Introduction to continuum mechanics-ii

Transcription

1 Ioan R. IONESCU LSPM, Universiy Paris 3 (Sorbonne-Paris-Cié) July 3, 27

2 Coninuum kinemaics B uneforme/reference configuraion, B eforme/acual configuraion Moion: x(, X ) = X + u(, X ) wih x(, B ) = B. Maerial/Lagrangian coorinaes: X = (X, X 2, X 3 ) B Spaial/Eulerian coorinaes: x = (x, x 2, x 3 ) B u(, X ) = x(, X ) X he isplacemen fiel

3 Velociy-acceleraion velociy (in Lagrange variables) v(, X ) = x (, X ) Suppose X x(, X ) one o one funcion, hen x X (, x) velociy (in Euler variables) v(, x) = v(, X (, x)). acceleraion (in Lagrange variables) a(, X ) = 2 x 2 (, X ) acceleraion (in Euler variables) a(, x) = a(, X (, x)). Example. Dilaaion: x = X + α X, x = X 2 + α 2 X 2, x 3 = X 3 + α 3 X 3 velociy (in Lagrange variables) v(, X ) = (α X, α 2 X 2, α 3 X 3 ) velociy (in Euler variables) v(, x) = ( α +α x, α 2 +α 2 x 2, α 3 +α 3 x 3).

4 Paricular (maerial, oal) erivaive Paricular (oal) erivaive of fiel K (he paricle is followe in is movemen) : K(, X ) in Lagrange escripion, K(, x) = K(, X (, x)) in Eulerian escripion if K is in Lagrange variables K K (, X ) = (, X ) if K is in Euler variables K K (, x) = (, x) + v(, x) xk(, x) Examples x K = x: (, x) = v(, x) v v K = v(, x): (, x) = a(, x) = (, x) + v(, x) xv(, x)

5 Reynols s ranspor heorem Paricular (maerial, oal) erivaive of a volume inegral Le ω B an ω = x(, ω ) B (he subse ω B is followe in is movemen) an K(, x) a fiel in Eulerian escripion = ( ) K K(, x) x = ω ω (, x) + iv x(k(, x)v(, x)) x ω K (, x)+k(, x)iv xv(, x) x = ω K (, x)+ K(, x)v(, x) n S ω Examples K : Vol(ω ) = iv x v(, x) x = v(, x) n S ω ω if iv x v(, x) = hen he volume is incompressible

6 of a coninuous boy infiniesimal elemen of a coninuous boy X = (X, X 2, X 3 ) I is possible o show ha (using Taylor expansion aroun a poin of eformaion) x = F (, X )X = (I + u(, x))x x i = F ik X k = (δ ik + u i X k )X k

7 Volume change Consier a ifferenial maerial volume V a some maerial poin ha goes o v afer eformaion How o measure he volume change? Reference volume: V = Z (X Y) Deforme volume: v = W (R V) J(, X ) = e(f (, X )) Jacobien of he rasformaion I is easy o show ha he volume change: v = JV

8 Polar ecomposiion heorem A roaion marix: R such ha RR T = R T R=I (er = I ). Polar ecomposiion heorem: For any marix F wih ef >, here exiss an unique roaion R an an unique posiive-efinie symmeric marix U such ha F = RU how o calculae i? Calculae he Cauhy-Green srain ensor C = F T F an hen U = C, i.e. Fin he eigenvalues {γ, γ 2, γ 3 } an eigenvecors {u, u 2, u 3 } of C calculae µ i = γ i an hen U is he marix wih eigenvalues {µ, µ 2, µ 3 } an he corresponing eigenvecors such ha U = µ u u + µ 2 u 2 u 2 + µ 3 u 3 u 3 R = F U

9 Polar ecomposiion heorem U R ~ X F ~x ~X R V ~x ~I3 ~i3 ~I ~I2 ~i ~i2

10 Velociy graien Velociy graien ensor L = L(, x) = x v(, x), L ij = v i x j L = Ḟ F = ( F )F L = ṘR T + R UU R T Srain rae (sreching, rae of eformaion) ensor D = D(v) D = D(, x) = 2 ( xv(, x) + T x v(, x)), Spin ensor W = W (v) W = W (, x) = 2 ( xv(, x) T x v(, x)), D ij = 2 ( v i x j + v j x i ) W ij = 2 ( v i x j v j x i ) L = D + W, D T = D, W T = W ω(, x) = curl x v(, x), W c = ω c, c 2

