Nonuniqueness of the solution of the problem of viscous interaction on an axisymmetric body

The article discusses solutions of the equations of the hypersonic boundary layer on an axisymmetric offset slender body (with a power exponent equal to 3/4), taking account of interactions with a nonviscous flow. It is shown that, in this case, the equations of the boundary layer have solutions differing from the self-similar solution corresponding to flow around a semi-infinite body. The solutions obtained are analogous to solutions for a strong interaction on a plate with slipping and triangular vanes [1–4], but are obtained over a wide range of values of the parameter of viscous interaction. An asymptotic solution is given to the problem with the approach to zero of the interaction parameter.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 41–47, September–October, 1973.The authors thank V. V. Mikhailova for discussion of the work and useful advice.     

Similarity solutions of energy and momentum boundary layer equations for a power-law shear driven flow over a semi-infinite flat plate

The problem of a steady forced convection thermal boundary-layer driven by a power-law shear is investigated. The search for similarity solutions reduces the problem to a couple of ordinary differential equations containing three parameters: the exponent of the decaying exterior velocity profile, the exponent of the power-law prescribing the thermal condition on the wall and Prandtl number. The effects of these parameters on the existence and form of similarity solution are investigated and the functional dependence of the local Nusselt number on these parameters is reported and discussed. An analysis of the assumptions usually accepted to derive similarity solutions is also reported in order to show the range of values of the exterior velocity power-law exponent for which such solutions may exist.     

Hidden symmetry concepts in the elastic buckling of axially-loaded cylinders

The recognized harmonic buckling modes for the elastic axially-loaded cylinder are by nature symmetric, equal and opposite amplitudes giving equal energy levels. On the other hand, in combination they account for considerable asymmetry, inwards deflection being fundamentally different from outwards; this asymmetry is also apparent in the underlying differential equations, and in the final large-deflection Yoshimura pattern. Taking the view that underlying symmetries are closely linked to the form of bifurcation experienced on buckling, and hence to the gross instability of the phenomenon, the paper thus explores a classic problem in a new light. The wellknown Donnell equations are first employed, the analysis being neatly written in terms of the two variables radial displacement w and stress function Φ. An extension is then presented, derived from the full strain-displacement equations appropriate to genuinely large deflections. This makes use of a suite of computer programs, written for a small micro, which handle the required manipulations of multiplication and integration of harmonic functions in algebraic rather than numerical fashion.     

黏弹性板的大挠度蠕变屈曲

研究了具有初始小挠度受轴向压载黏弹性板的蠕变屈曲问题,在建立控制方程时,利用了von Karman非线性应变-位移关系,并考虑了初始挠度,用标准线性固体模型描述材料的黏弹性特性,在求解非线性积分方程时,利用梯形公式计算记忆积分式,将非线性积分方程化为非线性代数方程进行数值求解,得到了结构的蠕变变形过程,又将问题退化到小挠度情况进行研究,得到了挠度随时间扩展的解析解,分析了瞬时失稳临界载荷、持久临界载荷的物理意义,讨论了考虑几何非线性对黏弹性板蠕变屈曲的影响。     

考虑在地基固结和流变的上部结构和地基基础的共同作用问题

本文提出一种能同时考虑地基的瞬时变形,团结变形和流变的地基基础与上部结构共同作用的一种新的计算方法,在这种方法中,土的计算考虑了Biot固结理论和流变,上部结构利用子结构法进行凝聚可得边界刚度矩阵和边界力,利用位移协讯条件则可以得到共同作用的控制方程,对上述方程进行Laplace变换,可以得到Laplace变换域内的控制方程,在Laplace变换域内对上述方程进行数值求解并进行相应的Laplace逆变换则可以得到时间境内任意时刻的解。     

Consistent formulation of the power-law rheology and its application to the spreading of non-Newtonian droplets

In this work, we introduce a general form of the Navier-Stokes equations for Generalized Newtonian fluids with an Ostwald power-law. The derivation, based on the covariant formalism, is frame-independent and gives rise to a source term in the Navier-Stokes equations referred to as the Ostwald vector which is characterized by the power-law exponent. The governing equations are then simplified in the long-wave approximation framework and applied to the spreading of an axisymmetric gravity current in the creeping flow regime. Well-known spreading laws are recovered through similarity solutions and a new derivation based on scaling arguments is proposed. Experimental results related to the spreading of gravity current are then presented and the potential to infer unknown rheological parameters from spreading rates is critically discussed in the context of a thorough error analysis.     

