Effect of an orientation-dependent non-linear grain fluidity on bulk directional enhancement factors

Bulk directional enhancement factors are determined for axisymmetric (girdle and single-maximum) orientation fabrics using a transversely isotropic grain rheology with an orientation-dependent non-linear grain fluidity. Compared to grain fluidities that are simplified as orientation independent, we find that bulk strain-rate enhancements for intermediate-to-strong axisymmetric fabrics can be up to a factor of ten larger, assuming stress homogenization over the polycrystal scale. Our work thus extends previous results based on simple basal slip (Schmid) grain rheologies to the transversely isotropic rheology, which has implications for large-scale anisotropic ice-flow modelling that relies on a transversely isotropic grain rheology. In order to derive bulk enhancement factors for arbitrary evolving fabrics, we expand the c-axis distribution in terms of a spherical harmonic series, which allows the rheology-required structure tensors through order eight to easily be calculated and provides an alternative to current structure-tensor-based modelling.

Direct Eulerian formulation of anisotropic hyperelasticity

Abstract Many soft materials and biological tissues comprise isotropic matrix reinforced by fibers in the characteristic directions. Hyperelastic constitutive equations for such materials are usually formulated in terms of a Lagrangean strain tensor referred to the initial configuration and Lagrangean structure tensors defining characteristic directions of anisotropy. Such equations are “pushed forward” to the current configuration. Obtained in this way, Eulerian constitutive equations are often favorable from both theoretical and computational standpoints. Abstract In the present note, we show that the described two-step procedure is not necessary and anisotropic hyperelasticity can be introduced directly in terms of an Eulerian strain tensor and Eulerian structure tensors referring to the current configuration. The newly developed constitutive equation is further applied to the particular case of the transverse isotropy for the sake of illustration.

Vanishing Hessian, wild forms and their border VSP

Wild forms are homogeneous polynomials whose smoothable rank is strictly larger than their border rank. The discrepancy between these two ranks is caused by the difference between the limit of spans of a family of zero-dimensional schemes and the span of their flat limit. For concise forms of minimal border rank, we show that the condition of vanishing Hessian is equivalent to being wild. This is proven by making a detour through structure tensors of smoothable and Gorenstein algebras. The equivalence fails in the non-minimal border rank regime. We exhibit an infinite series of minimal border rank wild forms of every degree $$d\ge 3$$ d ≥ 3 as well as an infinite series of wild cubics. Inspired by recent work on border apolarity of Buczyńska and Buczyński, we study the border varieties of sums of powers $$\underline{{\mathrm {VSP}}}$$ VSP ̲ of these forms in the corresponding multigraded Hilbert schemes.

On an Euler-Lagrange equation for color to grayscale conversion

In this paper we give a new method to find a grayscale image from a color image. The idea is that the structure tensors of the grayscale image and the color image should be as equal as possible. This is measured by the energy of the tensor differences. We deduce an Euler-Lagrange equation and a second variational inequality. The second variational inequality is remarkably simple in its form. Our equation does not involve several steps, such as finding a gradient first and then integrating it. We show that if a color image is at least two times continuous differentiable, the resulting grayscale image is not necessarily two times continuous differentiable.

Some characterizing results for hemi-slant warped product submanifolds of a Kaehler manifold

Many differential geometric properties of submanifolds of a Kaehler manifold are looked into via canonical structure tensors P and F on the submanifold. For instance, a CR-submanifold of a Kaehler manifold is a CR-product (i.e. locally a Riemannian product of a holomorphic and a totally real submanifold) if and only if the canonical tensor P is parallel on the submanifold. Since, warped product manifolds are generalized version of Riemannian product of manifolds, in this article, we consider the covariant derivatives of the structure tensors on a hemi-slant submanifold of a Kaehler manifold. Our investigations have led us to characterize hemi-slant warped product submanifolds.

A stochastic model of stroma: interweaving variability and compressed fibril exclusion

Hyperelastic constitutive models of the human stroma accounting for the stochastic architecture of the collagen fibrils and particularly suitable for computational applications are discussed. The material is conceived as a composite where a soft ground matrix is embedded with collagen fibrils characterized by non-homogeneous spatial distributions typical of reinforcing stromal lamellae. A multivariate probability density function of the spatial distribution of the fibril orientation is used in the formulation of the lamellar branching observed on the anterior third of the stroma, selectively excluding the contribution of compressed fibrils. The physical reliability and the computational robustness of the model are enhanced by the adoption of a second order statistics approximation of the average structure tensors typically employed in fiber reinforced models.