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_Journal of Structural Geology 32 (2010) 136–150_ _Contents lists available at ScienceDirect_ _Journal of Structural Geology_ _journal homepage: www.elsevier.com locate jsg _Determination of amphibole fracture strength for quantitative palaeostress analysis using microboudinage structures_ _Nozomi Kimura a,*, Shotaro Nakayama b, Katsuhiro Tsukimura a, Shinko Miwa b, Atsushi Okamoto c, Toshiaki Masuda b AIST, Geological Survey of Japan, Tsukuba 305-8567, Japan; b Institute of Geosciences, Shizuoka University, Shizuoka 422-8529, Japan; c Graduate School of Environmental Studies, Tohoku University, Sendai 980-8579, Japan_ _article info_ _Article history: Received 30 June 2008; Received in revised form 26 September 2009; Accepted 14 October 2009; Available online 21 October 2009_ _Keywords: Fracture strength Amphibole Size effect Absolute magnitude of palaeodifferential stress Microboudin_ _abstract_ _The fracture strength of amphibole is estimated with the aim of extending the applicability of the microboudin method to palaeostress analysis. Before estimating the fracture strength of amphibole, it is necessary to evaluate the influence of the size effect on fracturing. Among the three models of size effect (effective-length, effective-area, and effective-volume), the effective-length model is found to be the most suitable in our microboudinage analysis of a metachert from Turkey that contains numerous microboudinaged amphibole grains of variable width embedded within a quartz matrix. Taking into account the influence of the size effect on fracturing, we undertook a comparative microboudinage analysis of the metachert sample, which also contains coexisting microboudinage structures of tourmaline and epidote. The analysis revealed an instantaneous-fracture strength for amphibole of 80 MPa for a 1 mm cube. The far-field differential stress \(\sigma_0\); calculated based on amphibole microboudinage structures and considering the effect of the fatigue limit, is \(\sigma_0 = 8l^{1/2}\), where \(l\) is the dimensionless stress parameter determined by microboudinage analysis and \(w\) is the mean width of amphibole grains._ \(_{2009} Elsevier Ltd. All rights reserved.\) _1. Introduction_ _The nature of palaeostress within orogenic belts is a fundamental parameter in gaining a quantitative understanding of geodynamic processes within the earth; however, there exists no reliable and practical method for estimating the components of the absolute magnitude of the palaeostress tensor. In the 1970s, three metallurgical methods of stress analysis were introduced, based on grain size, subgrain size, and dislocation density; these methods have since been widely used to estimate palaeodifferential stress during plastic deformation within orogenic belts (e.g., Mercier et al., 1977; Twiss, 1977, 1986; Weathers et al., 1979; Kohlstedt and Weathers, 1980; Ord and Christie, 1984; Küstner and Stöckhert, 1999; Stipp and Tullis, 2003). These metallurgical methods assume steady-state deformation and no post-tectonic annealing, which are likely to be invalid assumptions when considering orogenic processes (e.g., White et al., 1980; Passchier and Trouw, 2005; Masuda et al., 2007). Methods based on calcite twins and microboudins have subsequently been proposed for palaeostress analysis based on observations of microstructures in deformed rocks (e.g., Jamison and Spang, 1976; Masuda et al., 1989, 2003; Burkhard, 1993; Kimura et al., 2006; Lacombe, 2007). The advantage of these methods is their applicability to cases involving nonsteady-state deformation._ _The microboudin method is based on the proportion of boudinaged mineral grains with respect to applied differential stress. As this proportion is not linearly related to the applied stress, the method involves some complexities. In the microboudin method, it is essential to determine the magnitude of fracture strength for microboudinaged minerals in estimating palaeodifferential stress (e.g., Masuda et al., 2003). Kimura et al. (2006) experimentally determined the magnitude of instantaneous-fracture strength for tourmaline and epidote, meaning that these two minerals can now be used for quantitative palaeostress analysis. Amphibole