Relative density and particle shape effects
This section starts analyzing the impacts of thermal cycles on the mechanics of granular materials with reference to the influence of relative density and particle shape. As in the remainder of this work, such an endeavor resorts to two types of tests: (i) isothermal tests involving mechanical loading up to a vertical stress of \({\sigma }_{v}=1\) MPa, followed by unloading at constant ambient temperature; and (ii) non-isothermal tests involving an equivalent path of mechanical loading and unloading, with the additional application of 50 thermal cycles with a temperature amplitude of \({\Delta }T=60\) °C under a constant stress of \({\sigma }_{v}=60\) kPa. Loose and dense sands with rounded, subangular, and angular particle shapes are considered. See Methods for details.
Figure 1 shows the impacts of 50 thermal cycles on the volumetric strain and porosity of granular materials with variable relative densities and particle shapes. The results show that thermal cycling generates a hysterical mechanical response of granular materials (Fig. 1(a)), which corresponds to thermal ratcheting. In agreement with previous evidence referring to sands subjected to one heating-cooling cycle 35,36, the magnitude of contractive deformations affecting granular materials subjected to individual heating-cooling cycles is relatively limited. However, such deformations accumulate cycle after cycle (Fig. 1(b)), resulting in significant bulk irreversible volumetric contractions of granular materials. Such deformations reach a cumulative value of about 0.9% after 50 thermal cycles with an amplitude of 60°C.
The results unveil that deformations caused by thermal ratcheting are strongly influenced by relative density of granular materials (Fig. 1(a-b)), even though the porosities of such materials differ dramatically at a given relative density (Fig. 1(c)). The porosity changes produced by 50 thermal cycles particularly indicate that irreversible deformations depend on the initial relative density, rather than the initial porosity (Fig. 1(d)). Dense sands exhibit smaller irreversible deformations than loose sands. This phenomenon arguably results from the smaller potential for microstructural reorganizations that characterizes denser compared to looser granular materials.
The results further uncover that deformations caused by thermal ratcheting are strongly influenced by particle shape (Fig. 1(a-b)), with more significant deformations characterizing granular materials with increasingly rounded particles. Granular materials with rounded particles arguably undergo larger deformations compared to materials with angular particles due to the larger translational and rotational freedom of the rounded particles compared to the angular particles upon thermal cycling. Such a greater freedom is supported by the magnitude of plastic strains and porosity changes induced by thermal ratcheting for each relative density (Fig. 1(b-d)). Exceptions apply to the first few thermal cycles, where granular materials with rounded particles show smaller plastic strains compared to the materials with subangular and angular particles, in agreement with previous results addressing the impacts of one heating-cooling cycle on the same materials36. In these circumstances, the larger initial porosity characteristic of angular and subangular sands is considered to overcome the inhibited freedom caused by particle angularity.
Deformations of about 1%, which are thus comparable to those observed in this work, are significant and can lead to problematic consequences for the operational performance of geotechnologies, such as geotechnical structures, geothermal systems, and underground thermal energy storage systems 13,38–40. These deformations are also expected to contribute to changes in the morphology of landforms. Experimental 4 and numerical 9 evidence referring to granular materials with spherical particles supports that the irreversible thermally induced strains associated with thermal cycling continue to accumulate even for a myriad of thermal cycles (i.e., up to 1000 cycles), making thermal ratcheting worthy of attention for the evolutionary study of both engineered and natural systems involving granular materials subjected to thermal cycles. A similar ratcheting response has been reported in sands subjected to cyclic mechanical loads 41,42, which undergo mechanical ratcheting 43. However, thermal ratcheting involves the key difference that particle rearrangements result from thermally induced deformations.
Figure 2 shows the mechanical response of granular materials with irregular particle shapes before, during, and after thermal cycling by comparing the compression curves associated with the non-isothermal and isothermal tests. Both the isothermal and non-isothermal tests indicate that granular materials characterized by smaller relative densities exhibit larger deformations under the same applied mechanical load (Fig. 2(a, c)) or thermal load (Fig. 2(b, d)). Thermal cycling causes a progressive increase in contractive deformations under constant applied vertical stress (Fig. 2(b, d)). 50 thermal cycles of \({\Delta }T=60\) °C cause contractive strains whose magnitudes can vary from about half to a similar value of strains caused by the application of a vertical stress of \({\sigma }_{v}=1\) MPa. This result is observed for sands under both loose conditions (Fig. 2(a-b)) and dense conditions (Fig. 2(c-d)).
