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There Is Way More to This Than We Can Cover Advanced CM4650 4/23/2018 Chapter 9: Advanced Constitutive Models CM4650 Polymer Rheology Professor Faith A. Morrison Department of Chemical Engineering Michigan Technological University 1 © Faith A. Morrison, Michigan Tech U. Chapter 9: Advanced Constitutive Models CM4650 Polymer Rheology WARNING: There is way more to this than we can Professor Faith A. Morrisoncover Department of Chemical Engineering Michigan Technological University 2 © Faith A. Morrison, Michigan Tech U. 1 Advanced CM4650 4/23/2018 Advanced Constitutive Modeling – Chapter 9 Fluids with Memory – Chapter 8 We have learned that the GLVE Summary: Generalized Linear-Viscoelastic Constitutive is a good model for: Equations • Deformations at low rates PRO: •A first constitutive equation with memory • Deformation to low strains •Can match SAOS, step-strain data very well •Captures start-up/cessation effects (linear regime) •Simple to calculate with •Can be used to calculate the LVE spectrum •Fails to predict shear normal stresses CON: The GLVE does not work for the •Fails to predict shear-thinning/thickening nonlinear regime, due to the problem •Only valid at small strains, small rates of lack of frame invariance. GLVE: •Not frame-invariant Fluids with Memory – Chapter 8 Shear flow in a rotating frame of reference y fluid x x, y, z Turntable problem GLVE: x, y, z 3 © Faith A. Morrison, Michigan Tech U. Advanced Constitutive Modeling – Chapter 9 Fluids with Memory – Chapter 8 Strain-related issues? Shear flow in a rotating frame of reference y fluid ⇒ perhaps the problem with the GLVE x model is associated with how strain is x, y, z mathematically described. GLVE: x, y, z The question then becomes, how is strain mathematically described in the GLVE Model? ′ strain rate 4 © Faith A. Morrison, Michigan Tech U. 2 Advanced CM4650 4/23/2018 Advanced Constitutive Modeling – Chapter 9 What is the strain measure that is used in the GLVE model? ′ strain rate (use integration by parts; see hand calculations) 5 © Faith A. Morrison, Michigan Tech U. Advanced Constitutive Modeling – Chapter 9 The infinitesimal Generalized Linear- strain tensor is the Viscoelastic Model: strain measure of the GLVE (strain version) ′ , ≡ ′ memory function Infinitesimal Strain Tensor Turns out: It is the use of the infinitesimal strain , ′′ tensor as the strain measure that causes the frame-variance in the GLVE model. 6 © Faith A. Morrison, Michigan Tech U. 3 Advanced CM4650 4/23/2018 Advanced Constitutive Modeling – Chapter 9 We have seen the infinitesimal strain tensor before: when we first defined strain (when we discussed material functions). Infinitesimal strain tensor ≡ Displacement When formally ut(,)()()ref t rt rt ref developed, , is function related to the displacement function, x (t) 1 , ′ . Particle r(t) x2 (t) tracking vector x (t) 3 123 7 © Faith A. Morrison, Michigan Tech U. What is strain? Summary (from Ch5) Strain is our measure of deformation (change of shape) For shear flow (steady or unsteady): Strain Strain is the , accumulates as integral of strain the flow rate progresses The time The strain rate is derivative of the rate of strain is the strain instantaneous Deformation rate rate shape change Applying this to each Infinitesimal component of and strain tensor ≡ generalizing to all flows: 8 © Faith A. Morrison, Michigan Tech U. 4 Advanced CM4650 4/23/2018 Advanced Constitutive Modeling – Chapter 9 Fluids with Memory – Chapter 8 Shear flow in a rotating frame of reference In addition to the turntable y fluid example, another “flow” we x can use to test the GLVE x, y, z Turntable model is rigid body problem rotation (no strain). GLVE: x, y, z Counter clockwise rigid body rotation (no strain). No strain should produce no stress. 9 © Faith A. Morrison, Michigan Tech U. Advanced Constitutive Modeling – Chapter 9 No stress is generated when a fluid is rotated CCW through (from position at time to position at time ′, what does the GLVE predict? (Warning: later, we are going to consider CCW rotation from ′ to through an angle ; see Table 9.3) t t' y y P(t') r' r cos P(t) r r r sin x x • calculate the infinitesimal strain tensor for rigid body rotation • use the strain-evident version of the GLVE 10 (note: we need , in the GLVE) © Faith A. Morrison, Michigan Tech U. 5 Advanced CM4650 4/23/2018 Advanced Constitutive Modeling – Chapter 9 What does the GLVE Predict for CCW Rigid-Body Rotation around the z-axis from to ? ′ ′ ′ ′ , , ′ (see book for details) 11 © Faith A. Morrison, Michigan Tech U. Advanced Constitutive Modeling – Chapter 9 What does the GLVE Predict for CCW Rigid-Body Rotation around the z-axis from to ? From geometry ̅ sin ̅ cos From trigonometry ′ ̅ sin ̅ sin cos sin cos cos sin ′ ̅ cos ̅ cos cos sin sin cos sin ′ From definitions of and cos sin ̅ ̅ cos sin 0 , ′ 12 © Faith A. Morrison, Michigan Tech U. 6 Advanced CM4650 4/23/2018 Advanced Constitutive Modeling – Chapter 9 GLVE Prediction for CCW Rigid-Body Rotation around the z-axis from to ′: 2 cos 1 0 0 02cos 1 0 ′ 000 13 © Faith A. Morrison, Michigan Tech U. Advanced Constitutive Modeling – Chapter 9 GLVE Prediction for CCW Rigid-Body Rotation around the z-axis from to ′: WRONG 2 cos 1 0 0 02cos 1 0 ′ 000 Stress depends on angle of rotation! Why does GLVE make this erroneous prediction? (t,t) u(t,t) u(t,t) T u(t,t) r(t) r(t) Because this vector, while accounting for deformation, also accounts for changes in orientation. 