[1] Members of a wide class of geomaterials are known to display complex and fascinating nonlinear and nonequilibrium dynamical behaviors over a wide range of bulk strains, down to surprisingly low values, e.g., 10À7 . In this paper we investigate two sandstones, Berea and Fontainebleau, and characterize their behavior under the influence of very small external forces via carefully controlled resonant bar experiments. By reducing environmental effects due to temperature and humidity variations, we are able to systematically and reproducibly study dynamical behavior at strains as low as 10 À9 . Our study establishes the existence of two strain regimes separated by M . At strains below M the material is nonlinear and quasi-equilibrium thermodynamics applies as evidenced by the success of Landau theory and a simple macroscopic description based on the Duffing oscillator. At strains above M the behavior becomes truly nonequilibrium, as demonstrated by the existence of material conditioning, and Landau theory no longer applies. The main focus of this paper is the study of the first region, but we also comment on how our work clarifies and resolves previous experimental conflicts, as well as suggest new directions of research.

The transition from linear to nonlinear dynamical elasticity in rocks is of considerable interest in seismic wave propagation as well as in understanding the basic dynamical processes in consolidated granular materials. We have carried out a careful experimental investigation of this transition for Berea and Fontainebleau sandstones. Below a well-characterized strain, the materials behave linearly, transitioning beyond that point to a nonlinear behavior which can be accurately captured by a simple macroscopic dynamical model. At even higher strains, effects due to a driven nonequilibrium state, and relaxation from it, complicate the characterization of the nonlinear behavior. PACS numbers: 62.40.+i, 62.65.+k, 91.60.Lj Rocks possess a variety of remarkable nonlinear elastic properties including hysterisis with end-point memory [1], variation of attenuation and sound velocity with strain [2], strong dependence of elastic and loss constants on pressure, humidity, and pore fluids [3], long-time relaxation phenomena ('slow dynamics') [4], and nontrivial variation of resonance frequency with strain [5,6]. Significantly, materials as diverse as sintered ceramics and damaged steels are now known to display similar effects [6]. Thus rocks may be viewed as representative members of a class of fascinating, but poorly understood, nonlinear elastic materials: Fundamental questions still to be resolved relate not only to the underlying causes of the nonlinear phenomena but also to the conditions under which they occur.In this Letter, we focus on delineating two strain thresholds, one below which the rocks behave effectively as linear elastic materials, ǫ L , the other beyond which memory and conditioning effects occur, ǫ M , and the dynamic elastic behavior straddling the region of these thresholds. While in Ref. [2] it was argued that ǫ L ∼ 10 −6 (albeit with some uncertainty), more recent data [5,6,7] have been used to support an extension of the nonlinear region to substantially lower strains; doubt has been cast even on the very existence of a threshold [8]. In addition, results from resonant bar experiments [5,7] have been interpreted to exhibit a 'nonclassical' frequency and loss dependence on the drive amplitude, i.e., frequency and Q softening linearly with drive amplitude rather than quadratically as predicted by Landau theory [9], even at strains as small as 10 −8 . (The importance of ǫ M in interpreting resonant bar data is emphasized below.)We have carried out a new set of well-characterized experiments, over a wide dynamic range, to unambiguously settle these questions: While longitudinal resonant excitation of bars is a classic measurement technique [10], rock samples require substantial care in terms of controlling the temperature and humidity and characterizing possible systematic effects, especially those due to conditioning of the sample by the external drive [4].Our major conclusions are as follows. For Berea and Fontainebleau sandstone samples, below a threshold strain ǫ L ∼ 10 −8 −10 −7 (lower end for ...

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