Mechanics Of Continuous Media Research Articles

A slender-body theory for interfacial failure in unidirectional fiber-reinforced composites

Based on asymtotic techniques which have been recently developed for the mechanics of linear continuous media containing slender inclusions, a theory is presented for the nonlinear problem of interfacial failure in a fiber-reinforced composite. Based on the lowest order slender-body theory, and a correspondingly simple model of interfacial slip or plastic yielding, a constitutive equation is, derived for unidirectional, dilute-fiber composites. This equation provides a tensile stress-strain relation which exhibits microscopic, ductile yield arising from the microscopic failure process. Approximate formulae are proposed to account for fiber interactions among closely-spaced parallel fibers and for interfacial slip with sliding friction. It is shown that all the results can be represented in terms of a “reduced-variable” plot, which suggests that experirniental data for various fiber aspect ratios and concentrations might he reducible to a single curve, depending only on tine mode of interfacial failure.

Seminars of the Institute for problems of mechanics of the USSR (seminar on the mechanics of continuous media under the direction of L. A. Galin)

Seminars of the Institute for problems of mechanics of the USSR (seminar on the mechanics of continuous media under the direction of L. A. Galin)

Some examples of invariant solutions of the liquid mechanics equations

Invariant solutions of the equations of planar motion written in a coordinate system related to previously unknown flow lines are considered. The models chosen for the continuous medium are an incompressible liquid which may be viscous and conductive, a compressible gas, and an electron plasma with zero temperature. At the present state of mechanics of continuous media, the method of [1, 2] for construction of exact solutions is widely known, based on study of the group properties of the corresponding equations. Over the past two decades this method has been used to obtain new results and to systematize previously known facts [3]. In this case within the space (x, u, p) the system of equations under study forms a hypersurface, upon the form of which all conservative transformations depend. The set of independent variables x is usually referred to some specified reference frame (most often, a Cartesian one). In [4] the study of group properties of a three-dimensional boundary layer on an arbitrary surface was reduced to the problem of specialization of an arbitrary element, which, in the formulation used, is analogous to the problem of a nonlinear thermal conductivity equation [2]. In [5] the problem of calculating a spatially viscous supersonic jet was formulated in the coordinates xi, related to a previously unspecified flow geometry. In such an approach the liquid mechanics equations should be supplemented by conditions for a Euclidean space, while the components of the metric tensor gik are the functions to be found together with the normal hydrodynamic variables. The system also includes differential equations relating the Cartesian and curvilinear coordinates. Reference [5] presented a brief review of studies employing “geometrization” in one form of another. We will demonstrate below, using the example of planar motion of a liquid, that such an approach leads to a new system of equations for description of well known phenomena, which system defines a hypersurface with certain new properties and has a number of new exact solutions.

Mechanics of Continuous Media. By S. C. HUNTER. Halsted Press, 1977. 567 pp. £15 (hardback) or £6.90 (paperback).

<b>Mechanics of Continuous Media.</b> By S. C. H<scp>UNTER</scp>. Halsted Press, 1977. 567 pp. £15 (hardback) or £6.90 (paperback).

Interpolational equation of state of NaCl, taking account of processes of melting, vaporization, dissociation, and ionization

With the investigation of many problems in the mechanics of continuous media, geophysics, etc., for the mathematical closure of the problems under consideration there are needed equations ofstate of the continuous media, describing their thermodynamic properties in a wide region of states, including the crystalline, molten, vaporized, dissociated, and ionized states. In [1-3], methods are developed for the construction of interpolational equations of state, taking in the above regions, with the exception of the region of melting. For a number of substances, the melting process has a considerable energy consumption and cannot be neglected in applied calculations. The present article, using the example of NaCI, develops the basic concepts of [I, 2] and constructs an equation of state which takes account of all the abovementioned processes, including melting.

Admissible groups of transformations for certain third-order quasilinear equations

We construct the infinitesimal operators of Lie algebras for the broadest group of transformations leaving invariant some quasilinear third-order equations with two independent variables occurring in the mechanics of continuous media.

