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BIOMECHANICS

BIOMECHANICS. Infla tation of latex tube – materi a l paramet er identification. NONLINEARITIES. s. e. In continuum mechanics. Geometric al n onlinearity Large displacements Large deformation. Material nonlinearity N onlinear constitutive equation. Constraints-contact

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BIOMECHANICS

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  1. BIOMECHANICS Inflatation of latextube – material parameter identification

  2. NONLINEARITIES s e In continuum mechanics Geometrical nonlinearity Large displacements Large deformation Material nonlinearity Nonlinear constitutive equation Constraints-contact Boundary conditions

  3. Inflation test Load • Transmural pressure • Axial force Goal - Constitutive equation of material

  4. Modelling Tensors of deformation Gradient of deformation F X(X1,X2,X3,t) – reference configurationx(X1,X2,X3,t) – loaded state. Displacement U(X,t)=x(X,t)-X. In terms of displacementsx=X+U

  5. Modelling Example: One-dimenional homogeneous deformationí x1=lX1, x2=X2, x3=X3. Displacement U :U1=x1-X1, U1=lX1-X1. Tensors of deformation Green–Lagrange tenzor

  6. Modelling Example: One-dimenional homogeneous deformationí x1=lX1, x2=X2, x3=X3. Displacement U :U1=x1-X1, U1=lX1-X1. Tensors of deformation Engineering deformation

  7. Hyperelastic material Ronald S. Rivlin (1915-2005) Melvin Mooney(1893-1968) Functiony– density of deformation energyW.  - true stress  - engineering deformation I1, I2 first and second invariant of deformation tensor deviator, Jchange of volume, limain stretches

  8. Hyperelastic material adiabatic incompressible li…stretchesp…Lagrange multiplicator (pressure)

  9. Experiment Inflation test Measured quantities 1. Outer radiusro2. Length of tubel3. Axial forceF4. Internal pressure pi

  10. Experimental setup Sample Flanges(3.) Weights Tank Syringe – pressure generator Syringe – weight adjustment T-cock Valve Pressure transducer Scale Stand Camera

  11. Model - deformations Cylindrical coordinate system Stretchesli tangential axial radial Deformation gradient F

  12. Model - deformations Incompressibility constraint Ro…initial outer radius Ri…initial inner radius ro…actual outer ri…and inner radius L…initial length l…actual length

  13. Model-membrane stt dz G szz Balance of forces

  14. Stress from loads (p,G) Outer radius is measured, inner radius is calculated from the incompressibility constraint

  15. Stress from Const. Equat. Stresses from deformation energy Mooney–Rivlin model W Using incompressibility W is expressed as function oflt, lr

  16. Material parameter identification Regrese Model prediction Experiment Goal function – least squares n is number of measurement (measured points)

  17. Material parameter identification Linear regression Stationary point (minimum)[c1*,c2*]: Q([c1*,c2*])=minQ

  18. Experiment Assembly of measuring instruments Measure dimensions in reference configuration (Ro,Ri,L,H,m) Adjust camera Flood pipe Several test cycles (preconditioning) without records At least 3 measuring cycles, recorded Disassembly and cleaning

  19. Experiment Measuring cycle At least 6 measuring points ([ro,l,p,G]-loads and corresponding dimensions) Upper limit for load~ 20 kPa Close the valve in each measuring point! It is not necessary to measure in the region where the model assumptions are viaolated (buckling at higher loads)

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