## Abstract

Experimental results concerning mechanical hysteresis in sandstone samples, which were obtained by means of either dynamic acousto-elastic or quasi-static techniques, are mathematically modelled assuming that the critical stress at which hysteresis is activated is not an intrinsic material property, but decreases with increasing stress rate. A macroscopic strain–stress constitutive relationship is derived from this assumption, which leads the main features characterizing mechanical hysteresis in sandstones to be recovered, at least in a qualitative sense. In particular, for sinusoidal loading used in dynamic acousto-elastic experiments, the model predicts a vanishing anelastic strain and a continuous variation of the modulus defect during the entire loading cycle. Furthermore, hysteresis is shown to disappear when the frequency of the excitation approaches the static limit. Experimental results and theoretical models concerning other Earth materials and metal alloys are also considered. Although in some case, the latter results have been acquired under dramatically different experimental conditions, they are better understood, and, for this reason, are used as reference to discuss those obtained by exploiting acousto-elasticity of sandstones. The striking difference between reference findings and results in sandstone suggests that equally strikingly different mechanisms are responsible for hysteresis in the latter material system.

## 1. Introduction

It is well known that rocks and similar man-made materials display mechanical hysteresis with endpoint memory when they are subjected to an externally applied stress. Hysteresis manifests itself also through an acoustic response characterized by preferred generation of odd higher harmonics and material softening [1,2]. Furthermore, and of particular interest to the present communication, the modulus defect of Earth materials subjected to cyclic loading displays hysteresis with strength decreasing as the loading rate approaches zero [3–5]. Several mechanisms may be called upon to justify these observations. In metals and metal alloys, dislocations interacting with point defects have been investigated in this regard and their role in the quasi-static acousto-elastic effect appears to be broadly understood [6]. In minerals, however, the material’s complex microstructure renders the identification of potential mechanisms responsible for hysteresis and dynamics-induced material softening more problematic, especially when the frequency of the excitation is well above those of seismic waves. Therefore, while several such mechanisms have been proposed (e.g. [7]), the identification of the culprit(s) still remains an open problem.

Dynamic acousto-elastic technique (DAET) [8–10] allows the modulus defect of a material sample to be measured with accuracy two orders of magnitude higher than that of alternative methods used for the same purpose (e.g. [11,12]). This is to say, values of the modulus defect of the order of 0.01% can be observed by means of DAET. This technique employs a longitudinal resonant mode of the sample to excite a stationary stress field that modulates in time the local mechanical properties of the material. Strain of the order of 10^{−5} can be generated within a sample. For its role in DAET, this mode is also referred to as the ‘pump’ wave. The typical geometry of these samples is that of a cylindrical rod with length of about 15 cm and diameter 2.5 cm. The resonant mode employed to produce the stationary stress field has a resonance frequency of the order of a few kilohertz. A sequence of pulses with central frequency in the megahertz region is used to monitor the variation of the local material’s stiffness during a cycle of the resonance mode. The lateral size of the sample ensures that, during the time taken by the each high-frequency pulse to travel from the emitter to the receiver (figure 1), the stationary stress field does not vary in an appreciable way. The pulse repetition frequency is chosen so that it is not commensurate to the frequency of the pump wave. This choice is made so that the sample is monitored during an entire period and not only at a few points in time. Thanks to the dynamic nature of the excitation, DAET allows the local material’s stiffness to be monitored both in compression and in tension. Figure 1 schematically illustrates the set-up of a DAET test, in which a vibration source excites a resonance mode of the bar, and two high-frequency ultrasonic transducers, a transmitter (T) and a receiver (R), emit and receive pulses which monitor the instantaneous state of the material at the location where the stress field has maximum amplitude.

