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TECHNICAL PAPERS

New Strain Energy Function for Acoustoelastic Analysis of Dilatational Waves in Nearly Incompressible, Hyper-Elastic Materials

[+] Author and Article Information
H. Kobayashi

Department of Engineering Physics, Department of Biomedical Engineering, Department of Orthopedics and Rehabilitation, University of Wisconsin—Madison, 600 Highland Avenue, Madison, WI 53792

R. Vanderby1

Department of Engineering Physics, Department of Biomedical Engineering, Department of Orthopedics and Rehabilitation, University of Wisconsin—Madison, 600 Highland Avenue, Madison, WI 53792vanderby@surgery.wisc.edu

1

To whom correspondence should be addressed.

J. Appl. Mech 72(6), 843-851 (Feb 11, 2005) (9 pages) doi:10.1115/1.2041661 History: Received October 06, 2003; Revised February 11, 2005

Acoustoelastic analysis has usually been applied to compressible engineering materials. Many materials (e.g., rubber and biologic materials) are “nearly” incompressible and often assumed incompressible in their constitutive equations. These material models do not admit dilatational waves for acoustoelastic analysis. Other constitutive models (for these materials) admit compressibility but still do not model dilatational waves with fidelity (shown herein). In this article a new strain energy function is formulated to model dilatational wave propagation in nearly incompressible, isotropic materials. This strain energy function requires four material constants and is a function of Cauchy–Green deformation tensor invariants. This function and existing (compressible) strain energy functions are compared based upon their ability to predict dilatational wave propagation in uniaxially prestressed rubber. Results demonstrate deficiencies in existing functions and the usefulness of our new function for acoustoelastic applications. Our results also indicate that acoustoelastic analysis has great potential for the accurate prediction of active or residual stresses in nearly incompressible materials.

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Copyright © 2005 by American Society of Mechanical Engineers
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Figures

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Figure 1

Three potential waves propagate in x2 direction in prestretched media

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Figure 2

Before and after the deformation of a unit block

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Figure 3

Experiment setting to stretch a rubber specimen and to measure dilatational wave travel time as a function of stretch

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Figure 4

Stress-strain relations for a simple tension test. Solid line (-) shows the least squared experimental results. The dot (.), circle (엯), cross (×), and plus (+) effectively superimpose to show the similar analytical results evaluated from functions 9,11,12,13, respectively.

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Figure 5

Relationship between the strain energy function 13 and J=(IIIC)1∕2. Solid line: Volumetric part of function 13 as a function of J. Dashed line: The entire strain energy function 13 as a function of J.

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Figure 6

Relation between E11 and ξ22 for shear wave in function 9

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Figure 7

Stress-strain relation of rubber from a simple tension test. Crosses (×): Normalized travel time from experiment. Circles (엯): Normalized thickness from experiment. Dashed line (- -): Normalized travel times predicted by functions 11,12. Solid line: Normalized travel time predicted by function 13.

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Figure 8

Velocity of two shear waves evaluated by functions 6,13. Solid line: Shear wave 01 by function 6. Dashed line (- -): Shear wave 02 by function 6. Pluses (+): Shear wave 01 by function 13. Crosses (×): Shear wave 02 by function 13.

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