Mechanics of Continuous Media

Mechanics of Continuous Media

Two effective methods of solving mixed linear problems of mechanics of continuous media: PMM vol. 41, n≗4, 1977, pp. 688–698

Two effective methods of solving mixed linear problems of mechanics of continuous media: PMM vol. 41, n≗4, 1977, pp. 688–698

On prospective trends and problems in mechanics of continuous media: PMM vol.40, n≗ 6, 1976, pp. 963–980

On prospective trends and problems in mechanics of continuous media: PMM vol.40, n≗ 6, 1976, pp. 963–980

Methods of statistical physics as a new approach to the mathematical theory of epidemics

t In accordance with government policy, the USSR is currently developing the so-called Epidsluzhba automatic control system. An important element in the software package for this system is the creation o f mathematical processes describing epidemic processes. A large number o f models are used in the mathematical theory of epidemics [2, 13], but, with rare exceptions, the results are not adequate to reality. This is primarily to be explained by the fact that epidemics generally run their course in large groups and are hence described by a dynamic system with a large number n o f degrees o f freedom. For n >> 1 the solution of the resultant equations is not only an extremely laborious problem, but one that is virtually useless in practice. In mathematical epidemiology, therefore, the a t tempt is not to use equations that yield a detailed "microdescript ion" o f the process, but rather a "macrodescript ion" of the process, as a whole, i.e., certain average characteristics such as the percentage o f those affected or the proport ion o f individuals for whom certain parameter values (e.g., antibody or antigen count) lie within certain limits. At the same tinie, it was the use o f "microequations" that made it possible for Rvachev to obtain important results in describing certain epidemic processes [1, 4] . An irremediable shortcoming o f his theory, however, was the theoretical impossibility o f using it effectively in nondeterministic (probabilistic) models, since the basis for the theory is provided by the mechanics of continuous media. Like direct recourse to macrodescription of epidemic processes, this approach involves ignoring individual differences, and therefore the problem of deriving "macroequations" from "microequations," allowing for such differences, remains a crucial one. To solve this problem, it is convenient to use the following fundamental notion of statistical physics [11, 12]: we should study not just one dynamic system taken individually, but an ensemble o f such systems with a probabil i ty measure introduced on this ensemble. Thus, the problem of describing the behavior of a specific dynamic system is replaced by one o f describing the behavior o f the overwhelming majority (in probabil i ty) o f the systems in the ensemble. This approach to mathematical epidemiology, which allows for individual differences, on the one hand, and statistical similarities, on the other, makes it possible to consider a great variety of such problems from a unified viewpoint. One o f the advantages of the approach is that it is possible to utilize formal analogies with problems in other fields of mathematics ( theory of finite automata, mathematical economics, mathematical neurobiology).

On the existence of periodic solutions in the nonlinear theory of oscillations of shallow shells, taking damping into account: PMM vol. 40, n≗ 4, 1976, pp. 699–705

Problems of existence of periodic solutions for various nonlinear equations of the continuous media mechanics are investigated in a number of papers, e.g. in [1, 2], The present paper proves the existence of an ω-periodic solution for nonlinear equations of anisotropic inhomogeneous shallow shells of variable thickness, with damping taken into account.

Stress tensor and averaging in mechanics of continuous media: PMM vol. 39, n≗2, 1975, pp. 374–379

The problem of determining the mean stress and the other macrovariables originates upon passing from the equations of motion which are valid in the microscale, to the macroscopic equations which describe the motion of continuous media (such as a turbulized fluid, an elastic medium with microdefects, the suspension of gas bubbles or solid particles in a fluid, etc.). The mean value of the stress tensor over a volume was introduced in the monograph [1], and precisely this quantity was used in the governing relations to compute the Einstein viscosity of suspensions. Moreover, some effective representation in terms of integrals over surfaces [1] was used in specific calculations of these means with respect to the volume. Later, Batchelor [2], and some other authors after him [3], used precisely these means with respect to the volume as the stresses in the macroequations of motion by assuming the equivalence between the average with respect to a volume and with respect to a surface. Hence, in particular, the absolute symmetry of the macrostress tensor follows in the above-mentioned cases. In this paper it is shown that the average of the microstress tensor and the microflux of the momenta with respect to the volume according to the rule in [1] determines only some symmetric part of the complete macrostress tensor. For the simple case of viscous fluid moving inhomogenously over a microlevel, this mean of the tensor with respect to the volume is related linearly to the mean strain rates. Moreover, the presentation used in [1] permits clarification of the essential difference between the mean stresses with respect to the volume and with respect to the surfaces, in the general case. The method of integrating the microequations with respect to the vioume [4–6] naturally results in the appearance of stresses in the macroequations, which are the means with respect to the differential macroareas. It is essential that the macrostress tensory is hence generally nonsymmetric although the equations of motion in the microscale correspond to symmetric continuum mechanics. It is this consideration which permitted the development of the continuum equations of motion of a suspension, which reflects the effect of nonequllibrium intrinsic rotation of the suspended particles [7], and the case of a turbulized fluid with anisotropies of eddy character is set in conformity to the nonzero antisymmetric part of the Reynolds stresses [8].

