A multiscale computational model was generated for each strip test (
Fig. 2) that was tested under Load Protocol A. The computer model was fitted to the loading curve of each load cycle independently. The computer model consisted of a finite element mesh representing the macrostructure of the strip. The mesh was composed of 400 quadratic hexahedral elements of serendipity-type. Please note that only one-fourth of the model shown in
Figure 2 (bottom right) was used in the calculation, accounting for the symmetry of the boundary value problem. We used our microstructure-based, hyperelastic constitutive model to account for the hierarchical collagen structure of the sclera.
30,35,58 At the microscale, collagen fibrils are assumed to crimp and buckle when the sclera is unloaded. When tension is applied to the sclera, this crimp is removed and the sclera stiffens as the collagen fibrils straightened. At the mesoscale, collagen fibrils form scleral lamellae that are strongly interwoven. We assume that the lamellae are randomly oriented tangential to the scleral surface. The random orientation of the scleral lamellae are represented by a von Mises distribution with concentration parameter
b = ∞ in the computer model. We assume that the scleral lamellae are embedded in an isotropic tissue matrix, which represents all noncollagenous tissue components (e.g., elastin, GAGs, PGs, cells, and fluids). The matrix is mostly composed of tissue water and therefore assumed to be isotropic and incompressible, and its elastic contribution described by the shear modulus
μ and bulk modulus. Nearly incompressible deformation behavior is achieved by setting the bulk modulus to 1000 times the value of the tissue's shear modulus. The constitutive model consists of four unknown parameters: the elastic modulus of collagen fibrils
Efib; the crimp angle of collagen fibril in the unloaded sclera
θ0; the ratio between the crimp amplitude and the fibril cross-sectional radius
R0/
r0; and the shear modulus of the sclera
μ. We applied the experimentally recorded deformations as boundary conditions to the computer model and computed the reaction forces at the clamps. The four model parameters were independently fitted for each load cycle by minimizing the sum of squared residuals between the computationally predicted and experimentally measured boundary forces. The fitting was performed by using the Nelder-Mead simplex algorithm, which is freely available through the Open Source Library of Scientific Tools (SciPy). We varied the initial guess of unknown model parameters over a wide range of values (
Efib = [0.1, 300.0 MPa];
θ0 = [0.1°, 50°];
R0/
r0 = [1.0, 50.0];
μ = [0.0001, 1.0 MPa]) to verify that the fitting algorithm converged to the global minimum. Detailed derivations of the constitutive model and its implementation into a finite element code can be found in our previous publications.
30,35,58