Nonlinear Elliptic–Parabolic Problems

We introduce a notion of viscosity solutions for a general class of elliptic–parabolic phase transition problems. These include the Richards equation, which is a classical model in filtration theory. Existence and uniqueness results are proved via the comparison principle. In particular, we show existence and stability properties of maximal and minimal viscosity solutions for a general class of initial data. These results are new, even in the linear case, where we also show that viscosity solutions coincide with the regular weak solutions introduced in Alt and Luckhaus (Math Z 183:311–341, 1983).     

Generalized Strongly Nonlinear Quasi-Complementarity Problems

ＧＥＮＥＲＡＬＩＺＥＤＳＴＲＯＮＧＬＹＮＯＮＬＩＮＥＡＲＱＵＡＳＩ－ＣＯＭＰＬＥＭＥＮＴＡＲＩＴＹＰＲＯＢＬＥＭＳＬｉＨｏｎｇ－ｍｅｉ（李红梅）ＤｉｎｇＸｉｅ－ｐｉｎｇ（丁协平）（ＳｉｃｈｕａｎＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ），Ｃｈｅｎｇｄｕ（Ｒｅ...     

Nonlinear Bending of Circular Sandwich Plates

In this paper,fundamental equations and boundary conditions of non-linear axisymmetrical bending theory for the circular sandwich plates with asoft core are derived by means of the method of calculus of variations.Especial-ly in the case of very thin faces,the preceding fundamental epuations andboundary conditions simplity considerably.For example,a circular sandwichplate with edge clamped but free to slip under the action of uniform lateralload is considered.A more accurate solution of this problem has been ob-tained by means of the modified iteration method.     

Nonlinear Weakly Curved Rod by Γ-Convergence

We present a nonlinear model of weakly curved rod, namely the type of curved rod where the curvature is of the order of the diameter of the cross-section. We use an approach analogous to the one for rods and curved rods and start from the strain energy functional of three dimensional nonlinear elasticity. We do not impose any constitutional behavior of the material and work in a general framework. To derive the model, by means of ??-convergence, we need to set the order of strain energy (i.e., its relation to the thickness of the body h). We analyze the situation when the strain energy (divided by the order of volume) is of the order h ⁴. This is the same approach as the one used in F?ppl-von Kármán model for plates and the analogous model for rods. The obtained model is analogous to Marguerre-von Kármán for shallow shells and its linearization is the linear shallow arch model which can be found in the literature.     

Three-Dimensional Nonlinear Analysis of Drill String Structurein Annulus

ＴＨＲＥＥ－ＤＩＭＥＮＳＩＯＮＡＬＮＯＮＬＩＮＥＡＲＡＮＡＬＹＳＩＳＯＦＤＲＩＬＬＳＴＲＩＮＧＳＴＲＵＣＴＵＲＥＩＮＡＮＮＵＩＵＳＬｉｕＹａｎ－ｑｉａｎｇ（刘延强）ＬｕＹｉｎｇ－ｍｉｎ（吕英民）（ＤｅｐａｒｔｍｅｎｔｏｆＥｎｇｉｎｅｅｒｉｎｇｍｅｃｈ...     

Riemann–Cartan Geometry of Nonlinear Dislocation Mechanics

We present a geometric theory of nonlinear solids with distributed dislocations. In this theory the material manifold—where the body is stress free—is a Weitzenb?ck manifold, that is, a manifold with a flat affine connection with torsion but vanishing non-metricity. Torsion of the material manifold is identified with the dislocation density tensor of nonlinear dislocation mechanics. Using Cartan’s moving frames we construct the material manifold for several examples of bodies with distributed dislocations. We also present non-trivial examples of zero-stress dislocation distributions. More importantly, in this geometric framework we are able to calculate the residual stress fields, assuming that the nonlinear elastic body is incompressible. We derive the governing equations of nonlinear dislocation mechanics covariantly using balance of energy and its covariance.     

Reaction–Diffusion Equations with Nonlinear Boundary Delay

We consider dissipative scalar reaction–diffusion equations that include the ones of the form u _t–u=f(u(t)), subjected to boundary conditions that include small delays, that is, we consider boundary conditions of the form u/n _a=g(u(t), u(t–r)). We show the global existence and uniqueness of solutions in a convenient fractional power space, and furthermore, we show that, for r sufficiently small, all bounded solutions are asymptotic to the set of equilibria as t tends to infinity.     

