14 - hyperelastic materials. me338 - syllabus hyperelastic materials hyperelastic materials. inflation of a spherical rubber balloon

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1 me338 - syllabus holzapfel nonlinear solid mechanics [2000], chapter 6, pages isotropic hyperelastic materials inflation of a spherical rubber balloon cauchy stress invariants in terms of principal stretches principal cauchy stresses

2 inflation of a spherical rubber balloon inflation of a spherical rubber balloon % cauchy stress vs stretch plot % loop over all stretches lambda from 1.0 to 10.0 for i=1:901 lam(i) = 1.0+(i-1)/100; sig_og(i) = m1_og * (lam(i)^a1_og-lam(i)^(-2*a1_og))... + m2_og * (lam(i)^a2_og-lam(i)^(-2*a2_og))... + m3_og * (lam(i)^a3_og-lam(i)^(-2*a3_og)); sig_mr(i) = m1_mr * (lam(i)^a1_mr-lam(i)^(-2*a1_mr))... + m2_mr * (lam(i)^a2_mr-lam(i)^(-2*a2_mr)); sig_nh(i) = m1_nh * (lam(i)^a1_nh-lam(i)^(-2*a1_nh)); sig_vg(i) = m1_vg * (lam(i)^a1_vg-lam(i)^(-2*a1_vg)); plot(lam,sig_og/10^6,'-k','linewidth',2.0) plot(lam,sig_mr/10^6,'-k','linewidth',2.0) plot(lam,sig_nh/10^6,'-k','linewidth',2.0) plot(lam,sig_vg/10^6,'-k','linewidth',2.0) mooney rivlin ogden neo hooke varga 5 6 inflation of a spherical rubber balloon inflation of a spherical rubber balloon mooney rivlin ogden neo hooke varga % pressure vs stretch plot % loop over all stretches lambda from 1.0 to 10.0 for i=1:901 lam(i) = 1.0+(i-1)/100; p_og(i) = 2*H/R *(m1_og * (lam(i)^(a1_og-3)-lam(i)^(-2*a1_og-3))... + m2_og * (lam(i)^(a2_og-3)-lam(i)^(-2*a2_og-3))... + m3_og * (lam(i)^(a3_og-3)-lam(i)^(-2*a3_og-3))); p_mr(i) = 2*H/R *(m1_mr * (lam(i)^(a1_mr-3)-lam(i)^(-2*a1_mr-3))... + m2_mr * (lam(i)^(a2_mr-3)-lam(i)^(-2*a2_mr-3))); p_nh(i) = 2*H/R *(m1_nh * (lam(i)^(a1_nh-3)-lam(i)^(-2*a1_nh-3))); p_vg(i) = 2*H/R *(m1_vg * (lam(i)^(a1_vg-3)-lam(i)^(-2*a1_vg-3))); plot(lam,p_og/10^2,'-k','linewidth',2.0) plot(lam,p_mr/10^2,'-k','linewidth',2.0) plot(lam,p_nh/10^2,'-k','linewidth',2.0) plot(lam,p_vg/10^2,'-k','linewidth',2.0) 7 8

3 inflation of a spherical rubber balloon inflation of a spherical rubber balloon mooney rivlin ogden mooney rivlin ogden neo hooke varga neo hooke varga 9 10 mitral valve leaflet kinematic controversy ex vivo strains ~35% (left heart simulator) in vivo strains ~12% (sonomicrometry/videofluoroscopy) ex vivo strain vs time in vivo strain vs time 30% 12% 20% 8% 10% 4% 0% 0% jimenez et al. [2007] rausch et al. [2011] why are strains in vivo 3x smaller than ex vivo? 11 12

