Solving Elliptic (and Hyperbolic) Differential Equations in Nonlinear Viscoelasticity Elasticity − → Hyperelasticity − → Viscoelasticity Bhavesh Shrimali Department of Civil and Environmental Engineering CS598APK, Fall 2017 Bhavesh Shrimali (UIUC) Solving Elliptic (and Hyperbolic) Differential Equations in Nonlinear Viscoelasticity CS598APK, Fall 2017 1 / 12
Introduction: Continuum Mechanics Kinematics Deformation mapping ( χ ) and Deformation Gradient ( F ) = ∇ χ ≡ F ij = ∂χ i = ∂ x i F , 1 ≤ i , j ≤ 3 ∃ χ ( X ) ∈ C 2 (Ω 0 ) : ∂ X j ∂ X j J = det F > 0 also u = χ − X = ⇒ F = I + ∇ u It is difficult to analytically determine χ for most BVPs ( Semi-inverse method, Fourier) or ( FEM,BEM !) Newton’s 2 nd Law Stresses (Cauchy and Piola-Kirchoff) � � � ∃ T : t = Tn & b ( x , t ) d x + t ( x , t ) d x = ρ ( x , t ) ¨ χ ( x , t ) d x Ω ∂ Ω Ω Bhavesh Shrimali (UIUC) Solving Elliptic (and Hyperbolic) Differential Equations in Nonlinear Viscoelasticity CS598APK, Fall 2017 2 / 12
� � � � � � � More Continuum Mechanics... A T = A v e B T = B X x ∴ AB = BA 0 v Modeling Viscoelasticity – Two approaches S = J TF − T Hereditary Integrals: Stieltjes Integral Internal variables ( Increasingly popular! ) Two Potential Constitutive Framework: ψ and φ S ( F , F v ) = ∂ψ ∂ F ( F , F v ) Constitutive Model: & Div S + B = 0 (1) ∂ F v + ∂φ ∂ψ � �� � F v = 0 BLM ∂ ˙ Bhavesh Shrimali (UIUC) Solving Elliptic (and Hyperbolic) Differential Equations in Nonlinear Viscoelasticity CS598APK, Fall 2017 3 / 12
BVP Isotropy and Non-negativity ψ ( F , F v ) > 0 ψ ( F , F v ) = ψ ( QFK , F v ) ∀ , Q , K ∈ U A : AA T = A T A = I � � U = Given a free energy function ( ψ ) and dissipation potential ( φ ), a domain (Ω 0 ) with smooth boundary ( ∂ Ω 0 ), choose an internal variable ( F v ) and solve : Div S = 0 for X ∈ Ω 0 (2) ∂ F v + ∂φ ∂ψ F v = 0 at each time step (3) ∂ ˙ In general, the practice is to solve (3) at each time step ( discretization ) and then solve (2) using FEM Bhavesh Shrimali (UIUC) Solving Elliptic (and Hyperbolic) Differential Equations in Nonlinear Viscoelasticity CS598APK, Fall 2017 4 / 12
Hyperelasticity ( φ = 0) For now, consider no dissipation and the following ( ψ ) ( Convex !) ψ = µ 2 ( I 1 − 3) + κ 2 ( J − 1) 2 where I 1 = F · F ≡ F ij F ij (Neo-Hookean) ∂ I 1 = ∂ ∂ F ( F · F ) = 2 F ∂ F ⇒ S = µ F + κ ( J − 1) J F − T ← = − ∂ J = ∂ ∂ F (det F ) = J F − T ∂ F Underlying PDE By balance of linear momentum, we finally get the PDE ⇒ µ ∇ · F + κ J ( J − 1) ∇ · F − T = 0 Div S = 0 = � u = g on ∂ Ω x ⇒ µ ∇ 2 u + κ ∇ ( J ( J − 1)) F − T = 0 0 = with (4) on ∂ Ω t t = h 0 Equation (4) is the Cauchy-Navier equation for Hyperelasticity Bhavesh Shrimali (UIUC) Solving Elliptic (and Hyperbolic) Differential Equations in Nonlinear Viscoelasticity CS598APK, Fall 2017 5 / 12
BVP: Set up Quasi-static deformation of a spherical shell ( R = | X | ) For now, consider ( J > 0), later we will consider ( J = 1) Consider the domain on the left, given by Ω : X ∈ R 3 , A ≤ | X | ≤ B (5) × 10 − 3 1 . 0 � = 10 − 3 A 0 . 5 where (6) = 2 × 10 − 3 0 . 0 B z − 0 . 5 − 1 . 0 Assumption: Radially symmetric − 1 . 5 − 1 . 5 deformation × 10 − 3 − 1 . 0 − 2 . 0 − 0 . 5 Points move radially outward 0 . 0 − 1 . 5 − 1 . 0 − 0 . 5 0 . 0 0 . 5 1 . 0 1 . 5 x 0 . 5 × 10 − 3 1 . 0 Both Dirichilet and Neumann y 1 . 5 No bifurcations (Cavitation!) Bhavesh Shrimali (UIUC) Solving Elliptic (and Hyperbolic) Differential Equations in Nonlinear Viscoelasticity CS598APK, Fall 2017 6 / 12
