOUNDARY CONDITIONS
H. GOUIN
Universit´e d’Aix-Marseille, 13397 Marseille Cedex 20, FranceE-mail: [email protected] to Prof. Antonio M. Greco
Motions of continuous media presenting singularities are associated with phe-nomena involving shocks, interfaces or material surfaces. The equations repre-senting evolutions of these media are irregular through geometrical manifolds.A unique continuous medium is conceptually simpler than several media withsurfaces of singularity. To avoid the surfaces of discontinuity in the theory, wetransform the model by considering a continuous medium taking into accountmore complete internal energies expressed in gradient developments associatedwith the variables of state. Nevertheless, resulting equations of motion are of anhigher order than those of the classical models: they lead to non-linear modelsassociated with more complex integration processes on the mathematical levelas well as on the numerical point of view. In fact, such models allow a precisestudy of singular zones when they have a non negligible physical thickness. Thisis typically the case for capillarity phenomena in fluids or mixtures of fluids inwhich interfacial zones are transition layers between phases or layers betweenfluids and solid walls. Within the framework of mechanics for continuous me-dia, we propose to deal with the functional point of view considering globallythe equations of the media as well as the boundary conditions associated withthese equations. For this aim, we revisit the d’Alembert-Lagrange principle ofvirtual works which is able to consider the expressions of the works of forcesapplied to a continuous medium as a linear functional value on a space of testfunctions in the form of virtual displacements . At the end, we analyze exam-ples corresponding to capillary fluids. This analysis brings us to numerical orasymptotic methods avoiding the difficulties due to singularities in simpler -butwith singularities- models.
1. Introduction
A mechanical problem is generally studied through force interactions be-tween masses located in material points: this Newton point of view leadstogether to the statistical mechanics but also to the continuum mechanics.The statistical mechanics is mostly precise but is in fact too detailed and ovember 12, 2018 11:38 WSPC - Proceedings Trim Size: 9in x 6in 2007˙AMNLWP in many cases huge calculations crop up. The continuum mechanics is anasymptotic notion coming from short range interactions between molecules.It follows a loose of information but a more efficient and directly computabletheory. In the simplest case of continuum mechanics, residual informationcomes through stress tensor like Cauchy tensor . The concept of stresstensor is so frequently used that it has become as natural as the notionof force. Nevertheless, tensor of contact couples can be investigated as inCosserat medium or configuration forces like in Gurtin approach withedge interactions of Noll and Virga . Stress tensors and contact forces areinterrelated notions .A fundamental point of view in continuum mechanics is: the Newton sys-tem for forces is equivalent to the work of forces is the value of a linearfunctional of displacements. Such a method due to Lagrange is dual ofthe system of forces due to Newton and is not issued from a variationalapproach; the minimization of the energy coincides with the functional ap-proach in a special variational principle only for some equilibrium cases.The linear functional expressing the work of forces is related to the theoryof distributions; a decomposition theorem associated with displacements(as test functions whose supports are C ∞ compact manifolds) uniquely de-termines a canonical zero order form (separated form) with respect both tothe test functions and the transverse derivatives of contact test functions .As Newton’s principle is useless when we do not have any constitutiveequation for the expression of forces, the linear functional method is uselesswhen we do not have any constitutive assumption for the virtual work func-tional. The choice of the simple material theory associated with the Cauchystress tensor corresponds with a constitutive assumption on its virtual workfunctional. It is important to notice that constitutive equations for the freeenergy χ and constitutive assumption for the virtual work functional maybe incompatible : for any virtual displacement ζ of an isothermal medium,the variation − δχ must be equal to the virtual work of internal forces δτ int .The equilibrium state is then obtained by the existence of a solution mini-mizing the free energy.The equation of motion of a continuous medium is deduced from the d’Alembert-Lagrange principle of virtual works which is an extension ofthe principle in mechanics of systems with a finite number of degrees offreedom: The motion is such that for any virtual displacement the virtualwork of forces is equal to the virtual work of mass accelerations .Let us note: if the virtual work of forces is expressed in classical notations ovember 12, 2018 11:38 WSPC - Proceedings Trim Size: 9in x 6in 2007˙AMNLWP in the form δτ = Z Z Z D { f . ζ + tr [( − p + 2 µ ∇ V ) . ∇ ζ ] } dv + Z Z S T . ζ ds (1)from the d’Alembert-Lagrange principle, we obtain not only the equationsof balance momentum for a viscous fluid in the domain D but also theboundary conditions on the border S of D . We notice that expression (1)is not the Frechet derivative of any functional expression.If the free energy depends on the strain tensor F , then δτ must depend on ∇ ζ and leads to the existence of the Cauchy stress tensor. If the free energydepends on the strain tensor F and on the overstrain tensor ∇ F, then δτ must depend on ∇ ζ and ∇ ζ . Conjugated (or transposed ) mappings being denoted by asterisk, for any vec-tors a , b , we write a ∗ b for their scalar product (the line vector is multipliedby the column vector) and ab ∗ or a ⊗ b for their tensor product (the columnvector is multiplied by the line vector). The product of a mapping A by avector a is denoted by A a . Notation b ∗ A means the covector c ∗ definedby the rule c ∗ = ( A ∗ b ) ∗ . The divergence of a linear transformation A is thecovector div A such that, for any constant vector a , (div A ) a = div ( A a ) . We introduce a Galilean or fixed system of coordinates ( x , x , x ) which isalso denoted by x as Euler or spatial variables. If f is a real function of x , ∂f∂ x is the linear form associated with the gradient of f and ∂f∂x i = ( ∂f∂ x ) i ;consequently, ( ∂f∂ x ) ∗ = grad f . The identity tensor is denoted by .Now, we present the method and its consequences in different cases of gradi-ent theory. As examples, we revisit the case of Laplace theory of capillarityand the case of van der Waals fluids.
2. Virtual work of continuous medium
The motion of a continuous medium is classically represented by a con-tinuous transformation ϕ of a three-dimensional space into the physicalspace. In order to describe the transformation analytically, the variables X =( X , X , X ) which single out individual particles correspond to mate-rial or Lagrange variables. Then, the transformation representing the mo-tion of a continuous medium is x = ϕ ( X ,t ) or x i = ϕ i ( X , X , X , t ) , i = 1 , , t denotes the time. At t fixed the transformation possesses an inverseand continuous derivatives up to the second order except at singular sur- ovember 12, 2018 11:38 WSPC - Proceedings Trim Size: 9in x 6in 2007˙AMNLWP faces, curves or points. Then, the diffeomorphism ϕ from the set D of theparticles into the physical space D is an element of a functional space ℘ ofthe positions of the continuous medium considered as a manifold with aninfinite number of dimensions.To formulate the d’Alembert-Lagrange principle of virtual works, we in-troduce the notion of virtual displacements . This is obtained by lettingthe displacements arise from variations in the paths of the particles. Let aone-parameter family of varied paths or virtual motions denoted by { ϕ η } and possessing continuous derivatives up to the second order and expressedanalytically by the transformation x = Φ ( X ,t ; η )with η ∈ O, where O is an open real set containing 0 and such that Φ ( X ,t ; 0) = ϕ ( X ,t ) or ϕ = ϕ (the real motion of the continuous mediumis obtained when η = 0). The derivation with respect to η when η = 0 isdenoted by δ . Derivation δ is named variation and the virtual displacement is the variation of the position of the medium . The virtual displacementis a tangent vector to ℘ in ϕ ( δϕ ∈ T ϕ ( ℘ )). In the physical space, the virtual displacement δϕ is determined by the variation of each particle: the virtual displacement of the particle x is such that ζ = δ x when δ X = 0, δη = 1 at η = 0; we associate the field of tangent vectors to D x ∈ D → ζ = ψ ( x ) ≡ ∂ Φ ∂η | η =0 ∈ T x ( D )where T x ( D ) is the tangent vector bundle to D at x . The concept of virtual x X D o D ϕ η ϕ S S o Physical space ζ ℘ ϕ = ϕϕ o η (C) δϕ functionalspace of positions Fig. 1. The boundary S of D is represented by a thick curve and its variation by a thincurve. Variation δϕ of family { ϕ η } of varied paths belongs to T ϕ ( ℘ ), tangent space to( ℘ ) at ϕ . ovember 12, 2018 11:38 WSPC - Proceedings Trim Size: 9in x 6in 2007˙AMNLWP work is purposed in the form: The virtual work is a linear functional value of the virtual displacement, δτ = < ℑ , δϕ > (2)where < . , . > denotes the inner product of ℑ and δϕ ; then, ℑ belongs tothe cotangent space of ℘ at ϕ ( ℑ ∈ T ∗ ϕ ( ℘ )).In Relation (2), the medium in position ϕ is submitted to the covector ℑ denoting all the stresses; in the case of motion, we must add the inertialforces associated with the acceleration quantities to the volume forces.The d’Alembert-Lagrange principle of virtual works is expressed as: For all virtual displacements, the virtual work is null . Consequently, representation (2) leads to: ∀ δϕ ∈ T ϕ ( ℘ ) , δτ = 0 Theorem:
If expression (2) is a distribution in a separated form, thed’Alembert-Lagrange principle yields the equations of motions and boundaryconditions in the form ℑ = 0 .
