Peculiarities of the Structure and Mechanical Properties of Biological Tissues*

Five basic principles that determine the structure and mechanical behavior of almost any biological tissue have been formulated. They are the principle of hierarchy, principle of helicality, principle of feedback, principle of universality, and principle of an optimum. It is shown that there are also two types of factors – biological and mechanical – that must be taken into account in determining the mechanical properties of biological tissue. As an example of complexity of the mechanical behavior of a biological tissue the data on compact bone tissue are presented.     

Calculation of the shock-wave adiabatics for some heterogeneous mixtures

A method has been proposed [1] for calculating shock-wave adiabatics for two-phase mixtures which does not use the additivity principle, and this is here extended to a medium consisting of n phases; the shock-wave adiabatic is derived and compared with experiment for a three-phase mixture; an alloy consisting of copper, zinc, and lead (brass). It is shown for a paraffin-tungsten mixture that the adiabatics for the individual phases deviate from ones for the corresponding solid materials. The known equations of state for aluminum and epoxide resin have been used to determine the shock-wave adiabatic for such a mixture. A comparison is made with the analogous calculation made from the additivity principle.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 130–137, July–August, 1973.I am indebted to V. N. Nikolaevskii for suggesting the topics and for useful discussion.     

The pinch-off process in a starting buoyant plume

The vortex ring formation process of a starting buoyant plume was studied experimentally using digital particle image thermometry and velocimetry (DPITV). The vortex ring was observed to pinch-off, or become disconnected, from the trailing plume. Pinch-off occurred at non-dimensional times, or formation numbers, between 4.4 and 4.9. The observed pinch-off process is consistent with an explanation based upon the Kelvin–Benjamin variational principle. This is analogous to the pinch-off of a vortex ring generated using a piston–cylinder apparatus, suggesting that pinch-off is a general component of the vortex ring formation process for various generation mechanisms.     

Initiation and stable growth of penny-shaped crack in aging viscoelastic transversally isotropic material

Considered is the long-term cracking of an aging transversally isotropic material containing a Mode I penny-shaped crack under remotely applied tensile stress. The aging material properties are described by the Boltzmann–Volterra’s linear theory for integral operators with non-difference kernels. It applied to wood, concrete, some polymers and rocks. Only the symmetric case is considered where the crack lies in the plane of isotropy. The modified Leonov–Panasyuk–Dugdale’s crack model is used with a constant process zone assuming that the critical opening displacement is the fracture criterion. Volterra’s principle is applied to derive the equations of subcritical crack growth. Numerical calculations are made for subcritical crack growth for the specific example of transversally isotropic material simulating the behavior of reinforced concrete.     

Variational methods for elliptical boundary-value problems with nonsymmetric tensor coefficients

Conclusions In addition to the methodological value in principle, the principle obtained has a practical application since it permits replacement of the domain during estimation of the resistance of Hall conductors by a simpler one. In particular, it is necessary to perform triangulation for a variational-difference realization of the upper bound such that the broken lines approximating the boundary with the insulator would pass within the domain while the approximating boundary to the ideal conductor would pass outside the domain. In this and only in this case, is the value obtained numerically is the true upper bound. For the lower bound the broken lines must be constructed in an opposite manner. The efficiency of the proposed symmetrization for the numerical solution of the problems is demonstrated in [10, 11].The boundedness of the proofs does not permit asserting the validity of the formulated principle for variable coefficients; however, in the two-dimensional cases foundations for doubts are not seen.Let us note that the proof of the energetic principles in [4–7] is carried out for a sufficiently broad circle of problems. Piecewise continuity, uniform boundedness, and uniform positive-definiteness of its symmetric part were required from the conductivity tensor s.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 69–75, May–June, 1989.     

Development of a submerged turbulent isothermal jet and the minimum entropy-production principle

It is shown in the example of a submerged turbulent isothermal jet that the process of turbulent mixing of gas jets can evolve in accordance with, the principle of minimum entropy production.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 52–58, July–August, 1978.     

A Boussinesq–Cerruti Solution Set for Constant and Linear Distribution of Normal and Tangential Load over a Triangular Area

Starting from the elastic solution to a concentrated load on an elastic half space, this paper derives all the required Boussinesq–Cerruti equations for constant, linear and bilinear distributions of normal and tangential load over a triangle area, and presents a solution set to the equations. The surface displacement field in both the normal and tangential direction is obtained. The evaluations of Boussinesq–Cerruti equations are achieved by using various integration techniques. This paper also suggests a composition methodology to construct the solution due to more complicated loading profiles using the principle of superposition.     

