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Research Papers

On Formulas for the Velocity of Rayleigh Waves in Prestrained Incompressible Elastic Solids

[+] Author and Article Information
Pham Chi Vinh1

Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Street, Thanh Xuan, Hanoi, Vietnampcvinh@vnu.edu.vn

1

Corresponding author.

J. Appl. Mech 77(2), 021006 (Dec 10, 2009) (9 pages) doi:10.1115/1.3197139 History: Received February 09, 2009; Revised June 15, 2009; Published December 10, 2009; Online December 10, 2009

In the present paper, formulas for the velocity of Rayleigh waves in incompressible isotropic solids subject to a general pure homogeneous prestrain are derived using the theory of cubic equation. They have simple algebraic form and hold for a general strain-energy function. The formulas are concretized for some specific forms of strain-energy function. They then become totally explicit in terms of parameters characterizing the material and the prestrains. These formulas recover the (exact) value of the dimensionless speed of Rayleigh wave in incompressible isotropic elastic materials (without prestrain). Interestingly that, for the case of hydrostatic stress, the formula for the Rayleigh wave velocity does not depend on the type of strain-energy function.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

Dependence of dimensionless Rayleigh velocity xr on δ3∊(−1,3) for the case of hydrostatic stress

Grahic Jump Location
Figure 2

Dependence of the dimensionless Rayleigh velocity xr on λ∊[0.72 1.5] for different given values of σ¯2: σ¯2=0 (line a), σ¯2=0.2 (line b), σ¯2=−0.5 (line c), and σ¯2=−1 (line d) for the case δ3≠0; W=μ(λ2+λ−2−2)/2

Grahic Jump Location
Figure 3

Plot of xr on λ∊[η01/23] for neo-Hookean strain-energy function and σ2=0

Grahic Jump Location
Figure 4

Dependence of dimensionless Rayleigh velocity xr on λ∊[0.72 1.5] for different given values of σ¯2: σ¯2=0 (line a), σ¯2=0.2 (line b), σ¯2=−0.5 (line c), and σ¯2=−1 (line d) for the case δ3≠0; W=2μ(λ+λ−1−2)

Grahic Jump Location
Figure 5

Plot of ζ=ρc2/μ as a function of λ(2<λ<4) for the case δ3=0 and the Varga strain-energy function

Grahic Jump Location
Figure 6

Dependence of dimensionless Rayleigh velocity xr on σ¯2∊[−1 1] for different given values of λ: λ=1 (line a), λ=0.8 (line b), λ=1.5 (line c), and λ=1.7 (line d) for the case δ3≠0; W=8μ(λ1/2+λ−1/2−2)

Grahic Jump Location
Figure 7

Plots of ζ=ρc2/μ (solid line) and ζs=α/μ (dashed line) as functions of λ for σ2=0 and m=1/2 strain-energy function

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