4 equilibrium controversy constitutive controversy ex vivo failure stress ~900 kpa (biaxial testing) in vivo stress ~3,000 kpa (videofluoroscopy/fe analysis) ex vivo stiffness E circ 40kPa/4MPa and E rad 10kPa/1MPa in vivo stiffeness E circ 40MPa and E rad 10MPa [kpa] 800 ex vivo stress vs strain [kpa] 3200 in vivo stress vs strain [kpa] 800 ex vivo stress vs strain [kpa] 3200 in vivo stress vs strain 600 radial circ rad 2400 circ rad 600 circ rad 2400 circ rad grande allen et al. [2005] 0 krishnamurthy et al. [2009] 0 sacks et al. [2000], grande allen et al. [2005] 0 krishnamurthy et al. [2009] why are stresses in vivo 3x larger than ex vivo? 13 mitral valve leaflet why is stiffness in vivo 1000x larger than ex vivo? 14 hemodynamics - pressure 120 average left ventricular pressure [mmhg] ED EIVC ES EIVR normalized cardiac cycle figure. left ventricular pressure averaged over 57 animals. the simulation is performed at eight discrete time points during isovolumetric relaxation. the arrow indicates the direction of the simulation going backward in time from isovolumetric relaxation to systole

5 transversely isotropic incompressible transversely isotropic incompressible incompressible material volumetric part isochoric part fiber orientation transversely isotropic material structural tensor with transversely isotropic incompressible example 01 - neo hooke model volumetric part isochoric part with and 19 20

6 example 02 - may newman model example 03 - holzapfel model with with transversely isotropic incompressible uniaxial stretching of anisotropic sheet! neo hooke - isotropic c 0! may newman - anisotropic, coupled c 0, c 1, c 2! holzapfel - anisotropic, decoupled c 0, c 1, c 2 may newman, yin [1998], holzapfel, gasser, ogden [2000] 23 methods function [] = UniAxialTest() lambda1 = [1:0.001:2.0]; lambda2 = lambda1; %%% material parameters %%%%%%%%%%% % neo hooke model c0_neo = ; % may-newman model c0_may = ; c1_may = ; c2_may = ; % holzapfel model c0_hlz = ; c1_hlz = ; c2_hlz = 97.44; % experiment c0_exp = 52.0; c1_exp = 4.63; c2_exp = 22.6; 24

7 uniaxial stretching of anisotropic sheet uniaxial stretching of anisotropic sheet %% derivatives of free energy wrt invariants %%%%%%%% function [Ppsi1] = psi1_neo(c0,i1,i4) psi1 = c0; function [psi1] = psi1_may(c0,c1,c2,i1,i4) psi1 = c0.*exp(c1.*(i1-3).^2+c2.*(i4-1).^2)*2*c1.*(i1-3); function [psi4] = psi4_may(c0,c1,c2,i1,i4) psi4 = c0.*exp(c1.*(i1-3).^2+c2.*(i4-1).^2).*2.*c2.*(i4-1); function [psi1] = psi1_hlz(c0,c1,c2,i1,i4) psi1 = c0; function [psi4] = psi4_hlz(c0,c1,c2,i1,i4) psi4 = c1.*(i4-1).* exp(c2.*(i4-1).^2); 25 %% stress-stretch in fiber direction %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% I1 = lambda1.^2 + 2./lambda1; I4 = lambda1.^2; % neo-hooke model neo_sigma_11 = 2.* psi1_neo(c0_neo,i1,i4) * (lambda1.^2-1./lambda1); % may-newman model may_sigma_11 = 2.*psi1_may(c0_may,c1_may,c2_may,I1,I4).* (lambda1.^2-1./lambda1) + 2.* psi4_may(c0_may,c1_may,c2_may,i1,i4).* lambda1.^2; % holzapfel model hlz_sigma_11 = 2.* psi1_hlz(c0_hlz,c1_hlz,c2_hlz,i1,i4).* (lambda1.^2-1./lambda1) * psi4_hlz(c0_hlz,c1_hlz,c2_hlz,i1,i4).* lambda1.^2; % experiment exp_sigma_11 = 2.* psi1_may(c0_exp,c1_exp,c2_exp,i1,i4).* (lambda1.^2-1./lambda1) + 2.* psi4_may(c0_exp,c1_exp,c2_exp,i1,i4).* lambda1.^2; plot(lambda1, neo_sigma_11/10^9,'r-','linewidth',2.0) plot(lambda1, may_sigma_11/10^9,'b-','linewidth',2.0) plot(lambda1, hlz_sigma_11/10^9,'g-','linewidth',2.0) plot(lambda1, exp_sigma_11/10^9,'k-','linewidth',2.0) 26 uniaxial stretching of anisotropic sheet uniaxial stretching of anisotropic sheet %% stress-stretch relation in cross-fiber direction %%%%%%%%%%%%%%%%%%%%%% I1 = lambda2.^2 + 2./lambda2; I4 = 1./lambda2; % neo-hooke model neo_sigma_22 = 2.* psi1_neo(c0_neo,i1,i4).* (lambda2.^2-1./lambda2); % may-newman model may_sigma_22=2.*psi1_may(c0_may,c1_may,c2_may,i1,i4).* (lambda2.^2-1./lambda2); % holzapfel model hlz_sigma_22 = 2.* psi1_hlz(c0_hlz,c1_hlz,c2_hlz,i1,i4).* (lambda2.^2-1./lambda2); % experiment exp_sigma_22 = 2.* psi1_may(c0_exp,c1_exp,c2_exp,i1,i4).* (lambda2.^2-1./lambda2); plot(lambda1, hlz_sigma_22/10^9,'g-','linewidth',2.0) plot(lambda1, may_sigma_22/10^9,'b-','linewidth',2.0) plot(lambda1, neo_sigma_22/10^9,'r-','linewidth',2.0) plot(lambda1, exp_sigma_22/10^9,'k-','linewidth',2.0) 27 28