is also a potentially useful mineral for microboudinage analysis because it is a common rock-forming mineral (e.g., Ernst, 1968; Leake, 1968; Papike, 1969; Veblen, 1981); however, the fracture strength of amphibole has yet to be determined experimentally because of difficulties in preparing amphibole specimens of treatable size._ _N. Kimura et al. Journal of Structural Geology 32 (2010) 136–150_ _137_ _With the aim of enabling the use of amphibole microboudins in palaeostress analysis, the present study quantitatively evaluates the size effect for fracturing in terms of naturally occurring submillimetre-scale amphibole grains, and then indirectly determines the fracture strength of amphibole based on coexisting microboudinage structures of sodic amphibole, tourmaline, and epidote in a single sample of naturally deformed metachert from central Turkey._ _2. Sample description_ _The analysed sample is a pebble of metachert collected from a Cretaceous high-pressure metamorphic belt in Eskisehir, central Turkey (e.g., Okay et al., 1998; Okay, 2002). Peak metamorphic conditions for the belt are estimated to have been 430°C and 2.4 GPa (Okay, 2002). The sample is the same as that described in Masuda et al. (2004b, 2008)._ _The sample is dominantly composed of quartz, with subordinate sodic amphibole, tourmaline, epidote, garnet, muscovite, apatite, chlorite, and opaque minerals. The sample contains a clearly developed foliation defined by the shape-preferred orientation of muscovite and bands of contrasting colour. A weak lineation upon the foliation surface is defined by the alignment of columnar minerals (sodic amphibole, tourmaline, and epidote). Because the long axes of columnar minerals are variably oriented upon the foliation surface (Fig. 1), the orientation of the lineation is determined using a statistical method (Masuda et al., 1999, 2004b). The calculated orientations of mineral lineations defined respectively by sodic amphibole, tourmaline, and epidote fall within ±6°. As in Masuda et al. (2004b, 2008), the orientation of the lineation in the present sample is defined by the alignment of sodic amphibole grains. No folding is apparent in the sample._ _Columnar mineral grains of amphibole, tourmaline, and epidote are classified into three orientations: p- and c-orientations, and others. p-orientation grains are those with long axes oriented parallel to the lineation (±15°), and c-orientation grains are those with long axes oriented perpendicular to the lineation (±15°) (Fig. 1b,c). For a given mineral species, the number of p-orientation grains is 4–5 times greater than the number of c-orientation grains._ _Three columnar minerals (sodic amphibole, tourmaline, and epidote) exhibit microboudinage structures within the quartz matrix of the sample (Fig. 2). The microboudinage structures are similar to those described in previous papers (e.g., Masuda et al., 1989, 1995, 2003, 2004b, 2008), with columnar mineral grains being fractured nearly perpendicular to their long axes and microboudins being pulled apart parallel to the long axes, without any evidence of significant rotation during microboudin development. The lack of rotation indicates that a pure shear component was predominant on the foliation surface, suggesting coaxial bulk deformation during microboudinage: if non-coaxial deformation had been predominant, the boudins would have been arranged asymmetrically (e.g., Goscombe et al., 2004; Passchier and Trouw, 2005). In the following analysis, it is assumed that the maximum principal stress \(s_1\) acted perpendicular to the foliation, while the minimum principal stress \(s_3\) acted parallel to the lineation, as shown in Fig. 1a. The pure shear strain during microboudinage in the analysed sample is also discussed in detail in Masuda et al. (2004b)._ _Amphibole overgrowths occur locally on the fracture plane in the interboudin gap within sodic amphibole crystals (Fig. 2f–h), whereas no overgrowths are apparent within tourmaline or epidote microboudins. Electron probe microanalysis reveals that the host sodic amphibole is mostly glaucophane, while the overgrowth phase is magnesioriebeckite (Masuda et al., 2004b). The amphibole nomenclature employed in the present study follows that of