Traditionally, investigations of rocks’ mechanical hysteresis employ only quasi-static, compressive, cyclic loading. By examining the material’s response under both tension and compression important clues about the nature of mechanisms potentially responsible for the material’s hysteresis can be recovered. For instance, the broadly symmetric response of hysteresis of the modulus defect in compression and in tension, together with the increasing area of the loop with increasing excitation amplitude, suggest that friction between opposing faces of partially closed microcracks is an unlikely cause of the observed hysteresis in Berea sandstone [13]. This conclusion seems to be in contradiction with findings from several investigations, which attribute to contact micromechanics with or without adhesion the observed mechanical hysteresis [14–16]. In appendix A, additional insights into this issue are discussed.

The complexity of the response of sandstones in DAET investigations has been experimentally documented. In addition to the investigations by Renaud *et al.* [8,9], and Rivière *et al.* [10], see also Rivière *et al.* [17]. Key features of the experimental data are the already cited symmetry of the hysteresis loops displayed by the modulus defect with respect to the compressive and tensile phases of the loading cycle; the increase of the area enclosed by these loops and of the concomitant material softening with increasing amplitude of the excitation; the appearance of new structures in the central region of the stress cycle at high values of the excitation in Berea and Pietra Serena, but not in Red Meule and Sander sandstone [17]; the tendency of the curvature of the cycle averaged over the loading and unloading phases to decrease for increasing excitation amplitude.

These and other features of the experimental data can be recovered to some extent by a composite model which has been derived by combining four mechanisms that are characteristic of (but not exclusive to) the rheology of dislocations interacting with point defects and microcracks with finite stiffness [13]. Among the most notable drawbacks of the composite model, there is the lack of a frequency dependence of hysteresis. Claytor *et al.* [3], using a triangular loading protocol, and Rivière *et al.* [4,5], using DAET, have conclusively documented the tendency of mechanical hysteresis in rocks to decrease as the loading rate decreases. Furthermore, the composite model predicts discontinuities of the modulus defect at the turning points of the cycle. This prediction contrasts experimental evidence obtained by DAET which shows the modulus defect to behave continuously everywhere during a cycle, although rapid variations of this quantity in proximity of the turning points are observed in some samples.

The purpose of this communication is to report an improved model of DAET measurements that focuses on the mathematical description of hysteresis displayed by the modulus defect of sandstone samples. As in [13], a rheological representation of friction-like mechanisms is still a key ingredient of the new model. In this work, however, the onset of friction is no longer assumed to be controlled by stress. Instead, it is determined by the power of the loading. Building on this assumption, the model yields a new strain–stress constitutive relationship leading to a dependence of the material’s modulus defect on both loading and loading rate, which qualitatively agrees with experimental data. In particular, for sinusoidal loading typical of DAET, the model removes the discontinuities of the modulus defect at the turning points of a cycle, while it predicts their presence when a triangular loading protocol is used. This result seems to be in contrast with those by McKavanagh & Stacey [18], who experimentally showed that the anelastic strain dependence on stress displays cusps at the turning points regardless of whether the loading’s time dependence is triangular or sinusoidal. To reconcile theoretical results with experimental evidence, an alternative explanation is provided, which leads one to consider stress instead of strain as the true independent variable in DAET. Finally, DAET experimental results and the new model are discussed on the background offered by similar data and models [11,12,19] referring to Carrara marble and fine-grained olivine samples at high temperature and pressure, and by the present understanding of the interaction between seismic waves and silicate minerals [20]. Elastic nonlinearity, which does not lead to dissipation, is considered only in the section presenting a comparison between experimental results and a simple composite model.

## 2. Theory

The dynamics of microscopic defects can comprise friction-like mechanisms that lead to dissipative dynamics at the macro scale. For instance, it is well known that dislocations interact with points defects, which are distributed in proximity of the dislocations glide planes [6]. This is a long-range interaction leading to the generation of anelastic strain, *e*_{an}, which depends on the applied stress, *σ*. A schematic mathematical representation of this dependence is as follows:
*S*_{1} defines the strength of the interaction and its dimensions are [stress]^{−1}. The symbol *H*(⋅) represents the Heaviside step function, which is null when its argument is negative, and equal to 1 when the latter is positive. The quantities *σ*_{cr} represent the amplitude of the applied cyclic stress and the value of the latter at which the interaction is activated, respectively. Equation (2.1) implies that, once activated, hysteresis continues to affect the material’s response as long as *σ*_{cr}. The function *C*(⋅) versus *σ*.

