J. B. Haddow

Asymptotic analysis for dispersive fluid filled tubes

In a previous paper [J. Acoust. Soc. Am. 64, 522 (1978)] we obtained a new approximate equation governing the propagation of small perturbations to the pressure in a thin‐walled fluid filled elastomer tube. There, the spatial fluctuations in the pressure perturbation for fixed time were evaluated numerically by a procedure based on the Fast Fourier Transform (FFT) algorithm. Here, by means of an asymptotic analysis, based on a modified steepest descent method, we explain an apparent anomaly in the numerical results by exhibiting the existence of a forerunner wave and investigate the accuracy of the large time asymptotic results, for times which may be regarded as small or intermediate, by comparing them with the results from the FFT algorithm. It is found that the results obtained from the asymptotic analysis are in close agreement with those obtained from the FFT for moderate times and in qualitative agreement for relatively small times. For nondimensional times greater than 100 the positions of the zero...

Nonlinear axial shear wave propagation in a hyperelastic incompressible solid

SummaryThis paper is concerned with the propagation of axial shear waves in an incompressible isotropic hyperelastic solid, whose strain energy function is expressible as a power series in (I1-3) and (I2-3) whereI1 andI2 are the first and second basic invariants of the left Cauchy-Green tensorB. Numerical solutions are presented for problems of wave propagation produced by a step function application, or a finite duration pulse, of axial shear stress at the surface of a cylindrical cavity in an unbounded medium. A modification of MacCormacks finite difference scheme [1] is proposed and is used to obtain these solutions along with a procedure for the determination of the position of the shock front for the step function application.An estimate of the breaking time of a wave, obtained from a procedure proposed by Whitham [2], is compared with the numerical results. The dissipation of mechanical energy due to shock propagation is considered.

Non-linear combined axial and torsional shear wave propagation in an incompressible hyperelastic solid

Abstract Finite amplitude combined axial and torsional shear wave propagation in an incompressible isotropic hyperelastic solid is considered. When the strain energy function of the solid is a non-linear function off I 1 ,− 3) and ( I 2 − 3), where I 1 , and I 2 are the first and second basic invariants of the left Cauchy-Green tensor, the two second order partial differential equations governing the propagation of the axial and torsional waves are non-linear and coupled. These two coupled equations are equivalent to a hyperbolic system of first order partial differential equations and a modification of the MacCormack finite difference scheme is used to obtain numerical solutions of this system. Numerical results, which show the effect of the coupling, are presented for boundary-initial value problems of propagation into initially unstressed and initially stressed regions at rest.

Nearly isochoric finite torsion of a compressible isotropic elastic circular cylinder

SummaryFinite torsion of a circular bar of isotropic compressible hyperelastic material is considered. A procedure suggested by Truesdell is used to obtain solutions for a particular strain energy function although the method used is not restricted to this particular form. The procedure is applicable if the volume strain is small. Results for finite twist with the length prevented from changing and with the length allowed to change so that the resultant longitudinal force is zero are presented.ZusammenfassungBetrachtet wird die endliche Verdrehung eines Stabes mit Kreisquerschnitt aus einem isotropen, kompressiblen, hyperelastischen Werkstoff. Eine von Truesdell vorgeschlagene Vorgangsweise wird zur Bestimmung der Lösungen, für eine spezielle Verzerrungsenergiefunktion, verwendet, obwohl die verwendete Methode nicht auf diese spezielle Form beschränkt ist. Diese Vorgangsweise ist anwendbar, sofern die Volumsverzerrung klein ist. Ergebnisse für die endliche Verdrehung werden angegeben, wobei entweder die Längenänderung unterbunden ist oder die Länge sich ändern kann, so daß die resultierende Längskraft verschwindet.

