InroToMaFuncions.pdf 4 CM465 To proceed o beer-designed consiuive equaions, we need o know more abou maerial behavior, i.e. we need more maerial funcions o predic, and we need measuremens of hese maerial funcions. More non-seady maerial funcions (maerial funcions ha ell us abou memory Maerial funcions ha ell us abou nonlineariy (srain Faih A. Morrison, Michigan Tech U. Summary of shear rae kinemaics (par., a. Seady o o o., b. Sress Growh o o., c. Sress o Relaxaion Faih A. Morrison, Michigan Tech U.
InroToMaFuncions.pdf 4 CM465 The nex hree families of maerial funcions incorporae he concep of srain. Faih A. Morrison, Michigan Tech U. 3 Summary of shear rae kinemaics (par., d. Creep o., e. Sep Srain o o f. SAOS. o cos, o sin o sin( 4 Faih A. Morrison, Michigan Tech U.
InroToMaFuncions.pdf 4 CM465 Shear Creep Flow Consan shear sress imposed samples oven MASS Faih A. Morrison, Michigan Tech U. 5 Because shear rae is no prescribed, i becomes somehing we mus measure. Kinemaics: v ( x Maerial Funcions: Creep Shear Flow Maerial Funcions 3 I is unusual o prescribe sress raher han ( Since we se he sress in his experimen (raher han measuring i, he maerial funcions are relaed o he deformaion of he sample. We need o discuss measuremens of deformaion before proceeding. 6 Faih A. Morrison, Michigan Tech U. 3
InroToMaFuncions.pdf 4 CM465 Pause on Maerial Funcions We need o define and learn o work wih srain. Faih A. Morrison, Michigan Tech U. 7 Deformaion (srain We need a way o quanify change in shape The problem of change in shape is a difficul, 3-dimensional problem; we can sar simple wih unidirecional flow (shear. 8 Faih A. Morrison, Michigan Tech U. 4
InroToMaFuncions.pdf 4 CM465 Srain in Shear Δ Δ Relaive change in displacemen H H H The srain is relaed o he change of shape of he deformed paricle. Faih A. Morrison, Michigan Tech U. 9 Srain in Shear Δ Δ Relaive change in displacemen H H H H H H H H / The srain is relaed o he change of shape of he deformed paricle. There is no unique way o measure change of shape. Faih A. Morrison, Michigan Tech U. 5
InroToMaFuncions.pdf 4 CM465 Deformaion (srain r( x ( x( x3( x( r( x( 3( x 3 This vecor keeps rack of he locaion of a fluid paricle as a funcion of ime. 3 flow paricle pah u(, r( r( x 3 r( Curren posiion compared o erence posiion (, x Relaive change in displacemen P( u u(, r( Displacemen funcion Shear srain P( x x Faih A. Morrison, Michigan Tech U. Wha is he srain in he sandard flow seady shear? x r( r( x x v 3 dx d dx d dx3 d 3 We can inegrae his differenial equaion because is a consan. We obain. Faih A. Morrison, Michigan Tech U. 6
InroToMaFuncions.pdf 4 CM465 Deformaion in shear flow (srain seady r( x ( x( x3( x ( r( x( 3( x u( 3 3, r( r( x ( ( x ( x ( 3 ( x x 3 3 Displacemen funcion 3 Faih A. Morrison, Michigan Tech U. u( Deformaion in shear flow (srain, r( r( ( x 3 Displacemen funcion seady Our choice for measuring change in shape: u du (, x dx Shear srain (, ( (for seady shear or in unseady shear for shor ime inervals 4 Faih A. Morrison, Michigan Tech U. 7
InroToMaFuncions.pdf 4 CM465 For unseady shear, is a funcion of ime: v dx ( x d dx d dx3 3 d 3 This inegraion is less sraighforward. We can obain he unseady resul for srain by applying he seady resul over shor ime inervals (where may be approximaed as a consan and add up he srains. shor ime inerval beween and : u p, p ( p x ( 5 Faih A. Morrison, Michigan Tech U. For unseady shear: u p, p ( p x ( (shor ime inerval For a long ime inerval, we add up he srains over shor ime inervals. shor ime inerval: long ime inerval: p, p ( p ( N N (, ( p, p ( p p p Taking he limi as Δ, N (, lim p d ( ( p Srain a wih respec o fluid configuraion a in unseady shear flow. 6 Faih A. Morrison, Michigan Tech U. 8
InroToMaFuncions.pdf 4 CM465 Change of Shape For shear flow (seady or unseady: (, ( d Noe also, by Leibniz rule: d d ( d d d d d ( ( d ( d ( d ( Srain a wih respec o fluid configuraion a in shear flow (seady or unseady. d ( d Deformaion rae Now we can coninue wih maerial funcions based on srain. 7 Faih A. Morrison, Michigan Tech U. Because shear rae is no prescribed, i becomes somehing we mus measure. Kinemaics: v ( x Maerial Funcions: Creep Shear Flow Maerial Funcions 3 I is unusual o prescribe sress raher han ( Since we se he sress in his experimen (raher han measuring i, he maerial funcions are relaed o he deformaion of he sample.. 8 Faih A. Morrison, Michigan Tech U. 9
InroToMaFuncions.pdf 4 CM465 Kinemaics: v Creep Shear Flow Maerial Funcions ( x Maerial Funcions: (, J (, ( Shear creep compliance 3 ( ~ ~ ( ~ J (, R(, r r ( ~,, Recoverable creep compliance, 9 Faih A. Morrison, Michigan Tech U. Creep Recovery -Afer creep, sop pulling forward and allow he flow o reverse -In linear-viscoelasic maerials, we can calculae he recovery maerial funcion from creep measuremens ~ (, (, r ( Recoverable srain Recoil srain Srain a he end of he forward moion Srain a he end of he recovery ~ ~ r ( J r (, Recoverable creep compliance Faih A. Morrison, Michigan Tech U.
