Constitutive models and finite element implementations of compressible finite deformation are straightforwardly formulated by the general isotropic continuum stored energy (CSE) functional without the isochoric-volumetric split. Coupled stress and elasticity tensors in reference and current configurations are derived. Modeling and predicting capabilities of the general CSE functional are exhibited through multiaxial experimental tests of compressible NR and SBR rubbers. Characterization of kinematic relation, rather than pressure-volume relation, is emphasized in experimental tests of compressibility. The isochoric-volumetric split does not hold based on either theoretical analyses or experimental validations.
The magic angle of θm = arctan[(√ 5 + 1)/2] ≈ 58.2825 • , rather than θ = arccos(1/ √ 3) ≈ 54.7356 • , has been discovered through theoretical derivations for arteries to accommodate twist buckling optimally. The magic angle matches many published experimental results by others. As byproducts of the derivation, the stable deformation ranges for normal and shear stretches are defined. The anisotropic continuum stored energy (CSE) functional has been used to model the equibiaxial tension tests of porcine thoracic aortas and special simple normal tests of human abdominal aorta aneurysms. In CSE models, constitutive constants are determined by a trial-and-error-on-digit (TED) method and the linear least squares (LLSQ) method combined.
The general CSE model fits Treloar’s uniaxial extension test and predicts unfitted uniaxial compression, equibiaxial extension, biaxial extension, pure shear, and simple shear tests. As a newly proposed method, the general CSE model, along with the stress-softening ratio, the residualstretch ratio, and the weighted piecewise two-point interpolation function, fits the Cheng–Chen’s test and the Diani–Fayolle–Gilormini’s test in cyclic uniaxial extension at different pre-stretches and predicts corresponding responses at untested pre-stretches. Physical mechanisms of the Mullins effect have also been predicted based on the evolution of constitutive parameters.
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