Continuity violation trajectories in perfectly plastic bodies

The consistency equations for the increments of small strains in a triorthogonal isostatic coordinate system supplemented with additional relations between the physical components of the inconsistency tensor are considered. There are six significant consistency equations. It is proved that, for the stress states corresponding to the edge of the Coulomb-Tresca prism, there are only three independent consistency equations. Systems of independent consistency equations written in the isostatic coordinate mesh are explicitly indicated and studied. Sufficient conditions for the remaining three consistency equations to be satisfied if the three independent consistency equations are satisfied are obtained. It is shown that the continuity violations on the surface of a perfectly plastic body propagate into the depth of the body along asymptotic lines on the layers of a vector field indicating the directions of the maximum principal normal stress. Since the asymptotic lines are less curved than any other lines on the surface (in the sense that the normal curvature of the asymptotic lines is zero), the continuity violations propagate into a perfectly plastic body along the least curved trajectories, which permits one to speak of the minimum curving of crack propagation trajectories in perfectly plastic rigid bodies.     

Solution of a mixed dynamic problem on the displacements given on the boundary of a half-plane lying on the surface of an isotropic homogeneous elastic half-space

We consider the problem on the motion of an isotropic elastic body occupying the half-space z ≥ 0 on whose boundary, along the half-plane x ≥ 0, the horizontal components of displacement are given, while the remaining part of the boundary is stress-free. We seek the solution by the method of integral Laplace transforms with respect to time t and Fourier transforms with respect to the coordinates x, y; the problem is reduced to a system of Wiener-Hopf equations, which can be solved by the methods of singular-integral equations and circulants. We invert the integral transforms and reduce the solution to the Smirnov-Sobolev form. We calculate the tangential stress intensity coefficients near the boundary z = 0, x = 0, |y| < ∞ of the half-plane. The circulant method for solving the Wiener-Hopf system was proposed in [1]. A static problem similar to that considered in the present paper was solved earlier. The Hilbert problem was reduced to a system of Fredholm integral equations in [2]. In the present paper, we solve the above problem by reducing the solution to quadratures and a quasiregular system of Fredholm integral equations. We give a numerical solution of the Fredholm equations and calculate the integrals for the tangential stress intensity coefficients.     

Vortex motions of liquid in a narrow channel

Quasi-linear integrodifferential equations that describe vortex flows of an ideal incomparessible liquid in a narrow curved channel in the Eulerian-Lagrangian coordinate system are considered. The necessary and sufficient conditions for hyperbolicity of the system of equations of motion are obtained for flows with a monotonic velocity depth profile. The propagation velocities of the characteristics and the characteristic form of the system are calculated. A particular solution is given in which the system of integrodifferential equations changes type with time. The solution of the Cauchy problem is given for linearized equations. An example of initial data for which the Cauchy problem is ill-posed is constructed. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 4, pp. 38–49, July–August, 1998.     

Equations of viscous supersonic jets with a high degree of overexpansion

A complete system of equations determining a viscous laminar, strongly overexpanded jet is obtained; the system is formed by shortened Navier—Stokes equations, equations for the metric of a coordinate system related with the form of the jet, and equations of transition from curvilinear coordinates to Cartesian. The problem of calculating the jet is formulated as a Cauchy problem for this system. Two- and three-dimensional flows are examined. Possible swirling of the jet is taken into account.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 137–147, March–April, 1977.     

The Equations of Equilibrium in Orthogonal Curvilinear Reference Coordinates

Analytical solutions to problems in finite elasticity are most often derived using the semi-inverse approach along with the spatial form of the equations of motion involving the Cauchy stress tensor. This procedure is somewhat indirect since the spatial equations involve derivatives with respect to spatial coordinates while the unknown functions are in terms of material coordinates, thus necessitating the use of the chain rule. In this classroom note, we derive compact expressions for the components of the divergence, with respect to orthogonal material coordinates, of the first Piola-Kirchhoff stress tensor. The spatial coordinate system is also assumed to be an orthogonal curvilinear one, although, not necessarily of the same type as the material coordinate system. We show by means of some example applications how analytical solutions can be derived more directly using the derived results.     

Investigation of instability in vertical liquid films as an initial value problem

The instability of plane-parallel vertical viscous layer downflow is investigated. We solve not the classical eigenvalue problem for the Orr-Sommerfeld equation but a Cauchy problem with respect to time and a boundary-value problem with respect to the spatial variable for a linearized system of equations. The problem is solved by means of a Laplace transformation in time and a Fourier transformation in the spatial variable. Subsequently, using the residue theorem and the method of steepest descent makes it possible to predict asymptotically the perturbation behavior as time t → ∞. The system is convectively unstable and a localized perturbation spreads out at the velocities of the trailing and leading fronts. The packet behavior is investigated over a wide range of the flow parameters.     