Leake et al. (1997)._ _Fig. 1. (a) Schematic of the orientation of the principal stress axes \(s_1 > s_2 > s_3\) acting on the analysed metachert sample at the time of microboudinage. The \(s_1\) axis is assumed to be perpendicular to the foliation, and the \(s_3\) axis is parallel to the lineation on the foliation surface. (b) Definition of p- and c-orientation grains on the foliation surface. p-orientation grains (pink) are those with long axes oriented parallel to the lineation (\(±15°\)), and c-orientation grains (blue) are those with long axes perpendicular to the lineation (\(±15°\)). The orientation of the lineation can be determined using a statistical method (Masuda et al., 1999, 2004b). (c) Photomicrograph of the metachert sample on the foliation surface. The orientation of the mineral lineation is indicated by the dashed line. The long axes of sodic amphibole grains are variably oriented. p: p-orientation grains; c: c-orientation grains. Plane polarized light. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)_ _3. Principle of size-effect analysis_ _3.1. Relationship between \(S^*\) and \(l\) of different groups of grains_ _Microboudinage analysis undertaken by Masuda et al. (2003) provides the relationship between the far-field differential stress applied to the sample \(\sigma_0\) and the mean fracture strength (the modal-extension fracturing) of th_\_ *Note: The text was truncated at the end for formatting purposes.* Ключевые слова: sodic, amphibole grain, parkinson, rst approximation, ax, tourmaline epidote, fracture strength, weaver, nye, foliation, equation, journal, microboudinage, size-effect, wa wt, dissimilar response, mechanics, palaeostress analysis, measured proportion, la lt, error bar, c-orientation, lawn, microboudinage history, based, unknown, foliation surface, long, size-effect model, p-, fracture, log, mechanical, analysis, post-growth stage, leake, analysed sample, stress, host amphibole, parameter, fracturing occurred, academic, c-orientations, statistical, p-orientation epidote, blenkinsop, microboudinaged grain, grain size, central three-quarters, ferguson, measured grain, mineral, effective-length effective-area, microboudinage structure, structural, geometric, applied, kimura journal, twiss, misch, critical microcracks, epstein, microboudins, metamorphic, aspect ratio, journal structural, atkinson, sample, analysed mineral, society america, measurement point, calcite twin, journal structural geology, total number, microboudin method, considered, columnar grain, frequency distribution, quartz, lineation, representative, effective-length model, microboudinaged black, microboudinaged mineral, method, ri, orogenic belt, grain, microboudinage analysis, de?ned, davidge, main, standard deviation, single sample, tectonophysics, number, strain, geophysical, amphibole, quartz grain, kimura, instantaneous fracturing, p-orientation, stress parameter, respect, effective-volume, instantaneous-fracture strength, japanese, scholz, differential, appendix, growth stage, lacombe, interboudin gap, orientations, intact grain, epidote, society, structure, kanaori, effective-length, rock, white, grain width, p-orientation grain, epidote p-orientation, microcracks, differential stress, calculated, variably oriented, elsevier, plastic deformation, strength, fracturing, effective-volume model, lt, study, microboudin, sodic amphibole, nature, absolute magnitude, plastic strain, size-effect analysis, deviation, shear-lag model, palaeostress, mottana, c-orientation grain, standard, data, microboudinaged, aspect, columnar, tourmaline, elastic constant, gure legend, distribution, determined, equivalent, wa, structural geology, measured, america, ceramic, orientation, blueschists, amphiboles, model, stress concentration, calculated based, geology, loglx, circle, mineral specie, effectivelength model, ratio, stage, wi, masuda, metachert, matrix, deformation, intact, eq, la, dislocation density, helper, analysed, frequency, press, davidge awaji, awaji, quartz matrix, effective, l-value, clark, weibull, width, size, surface, experimentally determined, group, le, c-orientation tourmaline, amphibole tourmaline, determined independently, metachert sample, best-t curve, mineral grain, magnitude