Waves of Transverse Stresses under a Plane Impact on the Surface of a Sandwich Panel (Acoustic Approximation)

The wave process arising in a sandwich panel with a free back surface under the action of a short-term dynamic load on the front surface of the upper layer (plane deformation) is investigated. The calculation procedure for displacements, rates, and stresses under a rectangular short-time pulse, whose duration does not exceed the double time of wave travel within a layer, is based on the representation of the solution to the one-dimensional wave equation in terms of characteristics. The transmission and reflection coefficients of the pressure pulses on the contact surfaces of layers with different physical properties are determined. The expressions for tensile stresses in the panel face layers and filler, which are responsible for the material failure by spalling, are presented. The stresses in relation to the geometry and dynamic parameters of the sandwich structure are analyzed. In the case of a symmetric panel structure, the stress pattern in the midlayer and on its contact boundaries is given, which takes into account the branching and superposition of pulses.     

A Study on the Propagation of Plane Stress Waves across the Thickness of a Plate by the Method of Analytic Continuation in Time

The interaction of plane tension/compression waves propagating within a plate perpendicularly to its surface is considered. The analytic solution is obtained by a modified method of characteristics for the one-dimensional wave equation used in problems on an impact of a rigid body on the surface of a plate. The displacements, velocities, and stresses in the plate are determined by the edge disturbance caused by the initial velocity and the stationary force field of masses of the striker and the plate. The method of analytic continuation in time put forward allows a stress analysis for an arbitrary time interval by using finite expressions. Contrary to a stress analysis in the frequency domain, which is commonly used in harmonic expansion of disturbances, the approach advanced allows one to analyze the solution in the case of discontinuous first derivatives of displacements without calculating jumps in summing series. A generalized closed-form solution is obtained for stresses in an arbitrary cycle n(t), which is determined by the multiplicity of the time of wave travel across the double thickness of the plate. A method of recurrent solution based on calculating the convolution of repeated integrals of the initial form of disturbance at t = 0 is elaborated. The procedure can be used for evaluating the maximum stress and the contact time in a plane impact on the surface of a plate.     

Influence of Lower Terms on the Well-Posedness of Characteristics Problems for Third-Order Hyperbolic Equations

In this paper, we consider a general problem of Goursat type for third-order hyperbolic equations of general form with dominated lower terms. We study the influence of the lower terms contained in the equation, as well as those in the boundary conditions, on the well-posedness of the problem under consideration.     

Computation of the scattering amplitude for a scattering wave produced by a disc — Approach by a fundamental solution method

Our fundamental solution method gives an analytic representation of the approximate solution for the reduced wave problem in the exterior region of a disc. The asymptotic behavior of this representation yields an approximate formula for the scattering amplitude. An error estimate for this formula is given. We add two numerical tests: the numerical estimate of errors; and profiles of scattering cross sections and the far-field coefficient. Both tests include cases of high wave numbers.     

Convergence in the form of a solution to the Cauchy problem for a quasilinear parabolic equation with a monotone initial condition to a system of waves

The time asymptotic behavior of a solution to the initial Cauchy problem for a quasilinear parabolic equation is investigated. Such equations arise, for example, in traffic flow modeling. The main result of this paper is the proof of the previously formulated conjecture that, if a monotone initial function has limits at plus and minus infinity, then the solution to the Cauchy problem converges in form to a system of traveling and rarefaction waves; furthermore, the phase shifts of the traveling waves may depend on time. It is pointed out that the monotonicity condition can be replaced with the boundedness condition.     

Global structure instability of Riemann solutions for general quasilinear hyperbolic systems of conservation laws in the presence of a boundary

This work is a continuation of our previous work [Z.-Q. Shao, D.-X. Kong, Y.-C. Li, Shock reflection for general quasilinear hyperbolic systems of conservation laws, Nonlinear Anal. TMA 66 (1) (2007) 93-124]. In this paper, we study the global structure instability of the Riemann solution containing shocks, at least one rarefaction wave for general n×n quasilinear hyperbolic systems of conservation laws in the presence of a boundary. We prove the nonexistence of global piecewise C¹ solution to a class of the mixed initial-boundary value problem for general n×n quasilinear hyperbolic systems of conservation laws on the quarter plane. Our result indicates that this kind of Riemann solution mentioned above for general n×n quasilinear hyperbolic systems of conservation laws in the presence of a boundary is globally structurally unstable. Some applications to quasilinear hyperbolic systems of conservation laws arising from physics and mechanics are also given.