Threshold-based Quasi-static Brittle Damage Evolution

We introduce models for static and quasi-static damage in elastic materials, based on a strain threshold, and then investigate the relationship between these threshold models and the energy-based models introduced in Francfort and Marigo (Eur J Mech A Solids 12:149–189, 1993) and Francfort and Garroni (Ration Mech Anal 182(1):125–152, 2006). A somewhat surprising result is that, while classical solutions for the energy models are also threshold solutions, this is shown not to be the case for nonclassical solutions, that is, solutions with microstructure. A new and arguably more physical definition of solutions with microstructure for the energy-based model is then given, in which the energy minimality property is satisfied by sequences of sets that generate the effective elastic tensors, rather than by the tensors themselves. We prove existence for this energy-based problem, and show that these solutions are also threshold solutions. A by-product of this analysis is that all local minimizers, in both the classical setting and for the new microstructure definition, are also global minimizers.     

The Method of R-Functions in the Solution of Elastic Problems on the Basis of Reissner's Mixed Variational Principle

A method is presented for solving boundary-value elastic problems on the basis of the variational–structural method of R-functions and Reissner's mixed variational principle. A mathematical formulation is given to problems on the deformation of elastic bodies under mixed boundary conditions and bodies interacting with smooth rigid dies. Solutions satisfying all the boundary conditions are proposed. For undetermined components of these solutions, the resolving equations are derived and their properties are studied. A posteriori estimation of numerical solutions is made. As examples, solutions are found to a problem on the stress–strain state of a short cylinder and to a contact problem on a cylinder interacting with a smooth die. A numerical method of solving such problems is analyzed for convergence, and the accuracy of the solutions is estimated.     

Class of differentially invariant solutions of the submodel of axisymmetric flows of an ideal gas

A class of differentially invariant solutions of a problem with the pressure independent of the radial coordinate is considered for a submodel of steady axisymmetric flows of a polytropic gas. The overdetermined system turns out to be compatible and is integrated. All solutions defining transonic and supersonic flows with a limiting surface are found. These solutions are compared with invariant solutions obtained previously.     

Regular, partially invariant solutions of rank 1 and defect 1 of equations of plane motion of a viscous heat-conducting gas

A system of the Navier-Stokes equations of two-dimensional motion of a viscous heat-conducting perfect gas with a polytropic equation of state is considered. Regular, partially invariant solutions of rank 1 and defect 1 are studied. A sufficient condition of their reducibility to invariant solutions of rank 1 is proved. All solutions of this class with a linear dependence of the velocity-vector components on spatial coordinates are examined. New examples of solutions that are not reducible to invariant solutions are obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 23–33, November–December, 2006.     

Response and stability analysis of piecewise-linear oscillators under multi-forcing frequencies

Steady-state solutions of a piecewise-linear oscillator under multi-forcing frequencies are obtained using the fixed point algorithm (FPA). Stability analysis is also performed using the same technique. For the periodic solutions of a piecewise-linear oscillator with single forcing frequency, the harmonic balance method (HBM) is also used along with the FPA. Although both FPA and HBM generate accurate solutions, it is observed that the HBM failed to converge to solutions in the superharmonic range of the forcing frequency.The fixed point algorithm was also applied to the oscillator under multifrequency excitation. The algorithm proved to be very effective in obtaining torus solutions and in locating corresponding bifurcation thresholds. A piecewise-linear oscillator model of an offshore articulated loading platform (ALP) subjected to two incommensurate wave frequencies is found to exhibit chaotic behavior. A second order Poincaré mapping technique reveals the hidden fractal-like nature of the resulting chaotic response. A parametric study is performed for the response of the ALP.     

Water Redistribution in Irrigated Soil Profiles: Invariant Solutions of the Governing Equation

Exact solutions for a class of nonlinear partial differential equations modelling soil water infiltration and redistribution in irrigation systems are studied. These solutions are invariant under two-parameter symmetry groups obtained by the group classification of the governing equation. A general procedure for constructing invariant solutions is presented in a way convenient for investigating numerous new exact solutions.     

Two-dimensional flows of a viscous binary fluid between moving solid boundaries

A class of exact solutions of hydrodynamic equations with additional Korteweg stresses is obtained which is characterized by a linear dependence of part of the velocity components on the space variable. In this class, exact solutions of two problems of binary fluid flow between moving flat solid boundaries was found. A family of particular exact solutions is obtained for the problem of viscous fluid flow between planes which approach or move away from each other according to a special law.     

Exact and soliton solutions to nonlinear transmission line model

A nonlinear transmission line (NLTL) is comprised of a transmission line periodically loaded with varactors, where the capacitance nonlinearity arises from the variable depletion layer width, which depends both on the DC and AC voltages of the propagating wave. An equivalent circuit model of NLTL is discussed analytically, in this article, and different type of solutions are celebrated. The improved extended tanh-function method has been applied successfully to extract the solutions. The obtained solutions are solitary wave solutions, singular periodic solutions, singular soliton solutions, Jacobi elliptic doubly periodic type solutions and Weierstrass elliptic doubly periodic type solutions. It is a very convenient tool to study the propagation of electrical solitons which propagate in the form of voltage waves in nonlinear dispersive media.     

Exact solutions for evolutionary submodels of gas dynamics

A new class of exact solutions with functional arbitrariness describing the motion of a polytropic gas is constructed on the basis of invariant submodels of rank two of the evolutionary type. In the solutions obtained, the velocity is a linear function of some spatial coordinates. These solutions describe continuous gas dispersion and the motion with density collapse at a finite time.     