*This rheological model can be thought of as being representative of a class of mechanisms describing friction-like dynamics. Therefore, its use in this work should not be regarded as implying that dislocation dynamics is the only possible, or the most probable explanation for the phenomena of interest*.

In an earlier model [13], the critical stress *σ*_{cr} was treated as an intrinsic property of the material, which is distributed according to the following density function
*σ*_{cr}. The product *φ*(*σ*_{cr})*dσ*_{cr} represents the fraction of dislocations breaking away from point defects when the applied stress, *σ*, is between *σ*_{cr} and (*σ*_{cr}+*dσ*_{cr}). The total fraction, *f*, of dislocations which have broken away as a function of *σ* is obtained by integrating *φ*(*σ*_{cr}) over [0,*σ*]. The result is
*σ*_{cr} is replaced by the critical value of the power density, *P*_{cr}, as the system’s parameter controlling the activation of friction. One reason for introducing *P*_{cr} as characteristic property of the mechanism of interest resides in the fact that no realistic loading is actually static. More importantly, the use of *P*_{cr} in the model introduces dispersion in the mechanical response of the system of interest. In fact, *P*_{cr} is also the *rate* of work a source is supposed to perform of the system in order to activate dissipative mechanisms which are characteristic of the material. In this work, equation (2.3) is then substituted by
*α* is a real parameter greater than 1, *α*>1.

The critical power density can also be written as *P*_{cr} to ensure that *σ*_{cr}>0. The symbol *S*_{o} represents the linear compliance of the material. Here, strain and loading rates are considered to be independent of *σ* in order to allow for different loading protocols to be implemented over the same stress range.

Once the loading rate, *P*_{cr} can be used to define the critical stress, *σ*_{cr}, required to trigger hysteresis for the given loading rate: *σ*_{cr}, which turns out to be
*δ*=2Δ/*S*_{o}, and *G*_{o} is a normalization constant.

The physical picture portrayed by equation (2.6) is one in which the barrier that must be overcome by the applied stress to activate hysteresis increases as the stress rate decreases. The large, but still finite value of the applied stress at which a material fractures clearly imposes limits to the range of validity of equation (2.6). The new function giving the fraction of friction-like events (for instance, dislocations that have broken away from their pinning points, or grain boundaries that have begun sliding against each other, or something else altogether) due to the stress *σ* is found to be
*γ*(*α*,*z*) is the incomplete gamma function. The normalization constant *G*_{o} is the value attained by the integral when its upper limit of integration increases indefinitely. Note the symmetrical role played by |*σ*| and

Presently, not having called upon any fundamental physical knowledge to justify the rheological model of equation (2.1) and the introduction of *P*_{cr}, equation (2.6) should be considered as being not more than a mere working hypothesis, which, in the best of the circumstances, provides a mathematical link lending itself to easy analytical manipulations. Eventually, its use and implications about its physical meaning may be justified only by the extent to which theoretical predictions based upon it and experimental data agree with each other.

Integration of the function *S*_{o}*σ* to equation (2.8), the constitutive relationship linking the total strain, *e*, to the applied stress, *σ*, and including distributed friction-like forces is obtained.

The effective compliance, *S*_{eff}, of the material can be recovered using equation (2.8) by deriving the total strain, *e*, with respect to the applied stress, *σ*. The result is
*ε*=*S*_{1}/*S*_{o}≪1, *E* is the elastic modulus of the material, Δ*E* is its variation, and the use of the sign ≅ implies that the above equation has been obtained by truncating a Taylor series expansion of an exact result.

