Role of Material Functions in Rheological Analysis. Role of Material Functions in Rheological Analysis
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1 Role of Maerial Funcions in Rheological Analysis QUALIY CONROL compare wih oher in-house daa on qualiaie basis unknown maerial QUALIAIVE ANALYSIS compare daa wih lieraure repors on arious fluids conclude wheher or no a maerial is appropriae for a specific applicaion measure maerial funcions, e.g. η, G'(ω, G"(w, G( conclude on he probable physical behaior of he fluid based on comparison wih known fluid behaior MODELING WORK compare measured wih prediced conclude which consiuie equaion is bes for furher modeling calculaions calculae predicions of maerial funcions from arious consiuie equaions Role of Maerial Funcions in Rheological Analysis QUALIY CONROL compare wih oher in-house daa on qualiaie basis unknown maerial QUALIAIVE ANALYSIS compare daa wih lieraure repors on arious fluids conclude wheher or no a maerial is appropriae for a specific applicaion measure maerial funcions, e.g. η, G'(ω, G"(w, G( conclude on he probable physical behaior of he fluid based on comparison wih known fluid behaior We will focus here firs MODELING WORK compare measured wih prediced conclude which consiuie equaion is bes for furher modeling calculaions calculae predicions of maerial funcions from arious consiuie equaions
2 Maerial funcion definiions. Choice of flow (shear or elongaion kinemaics ς ( x 3. Choice of deails of ε( ( + b x ε ( ( b x ε ( x 3 3 ς ( or ε(. Elongaional flow: b, Biaxial sreching: b, Planar elongaion: b, ε ( > ε ( < ε ( > 3. Maerial funcions definiions: will be based on τ, N N in shear or, in elongaional flows. τ 33 τ, τ τ Kinemaics: ς ( x Seady Shear Flow Maerial Funcions 3 ς ( consan Maerial Funcions: τ η Viscosiy Firs normal-sress coefficien Second normalsress coefficien Ψ Ψ ( τ τ ( τ τ 33
3 How do we predic maerial funcions? ANSWER: From he consiuie equaion. τ f ( Wha does he Newonian Fluid model predic in seady shearing? τ µ µ [ + ( ] Wha do we measure for hese maerial funcions?
4 Seady shear iscosiy and firs normal sress coefficien.e+5 o Ψ η, Poise.E+4 η o Ψ, (dyn/cm s.e+3 Ψ ( η (.E+.. Figure 6., p. 7 Menzes and, s Graessley conc. PB soluion SOR Shor Course Beginning Rheology Seady shear iscosiy and firs normal sress coefficien.e+6.e+5 η.e+9.e+8.e+4.e+7 83 kg/mol 57 kg/mol 35 kg/mol kg/mol η, Poise.E+3.E+.E+6.E+5 Ψ, dynes/cm.e+.e+4 Ψ.E+.E+3 Figure 6., p. 7 Menzes and Graessley conc. PB soluion; c.676 g/cm 3.E-.E+.., s SOR Shor Course Beginning Rheology
5 Seady shear iscosiy for linear and branched PDMS + linear 3 kg/mole branched 56 kg/mole linear 48 kg/mol branched 48 kg/mol Figure 6.3, p. 7 Piau e al., linear and branched PDMS SOR Shor Course Beginning Rheology Wha hae maerial funcions augh us so far? Newonian consiuie equaion is inadequae. Predics consan shear iscosiy (no always rue. Predics no shear normal sresses (hese sresses are generaed for many fluids Behaior depends on he maerial (chemical srucure, molecular weigh, concenraion
6 Can we fix he Newonian Consiuie Equaion? τ µ [ + ( ] Le s replace µ wih a funcion of shear rae because we wan o predic a non-consan iscosiy in shear τ M [ ] ( + ( Wha does his model predic for seady shear iscosiy? τ M [ ] ( + ( Answer: η M (
7 If we choose: ( < c n c m M M logη log slope (n- log c Problem soled! Bu wha abou he normal sresses? ( ( [ ] M + τ 3 3 I appears ha should no be simply proporional o τ ry somehing else... ( ( [ ] ( L C B A f f I τ τ µ τ ( (
8 Bu which ones? o sor ou how o fix he Newonian equaion, we need more obseraions (o gie us ideas. Le s ry anoher maerial funcion ha s no a seady flow (bu sick o shear. Sar-up of Seady Shear Flow Maerial Funcions Kinemaics: ς ( x 3 ς ( < Maerial Funcions: + τ η ( Shear sress growh funcion Firs normal-sress growh funcion Second normalsress growh funcion Ψ + Ψ + ( τ τ ( τ τ 33
9 Wha does he Newonian Fluid model predic in sar-up of seady shearing? τ µ µ [ + ( ] Again, since we know, we can jus plug i in and calculae he sresses. Maerial funcions prediced for sar-up of seady shearing of a Newonian fluid + η ( µ < + η ( µ ( τ τ Ψ + ( τ τ 33 Ψ + Do hese predicions mach obseraions?
