14 - hyperelastic materials me338 - syllabusbiomechanics.stanford.edu/me338_13/me338_s14.pdf · 14...

1 14 - hyperelastic materials

14 - hyperelastic materials

holzapfel �nonlinear solid mechanics� [2000], chapter 6, pages 205-305


me338 - syllabus


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


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)â1_og-lam(i)^(-2*a1_og)) ... + m2_og * (lam(i)â2_og-lam(i)^(-2*a2_og)) ... + m3_og * (lam(i)â3_og-lam(i)^(-2*a3_og)); sig_mr(i) = m1_mr * (lam(i)â1_mr-lam(i)^(-2*a1_mr)) ... + m2_mr * (lam(i)â2_mr-lam(i)^(-2*a2_mr)); sig_nh(i) = m1_nh * (lam(i)â1_nh-lam(i)^(-2*a1_nh)); sig_vg(i) = m1_vg * (lam(i)â1_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)





ogden neo hooke

varga

mooney rivlin


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)





ogden neo hooke

varga

mooney rivlin



ogden

neo hooke

varga

mooney rivlin


mitral valve leaflet


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]


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


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


mitral valve leaflet


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


transversely isotropic incompressible

incompressible material

volumetric part isochoric part

transversely isotropic material structural tensor fiber orientation


with



volumetric part isochoric part



example 01 - neo hooke model

with

and


example 02 - may newman model

with


example 03 - holzapfel model

with

23 methods may newman, yin [1998], holzapfel, gasser, ogden [2000]


! 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;


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


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)





%% 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)






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


sensitivity - discretization

belly deflection vs no of elems

30 elements 120 elements 480 elements 1920 elements 7680 elements

convergence upon mesh refinement 1920


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


sensitivity - chordae stiffness

moderately sensitive to chordae stiffness

leaflet stiffness [MPa] vs chord stiffness [MPa]

20.0


sensitivity - leaflet thickness

sensitive to leaflet thickness

leaflet stiffness [MPa] vs leaflet thickness[mm]

1.0


sensitivity - leaflet thickness

constant thickness deformation error [mm]

optimized thickness [mm]

optimized bending stiffness[Nmm2]

leaflet thickness is physiologically optimized


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



rausch, famaey, shultz, bothe, miller, kuhl [2013]



begley & macking [2004], zamir & taber [2004], rausch & kuhl [2013]


stiffening effect of prestrain

begley & macking [2004], zamir & taber [2004], rausch & kuhl [2013]



70%

100%

44 14 - hyperelastic materials may newman, yin [1998], rausch, kuhl [2013]

parameter identification w/prestrain


stress vs elastic stretch

rausch & kuhl [2013]


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


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

14 - hyperelastic materials me338 - syllabusbiomechanics.stanford.edu/me338_13/me338_s14.pdf · 14...

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