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Title: Nonlinear elasticity in rocks: A comprehensive three-dimensional description

Abstract

Here we study theoretically and experimentally the mechanisms of nonlinear and nonequilibrium dynamics in geomaterials through dynamic acoustoelasticity testing. In the proposed theoretical formulation, the classical theory of nonlinear elasticity is extended to include the effects of conditioning. This formulation is adapted to the context of dynamic acoustoelasticity testing in which a low-frequency “pump” wave induces a strain field in the sample and modulates the propagation of a high-frequency “probe” wave. Experiments are conducted to validate the formulation in a long thin bar of Berea sandstone. Several configurations of the pump and probe are examined: the pump successively consists of the first longitudinal and first torsional mode of vibration of the sample while the probe is successively based on (pressure) $P$ and (shear) $S$ waves. The theoretical predictions reproduce many features of the elastic response observed experimentally, in particular, the coupling between nonlinear and nonequilibrium dynamics and the three-dimensional effects resulting from the tensorial nature of elasticity.

Authors:
 [1]; ORCiD logo [2];  [1];  [2];  [2];  [1]
  1. Centre National de la Recherche Scientifique (CNRS), Marseille (France). Laboratoire de Mecanique et d'Acoustique (LMA)
  2. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Publication Date:
Research Org.:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org.:
USDOE Office of Fossil Energy (FE); French National Research Agency (ANR)
OSTI Identifier:
1371610
Report Number(s):
LA-UR-17-23120
Journal ID: ISSN 2475-9953; TRN: US1703288
Grant/Contract Number:
AC52-06NA25396; ANR-11 RSNR 0009
Resource Type:
Journal Article: Published Article
Journal Name:
Physical Review Materials
Additional Journal Information:
Journal Volume: 1; Journal Issue: 2; Journal ID: ISSN 2475-9953
Publisher:
American Physical Society (APS)
Country of Publication:
United States
Language:
English
Subject:
58 GEOSCIENCES; Nonlinear Elasticity; Dynamic Acousto-Elasticity; Geomaterials; Rocks; Hyperelasticity; Slow Dynamics

Citation Formats

Lott, Martin, Remillieux, Marcel, Garnier, Vincent, Le Bas, Pierre-Yves, Ulrich, Timothy James II, and Payan, Cedric. Nonlinear elasticity in rocks: A comprehensive three-dimensional description. United States: N. p., 2017. Web. doi:10.1103/PhysRevMaterials.1.023603.
Lott, Martin, Remillieux, Marcel, Garnier, Vincent, Le Bas, Pierre-Yves, Ulrich, Timothy James II, & Payan, Cedric. Nonlinear elasticity in rocks: A comprehensive three-dimensional description. United States. doi:10.1103/PhysRevMaterials.1.023603.
Lott, Martin, Remillieux, Marcel, Garnier, Vincent, Le Bas, Pierre-Yves, Ulrich, Timothy James II, and Payan, Cedric. 2017. "Nonlinear elasticity in rocks: A comprehensive three-dimensional description". United States. doi:10.1103/PhysRevMaterials.1.023603.
@article{osti_1371610,
title = {Nonlinear elasticity in rocks: A comprehensive three-dimensional description},
author = {Lott, Martin and Remillieux, Marcel and Garnier, Vincent and Le Bas, Pierre-Yves and Ulrich, Timothy James II and Payan, Cedric},
abstractNote = {Here we study theoretically and experimentally the mechanisms of nonlinear and nonequilibrium dynamics in geomaterials through dynamic acoustoelasticity testing. In the proposed theoretical formulation, the classical theory of nonlinear elasticity is extended to include the effects of conditioning. This formulation is adapted to the context of dynamic acoustoelasticity testing in which a low-frequency “pump” wave induces a strain field in the sample and modulates the propagation of a high-frequency “probe” wave. Experiments are conducted to validate the formulation in a long thin bar of Berea sandstone. Several configurations of the pump and probe are examined: the pump successively consists of the first longitudinal and first torsional mode of vibration of the sample while the probe is successively based on (pressure) $P$ and (shear) $S$ waves. The theoretical predictions reproduce many features of the elastic response observed experimentally, in particular, the coupling between nonlinear and nonequilibrium dynamics and the three-dimensional effects resulting from the tensorial nature of elasticity.},
doi = {10.1103/PhysRevMaterials.1.023603},
journal = {Physical Review Materials},
number = 2,
volume = 1,
place = {United States},
year = 2017,
month = 7
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record at 10.1103/PhysRevMaterials.1.023603

