Weak shock waves and shear bands in compressible,inextensible thermoelastic solids

A study of weak shock waves propagating into a solid, which is compressible but temperature-dependent extensible in a specified direction is presented. The inextensible solid is also considered. The constitutive equations of constrained thermoelastic material are written as the summation of constrained and unconstrained counterparts of the relevant quantities. The equation of motion of weak shock waves, which is recovered by the theory of singular surfaces, reduces to an eigenvalue problem. The solution of this eigenvalue problem yields the speeds of propagation of weak shock waves. In the case of an undeformed solid, the speeds of these waves are explicitly expressed. Additionally, a discussion on the ductility limits of constrained thermoelastic material subjected to the uniaxial and biaxial extensions is presented.     

The cumulative interaction of finite amplitude waves in strings

A “two time scale” asymptotic expansion procedure describing the modulation of a propagating simple wave governed by a system of non-linear partial differential equations is applied to the deflection waves of non-linear elastic strings. Rapid deflection signals propagating into a general slowly varying disturbance are modulated. In addition, they themselves affect the equations for that disturbance. The two effects are separated naturally when, to prevent the cumulative growth inherent in most “high frequency” procedures, an averaging technique is introduced. The interaction of two deflection waves is given as a specific example.     

Propagation of discontinuities in vibrating strings subject to solid friction

Summary We study the propagation, in a vibrating string subject to solid friction, of discontinuities of acceleration and curvature, which correspond to the starting and stopping, for finite or infinite waiting time, of the string. Those discontinuities related to the discontinuity of the friction law, propagate with a velocity which, in general, differs from the characteristic velocities. After a general study of the problem, examples are given.

---

Sommario Si studia la propagazione, in una corda vibrante soggetta ad attrito secco, di discontinuità di accelerazione e curvatura, che corrispondono all'avviarsi e al fermarsi della corda, in un tempo finito o infinito. Le discontinuità collegate con la discontinuità della legge d'attrito si propagano con una velocità che, in generale, differisce dalle velocità caratteristiche. Dopo uno studio generale del problema si danno esempi.

     

Propagation of weak shock waves

An asymptotic method is developed in order to treat the evolution of weak shock waves. One obtains a geometrical theory according to which weak shock waves propagate along rays and satisfy a transport law.     

Propagation of periodic long waves

Summary A method is developed for the computation of the steady solution of the shallow water equations with quasi periodic boundary conditions. Because of dissipation the influence of the initial conditions becomes negligible with increasing time and the solution finally depends on the boundary conditions. The unknowns variables (velocity and surface elevation) and the boundary conditions are developed in power series of a small perturbation parameter. The problem is then transformed in a sequence of linear problems which have the same associated homogeneous problem. By separating the time and space variables in the homogeneous problem we obtain an homogeneous elliptic problem of which we compute the first eigenvalues and eigensolutions. These are related to the characteristic oscillations of the water in the basin. The solution of each linear problem is then obtained as an eigensolution expansion with time dependent coefficients. These coefficients are solutions of ordinary differential equations which can be solved directly without proceeding step by step in time. In this way we are reduced to a stationary problem i.e. the determination of the eigenvalues and eigensolutions of the elliptic problem and to the computation of several integrals needed for the determination of the time dependent coefficients. A first test of the method has been carried out for a one-dimensional problem i.e. the tidal wave in a canal of finite length and constant depth. In this case the various steps of the procedure outlined above can be performed analitically. The results have been compared with those obtained by a step by step numerical integration of the shallow water equations. The agreement between these sets of results is good for the range of values of the parameters currently used in the applications.

