Dynamic measurements of the nonlinear elastic parameter α in rock under varying conditions

Abstract

[1] Since the exhaustive work by Adams and Coker at the Carnegie Institute in the early 1900s and the work of F. Birch's group at Harvard University conducted in the 1940s-1950s, it has been well documented that the quasi-static stress-strain behavior of rock is nonlinear and hysteretic. Over the past 20 years, there has been an increasing body of evidence suggesting that rocks are highly elastically nonlinear and hysteretic in their dynamic stress-strain response as well, even at extremely small strain amplitudes that are typical of laboratory measurements. In this work we present a compendium of measurements of the nonlinear elastic parameter a extracted from longitudinal (Young's mode) and flexural-mode resonance experiments in eight different rock types under a variety of saturation and thermal conditions. The nonlinear modulus a represents a measure of the dynamic hysteresis in the wave pressure-strain behavior. We believe that hysteresis is the primary cause of nonclassical nonlinear dynamics in rock, just as it is responsible for elastic nonlinear behavior in quasi-static observations. In dynamics, a is proportional to the wave speed and modulus reduction as a function of wave strain amplitude due to the hysteresis, based on our current model. The rocks tested include pure quartz sandstone (Fontainebleau), two sandstones that contain clay and other secondary mineralization (Berea and Meule), marble (Asian White), chalk, and three limestones (St. Pantaleon, Estaillades, and Lavoux). The values of a range from $500 to >100,000, depending on the rock type, damage, and/or water saturation state. Damaged samples exhibit significantly larger a than intact samples (hysteresis increases with damage quantity), and water saturation has an enormous influence on a from 0 to 15-30% water saturation.

Soil Dynamic Constitutive Model for Characterizing the Nonlinear-Hysteretic Response

“…Particulate materials (i.e., rock, sandstone, sand, and sediment) exhibit strong dynamic behaviors, such as end-point memory and a closed loop (hysteresis) in the stress-strain curve, higher harmonic generation in propagating waves, a resonance frequency shift and nonlinear damping in standing waves, and slow dynamics [22][23][24][25][26][27]. These macroscopic nonlinear responses of soils are the consequences of complex frictional mechanisms involving contact between the grains and grain rearrangement under loading-unloading cycles.…”

Characterization of sandwich beams with debonding by linear and nonlinear vibration method

“…We can observe from this table that 64% debonding length for sandwich with 60 kg m À3 and 48% debonding length for sandwich with 100 kg m À3 have respectively, a R = 0.39 ± 0.005 and R = 0.24 ± 0.007. If we compare these values with different materials [26], we will find that the sandwich with 60 kg m À3 with 64% debonding manifest a behavior nonlinear hysteretic close to the Marble and Perlite/graphite-metal., while the sandwich with 100 kg m À3 with 48% debonding the behavior is close to the quartzite. Fig.…”

“…Johnson et al [26] measured the ratio of nonlinear elastic parameter and nonlinear dissipative of different materials, it is defined as:…”

Effect of carbonation on the linear and nonlinear dynamic properties of cement-based materials

“…(3). 41 A related discussion on error in estimation of α due to incorrect or inaccurate selection of f o can be found in Ref. 41.…”

“…41 A related discussion on error in estimation of α due to incorrect or inaccurate selection of f o can be found in Ref. 41. Alternatively, changes in resonant frequency with varying spectral amplitude can be evaluated through a least-squares minimization of the model such that no previous selection of f o is required:…”