3. Some examples of linear functional of forces
Among all possible choices of linear functional of virtual displacements, weclassify the following ones:
Model of zero gradient
Model A.0
The medium fills an open set D of the physical space and the linear func-tional is in the form δτ = Z Z Z D F i ζ i dv where F i ( i = 1 , ,
3) denote the covariant components of the volume force F (including the inertial force terms) presented as a covector. The equationof the motion is ∀ x ∈ D, F i = 0 ⇔ F = 0 (3)3.1.2. Model B.0
The medium fills a set D and the surface S is the boundary of D belongingto the medium; with the same notations as in section , the linear ovember 12, 2018 11:38 WSPC - Proceedings Trim Size: 9in x 6in 2007˙AMNLWP functional is in the form δτ = Z Z Z D F i ζ i dv + Z Z S T i ζ i ds (4) T i are the components of the surface forces (tension) T . From Eq. (4), weobtain the equation of motion as in Eq. (3) and the boundary condition, ∀ x ∈ S, T i = 0 ⇔ T = 0, Model of first gradient
Model A.1
With the previous notations, the linear functional is in the form δτ = Z Z Z D (cid:16) F i ζ i − σ ji ζ i,j (cid:17) dv where σ ji ( i, j = 1 , ,
3) are the components of the stress tensor σ. Stokesformula gets back to the model B . in the separated form δτ = Z Z Z D (cid:16) F i + σ ji,j (cid:17) ζ i dv − Z Z S n j σ ji ζ i ds where n j ( j = 1 , ,
3) are the components of a covector which is the annu-lator of the vectors belonging to the tangent plane at the boundary S . It isnot necessary to have a metric in the physical space; nevertheless, for thesake of simplicity it is convenient to use the Euclidian metric; the vector n of components n j ( j = 1 , ,
3) represents the external normal to S relativelyto D ; the covector n ⋆ is associated with the components n j . We deducethe equation of motion ∀ x ∈ D, F i + σ ji,j = 0 ⇔ F + div σ = 0 (5)and the boundary condition ∀ x ∈ S, n j σ ji = 0 ⇔ n ⋆ σ = 03.2.2. Model B.1/0: (Mixed model with first gradient in D and zerogradient on S ) The linear functional is expressed in the form δτ = Z Z Z D (cid:16) F i ζ i − σ ji ζ i,j (cid:17) dv + Z Z S T i ζ i ds ovember 12, 2018 11:38 WSPC - Proceedings Trim Size: 9in x 6in 2007˙AMNLWP Stokes formula yields the separated form δτ = Z Z Z D (cid:16) F i + σ ji,j (cid:17) ζ i dv + Z Z S (cid:16) T i − n j σ ji (cid:17) ζ i ds ( S. )and we deduce the equation of motion in the same form as Eq. ( 5) and theboundary condition ∀ x ∈ S, n j σ ji = T i ⇔ n ⋆ σ = T Model B . / is the classical theory for elastic media and fluids in contin-uum mechanics.3.2.3. Model B.1
The linear functional is expressed in the form δτ = Z Z Z D (cid:16) F i ζ i − σ ji ζ i,j (cid:17) dv + Z Z S (cid:16) T i ζ i + γ ji ζ i,j (cid:17) ds (6)where the tensor γ of components γ ji is a new term. The boundary of D isa surface S shared in a partition of N parts S p of class C , ( p = 1 , ..., N )(Fig. 2). We denote by ( R m ) − the mean curvature of S ; the edge Γ p of S p is the union of the limit edges Γ pq between surfaces S p and S q assumed tobe of class C and t is the tangent vector to Γ p oriented by n ; n ′ is the unitexternal normal vector to Γ p in the tangent plane to S p : n ′ = t × n . Let usnotice that: γ ji ζ i,j = − γ ji,j ζ i + V j,j (7)where V j = γ ji ζ i ; consequently, from integration of the divergence of DS S Γ p q