Nonlinear shear wave interaction in soft solids

Jacob, Xavier; Catheline, Stéfan; Gennisson, Jean‐Luc; Barrière, Christophe; Royer, D.; Fink, Mathias

doi:10.1121/1.2775871

Cited by 41 publications

(45 citation statements)

References 17 publications

(14 reference statements)

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“…Landau and Lifshitz 42 write the strain energy W 3 up to third order by including terms in I 3 , I 1 I 2 , and I 3 1 . There are 4 combinations of the invariants in the strain energy at the fourth order: I 1 I 3 ; I 2 1 I 2 , I 2 2 , and I 4 1 , thus the fourth-order strain energy is 36,[43][44][45] …”

Section: Appendix: Fourth Order Elasticity Theorymentioning

confidence: 99%

Characterizing the nonlinear interaction of S- and P-waves in a rock sample

Gallot

Malcolm

Szabo

et al. 2015

Journal of Applied Physics

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The nonlinear elastic response of rocks is known to be caused by the rocks' microstructure, particularly cracks and fluids. This paper presents a method for characterizing the nonlinearity of rocks in a laboratory scale experiment with a unique configuration. This configuration has been designed to open up the possibility of using the nonlinear characterization of rocks as an imaging tool in the field. In our experiment, we study the nonlinear interaction of two traveling waves: a low-amplitude 500 kHz P-wave probe and a high-amplitude 50 kHz S-wave pump in a room-dry 15 Â 15 Â 3 cm slab of Berea sandstone. Changes in the arrival time of the P-wave probe as it passes through the perturbation created by the traveling S-wave pump were recorded. Waveforms were time gated to simulate a semi-infinite medium. The shear wave phase relative to the P-wave probe signal was varied with resultant changes in the P-wave probe arrival time of up to 100 ns, corresponding to a change in elastic properties of 0.2%. In order to estimate the strain in our sample, we also measured the particle velocity at the sample surface to scale a finite difference linear elastic simulation to estimate the complex strain field in the sample, on the order of 10 À6 , induced by the S-wave pump. We derived a fourth order elastic model to relate the changes in elasticity to the pump strain components. We recover quadratic and cubic nonlinear parameters:b ¼ À872 and d ¼ À1:1 Â 10 10 , respectively, at room-temperature and when particle motions of the pump and probe waves are aligned. Temperature fluctuations are correlated to changes in the recovered values ofb andd, and we find that the nonlinear parameter changes when the particle motions are orthogonal. No evidence of slow dynamics was seen in our measurements. The same experimental configuration, when applied to Lucite and aluminum, produced no measurable nonlinear effects. In summary, a method of selectively determining the local nonlinear characteristics of rock quantitatively has been demonstrated using traveling sound waves. V C 2015 AIP Publishing LLC.

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Section: Appendix: Fourth Order Elasticity Theorymentioning

confidence: 99%

Characterizing the nonlinear interaction of S- and P-waves in a rock sample

Gallot

Malcolm

Szabo

et al. 2015

Journal of Applied Physics

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“…Porubov and Maugin [12] applied nonlinear strain waves to study the growth of long bones. Jacob et al [13] conducted nonlinear shear wave experiments in soft solids with the transient elastography technique. Zabolotskaya et al [14] showed that the material model in Ref.…”

Section: Introductionmentioning

confidence: 99%

Adjustable solitary waves in electroactive rods

Wang

Zhang

Dai

et al. 2015

Journal of Sound and Vibration

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“…Transient elastography is one of the most efficient approaches, as it images in real-time the transient propagation of shear waves in soft media using ultrafast echographic scanners. Based on this concept, there are three possible ways to quantify nonlinearity in soft solids: propagation of shock shear waves [1], nonlinear interaction between shear waves [2] and acoustoelasticity [3]. In a recent paper, shear wave propagation theory in soft solids brought new ideas in the field of elasticity theory.…”

Section: Introductionmentioning

confidence: 99%

7B-2 Nonlinear Shear Elastic Moduli in Quasi-Incompressible Soft Solids

Rénier

Gennisson

Tanter

et al. 2007

2007 IEEE Ultrasonics Symposium Proceedings

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Dynamic elastography holds great promise for biological tissues characterization. Resulting from the radiation force induced by focused ultrasound beam, plane shear waves are generated within the medium and imaged with an ultrafast ultrasound scanner. Known as Supersonic Shear Imaging (SSI) technique, this method allows, from the measurements of shear wave velocities, to compute shear modulus (µ) maps. Beside, in order to improve tissue diagnostic, the evaluation of the nonlinear elastic moduli could be of some interest. Recently a new formulation of the nonlinear equation describing the propagation of plane shear waves in isotropic soft incompressible solids have been developed using a new expression, up to the fourth order, of the strain energy density (e): 2 2 3 2 3 DI I A µI e + + = . Where I 2 , I 3 are invariants defined by Landau of the strain tensor and A, D the third and fourth order shear elastic constants. It has been shown that the nonlinearity parameter depends only on three coefficients S = S (µ, A, D). To date, no measurement of the parameter D have been carried out in incompressible media. In order to estimate the nonlinear parameter A, this theoretical background on soft incompressible solids is applied to the acoustoelasticity theory. Such analysis gives the variations of shear wave speed as a function of the applied stress and leads to measure both the linear shear modulus (µ) and the third order shear modulus (A). Taking advantages of the SSI technique, an acoustoelasticity experiment is performed in different incompressible soft media (agar-gelatin based phantoms). In addition, to create finite amplitude plane shear waves, the SSI technique is replaced by a vibrator applied at the surface of the phantoms. Thanks to the ultrasound ultrafast imaging system, the third harmonic component is generated by nonlinearity is measured as a function of the propagation distance. Then by comparing experiments and analytical expression of the third harmonic component given by a perturbation method, the nonlinear parameter S is deduced. Finally, the combination of these experiments with results obtained in acoustoelasticity leads to the determination of the fourth order elastic modulus (D). First, measurements of the A modulus reveal that while the behavior of phantoms is quite close from a linear point of view, their nonlinear modulus A are quite different. Applied to acoustoelasticity, the SSI technique provides potential medical applications in in vivo conditions for nonlinear characterization of biological tissues. Second, results from the complete procedure reveal a variation of the nonlinear behavior as a function of the gelatin concentration increasing. This set of experiments provides the characterization, up to the fourth order, of the nonlinear shear elastic moduli in incompressible soft media.

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Nonlinear shear wave interaction in soft solids

Cited by 41 publications

References 17 publications

Characterizing the nonlinear interaction of S- and P-waves in a rock sample

Characterizing the nonlinear interaction of S- and P-waves in a rock sample

Adjustable solitary waves in electroactive rods

7B-2 Nonlinear Shear Elastic Moduli in Quasi-Incompressible Soft Solids

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