Kinematics ( ) Deformation Mapping. Eulerian/Lagrangian descriptions of motion. Deformation Gradient. x const. i ik k. u(x) e 2. e 1. e 3.

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1 Knematcs Deformaton Mappng y = x + u ( x, x, x,) t 3 Euleran/Lagrangan descrptons of moton = + = = t x = const t e y u y x u ( x, t) v ( x, t) y y = + = = t x = const t x = const x Orgnal Confguraton y x u ( y, t) v ( y, t) a ( y, t) y u(x) Confguraton δ u yk u y v (,) v (,) k = = a y t = + vk y t yk t t x const y= const t t x const y= const y = = k Deformaton Gradent y ( ( )) y= x+ ux = F ( ) u or = x + u = δ + = F x x x dy = F dx dy = F dx k k e dx x Orgnal Confguraton u(x+dx) u(x) dy Confguraton

2 Knematcs Sequence of deformatons e dx x u () (x) y dy u () (y) z dz Orgnal Confguraton After frst deformaton After second deformaton () () () () dz = F dx wth F= F F or dz = Fdx F = Fk Fk Lagrange Stran T E= ( F F I) or E = ( Fk Fk δ ) ( δl) 0 δ l l 0 0 l0 l l l mem = E mm = = + e l 0 m Orgnal Confguraton l Confguraton

3 Knematcs Volume Changes dv det ( ) dv = F = J 0 e dy dv 0 dx dz dv dr dv dw Undeformed Area Elements T k k 0 dan= JF da n dan = JF n da e da 0 n 0 dp (n) 0 u(x) n dp (n) da x Orgnal Confguraton t Confguraton e Infntesmal Stran Approxmates L-stran T ( ( ) ) or u u ε= u + u ε = + x x y B u u O uk u k u u E = ε x x x x x x l 0 C l 0 A x y l 0 (+ε ) yy O l 0 B π/ γ xy (+ε ) xx A x C elated to Engneerng Strans ε = ε xx = εyy ε ε = ε = γxy / = γyx /

4 Prncpal values/drectons of Infntesmal Stran or ε () () en () () klnl = en l ε n = Knematcs e n () n () n () Orgnal Confguraton Ι+ε n () Infntesmal rotaton T ( ( ) ) or u u w = u u w = x x Decomposton of nfntesmal moton n () Ι+ε n () I+w u x = ε + w e n () Orgnal Confguraton n () Confguraton

5 Knematcs Left and ght stretch tensors, rotaton tensor F= U F= V U,V symmetrc, so U= λu u + λ u u + λu u () () () () (3) (3) 3 V = λv v + λ v v + λ v v () () () () (3) (3) 3 λ prncpal stretches U u () v () u () u () u () e u () Orgnal Confguraton u (3) u (3) λ u () v () λ λ 3 u () Confguraton u () Left and ght Cauchy-Green Tensors T C= F F= U T B= F F = V

6 Generalzed stran measures Knematcs Lagrangan Nomnal stran: ( λ ) u 3 = 3 Lagrangan Logarthmc stran: log( λ ) u = () () u () () u Euleran Nomnal stran: ( λ ) v 3 = 3 Euleran Logarthmc stran: log( λ ) v = () () v () () v Euleran stran * T * E = ( I F F ) or E = ( δ Fk Fk )

7 Knematcs u(x) Velocty Gradent L v = yv L = y e x Orgnal Confguraton y Confguraton v dv = v ( y+ dy) v ( y) = dy y dv = F d d d dv = dy ( Fdx ) Fdx Fk dy dt = dt = L= F F k dt Stretch rate and spn tensors dl l dt = ndn =nd n T T D= ( L+ L ) W= ( L L ) e l 0 l n Orgnal Confguraton Confguraton

8 Knematcs v ω= v = ω= dual( W) ω = k W k y Vortcty vector curl( ) ω k k Spn-acceleraton-vortcty relatons v v a= = + ( vv) + W v t t y k k k k x = const y = const k k v v a= = + ( vv) + ω v t t y k k k k x = const y = const k a ω v = + k k k Dω y t x= const yk k ω