Methods of continuous medium mechanics for defining polyphase multicomponent mixtures with chemical reactions and heat- and mass-transfer: PMM vol. 39, n≗ 3, 1975, pp. 485–496

Equations of mechanics of multicomponent two-phase compressible disperse mixture, in which heat and mass transfer processes and chemical reactions take place, are derived. Interphase energy transfer in particular during component transition from one phase to another is analyzed. Problems of allowance for phase imperfections and of conditions of thermodynamic equilibrium are considered. Explicit expression is derived for the dissipation function, linear phenomenological relationships are analyzed, and singularities of the structure of direct and cross effects which develop in the system are disclosed. The state of investigations related to mechanics of heterogeneous mixtures is presented in the survey [1], We would point out here that Rakhmatulin was the first to formulate a closed system of equations for determining mixtures of compressible phases [2], The system included equations of mass and momenta of phases and equations for the over-all pressure. The proposed in it scheme for defining the force of interaction between phases is peculiar to a polyphase mixture and not to the multicomponent one. These concepts were applied to a saturated porous medium consisting of a mixture of two compressible phases without phase transitions. A scheme for defining energy exchange between phases and thermodynamic equations were proposed. A system of hydromechanical equations for a two-phase single-component disperse mixture of compressible phases was considered in [4], Phase transformations, which complicate the interphase exchange of energy and momentum, were assumed to be present in the mixture. This model was later supplemented. by an allowance for surface effects and small scale flows around inclusions [5], The equations of balance of mass, momentum and energy of components in a multi-speed form were formulated in [6–10], No clear distinction between homogeneous and heterogeneous mixtures was made in a number of investigations of this kind [1], with the analysis reduced to that of equations of conservation of components. It should be noted that in the case of a homogeneous mixture it is not necessary to have separate equations of momentum and energy for each component. Such separation (hence, also, the setting of momentum and energy exchange) must be made by phases only i.e. in the case of a heterogeneous mixture in which in addition to other conditions it is necessary to take into account parts of volume and interphase surface of mixture occupied by each phase [1] (this problem does not arise in the case of homogeneous mixtures). An important aspect of the definition of a heterogeneous mixture is the correct determination of the structure of terms which define the interaction between phases which takes place at interphase surfaces.

On some variational principles in mechanics of continuous media: PMM vol. 37, n≗6, 1973, pp. 963–973

Some problems of the analytical mechanics of continuous media are considered. The kinematc constraints, restricting the motion of elements of a continuous medium are separated into internal and external constraints; the internal surface stresses in a continuous medium are treated as reactive forces of the internal constraints. Since the work of these latter on possible displacements of elements of the continuous medium is not zero in the general case, then the internal constraints in a continuous medium should be referred to the category of nonideal constraints. The general equation of the dynamics of a continuous medium expressing the d'Alembert-Lagrange variational principle and including all the dynamic laws is examined. The work of the internal surface stresses in this equation can be given by using the first and second laws of thermodynamics, whereupon the general equation of the dynamics of continuous media can be represented in two other forms. Furthermore, an extension of the Gauss and Chetaev principles to continuous media is given.

Continual mechanochemical model of muscular tissue: PMM vol. 37, n≗3, 1973, pp. 448–458