On some diffusive waves in non-linear heat conduction

We address the non-linear heat conduction in the presence of absorption for the case of spherical symmetry geometry. The non-linear model is based on both a temperature-dependent thermal conductivity and a non-linear generalization of the Fourier law. The governing equation belongs to a class of degenerate parabolic equations. We obtain similarity solutions in closed form for the Cauchy problem corresponding to an instantaneous point source problem. We investigate the non-linear effects on the propagation of the temperature distrubances. We find that in certain cases the temperature distribution displays travelling wave characteristics. The solution for the Cauchy problem is recovered by considering a suitable first boundary value problem.     

Asymptotic properties of solutions of an equation of order <Emphasis Type="Italic">n</Emphasis> with almost constant coefficients and pulse action

We investigate the asymptotic behavior of solutions of linear differential equations with almost constant coefficients and pulse action at fixed times as t tends to infinity. We establish conditions for the times of pulse action under which there exist values of pulse action for which the solution of the considered Cauchy problem with initial conditions that coincide with the initial conditions for a certain (arbitrary but fixed) solution of the original equation without pulse action is bounded, unbounded, or tending to infinity. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 4, pp. 444–455, October–December, 2005.     

Geometric elasticity problem of stress concentration in a plate with a circular hole

We use the geometric elasticity equations [1], which permit relating the medium stress state to the geometry of the Riemannian space generated by the stresses, to consider the plane problem of stress concentration near a circular hole in a thin unbounded plate loaded by normal and tangential stresses. The Riemannian space metric coefficient corresponding to the coordinate normal to the plate plane is treated as the variable thickness of the plate in three-dimensional Euclidean space, which determines the optimal law for the plate material distribution. We consider plates in uniaxial tension, biaxial tension, and shear. For the plate with thickness variation laws thus obtained, we construct direct numerical solutions of the corresponding classical elasticity problems and determine the stress concentration factors.     

正交各向异性功能梯度涂层-基底结构共线裂纹问题的断裂分析

论文研究了一正交各向异性功能梯涂层粘结到一均匀基底含共线裂纹的平面I型断裂问题.引入新的双参数指数函数模拟连续改变的材料性质,正交各向异性的主轴方向分别为平行和垂直于带的边界,采用积分变换技术,所求的问题转化为第一类的Cauchy奇异积分方程,获得了共线裂纹尖端应力场,结果显示了材料常数和几何参数对应力强度因子的影响.     

半平面多边缘裂纹反平面问题的奇异积分方程

利用复变函数和奇异积分方程方法,求解弹性范围内半平面多边缘裂纹的反平面问题．提出了满足半平面边界自由的由分布位错密度表示的单边缘裂纹的基本解,此基本解由主要部分和辅助部分组成．将半平面多边缘裂纹问题看作是许多单边缘裂纹问题的叠加,建立了一组Cauchy型奇异积分方程．然后,利用半开型积分法则求解该奇异积分方程,得到了裂纹端处的应力强度因子．最后,给出了几个数值算例．     

L 1 Convergence to the Barenblatt Solution for Compressible Euler Equations with Damping

We study the asymptotic behavior of compressible isentropic flow through a porous medium when the initial mass is finite. The model system is the compressible Euler equation with frictional damping. As t ?? ??, the density is conjectured to obey the well-known porous medium equation and the momentum is expected to be formulated by Darcy??s law. In this paper, we prove that any L ^?? weak entropy solution to the Cauchy problem of damped Euler equations with finite initial mass converges strongly in the natural L ¹ topology with decay rates to the Barenblatt profile of the porous medium equation. The density function tends to the Barenblatt solution of the porous medium equation while the momentum is described by Darcy??s law. The results are achieved through a comprehensive entropy analysis, capturing the dissipative character of the problem.     

Ad hoc exact solutions for the stress and velocity fields in rigid,perfectly plastic materials under plane-strain conditions

Making use of convenient ad hoc assumptions we construct analytical and closed-form solutions for the problem of stress and velocity states in statically determinate rigid, perfectly plastic bodies under plane-strain conditions. The obtained solutions are expressed either implicitly including arbitrary functions, or explicitly in the form of specific functions. For the stresses they are extracted by means of the two equilibrium partial differential equations (PDEs) and the appropriate von Mises-Hencky condition; for the velocities we use the Saint Venant-von Mises theory of plasticity PDEs and the condition of incompressibility. The possibility of extending the developed solution technique for plane stress conditions is presented. Finally, several applications concerning the inverse, semi-inverse and direct problems are examined. The advantage of the proposed analytical solution methodology compared to the technique of characteristics is the general applicability delivering from the a priori construction of the slip-lines, as well as the demanded numerical solutions of the corresponding equations of characteristics.     

