The d'Alembert-lagrange principle for gradient theories and boundary conditions

0.25 MB·English

by  Henri Gouin

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THE D’ALEMBERT-LAGRANGE PRINCIPLE FOR

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GRADIENT THEORIES AND BOUNDARY CONDITIONS
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] H.GOUIN
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Universit´e d’Aix-Marseille, 13397 Marseille Cedex 20, France
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E-mail: [email protected]
d
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u Dedicated to Prof. Antonio M. Greco
l
f
s. Motions ofcontinuous mediapresenting singularitiesareassociated withphe-
c nomenainvolvingshocks,interfacesormaterialsurfaces.Theequationsrepre-
i sentingevolutionsofthesemediaareirregularthroughgeometricalmanifolds.
s
y Auniquecontinuous mediumisconceptually simplerthanseveralmediawith
h surfacesofsingularity.Toavoidthesurfacesofdiscontinuityinthetheory,we
p transformthemodelbyconsideringacontinuous mediumtakingintoaccount
[ morecompleteinternalenergiesexpressedingradientdevelopmentsassociated
withthevariablesofstate.Nevertheless,resultingequationsofmotionareofan

1 higherorderthanthoseoftheclassicalmodels:theyleadtonon-linearmodels
v associatedwithmorecomplexintegrationprocessesonthemathematicallevel
8 aswellasonthenumericalpointofview.Infact,suchmodelsallowaprecise
9 studyofsingularzoneswhentheyhaveanonnegligiblephysicalthickness.This
0 istypicallythecaseforcapillarityphenomenainfluidsormixturesoffluidsin
2 whichinterfacial zones aretransitionlayers between phases or layersbetween
. fluids and solidwalls.Within the frameworkof mechanics for continuous me-
1
dia, wepropose todeal withthefunctional pointofview consideringglobally
0
theequationsofthemediaaswellastheboundaryconditionsassociatedwith
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these equations. Forthisaim,werevisitthe d’Alembert-Lagrange principle of
0
virtual works which is able to consider the expressions of the works of forces
:
v appliedtoacontinuous mediumasalinearfunctionalvalueonaspaceoftest
i functions in the form of virtual displacements. At the end, we analyze exam-
X
ples corresponding to capillary fluids. This analysis brings us to numerical or
r asymptoticmethodsavoidingthedifficultiesduetosingularitiesinsimpler-but
a
withsingularities-models.
1. Introduction
A mechanical problem is generally studied through force interactions be-
tween masses located in material points: this Newton point of view leads
together to the statistical mechanics but also to the continuum mechanics.
The statistical mechanics is mostly precise but is in fact too detailed and
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in many cases huge calculations crop up. The continuum mechanics is an
asymptoticnotioncomingfromshortrangeinteractionsbetweenmolecules.
Itfollowsalooseofinformationbutamoreefficientanddirectlycomputable
theory. In the simplest case of continuum mechanics, residual information
comes through stress tensor like Cauchy tensor1,2. The concept of stress
tensor is so frequently used that it has become as natural as the notion
of force. Nevertheless, tensor of contact couples can be investigated as in
Cosserat medium3 or configuration forces like in Gurtin approach4 with
edge interactions of Noll and Virga5. Stress tensors and contact forces are
interrelated notions6.
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 linear
functional of displacements. Such a method due to Lagrange is dual of
the system of forces due to Newton7,8 and is not issued from a variational
approach;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 theory
of distributions; a decomposition theorem associated with displacements
∞
(astestfunctions whosesupports areC compactmanifolds)uniquelyde-
termines acanonicalzeroorderform(separated form) withrespectbothto
the testfunctions andthe transversederivativesofcontacttestfunctions9.
As Newton’s principle is useless when we do not have any constitutive
equationfortheexpressionofforces,thelinearfunctionalmethodisuseless
whenwedonothaveanyconstitutiveassumptionforthevirtualworkfunc-
tional.ThechoiceofthesimplematerialtheoryassociatedwiththeCauchy
stresstensorcorrespondswithaconstitutiveassumptiononitsvirtualwork
functional.Itis importanttonotice thatconstitutive equationsforthe free
energy χ and constitutive assumption for the virtual work functional may
beincompatible10:foranyvirtual displacementζ ofanisothermalmedium,
thevariation−δχ mustbeequaltothevirtual workofinternalforcesδτ .
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 of
the principle in mechanics of systems with a finite number of degrees of
freedom: The motion is such that for any virtual displacement the virtual
work 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
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in the form
δτ = {f.ζ+tr[(−p 1+2µ ∇V).∇ζ]}dv+ T.ζ ds (1)
Z Z Z Z Z
D S
from the d’Alembert-Lagrange principle, we obtain not only the equations
of balance momentum for a viscous fluid in the domain D but also the
boundary 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 andleadstotheexistenceoftheCauchystresstensor.Ifthefreeenergy
depends on the strain tensor F and on the overstrain tensor ∇F, then δτ
must depend on ∇ζ and ∇2ζ.