The obtained results allow to discover a peculiar role of particle shape on the deformation of granular materials, depending on whether mechanical or thermal loading is considered. Deformations caused by mechanical loading increase with the particle angularity (Fig. 2(a-c)). In other words, granular materials with angular particles deform more than materials with rounded particles under the same applied mechanical load, exhibiting a larger compressibility. In contrast, deformations caused by thermal loads decrease with the particle angularity (Fig. 2(b-d)). In other words, granular materials with angular particles deform less than materials with rounded particles upon thermal cycles of the same amplitude.
Particle shape appears to have complex effects on the mechanics of granular materials. An increasing particle angularity can lead to: (a) a higher porosity, (b) fewer but sharper contacts, and (c) particle interlocking due to concave surfaces. These aspects can lead to opposite consequences. Upon mechanical loading, effects (a) and (b) appear to govern the response of granular materials through a larger potential for plastic strains, as voids can be filled when significant rearrangements occur. These effects may also justify the exceptions associated with the first few thermal cycles. Upon thermal loading, effects (b) and (c) appear to govern the response of granular materials, with a progressively hampered influence of effect (a). Rough and concave surfaces generate larger friction and rolling resistances between particles, which hamper their relative sliding and rolling. Therefore, after the first thermal cycles, increasingly restrained rearrangements characterize angular sands, resulting in smaller consequent deformations for multiple thermal cycles when compared to rounded sands.
Stress level effects
This section further explores the impacts of thermal cycles on the mechanics of granular materials with reference to the influence of the stress level. This analysis resorts to the results of (i) isothermal tests and (ii) non-isothermal tests involving the application of 50 thermal cycles with an identical amplitude of \({\Delta }T=60\) °C at variable stress levels of \({\sigma }_{v}=60\) and \(500\) kPa. Results refer to the rounded sand. See Supplemental Material.
Figure 3 shows the impacts of 50 thermal cycles applied at different stress levels on the volumetric strain of granular materials at two relative densities. A larger applied stress prior to thermal cycling leads to smaller thermally induced irreversible strains (Fig. 3(a-b)). Consequently, a variable applied stress level involves a considerable difference in the magnitude of contractive volumetric strains caused by thermal cycling relative to those caused by mechanical loading (Fig. 3(c-d)).
The hampered impact of thermal ratcheting at a higher level of applied stress prior to thermal cycling is attributed to the increased structural stability that characterizes granular materials. This enhanced stability arguably arises from the strengthened force chains and larger restraint within the materials, which occur for increasing mechanical loads and material densities, respectively. The underlying rationale is that more heavily loaded and denser granular structures require more substantial perturbations to yield microstructural changes associated with thermal ratcheting.
Temperature amplitude effects
This section concludes the analysis of the impacts of thermal cycles on the mechanics of granular materials with reference to the influence of the temperature amplitude. This analysis resorts to the results of (i) isothermal tests and (ii) non-isothermal tests involving the application of 50 thermal cycles with variable amplitudes of \({\Delta }T=30\) and \(60\) °C at an identical stress level of \({\sigma }_{v}=60\) kPa. Results refer to the rounded sand. See Supplemental Material.
Figure 4 shows the impacts of 50 thermal cycles with different amplitudes on the volumetric strain of granular materials at two relative densities. Thermal cycles with a larger temperature amplitude generate more significant thermal ratcheting (Fig. 4(a-b)). Specifically, doubling the amplitude of thermal cycles results in more than proportional thermally induced strains. Under both loose and dense conditions, granular materials subjected to thermal cycles with an amplitude of \({\Delta }T=60\) °C show a more pronounced hysterical response (Fig. 4(a)) and larger plastic strains (Fig. 4(b)) compared to thermal cycles with an amplitude of \({\Delta }T=30\) °C. This result is consistent with previous evidence obtained for glass spheres 1 and originates from more significant thermally induced particle deformations and interconnected variations in interparticle forces 44.
A variable temperature amplitude involves a significant change in the magnitude of contractive volumetric strains caused by thermal cycling relative to mechanical loading (Fig. 4(c-d)). This result highlights, once again, the considerable influence of thermal loading compared to mechanical loading.