## 3. Numerical results

The numerical results to be presented next have been obtained using physical units in which all the relevant field quantities and model parameters are of the order of unity. This choice is reasonable in the context of the present investigation, which does not aim at reproducing specific quantitative results, but limits its scope to simulating the main qualitative features of the latter. Results will be presented for both triangular and sinusoidal loading protocols.

### (a) Friction-like mechanism

The first simulation concerns the effect of the loading rate on the anelastic strain–stress relationship (equation (2.8)). In these and all the following figures, the parameter *ε* (see equation (2.10)) that measures the strength of hysteresis is assumed to be equal to one. Figure 3*a* refers to the case in which the loading is triangular, whereas in figure 3*b* the loading is assumed to be sinusoidal. To generate figure 3*b* and similar ones that follow, stress and stress rate have been computed as functions of time first and then used in equation (2.8) to yield the anelastic strain. In the former case, the loading rate is labelled as *V* , while in the second figure the symbol ‘*f*’ represents the loading frequency. Figure 3*a* reproduces a well-known shape of hysteresis observed in experimentally derived strain–stress relationships (see [3,18], for instance) with the characteristic cusps at the turning points of the cycle. Figure 3*b*, on the other hand, illustrates a behaviour which, to the best of this author’s knowledge, has not yet been experimentally documented. At the turning points the anelastic strain, *e*_{an}, goes to zero as a consequence of the vanishing stress rate. As recalled earlier, McKavanagh & Stacey [18] found that mechanical hysteresis of a variety of Earth materials is characterized by cusps even at low strains and, more importantly for the present work, regardless of whether the loading protocol is triangular or sinusoidal. To understand this apparent conflict between experiments and theory, one must take note of the boundary conditions imposed on the system under inspection. In McKavanagh & Stacey’s investigation [18], the surfaces of the sample on which the external stress is applied are mechanically constrained (clamped). The stiffness of the loading machine being much larger than that of the samples transforms the character of the experiments from being load controlled into displacement controlled. Therefore, at the turning points of a sinusoidal cycle, the anelastic component of material’s deformation remains finite and changes discontinuously only when the applied stress begins the following half-cycle. On the contrary, in DAET experiments, the sample is subjected to a resonance mode in which one of the two ends is stress free. There and along the whole sample the response of the material is unconstrained, and, consequently, the anelastic strain follows the time evolution of the stress rate, as it is simulated by the proposed model. In conclusion, cusps are observed during sinusoidal cycling when the instrumentation employed to generate the driving stress constrains the geometry of the sample. An immediate consequence of these observations is that stress and strain cannot be regarded as physical variables playing equivalent roles. The former, rather than latter, should be used as independent variable contrary to the commonly accepted practice [8–10,17].

Last but not least, in agreement with experimental observations by Claytor *et al.* [3] and Rivière *et al.* [4,5], both figures clearly show that the area of the hysteresis loop decreases as the stress rate decreases. This is confirmed by figure 4, which reports values of the loop’s area as a function of frequency (sinusoidal loading) for three values of the parameter *α*, with *δ*=1. Results for triangular loading (constant loading velocity) display similar behaviour. These predictions show that the hysteretic contribution to nonlinear dissipation presents a maximum. Location of the maximum and width of the curve in the frequency domain vary with the parameter *α* (see equation (2.6)). The existence of this maximum is due to the competition between the increase of the fraction in hysteretic units that are activated and the strength of the critical stress at which these units are activated. The former increases with increasing *et al.* [3]. They show an initial increase of the loop’s area that begins to reverse its trend after reaching a maximum in a fashion that is closely resembled by the results of the present model.

An investigation similar to that leading to the results of figure 4 can be carried out on the role of *δ* on the frequency dependence of energy dissipation. Consistently with equation (2.6), the results show that *δ* is merely a scale factor.

