A theory of rods 2 or, equivalently, a one-dimensional theory of solids is a characterization of the behavior of slender three-dimensional solid bodies by a set of equations having the parameter of a certain curve and the time as the only independent variables. The formidable mathematical obstacles presented by three-dimensional continuum theories dictate the need for tractable, accurate, and illuminating one-dimensional models.

The Theory of Rods

Inertia and thermal effects on the localization of plastic flow

In the present article we examine the dynamics of single-phase, equilibrium, i.e., classical, fluids in the dense gas regime. The behavior of fluids of moderately large molecular weight is seen to differ significantly from that of air and water under normal conditions. New phenomena include the formation and propagation of expansion shocks, sonic shocks, double sonic shocks, and shock-splitting. The more complicated existence conditions for shock waves are described and related to the dissipative structure. We also give a brief description of transonic flows and show that the critical Mach number for conventional blade shapes can be increased by a factor of 30–50% for these fluids.

Nonclassical Dynamics of Classical Gases

The modified Burgers equation (MBE) $\frac{\partial V}{\partial X}+V^{2}\frac{\partial V}{\partial \tau}=\epsilon \frac{\partial ^{2}V}{\partial \tau ^{2}}$ has recently been shown by a number of authors to govern the evolution, with range X, of weakly nonlinear, weakly dissipative transverse waves in several distinct physical contexts. The only known solutions to the MBE correspond to the steady shock wave (analogous to the well-known Taylor shock wave in a thermoviscous fluid) or to a similarity form. It can, moreover, be proved that there can exist no Backlund transformation of the MBE onto itself or onto any other parabolic equation, and in particular, therefore, that no linearizing transformation of Cole-Hopf type can exist. Attempts to understand the physics underlying the MBE must then, for the moment, rest on asymptotic studies and direct numerical computation. Our aim in the present paper is to find asymptotic solutions to the MBE for small values of the dissipation coefficient $\epsilon $, but covering all values of the range and phase variables (X,$\tau $) of interest, this being achieved by systematic use of matched asymptotic expansion techniques. Two specific initial distributions are studied, in which V(0,$\tau $) is either an N-wave or a sinusoid. The corresponding problems for the ordinary Burgers equation, generalized to include cylindrical or spherical spreading effects, were treated in detail by this method elsewhere. A feature of particular interest in the present paper is that the steady Taylor-type shock waves, which separate the lossless portions of the wave form in the early stages after shock formation, develop an internal singularity at and beyond some finite range X$\_{1}$ (X$\_{1}$ = 10 for N-waves, X$\_{1}$ = 9.601$\ldots $ for the sinusoid) unless the condition H+2G = 0 for X $\geq $ X$\_{1}$ is imposed on the values H(X), G(X) of V on either side of the shock. This leads to a `refraction' of the characteristics of the lossless solution as they pass through a shock at ranges greater than X$\_{1}$, and to multiple refractions in the periodic problem. The structure of the shock waves is analysed at all ranges, and it is shown how, at large ranges (X = O ($\epsilon ^{-2}$) for N-waves, X = O ($\epsilon ^{-1}$) for the sinusoid) the shocks thicken and merge with the lossless portions, leading to a phase of the motion governed by the full MBE. This phase is followed at still larger ranges by transition into old-age decay under linear mechanisms, and the form of the old-age functions is given. Computations of the wave form for the sinusoidal initial distribution are given that support the imposition of the criterion H + 2G = 0 for X > X$\_{1}$. Appendix A gives a brief derivation of the MBE for the case of transverse electromagnetic waves in a nonlinear dielectric, and Appendix B provides a sketch of some interesting features and unresolved difficulties associated with higher-order calculations.

