Nonlinear elasticity and gels

Nonlinear elasticity and gels M. Carme Calderer School of Mathematics University of Minnesota New Mexico Analysis Seminar New Mexico State University April 4-6, 2008 1 / 23

Outline Balance laws for gels Free energy: elastic plus mixing Constrained elasticity Deformable porous media Applications 2 / 23

Gels We model a gel as incompressible, immiscible mixture of polymer and solvent. immisc Component 1: polymer; Component 2: solvent Ω 0 reference configuration of the gel; X Ω 0 Ω t domain occupied by the gel at time t > 0; x Ω t 3 / 23

Gels We model a gel as incompressible, immiscible mixture of polymer and solvent. immisc Component 1: polymer; Component 2: solvent Ω 0 reference configuration of the gel; X Ω 0 Ω t domain occupied by the gel at time t > 0; x Ω t φ i v i T i i volume fraction velocity field Cauchy stress tensor friction force Φ polymer deformation map:x = Φ(X, t) F polymer deformation gradient; F = X Φ, det F > 0 φ = φ(x, t), v = v(x, t)... For a survey on gels, see [Tanaka, 1981]; theory of mixtures, [Truesdell, 1984]; model, [Calderer-Chabaud, 2008] and [Calderer-Zhang, 2008] 3 / 23

Balance laws ρ 1 t + (v 1 )ρ 1 + ρ 1 v 1 = 0 ρ 2 t + (v 2 )ρ 2 + ρ 2 v 2 = 0 ρ 1 v 1 t + ρ 1(v 1 )v 1 = T 1 + f 1 ρ 2 v 2 t + ρ 2(v )v 2 = T 2 + f 2 φ 1 + φ 2 = 1 Add up equations of balance of mass: div(φ 1 v 1 + φ 2 v 2 ) = 0 Lagrangian form of balance of mass: φ det F = φ 0 5 / 23

Free energy Elastic stored energy function (per unit reference volume) Flory-Huggins mixing energy (per unit deformed volume) µ(φ 1 )W (F ) h(φ 1, φ 2 ) 6 / 23

Free energy Elastic stored energy function (per unit reference volume) µ(φ 1 )W (F ) Flory-Huggins mixing energy (per unit deformed h(φ 1, φ 2 ) volume) Total energy: E = {µ(φ 1 )W (F ) + det F h(φ 1, φ 2 )} dx Ω 0 Ψ(F, φ) := µ(φ)w (F ) + det F h(φ, 1 φ), φ := φ 1 6 / 23

Elastic and Flory-Huggins free energies Prototype of Flory-Huggins energy: h(φ 1, φ 2 ) = aφ 1 log φ 1 + bφ 2 log φ 2 + χφ 1 φ 2 7 / 23

Elastic and Flory-Huggins free energies Prototype of Flory-Huggins energy: h(φ 1, φ 2 ) = aφ 1 log φ 1 + bφ 2 log φ 2 + χφ 1 φ 2 Isotropic elasticity: W (F ) = µ(φ)w(i 1, I 2, I 3 ) + B(φ)((det F ) k (det F ) k ), k > 0, {I 1, I 2, I 3 } principal invariants of C = F T F solid limit: lim φ 1 µ(φ) = µ 0, shear modulus; fluid limit: lim φ 0 µ(φ) = 0 0 W (F ) K F T F β, det F > 0 Example: neo-heokean elasticity W (F ) = tr (FF T ). from statistical mechanics Derived 7 / 23

Shape of free energy function with respect to χ (1) Swollen (φ 0.3) 8 / 23

Shape of free energy function with respect to χ (1) Swollen (φ 0.3) (2) Swollen and collapsed 8 / 23

Shape of free energy function with respect to χ (1) Swollen (φ 0.3) (3) Swollen and collapsed (2) Swollen and collapsed 8 / 23

Shape of free energy function with respect to χ (1) Swollen (φ 0.3) (3) Swollen and collapsed (2) Swollen and collapsed (4) Collapsed (φ 0.7) 8 / 23

