Non-principal surface waves in deformed incompressible materials

Destrade, Michel; Otténio, Melanie; Pichugin, Aleksey V.; Rogerson, G. A.

doi:10.1016/j.ijengsci.2005.03.009

Cited by 20 publications

(23 citation statements)

References 23 publications

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“…Over the past five decades, such constrained material models have frequently been used in the study of acceleration waves, shock waves, and body waves [17][18][19][20][21][22][23][24][25]. They have also been used in the study of surface waves [26][27][28][29][30][31][32][33][34][35][36][37]. In this paper, we shall consider a generally constrained and prestressed elastic material.…”

Section: Accepted Manuscriptmentioning

confidence: 99%

Stroh formulation for a generally constrained and pre-stressed elastic material

Edmondson

2009

International Journal of Non-Linear Mechanics

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The Stroh formalism is essentially a spatial Hamiltonian formulation and has been recognized to be a powerful tool for solving elasticity problems involving generally anisotropic elastic materials for which conventional methods developed for isotropic materials become intractable. In this paper we develop the Stroh/Hamiltonian formulation for a generally constrained and prestressed elastic material. We derive the corresponding integral representation for the surface-impedance tensor and explain how it can be used, together with a matrix Riccati equation, to calculate the surface-wave speed. The proposed algorithm can deal with any form of constraint, pre-stress, and direction of wave propagation. As an illustration, previously known results are reproduced for surface waves in a pre-stressed incompressible elastic material and an unstressed inextensible fibre-reinforced composite, and an additional example is included analyzing the effects of pre-stress upon surface waves in an inextensible material.

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Section: Accepted Manuscriptmentioning

confidence: 99%

Stroh formulation for a generally constrained and pre-stressed elastic material

Edmondson

2009

International Journal of Non-Linear Mechanics

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“…Using the results of Destrade et al [14] (see also Chadwick [15]), we find that the incremental equations of motion can be written as a first-order differential system of six equations, namely 17) where the notation ξ is defined by 18) and the 6 × 6 matrix N has the block structure 19) in which the 3×3 matrices N 1 , N 2 , N 3 are real and their components depend on the material parameters γ ij and β ij given in (2.13), and the notation ρ = ρω 2 /k 2 has been introduced. Here I represents the 3 × 3 identity matrix.…”

Section: The Pre-stressed Elastic Half-spacementioning

confidence: 99%

Acoustic waves at the interface of a pre-stressed incompressible elastic solid and a viscous fluid

Otténio

Destrade

Ogden

2007

International Journal of Non-Linear Mechanics

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We analyze the influence of pre-stress on the propagation of interfacial waves along the boundary of an incompressible hyperelastic half-space that is in contact with a viscous fluid extending to infinity in the adjoining half-space.One aim is to derive rigorously the incremental boundary conditions at the interface; this derivation is delicate because of the interplay between the Lagrangian and the Eulerian descriptions but is crucial for numerous problems concerned with the interaction between a compliant wall and a viscous fluid. A second aim of this work is to model the ultrasonic waves used in the assessment of aortic aneurysms, and here we find that for this purpose the half-space idealization is justified at high frequencies. A third goal is to shed some light on the stability behaviour in compression of the solid half-space, as compared with the situation in the absence of fluid; we find that the usual technique of seeking standing waves solutions is not appropriate when the half-space is in contact with a fluid; in fact, a correct analysis reveals that the presence of a viscous fluid makes a compressed neo-Hookean half-space slightly more stable.For a wave travelling in a direction of principal strain, we obtain results for the case of a general (incompressible isotropic) strain-energy function. For a wave travelling parallel to the interface and in an arbitrary direction in a plane of principal strain, we specialize the analysis to the neo-Hookean strain-energy function.

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“…The extension of surface wave analysis and other wave propagation problems to anisotropic elastic materials has been the subject of many studies; see, for example, [Musgrave 1959;Anderson 1961;Stoneley 1963;Chadwick and Smith 1977;Royer and Dieulesaint 1984;Barnett and Lothe 1985;Mozhaev 1995;Nair and Sotiropoulos 1997;Destrade 2001a;2001b;Destrade et al 2002;Ting 2002a;2002c;2002b;Destrade 2003;Ogden and Vinh 2004]. For problems involving surface waves in a finitely deformed pre-stressed elastic solid (strain-induced anisotropy) we refer to [Hayes and Rivlin 1961;Flavin 1963;Chadwick and Jarvis 1979;Dowaikh and Ogden 1990;1991;Norris and Sinha 1995 (concerning a solid/fluid interface) ;Chadwick 1997;Prikazchikov and Rogerson 2004 (concerning prestressed transversely isotropic solids); Destrade et al 2005;Edmondson and Fu 2009]; see also [Song and Fu 2007]. As representatives of other works concerning waves in initially stressed elastic solids we cite [Norris 1983] on plane waves, the review [Guz 2002] and the analysis [Akbarov and Guz 2004] of waves in circular cylinders.…”

Section: Introductionmentioning

confidence: 99%

Propagation of waves in an incompressible transversely isotropic elastic solid with initial stress: Biot revisited

Ogden

Singh²

2011

J. Mech. Mater. Struct.

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In this paper, the general constitutive equation for a transversely isotropic hyperelastic solid in the presence of initial stress is derived, based on the theory of invariants. In the general finite deformation case for a compressible material this requires 18 invariants (17 for an incompressible material). The equations governing infinitesimal motions superimposed on a finite deformation are then used in conjunction with the constitutive law to examine the propagation of both homogeneous plane waves and, with the restriction to two dimensions, Rayleigh surface waves. For this purpose we consider incompressible materials and a restricted set of invariants that is sufficient to capture both the effects of initial stress and transverse isotropy. Moreover, the equations are specialized to the undeformed configuration in order to compare with the classical formulation of Biot. One feature of the general theory is that the speeds of homogeneous plane waves and surface waves depend nonlinearly on the initial stress, in contrast to the situation of the more specialized isotropic and orthotropic theories of Biot. The speeds of (homogeneous plane) shear waves and Rayleigh waves in an incompressible material are obtained and the significant differences from Biot's results for both isotropic and transversely isotropic materials are highlighted with calculations based on a specific form of strain-energy function.

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Non-principal surface waves in deformed incompressible materials

Cited by 20 publications

References 23 publications

Stroh formulation for a generally constrained and pre-stressed elastic material

Stroh formulation for a generally constrained and pre-stressed elastic material

Acoustic waves at the interface of a pre-stressed incompressible elastic solid and a viscous fluid

Propagation of waves in an incompressible transversely isotropic elastic solid with initial stress: Biot revisited

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