Figure 5*a*,*b* reports the simulation results on the dependence of the modulus defects associated with the strain–stress hysteresis of figure 3 for triangular and sinusoidal cycles, respectively. As expected, discontinuities at the turning points are predicted for a triangular loading protocol, while continuous transitions between loading and unloading phases of the cycles are found for sinusoidal loading, in agreement with DAET results. The reader can easily extra/interpolate these results, and verify that, contrary to predictions based on a quadratic dependence of hysteresis of the strain–stress relationship (see [13] for additional discussion), the intercept of the modulus defect cannot increase indefinitely, its limit value for large stress and/or stress rate being equal to 1 (when suitable units are employed). Furthermore, the dependence of the modulus defect obtained by averaging its values during the loading and unloading phases of a cycle and for the same stress is seen to be initially characterized by slightly positive curvature that changes its sign with increasing stress or stress rate amplitude.

Results that are similar to those in figure 5 are found by varying the applied stress amplitude, while maintaining the stress rate or frequency constant. Worthy of notice is that increasing *α* does not cause a shift of the minima of the modulus defect, which is attained when the instantaneous stress is equal to *α* leads to steeper transitions of the modulus defect between the two phases of the cycle. Transitions, which occur at different rates, are well-documented experimentally [17].

### (b) A composite model

Similar to Pecorari [13], a composite model can be assembled by adding the effects due to microcracks with faces in partial contact and classic nonlinearity to those due to friction-like mechanisms. These two contributions are able to account for the ‘slanting’ and curvature of the experimental results. Figure 7 illustrates predictions that were obtained by linearly combining the aforementioned contributions with weights reported in table 1. Additional parameters used in these simulations are: *ω*=2*πf*=2*π* (arb. units), where *f* if the frequency, *δ*=1 (arb. units), *α*=1.5, and *δ*_{crk}=0.7. The latter parameter controls the rate by which cracks close during the compressive phase of the loading cycle [13]. These plots are meant to be compared with those of fig. 3 in [17]. In that figure, experimental curves for four different sandstones are presented for increasing values of the excitation amplitude. Each type of sandstone displays distinctive features like the local maxima at low stress values and large excitation amplitude in Berea- and Pietra Serena sandstone. However, it appears that they all share properties, which are common also to the theoretical curves of figure 7. Therefore, once properties that are specific to a given type of sandstone are disregarded, the broad agreement between experimental results and theoretical predictions suggest that using such a composite model may be a valuable tool to guide further research about the micromechanical origins of sandstone’s mechanical hysteresis.

## 4. Discussion

The model presented in the previous section improves over an earlier one [13] in several ways. It does so by introducing a parametrized dependence of *σ*_{cr} on the loading rate, as it is given in equation (2.6). This result is achieved by assuming that the threshold at which hysteresis is activated is imposed on the power density generated the external source, rather than on the stress. While appealing for both its physical and mathematical simplicity, the relevance of this modification resides in the substantial, although still qualitative, agreement between the predictions of the model and DAET results. This agreement would appear to constrain the search of a plausible physical explanation of hysteresis in sandstones to mechanisms that resemble the interaction between dislocations and glide point defects or sliding of grain boundaries. However, as discussed next, there is evidence according to which the frequency dependence of these mechanisms may not be that observed in the DAET results, which is also predicted by the current model.

Gremaud [6] discusses the motion of dislocations interacting with point defects surrounding their glide planes at finite temperatures, and identifies four different combinations of temperature ranges and spatial distributions of point defects which require separate analysis. For two of these regions, an equation shows the critical stress, *σ*_{cr}, to be a monotonically increasing function of the excitation frequency. For example, the equation relating to the region characterized by high temperature and Mott’s statistics is
*k*_{B} is the Boltzmann constant, *T* is the absolute temperature, Δ*G*_{m} is the maximum interaction energy between dislocations and point defects, *ν* is the attempt frequency, *f* is the excitation frequency and *n* is a parameter equal to or greater than 2. If equation (2.6) were substituted by one in which *σ*_{cr} increases with increasing frequency, as shown by equation (4.1), the resulting strain–stress relationship would lead to theoretical predictions in striking contrast with experimental evidence obtained by DAET [4].