10 Sarup of Seady Shearing ς ( x 3 ς ( < + τ η ( Ψ + ( τ τ Figures 6.49, 6.5, p. 8 Menezes and Graessley, PB soln SOR Shor Course Beginning Rheology Wha abou oher non-seady flows?
11 Cessaion of Seady Shear Flow Maerial Funcions Kinemaics: ς ( x 3 ς ( < Maerial Funcions: τ η ( Shear sress decay funcion Firs normal-sress decay funcion Second normalsress decay funcion Ψ Ψ ( τ τ ( τ τ 33 Cessaion of Seady Shearing ς ( x 3 ς ( < τ η ( Figures 6.5, 6.5, p. 9 Menezes and Graessley, PB soln Ψ ( τ τ SOR Shor Course Beginning Rheology
12 Wha does he model we guessed a predic for sar-up and cessaion of shear? τ M [ ] ( + ( M ( M n m < c c τ M [ ] ( + ( Obseraions M ( M n m < c c he model predics an insananeous sress response, and his is no wha is obsered for polymers he prediced unseady maerial funcions depend on he shear rae, which is obsered for polymers η + η + (, No normal sresses are prediced Progress here
13 τ M [ ] ( + ( Obseraions M ( M n m < c c he model predics an insananeous sress response, and his is no wha is obsered for polymers Lacks memory he prediced unseady maerial funcions depend on he shear rae, which is obsered for polymers η η + (, No normal sresses are prediced + Progress here Relaed o nonlineariies o proceed o beer-designed consiuie equaions, we need o know more abou maerial behaior, 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
14 Summary of shear rae kinemaics (par. ς ( (, τ ( a. Seady o o τ o. ς ( (, τ ( b. Sress Growh o o. ς ( (, τ ( c. Sress Relaxaion o he nex hree families of maerial funcions incorporae he concep of srain.
15 Summary of shear rae kinemaics (par. ( (, τ ( d. Creep τ o. ς ( (, τ ( e. Sep Srain o o ε f. SAOS. ς ε ( o cosω ε (, o sinω ( τ ε τ o sin( ω + δ δ Shear Creep Flow Consan shear sress imposed samples oen MASS
16 Because shear rae is no prescribed, i becomes somehing we mus measure. Kinemaics: ( x Maerial Funcions: Creep Shear Flow Maerial Funcions 3 I is unusual o prescribe sress raher han. ς ( τ ( τ < 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. Deformaion (srain r( x( x( x3( x( r( x( 3( x 3 3 ( u( flow paricle pah, r( r( x 3 u, x r( P( u(, r( Shear srain Displacemen funcion P( x x
17 Physical inerpreaion of srain in shear fluid paricle a ime (, u x u ( P P x u ( x P x x u ( P fluid paricle a ime he srain is he inerse of he slope of he side of he deformed paricle. he srain is relaed o he change of shape of he deformed paricle. Deformaion in shear flow (srain x( r( x( x3( x ( r( x( 3( x u( 3 3, r( r( x ( 3 + ( x ( x ( ( (, x x u x 3 3 Shear srain Displacemen funcion
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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
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