11 Rae of Change of Lengh an Orienaion. x = F X, L = Ḟ F = (x) = Lx Orienaion an lengh of he infiniesimal vecors x = ns, x = n s, x 2 = n 2 s 2 Rae of change of orienaion (n) = Ln (Dn n)n (x x 2 ) = 2Dx x 2 Rae of change of lengh x = x 2 = ns = (ln s) = s Rae of change of angles (s) = Dn n (n n 2 ) = 2Dn n 2 ( Dn n + Dn 2 n 2) n n 2

12 Mass conservaion law Mass conservaion law Cauchy assumpion an sress vecors Momenum balance law Le ρ : B R +, an ρ(, ) : B R + be he mass ensiy such ha mass(ω ) = ρ (X ) X, ω mass(ω ) = ρ(, x) x ω for all ω B an ω = x(, ω ) B. Mass conservaion law: mass(ω ) = mass(ω ) for all ω B. Lagrangian escripion ρ(, x)j(, X ) = ρ (X ) for all X B Eulerian escripion ρ(, x) + ρ(, x)iv xv(, x) = for all x B Eulerian escripion ρ(, x) + iv x(ρ(, x)v(, x)) = Consequence: ρ(, x)k(, x) x = ρ(, x) K(, x) x ω ω

13 simplify furher an we make a noe of his here before proceeing furher. By using eiher of he ranspor equaions in (3.84) Z Z Z Vy = ( ) + iv v Vy = + ( + iv v) Vy. (4.7) D D D Mass conservaion law Cauchy assumpion an sress vecors Momenum balance law Forces acing on he boy The erm in parenheses on he righ han sie vanishes by he balance of mass fiel equaion DZ (4.6) an so we ge Z Vy = Vy. (4.8) D D y Equaion (4.8) will be use frequenly in whav follows. Noe ha in eriving i we have no 4.3. FORCE. gnore he fac ha D an are ime epenen even D hough he en resul appears o sugges his. 4.3 b Vy Force. D Ay Vy b Vy Ay D A Vy y D b Vy Ay Ay Vy 8 A A D2 D2 2 CHAPTER 4. MECHANICAL BALANCE LAWS AND FIELD EQUATIONS b Vy Ay Ay Ay nn n n A A D an D2 occupie by wo i eren pars of he boy wih he poin A common o Figure 4.2: Regions A boh bounariesa@d The figure on he lef has isolae D while ha on he righ has isolae R D D D2. The racion is applie a A byhe maerial ousie D. Similarly, D he racion 2 is applie area A area A area A Vy Vy b Vy D D D b Vy Ay Vy 7 D Ay a A by he maerial ousie D2. n Ay D2D2 D D2 Remark (a): In orer for he formulae (4.) - (4.3) o be useful, we mus specify he variables ha hese force ensiies epen on. We expec ha in general he boy force area A n D 2 ensiy maybounary epen on area boh posiion y an ime, an so we assume ha A Figure 4.: Forces on he par P: he racion is a force per uninarea acing a poins on he Assumpions b V Figure 4.3: Regions D an D2 occupie by wo isinc pars of he boy. The poin A is common o he y uni mass acing a poins in he inerior of of he subregion, an he boy force b is na force per b = b(y, ). (4.4) bounaries of boh hese regions. Moreover, he uni ouwar normal vecor a A o boh Ay an isp n. n o he racion. One migh assume ha he racion also We now urn our aenion o he forces ha ac on an arbirary he boy Remarkof(b): We now urn A y A o he forces ha ac on he region D area a ime. For simpliciy, we will someimes refer epens only on he same variables as he boy force, i.e. = (y, ). However some hey occupy regions D an D2 as shown in Figure 4.3. Noe ha he poin A is common raher han (more correcly) he par P. These forces are mos convenienly in his canno be so. To see his, consier wo pars of he boy, P an houghescribe shows ha o boh Moreover, noe ha he uni ouwar normal vecors o n D an D2 a ime as shown in Figure 4.2. Le ya be he erms of eniies ha ac on he region D R occupie by P in he currenpconfiguraion. Asregions 2, which occupy Boy forces ρbx: b = b(, x) Surface forces S acing on D : he acion of B \ D on D a A are he same. By Cauchy s hypohesis = (y,, n), an so he racion posiion of a poin ha is common o boh as shown. In general, on boh a A iscauchy he same; he racion oes no, for example, epen be replace by he isribuion ofconac he vecor we expec ha he force exere asress y (by he maerial ousie D ), o be A on he curvaure of he bounary when he Cauchy hypohesis is invoke. i eren o he conac force exere a ya (by he maerial ousie D2 ). However Remark I is worh racioncapure (y,, n) enoes he force if (): is a funcion of emphasizing y an only,ha henhe i canno his i erence since per bohuni of hese area applie he he parvalue of he which is ousie D on he maerial insie D. jus racions woulby have (yboy A, ). Thus he racion mus epen on more han area A Ofenhe we posiion speak of an he ime. sie ino whichalso n poins as on hehe posiive siesurface of he uner surfaceconsieraion (which is I mus epen specific as he ousie haorer, n poins away isfrom as he by negaive of henormal surfacevecor well: of =D To firs a surface escribe is unisie ouwar ) an ). sie coninuum mechanics-ii Ioan R. IONESCU LSPM, Universiy Paris 3 (Sorbonne-Paris-Cie ) (which hesoinsie of Dassume Then (y,, n) isohe force ensiy applie by he posiive sie n,isan we shall ha ). Inroucion Ay Cauchy s hypohesis: = (, x, n)

