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It is clear from Fig. 6 that the scatter in the compiled dataset (Fig. 5) cannot be explained by differences in rock type. The reasons for this are twofold. Although porosity exerts a first-order control on the uniaxial compressive strength of volcanic rocks (Figs. 5 and 6), (1) it does not consider differences in factors such as hydrothermal alteration, crystal content, and groundmass crystallinity and (2) porosity is a scalar and does not consider the nature of the void space: the proportion of pores and microcracks and their geometrical properties (e.g. pore radius and shape). The influence of these parameters on the uniaxial compressive strength of volcanic rocks is discussed below.
Microcracks may not contribute much to the porosity of a material, because they are volumetrically small compared to pores, but they can greatly reduce rock strength. The partitioning of the porosity is therefore an important factor not considered in plots of uniaxial compressive strength as a function of porosity (e.g. Figs. 5 and 6). For example, the strength of a volcanic rock with a porosity of 0.02 composed entirely of pores would likely be higher than a rock of the same porosity composed entirely of microcracks. However, while systematic studies on the influence of microcracks on the strength of granite exist, conducted by thermally stressing samples to different temperatures to create a suite of samples with different microcrack densities (e.g. Alm et al. 1985; David et al. 2012; Griffiths et al. 2017a), corresponding studies on volcanic rocks have been so far less insightful due to the difficulty in preparing samples characterised by different degrees of microcracking (e.g. Heap et al. 2014a; Schaefer et al. 2015; Coats et al. 2018; Heap et al. 2018a). These experimental studies have shown that some volcanic rocks may not form additional microcracks when thermally stressed in the laboratory. Indeed, the influence of thermal stressing on volcanic rocks may be linked to their original microstructure, as discussed in Daoud et al. (2020). Therefore, although we expect an increase in microcrack density to decrease the uniaxial compressive strength of volcanic rock, it is challenging at present to conclude as such with the available experimental data.
To conclude, the scatter in the data of Fig. 5 cannot be simply explained by differences in rock type (because a similar scatter is observed for a constant rock type; Fig. 6) and is therefore, primarily, the result of the high variability of volcanic rock samples in terms of hydrothermal alteration, crystallinity, microcrack density, pore radius, and pore shape, amongst other factors. It is unfortunately not possible to plot the uniaxial compressive strength data of Fig. 5 as a function of these parameters, or a variable that combines several of these parameters, primarily because these parameters are rarely quantified in published studies. To assist future data compilations, we urge future laboratory studies to provide as much information as possible on their experimental materials.
Duclos and Paquet (1991) found that the strength of basalt from the French Massif Central was reduced following exposure to high temperature (Fig. 10d). These authors found that uniaxial compressive strength decreased from 340 MPa (no thermal stressing) to 140 MPa following exposure to 1000 C (Duclos and Paquet 1991; Fig. 10d). However, the majority studies have found that the uniaxial compressive strength of lavas (basalt, andesite, and dacite) is largely independent of thermal stressing temperature, even for samples exposed to 900 C (e.g. Heap et al. 2009; Kendrick et al. 2013a; Heap et al. 2014a; Schaefer et al. 2015; Coats et al. 2018; Heap et al. 2018a). Uniaxial compressive strength as a function of thermal stressing temperature for andesite from Volcán de Colima (with a porosity of 0.07 to 0.09) is shown in Fig. 10c (the studies of Heap et al. (2014a), Schaefer et al. (2015), and Coats et al. (2018) only measured strength for one thermal stressing temperature). These data show that uniaxial compressive strength was unaffected by thermal stressing temperature. Reasons forwarded to explain this independence include (1) the mineral constituents of these volcanic rocks do not undergo chemical or phase transformations within the studied temperature range, (2) that these volcanic rocks adhere to the Kaiser temperature memory effect, which stipulates that a rock must be exposed to a temperature higher than it has previously seen to impart new microcrack damage, and (3) that thermal expansion is accommodated by the numerous pre-existing microcracks within these materials and, as a result, stresses at the microscale do not exceed the local strength of the mineral constituents.
The pore-crack and wing-crack models have previously been used to explore the mechanical behaviour of volcanic rocks (e.g. Zhu et al. 2011; Vasseur et al. 2013; Heap et al. 2014a, 2015a, 2016b; Zhu et al. 2016; Coats et al. 2018). The wing-crack model best suits rocks characterised by a microstructure composed of microcracks, usually low-porosity rocks, and the pore-crack model best suits rocks with a microstructure consisting of pores, usually high-porosity rocks. Recent studies that have measured the permeability of volcanic rocks suggest that the porosity at which the microstructure changes from microcrack-dominated to pore-dominated is about 0.15 (Heap et al. 2014a; Farquharson et al. 2015; Heap and Kennedy 2016; Kushnir et al. 2016) and so, from these data, we could infer that the wing-crack model should not be used for volcanic rocks with a porosity > 0.15 (further recommendations are described below).
