Implicitly constituted materials with fading memory V t Pr u sa - PowerPoint PPT Presentation

Implicitly constituted materials with fading memory V´ ıt Pr˚ uˇ sa prusv@karlin.mff.cuni.cz Mathematical Institute, Charles University 31 March 2012

Incompressible simple fluid T = − p I + F + ∞ C t ( t − s )) Truesdell and Noll (1965): s =0 (

T = − p I + f ( A 1 , A 2 , A 3 , . . . ) A 1 = 2 D Differential type models A n = d A n − 1 A n − 1 L + L ⊤ A n − 1 General form, Rivlin and Ericksen (1955): where T = − p I + F + ∞ C t ( t − s )) + d t Coleman and Noll (1960): These models can be understood as successive approximations of the history functional s =0 ( with “fading memory”.

Pressure dependent viscosity T = − p I + 2 µ ( p ) D

T = − p I + µ ( T ) D T δ | T ) = µ ∞ + ( µ 0 − µ ∞ ) e − | Stress dependent viscosity T ) = T δ | 2 + τ 2 Seely (1964): T ) = µ ( τ 0 T δ | n − 1 Blatter (1995): A µ ( � n − 1 � 2 | 0 Matsuhisa and Bird (1965): µ 0 µ ( 1 + α |

Implicit constitutive relation T , D ) = 0 Rajagopal (2003, 2006); Rajagopal and Srinivasa (2008): f (

T = − π I + S S + λ 1 S + λ 3 DS + SD ) + λ 5 S ) D + λ 6 2 ( S : D ) I D + λ 2 D + λ 4 D 2 + λ 7 2 ( D : D ) I Rate type models Oldroyd (1958): S + λ S + λξ 2 ( DS + SD ) = − µ D S ▽ 2 ( 2 (Tr � � ▽ = − µ d b ♭ b ♭ = def b ♭ − b ♭ [ ∇ v ] ⊤ Phan Thien (1978): ▽ Y Y = e − ε λ µ Tr Notation: ▽ d t − [ ∇ v ]

T , D ) = 0 T ( t − s ) , C t ( t − s )) = 0 Materials with fading memory Implicit algebraic relation: f ( Implicit relation between the histories: H + ∞ s =0 ( Questions: ◮ Are rate type and differential type models (and other known models) special instances or approximations of the material with fading memory? ◮ Is something like the celebrated retardation theorem by Coleman and Noll (1960) available for implicit type materials with fading memory?

Independent variables [. . . ] the properties of a material element may depend upon the previous rheological states through which that element has passed, but not in any way on the states of neighbouring elements and not on the motion of the element as a whole in the space. [. . . ] only those tensor quantities need to be considered which have a significance for the material element independent of its motion as a whole in space. Oldroyd (1950)

Convected coordinate system χ e ˆ z x = χ ( X , t ) e ˆ y ξ 3 e ˆ x τ = t X ξ 2 ξ 1 τ = t − s

Plan ◮ Formulate the constitutive relation in the convected coordinate system , Hencky (1925), Oldroyd (1950). ◮ Formulate the constitutive relation in an implicit form, Rajagopal (2003). ◮ Expand the functional using an analogue of the retardation theorem , Coleman and Noll (1960). ◮ Use representation theorems for isotropic linear and bilinear functions, Truesdell and Noll (1965). ◮ Transform the constitutive relation to a fixed-in-space coordinate system , Oldroyd (1950).

s =0 ( � ( ξ, t − s ) , � ( ξ, t − s )) = 0 , Constitutive assumptions General relation: H + ∞

s =0 ( � ( ξ, t − s ) , � ( ξ, t − s )) = 0 , Constitutive assumptions � = − π I + � 0 = G + ∞ s =0 ( � ( ξ, t − s ) − I , � ( ξ, t − s )) General relation: H + ∞ Special form of the general constitutive relation:

s =0 ( � ( ξ, t − s ) , � ( ξ, t − s )) = 0 , Constitutive assumptions � = − π I + � 0 = G + ∞ s =0 ( � ( ξ, t − s ) − I , � ( ξ, t − s )) General relation: H + ∞ � � ( ξ, t − s ) � S = O ( � � ( ξ, t − s ) − I � Γ ) as � � ( ξ, t − s ) − I � Γ → 0+ , Special form of the general constitutive relation: Constitutive assumption:

