A Theory of Anharmonic Lattice Statics for Analysis of Defective Crystals

This paper develops a theory of anharmonic lattice statics for the analysis of defective complex lattices. This theory differs from the classical treatments of defects in lattice statics in that it does not rely on harmonic and homogenous force constants. Instead, it starts with an interatomic potential, possibly with infinite range as appropriate for situations with electrostatics, and calculates the equilibrium states of defects. In particular, the present theory accounts for the differences in the force constants near defects and in the bulk. The present formulation reduces the analysis of defective crystals to the solution of a system of nonlinear difference equations with appropriate boundary conditions. A harmonic problem is obtained by linearizing the nonlinear equations, and a method for obtaining analytical solutions is described in situations where one can exploit symmetry. It is then extended to the anharmonic problem using modified Newton–Raphson iteration. The method is demonstrated for model problems motivated by domain walls in ferroelectric materials.      

Geometrical nonlinear analysis of thin-walled composite beams using finite element method based on first order shear deformation theory

Based on a seven-degree-of-freedom shear deformable beam model, a geometrical nonlinear analysis of thin-walled composite beams with arbitrary lay-ups under various types of loads is presented. This model accounts for all the structural coupling coming from both material anisotropy and geometric nonlinearity. The general nonlinear governing equations are derived and solved by means of an incremental Newton–Raphson method. A displacement-based one-dimensional finite element model that accounts for the geometric nonlinearity in the von Kármán sense is developed to solve the problem. Numerical results are obtained for thin-walled composite beam under vertical load to investigate the effects of fiber orientation, geometric nonlinearity, and shear deformation on the axial–flexural–torsional response.     

A velocity transformation method for the nonlinear dynamic simulation of railroad vehicle systems

The solution of the constrained multibody system equations of motion using the generalized coordinate partitioning method requires the identification of the dependent and independent coordinates. Using this approach, only the independent accelerations are integrated forward in time in order to determine the independent coordinates and velocities. Dependent coordinates are determined by solving the nonlinear constraint equations at the position level. If the constraint equations are highly nonlinear, numerical difficulties can be encountered or more Newton–Raphson iterations may be required in order to achieve convergence for the dependent variables. In this paper, a velocity transformation method is proposed for railroad vehicle systems in order to deal with the nonlinearity of the constraint equations when the vehicles negotiate curved tracks. In this formulation, two different sets of coordinates are simultaneously used. The first set is the absolute Cartesian coordinates which are widely used in general multibody system computer formulations. These coordinates lead to a simple form of the equations of motion which has a sparse matrix structure. The second set is the trajectory coordinates which are widely used in specialized railroad vehicle system formulations. The trajectory coordinates can be used to obtain simple formulations of the specified motion trajectory constraint equations in the case of railroad vehicle systems. While the equations of motion are formulated in terms of the absolute Cartesian coordinates, the trajectory accelerations are the ones which are integrated forward in time. The problems associated with the higher degree of differentiability required when the trajectory coordinates are used are discussed. Numerical examples are presented in order to examine the performance of the hybrid coordinate formulation proposed in this paper in the analysis of multibody railroad vehicle systems.     

Controlled random search technique for estimation of convective heat transfer coefficient

This paper is concerned with a method for solving inverse heat conduction problem. The method is based on the controlled random search (CRS) technique in conjunction with modified Newton–Raphson method. The random search procedure does not need the computation of derivative of the function to be evaluated. Therefore, it is independent of the calculation of the sensitivity coefficient for nonlinear parameter estimation. The algorithm does not depend on the future-temperature information and can predict convective heat transfer coefficient with random errors in the input temperature data. The technique is first validated against an analytical solution of heat conduction equation for a typical rocket nozzle. Comparison with an earlier analysis of inverse heat conduction problem of a similar experiment shows that the present method provides solutions, which are fully consistent with the earlier results. Once validated, the technique is used to investigate another estimation of heat transfer coefficient for an experiment of short duration, high heating rate, and employing indepth temperature measurement. The CRS procedure, in conjunction with modified Newton–Raphson method, is quite useful in estimating the value of the convective heat-transfer coefficient from the measured transient temperature data on the outer surface or imbedded thermocouple inside the rocket nozzle. Some practical examples are illustrated, which demonstrate the stability and accuracy of the method to predict the surface heat flux.     

On Singularity Formation of a Nonlinear Nonlocal System

We investigate the singularity formation of a nonlinear nonlocal system. This nonlocal system is a simplified one-dimensional system of the 3D model that was recently proposed by Hou and Lei (Comm Pure Appl Math 62(4):501–564, 2009) for axisymmetric 3D incompressible Navier–Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier–Stokes equations is that the convection term is neglected in the 3D model. In the nonlocal system we consider in this paper, we replace the Riesz operator in the 3D model by the Hilbert transform. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the nonlocal system for a large class of smooth initial data with finite energy. We also prove global regularity for a class of smooth initial data. Numerical results will be presented to demonstrate the asymptotically self-similar blow-up of the solution. The blowup rate of the self-similar singularity of the nonlocal system is similar to that of the 3D model.     

