G. A. Rogerson

Thermoelastic coupling effect on a micro-machined beam resonator

Abstract In this paper, the effect of thermoelastic coupling on a micro-machined resonator is studied. The calculated results show that the frequency shift ratio caused by thermoelastic coupling is of the order of 10−3, which is much larger than that of air-damping. Furthermore, the non-dimensional frequency is scale-dependent with thermoelastic coupling being considered. In contrast, the non-dimensional frequency only depends upon the ratio of thickness to length when the themoelastic coupling effect is disregarded.

The effect of finite primary deformations on harmonic waves in layered elastic media

The problem of extensional wave propagation in a pre-stressed, incompressible, 4-ply symmetric layered structure is considered. The high wave number behaviour of the harmonics is shown to fall into one of four distinct cases. Each of these are examined in detail and appropriate asymptotic expansions, giving phase speed as a function of wave number, are obtained. These are shown to provide excellent agreement with the numerical solution. A surface wave front arising from the combined influence of all harmonics is observed numerically. Corresponding plots of the eigenfunctions confirm that this is indeed a surface wave with the behaviour associated with each harmonic remarkably sensitive to changes in wave number. This paper concludes with a comparison of extensional and flexural waves.

An asymptotic analysis of the dispersion relation of a pre-stressed incompressible elastic plate

SummaryThis paper concerns an asymptotic analysis of the dispersion relation for wave propagation in a pre-stressed incompressible elastic plate. Asymptotic expansions for the wave speed as a function of wavenumber and pre-stress are obtained. These expansions have important potenatial applications to many dynamic problems such as impact problems. It is shown that in the large wavenumber limit the wave speed of the fundamental modes of both symmetric and anti-symmetric motions tends to the associated Rayleigh surface wave speed, on the other hand, the wave speeds of all the harmonics tend to a single limit which is the corresponding body wave speed. It is also shown that, whereas the fundamental modes are very sensitive to changes in the underlying pre-stress, the harmonics are little affected by such changes, espcially in the small and large wavenumber limits.

A low-frequency model for dynamic motion in pre-stressed incompressible elastic structures

An asymptotic one–dimensional theory, with minimal essential parameters, is constructed to help elucidate (two–dimensional) low–frequency dynamic motion in a pre–stressed incompressible elastic plate. In contrast with the classical theory, the long–wave limit of the fundamental mode of antisymmetric motion is non–zero. The occurrence of an associated quasi–front therefore offers considerable deviation from the classical case. Moreover, the presence of pre–stress makes the plate stiffer and thus may preclude bending, in the classical sense. Discontinuities on the associated leading–order wavefronts are smoothed by deriving higher–order theories. Both quasi–fronts are shown to be either receding or advancing, but of differing type. The problems of surface and edge loading are considered and in the latter case a specific problem is formulated and solved to illustrate the theory. In the case of antisymmetric motion, and an appropriate form of pre–stress, it is shown that the leading–order governing equation for the mid–surface deflection is essentially that of waves propagating along an infinite string, a higher–order equation for which is derived.

Some asymptotic expansions of the dispersion relation for an incompressible elastic plate

Abstract This paper is concerned with a general asymptotic analysis of the dispersion relation associated with waves propagating in a pre-stressed, incompressible elastic plate. In the high wave number limit it is well-known that, whenever a real surface wave speed exists, the fundamental modes of both symmetric and anti-symmetric motions tend to this surface wave speed, with all harmonics tending to a single shear wave speed limit. The character of the two dispersion curves in the moderate and high wave number regimes falls into one of two distinct cases, these being dependent on pre-stress. In the first case all the harmonics are monotonic decreasing functions and as such the asymptotic analysis in this case offers a modest generalisation of an earlier study, see Rogerson and Fu (Rogerson, G. A. and Fu, Y. B. (1995) An asymptotic analysis of the dispersion relation of a pre-stressed incompressible elastic plate. Acta Mechanica 111 , 59–77). In contrast, the second case is quite different in character with the passage to the high wave number limit accompanied by sinusoidal behaviour. This behaviour is fully elucidated by obtaining asymptotic expansions which give phase speed as a function of wave number, pre-stress and harmonic number, sinusoidal terms being found to occur at third order. Both these asymptotic expansions and ones obtained for high harmonic number are found to provide excellent agreement with numerical solutions for Varga materials in the appropriate regimes. It is envisaged that the expansions derived in this paper may well find important potential applications in the numerical inversion of the transform solution sometimes used in impact problems.