Technique for selecting the functions of the constitutive equations of creep and long-term strength with one scalar damage parameter

The strain-strength characteristics of aerostructures made of hardening materials under uniaxial tension in creep conditions are determined. The problem is reduced to a system of ordinary differential equations of the kinetic theory of creep with one scalar damage parameter. The approximate solutions of the problem are obtained with the help of the implicit Euler method and of the arc length method in combination with the explicit methods of the Runge–Kutta family for cylindrical St.45 steel samples and 3V titanium alloy plates.     

Some Results of the Finite Deflection Analysis of Clamped Skew Sandwich Plates

The suitability of Galerkin's method for the solution of the problem of the finite deflection analysis of clamped skew sandwich plates is studied. The five coupled nonlinear governing differential equations for sandwich plates are transformed into nonlinear algebraic equations by using Galerkin's method of error minimization. These equations are then solved using an iterative algorithm suggested by Brown. Comparisons of the results of the present analysis with available solutions show good agreement. Numerical results are presented for skew sandwich plates for a wide range of values of the core modulus for different skew angles and aspect ratios. Simplicity in formulation and computation is the advantage of the method as compared with other methods of nonlinear analysis. Computing time and memory requirements in a digital computer are relatively very small, which makes the method attractive.     

Energy version of the kinetic equations of isothermal creep and long-term strength

Energy-type kinetic equations of inelastic rheological deformation are proposed in which the elastic, plastic, and creep strains are the additive components of the total strain, and the damage parameter is taken into account. A model of viscoelastic material with a creep kernel of exponential type is considered. The Lyapunov stability of solutions under constant stress is studied. The stability range of the solutions of the differential equations of the mathematical model corresponding to asymptotically bounded creep is established. It is shown that the instability range of the solutions corresponds to the onset of the third stage of creep. The relationship is determined between the Lyapunov stability of the solutions and the stability of the computational algorithm for the numerical solution of the system of equations. The proposed model is experimentally verified. It is shown that the calculated and experimental data are in good agreement.     

Initial-value problem for coupled Boussinesq equations and a hierarchy of Ostrovsky equations

We consider the initial-value problem for a system of coupled Boussinesq equations on the infinite line for localised or sufficiently rapidly decaying initial data, generating sufficiently rapidly decaying right- and left-propagating waves. We study the dynamics of weakly nonlinear waves, and using asymptotic multiple-scale expansions and averaging with respect to the fast time, we obtain a hierarchy of asymptotically exact coupled and uncoupled Ostrovsky equations for unidirectional waves. We then construct a weakly nonlinear solution of the initial-value problem in terms of solutions of the derived Ostrovsky equations within the accuracy of the governing equations, and show that there are no secular terms. When coupling parameters are equal to zero, our results yield a weakly nonlinear solution of the initial-value problem for the Boussinesq equation in terms of solutions of the initial-value problems for two Korteweg-de Vries equations, integrable by the Inverse Scattering Transform. We also perform relevant numerical simulations of the original unapproximated system of Boussinesq equations to illustrate the difference in the behaviour of its solutions for different asymptotic regimes.     

The influence of magnetic field upon the collapse of a cylindrical shock wave

In the present paper, the problem of propagation of collapsing cylindrical shock wave in an ideal gas permeated by a transverse magnetic field with infinite electrical conductivity is investigated. Here it is assumed that the medium ahead of the shock front is uniform and at rest. Also, its counter pressure concerning the motion of the wave front is neglected. This problem admits a self similar solution of second kind. The similarity exponent has been computed by solving a nonlinear eigenvalue problem and integrating numerically the self-similar equations for various values of adiabatic heat exponent and Cowling number. Numerical computations have been performed to determine the flow field behind the shock wave. The influence of magnetic field strength and adiabatic heat exponent on the flow parameters for various cases is presented.     

Solving initial–boundary-value creep problems

The Galerkin–Bubnov method with global approximations is used to find approximate solutions to initial–boundary-value creep problems. It is shown that this approach allows obtaining solutions available in the literature. The features of how the solutions of initial–boundary-value problems for oneand three-dimensional models are found are analyzed. The approximate solutions found by the Galerkin–Bubnov method with global approximations is shown to be invariant to the form of the equations of the initial–boundary-value problem. It is established that solutions of initial–boundary-value creep problems can be classified according to the form of operators in the mathematical problem formulation     

Self-similar collapse of a circular cavity of a power-law liquid

Using the lubrication approximation we investigate the self-similar axisymmetric flow of a power-law liquid towards a central circular cavity. It is shown that this problem has a self-similar solution of the second kind. The self-similarity exponent is found by solving a non-linear eigenvalue problem arising from the requirement that the integral curve that represents the solution must join the appropriate singular points in the phase plane of the governing equation. The eigenvalues for different values of the rheological index are computed. Numerical integration of the equations allows us to determine the shape of the solution in terms of the physical variables. We make a detailed analysis of the influence of the rheology on the properties of the solutions.     