Studies of quality of geometrically nonlinear elasticity theory for small strains and arbitrary displacements

Earlier it was shown in [1, 2] that the equations of classical nonlinear elasticity constructed for the case of small strains and arbitrary displacements are ill posed, because their use in specific problems may result in the appearance of “spurious” bifurcation points. A detailed analysis of these equations and the construction, in their stead, of consistent equations of geometrically nonlinear theory of elasticity can be found in [3]. Certain steps in this direction were also made in [4, 5]. In [3], it was also stated that the methods and applied program packages (APPs) based on the use of the classical relations of nonlinear elasticity require some revision and correction. In the present paper, this conclusion is justified and confirmed by numerical finite-element solutions of several three-dimensional geometrically nonlinear deformation problems and linearized problems on the stability of equilibrium of a rectilinear beam. These solutions were obtained by using two APPs developed by the authors and the well-known APP “ANSYS.” It is shown that the classical equations of the geometrically nonlinear theory of elasticity, which underly the first of the developed APP and the well-known APP “ANSYS,” often lead to overestimated buckling loads for structural members as compared with the consistent equations proposed in [1–3].     

Permeability identification of a weakly permeable partially saturated porous rock

The present paper deals with the determination of permeability in partially saturated conditions for weakly permeable porous continua such as argillites or deep clayey formations. The permeability can be deduced from measurements of transient weight loss of a sample submitted to a laboratory drying test: a decrease of relative humidity is imposed by saline solution in an hermetic chamber. Assumptions of constant gas pressure equal to atmospheric pressure and of negligible Fickean diffusive transport of vapour are adopted. The only transport phenomenon taken into account inside the sample is the Darcean advective transport of the water liquid. The forward problem is solved by following two modelling approaches: a linear one and a nonlinear one. The parameter identification procedure is based upon the solution of corresponding inverse problems. In the two cases, the Levenberg–Marquardt algorithm has been used for the minimization problem. In the linear approach, the solution of the forward problem is explicit. In the non linear approach, finite volume method for the spatial discretization combined with a Newton–Raphson algorithm has been used to solve the non linear forward problem. The identification method enables variations of permeability and capillary capacity to be estimated. Comparisons between linear and non linear approaches show that the first one is useful to give mean values and order of magnitude of permeability and capacity. A more complete information is deduced from the non linear approach as variations of equivalent capacity and permeability during a test are significant in most cases. The analysis of the obtained results shows that the basic modelling assumption of constant gas pressure inside the sample would not be relevant for lower range of relative humidities and liquid permeability than those investigated.     

A comparison of two formulations of the fin efficiency for straight fins

A formulation of the fin efficiency based on Newton’s law of cooling is compared with a formulation based on a ratio of heat transferred from the fin surface to the surrounding fluid to the heat conducted through the base.The first formulation requires that the solution of the nonlinear fin equations for constant heat transfer coefficient and constant thermal conductivity is known,whilst the second formulation of the fin efficiency requires only that a first integral of the model equation is known.This paper shows the first formulation of the fin efficiency contains approximation errors as only power series and approximate solutions to the nonlinear fin equations have been determined.The second formulation of the fin efficiency is exact when the first integrals can be determined.     

Finite-Amplitude Equilibrium Solutions for Plane Poiseuille–Couette Flow

Two-dimensional nonlinear equilibrium solutions for the plane Poiseuille–Couette flow are computed by directly solving the full Navier–Stokes equations as a nonlinear eigenvalue problem. The equations are solved using the two-point fourth-order compact scheme and the Newton–Raphson iteration technique. The linear eigenvalue computations show that the combined Poiseuille–Couette flow is stable at all Reynolds numbers when the Couette velocity component σ₂ exceeds 0.34552. Starting with the neutral solution for the plane Poiseuille flow, the nonlinear neutral surfaces for the combined Poiseuille–Couette flow were mapped out by gradually increasing the velocity component σ₂. It is found that, for small σ₂, the neutral surfaces stay in the same family as that for the plane Poiseuille flow, and the nonlinear critical Reynolds number gradually increases with increasing σ₂. When the Couette velocity component is increased further, the neutral curve deviates from that for the Poiseuille flow with an appearance of a new loop at low wave numbers and at very low energy. By gradually increasing the σ₂ values at a constant Reynolds number, the nonlinear critical Reynolds numbers were determined as a function of σ₂. The results show that the nonlinear neutral curve is similar in shape to a linear case. The critical Reynolds number increases slowly up to σ₂∼ 0.2 and remains constant until σ₂∼ 0.58. Beyond σ₂ > 0.59, the critical Reynolds number increases sharply. From the computed results it is concluded that two-dimensional nonlinear equilibrium solutions do not exist beyond a critical σ₂ value of about 0.59. Received: 26 November 1996 and accepted 12 May 1997     

