Fracture and secondary flow of elastic liquids

Summary In the absence of disturbances at the free surfaces, secondary flow due to elastic forces occurs in elastic liquids sheared in cone-and-plate rheometers but not in parallel-plate rheometers. In both types of apparatus an instability is observed. Using a theory based on a fracture mechanism, the critical normal stress differences for the onset of the instability can be correctly predicted for both types of apparatus. It is concluded that the fracture instability is a distinctly different effect from the secondary flow considered.     

Experimental investigations of the stability limit of the helical flow of pseudoplastic liquids

The stability of the laminar helical flow of pseudoplastic liquids has been investigated with an indirect method consisting in the measurement of the rate of mass transfer at the surface of the inner rotating cylinder. The experiments have been carried out for different values of the geometric parameter = R ₁/R ₂ (the radius ratio) in the range of small values of the Reynolds number,Re < 200. Water solutions of CMC and MC have been used as pseudoplastic liquids obeying the power law model. The results have been correlated with the Taylor and Reynolds numbers defined with the aid of the mean viscosity value. The stability limit of the Couette flow is described by a functional dependence of the modified critical Taylor number (including geometric factor) on the flow indexn. This dependence, general for pseudoplastic liquids obeying the power law model, is close to the previous theoretical predictions and displays destabilizing influence of pseudoplasticity on the rotational motion. Beyond the initial range of the Reynolds numbers values (Re>20), the stability of the helical flow is not affected considerably by the pseudoplastic properties of liquids. In the range of the monotonic stabilization of the helical flow the stability limit is described by a general dependence of the modified Taylor number on the Reynolds number. The dependence is general for pseudoplastic as well as Newtonian liquids.Nomenclature C _i concentration of reaction ions, kmol/m³ - d = R ₂ –R ₁ gap width, m - F _M () Meksyn's geometric factor (Eq. (1)) - F ₀ Faraday constant, C/kmol - i _l density of limit current, A/m³ - k _c mass transfer coefficient, m/s - n flow index - R ₁,R ₂ inner, outer radius of the gap, m - Re = V _m ·2d·/µ _m Reynolds number - Ta _c = _c ·d^3/2·R ₁ ^1/2 ·/µ _m Taylor number - Z _i number of electrons involved in electrochemical reaction - = R ₁/R ₂ radius ratio - µ apparent viscosity (local), Ns/m² - µ _m mean apparent viscosity value (Eq. (3)), Ns/m² - µ _i apparent viscosity value at a surface of the inner cylinder, Ns/m² - density, kg/m³ - _c angular velocity of the inner cylinder (critical value), 1/s     

Upper limit of the linear law of gas flow through a porous medium

The dependence of the critical Reynolds number Re₁ characterizing the upper limit of applicability of the linear flow law, on the other characteristics of the porous medium is considered. It is shown that Re₁ decreases with increase in the dimensionless inertial resistance coefficient on the developed inertial flow interval _*. Most of the known experimental data can be quite closely approximated by the expression Re₁=7_* ^–1.16. The effect of the error in determining Re₁ by means of the relation proposed on the error in finding the resistance coefficient of the porous medium is analyzed. It is concluded that the relation obtained can be used for determining Re₁ in engineering calculations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 186–190, January–February, 1991.     

Spontaneous Fracture of Solids as Affected by Surfactants

Fracture of solids under the action of surfactants is studied. The limiting case of manifestation of chemisorption effects without significant effect of mechanical stresses is considered. The growth of plane cracks in solids with a regular structure upon contact with surfactants is described for the case where the model of the process is molecular wedge–assisted crack wedging. The critical stress–intensity factors are estimated in terms of the critical crack opening in wedging by a semi–infinite wedge of constant thickness. An anomalously low resistance to breaking of a solid into parts is observed under certain conditions (spontaneous fracture).     

A general algorithm for plastic flow simulation by finite element limit analysis

Limit analysis has been rendered versatile in many structural and metal forming problems. In metal forming analysis, the slip-line method and the upper bound method have filled the role of limit analysis. As a breakthrough of the previous work, a computational approach to limit solutions is considered as the most challenging area.In the present work, a general algorithm for limit solutions of plastic flow is developed with the use of finite element limit analysis. The algorithm deals with a generalized Hölder inequality, a duality theorem, and combined smoothing and successive approximation in addition to a general procedure for finite element analysis. The algorithm is robust such that from any initial trial solution, the first iteration falls into a convex set which contains the exact solution (s) of the problem. The idea of the algorithm for limit solutions is extended from rigid⧹perfectly plastic materials to work-hardening materials by the nature of the limit formulation, which is also robust with numerically stable convergence and highly efficient computing time.     

Elastostatics as a limit of elastodynamics — a matrix formulation

A T matrix formulation of the three-dimensional elastostatic single inhomogeneity problem is presented. It is shown how it can be obtained as a zero frequency limit of the T matrix formulation of a corresponding time-harmonic scattering problem. Numerical results for some rotationally symmetric cavities are given.     

