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.

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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.

The uses of a weighting factor along with a time step in a single-step trapezoidal method to solve a first-order parabolic system have been systematically studied. The weighting factors are used in two main types: constants and variables. The most commonly used constant weighting factors can be defined by the ratio of the Fibonacci sequence. Among them, the optimal weighting factor is 0.618, resulting in a balance between the overall accuracy and efficiency. With the finite element formulation, the space and time dimensions can be discretized separately. For the time discretization only, there exists a zero-error dimensionless time step if a weighting factor is within the range of 0.5–1.0. By taking advantage of the zero-error condition, the weighting factor can be correlated with a time step. The influence of spatial dimensions is lumped into a nonzero eigenvalue of the system. Through validity tests of two benchmark linear problems, the variable weighting factor for a single-step trapezoidal method is shown to be accurate, efficient, and stable. The relevant features have been captured.