BIOMECHANICS Inflatation of latextube – material parameter identification
NONLINEARITIES s e In continuum mechanics Geometrical nonlinearity Large displacements Large deformation Material nonlinearity Nonlinear constitutive equation Constraints-contact Boundary conditions
Inflation test Load • Transmural pressure • Axial force Goal - Constitutive equation of material
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
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
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
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
Hyperelastic material adiabatic incompressible li…stretchesp…Lagrange multiplicator (pressure)
Experiment Inflation test Measured quantities 1. Outer radiusro2. Length of tubel3. Axial forceF4. Internal pressure pi
Experimental setup Sample Flanges(3.) Weights Tank Syringe – pressure generator Syringe – weight adjustment T-cock Valve Pressure transducer Scale Stand Camera
Model - deformations Cylindrical coordinate system Stretchesli tangential axial radial Deformation gradient F
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
Model-membrane stt dz G szz Balance of forces
Stress from loads (p,G) Outer radius is measured, inner radius is calculated from the incompressibility constraint
Stress from Const. Equat. Stresses from deformation energy Mooney–Rivlin model W Using incompressibility W is expressed as function oflt, lr
Material parameter identification Regrese Model prediction Experiment Goal function – least squares n is number of measurement (measured points)
Material parameter identification Linear regression Stationary point (minimum)[c1*,c2*]: Q([c1*,c2*])=minQ
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
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)