A mpq S p Γ pq n t n' Fig. 2. The set D has a surface boundary S divided in several parts. The edge of S isdenoted by Γ which is also divided in several parts with end points A m . ovember 12, 2018 11:38 WSPC - Proceedings Trim Size: 9in x 6in 2007˙AMNLWP vector V on surfaces S p we obtain, Z Z S p V j,j ds = − Z Z S p n j (cid:18) V j R m − V j,l n l (cid:19) ds + Z Γ p n ′ j V j dℓ (8)We emphasize with the fact that V j,l n l corresponds to the normal derivativeto S p denoted dV j dn . An integration by parts of the term σ ji ζ i,j in relation(6) and taking account of relations (7-8) implies δτ = Z Z Z D F i ζ i dv + Z Z S T i ζ i ds + Z Z S L i dζ i dn ds + N X p =1 Z Γ p R pi ζ i dℓ ( S. )with the following definitions F i ≡ F i + σ ji,j , L i ≡ n j γ ji T i ≡ T i − n j (cid:18) σ ji − ddn ( γ ji ) + 1 R m γ ji (cid:19) − γ ji,j , R pi ≡ n ′ j γ ji Due to theorem in , the distribution (S. ) has a unique decomposition indisplacements and transverse derivatives of displacements on the manifoldsassociated with D and its boundaries: expression ( S. ) is in a separatedform. Consequently, the equation of motion is ∀ x ∈ D, F i = 0 ⇔ F = 0and the boundary conditions are ∀ x ∈ S, T i = 0 , L i = 0 ⇔ T = 0 , L = 0 ∀ x ∈ Γ pq , R pi + R qi = 0 ⇔ R p + R q = 0Term L is not reducible to a force: its virtual work L i dζ i dn is not the productof a force with the displacement ζ ; the term L is an embedding action . Model of second gradient
Model A.2
The linear functional is in the form δτ = Z Z Z D (cid:16) F i ζ i − σ ji ζ i,j + S jki ζ i,jk (cid:17) dv Tensor S with S jki = S kji is an overstress tensor . An integration by partsof the last term brings back to the model B . , δτ = Z Z Z D (cid:16) F i ζ i − (cid:16) σ ji + S jki,k (cid:17) ζ i,j (cid:17) dv + Z Z S n k S jki ζ i,j ds ovember 12, 2018 11:38 WSPC - Proceedings Trim Size: 9in x 6in 2007˙AMNLWP and the virtual work gets the separated form ( S. ) with: F i = F i + σ ji,j + S jki,jk volume force T i = − n j (cid:18) σ ji + S jki,k − ddn (cid:16) n k S jki (cid:17) + 1 R m n k S jki (cid:19) surface force R pi = n ′ j n k S jki line force L i = n j n k S jki embedding actionand consequently yields the same equation of motion and boundary condi-tions as in case B . .3.3.2. Model B.2
The linear functional is in the form δτ = Z Z Z D (cid:16) F i ζ i − σ ji ζ i,j + S jki ζ i,jk (cid:17) dv + Z Z S (cid:16) T i ζ i + γ ji ζ i,j + U jki ζ i,jk (cid:17) ds This functional yields two integrations successively on S p and on Γ pq withterms at the points A m . With obvious notations, for the same reasons asin section , the virtual work gets the separated form δτ = Z Z Z D F i ζ i dv + Z Z S (cid:18) T i ζ i + L i dζ i dn + L i d ζ i dn (cid:19) ds + X p Z Γ p (cid:18) R pi ζ i + M pi dζ i dn ′ (cid:19) dℓ + X m φ mi ζ i A m ( S. )where ζ i A m ( i = 1 , ,
3) are the components of ζ at point A m . The calcula-tions are not expended. They introduce the curvature tensor on S p and thegeodesic curvature of Γ pq. . Consequently, F i , T i , R pi , φ mi are associatedwith volume, surface, line and forces at points; L i , L i , M pi are embeddingefforts of order 1 and 2 on S and of order 1 on the edge Γ . The equa-tion of motion and boundary conditions express that these seven tensorialquantities are null on their domains of values D , S , Γ p and A m .