用相似接近度法从多目标优化非劣解中求最优解

本文讨论了从多目标优化非劣解中求最优解的各种方法及其不足,提出了用灰色关联度解决非劣解各序列间的形状差异问题,用模糊数学中海明距离和贴近度解决非劣解各序列间位置差异问题,然后加以合并,综合衡量两差异得到相似接近度概念,并用以确定最优解。     

A theoretical study of a thin-film delamination using shaft-loaded blister test: Constitutive relation without delamination

This paper presents exact solutions for nonlinear large deflection of thin circular membrane loaded by a central point force using blister test with two types of boundary conditions and with or without residual stress cases. A comparison with existing solutions is presented, and a geometrically nonlinear finite element analysis (FEA) is conducted to verify our analytical solutions.     

Exact solution of the Navier-Stokes equations in a fluid layer between the moving parallel plates

This paper studies exact solutions of the Navier-Stokes equations for a layer between parallel plates the distance between which increases proportionally to the square root of time. A countable set of exact solutions and their derived countable set of continuous families of exact solutions are obtained. It is shown that certain intervals of the Reynolds parameter have two solutions and some of them one solution.     

Double Hopf Bifurcations and Chaos of a Nonlinear Vibration System

A double pendulum system is studied for analyzing the dynamic behaviour near a critical point characterized by nonsemisimple 1:1 resonance. Based on normal form theory, it is shown that two phase-locked periodic solutions may bifurcate from an initial equilibrium, one of them is unstable and the other may be stable for certain values of parameters. A secondary bifurcation from the stable periodic solution yields a family of quasi-periodic solutions lying on a two-dimensional torus. Further cascading bifurcations from the quasi-periodic motions lead to two chaoses via a period-doubling route. It is shown that all the solutions and chaotic motions are obtained under positive damping.     

用离散速度方法计算浅水长波方程

用离散速度法计算浅水波方程，将空气动力学方程和浅水波方程作了比较，用Nadiga提出的近平衡流动方法模拟浅水波方程的连续和间断解。计算了一维的溃坝波问题和Thacker提出的连续解问题，结果与精确解作了比较，并且计算了水流跃过障碍物的问题。     

Finite-dimensional approximation to global minimizers in functional spaces with R-convergence

A new concept of convergence (R-convergence) of a sequence of measures is applied to characterize global minimizers in a functional space as a sequence of approximate solutions in finite-dimensional spaces. A deviation integral approach is used to find such solutions. For a constrained problem, a penalized deviation integral algorithm is proposed to convert it to unconstrained ones. A numerical example on an optimal control problem with non-convex state constraints is given to show the effectiveness of the algorithm.     

On the Passage of a Shock Wave through a Layer of Charged Gas

A class of exact solutions of the ideal electrohydrodynamics equations is presented. These solutions describe the propagation of a plane shock wave along a static background with decreasing density in the presence of gravity and longitudinal electric fields. This class of solutions contains an arbitrary function of the Lagrangian variable which makes it possible to consider many physically different cases.     

Solution with a linear velocity field for a submodel of one-dimensional gas motions

A method for finding exact solutions of the equations of gas dynamics with a linear velocity field is proposed. This method was used to find exact solutions for one submodel of the evolutionary type which was fully integrated for the case of a polytropic gas. Examples of particle motion for the obtain exact solutions are given.     

Multi-type Entire Solutions in a Nonlocal Dispersal Epidemic Model

This paper deals with entire solutions of a nonlocal dispersal epidemic model. Unlike local (random) dispersal problems, a nonlocal dispersal operator is not compact and the solutions of nonlocal dispersal system studied here lack regularity in suitable spaces, which affects the uniform convergence of the solution sequences and the technique details in constructing the entire solutions. In the monostable case, some new types of entire solutions are constructed by combining leftward and rightward traveling fronts with different speeds and a spatially independent solution. In the bistable case, the existence of many different entire solutions with merging fronts are proved by constructing different sub- and super-solutions. Various qualitative features of the entire solutions are also investigated. A key idea is to characterize the asymptotic behaviors of the traveling wave solutions at infinite in terms of appropriate sub- and super-solutions. Finally, we also obtain the smoothness of the entire solutions in space, i.e., the solutions established in our paper are global Lipschitz continuous in space.     

Start-up flows of second grade fluids in domains with one finite dimension

A number of unidirectional transient flows of a second grade fluid in a domain with one finite dimension are studied. The method of integral transforms (Fourier, Hankel or Laplace) is applied to obtain exact solutions. A general theorem on start-up flows for second grade fluids is presented that allows us to determine unidirectional flows of second grade fluids once the corresponding solution is known within the context of the Navier-Stokes theory. In the process of obtaining solutions for the fluid of second grade, we find several new exact solutions within the context of the classical Navier-Stokes theory.     

Exact solutions of the kinetic-moment equations of a mixture of monatomic gases

An extension is given of the class of exact solutions of the kinetic-moment equations for a monatomic gas in the absence of external forces [1] to the case of a mixture of monatomic Maxwellian gases with account for external forces. Very simple solutions of this class are obtained which are examples of the normal solutions of the Boltzmann equations in the Chapman-Enskog sense [2]. Conclusions are summarized concerning the applicability of the various methods of solving the Boltzmann equations and their properties, obtained on the basis of an analysis of the indicated exact solutions.The author wishes to thank M. N. Kogan and A. A. Nikol'skii for their interest in the study.