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

Pichugin, Aleksey V.; Rogerson, G. A.

doi:10.1098/rspa.2001.0932

Cited by 26 publications

(17 citation statements)

References 11 publications

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“…These correspond to the classical plate extension and bending theories, respectively. The anti-symmetric transient response of a semi-infinite pre-stressed incompressible plate subjected to an instantaneous out-of-plane edge impulse loading has already been studied by Kaplunov & Pichugin (2005), using the asymptotic theory derived by Pichugin & Rogerson (2002). Thus, this investigation is focused on long-wave low-frequency extensional motion, which may be described by the refined asymptotic theory 1…”

Section: Long-wave Low-frequency Theory For Plate Extensionmentioning

confidence: 99%

On a Lamb-type problem for a bi-axially pre-stressed incompressible elastic plate†

Kaplunov¹,

Pichugin²,

Rogerson³

2006

IMA Journal of Applied Mathematics

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The far-field response of a bi-axially pre-stressed incompressible elastic plate, subjected to an instantaneous edge impulse loading, is studied using a refined long-wave low-frequency theory. The second order correction introduced by the refined theory is demonstrated to smooth the discontinuity associated with one of the wave fronts predicted by the leading order hyperbolic theory. The character of so-called quasi-front is shown to depend greatly both on the material parameters and pre-stress and may be either classical receding or advancing. Additionally, and in contrast to the analogous problem in linear isotropic elasticity, in a pre-stressed plate the dilatational quasi-front may propagate slower than the shear wave front. This situation is demonstrated to lead to the formation of a head-wave quasi-front.

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Section: Long-wave Low-frequency Theory For Plate Extensionmentioning

confidence: 99%

On a Lamb-type problem for a bi-axially pre-stressed incompressible elastic plate†

Kaplunov¹,

Pichugin²,

Rogerson³

2006

IMA Journal of Applied Mathematics

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“…Indeed, open questions about the relationship between such theories and three-dimensional elasticity have furnished the impetus for ongoing research (see, for example, Pichugin & Rogerson, 2002;Kaplunov et al, 2006;Paroni, 2006;Steigmann, 2007). Efforts to address these issues, or, more accurately, to side-step them, have been based in the present context on models in which the film is regarded as an elastic boundary with essentially zero thickness, as in the classical theory of capillary surfaces.…”

Section: Introductionmentioning

confidence: 99%

Surface waves supported by thin-film/substrate interactions

Steigmann

Ogden

2007

IMA Journal of Applied Mathematics

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A systematic approximation to the linear equations for small-amplitude surface waves in an elastic half space, interacting with a residually-stressed thin film of different material bonded to its plane boundary, is developed in powers of the film thickness, assuming the latter to be small compared to the wavelength of the disturbance. The theory is illustrated by calculating asymptotic expansions of the wave speeds for Love and Rayleigh waves valid to second order in the dimensionless film thickness for a transversely isotropic film bonded to an isotropic substrate.

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“…After the numerical investigations in the plane strain problem were carried out in [7] and [8], an asymptotic long and short wave analysis was carried out for a variety of layered pre-stressed structures, see [9] and [10]. Three dimensional motion in the form of a plane wave travelling parallel to the plane of the layer was also examined in [11]. Long wave approximations of the dispersion relation have also been used to help derive specific models for long wave motion in pre-stressed elastic plates, see [12] and [13].…”

Section: Introductionmentioning

confidence: 99%

Small amplitude waves in a pre-stressed compressible elastic layer with one fixed and one free face

Lashhab

Rogerson

Prikazchikova

2015

Z. Angew. Math. Phys.

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Abstract. We address the problem of wave propagation in a pre-stressed elastic layer with mixed boundary conditions, the layer having one fixed the one free face. Numerical analysis provides a good initial insight into the influence of these boundary conditions on dispersion characteristics. In the long wave regime, there is clearly no evidence of low frequency motion and thus an absence of any long wave fundamental mode-like features. In the short wave regime however, the dispersion relations does show evidence of low frequency dispersion phenomena. The first harmonic's short wave phase speed limit is shown to be distinct from that of all other harmonics; this coincides with the associated Rayleigh surface wave speed. The short wave analysis is completed with the derivation of approximate solutions for the higher harmonics. Asymptotic long wave approximations of the dispersion relation are then obtained for motion within the vicinity of the thickness stretch and thickness shear resonance frequencies. These approximations are required to obtain the relative asymptotic orders of the displacement components for frequencies within the vicinity of either the shear or stretch resonance frequencies. This enables an analogue of the asymptotic stress-strainstain to be established through asymptotic integration.

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

Cited by 26 publications

References 11 publications

On a Lamb-type problem for a bi-axially pre-stressed incompressible elastic plate†

On a Lamb-type problem for a bi-axially pre-stressed incompressible elastic plate†

Surface waves supported by thin-film/substrate interactions

Small amplitude waves in a pre-stressed compressible elastic layer with one fixed and one free face

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