Conjugated(ortransposed)mappingsbeingdenotedbyasterisk,foranyvec-
torsa,b,wewritea∗bfortheirscalar product(thelinevectorismultiplied
bythecolumnvector)andab∗ ora⊗bfortheirtensorproduct (thecolumn
vector is multiplied by the line vector). The product of a mapping A by a
vector a is denoted by Aa. Notation b∗A means the covector c∗ defined
bytherulec∗ =(A∗b)∗.ThedivergenceofalineartransformationAisthe
covector divA such that, for any constant vector a, (divA)a=div(A a).
We introduceaGalileanorfixedsystemofcoordinates(x1,x2,x3)whichis
also denoted by x as Euler or spatial variables. If f is a real function of x,
∂f ∂f ∂f
is the linear form associated with the gradient of f and =( ) ;
∂x ∂xi ∂x i
∂f
∗
consequently, ( ) =gradf. The identity tensor is denoted by 1.
∂x
Now,wepresentthemethodanditsconsequencesindifferentcasesofgradi-
enttheory.As examples,werevisitthe caseofLaplacetheoryofcapillarity
and 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 physical
space. In order to describe the transformation analytically, the variables
X=(X1,X2,X3)whichsingleoutindividualparticlescorrespondtomate-
rial or Lagrange variables. Then, the transformation representing the mo-
tion of a continuous medium is
x=ϕ(X,t) or xi =ϕi(X1,X2,X3,t), i=1,2,3
wheretdenotesthetime.Attfixedthetransformationpossessesaninverse
and continuous derivatives up to the second order except at singular sur-
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faces, curves or points. Then, the diffeomorphism ϕ from the set D of the
0
particles into the physical space D is an element of a functional space ℘ of
the positions of the continuous medium considered as a manifold with an
infinite 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 letting
the displacements arise from variations in the paths of the particles. Let a
one-parameter family of varied paths or virtual motions denoted by {ϕ }
η
andpossessingcontinuousderivativesuptothesecondorderandexpressed
analytically 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 medium
0
is obtained when η = 0). The derivation with respect to η when η = 0 is
denotedbyδ.Derivationδ isnamedvariation andthe virtual displacement
is the variation of the position of the medium11. The virtual displacement
is a tangent vector to ℘ in ϕ (δϕ ∈ T (℘)). In the physical space, the
ϕ
virtual displacement δϕ is determined by the variationof eachparticle: the
virtual displacement of the particle x is such that ζ = δx when δX = 0,
δη =1 at η =0; we associate the field of tangent vectors to D
∂Φ
x∈D →ζ =ψ(x)≡ |η=0 ∈Tx(D)
∂η
whereTx(D)isthetangentvectorbundletoD atx.Theconceptofvirtual
Physical space
δϕ
ϕ
D η
S D
ζ ϕη o ϕ =ϕo
x ϕ X So ℘
functional
(C) space of
positions
Fig.1. TheboundarySofDisrepresentedbyathickcurveanditsvariationbyathin
curve. Variation δϕ of family {ϕη} of varied paths belongs to Tϕ(℘), tangent space to
(℘)atϕ.
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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 to
∗
the 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 inertial
forces 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 (℘), δτ =0
ϕ
Theorem: If expression (2) is a distribution in a separated form, the
d’Alembert-Lagrangeprincipleyields theequationsofmotionsandboundary
conditions in the form ℑ=0.