Hydrogel as a Medium for Fluid-Driven Fracture Study

In this paper we describe how to construct polyacrylamide hydrogels to study the processes linked with hydraulic fracturing. These transparent, linearly elastic and brittle gels permit fracturing at low pressures and speeds allowing accurate measurements to be obtained. In the context of hydraulic fracturing, the broad range of modulus and fracture energy values that are attainable allow experimental exploration of particular regimes of importance. We also describe how material properties may be deduced from hydraulic fracturing experiments. Lastly, we analyse the fracture surface patterns that emerge from fluid-driven cracks occurring within the medium. These patterns are similar to those that have been observed in other materials and we comment on their fractal-like nature.     

Dynamic Brittle Fracture as a Small Horizon Limit of Peridynamics

We consider the nonlocal formulation of continuum mechanics described by peridynamics. We provide a link between peridynamic evolution and brittle fracture evolution for a broad class of peridynamic potentials associated with unstable peridynamic constitutive laws. Distinguished limits of peridynamic evolutions are identified that correspond to vanishing peridynamic horizon. The limit evolution has both bounded linear elastic energy and Griffith surface energy. The limit evolution corresponds to the simultaneous evolution of elastic displacement and fracture. For points in spacetime not on the crack set the displacement field evolves according to the linear elastic wave equation. The wave equation provides the dynamic coupling between elastic waves and the evolving fracture path inside the media. The elastic moduli, wave speed and energy release rate for the evolution are explicitly determined by moments of the peridynamic influence function and the peridynamic potential energy.     

Enhance limit cycle oscillation in a wind flow energy harvester system with fractional order derivatives

Advances in material science and mathematics in conjunction with technological needs have triggered the use of material and electric components with fractional order physical properties. This paper considers the mathematical model of a piezoelectric wind flow energy harvester system for which the capacitance of the piezoelectric material has fractional order current-voltage characteristics. Additionally the mechanical element is assumed to have fractional order damping. The analysis is focused on the effects of order of derivatives on the appearance and characteristics of limit circle oscillations (LCO). It is obtained that, the order of derivatives to enhance the amplitude of LCO and lower the threshold condition leading to LCO. The domains of efficiency of the system are illustrated in various parameters spaces.     

Transition flow of a fibrous suspension in a pipe as the flow of an anisotropic fluid

The transition flow is considered of a fibrous suspension in a pipe. The flow region consists of two subregions: at the center of the flow a plug formed by interwoven fibers and fluid moves as a rigid body; between the solid wall and the plug is a boundary layer in which the suspension is a mixture of the liquid phase and fibers separated from the plug [1–3]. In the boundary region the suspension is simulated as an anisotropic Ericksen—Leslie fluid [4, 5] which satisfies certain additional conditions. Equations are obtained for the velocity profile and drag coefficient of the pipe, which are both qualitatively and quantitatively in good agreement with the experimental results [6–8]. Within the framework of the model, a mechanism is found for reducing the drag in the flow of a fibrous suspension as compared to the drag of its liquid phase.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 91–98, September–October, 1985.     

A consistent thermodynamical model of incompressible media as limit case of quasi-thermal-incompressible materials

In this paper we extend the conditions on quasi-thermal-incompressible materials presented in Gouin et al. (2011) [1] so that they satisfy all the principles of thermodynamics, including the stability condition associated with the concavity of the chemical potential. We analyze the approximations under which a quasi-thermal-incompressible medium can be considered as incompressible. We find that the pressure cannot exceed a very large critical value and that the compressibility factor must be greater than a lower limit that is very small. The analysis is first done for the case of fluids and then extended to the case of thermoelastic solids.     

On the dynamic elastic limit

The fact that the elastic limit of some solids increases with increasing stress rate has been qualitatively and semiquantitatively established for many decades. Well known experimental difficulties have impeded reliable quantitative measurements of the magnitude or, in some solids, even the existence of such an increase of the elastic limit with stress rate. The present paper describes a simple method for accurately measuring the dynamic elastic limit in any solid which has a linear-elastic domain at small strain, including high-strength structural metal alloys. This method has the advantages of laboratory simplicity, a minimum of complex assumptions, and a close parallel with the manner in which the quasistatic elastic limit generally is determined. Although it is subsidiary to the main focus of this paper, evidence is presented here that a knowledge of the dynamic elastic limit firmly established by experiment, can be of considerable value for subsequent research in the continuum mechanics of solids, particularly with respect to the existence and properties of two distinct yield surfaces during impact loading.     

The unique limit of the Glimm scheme

We introduce the definitions of a standard Riemann semigroup and of a viscosity solution for a nonlinear hyperbolic system of conservation laws. For a class including general 2×2 systems, it is proved that the solutions obtained by a wavefront tracking algorithm or by the Glimm scheme are precisely the semigroup trajectories. In particular, these solutions are unique and depend Lipschitz continuously on the initial data in the L ¹ norm.