Boundary conditions Let Ω = Γ 1 Γ 2, Γ 1 Γ 2 = 0 Elasticity 1. Displacement: Φ = Φ 0, on Γ 1 2. Traction: (T 1 + T 2 )ν = t 0, on Γ 2 Permeability of membrane φ 1. impermeable: ν = 0 on Ω (or part of it) 2. fully permeable: φ 2 p + Π 2 (φ 1, φ 2 ) = P 0, P 0 pressure of surrounding solvent Π 2 osmotic pressure of in-gel solvent 3. semi-permeable: P ( p + Π 2 (x, t) ) = κ(v 2 v 1 ) ν, κ > 0 permeability constant 9 / 23

Equilibrium states: convex mixing energy X 0 = {u : u u 0 + W 1,2β 0, det F > 0 a.e.} X Γ = {u : u u 0 + W 1,2β Γ, det F > 0 a.e.} W 1,2β Γ = {u W 1,2β, u = 0 on Γ Ω 0 } Minimize E = {µ(φ 1 )W (F ) + det F h(φ 1, φ 2 )} dx Ω 0 subject to φ det F = φ 0, 0 < φ 0 < 1, u X 0 X Γ 10 / 23

Equilibrium states: convex mixing energy X 0 = {u : u u 0 + W 1,2β 0, det F > 0 a.e.} X Γ = {u : u u 0 + W 1,2β Γ, det F > 0 a.e.} W 1,2β Γ = {u W 1,2β, u = 0 on Γ Ω 0 } Minimize E = {µ(φ 1 )W (F ) + det F h(φ 1, φ 2 )} dx Ω 0 subject to φ det F = φ 0, 0 < φ 0 < 1, u X 0 X Γ Theorem (Zhang-2007) Let Ω 0 be bounded and with Lipschitz boundary. Let β > 3 2. Suppose that g(s) = sh( 1 s, 1 1 s ) is a convex monotonically decreasing function of s. Assume that W (F ) is polyconvex. Then there exists at least one minimizer of E in X 0 and in X Γ. Existence theorems in nonlinear elasticity, see [Ball, 1977] and [Ciarlet, 1987] 10 / 23

Nonconvex free energy Suppose that h is nonconvex with respect to φ. 11 / 23

Nonconvex free energy Suppose that h is nonconvex with respect to φ. Modify the energy to include φ 2, and keep balance of mass constraint. X = {(u, φ) : φ W 1,2, u u 0 + W 1, 0, φ det F = φ 0, a.e 0 < φ < 1, u L < C < } Minimize (u,φ) X E = + {µ(φ 1 )W (F ) + det F h(φ 1, φ 2 } dx Ω 0 δ φ 2 dx Ω Ω δ φ 2 dx = (det( u)) 1 2 X adj ( u) 2 Ω 0 11 / 23

Existence theorem Theorem (Zhang, 2007) Let β > 0. Then for every C > 0 there exists a minimizer of the regularized energy in X. Proof: 1. u L < C implies det u 9C 3 2. There is a minimizing sequence {φ h, u h } X 3. Poincare inequality allows us to extract a subsequence (same label) u ū weak* in W 1, 4. 0 < φ h < 1, det u h > 1 and φ h > 1 9C 3 5. Obtain bound for R Ω 0 X φ h 2 6. u h ū weak* in W 1, and φ h φ weakly in W 1,2 7. Show that { φ, ū} X. Use the weak continuity of determinants 8. Proof of weak lower semicontinuity of last term in energy analogous to the case of liqud crystal elastomers [Calderer-Liu-Yan, 2006; 2008] 12 / 23

Mechanical dissipation and constitutive equations Postulate Second Law of Thermodynamics in form of Clausius-Duhem inequality (isothermal case): a tr(t T a v a ) φ a ψ a f a v a 0. 13 / 23

Mechanical dissipation and constitutive equations Postulate Second Law of Thermodynamics in form of Clausius-Duhem inequality (isothermal case): a Reversible components of the stress tr(t T a v a ) φ a ψ a f a v a 0. T r 1 = φ 1 Ψ F F T ( φ 1 p + π 1 ) I T r 2 = ( φ 2 p + π 2 ) I π i = h(φ 1,φ 2 ) φ i : osmotic pressures T i = Ti r + η i 2 ( v i + v T i ), f 1 = φ 1 p + β(v 1 v 2 ) = f 2 η i > 0 represents Newtonian viscosity 13 / 23