Nonlinear Oscillation Analysis by an Orthogonal Function Method

ＮＯＮＬＩＮＥＡＲＯＳＣＩＬＩＡＴＩＯＮＡＮＡＬＹＳＩＳＢＹＡＮＯＲＴＨＯＧＯＮＡＬＦＵＮＣＴＩＯＮＭＥＴＨＯＤＳｕｎＰｉ，ｈｏｎｇ（孙丕忠）ＴａｎｇＰｉａｎｇａｎｇ（唐乾刚）ＳｕｎＳｈｉｘｉａｎ（孙世贤）（ＮａｔｉｏｎＵｎｉｖｅｒｓｉｔｙｏｆＤｅｆ...     

Nonlinear Response of Double Wall Sandwich Panels∗

An analytical study is presented to predict the nonlinear response of a double wall sandwich panel system that is subjected to random-type loading. Viscoelastic and nonlinear spring-dashpot models are chosen to characterize the behavior of the core. The nonlinear panel response is obtained by utilizing modal analyses and Monte Carlo simulation techniques. Numerical results include the response spectral densities, root-mean-square responses, and probability density function histograms. It is found that by proper selection of the dynamic parameters and damping characteristics, the structural response can be significantly reduced.     

Nonlocal Nonlinear Schrödinger Equations in R 3

This paper studies a class of nonlocal nonlinear Schrödinger equations in R ³, which occurs in the infinite ion acoustic speed limit of the Zakharov system with magnetic fields in a cold plasma. The magnetic fields induce some nonlocal effects in these nonlinear Schrödinger systems, and the main goal of this paper is to understand these effects. The key is to establish some a priori estimates on the nonlocal terms generated by the magnetic field, through which we obtain various conclusions including finite time blow-ups, sharp threshold of global existence and instability of standing waves for these equations.     

Kinetic Stabilization¶of a Nonlinear Sonic Phase Boundary

We consider a system arising in the study of phase transitions in elastodynamics – a system of two conservation laws, in a single space dimension. The system has two hyperbolic regions with an elliptic zone in between. A phase boundary is a strong discontinuity in a solution, with left and right states belonging to different hyperbolic regions. We call such a solution a phase wave. We first address the Riemann problem for initial states close to a fixed sonic phase wave, in the genuinely nonlinear case. This problem is naturally underdetermined. We propose two essentially different types of Reimann problems: a sonic one, which is smooth, and a kinetic one, which is only Lipschitz-continuous. Both problems are well posed owing to a shared stability condition that is of a purely sonic nature. In the kinetic case we prove the global existence of solutions to the Cauchy problem for initial data having small variation and close to a sonic kinetic wave. The crucial issue is the interaction of the phase boundary with a small wave of the same mode. The introduction of a pertinent quantity, called here detonation potential, ensures a balance between ingoing and outgoing waves. The proof is based on a Glimm-type scheme; we define a potential, which includes the detonation potential, along the strong discontinuity, and this potential controls the outbreak of unusual shocks. Accepted: June 9, 1999     

Supercritical Geometric Optics for Nonlinear Schrödinger Equations

We consider the small time semi-classical limit for nonlinear Schrödinger equations with defocusing, smooth, nonlinearity. For a super-cubic nonlinearity, the limiting system is not directly hyperbolic, due to the presence of vacuum. To overcome this issue, we introduce new unknown functions, which are defined nonlinearly in terms of the wave function itself. This approach provides a local version of the modulated energy functional introduced by Y. Brenier. The system we obtain is hyperbolic symmetric, and the justification of WKB analysis follows.     

Nonlinear flexural waves in fluid–filled elastic channels

Nonlinear waves on liquid sheets between thin infinite elastic plates are studied analytically and numerically. Linear and nonlinear models are used for the elastic plates coupled to the Euler equations for the fluid. One-dimensional time-dependent equations are derived based on a long-wavelength approximation. Inertia of the elastic plates is neglected, so linear perturbations are stable. Symmetric and mixed-mode travelling waves are found with the linear plate model and symmetric travelling waves are found for the nonlinear case. Numerical simulations are employed to study the evolution in time of initial disturbances and to compare the different models used. Nonlinear effects are found to decrease the travelling wave speed compared with linear models. At sufficiently large amplitude of initial disturbances, higher order temporal oscillations induced by nonlinearity can lead to thickness of the liquid sheet approaching zero.     