---

Sommario Viene presentato un metodo per il calcolo della soluzione a regime delle equazioni delle onde lunghe, dipendente dalle sole condizioni al contorno quasi-periodiche, dopo aver mostrato che l'influenza delle condizioni iniziali diventa col tempo trascurabile a causa del termine di resistenza. Il metodo si basa sullo sviluppo in serie di potenze di un piccolo parametro sia delle incognite (velocità ed elevazioni del pelo libero) sia delle condizioni al contorno al fine di trasformare il problema non lineare in una serie di problemi lineari aventi lo stesso problema omogeneo associato. Con la separazione delle variabili spazio, tempo quest'ultimo problema viene ricondotto ad un problema ellittico omogeneo di cui si calcolano i primi autovalori ed autosoluzioni. Da ultimo la soluzione di ciascun problema lineare è ottenuta come sviluppo in serie di autosoluzioni a coefficienti dipendenti dal tempo: questi si ricavano risolvendo analiticamente delle equazioni differenziali ordinarie. Si elimina cosi la necessità di procedere passo passo nel tempo analogamente ai classici metodi armonici di soluzione di sistemi lineari. Riassumendo, l'applicazione del metodo riconduce alla soluzione di un problema stazionario (determinazione di autovalori ed autosoluzione del problema ellittico) e quindi al calcolo dei vari integrali necessari per la determinazione dei coefficienti temporali. Il metodo è stato provato nel caso semplice della propagazione di onde di marea in un canale di lunghezza finita e sezione costante. Per questo esempio i vari passi di calcolo possono essere svolti analiticamente. I risultati sono stati confrontati con quelli ottenuti dalla integrazione numerica delle equazioni col metodo delle caratteristiche, ottenendo un buon accordo.

---

---

Work sponsored by the CNR (National Research Council) in the framework of Project ?Conservazione del suolo? Subproject ?Dinamica dei Litorali? Publ. n° 46.     

The attenuation of monochromatic surface waves due to the presence of an inextensible cover

The attenuation of surface gravity waves is an important process associated with air–sea and wave–current interactions. Here we investigate experimentally the attenuation of monochromatic surface gravity waves due to the presence of various surface covers. The surface covers are fixed in space such that they do not advect with the wave motion and are selected such that the bending modulus is negligible for the wave frequencies used in the experiment in order to minimize any flexural effects. Wave attenuation rates are found to be independent of wave steepness and the type of cover used over the tested parameter range. Results are consistent with the theoretical attenuation rate for an inextensible surface cover.     

Propagation of elastic waves in water-saturated soils

A brief review of the literatures on the titled subject is given. A set of wave equations, taking the inertial coupling effect between soil skeleton and pore water into account, are established for saturated soils. The preliminary analysis shows that the nature of wave propagation is mainly influenced by permeability coefficient,k. There are three types of waves, two (P-and S-wave) propagating through soil skeleton and one(P-wave) through pore water. For a soil with large value ofk, compression wave velocity through pore water will be greater than that through single-phased water, and ask→∞, the former could be times as great as the latter. For a soil with extremely low permeability, the compression wave velocity could be either less or greater than that through single-phased water, depending on the rigidity of the soil passing through. Some phenomena observed from tests presented in the literature may be reasonably explained by the proposed theory herein, and thus more reliable parameters of soil could be obtained from wave velocity measurements. Further studies on this subject are still needed. This paper is a part of the dissertation of the first author for the Ph.D. degree, the second author is his advisor.     

Propagation of combustion waves in bubbly media

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 2, pp. 165–166, March–April, 1992.     

Propagation of pressure waves in two-phase flow

We deal with a pressure wave of finite amplitude propagating in a gas and liquid medium or in the fluid in an elastic tube. We study the effects of pipe elasticity on the propagation velocity of the pressure wave. Pressure waves of finite amplitude progressing in the two-phase flow are treated considering the void fraction change due to pressure rise. The propagation velocity of the two-phase shock wave is also investigated, and the behavior of the reflection of the pressure wave at the rigid wall is analyzed and compared to that in a pure gas or liquid. The results are compared to experimental data of a pressure wave propagating in the two-phase flow in a vertical shock tube.     

Propagation of shock waves in free space

The problem of the exit of a shock wave from an axisymmetric channel and its propagation in a free space occupied by an ideal gas is examined. This problem has been studied earlier in [1], in which the shock wave front was considered planar, as well as in [2], in which the wave front was regarded as a surface of an ellipsoid of revolution. The solutions obtained in these studies assumed the presence of two regions in the wave-front surface: the region of the original shock wave and a region stemming from the decomposition of an infinitesimally thin annular discontinuity of the gas parameters, with the wave intensity over the front surface in each region being considered constant, i.e., the wave character of the process over the front was not considered. In this study a solution will be achieved by the method of characteristics [3–5] of the equations of motion of the shock-wave front, as obtained in [6, 7]. Flow fields are determined for the region immediately adjacent to the shock-wave front for a wide range of shock-wave Mach numbers M _a =1.6–20.0 for = 1.4. On the basis of the data obtained, by introduction of variables connected with the length of the undisturbed zone, as calculated from the channel cross-section along the x axis, together with the pressure transition at the wave front, relationships are proposed which approach self-similarity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 163–166, September–October, 1971.In conclusion, the author thanks S. S. Semenov for his valuable advice on this study.     