9 Knetcs S 0 S b t = lm da 0 Surface tracton dp da da dp e ρb = 0 Orgnal Confguraton lm dv 0 dp dv t Confguraton Body Force dv dp Internal Tracton dp n T(n) t Tn ( ) = lm da 0 ( n) dp da da t e T(-n) -n esultant force on a volume P = Tn ( ) da + ρ b dv A V S 0 e 0 Orgnal Confguraton t S V b n Confguraton T(n)

10 Knetcs estrctons on nternal tracton vector T(n) Newton II T( n) = Tn ( ) n T(-n) -n Newton II&III da e da (n) n T(n) da (n) Tn ( ) = n σ or T ( n) = n σ - T(- ) da Cauchy Stress Tensor σ e σ 3 σ 33 σ 3 σ 3 σ 3 σ σ σ

11 Other Stress Measures S 0 S b F I u u = + F = δ + J = det( F) x e 0 Orgnal Confguraton t Confguraton n 0 dp (n) 0 n dp (n) e da 0 u(x) da Krchhoff τ σ τ = J = Jσ x Orgnal Confguraton t Confguraton Nomnal/ st Pola-Krchhoff Materal/ nd Pola-Krchhoff S JFk σ k ( n) 0 = da0 n S S= JF σ = dp T σ Σ k σ kl l Σ= JF F = JF F ( 0) ( ) n = dp n ( n dp 0) = da0 n 0 Σ F dp

12 eynolds Transport elaton e x 0 S 0 C 0 y Orgnal Confguraton u(x) S n C Confguraton d φ v φ φv dv dv dv dt t y t y φ = + φ = + V V = const V = const x y

13 Conservaton Laws for Contnua V Mass Conservaton ρ v ρ v dv 0 + ρ = + ρ = 0 t x= const y t x= const y ρ ρv ρ + = 0 + y ( ρv) = 0 t y t y= const y= const e S 0 x 0 V 0 y Orgnal Confguraton S u(x) V n Confguraton Lnear Momentum Conservaton σ y σ + ρ b = ρ a or + ρb = ρa y e S 0 x 0 y Orgnal Confguraton S u(x) t V b n Confguraton T(n) Angular Momentum Conservaton σ = σ

14 Work-Energy elatons ate of mechancal work done on a materal volume ( ) d n r = T vda + ρbv dv = σddv + ρvv dv dt A V V V e S 0 x 0 y Orgnal Confguraton S u(x) t V b n Confguraton T(n) Conservaton laws n terms of other stresses S S+ ρ b= ρ a + ρ = ρ a 0 0 0b 0 x ( k Fk ) Σ T Σ F + ρ b= ρ a + ρ b = ρ a x Mechancal work n terms of other stresses ( n) d r = T vda + ρbv dv = SF dv0 + ρ0vv dv0 dt A V V V 0 0 ( n) d r = T vda + ρbv dv = Σ E dv0 + ρ0vv dv0 dt A V V V 0 0

15 Prncple of Vrtual Work (alternatve statement of BLM) δl δv δv δ v = δd = + y y y e S 0 x 0 y Orgnal Confguraton S u(x) b S t V Confguraton dv If σδ D dv + ρ δv dv ρbδv dv tδ v da = 0 dt for all δ v V V V S Then σ y + ρb = dv ρ dt nσ = t on S

16 Thermodynamcs S 0 S b Temperature θ Specfc Internal Energy ε Specfc Helmholtz free energy Heat flux vector q External heat flux q Specfc entropy s ψ = ε θs e 0 Orgnal Confguraton t Confguraton d Frst Law of Thermodynamcs ( Ε+ KE) = Q + W dt ε ρ t x= const ds dη Second Law of Thermodynamcs 0 dt dt q = σ D + q y s ( q / θ ) q ρ + 0 t y θ Dsspaton Inequalty D q θ ψ σ ρ s θ + 0 θ y t t