8 uniaxial stretching of anisotropic sheet parameter identification initialize current parameter set update animal experiment finite element analysis objective function genetic algorithm LVP convergence? yes final parameter set no rausch, famaey, shultz, bothe, miller, kuhl [2013] sensitivity - discretization sensitivity - chord position & number 30 elements 120 elements 480 elements 1920 elements 7680 elements belly deflection vs no of elems close to annulus and commissures = 77.5 MPa close to annulus and midline = 85.0 MPa chordae close to free edge = 63.7 MPa single cord close to free edge = 65.4 MPa 1920 convergence upon mesh refinement insensitive to chordae position 31 32

9 sensitivity - chordae stiffness sensitivity - leaflet thickness leaflet stiffness [MPa] vs chord stiffness [MPa] leaflet stiffness [MPa] vs leaflet thickness[mm] moderately sensitive to chordae stiffness sensitive to leaflet thickness sensitivity - leaflet thickness coupled anisotropic model constant thickness deformation error [mm] optimized thickness [mm] optimized bing stiffness[nmm 2 ] nodal error -1.0mm +1.0mm c 0 = 119,020.7kPa c 1 = c 2 = leaflet thickness is physiologically optimized why is stiffness in vivo 1000x larger than ex vivo? may newman, yin [1998], rausch, famaey, shultz, bothe, miller, kuhl [2013] 35 36

10 decoupled anisotropic model what s the influence of prestrain? nodal error in vivo min LVP in vivo max LVP -1.0mm +1.0mm c 0 = 18,364.4kPa c 1 = 2,499,419.2kPa c 2 = 97.4 why is stiffness in vivo 100x larger than ex vivo? holzapfel, gasser, ogden [2000], rausch, famaey, shultz, bothe, miller, kuhl [2013] what s the influence of prestrain? 37 rad circ ex vivo amini, eckert, koomalsingh, mcgarvey, minakawa, gorman, gorman, sacks [2012] what s the influence of prestrain? 38 in vivo min LVP λ = 1.21 in vivo max LVP ex vivo rad circ λ p = 1.32 λ e = 1.60 amini, eckert, koomalsingh, mcgarvey, minakawa, gorman, gorman, sacks [2012] rausch, famaey, shultz, bothe, miller, kuhl [2013] 39 40

11 what s the influence of prestrain? stiffening effect of prestrain begley & macking [2004], zamir & taber [2004], rausch & kuhl [2013] begley & macking [2004], zamir & taber [2004], rausch & kuhl [2013] what s the influence of prestrain? parameter identification w/prestrain 70% 100% may newman, yin [1998], rausch, kuhl [2013] 43 44

12 stress vs elastic stretch stress vs total stretch rausch & kuhl [2013] rausch & kuhl [2013] parameter identification w/prestrain what s the effect of prestrain? stiffness is significantly larger in vivo than ex vivo concept of prestrain may explain this controversy prestrain is conceptually simpler than residual stress ex vivo testing alone tells us little about in vivo behavior likely true for thin biological membranes in general in vivo stiffness = ex vivo stiffness may newman, yin [1998], rausch, kuhl [2013] 47 48