Spontaneous swirling in axisymmetric MHD flows of an ideally conducting fluid with closed streamlines

The region of instability of the Hill-Shafranov viscous MHD vortex with respect to azimuthal axisymmetric perturbations of the velocity field is determined numerically as a function of the Reynolds number and magnetization in a linear formulation. An approximate formulation of the linear stability problem for MHD flows with circular streamlines is considered. The further evolution of the perturbations in the supercritical region is studied using a nonlinear analog model (a simplified initial system of equations that takes into account some important properties of the basic equations). For this model, the secondary flows resulting from the instability are determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 40–50, May–June, 2007.     

Optimum Suction Distribution for Transition Control

The optimum suction distribution which gives the longest laminar region for a given total suction is computed. The goal here is to provide the designer with a method of finding the best suction distribution subject to some overall constraints applied to the suction. We formulate the problem using the Lagrangian multiplier method with constraints. The resulting nonlinear system of equations is solved using the Newton–Raphson technique. The computations are performed for a Blasius boundary layer on flat-plate and crossflow cases. For the Blasius boundary layer, the optimum suction distribution peaks upstream of the maximum growth rate region and remains flat in the middle before it decreases to zero at the end of the transition point. For the stationary and travelling crossflow instability, the optimum suction peaks upstream of the maximum growth rate region and decreases gradually to zero. Received 8 May 1997 and accepted 5 November 1998     

Fluid flow in an arbitrary cascade on an axisymmetric stream surface in a variable thickness layer

A solution is given for the problem of flow past a cascade on an axisymmetric stream surface in a layer of variable thickness, which is a component part of the approximate solution of the three-dimensional problem for a three-dimensional cascade. Generalized analytic functions are used to obtain the integral equation for the potential function, which is solved via iteration method by reduction to a system of linear algebraic equations. An algorithm and a program for the Minsk-2 computer are formulated. The precision of the algorithm is evaluated and results are presented of the calculation of an example cascade.In the formulation of [1, 3] the problem of flow past a three-dimensional turbomachine cascade is reduced approximately to the joint solution of two-dimensional problems of the averaged axisymmetric flow and the flow on an axisymmetric stream surface in an elementary layer of variable thickness.In the following we solve the second problem for an arbitrary cascade with finite thickness rotating with constant angular velocity in ideal fluid flow: the solution is carried out on a Minsk-2 computer.Many studies have been devoted to this problem. A method for solving the direct problem for a cascade of flat plates in a hyperbolic layer was presented in [2]. Methods were developed in [1, 3] for constructing the flow for the case of a channel with variable thickness; these methods are approximately applicable for dense cascades but yield considerable error for small-load turbomachine cascades. The solution developed in [4], somewhat reminiscent of that of [2], is applicable for thin, slightly curved profiles in a layer with monotonically varying thickness. A solution has been given for a circular cascade for layers varying logarithmically [5] and linearly [6]. Approximate methods for slightly curved profiles in a monotonically varying layer with account for layer variability only in the discharge component were examined in [7–9]. A solution is given in [10] for an arbitrary layer by means of the relaxation method, which yields a roughly approximate flow pattern. The general solution of the problem by means of potential theory and the method of singularities presented in [11] is in error because of neglect of the crossflow through the skeletal line. The computer solution of [12] contains an unassessed error for the calculations in an arbitrary layer. The finite difference method is used in [13] to solve the differential equation of flow, which is illustrated by numerical examples for monotonie layers of axial turbomachines. The numerical solution of [13] is very complex.The solution presented below is found in the general formulation with respect to the geometric parameters of the cascade and the axisymmetric surface and also in terms of the layer thickness variation law.The numerical solution requires about 15 minutes of machine time on the Minsk-2 computer.     

The blunt-body problem in a nonuniform hypersonic viscous gas flow

The laws of heat transfer associated with the interaction of underexpanded supersonic gas jets and obstacles or blunt bodies have been investigated, for example, in [1–3]. Similar problems of nonuniform flow occur when bodies move in the wake behind other bodies; however, in this case the laws of heat transfer have so far received little attention [4–8]. It has been established that for a certain Reynolds number and flow nonuniformity parameters a zone of reverse-circulatory flow develops near the front of the blunt body. However, the conditions of transition to separated flow have not been determined. This paper presents a self-similar solution of the equations of the viscous shock layer near the stagnation line in supersonic flow past an axisymmetric blunt body located behind another body. On the basis of this solution a separationless flow criterion is proposed. The effect of the nonuniformity and the Reynolds number on the shock standoff distance, the convective heat flux and the friction drag of the blunt body is investigated. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 120–125, November–December, 1986. In conclusion the authors wish to thank I. G. Eremeitsev for useful suggestions and G. A. Tirskii for discussing their work.     