17 Transformatons under observer changes Transformaton of space under a change of observer y = y () t + Q()( t y y ) * * 0 0 d Ω= Q Q dt All physcally measurable vectors can be regarded as connectng two ponts n the nertal frame These must therefore transform lke vectors connectng two ponts under a change of observer * * * * b = Qb n = Qn v = Qv a = Qa * Inertal frame e e * e * Observer frame Note that tme dervatves n the observer s reference frame have to account for rotaton of the reference frame * * * dy d T * * dy dy0 * * = = = 0 t = 0 T y* y n Confguraton b Confguraton v Qv Q Q Q ( y y ( )) Ωy ( y ( t)) dt dt dt dt * * * * * d y d T * * d y d y0 dω * * dy dy0() t a = Qa = Q = Q Q ( y y 0( t)) = + ( Ω y y0( t)) Ω( ) dt dt dt dt dt dt dt b * n *

18 Some Transformatons under observer changes Obectve (frame ndfferent) tensors: map a vector from the observed (nertal) frame back onto the nertal frame t = n σ Inertal frame n * T * σ = QσQ D = QDQ T e y b Invarant tensors: map a vector from the reference confguraton back onto the reference confguraton T 0 = m Σ * = Σ Σ * * e * e y* Confguraton b * n * Mxed tensors: map a vector from the reference confguraton onto the nertal frame dy = Fdx * = F QF Observer frame Confguraton

19 Some Transformatons under observer changes * * 0 0 The deformaton mappng transforms as y( X,) t = y () t + Q() t ( yx (,) t y ) * y y The deformaton gradent transforms as F = = Q = QF X X The rght Cauchy Green stran Lagrange stran, the rght stretch tensor are nvarant * * * T * T T * * C = F F = F Q QF = C E = E U = U The left Cauchy Green stran, Euleran stran, left stretch tensor are frame ndfferent * * * T T T T * T B = F F = QFF Q = QCQ V = QVQ The velocty gradent and spn tensor transform as * * * T T L = F F = QF + QF F Q = QLQ + Ω * * * T ( ) W = ( L L )/= QWQ + Ω The velocty and acceleraton vectors transform as * * * dy d T * * dy dy0 * * = = = 0 t = 0 v Qv Q Q Q ( y y ( )) Ωy ( y ( t)) dt dt dt dt * * * * * d y d T * * d y y0 dω * * dy y = = = 0 0 t = + 0 t d d () t a Qa Q Q Q ( y y ( )) Ω ( y y ( )) Ω( ) dt dt dt dt dt dt dt (the addtonal terms n the acceleraton can be nterpreted as the centrpetal and corols acceleratons) * T The Cauchy stress s frame ndfferent σ = QσQ (you can see ths from the formal defnton, or use the fact that the vrtual power must be nvarant under a frame change) The materal stress s frame nvarant T Σ * = Σ * T T T σ σ (note that ths The nomnal stress transforms as S = J( QF) Q Q = JF Q = SQ transformaton rule wll dffer f the nomnal stress s defned as the transpose of the measure used here )

20 Consttutve Laws Equatons relatng nternal force measures to deformaton measures are known as Consttutve elatons General Assumptons:. Local homogenety of deformaton (a deformaton gradent can always be calculated). Prncple of local acton (stress at a pont depends on deformaton n a vanshngly small materal element surroundng the pont) e Orgnal Confguraton Confguraton estrctons on consttutve relatons:. Materal Frame Indfference stress-stran relatons must transform consstently under a change of observer. Consttutve law must always satsfy the second law of thermodynamcs for any possble deformaton/temperature hstory. D q θ ψ σ ρ s θ + 0 θ y t t

21 Fluds Propertes of fluds No natural reference confguraton Support no shear stress when at rest Knematcs Only need varables that don t depend on ref. confg e y t S b Confguraton L v = + = + = v ω = k = y = L D W D ( L L )/ W ( L L )/ y W k k v v yk v v v a = = + = Lkvk + = ( Dk + Wk ) vk + t x = const yk t t y = const t y = const t y = const k v ( ) = vv k k + Wkvk = + ( vv k k ) + k ω vk y t y = const y Conservaton Laws k ρ ρ ρv + ρd 0 or kk = + = 0 t t y x= const y= const σ v v y y t + ρb = ρ vk + σ = σ k y = const ε q ρ = σ D + q t x= const y D q θ ψ σ ρ s θ + 0 θ y t t