Nonlinear Wave Motion Governed by the Modified Burgers Equation

In this paper, we investigate the evolution of N-waves in a medium governed by the modified Burgers' equation. It is shown that the general behaviour when the nonlinearity is of arbitrary odd integer order is the same as for the cubic case. For an N-wave of zero mean displacement, a shock is formed immediately to prevent a multi-valued solution and a second shock is formed at later times. At a finite time, the second shock satisfies a sonic condition and this state persists. The Taylor-type shock structure ceases to be the appropriate description, and instead we have a shock which matches only algebraically to the outer wave on one side. At a larger time still, the other shock is affected but the two shocks remain distinct until the wave dies under linear mechanisms. The behaviour of N-waves of non-zero mean is also examined and it is shown that in some cases, a purely one-signed profile remains.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0956792500002291

Sonic shocks governed by the modified Burgers' equation

Assuming a similarity hypothesis that achieves the balance of non-linearity, unsteadiness and dissipative effects, a new type of Burgers' equation is obtained. Consideration is limited to those cases where the singular surface theory predicts no growth of acceleration wave. Two examples are given. In these examples the interpretation of the evolutionary character of shock formation from acceleration waves for a non-dissipative system fails. The present work provides such an interpretation. Taylor's solution for a plane shock and self-similar solutions for plane and cylindrical shocks are discussed.



Unter Zugrundelegung einer Ahnlichkeitshypothese, die den Ausgleich zwischen Nichtlinearitat, Instationaritat und dissipativen Effekten sichert, wird ein neuer Typ einer Burgers-Gleichung gewonnen. Die Untersuchung ist auf solche Falle beschrankt, bei denen die Theorie der singularen Flachen kein Anwachsen der Beschleunigungswelle voraussagt. Zwei Beispiele werden gegeben, bei denen die Deutung des Entwicklungscharakters bei einer Stosbildung aus Beschleunigungswellen fur nichtdissipative Systeme versagt. Die vorliegende Arbeit liefert eine solche Interpretation. Dabei werden die Taylorsche Losung fur einen ebenen Stos und Ahnlichkeitslosungen fur offene und zylindrische Stose diskutiert.

A New Type of Burgers' Equation

A systematic procedure is developed for constructing dynamical theories of rods, plates, and shells, based on full three-dimensional equations. The title problem is chosen to bring out the various difficulties. The method of asymptotic expansion, used till now for static problems, is extended to time-dependent cases. Successive approximations give classical wave equations, the effect of lateral inertia, and the effect of transverse shear. However, two additional objectives are achieved. First, it has been possible to construct a complete and well-posed “inner problem”; this obviates the necessity of “matching” with “boundary layer” solutions at each stage. Second a modified approach is proposed to construct higher approximations; previously known methods have led to solutions where an initial singularity worsens with higher approximations. The procedure herein developed not only avoids it but gives a physically more realistic solution.



Auf der Grundlage vollstandiger dreidimensionaler Gleichungen wird ein systematisches Verfahren zur Aufstellung von Theorien fur das dynamische Verhalten von Staben, Platten und Schalen entwickelt. Das im Titel genannte Problem wurde ausgewahlt, um daran die verschiedenen Schwierigkeiten deutlich zu machen. Die bisher auf statische Probleme angewendete Methode der asymptotischen Entwicklung wird auf zeitabhangige Falle ausgedehnt. Die Approximationen liefern der Reihe nach die klassischen Wellengleichungen, die seitlichen Tragheitswirkungen und die Querschubwirkung. Daruber hinaus werden noch zwei weitere Ziele erreicht. Erstens konnte ein vollstandiges „inneres Problem” konstruiert und damit die Notwendigkeit umgangen werden, die Losungen in jedem Schritt den Grenzschichtlosungen anzupassen. Zweitens wird ein modifiziertes Verfahren zur Gewinnung hoherer Approximationen vorgeschlagen. Die von fruher her bekannten Methoden haben zu Losungen gefuhrt, bei denen sich eine anfanglich vorhandene Singularitat mit wachsendem Grad der Annaherung verschlimmert. Das hier entwickelte Verfahren vermeidet dies und liefert eine physikalisch realistischere Losung.

G. A. Nariboli

Papers

A New Type of Burgers' Equation

Asymptotic Theory of Wave‐Motion in Rods (Longitudinal Wave‐Motion)