InroToMaFuncions.pdf 4 CM465 Maerial funcions prediced for creep of a Newonian fluid Newonian: ( v v T Shear creep compliance Recoverable creep compliance J(,? ( J r ~ (,? Faih A. Morrison, Michigan Tech U. Maerial funcions prediced for creep of a Newonian fluid J ( J( ~ J ( r No recovery in Newonian fluids remove sress ( Faih A. Morrison, Michigan Tech U.
InroToMaFuncions.pdf 4 CM465 Shear Creep of a Viscoelasic liquid v ( x 3 ( J p J ( T T T (, J (, Daa have been correced for verical shif. Figure 6.53, p. Plazek; PS mel 3 Faih A. Morrison, Michigan Tech U. Characerisics of a Creep Curve A long imes he creep compliance J(, becomes a sraigh line (seady flow. dj d d d J ( seady sae seady sae ( ( We can define a seady-sae compliance C The slope a seady sae is he inverse of he seady sae viscosiy J s ( Seady-sae compliance 4 Faih A. Morrison, Michigan Tech U.
InroToMaFuncions.pdf 4 CM465 Shear creep maerial funcions Seady-sae compliance J s o consan slope = ( regime of seady flow a shear rae 5 Faih A. Morrison, Michigan Tech U. Characerisics of a Creep Recovery Curve A long recoil imes we can define an ulimae recoil funcion R ( ~, R( ~, J r ( ( R Ulimae recoil funcion ~ ~, recovery 6 Faih A. Morrison, Michigan Tech U. 3
InroToMaFuncions.pdf 4 CM465 In he Linear Viscoelasic (LVE Limi, i is easy o relae he wo shear creep maerial funcions J, ( o ~, J r (, ~ ~ creep, recovery 7 Faih A. Morrison, Michigan Tech U. Linear Viscoelasic Creep (no dependence on oal srain recoverable srain non-recoverable srain ( ( r ( ( r J ( J ( J r ( J ( r For LVE maerials, we can obain R( wihou a recovery experimen 8 Faih A. Morrison, Michigan Tech U. 4
InroToMaFuncions.pdf 4 CM465 Shear Creep - Recoverable Compliance Time-emperaure shifed ~ J r ( Figures 6.54, 6.55, p. Plazek; PS mel 9 Faih A. Morrison, Michigan Tech U. Shear creep maerial funcions Linear-viscoelasic (LVE limi consan slope = o J S J ( ~ J r ( R J s Ulimae recoil funcion LVE seady-sae compliance, ~ ~ creep, recovery Non-recoverable srain 3 Faih A. Morrison, Michigan Tech U. 5
InroToMaFuncions.pdf 4 CM465 Kinemaics: ( x v 3 Sep Shear Srain Maerial Funcions ( lim consan Maerial Funcions: (, G(, Relaxaion modulus Firs normal-sress relaxaion modulus Second normalsress relaxaion modulus G G 33 3 Faih A. Morrison, Michigan Tech U. Wha is he srain in his flow? (, ( d lim lim d d The srain imposed is a consan 3 Faih A. Morrison, Michigan Tech U. 6
InroToMaFuncions.pdf 4 CM465 Sep shear srain - srain dependence, G(, Pa, <.87 3.34 5. 6.68 3.4 8.7 5.4 Figure 6.57, p. Einaga e al.; PS soln ime, s 33 Faih A. Morrison, Michigan Tech U. Linear viscoelasic limi lim G(, G( A small srains he relaxaion modulus is independen of srain. The polysyrene soluions on he previous slide show ime-srain independence, i.e. he curves have he same shape a differen srains. Damping funcion, h G(, h( G( The damping funcion summarizes he non-linear effecs as a funcion of srain ampliude. 34 Faih A. Morrison, Michigan Tech U. 7