On small amplitude vibrations of pre-stressed laminates

In this paper small amplitude vibrations, in the form of extensional waves, of symmetric, pre-stressed elastic laminated plates are investigated. The dispersion relation is derived for an arbitrary strain energy function in the case of a four-ply plate, each layer being composed of an incompressible, pre-stressed elastic solid. This relation is investigated both numerically and analytically for a restricted class of strain energy function. The asymptotic short wave and long wave limits are fully investigated and some asymptotic expansions obtained which give phase speed as a function of scaled wave number and pre-stress. The short wave limiting wave speed of the fundamental mode is shown to be a Rayleigh surface wave speed, a Stonely interfacial wave speed or one of two associated shear wave speeds, whilst all harmonics tend to the least of the two shear wave speeds. In the long wave limit the fundamental mode is shown to be the only one which retains finite wave speed. Stability of the laminated plate is also discussed and the paper concludes with an investigation into the behaviour of the eigenfunctions as the scaled wave number increases and the asymptotic limit of the fundamental mode is either a surface or interfacial wave speed. In such cases clear localization of stress around the surface or interface is observed as scaled wave number increases.

An Asymptotically Consistent Model for Long-Wave High-Frequency Motion in a Pre-Stressed Elastic Plate

A one-dimensional asymptotic model is derived to elucidate the effect of pre-stress on long-wave high-frequency two-dimensional motion in an incompressible elastic plate. Solutions for the leading-order displacements and pressure increment are derived in terms of the long-wave amplitude; a governing equation for which is derived from the second-order problem. This equation is shown to become elliptic for certain states of pre-stress. Loss of hyperbolicity is shown to be synonymous with the existence of negative group velocity at low wavenumber. A higher-order theory is constructed, with solutions obtained in terms of both the long-wave amplitude and its second-order correction. An equation relating these is obtained from the third-order problem. The dispersion relations derived from the one-dimensional governing equations are also obtained by expansion of the corresponding exact two-dimensional relations, indicating asymptotic consistency. The model is highly relevant for stationary thickness vibration of, or transient response to high-frequency shock loading in, thin-walled bodies and also fluid-structure interaction. These are areas for which the effects of pre-stress have previously largely been ignored.

A Nonlinear Analysis of Instability of a Pre-Stressed Incompressible Elastic Plate

For an isotropic, incompressible elastic plate which is pre-stressed into a state of hydrostatic pressure, linear stability analysis predicts that when the value of the hydrostatic pressure p̄ is increased above p̄+ (say) or decreased below p̄- (say), the plate will become unstable in the sense that certain forms of travelling waves will become standing waves whose amplitude grows exponentially in time. The upper branch neutral value p̄+ is a function of the product of the wave number and plate thickness while the lower branch neutral value p̄- is a constant. Those waves which occur when the pressure deviates from the neutral values p̄+ or p̄- by a small amount are termed near-neutral waves. In this paper the nonlinear evolutionary behaviour of upper-branch near-neutral waves are investigated. It is shown that the amplitude A of such near-neutral waves satisfies an evolution equation of the form d2A/dז2 = – v0A – v1A|A|2, where v0 and v1 are constants and ז is a slow time scale. The general properties of the solution of this equation are studied. It is found that when v1 > 0 nonlinear effects are stabilizing in the sense that any exponential growth will be suppressed by nonlinearity and turned into an oscillation, and when v1 < 0 they are destabilizing in the sense that they help the wave amplitude grow. The values of v1 are calculated for a selection of the plate thicknesses (and wavenumbers) using two material models. We find that the sign of v1 is not definite and thus nonlinear effects are stabilizing only over a certain wavenumber régime while destabilizing over the remaining wavenumber régime. All the relevant factors which affect the stability of the plate are fully discussed.

Penetration of impact waves in a six-ply fibre composite laminate

Abstract In this paper the dynamic response of a six-ply fibre reinforced laminated plate to an impulsive line load acting on the upper surface is examined. Each layer of the laminate is modelled as a transversely isotropic elastic material for which the fibre direction is inextensible. The method of solution involves the use of integral transforms and the propagator matrix method. The governing equations are solved exactly within each layer and interface continuity conditions and upper and lower surface boundary conditions then satisfied. Numerical inversion of the normal displacement transform is carried out at two fixed times after impact. Results are presented for this displacement at the upper and lower surfaces as well as various interfaces through the plate.

An asymptotic membrane-like theory for long-wave motion in a pre-stressed elastic plate

An asymptotically consistent two–dimensional theory is developed to help elucidate dynamic response in finitely deformed layers. The layers are composed of incompressible elastic material, with the theory appropriate for long–wave motion associated with the fundamental mode and derived in respect of the most general appropriate strain energy function. Leading–order and refined higher–order equations for the mid–surface deflection are derived. In the case of zero normal initial static stress and in–plane tension, the leading–order equation reduces to the classical membrane equation, with its refined counterpart also being obtained. The theory is applied to a one–dimensional edge loading problem for a semi–infinite plate. In doing so, the leading– and higher–order governing equations are used as inner and outer asymptotic expansions, the latter valid within the vicinity of the associated quasi–front. A solution is derived by using the method of matched asymptotic expansions.