Abstract
Soft tissues, such as skin, myocardium, and chordae tendineae, typically display anisotropic mechanical behavior due to their fibrous nature. In constitutive modeling, fiber families frequently are assumed to be unidirectional. Recent numerical results, however, display the need to incorporate dispersion of fiber orientation. This evidence gets supplemented by new experimental results based on high-resolution second-harmonic imaging microscopy. Generalized structure-tensor (GST) models are frequently utilized to model fiber dispersion, as they are mathematically easy to treat and demand only a little effort to implement. They can be regarded as Taylor-series expansions of the numerically more challenging angular-integration (AI) method, which encompasses a distribution of fiber orientations together with the associated fiber stress. In this work, we show how low-order GST models give rise to numerical instabilities as they show strong sensitivity with regards to the mean fiber orientation. To overcome these instabilities, we propose a different class of GST models, termed squared GST (SGST), which computes faster, is easier to implement, and converges to the AI faster than previous GST models of similar order. The SGST models promise to be adaptable to generalized problems, such as functional decomposition of fiber density as well as coupling between different fiber families. Advanced simulations with the proposed models will shed light on the complex behavior of fiber reinforced soft materials.
Highlights
Whether it is neurons, myocytes, or collagen, most soft biological tissues comprise fibers leading to anisotropic mechanical behavior [1,2]
To prevent instabilities arising from sensitivities, we develop a class of Generalized structure-tensor (GST) models, termed squared generalized structure-tensor (SGST) models, which closely match the directional sensitivity of the AI model [cf. 2SGST in Fig. 1(b)], while still offering the framework of classical GST models, yielding an easy implementation and rapid numerical calculation
As fiber architecture has a strong influence on material mechanics, said analysis contains, both fibers aligned with principle directions of the deformation, as well as fibers rotated by 45◦ in the respective plane
Summary
Myocytes, or collagen, most soft biological tissues comprise fibers leading to anisotropic mechanical behavior [1,2]. The second school, on the other hand, follows the steps of Gasser et al [6], who translate the fiber architecture into a generalized structure tensor (GST), which describes collective fiber deformation. Both schools, expectedly, carry their own advantages and disadvantages. For example, the (fn1) shear mode with a shear of γ = 0.2, a dispersionless (HO) and a classical GST model (0GST) deviate in stress by roughly 25% if the fibers are rotated by only 1◦, contrary to the AI model, which varies only by roughly 5% An easy-to-use MATLAB library, comprising the different models up to sixth polynomial order, is provided as Supplemental Material [21]
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