3. Some examples of linear functional of forces
Among allpossible choicesof linear functionalofvirtualdisplacements,we
classify the following ones:
3.1. Model of zero gradient
3.1.1. Model A.0
The medium fills an open set D of the physical space and the linear func-
tional is in the form
δτ = F ζidv
i
Z Z Z
D
where F (i = 1,2,3) denote the covariant components of the volume force
i
F(includingtheinertialforceterms)presentedasacovector.Theequation
of the motion is
∀x∈D, F =0 ⇔ F=0 (3)
i
3.1.2. Model B.0
Themediumfills asetD andthe surfaceS isthe boundaryofD belonging
to the medium; with the same notations as in section 3.1.1, the linear
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functional is in the form
δτ = F ζidv+ T ζids (4)
i i
Z Z Z Z Z
D S
T are the components of the surface forces (tension) T. From Eq. (4), we
i
obtain the equation of motion as in Eq. (3) and the boundary condition,
∀x∈S, T =0 ⇔ T=0
i
,
3.2. Model of first gradient
3.2.1. Model A.1
With the previous notations, the linear functional is in the form
δτ = F ζi−σjζi dv
Z Z ZD(cid:16) i i ,j(cid:17)
where σj(i,j = 1,2,3) are the components of the stress tensor σ. Stokes
i
formula gets back to the model B.0 in the separated form
δτ = F +σj ζidv− n σj ζids
Z Z ZD(cid:16) i i,j(cid:17) Z ZS j i
where n (j = 1,2,3) are the components of a covector which is the annu-
j
latorof the vectorsbelonging to the tangentplane atthe boundary S.Itis
not necessary to have a metric in the physical space; nevertheless, for the
sake of simplicity it is convenient to use the Euclidian metric; the vector n
ofcomponentsnj(j =1,2,3)representstheexternalnormaltoS relatively
to D; the covector n⋆ is associated with the components n . We deduce
j
the equation of motion
∀x∈D, F +σj =0 ⇔ F+divσ =0 (5)
i i,j
and the boundary condition
∀x∈S, n σj =0 ⇔ n⋆σ =0
j i
3.2.2. Model B.1/0: (Mixed model with first gradient in D and zero
gradient on S)
The linear functional is expressed in the form
δτ = F ζi−σjζi dv+ T ζids
Z Z ZD(cid:16) i i ,j(cid:17) Z ZS i
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Stokes formula yields the separated form
δτ = F +σj ζidv+ T −n σj ζids (S.0)
Z Z ZD(cid:16) i i,j(cid:17) Z ZS(cid:16) i j i(cid:17)
andwededucethe equationofmotioninthe sameformasEq.(5)andthe
boundary condition
∀x∈S, n σj =T ⇔ n⋆σ =T
j i i
Model B.1/0 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 = F ζi−σjζi dv+ T ζi+γjζi ds (6)
Z Z ZD(cid:16) i i ,j(cid:17) Z ZS(cid:16) i i ,j(cid:17)
where the tensor γ of components γj is a new term. The boundary of D is
i
a surface S shared in a partition of N parts S of class C2, (p = 1,...,N)
p
(Fig. 2).We denote by (R )−1 the meancurvature of S;the edge Γ of S
m p p
is the union of the limit edges Γ between surfaces S and S assumed to
pq p q
beofclassC2 andtisthetangentvectortoΓ orientedbyn;n′ istheunit
p
external normal vector to Γ in the tangent plane to S : n′=t×n. Let us
p p
notice that:
γjζi =−γj ζi+Vj (7)
i ,j i,j ,j
where Vj = γjζi ; consequently, from integration of the divergence of
i
A
m
S
p Γ
pq
S n Γ
D S p pq
t
q
n'
Fig.2. ThesetD has asurfaceboundary S dividedinseveralparts. TheedgeofS is
denotedbyΓwhichisalsodividedinseveralpartswithendpointsAm.
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vector V on surfaces S we obtain,
p
Vj ds=− n Vj −Vjnl ds+ n′Vjdℓ (8)
Z ZSp ,j Z ZSp j(cid:18)Rm ,l (cid:19) ZΓp j
WeemphasizewiththefactthatVjnl correspondstothenormalderivative
,l
dVj
to S denoted . An integration by parts of the term σjζi in relation
p dn i ,j
(6) and taking account of relations (7-8) implies
dζi N
δτ = F1ζi dv+ T1ζi ds+ L ds+ R ζidℓ
Z Z Z i Z Z i Z Z idn Z pi
D S S Xp=1 Γp
(S.1)
with the following definitions
F1 ≡F +σj , L ≡n γj
i i i,j i j i
 Ti1 ≡Ti−nj(cid:18)σij − ddn(γij)+ R1 γij(cid:19)−γij,j, Rpi ≡n′j γij
m
Duetotheorem37in9 ,thedistribution(S.1)hasauniquedecompositionin
displacementsandtransversederivativesofdisplacementsonthemanifolds
associated with D and its boundaries: expression (S.1) is in a separated
form. Consequently, the equation of motion is
∀x∈D, F1 =0⇔ F1 =0
i
and the boundary conditions are
∀x∈S, T1 =0,L =0 ⇔ T1 =0, L=0
i i
∀x∈Γ , R +R =0⇔ R +R =0
pq pi qi p q
dζi
TermLisnotreducibletoaforce:itsvirtualworkL isnottheproduct
i
dn
of a force with the displacement ζ; the term L is an embedding action.