14 © Faith A. Morrison, Michigan Tech U. 7 Advanced CM4650 4/23/2018 Advanced Constitutive Modeling – Chapter 9 (t,t) u(t,t) u(t,t) T Accounts for changes in shape and orientation. u(t,t) r(t) r(t) 15 © Faith A. Morrison, Michigan Tech U. Advanced Constitutive Modeling – Chapter 9 We desire a strain tensor that accurately captures large-strain deformation without being affected by translation and rotation. time=t’ Consider: What change does the deformation time=t cause in the vector that separates two P Q dr very nearby material elements? P Q dr fixed coordinate system (xyz) 16 © Faith A. Morrison, Michigan Tech U. 8 Advanced CM4650 4/23/2018 Advanced Constitutive Modeling – Chapter 9 Define change-of-shape tensors that rely on How does dr map to dr’ along a particle path? relative location of two nearby particles particle label (reference time t) Particle position at t’ ′ location at time t’ of the particle labeled r is a function of past position, x d r dr r y f (r) z r r xyz dx df dy ? dz xyz 17 © Faith A. Morrison, Michigan Tech U. Advanced Constitutive Modeling – Chapter 9 We can relate ′ to using the chain rule. ′ ′ ,, ′ ? ? ? (see text) ′ ′ ′ 18 © Faith A. Morrison, Michigan Tech U. 9 Advanced CM4650 4/23/2018 Advanced Constitutive Modeling – Chapter 9 x' y' z' Combine answers from 3 directions: x x x x' y' z' dx dy dz dx dy dz xyz xyz y y y x' y' z' z z z xyz dr dr F x' y' z' Deformation-gradient tensor x x x x' y' z' r' r ' F(t,t) i eˆ eˆ p i y y y r rp x' y' z' In Einstein notation: z z z xyz ′ ′̂ , , ′ 19 © Faith A. Morrison, Michigan Tech U. Advanced Constitutive Modeling – Chapter 9 We can calculate as follows: 1 Define: F F I Then use: dr dr F ? ⋅ ⋅ ⋅⋅ ⋅ 20 © Faith A. Morrison, Michigan Tech U. 10 Advanced CM4650 4/23/2018 Advanced Constitutive Modeling – Chapter 9 Deformation-gradient x' y' z' tensor x x x x' y' z' r' r ' dr dr F F(t,t) i eˆ eˆ p i y y y r rp x' y' z' z z z xyz Inverse deformation- x y z gradient tensor x' x' x' 1 1 x y z r r dr dr F F (t',t) m eˆ eˆ j m y' y' y' r' rj ' x y z z' z' z' xyz These strain measures get rid of 21 the translation problem. © Faith A. Morrison, Michigan Tech U. Advanced Constitutive Modeling – Chapter 9 We desire a strain tensor that accurately captures large-strain deformation without being affected by translation and rotation. u These strain measures include translation, deformation and orientation F These strain measures include F1 deformation and orientation We can separate the deformation and orientation information in F 1 and F using a technique called polar decomposition. (en.wikipedia.org/wiki/Polar_decomposition) 22 © Faith A. Morrison, Michigan Tech U. 11 Advanced CM4650 4/23/2018 Advanced Constitutive Modeling – Chapter 9 Polar Decomposition Theorem (en.wikipedia.org/wiki/Polar_decomposition) Any tensor for which an inverse exists has two unique decompositions: ⋅ ⋅ Pure rotation tensor / ⋅ Right stretch tensor Orthogonal tensor / , symmetric, ⋅ Left stretch tensor nonsingular tensors ⋅ ⋅ / ⋅ 23 © Faith A. Morrison, Michigan Tech U. Advanced Constitutive Modeling – Polar Decomposition EXAMPLE: Calculate the right stretch tensor and rotation tensor for a given tensor. Calculate the angle through which rotates the vector u. 1 0 2 1 A 0 3 2 u 2 2 0 0 1 xyz xyz ⋅ u w ⋅ ⋅ ⋅ A U Pure All the stretch; R v rotation some of the rotation 24 © Faith A. Morrison, Michigan Tech U. 12 Advanced CM4650 4/23/2018 Advanced Constitutive Modeling – Polar Decomposition We have partially isolated the effect of rotation through polar decomposition. pure rotation tensor left stretch tensor A R U V R right stretch tensor original (strain) tensor We can further isolate stretch from rotation by considering the eigenvectors of and . 25 © Faith A. Morrison, Michigan Tech U. Advanced Constitutive Modeling – Polar Decomposition ⋅ ⋅ U ˆk kˆk eigenvectors V ˆj jˆj eigenvalues Physical Interpretation ˆ ⋅ i PATH I ⋅ ⋅ ⋅ ⋅ U R ⋅ (stretch first; R then rotate, or ˆ Pure i the reverse) Pure stretch rotation ˆ ˆ V i i PATH II 26 © Faith A.
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