1 14 - hyperelastic materials
14 - hyperelastic materials
holzapfel �nonlinear solid mechanics� [2000], chapter 6, pages 205-305
2 14 - hyperelastic materials
me338 - syllabus
3 14 - hyperelastic materials
isotropic hyperelastic materials
cauchy stress
invariants in terms of principal stretches
principal cauchy stresses
http://www.youtube.com/watch?v=yURomiwg9PE http://www.youtube.com/watch?v=zoFMUMWVHD0
4 14 - hyperelastic materials
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)); end
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)
5 14 - hyperelastic materials
inflation of a spherical rubber balloon
6 14 - hyperelastic materials
inflation of a spherical rubber balloon
ogden neo hooke
varga
mooney rivlin
7 14 - hyperelastic materials
inflation of a spherical rubber balloon ogden neo hooke
varga
mooney rivlin % 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))); end
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)
8 14 - hyperelastic materials
inflation of a spherical rubber balloon
9 14 - hyperelastic materials
inflation of a spherical rubber balloon
ogden neo hooke
varga
mooney rivlin
10 14 - hyperelastic materials
inflation of a spherical rubber balloon
ogden
neo hooke
varga
mooney rivlin
11 14 - hyperelastic materials
mitral valve leaflet
12 14 - hyperelastic materials
kinematic controversy
why are strains in vivo 3x smaller than ex vivo?
• ex vivo strains ~35% (left heart simulator) • in vivo strains ~12% (sonomicrometry/videofluoroscopy)
12%
8%
4%
0%
30%
20%
10%
0%
ex vivo strain vs time in vivo strain vs time
jimenez et al. [2007] rausch et al. [2011]
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equilibrium controversy
why are stresses in vivo 3x larger than ex vivo?
• ex vivo failure stress ~900 kPa (biaxial testing) • in vivo stress ~3,000 kPa (videofluoroscopy/fe analysis)
3200 800
ex vivo stress vs strain in vivo stress vs strain
grande allen et al. [2005] krishnamurthy et al. [2009]
400
600
200
0
[kPa]
2400
1600
0
1800
[kPa]
circ rad
circ rad radial
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constitutive controversy
why is stiffness in vivo 1000x larger than ex vivo?
• ex vivo stiffness Ecirc ≈40kPa/4MPa and Erad ≈10kPa/1MPa • in vivo stiffeness Ecirc ≈40MPa and Erad ≈10MPa
3200 800
ex vivo stress vs strain in vivo stress vs strain
sacks et al. [2000], grande allen et al. [2005] krishnamurthy et al. [2009]
400
600
200
0
[kPa]
2400
1600
0
1800
[kPa]
circ rad
circ rad
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mitral valve leaflet
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hemodynamics - pressure
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 end isovolumetric relaxation to end systole.
normalized cardiac cycle
average left ventricular pressure [mmHg]
ED EIVC ES
120
100
80
60
40
20
0 EIVR
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transversely isotropic incompressible
incompressible material
volumetric part isochoric part
transversely isotropic material structural tensor fiber orientation
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with
transversely isotropic incompressible
19 14 - hyperelastic materials
volumetric part isochoric part
transversely isotropic incompressible
20 14 - hyperelastic materials
example 01 - neo hooke model
with
and
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example 02 - may newman model
with
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example 03 - holzapfel model
with
23 methods may newman, yin [1998], holzapfel, gasser, ogden [2000]
transversely isotropic incompressible
! neo hooke - isotropic c0
! may newman - anisotropic, coupled c0, c1, c2
! holzapfel - anisotropic, decoupled c0, c1, c2
function [] = UniAxialTest() lambda1 = [1:0.001:2.0]; lambda2 = lambda1;
%%% material parameters %%%%%%%%%%% % neo hooke model c0_neo = 63700000; % may-newman model c0_may = 8958355943.52; c1_may = 0.89577484; c2_may = 1.79619884; % holzapfel model c0_hlz = 18364377.50; c1_hlz = 2499419166.42; c2_hlz = 97.44; % experiment c0_exp = 52.0; c1_exp = 4.63; c2_exp = 22.6;
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uniaxial stretching of anisotropic sheet
%% derivatives of free energy wrt invariants %%%%%%%%
function [Ppsi1] = psi1_neo(c0,I1,I4) psi1 = c0; end function [psi1] = psi1_may(c0,c1,c2,I1,I4) psi1 = c0.*exp(c1.*(I1-3).^2+c2.*(I4-1).^2)*2*c1.*(I1-3); end
function [psi4] = psi4_may(c0,c1,c2,I1,I4) psi4 = c0.*exp(c1.*(I1-3).^2+c2.*(I4-1).^2).*2.*c2.*(I4-1); end function [psi1] = psi1_hlz(c0,c1,c2,I1,I4) psi1 = c0; end
function [psi4] = psi4_hlz(c0,c1,c2,I1,I4) psi4 = c1.*(I4-1).* exp(c2.*(I4-1).^2); end
25 14 - hyperelastic materials
uniaxial stretching of anisotropic sheet %% 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) ... + 2 .* 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 14 - hyperelastic materials
uniaxial stretching of anisotropic sheet
27 14 - hyperelastic materials
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)
28 14 - hyperelastic materials
uniaxial stretching of anisotropic sheet
29 14 - hyperelastic materials
uniaxial stretching of anisotropic sheet
30 14 - hyperelastic materials
final parameter set
current parameter set
finite element analysis
objective function animal experiment genetic algorithm
rausch, famaey, shultz, bothe, miller, kuhl [2013]
no yes
update
initialize
LVP
convergence?