The pore-crack model has been more widely used in studies of the mechanical behaviour of volcanic rocks than the wing-crack model (e.g. Zhu et al. 2011; Vasseur et al. 2013; Heap et al. 2014a, 2015a, 2016b; Zhu et al. 2016; Coats et al. 2018). The reasons for this are twofold. First, studies of the mechanical behaviour of volcanic rocks have thus far largely focused on porous rocks. Furthermore, as highlighted by the studies of Heap et al. (2014a) and Zhu et al. (2016), and described above, even low-porosity volcanic rocks contain pores that influence their mechanical behaviour. Second, the pore-crack model (Eq. (2)) contains fewer variables than the wing crack model (Eq. (1)). Although Eq. (2) can be used to estimate the strength of a particular sample if the porosity, pore radius, and KIC are known, it is more common, and more useful, to see if the pore-crack model can explain the mechanical behaviour of, for example, a suite of rocks of the same rock type or from the same volcano. To do so, Eq. (2) can be used to provide a strength-porosity curve for a value of the term \( K_IC/\sqrt\pi r \) that best describes the data or, and more common, two curves for values of \( K_IC/\sqrt\pi r \) that bracket the experimental data. For example, the values of \( K_IC/\sqrt\pi r \) that bracket our data compilation are 1.5 and 55 MPa (Fig. 12a). The two main reasons why these data cannot be described by a single curve are as follows: (1) these data are for different rock types with different microstructures that are likely characterised by different values of KIC and (2) the samples within the compilation cannot be described by a single pore radius. However, information can still be gleaned from such analysis. For example, if we assume a reasonable minimum and maximum macropore radius of 5 and 500 μm, the range of possible values for KIC is 0.006 and 0.218 MPa.m0.5 and 0.059 and 2.180 MPa.m0.5, respectively. These ranges for KIC suggest, as stated above, that the representative KIC value for the model is that of the mineral, not the rock (see Table 3). However, since the largest variation in values of KIC is likely related to rock type, it is perhaps sensible to restrict this type of analysis to a single rock type. For example, the compiled data for andesite can be bracketed between \( K_IC/\sqrt\pi r \) values of 3.5 and 40 MPa (Fig. 12b). Possible ranges for values of KIC for andesite, assuming a minimum and maximum macropore radius of 5 and 500 μm, are 0.014 and 0.159 MPa.m0.5 and 0.139 and 1.59 MPa.m0.5, respectively. Alternatively, if we assume the relevant KIC is that of the mineral constituents (0.7 and 0.3 MPa.m0.5 for glass and feldspar, respectively), we can use the available data to provide a range of pore radii that can describe the dataset. If we assume the microcracks are propagating through glass (KIC = 0.7 MPa.m0.5), the range of possible pore radii is between 12.7 mm and 97.5 μm. If we assume the KIC of feldspar (KIC = 0.3 MPa.m0.5), the range of possible pore radii is between 2.3 mm and 17.9 μm. Even without a thorough microstructural investigation, we can confidently state that the upper limit pore radius estimates using this approach (12.7 and 2.3 mm) are overestimations. However, as stated above, and in published studies, even data for rocks of the same rock type from the same volcano cannot not described by a single curve defined by the pore-crack model. Therefore, if we assume that the KIC does not differ significantly for a particular rock type, the pore radius must not be constant over the range of porosity studied. This statement is confirmed by microstructural studies: as porosity increases, the pore size also generally increases. We plot in Fig. 12c curves defined by different characteristic pore radii, assuming a constant KIC of 0.3 MPa.m0.5. These curves are not inconsistent with andesite samples measured by Heap et al. (2014a) for which the pore radius is well constrained. In the examples provided herein, we have used the values for constants a and b (Eq. (2)) provided by Zhu et al. (2010) (i.e. a = 1.325 and b = 0.414). To conclude, despite the limitations presented by the pore-crack model, such as assumption of uniform, circular pores and the absence of pre-existing microcracks, insights into the mechanical behaviour of volcanic rocks can be gleaned using this model. Although the formulation of a more complex micromechanical model, described by an elastic medium populated with pores and microcracks or populated with pores with a nonuniform size distribution, is likely required to better capture the mechanical behaviour of volcanic rocks, the pore-crack model of Sammis and Ashby (1986) provides insight invoking only a small number of variables. 2ff7e9595c
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