The behavior of active muscular tissue is described with the help of a closed system of equations of motion of a two-phase, multicomponent, anisotropic continuous medium, with the mechanochemical processes occurring within it taken into account. The fundamental hypotheses are based on the information of general character concerning the structure and performance of the muscular tissue. It is assumed that the phase in which the mechanochemical reactions take place is viscoelastic, while the other phase is assumed elastic. The medium is assumed to have single velocity, although a passage of components between the phases is allowed. The laws of conservation are given and the rheological equations are written in accordance with the general principles of the mechanics of continuous medium and thermodynamics of irreversible processes [1–4]. It is shown that the model constructed describes, e.g., such characteristic properties of the muscle tissue as the existence of stresses in the absence of strains, zero-load deformations, and dissipation of energy in the state of mechanical equilibrium. The activity of the muscular tissue is governed by chemical processes taking place in the tissue, within the specific ordered structures called myofibrillae and, in the final count, by the mechanochemical reactions which affect the form or the relative distribution of the protein molecules [5–8]. Outside the myofibrillae we have various auxilliary systems, the connecting tissue and other structures, including capillary blood vessels which serve as the source of initial chemical compounds. The onset of active muscular contraction is connected with the arrival of specific reagents at the myofibrillae. The study of various physiological phenomena (such as the working of heart, propagation of an excitation in the tissues, organization of movements, regulation of the blood circulation and breathing) requires, on the one hand. a rheological equation of the active muscle to be available, on the other hand it requires the knowledge of the relation connecting the basic mechanical parameters of the muscle (such as length, load, etc.) with the parameters defined by the chemical processes, such as the energy needed by the muscle during contraction. The absence of a sufficiently general model of the muscle makes necessary the use, at the present time, of much simpler models [9–11]. The following problems are usually considered in the literature on their own: the propagation of excitation throughout the muscle tissue, mechanics of the muscle as a whole in terms of the load — length, the energetics and biochemistry of the muscular activity. All existing models of the muscular tissue are, with the exception of one given in [12], one-dimensional and are given, as a rule. in the form of relations connecting the load and extension and their derivatives with respect to time, with certain additional parameters which have the dimensions of the load or extension. These additional parameters have no explicit connection with the chemical or other internal processes taking place within the muscle, and their variation with respect to time is defined a priori, differently for the muscle in the active and in the passive state [12–18]. Thus the author of [12] proposes a rheological equation of the form P ij d = k ijln ( ε ln → N ln ') (0.1) where P ij d is the deviator of the stress tensor, ε ij is the strain tensor and N ij' is a tensor parameter (“biofactor”) characterizing the activity of the tissue; N ij' = 0 and N ij' ≠ 0 for the passive and the active state, respectively. The so-called three-element models (see e. g. [19]) contain practically the same additional parameters. Another group of one-dimensional models explains the meaning of the additional parameters contained in the rheological equation using the terms and quantities chatacterizing the microstructure of the muscle and its hypothetic “internal” mechanics (see e. g. [16–18, 20, 21]). We have already noted that these models are also based on the assumption that active deformations (contractions), i. e. zero-stress deformations, are possible. The purpose of this paper is to construct a model of a continuous medium with properties typical of a muscular tissue, without assumptions of existence of active deformations and without using the discussion concerning the micro-structure of the myofibrillae and the processes taking place within them.

Mixed problems of the mechanics of continuous media associated with hankel and mehler-fock integral transforms: PMM vol. 36, n≗3, 1972, pp. 494–504

On the basis of the Hankel and Mehler-Fock integral transforms we consider some types of dual integral equations and the corresponding types of integral equations of the second kind which occur in the study of a series of mixed problems of the mechanics of continuous media. By generalizing the asymptotic methods of [1, 2], effective approximate solutions are obtained. As an example we investigate the torsion, induced by a punch, of a truncated sphere, fixed along the spherical boundary.

Variational methods of constructing models of shells: PMM vol. 36, n≗5, 1972, pp. 788–804

The purpose herein is to derive the relationships of the theory of elastic shells from the variational equation of the mechanics of continuous media in the general case of physically and geometrically nonlinear models. The examination of this question is interesting in connection with the fact that all the hypotheses acquire the most compact and explicit formulation in the variational approach, and a logical basis appears for the comparison and estimation of the various models proposed in the theory of shells. Moreover, the shell models yield an interesting illustration of models of continuous media in which there are firstly higher derivatives, and secondly, internal degrees of freedom originate, as will be seen later. The appearance of the internal degrees of freedom requires the establishment of additional equations, in addition to the ordinary equations of mechanics, in order to determine new parameters, and to raise the order of the differential equations — additional boundary conditions and conditions on discontinuities. These relationships have been obtained by using methods developed for arbitrary models of continuous media with internal degrees of freedom and with higher derivatives in [1, 2]. Let us note that the extension of the theory to inelastic shells is associated only with complicating the functional δW ∗ in (1.1) and adding new degrees of freedom due to plastic deformations, viscous deformations, etc. Only the general part of the theory is contained herein. Specific shell models will be examined separately.