parameter identification
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sensitivity - discretization
belly deflection vs no of elems
30 elements 120 elements 480 elements 1920 elements 7680 elements
convergence upon mesh refinement 1920
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sensitivity - chord position & number
single cord close to free edge
close to annulus and
commissures
close to annulus and midline
chordae close to
free edge
= 85.0 MPa
= 65.4 MPa = 63.7 MPa
= 77.5 MPa
insensitive to chordae position
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sensitivity - chordae stiffness
moderately sensitive to chordae stiffness
leaflet stiffness [MPa] vs chord stiffness [MPa]
20.0
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sensitivity - leaflet thickness
sensitive to leaflet thickness
leaflet stiffness [MPa] vs leaflet thickness[mm]
1.0
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sensitivity - leaflet thickness
constant thickness deformation error [mm]
optimized thickness [mm]
optimized bending stiffness[Nmm2]
leaflet thickness is physiologically optimized
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coupled anisotropic model
c0 = 119,020.7kPa c1 = 152.4 c2 = 185.5 -1.0mm +1.0mm
nodal error
why is stiffness in vivo 1000x larger than ex vivo? may newman, yin [1998], rausch, famaey, shultz, bothe, miller, kuhl [2013]
37 14 - hyperelastic materials holzapfel, gasser, ogden [2000], rausch, famaey, shultz, bothe, miller, kuhl [2013]
decoupled anisotropic model
c0 = 18,364.4kPa c1 = 2,499,419.2kPa c2 = 97.4 -1.0mm +1.0mm
why is stiffness in vivo 100x larger than ex vivo?
nodal error
38 14 - hyperelastic materials amini, eckert, koomalsingh, mcgarvey, minakawa, gorman, gorman, sacks [2012]
what’s the influence of prestrain? in vivo
min LVP in vivo
max LVP
ex vivo
rad circ
39 14 - hyperelastic materials amini, eckert, koomalsingh, mcgarvey, minakawa, gorman, gorman, sacks [2012]
what’s the influence of prestrain?
ex vivo
rad circ
λp = 1.32
λ = 1.21
λe = 1.60
in vivo min LVP
in vivo max LVP
40 14 - hyperelastic materials
what’s the influence of prestrain?
rausch, famaey, shultz, bothe, miller, kuhl [2013]
41 14 - hyperelastic materials
what’s the influence of prestrain?
begley & macking [2004], zamir & taber [2004], rausch & kuhl [2013]
42 14 - hyperelastic materials
stiffening effect of prestrain
begley & macking [2004], zamir & taber [2004], rausch & kuhl [2013]
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what’s the influence of prestrain?
70%
100%
44 14 - hyperelastic materials may newman, yin [1998], rausch, kuhl [2013]
parameter identification w/prestrain
45 14 - hyperelastic materials
stress vs elastic stretch
rausch & kuhl [2013]
46 14 - hyperelastic materials
stress vs total stretch
rausch & kuhl [2013]
47 14 - hyperelastic materials may newman, yin [1998], rausch, kuhl [2013]
in vivo stiffness = ex vivo stiffness
parameter identification w/prestrain
48 14 - hyperelastic materials
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