Energy relation Theorem (Calderer-Zhang, 2008). Let {φ i, v i, p} be a smooth solution of the governing equations. Then it satisfies the following equation of balance of energy: d [( φ 1 dt Ω(t) 2 v 1 2 + φ 2 2 v 2 2 ) + Ψ] dx (t 1 v 1 + t 2 v 2 ) ds 0, Ω(t) 14 / 23

Energy relation Theorem (Calderer-Zhang, 2008). Let {φ i, v i, p} be a smooth solution of the governing equations. Then it satisfies the following equation of balance of energy: d [( φ 1 dt Ω(t) 2 v 1 2 + φ 2 2 v 2 2 ) + Ψ] dx (t 1 v 1 + t 2 v 2 ) ds 0, Ω(t) It is a consequence of the constitutive equations satisfying the second law of thermodynamics Applying the divergence theorem to the terms of the surface terms, we obtain an energy inequality used in proving weak solutions 14 / 23

Governing system revisited ρ 1 t + (v 1 )ρ 1 + ρ 1 v 1 = 0 ρ 2 t + (v 2 )ρ 2 + ρ 2 v 2 = 0 ρ 1 v 1 t + ρ 1(v 1 )v 1 = T 1 + f 1 ρ 2 v 2 t + ρ 2(v )v 2 = T 2 + f 2 φ 1 + φ 2 = 1 15 / 23

Governing system revisited ρ 1 t + (v 1 )ρ 1 + ρ 1 v 1 = 0 ρ 2 t + (v 2 )ρ 2 + ρ 2 v 2 = 0 ρ 1 v 1 t + ρ 1(v 1 )v 1 = T 1 + f 1 ρ 2 v 2 t + ρ 2(v )v 2 = T 2 + f 2 φ 1 + φ 2 = 1 F t + (v 1 )F = ( v 1 )F 15 / 23

Governing system revisited ρ 1 t + (v 1 )ρ 1 + ρ 1 v 1 = 0 ρ 2 t + (v 2 )ρ 2 + ρ 2 v 2 = 0 ρ 1 v 1 t + ρ 1(v 1 )v 1 = T 1 + f 1 ρ 2 v 2 t + ρ 2(v )v 2 = T 2 + f 2 φ 1 + φ 2 = 1 F t + (v 1 )F = ( v 1 )F Difficulty in proving existence due to the last equation [Liu-Walkington, 2001]; it is a conservation law 15 / 23

Governing system revisited ρ 1 t + (v 1 )ρ 1 + ρ 1 v 1 = 0 ρ 2 t + (v 2 )ρ 2 + ρ 2 v 2 = 0 ρ 1 v 1 t + ρ 1(v 1 )v 1 = T 1 + f 1 ρ 2 v 2 t + ρ 2(v )v 2 = T 2 + f 2 φ 1 + φ 2 = 1 F t + (v 1 )F = ( v 1 )F Difficulty in proving existence due to the last equation [Liu-Walkington, 2001]; it is a conservation law It becomes simple if v 1 = 0, not the case here 15 / 23

Evolution equation for the deformation gradient Take divergence of equation: F is,it + v k F is,ik + v k,i F is,k = v i,ij F js + v i,j F js,i If v = 0, it reduces to div(f T ) t + (v ) div(f T ) = 0 Prescribe appropriate initial and boundary values so that div(f T ) = 0 for all time. 16 / 23

Special class of problems: Linearized elasticity 1 φ 1 v 1,t = div T r (F, φ) φ 1 p + η 1 v 1 + β(v 1 v 2 ) φ 2 (v 2,t + (v 2 )v 2 ) = φ 2 (p + π 2 ) +η 1 v 2 + β(v 2 v 1 ) φ 1 = φ 0 (1 tr ( u)) div(φ 1 v 1 + φ 2 v 2 ) = 0 F T F = I + ( u + u T ) + o( u ), v 1 = u t u denotes displacement vector u = x X Coupling of dissipative linear elasticity equations for the solid with Navier-Stokes equations for the polymer; note that the constraint is not the standard one For experimental and modeling references on mechanics of gels, see, [Tanaka-Filmore, 1979], [Doi-Yamaue, 2004, a, b] 1 MCC, Micek, Rognes; work in progress 17 / 23