Grain boundary sliding has been called upon to account for the decrease of the shear modulus with decreasing frequency as documented by experimental results in fine-grained olivine samples subjected to high temperature and confining pressure, and Carrara marble [11,12,19]. These findings concern samples driven by sources operating at frequencies within the seismic range. Along the same line, experimental evidence has been recently produced by Farla *et al.* [12] according to which, in pre-deformed samples subjected to conditions that simulate the environment within the Earth’s mantle, dislocation glide is partly responsible for the rheology of this material at seismic frequency. As in the analysis by Gremaud [6], even these findings seem to be irreducible to those obtained in sandstones at ambient conditions, which show the modulus defect and dissipation disappearing with decreasing frequency. In short, these studies provide reasons to question whether the *same* mechanisms may be used to explain the nonlinear dissipative dynamics of sandstone at room temperature, under ambient pressure and at frequencies well above the seismic range.

The purpose of the above discussion is to point out discrepancies between the dynamics of two well-investigated mechanisms and experimental evidence produced by DAET. It does not intend to state that such mechanisms may not occur in sandstones. The possibility still exists that this contrasting evidence may be reconciled considering that DAET results are generally obtained under conditions that strongly differ from those simulated in the investigations just cited. This seems to be the direction partly suggested by still unpublished results by Rivière *et al.* [5]. By confining the inspected sample to increasingly higher compressive static stress fields, these authors have experimentally demonstrated that hysteresis tends to disappear even in sandstones. This work was carried out at room temperature. Whether raising the sample temperature at high confining pressure may re-activate the same mechanism(s) responsible for the DAET observations in sandstones at room temperature and atmospheric pressure, or produce a different rheology could be a subject worthy of attention in future investigations.

Worthy of notice is that data on olivine samples have been gathered on materials with grains having average diameter of a few micrometres, while grains in the sandstone samples used in DAET investigations are almost one hundred times larger. For results on different materials to be comparable with each other, samples with similar grain size should be employed to avoid the possibility that grain boundaries may affect dislocation dynamics differently in different materials, and display themselves a rheological behaviour that is sensitive to their own dimensions.

Finally, the model in equation (2.1) has been suggested in connection with rheology caused by dislocation gliding among a cloud of point defects in metals and metal alloys. Doubts can be raised on whether the same type of deformation at the atomic scale may occur in silicates and other materials where covalent bonds between constituent atoms prevail (high Peierls potential). However, as discussed by Karato [20], motion of dislocations via displacement of (double) kinks or jogs might still provide potentially viable mechanisms to explain the rheology of Earth materials at seismic or higher frequencies, especially in the view of equation (2.6). Recall that, in DAET testing at ambient conditions, sandstones respond to excitations at frequencies that are *three orders of magnitude* higher than seismic frequencies. Should equation (2.6) (or equivalent) be confirmed by additional and independent experimental evidence, Karato’s hypothesis would acquire considerable more weight. To this day, however, it would appear that these mechanisms have not yet been sufficiently investigated either theoretically or experimentally under loading conditions characterized by high stress rates, as those used in DAET. Therefore, the interpretation of equation (2.6) in terms of an actual physical mechanism at the microscopic scale remains an open problem.