22 General Consttutve Models for Fluds Obectvty and dsspaton nequalty show that consttutve relatons must have form Internal Energy ε= ˆ( ε ρθ, ) Entropy s= sˆ( ρθ, ) Free Energy ˆ (, ) Stress response functon Heat flux response functon ψ= ψ ρθ = ε θs ˆ (,, ) ˆ (, ) ˆ vs σ = σ θ ρd = πeq ρθδ + σ ( ρθ,, D ) θ q = qˆ θ, ρ,, D y In addton, the consttutve relatons must satsfy ψˆ ψ ˆ πeq = ρ sˆ= ρ θ ˆ πeq ˆ sˆ πeq ˆ ε = ρ ˆ πeq = θ + ρ θ ρ θ ρ c v ψˆ c ˆ v θ πeq = θ = θ ρ ρ θ vs θ θ σ ( ρθ,, D ) D 0 q ρθ,, 0 y y where ˆ ε c v ( θ, ρ) = θ

23 Consttutve Models for Fluds ψ= ψˆ ( ρθ, ) = ε θs vs eq σ = ˆ σ ( θ, ρ, D ) = ˆ π ( ρθδ, ) + ˆ σ ( ρθ,, D ) Elastc Flud ψ= ψˆ ( ρ) σ = π ( ρδ ) eq Ideal Gas p ε = c θ = ψ = c θ θ ( c logθ log ρ s ) σ = pδ = ρθδ ( γ ) ρ v v v 0 Newtonan Vscous ψ= ψˆ ( ρθ, ) σ = ( π ( ρθ, ) κ( ρθ, ) D ) δ + η( ρθ, )( D D δ /3) eq kk kk Non-Newtonan ψ= ψˆ ( ρθ, ) σ = π ( ρθδ, ) + η( I, I, I, ρθδ, ) + η ( I, I, I, ρθ, ) D + η ( I, I, I, ρθ, ) D D eq k k

24 Unknowns: Derved Feld Equatons for Newtonan Fluds pv, Must always satsfy mass conservaton ρ ρ ρv + ρd 0 or kk = + = 0 t t y x= const y= const Combne BLM σ v v v + ρb= ρa a= v + = + ( vv) + ω v y y t t y Wth consttutve law. Also recall k k k k k k y = const yk= const v v D = + y y p Compressble Naver-Stokes ( ) Wth densty ndep vscosty + η( ρθ, ) D D δ /3 + ρb = ρa p= π ( ρθ, ) κ( ρθ, ) D y y kk eq kk πeq η v κ η v b= a ρ y ρ y y ρ 3ρ y y For an ncompressble Newtonan vscous flud For an elastc flud (Euler eq) πeq v + ρb= ρ + ρ ( vv) + ρ ω v y t y p η v + + b = a ρ y ρ y y Incompressblty reduces mass balance to k k k k y = const k v y = 0

25 Derved Feld Equatons for Fluds ecall vortcty vector ω v k = k = k W y a ω v = + k k k Dω y t x= const yk ω Vortcty transport equaton (constant temperature, densty ndependent vscosty) η ρ η = ρ y y ρ y y y y y y y t x= ω v k v l v ( ) k ω k η κ k bk Dω ω l l 3 l k k const For an elastc flud v ( ) k ω k bk + Dω ω = x y t x k = const For an ncompressble flud η ω ω + + ( ) k bk + Dω = y y x t x ρ = const If flow of an deal flud s rrotatonal at t=0 and body forces are curl free, then flow remans rrotatonal for all tme (Potental flow)

26 Derved feld equatons for fluds For an elastc flud Bernoull πeq H= ψ + + vv +Φ= constant along streamlne ρ For rrotatonal flow H πeq = ψ + + vv +Φ= constant everywhere ρ For ncompressble flud p ρ + vv +Φ= constant

27 Solvng fluds problems: control volume approach Governng equatons for a control volume (revew) B

28 Example Steady D flow, deal flud Calculate the force actng on the wall Take surroundng pressure to be zero A 0 v 0 ρ 0 A 4 α v A 3 A A 5 d n σda + ρbdv = ρvdv + ( ρvvn ) da dt B B A v A 0 0ρ0 0 α 0 A ( p n da) A v sn + ( p p ) da= 0 3 F = A 0 ρ 0 v 0 sn α