InroToMaFuncions.pdf 4 CM465 Wha ypes of maerials generae sress in proporion o he srain imposed? Answer: elasic solids Hooke s Law for elasic solids G v x x u iniial sae, no flow, no forces deformed sae, u G x Hooke's law for elasic solids x x f spring resoring force iniial sae, no force deformed sae, f k x Hooke's law for linear springs Similar o he linear spring law 35 Faih A. Morrison, Michigan Tech U. Small-Ampliude Oscillaory Shear Maerial Funcions Kinemaics: ( x v 3 ( cos Maerial Funcions: (, Gsin G cos G( cos Sorage modulus is he phase difference beween sress and srain waves G ( sin Loss modulus 36 Faih A. Morrison, Michigan Tech U. 8
InroToMaFuncions.pdf 4 CM465 Wha is he srain in his flow? (, ( d cos d sin The srain ampliude is The srain imposed is sinusoidal. 37 Faih A. Morrison, Michigan Tech U. Generaing Small Ampliude Oscillaory Shear (SAOS seady shear b ( V h o h x V consan x small-ampliude oscillaory shear b( h o sin h o sin h x V periodic x 38 Faih A. Morrison, Michigan Tech U. 9
InroToMaFuncions.pdf 4 CM465 In SAOS he srain ampliude is small, and a sinusoidal imposed srain induces a sinusoidal measured sress. ( sin( ( sin( sin cos cos sin cos sin sin cos porion in-phase wih srain porion in-phase wih srain-rae 39 Faih A. Morrison, Michigan Tech U. 3 is he phase difference beween he sress wave and he srain wave 4 6 8 (, ( - - ( -3 4 Faih A. Morrison, Michigan Tech U.
InroToMaFuncions.pdf 4 CM465 SAOS Maerial Funcions ( cos sin sin cos porion in-phase wih srain G porion in-phase wih srain-rae G For Newonian fluids, sress is proporional o srain rae: G is hus known as he viscous loss modulus. I characerizes he viscous conribuion o he sress response. 4 Faih A. Morrison, Michigan Tech U. SAOS Maerial Funcions ( cos sin sin cos porion in-phase wih srain G porion in-phase wih srain-rae G For Hookean solids, sress is proporional o srain : G G is hus known as he elasic sorage modulus. I characerizes he elasic conribuion o he sress response. (noe: SAOS maerial funcions may also be expressed in complex noaion. See pp. 56-59 of Morrison, 4 Faih A. Morrison, Michigan Tech U.
InroToMaFuncions.pdf 4 CM465.E+7.E+6 SAOS Moduli of a Polymer Mel G' (Pa G'' (Pa.E+5 Loss Modulus, G (; viscous characer.e+4.e+3.e+ Sorage Modulus, G (, elasic characer kthe k SAOS (s moduli as a g k (kpa funcion of frequency may.3e-3 be correlaed 6 wih 3.E-4 4 maerial composiion and 3 3.E-5 9 used like a mechanical 4 3.E-6 4 5 3.E-7 specroscopy. 4.E+.E+.E+.E+.E+3.E+4.E+5.E+6.E+7 a T, rad/s Figure 8.8, p. 84 daa from Vinogradov, PS mel Faih A. Morrison, Michigan Tech U.., a. Seady o o o., We have discussed six shear maerial funcions; Now, he equivalen elongaional maerial funcions b. Sress Growh. o c. Sress Relaxaion o d. Creep., o, o., e. Sep Srain f. SAOS o. o cos, o o sin o sin( 44 Faih A. Morrison, Michigan Tech U.
