Vol 23, No 1 (2019)

In the present paper, the main force parameter is estimated under the static tests. This parameter is the specific penetration resistance and it is generally believed that the parameter is a constant in case of static penetration. However, a number of experimental and analytical data illustrate varying the specific penetration resistance according to the current depth of penetration. It is noted that the weakening free-surface effect decreases the specific penetration resistance near the facial or rear edge. Consequently, the relevance of the topic is emphasized by the influence of the facial and rear weakening free-surface effect on the penetration parameters detected in the experimental studies and engineering calculations.
The refined approximation of the specific penetration resistance presented in this paper is taking account of the penetration of the sharp indenter into the plate of middle thickness within the framework of the viscous crater formation and the facial and rear weakening free-surface effect. Also this article contains data processing technique.
For carrying out the tests a number of experimental samples were made. It is plates of different thicknesses, it must be emphasized that test sample materials are technical plasticine, plumbum and Wood's metal. It should also be noted that for the static tests three cone-nose indenters were made. Indenter sizes: the diameter of the cylindrical part is 7 mm in all cases and the lengths of the conical nose are 3.2 mm, 5.6 mm, and 8.4 mm. The test were carryed out on a testing machine Zwick/Roell Z-250. The key parameters derived from the experiment are the specific penetration resistance of the deep layers, the friction coefficient and the parameters of the weakening free-surface effect.
The results obtained in the experiment lead to the approximation of the resistance force from more general parameters. These parameters are the specific penetration resistance of the deep layers and the friction coefficient of a sample, geometric parameters of indenter and plate. An approximation error does not exceed 25 % for the technical plasticine, 16 % for the Wood's metal, and 25 % for the plumbum. These errors are given for “sharp” (the length of the cone-nose is 8.4 mm) and “middle” (the length of the cone-nose is 5.6 mm) indenter because of a problem has been in depth considered in the investigation. This problem is that penetration of the “blunt” indenter is not follow to condition of viscous crater formation. Therefore, different versions should be used to describe the penetration process (for example, plugging mechanism).
It is proposed in penetration models for the estimation of the penetration resistance force of sharp indenters into the plate of middle thickness.

On the basis of the time-steps algorithm the structural model is constructed for elastic-plastic deformation of bended plates with spatial reinforcement structures. The inelastic behavior of the composition phase materials is described by equations of the theory of plastic flow with isotropic hardening. The possible weakened resistance of the reinforced plates to the transverse shear is taken into account on the basis of the refined theory, from which the relations of the Reddy theory are obtained in the first approximation. The geometric nonlinearity of the problem is considered in the Karman approximation. The solution of the formulated initial boundary value problems is based on an explicit numerical “cross” scheme. The dynamic inelastic deformation of spatially- and flat-cross-reinforced metal-composite and fiberglass flexible plates of different relative thickness is investigated in the case of the load caused by an air blast wave. It is demonstrated that for relatively thick fiberglass plates, the replacement of the flat-cross reinforcement structure by the spatial structure with the preservation of the total fiber consumption leads to a decrease in the structural flexibility in the transverse direction by almost 1.5 times, as well as to a decrease of the maximum of intensity of deformation in the binder by half. For relatively thin both fiberglass and metal-composite plates, the replacement of flat-cross 2D reinforcement structure with 3D and 4D spatial structures does not lead to a noticeable decrease in their deflections, but allows to reduce the intensity of deformations in the binder by 10 % or more. It is shown that the widely used non-classical Reddy theory does not allow obtaining reliable results of calculations of the elastic-plastic dynamic behavior of the bended plates, both with plane and spatial reinforcement structures, even with a small relative thickness of the structures and weak anisotropy of the composition.

The paper deals with applying mathematical modeling to study tumor growth process and optimizing cancer treatment. A structured review of the studies devoted to this problem is given. The role of the cell life cycle in understanding the tumor growth and the mechanisms of cancer treatment is discussed. It is important that modern cancer treatment methods, in particular, chemotherapy and radiation therapy, affect both normal and tumor cells in certain stages of the life cycle and do not influence on cells in other stages. Cell life cycle description is given as well as the mechanisms that maintain and restore normal density of the cell population. A graph of cell life cycle stages and transitions is demonstrated. Dynamic mathematical model of proliferative homeostasis in the cell population is proposed, which takes into account the heterogeneity of cell populations by life cycle stages. The model is a system of differential equations with delays. The stationary state of the model is investigated, which allows to determine the parameters values for the normal cell population. The results of a numeric experiment is obtained, which is focused on the process of cell population density recovery after mass death of cells. As the experiment shows, after cell death, the densities of cells in different life cycle stages are restored to normal values, which corresponds to the concepts of proliferative homeostasis in cell populations.

In the first part of this study, the well-posed Goursat-type problem is considered for the hyperbolic differential equation of the third order with non-multiple characteristics. The example illustrating the non-well-posed Goursat-type problem for the hyperbolic differential equation of the third order is discussed. The regular solution of the Goursat-type problem for the hyperbolic differential equation of the third order with the non-multiple characteristics is obtained in an explicit form.
In the second part, the well-posed Goursat-type problem is considered for a system of the hyperbolic differential equations of the third order. The regular solution of the Goursat-type problem for this system is also obtained in an explicit form.
The theorems for the Hadamard's well-posedness of Goursat-type problem for the hyperbolic differential equation and for a system of the hyperbolic differential equations is formulated as the result of the research.