Analysis of creep deformation and creep damage in thin-walled branched shells from materials with different behavior in tension and compression

A constitutive model for describing the creep and creep damage in initially isotropic materials with different properties in tension and compression has been applied to the modeling of creep deformation and creep damage growth in thin-walled shells of revolution with the branched meridian. The approach of establishing the basic equations for axisymmetrically loaded branched shells under creep deformation and creep damage conditions has been introduced. To solve the initial/boundary-value problem, the fourth-order Runge–Kutta–Merson’s method of time integration with the combination of the numerically stable Godunov’s method of discrete orthogonalization is used. The solution of the boundary value problem for the branched shell at each time instant is reduced to integration of the series of systems of ordinary differential equations describing the deformation of each branch and the shell with basic meridian. Some numerical examples are considered, and the processes of creep deformation and creep damage growth in a shell with non-branched meridian as well as in a branched shell are analyzed. The influence of the tension–compression asymmetry on the stress–strain state and damage evolution in a shell with non-branched meridian as well as in a branched shell with time are discussed.     

Creep buckling of axially compressed circular cylindrical shells of a zirconium alloy: Experiment and computer simulation

Experiments were performed to study the deformation and buckling of axially compressed circular cylindrical shells of Zr2.5Nb zirconium alloy under creep conditions. Computer simulation using the MSC.Marc 2012 software was conducted by step-by-step integration of the equations of quasistatic deformation of thin shells using Norton’s law of steady creep. The results of the experiment and computer simulation show that the buckling modes are a combination of axisymmetric bulges located near one end or both ends of the shell and axisymmetric buckling modes with the formation of three or four waves in the circumferential direction. A comparison is made of the time dependences of the axial strain of the shells obtained in the experiment and by computer simulation. It is shown that for large axial compressive stresses, these dependences are in satisfactory agreement. For lower values of these stresses, the difference between the theoretical and experimental dependences is greater.     

Three-dimensional flow of a viscoelastic fluid on an exponentially stretching surface

An analysis of a three-dimensional viscoelastic fluid flow over an exponentially stretching surface is carried out in the presence of heat transfer. Constitutive equations of a second-grade fluid are employed. The governing boundary layer equations are reduced by appropriate transformations to ordinary differential equations. Series solutions of these equations are found, and their convergence is discussed. The influence of the prominent parameters involved in the heat transfer process is analyzed. It is found that the effects of the Prandtl number, viscoelastic parameter, velocity ratio parameter, and temperature exponent on the Nusselt number are qualitatively similar.     

无铅焊料合金压入蠕变性能参数表征研究

微互连焊点高温下的服役可靠性很大程度上决定了电子产品性能的优劣,本文通过微纳米压入法开展了典型无铅焊料合金Sn-3.0Ag-0.5Cu的高温蠕变性能研究,采用"快速加载–保载–快速卸载"的测试方式获取不同温度下完备的蠕变行为,分析发现温度升高焊料软化加剧、蠕变变形更为明显;得到了焊料合金不同温度下的稳态蠕变应力指数n与不同硬度下的蠕变激活能Q,分析了温度与硬度对二者的影响并给出激活能Q与ln H之间的关系.     

Global Existence and Large-Time Behavior of Solutions to the Three-Dimensional Equations of Compressible Magnetohydrodynamic Flows

The three-dimensional equations of compressible magnetohydrodynamic isentropic flows are considered. An initial-boundary value problem is studied in a bounded domain with large data. The existence and large-time behavior of global weak solutions are established through a three-level approximation, energy estimates, and weak convergence for the adiabatic exponent ${\gamma > \frac 32}${\gamma > \frac 32} and constant viscosity coefficients.     

Creep analysis with a stress range dependent constitutive model

Many materials exhibit the stress range dependent creep behavior. The power law creep observed for a certain stress range changes to the viscous type creep as the stress value decreases. Recently published experimental data for advanced heat resistant steels indicate that the high creep exponent (in the range 7–12) may decrease to the low value of approximately unity within the stress range relevant for engineering structures. The aim of this paper is to analyze the influence of the stress range dependent power-law-viscous creep transition on the behavior of structures at elevated temperature. A constitutive model for the minimum creep rate is introduced to describe both the linear and the power law creep depending upon the stress level. To demonstrate basic features of the stress range dependent creep modeling, several elementary examples from structural mechanics are presented. They include a stress relaxation problem, a beam subjected to pure bending and a pressurized thick-walled cylinder. Based on the uni-axial transition stress the transition value of the external load is estimated such that above this value the power law can be applied. For the loading levels below this value the character of the stress distribution as well as the stress values are essentially influenced by the viscous creep.