InroToMaFuncions.pdf 4 CM465 Seady Elongaional Flow Maerial Funcions Kinemaics: ( ( b x v ( ( b x ( x 3 3 Maerial Funcions: ( consan Elongaional flow: b=, ( Biaxial sreching: b=, ( Planar elongaion: b=, ( or or B P 33 p Uniaxial or Biaxial or Firs Planar Elongaional Viscosiy Second Planar Elongaional Viscosiy 45 Faih A. Morrison, Michigan Tech U. Wha is he srain in his flow? (o answer, review how srain was developed/defined for previous flows... Faih A. Morrison, Michigan Tech U. 3
InroToMaFuncions.pdf 4 CM465 Recall, for shear... How did we do ha before..? r( Deformaion (srain x ( x( x3( x ( r( x( 3( x 3 This vecor keeps rack of he locaion of a fluid paricle as a funcion of ime. 3 flow paricle pah u(, r( r( x3 (, r( Curren posiion compared o erence posiion P( u x u(, r( Relaive change in displacemen Displacemen funcion Shear srain P( x ( r( x( 3( x u( 3, r( r( x ( ( x ( x ( 3 x ( x 3 3 Displacemen funcion x x Our choice for measuring change in shape: Shear srain ( ( u, x, ( (for seady shear or in unseady shear for shor ime inervals Faih A. Morrison, Michigan Tech U. Pah o srain for shear:,, Try o follow for elongaion. x ( x ( r ( x( r( x(? 3( x 3( x 3 3 u( u, r( r( T u?? Faih A. Morrison, Michigan Tech U. 4
InroToMaFuncions.pdf 4 CM465 v x dx d dx Shear x x d x x x, ( x x, x v 3 dx d dx x x 3 3 3 x ln x Elongaion x d 3 3, 3 3 v consan x Piece of deformaion over ime inerval Δ Noes: The way we quanified deformaion for shear, du /dx, is no so appropriae for elongaion. Velociy gradien consan in boh flows (bu no he same gradien (Velociy gradien ( is a measure of deformaion ha accumulaes linearly wih flow Faih A. Morrison, Michigan Tech U. Press on: srain = velociy gradien d homogeneous flows (velociy gradien he same everywhere in he flow Shear: Elongaion: (, ( d (, ( d Noe: Need a beer definiion of srain for he general case Faih A. Morrison, Michigan Tech U. 5
InroToMaFuncions.pdf 4 CM465 Hencky srain (, ( d (choose = ln l l The srain imposed is proporional o ime. The raio of curren lengh o iniial lengh is exponenial in ime. 5 Faih A. Morrison, Michigan Tech U. Wha does he Newonian Fluid model predic in uniaxial seady elongaional flow? v v T Again, since we know v, we can jus plug i in o he consiuive equaion and calculae he sresses. 5 Faih A. Morrison, Michigan Tech U. 6
InroToMaFuncions.pdf 4 CM465 Seady Sae Elongaion Viscosiy Boh ension hinning and hickening are observed. Figure 6.6, p. 5 Munsed.; PS mel Trouon raio: Tr 53 Faih A. Morrison, Michigan Tech U. Wha does he model we guessed a predic for seady uniaxial elongaional flow? M v v T M M n m c c 54 Faih A. Morrison, Michigan Tech U. 7
InroToMaFuncions.pdf 4 CM465 Wha if we make he following replacemen? v x This a leas can be wrien for any flow and i is equal o he shear rae in shear flow. 55 Faih A. Morrison, Michigan Tech U. M v v T Observaions M M n m c c The model conains parameers ha are specific o shear flow makes i impossible o adap for elongaional or mixed flows Also, he model should only conain quaniies ha are independen of coordinae sysem (i.e. invarian We will ry o salvage he model by replacing he flow-specific kineic parameer wih somehing ha is frame-invarian and no flow-specific. 56 Faih A. Morrison, Michigan Tech U. 8
InroToMaFuncions.pdf 4 CM465 We will ake ou he shear rae and replace wih he magniude of he rae-of-deformaion ensor (which is relaed o he second invarian of ha ensor. M v v T M M m n c c 57 Faih A. Morrison, Michigan Tech U. The oher elongaional experimens are analogous o shear experimens (see ex Elongaional sress growh Elongaional sress cessaion (nearly impossible Elongaional creep Sep elongaional srain Small-ampliude Oscillaory Elongaion (SAOE 58 Faih A. Morrison, Michigan Tech U. 9
InroToMaFuncions.pdf 4 CM465 Sar-up of Seady Elongaion Figure 6.64, p. 8 Kurzbeck e al.; PP Srainhardening Fi o an advanced consiuive equaion ( mode pom-pom model Figure 6.63, p. 7 Inkson e al.; LDPE 59 Faih A. Morrison, Michigan Tech U. Wha s nex? Underlying physics (mass, momenum balances, sress ensor Sandard flows Maerial funcions? Consiuive equaions Model flows/solve engineering problems We wan o design consiuive equaions based on he maerial behavior of real non- Newonian fluids. Wha is he behavior of non-newonian fluids? 6 Faih A. Morrison, Michigan Tech U. 3