Aerodynamic equivalence of asymmetric bodies with allowance for nonlinear body shape influence factors and angle of attack

In order to determine the aerodynamic coefficients of asymmetric bodies the principle of aerodynamic equivalence is used. In the gas dynamic function expansions in the asymmetry parameter and the angle of attack both linear and second-order terms are taken into account. This makes it possible to establish theoretically the principle of aerodynamic equivalence with allowance for nonlinear body shape influence factors and angle of attack. A universal structure of the aerodynamic characteristics of asymmetric bodies with an arbitrary cross-sectional shape is obtained. The theoretical conclusions are confirmed by the results of a numerical solution of the three-dimensional gas dynamic problem.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 116–122, November–December, 1993.     

Optimal Bounded Control for Minimizing the Response of Quasi Non-Integrable Hamiltonian Systems

A strategy for designing optimal bounded control to minimize theresponse of quasi non-integrable Hamiltonian systems is proposed basedon the stochastic averaging method for quasi non-integrable Hamiltoniansystems and the stochastic dynamical programming principle. Theequations of motion of a controlled quasi non-integrable Hamiltoniansystem are first reduced to an one-dimensional averaged Itô stochasticdifferential equation for the Hamiltonian by using the stochasticaveraging method for quasi non-integrable Hamiltonian systems. Then, thedynamical programming equation for the control problem of minimizing theresponse of the averaged system is formulated based on the dynamicalprogramming principle. The optimal control law is derived from thedynamical programming equation and control constraints without solvingthe equation. The response of optimally controlled systems is predictedthrough solving the Fokker–Planck–Kolmogrov (FPK) equation associatedwith completely averaged Itô equation. Finally, two examples are workedout in detail to illustrate the application and effectiveness of theproposed control strategy.     

Almost Periodic Solutions and Global Attractors of Non-autonomous Navier–Stokes Equations

The article is devoted to the study of non-autonomous Navier–Stokes equations. First, the authors have proved that such systems admit compact global attractors. This problem is formulated and solved in the terms of general non-autonomous dynamical systems. Second, they have obtained conditions of convergence of non-autonomous Navier–Stokes equations. Third, a criterion for the existence of almost periodic (quasi periodic, almost automorphic, recurrent, pseudo recurrent) solutions of non-autonomous Navier–Stokes equations is given. Finally, the authors have derived a global averaging principle for non-autonomous Navier–Stokes equations.     

Use of compression wave focusing for the production of a superjet

The principle of focusing compression waves by their reflection from nonflat surfaces is well-known. The invention in [1] postulates for this purpose the use in a pulsed hydrojet of a piston with a specially shaped surface. In this paper, the use of focusing is investigated numerically for the production of superjets.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 153–156, September–October, 1976.     

Bounds for the effective properties of heterogeneous plates

This paper presents new bounds for heterogeneous plates which are similar to the well-known Hashin–Shtrikman bounds, but take into account plate boundary conditions. The Hashin–Shtrikman variational principle is used with a self-adjoint Green-operator with traction-free boundary conditions proposed by the authors. This variational formulation enables to derive lower and upper bounds for the effective in-plane and out-of-plane elastic properties of the plate. Two applications of the general theory are considered: first, in-plane invariant polarization fields are used to recover the “first-order” bounds proposed by Kolpakov [Kolpakov, A.G., 1999. Variational principles for stiffnesses of a non-homogeneous plate. J. Meth. Phys. Solids 47, 2075–2092] for general heterogeneous plates; next, “second-order bounds” for n-phase plates whose constituents are statistically homogeneous in the in-plane directions are obtained. The results related to a two-phase material made of elastic isotropic materials are shown. The “second-order” bounds for the plate elastic properties are compared with the plate properties of homogeneous plates made of materials having an elasticity tensor computed from “second-order” Hashin–Shtrikman bounds in an infinite domain.     

Shock compression of two-component paraffin-tungsten mixtures

Up to pressures of 2 mbar, curves have been recorded experimentally for the shock compression of paraffin and for two paraffin-tungsten mixtures, containing 66.2 and 84.0 wt.% tungsten. It is shown that over the whole range of pressures investigated experimentally, the mixture shock adiabatic curves satisfy the principle of additivity. Attention is called to the great potential of the method of weighting additives for the investigation of light media at high shock pressures. The criteria for the applicability of the principle of additivity have been analyzed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 152–155, July–August, 1971.     