29 Exact solutons: potental flow If flow rrotatonal at t=0, remans rrotatonal; Bernoull holds everywhere Irrotatonal: curl(v)=0 so v Ω = y Mass cons v y Ω = 0 = 0 y y Bernoull p Ω + vv constant ρ +Φ+ = t Example: flow surroundng a movng sphere e a V avα( yα Vt) Ω= α r = ( y V t)( y V t) α α α α r

30 Exact solutons: Stokes Flow Steady lamnar vscous flow between plates Assume constant pressure gradent n horzontal drecton η v v b η y const p p f = 0 ρ y ρ y v t = L y k y h V y Solve subect to boundary condtons y p v= V y ( h y ) h L e ηv p h σ = + y h L

31 Exact Solutons: Acoustcs Assumptons: Small ampltude pressure and densty fluctuatons Irrotatonal flow Neglgble heat flow Neglect body forces Irrotatonal: v Ω = y Approxmate N-S as: p y v ρ t y = const k p y y v ρ y t y k = const p y t v ρ t y k = const For small perturbatons: Mass conservaton: p δρ = cs t t δρ t x= const c s v + ρ = 0 y p = ρ s = const Combne: c s t Ω Ω = 0 y y δ p = ρ Ω 0 t (Wave equaton)

32 Wave speed n an deal gas Assume heat flow can be neglected Entropy equaton: s θ t x= const q = + q s = const y v v v ( log log ) ε = c θ ψ = c θ θ c θ ρ + s p= ρθ ( ) c / v 0 0 s= c logθ log ρ + s θ = ρ exp[( s s ) / c ) v / c = γ so p= kρ γ v 0 v Hence: p γ p cs = = kγρ = γ = γ θ ρ ρ s= const

33 Applcaton of contnuum mechancs to elastcty S 0 e x 0 y u Orgnal Confguraton t S b Confguraton Modulus G' (N/m ) Glassy Glass Transton temperature T g Vscoelastc ubbery Melt (frequency) - Materal characterzed by

34 S 0 e x 0 General structure of consttutve relatons y u Orgnal Confguraton t S b Confguraton F C B u = δ + = F F x k k = F F k k = k k Q JF q S JFk σ k S= JF σ = T σ Σ k σ kl l Σ= JF F = JF F S x v + ρ 0b = ρ0 t x = const θ ψ θ Σ C Q ρ0 + s 0 θ x t t * = F QF * * * T B = F F = QBQ * * T * C = F F = C T * = Σ Σ Frame ndfference, dsspaton nequalty ˆ ψˆ ψˆ Σ = ρ0 sˆ = C θ Σˆ sˆ = ρ0 θ C Q k θ 0 x k

35 Forms of consttutve relaton used n lterature I 3 = trace( B) = B ( B B) ( k k ) I = I = I B B I = det B = J Stran energy potental 0 kk W = ρψ I Bkk I = = /3 /3 J J I B B Bk Bk I = = I 4/3 I 4/3 = 4/3 J J J J = det B B= λ b b + λ b b + λ b b () () () () (3) (3) 3 W( F) = Wˆ ( C) = U( I, I, I ) = U( I, I, J) = U ( λ, λ, λ ) 3 3 σ = J F k W F k σ σ U I U B U = + B B + I U δ k k 3 I3 I I I I3 U U U U δ U U = /3 + I B I + I B 4/3 k Bk + δ J J I I I I 3 J I J σ λ U λ U λ U () () () () 3 (3) (3) = b b + b b + b b λλ λ3 λ λλ λ3 λ λλ λ3 λ3

36 Specfc forms for free energy functon Neo-Hookean materal µ K U = ( I 3) + ( J ) µ σ δ ( ) δ J 3 = 5/3 B Bkk + K J Mooney-vln µ µ K U = ( I 3) + ( I 3) + ( J ) µ µ σ δ δ δ ( ) δ J 3 J 3 3 = [ ] 5/3 B Bkk + 7/3 Bkk B Bkk Bk Bk + Bkn Bnk + K J N N Generalzed polynomal functon K ( 3) ( 3) ( ) + = = U = C I I + J Ogden U N µ α α α K = ( λ + λ + λ 3 3) + ( J ) α = Arruda-Boyce 3 K U = µ ( I 3) + ( I 9) + ( I 4 7) J 0β 050β ( )