Norm � � � L 2 � ( s ) | 2 h ( s ) d s [...] the deformations that occurred in the distant past � , � ] ⊤ � � � � 2 h + � � � 2 should have less influence in determining the present stress than those that occurred in the recent past. Truesdell and Noll (1965) �� + ∞ � 1 2 h = def | s =0 � � 1 � 2 � � � [ = def L 2 L 2 � L 2 h × L 2 h h

� = − π I + � 0 = G + ∞ s =0 ( � ( ξ, t − s ) − I , � ( ξ, t − s )) � ( ξ, t − s ) − I � ( ξ, t − s ) − I Taylor series for the functional � ( ξ, t − s ) � ( ξ, t − s ) 0 0 0 � ( ξ, t − s ) − I 0 � ( ξ, t − s ) �� � �� �� �� � 2 � � G + ∞ 0 � ( ξ, t − s ) − I = A + B + C + o � � � ( ξ, t − s ) − I , � ( ξ, t − s ) s =0 � � 0 � ( ξ, t − s ) L 2 h × L 2 h �� �� A = G + ∞ s =0 �� �� � � B = δ G + ∞ s =0 �� �� � � � ⊤ δ 2 G + ∞ � C = s =0

� ( x , t − s ) Slow history t − s t − α ¯ s t

� ( ξ, t − α ¯ I � ( ξ, t − α ¯ � � � ( ξ, t ) − I � � � ( ξ, t ) Taylor series for the metric tensor–stress tensor history � � 0 � � � ( ξ, t ) Informal expansion: � � s ) − s ) � � � d � � � d 2 + 1 d t ( ξ, t ) d t 2 ( ξ, t ) � α 2 � 2 α 2 ¯ s 2 = − α ¯ s + o � ( ξ, t − α ¯ I d d 2 d t ( ξ, t ) d t 2 ( ξ, t ) � ( ξ, t − α ¯ � � � d � � � d 2 s + 1 d t ( ξ, t ) d t 2 ( ξ, t ) s 2 + o � α 2 � 2 α 2 = − α ¯ ¯ d 2 d d t ( ξ, t ) d t 2 ( ξ, t ) � �� � � �� � � �� � (0) (1) (2) g g g Rigorous result: � �� � � � 1 s + 1 s ) − � ( 0 ) ( 1 ) 2 α 2 ( 2 ) � s 2 lim − g − α g ¯ g ¯ = 0 � � α 2 s ) � � α → 0+ L 2 h × L 2 h

� � ( ξ, t − s ) � S = O ( � � ( ξ, t − s ) − I � Γ ) as � � ( ξ, t − s ) − I � Γ → 0+ , Approximation for slow histories � ( ξ, t − α ¯ I � ( ξ, t − α ¯ � ( ξ, t ) Use constitutive assumption � ( ξ, t − α ¯ I � ( ξ, t − α ¯ � ( ξ, t ) and substitute to the Taylor formula for the functional. First order: �� �� s ) − � ( � � ( � 0 ) 1 ) G + ∞ = f 0 + f 1 + o ( α ) g g s =0 s ) − Second order: �� �� � ( � � ( � � ( � s ) − 0 ) 1 ) 2 ) G + ∞ = f 0 + f 1 + f 2 g g g s =0 s ) − � ( � � ( � � ( � � ( � 0 ) ( 0 ) 1 ) ( 0 ) 0 ) ( 1 ) 1 ) ( 1 ) � α 2 � + g 00 g , + g 10 g , + g 01 g , + g 11 g , + o g g g g