Equations of motion of nematic liquid-crystal media: PMM vol. 35, n≗ 5, 1971, pp. 879–891

Equations of motion for nematic liquid-crystal media in a magnetic field and also the equations of thermal conductivity are obtained. Together with the condition of continuity and the equation of state, these relationships determine the fields of nine quantities which characterize the nematic fluid: densities, pressures, temperatures, basis vectors of the local axis of anisotropy, rates of collective rotations of molecules near their “long” axes, and vectors of translational velocity. Initial conditions and boundary conditions are formulated. Special cases are examined: equilibrium of the medium in a homogeneous magnetic and temperature field, disinclinations, orientational boundary layer, and also the flow in a flat capillary in a magnetic field and the drag of fluid by a rotating magnetic field. Based on obtained results, an explanation is given for a number of effects which have been discovered experimentally earlier. Liquid crystals occupy on the thermodynamic scale of states an intermediate (mesophase) position between anisotropic crystals and isotropie liquids. Two fundamental varieties of mesophases exist: the smectic and the nematic. In the liquid crystal medium of the smectic type the one-dimensional long-range coordination structure is preserved. The molecules are organized in regularly spaced parallel monolayers. In the medium of the nematic type the long-range order is completely absent in the spatial arrangement of molecules, just as in the ordinary liquid. However, in contrast to a liquid and in similarity to a solid crystal the long-range order of orientation for the “long” molecular axes is preserved. The orientational order is characterized in each point of the medium by the axis of mean molecular orientation. This axis is simultaneously the local axis of symmetry of the medium. In their mechanical properties the nematic media are quite close to liquids. Experiments show [1, 2] that the behavior of nematic liquids in a force field, a temperature field, a magnetic field, and an electrical field has a number of anomalies (anisotropy of viscosity, scale effect, orientation in hydrodynamic flow, drag of the medium by a rotating magnetic field, and others.) The peculiar combination of mechanical properties makes liquid crystal media interesting objects for investigation from the point of view of continuum mechanics. At the present time the hydrostatic theory [3 – 9] is the most developed. In papers [10 – 12] linear hydrostatics is examined with consideration of thermal conductivity and effects of rotational viscosity. The hydrodynamic theory which takes into account elastic and thermal effects in a magnetic field is just being developed [13, 14]. Some results are available in paper [9]. Liquid crystals belong to so-called media with moments or media with rotational degrees of freedom [15]. At the phenomenological level these degress of freedom are taken into account in asymmetric mechanics of continuous media. On the basis of ideas of asymmetric mechanics [16, 17] the general hydrodynamic theory for nematic liquids is developed in [13, 14] taking into account elastic, thermal and magnetic effects. Based on this development, the goal of the present paper is to obtain a closed system of equations of motion, to formulate boundary conditions and initial conditions, to clarify the most essential characteristics of the equations, and to examine the simplest cases of motion of nematic liquids.

Approximate solution of Cauchy's problem for laplace's equation applicable to the problem of shaping of dense spatially inhomogeneous beams of charged particles: PMM vol. 35, no. 4, pp. 656–668

A solution of the problem of shaping (Sect. 1) of a number of three-dimensional beams is given in the form of asymptotic expansions. The results are compared with exact expressions which determine the shaping electrodes for a plane flow along circular trajectories (Sect. 2). From the paraxial approximation for the electrostatic beams, cases which satisfy the conditions of a full spatial charge on the emitter, without disturbing the regularity of this approximation (Sect. 3) are selected. Quasi-axially symmetric beams (Sect. 4) and a quasi-cylindrical domain of arbitrary section (Sect. 5) are considered. The hydrodynamic theory of intense beams of charged particles represents one of the branches of the mechanics of continuous medium. However, the asymptotic methods, although used widely and for a long time in other branches of mechanics, began to find application in this field only recently. The inverse problem or the problem of synthesis which appears when a system with desired characteristics is constructed, consists of two parts; the internal problem, which deals with solutions of the equations of the beam, and the external problem, connected with determining the shaping electrodes which provide the realization of the computed flow. The Cauchy problem for the Laplace equation represents the mathematical expression for the latter. For the solution of the internal problem for narrow beams the asymptotic method of the extension or the narrow strip type [4,5] have been successfully used in [1–3], although the study of the problem of shaping was complicated by the existence of singularities at the flow boundary. An approximate solution of this problem with singularities present in the Cauchy conditions based on the multiscale and factorization method, is given below.

The principle of gauss in the mechanics of continuous media

The principle of gauss in the mechanics of continuous media