Wave propagation in a fluid-filled elastic tube

SummaryIn a recent paper [1] the present authors (T.B.M. and J.B.H.) studied dispersive wave motions in a tethered, fluid-filled elastomer tube. There the radial inertia of the fluid was taken into account by employing an approximation similar to that proposed by Love [2] for analysis of wave propagation in bars and a simple bending theory of shells was employed for the tube wall. Here, by solving the fluid equations exactly we determine conditions under which the Love approximation is valid. We then extend our previous results to include the effect of shear deformation of the tube wall and analyze this extended theory to ascertain the relative importance of including shear in fluid-filled tube models designed for biological applications.ZusammenfassungIn einer vorangegangenen Arbeit [1] behandelten die beiden letztgenannten Autoren dispersive Wellenbewegungen in einem axial festgehaltenen, fluidgefüllten, elastomeren Rohr. Dort wurde die Radialträgheit des Fluids mitberücksichtigt durch Anwendung einer ähnlichen Näherung, wie sie von Love [2] für die Behandlung der Wellenausbreitung in Stäben vorgeschlagen wurde, wobei eine einfache Schalenbiegetheorie für die Rohrwand verwendet wurde. In der vorliegenden Arbeit werden durch exaktes Lösen der Gleichungen für das Fluid Bedingungen bestimmt, unter welchen die Näherung von Love gültig ist. Es werden dann die vorhergehenden Ergebnisse erweitert um Einflüsse der Schubverformung der Rohrwand mit einzuschließen und diese erweiterte Theorie wird untersucht, um die relative Bedeutung der Berücksichtigung des Schubs in fluidgefüllten Rohrmodellen, wie sie für Anwendungen in der Biologie entworfen wurden, festzustellen.

Waves in thin‐walled, nonuniform, perfectly elastic tubes containing incompressible inviscid fluid

Wave propagation in a semi‐infinite, nonuniform, elastic thin‐walled tube filled with incompressible, inviscid fluid is considered. A one‐dimensional problem, with the tube and fluid initially at rest and a pressure disturbance applied at the end, is solved for different axial variations of the tube properties. Formal asymptotic techniques are applied to the linearized partial differential equation governing the pressure and simple progressing wave and high‐frequency solutions are obtained for Heaviside and oscillatory‐pressure boundary conditions, respectively. These asymptotic series solutions terminate for certain parameter variations to give exact solutions for the pressure. Numerical results are presented and some interesting observations concerning the accuracy of the formal results are made.

Finite Oscillation of Viscoelastic Cantilever Strip

SummaryLarge amplitude free vibration of an inextensible initially straight thin viscoelastic cantilever, which is released from rest, from a relaxed deflected form is analysed. The cantilever, which is a thin strip of rectangular cross section is assumed to be composed of standard viscoelastic material. Although large deflections and rotations are considered the strains are small so that linear viscoelastic theory can be incorporated into a non-linear bending theory. It is shown how approximate solutions can be obtained by using Galerkins method and numerical results are presented graphically.

Wave propagation in a thin walled fluid filled viscoelastic tube

SummaryThe dynamic response of a thin walled, fluid filled, viscoelastic tube, subjected to the sudden release of a uniformly distributed circumferential line loading, is analyzed. It is assumed that the fluid is incompressible and inviscid and that the behavior of the tube material is represented by the standard viscoelastic model. A simple approximate shell theory, for tethered tubes, is employed. Results, for parameters appropriate to biological applications, are obtained by numerical inversion of Fourier transforms.

Analysis of expansion of spherical cavity in unbounded hyperelastic medium by method of characteristics

SummaryThe finite spherically symmetric dynamic expansion of a spherical cavity in an unbounded isotropic compressible hyperelastic solid is analysed by the method of characteristics. A spatially uniform step function application of pressure at the cavity wall is considered and a numerical procedure is given for the determination of the field of characteristics and the shock front path. Results are presented for a particular strain energy function.ZusammenfassungDie endliche, kugelsymmetrische, dynamische Ausdehnung eines kugelförmigen Hohlraumes in einem unendlichen, isotropen, kompressiblen, hyperelastischen festen Körper wird mit Hilfe der Charakteristikenmethode untersucht. Betrachtet wird ein räumlich konstanter Sprung des Druckes an der Hohlraumoberfläche und es wird ein numerisches Verfahren zur Bestimmung des Feldes der Charakteristiken und des Weges der Stoßfront angegeben. Für eine bestimmte Verzerrungsenergiefunktion werden Resultate angegeben.

Finite amplitude elastic shear wave propagation

Abstract We consider the propagation of shear waves in an incompressible, isotropic, hyperelastic hollow cylinder, where the outer radius is much larger than the inner radius, when shears are applied to the inner face. The cylindrical geometry introduces additional difficulties. We analyse the propagation of acceleration and shock waves. If the perturbations from the steady state deformed or underformed configuration are small we also consider a linearised approach and shock formation in this case. A few numerical results are included.