T. Bryant Moodie

Waves in nonlinear fluid filled tubes

Abstract Wave propagation and shock formation in nonlinear elastic and viscoelastic fluid filled tubes is discussed. For a Mooney-Rivlin material a simple exact solution exhibiting distortionless propagation is found.

Complex variable methods and closed form solutions to dynamic crack and punch problems in the classical theory of elasticity

Abstract Closed form solutions to dynamical problems in the classical theory of elasticity are rare. In this paper we demonstrate how complex variable techniques may be used to obtain closed form solutions to several problems of practical importance. We consider an elastic strip, and firstly obtain closed form solutions when the surface of the strip is subjected to mixed moving boundary conditions. Next we consider the problem of a pair of punches moving along the lateral boundaries of the strip and opening a crack along the mid surface. To the best of our knowledge the closed form solutions presented are new.

Closed-Form Solutions for Finite Length Crack Moving in a Strip Under Anti-Plane Shear Stress

SummaryIn this paper exact expressions for the anti-plane dynamic stress distributions around finite length cracks propagating with constant velocity in infinitely long finite width strips are determined. Two cases of practical importance are investigated. Firstly, the lateral boundaries of the strip are clamped and displaced in equal and opposite directions, to produce anti-plane shear resulting in a tearing motion along the leading edge of the crack and, secondly, the lateral boundaries of the strip are subjected to shearing stresses. Employing Fourier transforms the solution of each problem is reduced to solving a pair of dual integral equations. Closed-form solutions of these integral equations are obtained leading to exact expressions for the stress intensity factors. Numerical results are presented in graphical form.ZusammenfassungIn dieser Arbeit werden exakte Ausdrücke für die antiplanen, dynamischen Spannungsverteilungen um Risse endlicher Länge, die sich mit konstanter Geschwindigkeit in einem unendlich langen Streifen begrenzter Breite ausbreiten, bestimmt. Zwei Fälle von praktischer Bedeutung werden untersucht. Erstens werden die Seitenränder des Streifens eingespannt und sowohl in dieser als auch in entgegengesetzter Richtung versetzt, um einen antiplanen Schub zu erzeugen, der eine Aufreißbewegung längs der Vorderkante des Risses bewirkt und zweitens werden die Seitenränder des Streifens einer Schubspannung unterworfen. Die Lösung jedes Problems wird durch die Verwendung der Fouriertransformationen auf die Lösung zweier dualer Integralgleichungen reduziert. Es werden Lösungen dieser Integralgleichungen in geschlossener Form erhalten, die auf exakte Ausdrücke für den Spannungsintensitätsfaktor führen. Numerische Ergebnisse werden in graphischer Form gezeigt.

On the constitutive relations for second sound in elastic solids

We give a derivation of the constitutive relations of an elastic heat conductor for which the heat flux and the temperature obey a frame-invariant form of Cattaneos equation.

Waves in thin-walled elastic tubes containing an incompressible inviscid fluid

Abstract The effect of the non-linearity of the governing equations on the propagation of waves in fluid filled elastic tubes is investigated. Results are obtained by the method of characteristics for a particular form of pressure pulse applied at the end of a semi-infinite initially uniform tube. An expression is obtained for the distance along the tube at which shock formation is predicted. Two different hyperelastic materials whose elastic properties model those of biological tissue are considered for the tube walls. Numerical results are presented in graphical form.

Desert margins near the Chinese Loess Plateau during the mid-Holocene and at the Last Glacial Maximum: a model-data intercomparison

Abstract Expansion and contraction of desert margins around the globe have been inferred from a variety of proxy data and have since been linked, particularly in northern China and in the sub-Sahel, to changes in freshwater flux, vegetation cover, sea surface temperatures and, perhaps most importantly, monsoon circulations. We present a direct comparison of results from numerical general circulation model experiments for the mid-Holocene and for the Last Glacial Maximum (LGM) with the climatic conditions that have been inferred from loess–paleosol sequences taken from the Chinese Loess Plateau. During the mid-Holocene in northern China, the northwestward migration of the southeast desert margin that has been suggested by grain size analysis is also expressed in the model results. There is a statistically significant wetting of the Plateau region, and increased soil moisture is a consequence of an enhanced summer monsoon whose latent heat release deepens the cyclonic Tibetan low and brings increased low-level convergence and precipitation to the area. North of the desert region, this circulation dries the soil through enhanced atmospheric subsidence, although the northern margin of the desert does not migrate significantly. Expansion of the desert margin toward the southeast at the LGM is small, but there is a statistically significant drying of the Plateau. The local hydrological cycle is reduced, and there is an increase in large-scale atmospheric subsidence over the region that is caused by the presence of the Fennoscandian ice sheet upwind. Model results therefore suggest that, in addition to local micro- and mesoclimatic conditions, regional effects, such as monsoon circulations and distal orography, are also important factors in determining the location of desert margins.

Padé extensions to ray series methods and analysis of transients in inhomogeneous viscoelastic media of hereditary type

Abstract In this paper we demonstrate that wavefront expansions for the analysis of transient phenomena are far from adequate when numerical information back of the wavefront is required. However, by employing Pade approximants together with ray series methods, we can obtain directly and greatly extend the range of validity of these expansions. The procedure, which is straightforward and requires very little computing time, is here applied to a non-trivial problem involving impact-generated shear transients in inhomogenoues viscoelastic media whose stress-strain laws are given in integral form. For a special combination of the material parameters an exact solution is recovered and used to check the validity of our approximate Pade-extended wavefront solution. We also compare our results from the extended wavefront solution with numerical solutions obtained using Bellmans approximate inversion scheme for Laplace transforms. A further advantage of our approach is that, unlike transform techniques, it does not depend upon being able to find tabulated special function equations for the transformed dependent variables. All numerical results are presented graphically for ease of comparison.

Wavefront expansions and nonlinear hyperbolic waves

Abstract Asymptotic wavefront expansions are here employed in the study of nonlinear hyperbolic waves. Numerical results based upon these methods are obtained for a particular case of interest and both these results and those obtained by the method of characteristics are presented graphically. The wavefront-based method is accurate, requires an order of magnitude less computer time, and offers a clearer understanding of the underlying wave process.

A boundary value problem for fluid-filled viscoelastic tubes

Abstract A realistic boundary value problem designed to study the generation, propagation and dissipation of transient pressure disturbances in fluid-filled viscoelastic tubes is formulated and solved. The tube equation is based upon a thin-walled shell theory for tethered tubes, the fluid is assumed inviscid, and a one-dimensional theory is extracted by averaging quantities over the tube cross section. The standard linear solid is chosen to model the viscoelastic response of the tube and comparisons with a purely elastic analysis are displayed graphically.

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.