Nonlinear Electrohydrodynamic Stability of a Poiseuille Two–Layer Flow

The long–wave stability of the Poiseuille two–layer flow of homogeneous viscous dielectrics between plate electrodes under a constant potential difference is studied in an electrohydrodynamic approximation. A linear asymptotic stability analysis shows that surface polarization forces are a destabilizing factor, in addition to viscous stratification. The method of many scales is used to obtain the Kuramoto—Sivashinsky equation governing the weakly nonlinear evolution of the interface between the dielectrics. Within the framework of the approaches used, it is shown that nonlinear interactions limit perturbation growth and the interface does not fail even for a rather large potential difference.     

Nonlinear dynamics of hardware-in-the-loop experiments on stick–slip phenomena

A single degree-of-freedom nonlinear mechanical model of the stick–slip phenomenon is studied when the Stribeck-type friction force is emulated by means of a digitally controlled actuator. The relative velocity of the slipping contact surfaces is considered as bifurcation parameter. The original physical system presents subcritical Hopf bifurcation with a wide bistable parameter region where stick–slip and steady-state slipping are both stable locally. Hardware-in-the-loop experiments are performed with a physical oscillatory system subjected to the emulated Stribeck forces. The effect of sampling time is studied with respect to the stability and nonlinear behavior of this experimental system. The existence of subcritical Neimark–Sacker bifurcations are proven in the digital system, the stability and bifurcation characteristics of the continuous and the digital systems are compared, and the counter-intuitive stabilizing effect of sampling time is shown both analytically and experimentally. The conclusions draw the attention to the limitations of hardware-in-the-loop experiments when the corresponding systems are strongly nonlinear.     

Nonlinear dynamics of a delayed Leslie predator–prey model

The present paper is concerned with a delayed Leslie predator–prey model. The conditions of boundedness of the solutions of the system, existence, and stability of the equilibrium of the system are investigated. Meanwhile, we find that the system can also undergo a Hopf bifurcation of nonconstant periodic solution at the positive equilibrium when the delay crosses through a sequence of critical values. The extensive simulations carried out show that the bifurcations arise around the positive equilibrium.     

Independent‐Stress Method for Analysis of Nonlinear Antiplane Strain

The stress field in a cylindrical body under antiplane strains is studied using the nonlinear theory of elasticity in actual variables under assumptions of the absence of body forces and weak nonlinearity of the elastic potential. The stresses are determined by solving the nonlinear boundaryvalue problem for two independent stresses in polar coordinates of the physical and stress planes. Analytical solutions of the nonlinear problems are obtained. The effect of potential nonlinearity is studied. It is shown that the nonlinear problem can be solved using the harmonicequation solution corresponding to the linear potential.     

Nonlinear analysis of a structure loaded by a stochastic excitation

For a non-linear system excited by a stochastic load which is expressed as a time series,a recursive method based on the Z-transform is presented.To identify the obtained response time series,a discrete wavelet transform (DWT) technique is proposed.     

Hyperbolicity of the Nonlinear Models of Maxwell’s Equations

We consider the class of nonlinear models of electromagnetism that has been described by Coleman & Dill [7]. A model is completely determined by its energy density W(B,D). Viewing the electromagnetic field (B,D) as a 3×2 matrix, we show that polyconvexity of W implies the local well-posedness of the Cauchy problem within smooth functions of class H^s with s>1+d/2. The method follows that designed by Dafermos in his book [9] in the context of nonlinear elasticity. We use the fact that B×D is a (vectorial, non-convex) entropy, and we enlarge the system from 6 to 9 equations. The resulting system admits an entropy (actually the energy) that is convex. Since the energy conservation law does not derive from the system of conservation laws itself (Faradays and Ampères laws), but also needs the compatibility relations divB=divD=0 (the latter may be relaxed in order to take into account electric charges), the energy density is not an entropy in the classical sense. Thus the system cannot be symmetrized, strictly speaking. However, we show that the structure is close enough to symmetrizability, so that the standard estimates still hold true.