Fitting Hyperelastic Ogden Model Parameters to Biaxial Stretch Test Data
Simultaneous fitting of multi-axial stretch data to Ogden parameters prevents FEA solver instability and accurately predicts material thinning under strain.
Strain Invariants
Meaning
Scalar algebraic quantities derived from the deformation gradient tensor remain completely unchanged by coordinate system rotations during three-dimensional strain evaluations. In non-linear solid mechanics and hyperelastic material modeling, strain invariants describe the state of physical deformation independently of the spatial orientation of the loaded component. Three primary invariants define the kinematic state, representing stretch along principal axes, surface area changes and volumetric alteration.
Hyperelastic strain energy density functions use these invariant quantities as fundamental state variables to formulate mechanical responses under multi-axial stress states. The invariant framework ceases to apply directly when material structures exhibit directional anisotropy, such as long-fiber reinforced polymers, requiring alternative directional tensor formulations.
Tensor Derivation
Mathematical extraction of these values begins with the right or left Cauchy-Green deformation tensors computed from the deformation gradient. The first invariant equates to the trace of the tensor, capturing the sum of the squared principal stretch ratios across the three spatial dimensions. Combining second-order minors produces the second invariant, which relates to changes in surface area throughout the deformed continuum.
The third invariant equals the determinant of the deformation tensor, quantifying the local volume ratio between initial and deformed states. Incompressible polymers such as unfilled elastomers maintain a constant third invariant equal to unity, simplifying the constitutive equations by removing volumetric work terms.
Constitutive Application
Finite element codes use invariant formulations to predict stress distributions in complex moulded components under arbitrary loading. Classic constitutive relationships like the Mooney-Rivlin or Yeoh models write strain energy density as explicit polynomial expansions of the first and second invariants. If a material card calibrates these invariant parameters solely on uniaxial tension data, simulations generate massive errors in shear and biaxial load cases.
Injection moulders encounter this failure when testing virtual snap fits, where multi-axial stress states cause unexpected fracture despite safe uniaxial predictions. Virgin polymers follow classic invariant formulations cleanly, whereas filled polymers or regrind blends introduce anisotropic filler alignments that distort pure invariant assumptions.
Processing Implications
Processing operations subject hot polymer melts to complex flow fields that stretch macromolecular networks along multiple directions simultaneously. In blow moulding and thermoforming, wall thickness distribution depends on the polymer resisting biaxial extension, which is governed by the second invariant. Tool designers adjust cavity draft angles and wall transitions to avoid localized zones of extreme invariant magnitude that promote stress crazing.
Recycled polymers with broken molecular chains exhibit premature yield when invariant thresholds are reached, lowering the burst strength of blow-moulded containers. Characterizing invariant responses across multiple deformation modes ensures structural models accurately forecast finished part performance under real-world mechanical stress.