Special class of problems: Deformable porous media 2 In applications η 1 >> η 2, polymer dissipation much larger than solvent s and also greater than elastic effects φ 0 ((K 2 3 G)( u)i + 2GE) = (p + Π 1 ) ( K B T N 2 V m ((1 φ 0 ) + φ 0 u) + (1 φ 0 )p ) = β(v 1 v 2 ), (φ 0 v 1 + (1 φ 0 )v 2 ) = 0, v 1 = u t. Coupling of steady state elasticity with Stokes problem for fluids, although constraint is non-standard. K, G are elastic moduli The second equation corresponds to Darcy s law Mathematical analogs found in geology in dealing with soil media and clays [Bennethum-Murad-Cushman, 2000] 2 MCC, Chabaud, Luo; work in progress 18 / 23

Special problems Application of finite element analysis to nonlinear problem (Rognes, Micek, MCC; work in progress) Analysis of nonlinear problem in one-dimensional geometry (Ming Chen, MCC; work in progress) 19 / 23

References I [Ball-1977]. J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal., 63, 337-403, 1977. [Bennethum-Murad-Cushman, 2000], Macroscale Thermodynamics and the Chemical Potential of Swelling Porous Media, Transport in Porous Media, 39, 187-225, 2000. [Calderer-Liu-Yan, 2006] M.C. Calderer, C. Liu and B. Yan, A model for total energy of nematic elastomers with non-uniform prolate spheroid s, Advances in applied and computational mathematics, 245 259, Nova Sci. Publ., Hauppauge, NY, 2006. [Calderer-Liu-Yan, 2008]. M.C. Calderer, C. Liu and B. Yan, A Mathematical Theory for Nematic Elastomers with Non-uniform Prolate Spheroids, submitted, 2008. [Calderer-Zhang, 2008]. M.C. Calderer and Hang Zhang, Incipient dynamics of swelling of gels, SIAM J. Appl. Math., in press; IMA preprint no. 2188, February 2008; http://www.ima.umn.edu/preprints/feb2008/feb2008.html 20 / 23

References II [Calderer-Chabaud, 2008]. M.C. Calderer, Brandon Chabaud, Suping Lyu and Hang Zhang, Modeling approaches to the dynamics of hydrogel swelling, submitted, IMA preprint no. 2189, February 2008; http://www.ima.umn.edu/preprints/feb2008/feb2008.html [Ciarlet-1987]. P.G. Ciarlet, Mathematical Elasticity, Vol 1, North-Holland, 1987. [Doi-Yamaue, 2004]. T. Yamaue and M. Doi, Swelling dynamics of constrained thin-plate gels under an external force, Phys. Rev. E, 70, 011401, 2004. [Doi-Yamaue, 2004]. T. Yamaue and M. Doi, Swelling dynamics of constrained thin-plate gels under an external force, Phys. Rev. E, 70, 011401, 2004. [Doi-Yamaue, 2004]. T. Yamaue and M. Doi, Swelling dynamics of constrained thin-plate gels under an external force, Phys. Rev. E, 70, 011401, 2004. 21 / 23

References III [Doi-Yamaue, 2004a]. T. Yamaue and M. Doi, Theory of one-dimensional swelling dynamics of polymer gels under mechanical constraint, Phys. Rev. E, 69, 011402, 2004. [Liu-Walkington, 2001]. C. Liu and N. J. Walkington, An Eulerian Description of Fluids Containing Visco-hyperelastic Particles, Arch. Rat. Mech. Anal., 159, 229-252, 2001. [Tanaka-Filmore, 1979]. T. Tanaka and D.J. Filmore, Kinetics of swelling gels, J. Chem. Phys., 70, 1214-1231, 1979. [Tanaka, 1981]. T. Tanaka, Gels, Scientific American, vol 244, 124-138. [Truesdell, 1984]. C. Truesdell, Rational Thermodynamics, Springer Verlag, second edition. [Zhang, 2007]. Hang Zhang, Ph.D. Thesis, Univeristy of Minnesota, July 2007. 22 / 23

Immiscibility and Incompressibility Immiscibility: the constitutive equations depend on volume fractions. It is always possible to distinguish between components Incompressibility: the intrinsic density is constant. Note that ρ = φγ ρ = mass of component, γ = mixture space So the incompressibility statement reduces to γ = constant mass of component component space Back 23 / 23