Retardation of a crack with connections between the faces using an induced thermoelastic stress field

This paper considers local temperature variations near the tip of a crack in the presence of regions in which the crack faces interact. It is assumed that these regions are adjacent to the crack tip and are comparable in size to the crack size. The problem of local temperature variations consists of delay or retardation of crack growth. For a crack with connections between the crack faces subjected to external tensile loads, an induced thermoelastic stress field, and the stresses at the connections preventing crack opening, the boundary-value problem of the equilibrium of the crack reduces to a system of nonlinear singular integrodifferential equations with a Cauchy kernel. The normal and tangential stresses at the connections are found by solving this system of equations. The stress intensity factors are calculated. The energy characteristics of cracks with tip regions are considered. The limiting equilibrium condition for cracks with tip regions is formulated using the criterion of limiting stretching of the connections.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 133–143, January–February, 2005     

Averaging of stochastic evolution equations of transport in porous media

A study is made of the problem of averaging the simplest one-dimensional evolution equations of stochastic transport in a porous medium. A number of exact functional equations corresponding to distributions of the random parameters of a special form is obtained. In some cases, the functional equations can be localized and reduced to differential equations of fairly high order. The first part of the paper (Secs. 1–6) considers the process of transport of a neutral admixture in porous media. The functional approach and technique for decoupling the correlations explained by Klyatskin [4] is used. The second part of the paper studies the process of transport in porous media of two immiscible incompressible fluids in the framework of the Buckley—Leverett model. A linear equation is obtained for the joint probability density of the solution of the stochastic quasilinear transport equation and its derivative. An infinite chain of equations for the moments of the solution is obtained. A scheme of approximate closure is proposed, and the solution of the approximate equations for the mean concentration is compared with the exactly averaged concentration.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 127–136, September–October, 1985.We are grateful to A. I. Shnirel'man for pointing out the possibility of obtaining an averaged equation in the case of a velocity distribution in accordance with a Cauchy law.     

On a supplementary conservation law for the energy equation in a rigid heat conductor

We consider a rigid heat conductor with specified constitutive equations and show that the internal energy equation may be written in the form of a symmetric and conservative hyperbolic system of first order quasi-linear equations for which the Cauchy problem is well-posed. Moreover, such a system is useful to study shocks. Several particular cases are examined.     

Cylindrical elastic-plastic waves due to discontinuous loading at a circular cavity

The plastic waves in rate-independent, isotropically work-hardening media obeying the von Mises yield condition generated by radial stress uniformly applied at a circular cavity of radius r = r₀, are studied. Both plane stress and plane strain motions are considered. The radial stress and its time derivative at the cavity may be discontinuous at time t = t₀. If the applied radial stress is continuous while its time derivative is not, the discontinuity at (r₀, t₀) propagates into r >r₀ along the characteristics and/or the elastic-plastic boundaries. If the applied radial stress itself is discontinuous, the discontinuity in stress may propagate into r >r₀ in the form of a shock wave, or a pseudo centered simple wave, or a combination of both. This is a systematic study on the nature of solutions in the neighborhood of (r₀, t₀) for all possible combinations of discontinuous loadings applied at (r₀, t₀). The special cases of linear work-hardening and perfectly-plastic media are also discussed. Finally, the corresponding problem for materials obeying the Tresca yield condition is studied briefly.     

Practical limitations to the influence of through-thickness normal stress on sheet metal formability

According to a recent (original) model, when hardening properties and the ratio of through-thickness normal stress to the first principal stress (γ≡σ₃/σ₁) are held constant, sheet metal formability can be increased dramatically through the introduction of a compressive through-thickness normal stress, σ₃. In practice, however, both the hardening properties and γ evolve with the progression of deformation. To manage most efficiently the evolution of the hardening properties and γ, the original model is cast into a more compact form and presented as a proposed alternative form (proposed model). When the evolution of the hardening properties and γ is considered, the proposed model is shown to be in very good agreement with observed data; the influence of through-thickness normal stress on sheet metal formability is quite small for all practical purposes. Because the structure of the original model is similar to that of the proposed model, the original model is also validated. Ultimately, it is verified that although the theory of the original and proposed model may be acceptable, the implications of such theories are less profound than those first proposed when practical limitations are considered. This work serves as a useful basis for: (1) further understanding the limitations of the influence of compressive through-thickness normal stress on sheet metal formability and (2) exploring opportunities for improving sheet metal formability.     

CAUCHY PROBLEM OF ONE TYPE OF ATMOSPHERE EVOLUTION EQUATIONS

One type of evolution atmosphere equations was discussed. It is found that according to the stratification theory, （i） the inertial force has no influence on the criterion of the well-posed Cauchy problem; （ii） the compressibility plays no role on the well-posed condition of the Cauchy problem of the viscid atmosphere equations, but changes the well-posed condition of the viscid atmosphere equations; （iii） this type of atmosphere evolution equations is ill-posed on the hyperplane t = 0 in spite of its compressibility and viscosity; （iv） the Cauchy problem of compressible viscosity atmosphere with still initial motion is ill-posed.