Radially symmetric mappings Consider deformation mapping ( χ ) of the form � � ⇒ F = ( Rf ′ ( R ) + f ) 1 I − 1 χ = f ( R ) X = R 2 X ⊗ X + f R 2 X ⊗ X (7) � �� � � �� � K 1 K 2 = ⇒ F = λ 1 K 1 + λ 2 K 2 ⇐ ⇒ S = σ 1 K 1 + σ 2 K 2 (8) Matrix Forms The spectral forms of S and F σ 1 0 0 λ 1 0 0 S = 0 σ 2 0 F = 0 λ 2 0 (9) 0 0 σ 2 0 0 λ 2 With some algebra, the BLM reduces to d R + 2 d σ 1 R ( σ 1 − σ 2 ) = 0 with f ( A ) = 1 , f ( B ) = 2 (10) Bhavesh Shrimali (UIUC) Solving Elliptic (and Hyperbolic) Differential Equations in Nonlinear Viscoelasticity CS598APK, Fall 2017 7 / 12
BVP... Finally Therefore, the entire problem reduces to a single nonlinear ODE of the form � µ + κ f 4 � + 2 R κ f 3 f ′ 2 + R � µ + κ f 4 � f ′′ = 0 4 (11) which reduces to � R K ( R , f ′ ( ξ ) , ξ ) F ( f ( ξ )) d ξ = G ( R ) f ( R ) + 2 κ (12) A � R � � R � � � G ( R ) = 1 + c ( R − A ) − 4 A log − 1 + 1 (13) A A K ( R , f ′ ( ξ ) , ξ ) = ( R − ξ ) f ′ 2 ( ξ ) (14) f 3 ( ξ ) F ( f ( ξ )) = (15) µ + κ f 4 ( ξ ) � B � B � � B � � � K ( B , f ′ ( ξ ) , ξ ) F ( f ( ξ )) d ξ (16) c = 1 + 4 A log − 1 + 1 + 2 κ A A A Bhavesh Shrimali (UIUC) Solving Elliptic (and Hyperbolic) Differential Equations in Nonlinear Viscoelasticity CS598APK, Fall 2017 8 / 12
Quadrature − → Nonlinear System Using ideas from Linear IEs Nystr¨ om discretization n-point Gauss-Legendre Quadrature to evaluate the integrals in (12) Nonlinear-Kernel N � ω j K ( R i , f ′ f n ( R i ) = G n ( R i ) − 2 κ n ( ξ j ) , ξ j ) F ( f n ( ξ j )) (17) j =1 Successive approximations N � � � � � f ( k +1) ( R i ) = G ( k ) R i , f ′ ( k ) f ( k ) ( R i ) − 2 κ ω j K ( ξ j ) , ξ j F ( ξ j ) n n n n j =1 (18) Bhavesh Shrimali (UIUC) Solving Elliptic (and Hyperbolic) Differential Equations in Nonlinear Viscoelasticity CS598APK, Fall 2017 9 / 12
Existence (and Uniqueness) Nature of f ′ ( R ) Exact form of the kernel not reported in the literature For equations of the following form � R K ( R , ξ ) ˆ f ( R ) + F ( f ( ξ )) d ξ = G ( R ) A K ( R , ξ ) satisfies the Lipchitz condition ˆ F satisfies Lipchitz condition f ( R ) bounded and integrable G ( R ) bounded and integrable In general for nonlinear equations existence and uniqueness is not straightforward. Linearize (?) Do it for the linear problem Bhavesh Shrimali (UIUC) Solving Elliptic (and Hyperbolic) Differential Equations in Nonlinear Viscoelasticity CS598APK, Fall 2017 10 / 12
Sample Results 4000 2.0 1.8 3000 1.6 2000 1.4 1000 1.2 0.0012 0.0014 0.0016 0.0018 0.0020 0.0012 0.0014 0.0016 0.0018 0.0020 (a) f ′ ( R ) vs R (b) f ( R ) vs R Calculations from FEM → A + The gradient is sharp as R − Need more points to evaluate the integral (?) Bhavesh Shrimali (UIUC) Solving Elliptic (and Hyperbolic) Differential Equations in Nonlinear Viscoelasticity CS598APK, Fall 2017 11 / 12
THANK YOU Bhavesh Shrimali (UIUC) Solving Elliptic (and Hyperbolic) Differential Equations in Nonlinear Viscoelasticity CS598APK, Fall 2017 12 / 12
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