Propagation of shock waves in nonuniform plasma

Results of numerical simulation of the propagation of one-dimensional magnetohydrodynamic shock waves in a nonuniform plasma containing a magnetic field are discussed. Possible uses for the production of high velocities and temperatures and astrophysical applications are considered. The essential effect of the magnetic field is shown; acceleration of a shock wave is intensified in a medium with decreasing density.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 22–26, March–April, 1976.In conclusion, the authors are grateful to S. K. Godunov for a detailed discussion of the computational aspects and of the results, and to A. E. Voitenko for a discussion of experimental possibilities and of the results.     

Propagation of waves in thin elasto-viscoplastic layers

International Applied Mechanics -     

Propagation of guided waves in pressure vessel

Pressure vessels usually operate under extremes of high/low temperatures and high pressures. Defect, such as crack and corrosion, can result in leakage or rupture failures, even catastrophic incidents. Guided wave-based structural health monitoring (SHM) technology is one of the most prominent options in non-destructive evaluation and testing (NDE/NDT) techniques. Propagation of guided waves in a typical pressure vessel is systematically investigated in this study for the application of guided wave-based SHM. Shape of the pressure vessel is a cylinder with two end caps. Because of geometric similarity, theory of guided wave propagation in the cylinderical structure is analyzed to study dispersive features of guided waves in the pressure vessel. Dispersion curves of three different types of guided wave modes, viz. the longitudinal, torsional and flexural modes, are calculated using theoretical method. Based on the analyses and experimental wave signals, central frequency and wave parameters of incident wave are optimized. Effect of contained liquid on propagation of guided waves, especially the L(0, 2) mode, in the pressure vessel is further investigated to minimize energy leakage of the wave to the contained liquid. The analytical method, finite element analysis (FEA) and experiments are applied to study propagation characteristics of guided waves in the pressure vessel, so as to demonstrate the feasibility of guided wave-based non-destructive evaluation and testing (NDE/NDT) for such kind of complex structures.     

Propagation of weak waves in elastic-plastic solids

Wave fronts admitting discontinuities only in the derivatives of the dependent variables are by convention called ‘weak’ waves. For the special case of discontinuous first-order derivatives, the fronts are customarily called ‘acceleration’ waves. If the governing equations are quasi-linear, then the weak waves are necessarily characteristic surfaces. Sometimes, these surfaces are also referred to as ‘singular surfaces’ of order r ? 1, where r stands for the order of the first discontinuous derivatives. This latter approach is adopted in this paper and applied to governing equations which form a set of first-order quasi-linear hyperbolic equations. When these equations are written in terms of singular surface coordinates, simplification occurs upon differencing equations written on the front and rear sides of the surface: a set of algebraic (‘connection’) equations is generated for the discontinuities in the normal derivatives of the dependent variables across the surface. When a similar operation is performed on the governing equations written for second-order derivatives, a set of first-order differential (‘transport’) equations is generated.     

Propagation of weakly nonlinear magnetoacoustic waves

Using the reductive perturbation method we show that the modified Burgers equation governs the propagation of a weakly nonlinear slow magnetoacoustic wave near the equilibrium state with zero transverse magnetic field.     

Propagation characteristics of shock waves driven by gaseous detonation waves

We experimentally investigated propagation characteristics of the shock wave driven by a gaseous detonation wave emerging from the open end of a cylindrical detonation tube. In the present study, we visualized the shock wave and exhaust flowfields using a shadowgraph optical system and we obtained peak overpressure in the tube axial direction and the continuous shape transformation of shock waves around the tube open end. We also obtained overpressure histories of the shock wave using piezo-pressure transducers within 201 m from the open end of the tube. We normalized and classified these results by four regions using non-dimensional pressure and distance which are independent of variety of mixture and tube diameter. In the vicinity of the open end of the tube, the shock wave is nearly planar and does not significantly attenuate, and the peak overpressure maintains approximately C–J pressure. Subsequently, the shock wave attenuates rapidly, transforming from quasi-spherical to spherical. Farther from the tube open end, the shock wave propagates with approximately sound characteristic so that the peak overpressure decreases proportional to 1/r. Eventually, the shock wave begins to attenuate more rapidly than ideal sound attenuation, which may be due to the viscous effect.