By Oxana Sadovskaya, Vladimir Sadovskii, Holm Altenbach

This monograph comprises unique leads to the sector of mathematical and numerical modeling of mechanical habit of granular fabrics and fabrics with diverse strengths. It proposes new types assisting to outline zones of the tension localization. The booklet exhibits find out how to examine methods of the propagation of elastic and elastic-plastic waves in loosened fabrics, and constructs versions of combined variety, describing the circulation of granular fabrics within the presence of quasi-static deformation zones. In a final half, the ebook experiences a numerical attention of the versions on multiprocessor computers.

The publication is meant for clinical researchers, academics of universities, post-graduates and senior scholars, who concentrate on the sector of the deformable fabrics mechanics, mathematical modeling and adjoining fields of utilized and calculus mathematics.

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**Additional info for Mathematical Modeling in Mechanics of Granular Materials**

**Example text**

If each function fl (u), where l = 1, . . , n, is convex then the set F = u ∈ Rm fl (u) ≤ 0, l = 1, . . , n is convex as well. To prove it, we consider two elements u and u˜ ∈ F. e. uλ ∈ F. Hence, and fl (u) F is convex. It turns out that any convex set can be described with the help of some convex function. The Minkowski function yields one way of such a description. Assume that F is a convex closed set in the space Rm (u) for which the point 0 (the origin of coordinates) is an interior point (with at least one interior point of the set, this requirement can be always satisfied due to translation of a coordinate system).

Let f (u) be convex. 2) holds. Using the Taylor expansion of its left-hand side in the neighborhood of the point u with the estimate of a remainder term in the Peano form, we obtain the inequality 1 ∂ 2 f (u) (u˜ − u) (u˜ − u) + o |u˜ − u|2 ≥ 0, 2 ∂ u2 which leads to the condition of non-negative definiteness of ∂ 2 f /∂ u2 at the point u ∈ F since u˜ is arbitrary. A proof of the converse statement is based on the Taylor ˜ with a remainder term in the Lagrange form: expansion of the function f (u) ˜ = f (u) + (u˜ − u) f (u) ∂ 2 f (uλ ) ∂ f (u) 1 + (u˜ − u) (u˜ − u), ∂u 2 ∂ u2 where λ ∈ (0, 1).

Later on a medium adapts itself to the periodic load and never achieves a compaction mode (Fig. 17). In the case of an increasing amplitude, the interval of a closed state of a contact is periodically repeated (Figs. 19). With increasing frequency (see Figs. 21), the curve 2 approaches to the curve 1. These curves can coincide exactly only in the case of a rheological scheme involving a single elastic element, hence, the influence of viscosity, plasticity, and heterostrength in comparison with elastic properties of a material becomes insignificant with increasing a loading frequency.