Deriving Ogden Constitutive Parameters from Biaxial Stretch Testing Data
Deriving Ogden constitutive parameters pairs non-linear biaxial stretch data with Drucker stability constraints to ensure reliable finite element simulations.
Drucker Stability
Meaning
A thermodynamic criterion evaluates strain energy functions to ensure non-negative work during incremental stress applications. Verification of drucker stability guarantees that hyperelastic material models maintain positive-definite stiffness matrices across all strain states encountered in numerical simulations. The requirement applies strictly to stable material definitions in non-linear FEA solvers, identifying strain boundaries where hyperelastic equations yield physically impossible softening or negative energy generation.
Mathematical Criterion
Physical consistency demands that incremental changes in stress produce positive incremental work along any deformation path. Evaluating drucker stability requires checking whether the material stiffness tensor remains positive definite across arbitrary combinations of principal stretch ratios. Higher-order polynomial hyperelastic formulations, such as third-order Yeoh or polynomial equations, often violate this condition at stretch ratios exceeding fitted laboratory data.
Solvers flag stability violations when the strain energy density function exhibits negative curvature, leading to non-physical material behavior.
Finite Element Divergence
Unstable constitutive equations trigger numerical collapse in implicit non-linear structural solvers. Stiffness matrices become singular, causing Newton-Raphson iterations to fail during complex compression or stretch blow moulding simulations.
Material Limit
Polymer strain hardening behavior usually satisfies stability requirements until molecular chain rupture limits approach. Modellers must cap strain ranges or select alternative potentials when stability checks fail at elevated strain levels.