37 Solvng problems for elastc materals (sphercal/axal symmetry) Assume ncompressbllty Knematcs σ rr 0 0 Frr 0 0 Brr 0 0 σ 0 σθθ 0 F 0 Fθθ 0 B 0 Bθθ σ φφ 0 0 F φφ 0 0 B φφ dr r dr r Frr = Fφφ = Fθθ = Brr = Bφφ = Bθθ = d d φ θ r e e e θ e φ Consttutve law dr r = d σ σ r a = A U U I U I U U = + I B B + p rr rr rr I I 3 I 3 I I U U I U I U U = = + I B B + p θθ σφφ θθ θθ I I 3 I 3 I I Equlbrum (or use PVW) dσ dr rr ( σrr σθθ σφφ ) ρ0 + + br = 0 r (gves ODE for p(r) Boundary condtons u ( a) = g u ( b) = g σ r a r b ( a) = t σ ( b) = t rr a rr b

38 Lnearzed feld equatons for elastc materals Approxmatons: Lnearzed knematcs All stress measures equal Lnearze stress-stran relaton S 0 e 0 Orgnal Confguraton t S b Confguraton u u σ u = + = Ckl ( kl kl ) + b = x x x t ε σ ε α θ ρ ρ * * = ( ) on σ = ( ) on u u t n t t Elastc constants related to stran energy/unt vol C kl ˆ σ ˆ U = = = = ε ε ε θ ε θ U σ β kl kl ε = Sσ + α T σ= Cε ( α T ) Isotropc materals: + ν ν ε = σ σkkδ + α Tδ E E E ν Eα T σ = ε + εkkδ δ + ν ν ν

39 Solvng lnear elastcty problems sphercal/axal symmetry Knematcs Consttutve law σ 0 0 ε 0 0 σ 0 σθθ 0 ε 0 εθθ σ φφ 0 0 ε φφ du u ε = εφφ = εθθ = d du σ E ν ν d Eα T σ = θθ ( + ν)( ν ) ν u ν θ e e e θ φ e e φ Equlbrum dσ d ( σ σθθ σφφ ) ρ0 + + b = 0 ( u) ( ) ( ν) ( )( ) ( ν) d u du u d d + d T + + = d d = d d d E α ν ν ν ρ 0b ( ) Boundary condtons u ( a) = g u ( b) = g σ r a r b ( a) = t σ ( b) = t rr a rr b

40 Some smple statc lnear elastcty solutons Naver equaton: uk u b ρ0 u + + ρ 0 = k k k ν x x x x µ µ t Potental epresentaton (statcs): ( + ν ) u = Ψ + Ψ E 4( ν ) x ( φ x ) k k φ = ρ0b = ρ0bx k k Ψ x x x x P Pont force n an nfnte sold: Ψ = φ = 0 4π ( + ν ) Px k kx u = + (3 4 ν ) P 8 ( ) πe ν ( + ν ) 3 Px k kxx Px k kδ Px + Px ε = ( ) + ν 3 8 πe( ν) 3 Px k kxx Px + Px δpx k k σ = ( ν) π( ν) Pont force normal to a surface: P ( νδ ) 3 ( ν)( ν) φ log( 3 ) Ψ = = + x π π u ( + ν) P xx 3 δ 3 ( ) (3 4 ) ν x = + ν δ π E + x 3 ν ( xx x3 x3( 3x 3x) ) ( 3 3 ) P xx x3 ( ν )( + x3) ( ) σ = δ δ + δ + δ δ δ π ( + x3) ( + x3)

41 Dynamc elastcty solutons Plane wave soluton u = a f ( ct x p ) k k Naver equaton uk u b ρ0 u + + ρ 0 = k k k ν x x x x µ µ t ( µ ρ ) µ + = ν 0c ak pap k 0 Solutons: 0 ap = 0 c = c = ρ / µ L a = η p c = c = µ ( ν) / ρ ( ν) 0