4. Conclusion
It is possible to extend the previous presentation by means of more complexmedium with gradient of order n . The models introduce embedding effectsof more important order on surfaces, edges and points. The (A.n) modelrefers to a (B.n-1) model: the fact that boundary surface S is (or is not) amaterial surface has now a physical meaning. Consequently, we can resumethe previous presentation as follows: ovember 12, 2018 11:38 WSPC - Proceedings Trim Size: 9in x 6in 2007˙AMNLWP a ) The choice of a model corresponds to specify the part G of the algebraicdual T ∗ ϕ ( ℘ ) in which the efforts are considered: ℑ ∈ G ⊂ T ∗ ϕ ( ℘ ). b ) In order to operate with the principle of virtual works and to obtainthe mechanical equations in the form ℑ = 0, it is no matter that the part G of the dual is separating ( ∀ ℑ ∈ G , < ℑ , δϕ > = 0 ⇒ δϕ = 0), but it isimportant the part G is separated ( ℑ ∈ G , ∀ δϕ ∈ T ϕ ( ℘ ) , < ℑ , δϕ > = 0 ⇒ℑ = 0). c ) The functionals A . , B . / , A . , B . are not separated: if ℑ consists inthe data of the fields F , σ , T , it is not possible to conclude that the fieldsare zero. d ) Functionals in A . , B . , S . , S . . . . are separated: if the fields S , T , R , L , . . . are continuous then, by using the fundamental lemma of variationcalculus, their values must be equal to zero. They are the only functionalswe must know for using the principle of virtual works; it is exactly as for asolid: the torque of forces is only known in the equations of motion. e ) When the fields are not continuous on surfaces or curves, we have toconsider a model of greater order in gradients and to introduce integrals oninner boundaries of the medium.For conservative medium, the first gradient theory corresponds to thecompressible case. The theory of fluid, elastic, viscous and plastic mediarefers to the model ( S . ). The Laplace theory of capillarity in fluids refersto the model ( S . ). To take into account superficial effects acting betweensolids and fluids, we use the model of fluids endowed with capillarity ( S . );the theory interprets the capillarity in a continuous way and contains theLaplace theory of capillarity; for solids, the model corresponds to ”elasticmaterials with couple stresses” indicated by Toupin in .
5. Example 1: The Laplace theory of capillarity
Liquid-vapor and two-phase interfaces are represented by a material surfaceendowed with an energy relating to Laplace surface tension. The interfaceappears as a surface separating two media with its own characteristic be-havior and energy properties (when working far from critical conditions,the capillary layer has a thickness equivalent to a few molecular beams ).The Laplace theory of capillarity refers to the model B . in the form ( S . )as following: for a compressible fluid with a capillary effect on the wallboundaries, the free energy is in the form χ = Z Z Z D ρ α ( ρ ) dv + N X p =1 a p Z Z S p ds ovember 12, 2018 11:38 WSPC - Proceedings Trim Size: 9in x 6in 2007˙AMNLWP where α ( ρ ) is the fluid specific energy, ρ is the matter density and coeffi-cients a p are the surface tensions of each surface S p . Surface integrationsare associated to the space metric; the virtual work of internal forces is δτ int = Z Z Z D − p ,i ζ i dv + N X p =1 Z Z S p n i (cid:18) p − a p R m (cid:19) ζ i ds + N X p =1 Z Γ p a p n ′ i ζ i dℓ where p ≡ ρ α ′ ( ρ ) is the fluid pressure. The external force (including inertialforces) is the body force ρ f defined in D , the surface force is T defined on S and the line force is R defined on Γ. D’Alembert-Lagrange principle yieldsthe equation of motion and boundary conditions: ∀ x ∈ D, − p ,i + ρ f i = 0 ⇔ − grad p + ρ f = 0 , ∀ x ∈ S, p n i + T i − a p n i R m = 0 ⇔ p n + T − a p n R m = 0 ∀ x ∈ Γ pq , a p n ′ pi + a q n ′ qi + R i = 0 ⇔ a p n ′ p + a q n ′ q + R = 0Boundary conditions are Laplace equation and
Young-Dupr´e condition .