3.3. Model of second gradient
3.3.1. Model A.2
The linear functional is in the form
δτ = F ζi−σjζi +Sjkζi dv
Z Z ZD(cid:16) i i ,j i ,jk(cid:17)
Tensor S with Sjk = Skj is an overstress tensor. An integration by parts
i i
of the last term brings back to the model B.1,
δτ = F ζi− σj +Sjk ζi dv+ n Sjk ζi ds
Z Z ZD(cid:16) i (cid:16) i i,k(cid:17) ,j(cid:17) Z ZS k i ,j
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and the virtual work gets the separated form (S.1) with:
F1 =F +σj +Sjk volume force
i i i,j i,jk
 d 1
Ti1 =−nj(cid:18)σij +RSij,kk=−nd′nn(cid:16)nkSSjkijk(cid:17)+ RmnkSijk(cid:19) sulrinfaecefofrocrece
pi j k i
and consequently yielLdsi =thenjsanmkeSeijqkuation of motion aenmdbbeodudnidnagraycctoionndi-
tions as in case B.1.
3.3.2. Model B.2
The linear functional is in the form
δτ = F ζi−σjζi +Sjkζi dv+ T ζi+γjζi +Ujkζi ds
Z Z ZD(cid:16) i i ,j i ,jk(cid:17) Z ZS(cid:16) i i ,j i ,jk(cid:17)
This functional yields two integrations successively on S and on Γ with
p pq
terms at the points A . With obvious notations, for the same reasons as
m
in section 3.2.3, the virtual work gets the separated form
dζi d2ζi
δτ = F1ζidv+ T1ζi+L1 +L2 ds
Z Z Z i Z Z (cid:18) i i dn i dn2(cid:19)
D S
dζi
+ R ζi+M dℓ+ φ ζi (S.2)
Z (cid:18) pi pidn′(cid:19) mi Am
Xp Γp Xm
where ζi (i=1,2,3) are the components of ζ at point A . The calcula-
A m
m
tionsarenotexpended.TheyintroducethecurvaturetensoronS andthe
p
geodesic curvature of Γ . Consequently, F1, T1, R , φ are associated
pq. i i pi mi
withvolume,surface,lineandforcesatpoints;L1, L2, M areembedding
i i pi
efforts 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 tensorial
quantities are null on their domains of values D, S, Γ and A .
p m
4. Conclusion
Itispossibletoextendthepreviouspresentationbymeansofmorecomplex
medium with gradient of order n. The models introduce embedding effects
of more important order on surfaces, edges and points. The (A.n) model
refers to a (B.n-1) model: the fact that boundary surface S is (or is not) a
materialsurfacehasnowaphysicalmeaning.Consequently,wecanresume
the previous presentation as follows:
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a) Thechoiceofamodelcorrespondsto specify the partGofthe algebraic
∗ ∗
dual T (℘) in which the efforts are considered: ℑ∈G⊂T (℘).
ϕ ϕ
b) In order to operate with the principle of virtual works and to obtain
the 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 is
important the part G is separated (ℑ ∈ G,∀δϕ ∈ T (℘),< ℑ,δϕ >= 0 ⇒
ϕ
ℑ=0).
c)ThefunctionalsA.1,B.1/0,A.2,B.2 arenotseparated:ifℑconsistsin
the data of the fields F, σ, T, it is not possible to conclude that the fields
are zero.
d) Functionals in A.0, B.0, S.1, S.2...are separated:if the fields S1,T1,
R1,L,...arecontinuousthen,byusingthefundamentallemmaofvariation
calculus, their values must be equal to zero. They are the only functionals
we must know for using the principle of virtualworks;it is exactly as for a
solid: 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 to
consideramodelofgreaterorderingradientsandtointroduceintegralson
inner boundaries of the medium.
For conservative medium, the first gradient theory corresponds to the
compressible case. The theory of fluid, elastic, viscous and plastic media
refers to the model (S.0). The Laplace theory of capillarity in fluids refers
to the model (S.1). To take into account superficial effects acting between
solidsandfluids,weusethemodeloffluidsendowedwithcapillarity(S.2);
the theory interprets the capillarity in a continuous way and contains the
Laplace theory of capillarity; for solids, the model corresponds to ”elastic
materials with couple stresses” indicated by Toupin in12.
5. Example 1: The Laplace theory of capillarity
Liquid-vaporandtwo-phaseinterfacesarerepresentedbyamaterialsurface
endowed with an energy relating to Laplace surface tension. The interface
appears as a surface separating two media with its own characteristic be-
havior and energy properties13 (when workingfar from critical conditions,
the capillary layer has a thickness equivalent to a few molecular beams14).
TheLaplacetheoryofcapillarityreferstothemodelB.1 intheform(S.1)
as following: for a compressible fluid with a capillary effect on the wall
boundaries, the free energy is in the form
N
χ= ρ α(ρ) dv+ a ds
p
Z Z Z Z Z
D Xp=1 Sp