1. Limiting fiber extensibility as parameter for damage in venous wall Lukas Horny, Rudolf Zitny, Hynek Chlup, Tomas Adamek and Michal Sara Faculty of Mechanical Engineering Czech Technical University in Prague Czech Republic

2. Introduction  Constitutive modeling of a wall of human vena cava inferior.  Constitutive model is fundamental information about material behavior  Constitutive model is needed for every engineering simulation which includes displacements and deformations

3. Goals  Suggestion of new constitutive model incorporation of structural information possibility of clear interpretation for parameters  Description of mechanical response within elastic behavior within inelastic behavior

4. Blood vessel mechanics  Geometric nonlinearity ̶ large strains  Physical nonlinearity ̶ nonlinear stress – strain relationship large strain stiffening  Anisotropy

5. Blood vessel mechanics  Inelastic behavior  Preconditioning  Viscoelasticity  Pseudoelasticity Loading and unloading curves are always different

6. Constitutive equations  Stored energy function ψ  1 st law of Thermodynamics  Zero energy under reference configuration  Zero stress under reference configuration  Constitutive equation  Terms for stored energy

7. Limiting fiber extensibility Reference configuration Deformed configuration Limiting configuration stress stretch  Limiting fiber extensibility mimics idea of limiting chain extensibility in polymer physics  Limiting fiber extensibility can capture large strain stiffening Reference Deformed Limiting

8. Limiting chain extensibility  Gent model - isotropy AN Gent 1996 - rubbers …material parameter - shear modulus …material parameter – limiting extensibility parameter …1st invariant of a deformation tensor stress I1I1 I 1 = J m + 3 Limiting

9. Limiting fiber extensibility  Suggested model – local orthotropy …shear modulus …limiting extensibility parameter …4th invariant of a deformation tensor …angle between fibers and circumferential axis Blood vessel wall as a fiber reinforced composite stress I4I4 I 4 = J m + 1 Limiting z t 

10. Material parameters estimation Material parameters must be identified experimentally  Inflation–extension test  Computational model  Thick walled – tube  Hyperelastic fiber reinforced material  Matrix + fibers  Incompressibility  No shear strains  No residual strains

11. Regression  Measured data  Model predictions  Radial displacement (image analysis of photographs)  Axial displacement (image analysis of photographs)  Internal pressure (pressure probe recording)  Axial force (defined weight + pressure onto bottom) p …internal pressure F …axial force r …deformed radius t, z …stretch – z axial; t circumferential

12. Results – vena cava inferior Internal pressure p [kPa] Circumferential and axial stretch t, z [1] Axial force F [N] 1 st overloading cycle 2 nd overloading cycle 3 rd overloading cycle 4 th overloading cycle  Axial force Internal pressure o Physiological loading Supra-physiological

13. Results – vena cava inferior Axial force F [N] Internal pressure p [kPa] Representative cycle of supra-physiological loading Circumferential and axial stretch t, z [1]  Axial force Internal pressure Model predictions

14. Damage evolution 1 st overloading cycle 2 nd overloading cycle 3 rd overloading cycle 4 th overloading cycle  Axial force Internal pressure Supra-physiological loading only Axial force F [N] Internal pressure p [kPa] Circumferential and axial stretch t, z [1] Model predictions

15. Conclusion  Stored energy function based on limiting fiber extensibility assumption fits experimental data (inflation –extension test) successfully  Damage can be related to evolution of the limiting extensibility parameter J m  The only history dependent parameter J m is capable to explain different trends of stretches in axial and circumferential directions

16. Limiting fiber extensibility as parameter for damage in venice walls The End