Joukowsky formula for the lift for a cylinder in an arbitrary steady stream of an ideal incompressible fluid

Joukowsky's formula is generalized to the case of a circular cylinder in an arbitrary stream of ideal fluid whose velocity on the cylinder can be expanded in a Fourier series. The principle of reversibility of the motion in the case of an arbitrary steady displacement of a profile is discussed. Finally, Joukowsky's formula is generalized for any profile in a flow with homogeneous (uniform) vorticity. Illustrations are given. It is shown that the results described in the literature follow from the generalized Joukowsky formula derived in the present paper as special cases.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 6–18, March–April, 1981.I should like to express my thanks to G. Yu. Stepanov, who carefully read the draft, made a number of constructive comments about it, and gave much encouragement aimed at deepening the content and improving the exposition.     

Control of the convective equilibrium stability of a fluid heated from below

Potentially unstable equilibrium is stabilized by means of a method of control based on the single-channel feedback principle using the characteristics of the solution of the complete nonlinear thermal convection equations. The convection equations are solved by a finite-difference method.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 6–11, July–August, 1990.     

Determining the radius of the air vortex during the laminar flow of a liquid in a centrifugal atomizer

In existing theories of centrifugal atomizers, such as that of Abramovich [1], in order to determine the radius r₀ of the air vortex the conditions of the maximum rate of flow or some other extremal principle are conventionally employed. In this paper the radius of the air vortex will be determined from the equations of motion of a viscous incompressible liquid.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 54–57, May–June, 1976.     

The integral representation of fractionally exponential functions and their application to dynamic problems of linear visco-elasticity

Fractionally exponential functions are written in the integral form and distribution functions with an Abelian singularity are obtained for the corresponding relaxation and retardation spectra. A principle is stated, defining the dynamic problems for which weakly singular functions can be used as the kernels of the integral operators. A one-dimensional sound wave traveling in a semiinfinite visco-elastic medium is considered. The generalized exponential functions of fractional order, proposed by Yu. N. Rabotnov [1, 2] as the kernels of Boltzmann-Volterra integral relations, have found wide applications in theory of linear visco-elasticity. This is explained partly by the great mathematical flexibility of the F-operators when applying the Volterra principle to the solution of elastically hereditary problems and partly by the fact that almost all weakly singular kernels possessing an Abelian singularity are connected in some way or other with the F-functions. For example, the resolvent of the elementary weakly singular Abelian kernel is an F-function. The product of an exponential function with an Abelian kernel represents a particular case of the product of two F-functions with different fractional parameters, while the resolvent of such a kernel is the product of an exponential function with an F-function [3, 4]. Since the e-functions are defined by slowly convergent series, their various asymptotic forms [2, 5–8] are commonly used in practical calculations. The theory of F-functions can be developed further in the context of their integral representations, which enables a more exact physical interpretation to be given to their parameters and on occasion simplifies computational operations.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 1, pp. 103–110, January–February, 1970.In conclusion, the author thanks Yu. N. Rabotnov for discussion.     

Stability of two-dimensional curvilinear flows of an ideal barotropic fluid

It was established by Arnol'd [1] that the conservation laws for the energy and vorticity can be used to establish sufficient conditions of stability of two-dimensional curvilinear flows of an ideal incompressible fluid in the exact nonlinear formulation. It is shown below that one can obtain similarly conditions of stability of two-dimensional curvilinear steady flows of an ideal barotropic fluid in the linear approximation. One of the conditions has a significance similar to Rayleigh's criterion and its generalization by Arnol'd [1]; the other is the condition of subsonic flow. In addition, a variational principle is established and an expression found for the second variation of the corresponding functional; these can be used to prove the stability of these flows in the exact nonlinear formulation.Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 5, pp. 19–25, September–October, 1981.I am sincerely grateful to V. L. Berdichevskii and A. G. Kulikovskii for constructive advice.     

Subcritical Growth of an Internal Circular Crack in an Aging Viscoelastic Laminated Composite

Delayed fracture of a laminated composite under tensile loads applied at infinity is studied. The composite consists of alternating elastic and aging viscoelastic layers and contains an internal penny-shaped mode I macrocrack located in parallel to the layers. A modified Leonov–Panasyuk–Dugdale crack model and the critical crack-tip opening criterion constitute a fracture model. The subcritical crack growth equations are derived using the Volterra principle and the method of operator continued fractions. The laws governing delayed fracture are studied for a specific composite material     

Nonlinear Oscillations of Viscoelastic Rectangular Plates

Nonlinear oscillations of viscoelastic simply supported rectangular plates are studied by assuming the Voigt–Kelvin constitutive model. Using Hamilton's principle in conjunction with the kinematics associated with Kirchhoff's plate model, the governing equations of motion including the effect of damping are represented in terms of the transversal deflection and a stress function. Utilizing the Bubnov–Galerkin method, the nonlinear partial differential equations are reduced to an ordinary differential equation which is studied geometrically by approximate construction of the Poincaré maps. Explicit expressions are given for periodic solutions.