## 5. Summary and concluding remarks

A new constitutive relationship linking anelastic strain to stress was proposed, which overcomes drawbacks of an earlier model to the extent that all the main qualitative features characterizing hysteresis in sandstone documented by either quasi-static or DAET experiments are recovered—the only notable exception being slow dynamics [1], which is a phenomenon concerning the approach of a system to its thermo-dynamical equilibrium, and, therefore, cannot be described by means of a constitutive relationship. In particular, the following properties of experimentally observed mechanical hysteresis in sandstone have been reproduced or predicted for the first time: (i) a qualitatively correct dependence of hysteresis on frequency as the latter approaches zero, (ii) the appearance of cusps in the strain–stress loops and corresponding discontinuities of the material modulus defect when loading is triangular, (iii) a novel hysteretic anelastic strain–stress loop and continuous evolution of the associated modulus defect when the load is sinusoidal and alternating compressive and tensile phases during a cycle, (iv) the parameter *α* appearing in equation (2.6) is shown to control both the rate of change of the modulus defect in proximity of the turning point for sinusoidal loadings, and the nonlinear dissipation of energy by hysteresis. Finally, this model prompted the explanation of the appearance of cusps in the strain–stress loops for sinusoidal load cycles [3] to be searched in the mechanical constraints imposed by the experimental technique on the material. A relevant consequence of this explanation is that the applied stress, and not the strain, must be used as independent variable in DAET measurements.

This progress was achieved by introducing a dependence on the loading rate of the critical power density at which mechanisms responsible for the material’s hysteresis are activated. The physical meaning of the dependence at macroscopic level is explicit: hysteresis can be activated even at low stress level provided that the latter is applied at sufficiently high rate; its actual origin at micro-scale is not. Regardless of the present lack of understanding, the substantial agreement between the model’s predictions and DAET results should suffice to credit equation (2.6) with the status of additional macroscopic fingerprint of the micromechanics of sandstone materials. In this role, equation (2.6) may (and should) be used as test bench for models of rheology of sandstones at the microscopic scale. Finally, even though DAET is used to investigate material’s properties under conditions that do not simulate those found in the Earth’s mantle, mechanical properties displayed by these materials under ambient conditions may reveal dynamical properties which could be of considerable relevance to attain a complete physical picture of the microstructure of these Earth materials.

## Data accessibility

Data used to support the content of this work can be found in the publications given in the references list.

## Authors' contributions

I am the sole author.

## Competing interests

I have no competing interests.

## Funding

I have received no funding for this work.

## Appendix A

Cracks tend to open during the tensile phase of the DAET loading, and in doing so they deactivate friction forces between asperities in contact. Under compression, cracks tend to close, eventually causing asperities to become mechanically locked to each other. If a crack is not initially entirely open or closed, both in compression and in tension, it is reasonable to assume that there are asperities that may slide again each other.

For a micromechanical model based on friction forces between asperities in contact to reproduce the symmetrical hysteretic modulus defect measured in a DAET test, the following must occur. First, the number of asperities sliding against each other must depend only on the absolute value of the applied stress. Second, this number must increase as the amplitude of the stress increases, irrespectively of the rate at which asperities become mechanically locked in compression, or are brought out of contact in tension. Third, this increase must take place at the same rate in compression and in tension. To summarize the number of sliding contacts, like the hysteretic component of the modulus defect, must be an even function of the applied stress.

Recall that the elastic modulus of cracked solids increases in compression, while it decreases in tension. In both cases, the elastic modulus approaches asymptotic values as the strength of the applied stress increases. As a function of the applied stress, the elastic modulus seems to be better approximated by an odd function, and this seems to be the case even for the total number of asperities in contact.

In conclusion, the total number of asperities in contact must be a decreasing function of the applied stress, while that of those in contact and sliding against each other must increase when the stress increases both in compression and in tension. While no mathematical proof is available to show the logical impossibility that a model based on the micromechanics of crack’s asperities in contact may account for the DAET response by sandstones, physical intuition (and Occam’s razor) casts serious doubts about the likelihood that this may be the case. If at all possible, a DAET experiment in which a sample is subjected to an increasingly higher tensile static stress may shed further light on this issue. It could eliminate mechanisms that have zero strength in tension, like cracks do, as possible causes of hysteresis in DAET tests.

- Received June 3, 2015.
- Accepted November 18, 2015.

- © 2015 The Author(s)