1. No category

14 - hyperelastic materials
me338 - syllabus
holzapfel nonlinear solid mechanics [2000], chapter 6, pages 205-305
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isotropic hyperelastic materials
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inflation of a spherical rubber balloon
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
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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));
end
mooney rivlin
ogden
neo hooke
varga
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)
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inflation of a spherical rubber balloon
mooney rivlin
ogden
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inflation of a spherical rubber balloon
% pressure vs stretch plot
neo hooke
% 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
varga
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)
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inflation of a spherical rubber balloon
inflation of a spherical rubber balloon
mooney rivlin
ogden
mooney rivlin
ogden
neo hooke
varga
neo hooke
varga
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kinematic controversy
mitral valve leaflet
• ex vivo strains ~35% (left heart simulator)
• in vivo strains ~12% (sonomicrometry/videofluoroscopy) ex vivo strain vs time 30%
in vivo strain vs time 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?
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equilibrium controversy
constitutive controversy
• ex vivo failure stress ~900 kPa (biaxial testing)
• in vivo stress ~3,000 kPa (videofluoroscopy/fe analysis) • ex vivo stiffness Ecirc ≈40kPa/4MPa and Erad ≈10kPa/1MPa
• in vivo stiffeness Ecirc ≈40MPa and Erad ≈10MPa
[kPa]
ex vivo stress vs strain
800
600
radial
circ
rad
[kPa]
3200
in vivo stress vs strain
[kPa]
ex vivo stress vs strain
800
2400 circ
circ
rad
600
rad
[kPa]
3200
2400 circ
400
1600
400
1600
200
1800
200
1800
0
0
grande allen et al. [2005]
0
sacks et al. [2000], grande allen et al. [2005]
krishnamurthy et al. [2009]
why are stresses in vivo 3x larger than ex vivo?
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in vivo stress vs strain
rad
0
krishnamurthy et al. [2009]
why is stiffness in vivo 1000x larger than ex vivo?
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mitral valve leaflet
hemodynamics - pressure
average left ventricular pressure [mmHg]
120
100
80
60
40
20
0
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 end isovolumetric relaxation to end systole. 14 - hyperelastic materials
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transversely isotropic incompressible
transversely isotropic incompressible
incompressible material
volumetric part
isochoric part
transversely isotropic material
fiber orientation
structural tensor
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transversely isotropic incompressible
volumetric part
with
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example 01 - neo hooke model
isochoric part
with
and
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example 02 - may newman model
example 03 - holzapfel model
with
with
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transversely isotropic incompressible
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uniaxial stretching of anisotropic sheet
function [] = UniAxialTest()
lambda1 = [1:0.001:2.0]; lambda2 = lambda1; ! neo hooke - isotropic c0
! may newman - anisotropic, coupled c0, c1, c2
! holzapfel - anisotropic, decoupled c0, c1, c2
may newman, yin [1998], holzapfel, gasser, ogden [2000]
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methods
%%% 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
uniaxial stretching of anisotropic sheet
%% derivatives of free energy wrt invariants %%%%%%%%
%% stress-stretch in fiber direction %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
I1 = lambda1.^2 + 2./lambda1;
I4 = lambda1.^2; 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
% 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; 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
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)
function [psi4] = psi4_hlz(c0,c1,c2,I1,I4)
psi4 = c1.*(I4-1).* exp(c2.*(I4-1).^2);
end
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uniaxial stretching of anisotropic sheet
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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)
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uniaxial stretching of anisotropic sheet
parameter identification
initialize
current
parameter set
update
finite element
analysis
LVP
animal experiment
objective
function
convergence?
yes
genetic algorithm
no
final
parameter set
rausch, famaey, shultz, bothe, miller, kuhl [2013]
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sensitivity - chord position & number
sensitivity - discretization
30 elements 120 elements 480 elements 1920 elements 7680 elements
close to
annulus and
commissures
= 77.5 MPa
belly deflection vs no of elems
chordae
close to
free edge
= 63.7 MPa
close to
annulus and
midline
= 85.0 MPa
single cord
close to
free edge
= 65.4 MPa
1920
convergence upon mesh refinement
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insensitive to chordae position
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sensitivity - chordae stiffness
sensitivity - leaflet thickness
leaflet stiffness
leaflet stiffness
[MPa] vs chord stiffness [MPa]
[MPa] vs leaflet thickness[mm]
1.0
20.0
moderately sensitive to chordae stiffness
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sensitive to leaflet thickness
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optimized
thickness [mm]
optimized
bending stiffness[Nmm2]
nodal error
-1.0mm
c0 = 119,020.7kPa
+1.0mm
c1 = 152.4
c2 = 185.5
why is stiffness in vivo 1000x larger than ex vivo?
leaflet thickness is physiologically optimized
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coupled anisotropic model
sensitivity - leaflet thickness
constant thickness
deformation error [mm]
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may newman, yin [1998], rausch, famaey, shultz, bothe, miller, kuhl [2013]
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decoupled anisotropic model
what’s the influence of prestrain?
in vivo
min LVP
in vivo
max LVP
nodal error
-1.0mm
c0 = 18,364.4kPa
ex vivo
+1.0mm
c1 = 2,499,419.2kPa
c2 = 97.4
rad
circ
why is stiffness in vivo 100x larger than ex vivo?
amini, eckert, koomalsingh, mcgarvey, minakawa, gorman, gorman, sacks [2012]
holzapfel, gasser, ogden [2000], rausch, famaey, shultz, bothe, miller, kuhl [2013]
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what’s the influence of prestrain?
in vivo
min LVP
λ = 1.21
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what’s the influence of prestrain?
in vivo
max LVP
ex vivo
rad
circ
λp =
1.32
λe = 1.60
amini, eckert, koomalsingh, mcgarvey, minakawa, gorman, gorman, sacks [2012]
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rausch, famaey, shultz, bothe, miller, kuhl [2013]
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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]
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what’s the influence of prestrain?
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parameter identification w/prestrain
70%
100%
may newman, yin [1998], rausch, kuhl [2013]
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stress vs elastic stretch
stress vs total stretch
rausch & kuhl [2013]
rausch & kuhl [2013]
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parameter identification w/prestrain
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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]
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