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  • Here we study theoretically and experimentally the mechanisms of nonlinear and nonequilibrium dynamics in geomaterials through dynamic acoustoelasticity testing. In the proposed theoretical formulation, the classical theory of nonlinear elasticity is extended to include the effects of conditioning. This formulation is adapted to the context of dynamic acoustoelasticity testing in which a low-frequency “pump” wave induces a strain field in the sample and modulates the propagation of a high-frequency “probe” wave. Experiments are conducted to validate the formulation in a long thin bar of Berea sandstone. Several configurations of the pump and probe are examined: the pump successively consists ofmore » the first longitudinal and first torsional mode of vibration of the sample while the probe is successively based on (pressure) $P$ and (shear) $S$ waves. The theoretical predictions reproduce many features of the elastic response observed experimentally, in particular, the coupling between nonlinear and nonequilibrium dynamics and the three-dimensional effects resulting from the tensorial nature of elasticity.« less
  • Dynamic acousto-elastic (DAE) studies are performed on a set of 6 rock samples (four sandstones, one soapstone, and one granite). From these studies, at 20 strain levels 10 -7 < ϵ < 10 -5, four measures characterizing the nonlinear elastic response of each sample are found. Additionally, each sample is tested with nonlinear resonant ultrasonic spectroscopy (NRUS) and a fth measure of nonlinear elastic response is found. The ve measures of the nonlinear elastic response of the samples (approximately 3 x 6 x 20 x 5 numbers as each measurement is repeated 3 times) are subjected to careful analysis usingmore » model independent statistical methods, principal component analysis and fuzzy clustering. This analysis reveals di erences among the samples and di erences among the nonlinear measures. Four of the nonlinear measures are sensing much the same physical mechanism in the samples. The fth is seeing something di erent. This is the case for all samples. Although the same physical mechanisms (two) are operating in all samples there are distinctive features in the way the physical mechanisms present themselves from sample to sample. This suggests classi cation of the samples into two groups. The numbers in this study and the classi cation of the measures/samples constitute an empirical characterization of rock nonlinear elastic properties that can serve as a valuable testing ground for physically based theories that relate rock nonlinear elastic properties to microscopic elastic features.« less
  • We introduce a transfer-matrix method for calculating the thermodynamic properties of random-tiling models of quasicrystals in any number of dimensions, and describe how it may be used to calculate the phason elastic properties of these models, which are related to experimental measurables such as phason Debye-Waller factors, and diffuse scattering wings near Bragg peaks. We apply our method to the canonical-cell model of the icosahedral phase, making use of results from a previously presented calculation in which the possible structures for this model under specific periodic boundary conditions were cataloged using a computational technique. We give results for the configurationalmore » entropy density and the two fundamental elastic constants for a range of system sizes. The method is general enough to allow a similar calculation to be performed for any other random tiling model.« less
  • A parallel domain decomposition boundary integral algorithm for three-dimensional exponentially graded elasticity has been developed. As this subdomain algorithm allows the grading direction to vary in the structure, geometries arising from practical FGM applications can be handled. Moreover, the boundary integral algorithm scales well with the number of processors, also helping to alleviate the high computational cost of evaluating the Green's function. Numerical results for cylindrical geometries show excellent agreement with the new analytical solution deduced for axisymmetric plane strain states in a radially graded material.
  • The numerical implementation of the Green's function for an isotropic exponentially graded three dimensional elastic solid is reported. The formulas for the nonsingular {\lq}grading term{\rq} in this Green's function, originally deduced by Martin et al., \emph{Proc. R. Soc. Lond. A, 458, 1931-1947, 2000}, are quite complicated, and a small error in one of the formulas is corrected. The evaluation of the fundamental solution is tested by employing indirect boundary integral formulation using a Galerkin approximation to solve several problems having analytic solutions. The numerical results indicate that the Green's function formulas, and their evaluation, are correct.