14 Mass conservaion law Cauchy assumpion an sress vecors Momenum balance law The Balance of Momenum Principles Balance principle for linear momenum (Newon s law): ρ(, x)v(, x) x = ρ(, x)b(, x) x + (, x, n) S ω ω ω Balance principle for angular momenum (Newon s law): ρ(, x)x v(, x) x = ρ(, x)x b(, x) x+ x (, x, n) S ω ω ω for all ω B.

15 Mass conservaion law Cauchy assumpion an sress vecors Momenum balance law Consequences of balance principles an sress ensor Consequences of linear momenum balance principle + Cauchy s hypohesis n (, x, n) is linear an here exiss σ(, x) Cauchy sress ensor such ha (, x, n) = σ(, x)n Equaion of moion ρ(, x) v(, x) = iv xσ(, x) + ρ(, x)b(, x) Consequence of angular momenum balance principle Cauchy sress ensor is symeric σ T (, x) = σ(, x)

16 e3 e3 Mass conservaion law e3 Cauchy assumpion an sress vecors Momenum balance law n n e n Equaion of moion in Lagrange formulaion S e e e S e2 S R e2 e R!) R Firs Piola-Kirchhoff sress ensor (non-symmeric 3 e2 e2 4. MECHANICAL BALANCE LAWS AND FIELD EQUATIONS CHAPTER x e3 y en T x x bu, if wrien erms of some oher suiable sress ensor S(x, ) has a simple form. e3 ein 3 S n for fining We se ourself he ask such a sress ensor S. S es2 S S n n n Conac force S R y n e3 S Conac Area of( n force = S S R x y S = Tn Area of( e S = Sn Area of( n R y ConacRforce = s Area of( R S S x S e2 S x Conac force Area of( S ) S = e x Conac force R= Tn Area of( S ) n S Conac force of( = Conac force = Area S )Sn Area of( S ) e3 S R = Tn Area of( = S ) s Area of( S ) S e2 S Area of( S ) Conac force = = Sn Area of( S ) T n = Tn Area of( S ) n n e = s Area of( S ) x = Sn Area of( S ) n e3 S = s Area of( S ) Figure 4.: Surface S an surface elemen S in curren configuraion, an heir images S an S S S e2 in he reference configuraion. ways for characerizing he conac force are shown. S n Di erens(equivalen) X Noe ha he conac force acsron he curren configuraion. 2 Π(, X ) = J(, X )σ(, x(, X ))F n S = σ(, x)ns = Π(, X )n S Nanson s formula ns = JF ρ (X ) (, X ) v (, X ) = iv Π(, X ) + ρ (X )b(, X ) R e n he moion. Le S be a surface in R an le S be 3Sinsan Consier some fixe uring is image reference y be a poin S an Equilibrium equaion ivxin he Π(, X )configuraion; + ρ (Xle)b(, X )on = le x be is image on S ; le n be a uni x normalnvecor o S Sa y, an le n be he corresponing uni normal vecor R Secon Piola-Kirchhoff ensor!) surface S =elemen F onπ o S a x; an finally, le(symmeric S be an infiniesimal S a y whose area is Ay, an le S S be is image in he reference configuraion whose area is Ax. This is S x illusrae in Figure 4.. Ioan R. IONESCU LSPM, Universiy Paris 3 (Sorbonne-Paris-Cie ) 2 S ) S ) S ) S )

17 Examples of sress ensors Mass conservaion law Cauchy assumpion an sress vecors Momenum balance law