6. Example 2: Fluids endowed with internal capillarity
For interfacial layers, kinetic theory of gas leads to laws of state associatedwith non-convex internal energies . This approach dates back to vander Waals , Korteweg , corresponds to the Landau-Ginzburg theory and presents two disadvantages. First, between phases, the pressure maybecome negative; simple physical experiments can be used to cause tractionthat leads to these negative pressure values . Second, in the field betweenbulks, internal energy cannot be represented by a convex surface associatedwith density and entropy; this fact seems to contradict the existence ofequilibrium states; it is possible to eliminate this disadvantage by writingin an anisotropic form the stress tensor of the capillary layer which allowsto study interfaces of non-molecular size near a critical point.One of the problems that complicates this study of phase transformationdynamics is the apparent contradiction between Korteweg classical stresstheory and the Clausius-Duhem inequality . Proposal made by Eglit ,Dunn and Serrin , Casal and Gouin and others rectifies this anomalyfor liquid-vapor interfaces. The simplest model in continuum mechanicsconsiders a free energy as the sum of two terms: a first one correspond-ing to a medium with a uniform composition equal to the local one and asecond one associated with the non-uniformity of the fluid . The sec-ond term is approximated by a gradient expansion, typically truncated to ovember 12, 2018 11:38 WSPC - Proceedings Trim Size: 9in x 6in 2007˙AMNLWP the second order. The model is simpler than models associated with therenormalization-group theory but has the advantage of easily extendingwell-known results for equilibrium cases to the dynamics of interfaces .We consider a fluid D in contact with a wall S . Physical experiments provethat the fluid is nonhomogeneous in the neighborhood of S . The internalenergy ε is also a function of the entropy. In the case of isothermal motions,the internal energy is replaced by the free energy. In the mechanical case,the entropy and the temperature are not concerned by the virtual displace-ments of the medium. Consequently, for isentropic or isothermal motions, ε = f ( ρ, β ) where β = (grad ρ ) . The fluid is submitted to external forcesrepresented by a potential Ω as a function of Eulerian variables x . To obtainboundary conditions it is necessary to know the wall effect. An explicit formfor the energy of interaction between surfaces and liquids is proposed in .We denote by B the surface density of energy at the wall. The total energy E of the fluid is the sum of three potential energies: E f (bulk energy), E p (external energy) and E S (surface energy). E f = Z Z Z D ρ ε ( ρ, β ) dv, E p = Z Z Z D ρ Ω( x ) dv, E S = Z Z S B ds
We have the results (see Appendix): δE f = Z Z Z D ( − div σ ) ζ dv + Z Z S (cid:26) − A dζ n dn + (cid:18) AR m n ∗ + grad ∗ tg A + n ∗ σ (cid:19) ζ (cid:27) ds − Z Γ A n ′∗ ζ dℓ with σ = − P − C grad ρ ⊗ grad ρ ≡ − P − C ( ∂ρ∂ x ) ∗ ∂ρ∂ x , where C = 2 ρ ε ′ β , P = ρ ε ′ ρ − ρ div( C grad ρ ), ε ′ ρ (or ε ′ β ) denoting the partial derivative of ε with respect to ρ (or β ), ζ n = n ∗ ζ ; A = Cρ dρdn where dρdn = ∂ρ∂ x n andgrad tg denotes the tangential part of the gradient relatively to S . δE p = Z Z Z D ρ ∂ Ω ∂ x ζ dv ≡ Z Z Z D ρ (grad ∗ Ω) ζ dv ; and ,δE S = Z Z S (cid:26) δB − (cid:18) BR m n ∗ + grad ∗ tg B (cid:19) ζ (cid:27) ds + Z Γ B n ′∗ ζ dℓ The density in the fluid has a limit value ρ S at the wall S and B is assumedto be a function of ρ S only . Then, δB = B ′ ( ρ S ) δρ S = − ρ S B ′ ( ρ S ) div ζ ,where div ζ is computed on S . Let us denote G = − ρ S B ′ ( ρ S ); Appendixyields Z Z S δB ds = Z Z S G div ζ ds = ovember 12, 2018 11:38 WSPC - Proceedings Trim Size: 9in x 6in 2007˙AMNLWP = Z Z S (cid:26) G dζ n dn − (cid:18) GR m n ∗ + grad ∗ tg G (cid:19) ζ (cid:27) ds + Z Γ G n ′∗ ζ dℓδE S = Z Z S (cid:26) G dζ n dn − (cid:18) HR m n ∗ + grad ∗ tg H (cid:19) ζ (cid:27) ds + Z Γ H n ′∗ ζ dℓ with H = B ( ρ S ) − ρ S B ′ ( ρ S ). Then, δE = Z Z Z D ( ρ grad ∗ Ω − div σ ) ζ dv − Z Γ ( A − H ) n ′∗ ζ dℓ + Z Z S ( G − A ) dζ n dn + (cid:18) A − H ) R m n ∗ + grad ∗ tg ( A − H ) + n ∗ σ (cid:19) ζ ds (9)At equilibrium, δτ ≡ − δE = 0. The fundamental lemma of variation cal-culus associated with separated form (9) corresponding to ( S. ), yields: Equation of equilibrium
From any arbitrary variation x ∈ D → ζ ( x ) such that ζ = on S , we get Z Z Z D ( ρ grad ∗ Ω − div σ ) ζ ds = 0 . Then ,ρ grad ∗ Ω − div σ = 0 (10)This equation is written in the classical form of equation of equilibrium . It is not the same for the boundary conditions.
