A stereomicroscope system is adapted to make accurate, quantitative displacement, and strain field measurements with microscale spatial resolution and nanoscale displacement resolution on mouse carotid arteries. To perform accurate and reliable calibration for these systems, a two-step calibration process is proposed and demonstrated using a modification to recently published procedures. Experimental results demonstrate that the microscope system with three-dimensional digital image correlation (3D-DIC) successfully measures the full 3D displacement and surface strain fields at the microscale during pressure cycling of 0.40-mm-diameter mouse arteries, confirming that the technique can be used to quantify changes in local biomechanical response which may result from variations in extracellular matrix composition, with the goal of quantifying properties of the vessel.
The partial differential equation for isotropic hyperelastic constitutive models has been postulated and derived from the balance between stored energy and stress work done. The partial differential equation as a function of three invariants has then been solved by Lie group methods. With geometric meanings of deformations, the general solution boils down to a particular threeterm solution. The particular solution has been applied for several isotropic hyperelastic materials. For incompressible materials, vulcanized rubber containing 8% sulfur and Entec Enflex S4035A thermoplastic elastomer, three coefficients have been determined from uniaxial tension data and applied to predict the pure shear and equibiaxial tension modes. For a slightly compressible rubber material, the coefficients have also been extracted from the confined volumetric test data.
An anisotropic continuum stored energy (CSE), which is essentially composed of invariant component groups (ICGs), is postulated to be balanced with its stress work done, constructing a partial differential equation (PDE). The anisotropic CSE PDE is generally solved by the Lie group and the ICGs through curvatures of elasticity tensor are particularly grouped by differential geometry, representing three general deformations: preferred translational deformations, preferred rotational deformations, and preferred powers of ellipsoidal deformations. The anisotropic CSE constitutive models have been curve-fitted for uniaxial tension tests of rabbit abdominal skins and porcine liver tissues, and biaxial tension and triaxial shear tests of human ventricular myocardial tissues. With the newly defined second invariant component, the anisotropic CSE constitutive models capture the transverse effects in uniaxial tension deformations and the shear coupling effects in triaxial shear deformations.
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.
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.
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