Determination of the ultimate states of elastoplastic bodies

The equations of quasistatic deformation of elastoplastic bodies are considered in a geometrical linear formulation. After discretization of the equations with respect to spatial variables by the finite-element method, the problem of determining equilibrium onfigurations reduces to integration of a system of nonlinear ordinary differential equations. In the ultimate state of a body of an ideal elastoplastic material, the matrix of the system degenerates and the problem becomes singular. A regularization algorithm for determining solutions of the problems for the ultimate states of bodies is proposed. Numerical solutions of test problems of determining the ultimate loads and equilibrium configurations for ideal elastoplastic bodies confirm the reliability of the regularization algorithm proposed. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 5, pp. 196–204, September–October, 2000.     

On stability of orthotropic cylindrical shells

In contrast to [1–3], the present paper obtains a system of stability equations and the corresponding resolving equation for orthotropic cylindrical shells of any but very short length in the case where the precritical stress state cannot be treated as the zero-moment state. These equations are a generalization of the results obtained in [4]. On the basis of these equations, one can obtain both the well-known formulas [1–3] and, for medium-length shells, some new expressions of the critical load in longitudinal compression and that under the joint action of torsionalmoments, normal pressure, and longitudinal compression. Some estimates are performed and the determination of the domain of application of some formulas given in [2] and in the present paper is attempted. For an orthotropic shell, a relationship between the elastic parameters and the shear modulus is established for axisymmetric and nonaxisymmetric buckling mode shapes in longitudinal compression.     

The stress state of a laminated nonlinearly elastic thick-walled spherical shell

This paper deals with spatial axisymmetric boundary-value problems of the physically nonlinear theory of elasticity for piecewise-homogeneous spherical bodies. The passage to dimensionless characteristics of the stress-strain state allows us to extract a physical dimensionless small parameter in the nonlinear state equations. The solution of nonlinear equilibrium equations and boundary-value problems is searched for in the form of series in positive degrees of the small parameter. This approach allows reducing the stated physically nonlinear boundary-value problem to a sequence of corresponding linear nonhomogeneous problems. A specific analytical solution and numerical results are obtained for a two-layer nonlinearly elastic spherical shell under bilateral pressure. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 12, pp. 26–32, December, 1999.     

Experimental and numerical investigations of impacting cantilever beams part 1: first mode response

In this paper, the dynamic behavior of a cantilever beam impacting two flexible stops as well as rigid stops is studied both experimentally and numerically. The effect of contact stiffness, clearance, and contacting materials is studied in detail. For the numerical study of the system, a finite element model is created and the resulting differential equations are solved using a Time Variational Method (TVM). To achieve higher computational efficiency, the Newton–Krylov method is used along with TVM. Experimental results validate the contact model proposed for predicting the first mode system dynamics. A new nonlinear force estimation function has been proposed based on measured accelerations, which enables the understanding of the impact dynamics.     

Digital Image Correlation for Specimens with Multiple Growing Cracks

Measuring the surface displacements of specimens having multiple, growing cracks is difficult with most implementations of the digital image correlation (DIC) method. This difficulty arises from the need to exclude the cracked area from the analysis, a process that oftentimes requires significant and time-consuming user input to achieve successful results. This work presents a set of modifications to the Newton–Raphson based DIC process that allows the method to automatically analyze specimens with multiple growing cracks. The modifications combine a relatively simple crack identification process that takes advantage of the consistency of quasi-regular speckle patterns with a method to reestablish the analysis in areas segregated by the crack growth. The use of a regular dot pattern does, however, introduce a greater chance for registration error in the correlation process. A method to minimize possible registration problems is also presented. Finally, the effectiveness of the method is demonstrated using images of concrete specimens with a complex and growing crack pattern.     

Initial postbuckling behavior of cylindrical composite shells under axisymmetric deformation

A solution is obtained that describes the postbuckling behavior of cylindrical shells in the case of axisymmetric buckling. The basis for this solution is Koiter’s asymptotic method and the nonlinear equations of the third-order Timoshenko theory of shells. It is shown that the bifurcation point in this case is a symmetrically unstable one. The effect of the initial axisymmetric deflections on the buckling loads is weaker when buckling is axisymmetric. The results obtained by Koiter’s special theory evidence this __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 4, pp. 108–118, April 2006.