Boundary conditions
Case of a rigid wall
We consider a rigid wall; on S , the virtual displacements satisfy the condi-tion n ∗ ζ = 0. Then, at the rigid wall, ∀ x ∈ S → ζ ( x ) such that n ∗ ζ = 0, Z Z S ( G − A ) dζ n dn + (cid:18) A − H ) R m n ∗ + grad ∗ tg ( A − H ) + n ∗ σ (cid:19) ζ ds = 0Due to σ = σ ∗ , we deduce the boundary conditions (11-12) ∀ x ∈ S, G − A = 0 , (11)and there exists a Lagrange multiplier x ∈ S → λ ( x ) ∈ R such that, ∀ x ∈ S, A − H ) R m n + grad tg ( A − H ) + σ n = λ n (12)The edge Γ of S belongs to the solid wall and consequently on Γ, ζ = η t :the integral on Γ is null and does not yield any additive condition. ovember 12, 2018 11:38 WSPC - Proceedings Trim Size: 9in x 6in 2007˙AMNLWP Case of an elastic wall
The equilibrium equation (10) is unchanged. On S, the condition (11) isalso unchanged. The only different condition comes from the fact that wedo not have anymore the slipping condition for the virtual displacementon S , ( n ∗ ζ = 0). Due to the possible deformation of the wall, the virtualwork of stresses on S is δE e = Z Z S κ ∗ ζ ds + Z Γ R ∗ ζ dℓ where κ = σ e n is the stress (loading) vector associated with stress tensor σ e of the elasticwall and R is the line force due to the elasticity of the line. Relation (12)is replaced by ∀ x ∈ S, A − H ) R m n + grad tg ( A − H ) + σ n + κ = 0We obtain an additive condition on Γ in the form ( H − A ) n ′ + R = 0 anddue to condition (11), ∀ x ∈ Γ , B n ′ + R = 0 (13)(If Γ is the union of edges Γ p , B n ′ is replaced by X p B p n ′ p ). Analysis of the boundary conditions
Eq. (11) yields
C dρdn + B ′ ( ρ S ) = 0; the definition of σ implies: σ n = − P n − C dρdn grad ρ Due to the fact that the tangential part of Eq. (12) is always verified,the only condition comes from Eq. (11); Eq. (12) yields the value of theLagrange multiplier λ and Eq. (13) the value of R . For an elastic (non-rigid) wall we obtain, κ tg = 0 and κ n = P + 2 BR m − B ′ ( ρ S ) dρdn (14)where κ tg and κ n are the tangential and the normal components of κ .Taking into account of Eq. (14) we obtain the stress values at the non rigidelastic wall. The surface energy is : B ( ρ S ) = − γ ρ S + γ ρ S where γ and γ are two positive constants and the fluid density condition at the wall is C dρdn = γ − γ ρ S ovember 12, 2018 11:38 WSPC - Proceedings Trim Size: 9in x 6in 2007˙AMNLWP If we denote by ρ B = γ /γ the bifurcation fluid density at the wall, due tothe fact C is positive constant , we obtain: if ρ S < ρ B , ( or ρ S > ρ B ), dρdn is positive ( or negative) and we have a lack ( or excess) of fluid density atthe wall. Such media allow to study fluid interfaces and interfacial layersbetween fluids and solids and lead to numerical and asymptotic methods .The extension to the dynamic case is straightforward: Eq. (10) yields ρ Γ ∗ − div σ + ρ grad ∗ Ω = 0Vector Γ is the acceleration; boundary conditions (11-14) are unchanged. Acknowledgments
I am grateful to Professor Tommaso Ruggeri for helpful discussions.
