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• 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]
13 14 - hyperelastic materials
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
14 14 - hyperelastic materials
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
15 14 - hyperelastic materials
mitral valve leaflet
16 14 - hyperelastic materials
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
17 14 - hyperelastic materials
transversely isotropic incompressible
incompressible material
volumetric part isochoric part
transversely isotropic material structural tensor fiber orientation
18 14 - hyperelastic materials
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
21 14 - hyperelastic materials
example 02 - may newman model
with
22 14 - hyperelastic materials
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;
%% 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;
• 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