A ) = a 1 (Tr A ) I + a 2 A Linear and bilinear tensor functions h ( A , B ) = ( c 1 Tr A Tr B + c 2 Tr ( AB )) I A ) B + c 4 (Tr B ) A + c 5 ( AB + BA ) Representation for isotropic linear functions: h ( Representation for isotropic bilinear functions: + c 3 (Tr

Time derivatives with respect to fixed-in-space coordinate system Oldroyd (1950): � ∂ b k � d b k i i + v m b k − v k | m b m i + b k m v m | i , = def i | m d t ∂ t � ∂ b ki � d b ki + v m b ki | m + v m | k b mi + b km v m | i , = def d t ∂ t � ∂ b ki � d b ki + v m b ki | m − v k | m b mi − b km v i | m , = def d t ∂ t

d t = d b b b + b [ ∇ v ] , Time derivatives with respect to fixed-in-space coordinate d t = d b ♯ b ♯ system b ♯ + b ♯ [ ∇ v ] , d t = d b ♭ b ♭ b ♭ − b ♭ [ ∇ v ] ⊤ , d d t − [ ∇ v ] d d t + [ ∇ v ] ⊤ d d t − [ ∇ v ]

� ( ξ, t ) �→ S ( x , t ) d � D ( x , t ) Identification of the derivatives � D ( x , t ) d � S ( x , t ) � S ( x , t ) d t ( ξ, t ) �→ d 2 ▽ d t 2 ( ξ, t ) �→ ▽ d t ( ξ, t ) �→ d 2 ▽ ▽ d t 2 ( ξ, t ) �→

� ( ξ, t − α ¯ I Approximation formulae � ( ξ, t − α ¯ � ( ξ, t ) S ) I + b 1 S + 2 b 3 D + b 4 S I + b 5 S + o ( α ) First order: �� �� s ) − G + ∞ s =0 s ) − � � ▽ ▽ �→ b 0 (Tr Tr

� ( ξ, t − α ¯ I � ( ξ, t − α ¯ � ( ξ, t ) S ) + b 4 S S + ( b 15 − 2 b 8 ) Tr ( D ) 2 S ) 2 + b 11 Tr ( S ) 2 � S S Approximation formulae Second order: D S S Tr S + b 28 Tr S S + b 33 Tr ( SD ) I �� �� S ) + b 30 s ) − S S + S S ) D G + ∞ s =0 s ) − S ) 2 + b 17 ( D ) 2 + b 36 ( SD + DS ) � � � � � ▽ ▽ ▽ �→ b 0 (Tr Tr + b 6 Tr D + S S ) S + b 21 S � � 2 � � � � 2 � ▽ ▽ + b 10 (Tr + b 18 Tr + b 19 Tr D S + SD S S + SS S + o � � � � �� � ▽ ▽ ▽ + b 23 Tr + b 27 Tr � � �� � � � � ▽ ▽ + b 1 + b 12 (Tr Tr b 3 + b 25 Tr + b 34 (Tr + b 13 ( � � � � ▽ � ▽ � 2 ▽ ▽ + b 9 b 5 + b 20 Tr + b 29 (Tr � � � � ▽ ▽ ▽ ▽ ▽ ▽ � α 2 � + b 26 + b 31 + b 7

T = − π I + S S + λ 1 S + λ 3 DS + SD ) + λ 5 S ) D + λ 6 2 ( S : D ) I D + λ 2 D + λ 4 D 2 + λ 7 2 ( D : D ) I Rate type models Oldroyd (1958): S + λ S + λξ 2 ( DS + SD ) = − µ D S ▽ 2 ( 2 (Tr � � ▽ = − µ d b ♭ b ♭ = def b ♭ − b ♭ [ ∇ v ] ⊤ Phan Thien (1978): ▽ Y Y = e − ε λ µ Tr Notation: ▽ d t − [ ∇ v ]