Appendix
Let S be a surface in the 3-dimensional space and n its external normalextended locally in the vicinity of S by the expression n ( x ) = grad d ( x ) , where d is the distance of a point x to S ; for any vector field w , we obtain :rot( n × w ) = n div w − w div n + ∂ n ∂ x w − ∂ w ∂ x n From n ∗ ∂ n ∂ x = 0 and div n = − R m we deduce on S , n ∗ rot( n × w ) = div w + 2 R m n ∗ w − n ∗ ∂ w ∂ x n (15)We deduce: for any scalar field A and w = A ζ , A div ζ = A dζ n dn − AR m ζ n − (grad ∗ tg A ) ζ + n ∗ rot ( A n × ζ ) (16) Let us calculate δE f ; D is a material volume, then δE f = Z Z Z D ρ δε dv with δε = ∂ε∂ρ δρ + ∂ε∂β δβ . From δ ∂ρ∂ x = ∂δρ∂ x − ∂ρ∂ x ∂ ζ ∂ x (see ), ρ ε ′ β δβ = 2 ρ ε ′ β δ ∂ρ∂ x (cid:18) ∂ρ∂ x (cid:19) ∗ = C (cid:18) ∂δρ∂ x − ∂ρ∂ x ∂ ζ ∂ x (cid:19) (cid:18) ∂ρ∂ x (cid:19) ∗ = div( C grad ρ δρ ) − div( C grad ρ ) δρ − tr (cid:18) C grad ρ grad ∗ ρ ∂ ζ ∂ x (cid:19) ovember 12, 2018 11:38 WSPC - Proceedings Trim Size: 9in x 6in 2007˙AMNLWP Due to δ ρ = − ρ div ζ (see ), δE f = Z Z Z D (cid:18) ∂P∂ x + div( C grad ρ grad ∗ ρ ) (cid:19) ζ dv − Z Z Z D div ( C ρ grad ρ div ζ + C grad ρ grad ∗ ρ ζ + P ζ ) dv = Z Z Z D − (div σ ) ζ dv + Z Z S ( − A div ζ + n ∗ σ ζ ) ds From Eq. (16), we deduce immediatly: δE f = Z Z Z D ( − div σ ) ζ dv + Z Z S (cid:26) − A dζ n dn + (cid:18) AR m n ∗ + grad ∗ tg A + n ∗ σ (cid:19) ζ (cid:27) ds − Z Γ A n ′∗ ζ dℓ Let us calculate δE S ; due to E S = Z Z S B det ( n , d x , d x ) where d x and d x are two coordinate lines of S, we get: E S = Z Z S B det F det ( F − n , d X , d X )where S is the image of S in a reference space with Lagrangian coordinates X and F is the deformation gradient tensor ∂ x ∂ X of components (cid:26) ∂x i ∂X j (cid:27) .Then, δE S = Z Z S δB det F det ( F − n , d X , d X )+ Z Z S B δ (cid:0) det F det ( F − n , d X , d X ) (cid:1) . with Z Z S B δ (cid:0) det F det ( F − n , d X , d X ) (cid:1) = Z Z S B div ζ det( n , d x , d x ) + B det (cid:18) ∂ n ∂ x ζ , d x , d x (cid:19) − B det (cid:18) ∂ ζ ∂ x n , d x , d x (cid:19) = Z Z S (cid:18) div( B ζ ) − (grad ∗ B ) ζ − B n ∗ ∂ ζ ∂ x n (cid:19) ds Relation (15) yields: div ( B ζ ) + 2 BR m n ∗ ζ − n ∗ ∂B ζ ∂ x n = n ∗ rot ( B n × ζ ), Z Z S B δ (cid:0) det F det ( F − n , d X , d X ) (cid:1) = Z Z S (cid:18) − BR m n ∗ + grad ∗ B ( nn ∗ − ) (cid:19) ζ ds + Z Z S n ∗ rot ( B n × ζ ) ds where grad ∗ B ( nn ∗ − ) belongs to the cotangent plane to S ; we obtain δE S = Z Z S (cid:18) δB − (cid:18) BR m n ∗ + grad ∗ tg B (cid:19) ζ (cid:19) ds + Z Γ B n ′∗ ζ dℓ. ovember 12, 2018 11:38 WSPC - Proceedings Trim Size: 9in x 6in 2007˙AMNLWP References
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