R.J. Tait

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.

On thermal transients with finite wave speeds

SummaryThe linear Gurtin-Pipkin theory of heat conduction is invoked to study the problem of an inhomogeneous half space whose boundary is subjected to step inputs of temperature. A ray series approach is employed to reduce the governing integro-differential equation to a set of differential-difference equations which may be solved. Various general properties of the propagation process are derived in a simple and direct fashion and the solution constructed for particular choices of the heat-flux and energy relaxation functions.

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.

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.

Azimuthal shearing and transverse deflection of an annular elastic membrane

Abstract We consider an annular membrane which has its outer rim fixed and its inner rim displaced normal to the reference plane and twisted. The resulting deformation is assumed to be axisymmetric and a direct two-dimensional formulation is adopted. The analysis is based on the Mooney-Rivlin strain energy function for isotropic elastic solids. The formation of wrinkles is accommodated in an approximate way by introducing a relaxed strain energy function. We first study the problem without twist, including a case in which wrinkling occurs, and then we investigate the full problem with twisting present. In both cases, the equations of equilibrium are reduced to a first order system of ordinary differential equations and solved numerically. Using convexity conditions, we discuss the local stability of the computed deformations.

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.

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.

Ray analysis and punching problems for stretched elastic plates

The present paper presents a ray analysis for a problem of technical importance in fragmentation studies. The problem is that of suddenly punching a circular hole in either isotropic or transversely isotropic plates subjected to a uniaxial tension field. The ray method, which involves only differentiation, integration, and simple algebra, is shown to be particularly useful in clarifying the propagation process of the resulting unloading waves and obtaining the attendant discontinuities of the various quantities involved. Numerical results obtained from the ray analysis are presented in graphical form and compared with those obtained by more elaborate schemes.

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.