Apparent fracture toughness in Mode I of microcracking materials such as rocks under confining pressure is analyzed based
on a cohesive crack model. In rocks, the apparent fracture toughness for crack propagation varies with the confining pressure.
This study provides analytical solutions for the apparent fracture toughness using a cohesive crack model, which is a model
for the fracture process zone. The problem analyzed in this study is a fluid-driven fracture of a two-dimensional crack with
a cohesive zone under confining pressure. The size of the cohesive zone is assumed to be negligibly small in comparison to
the crack length. The analyses are performed for two types of cohesive stress distribution, namely the constant cohesive stress
(Dugdale model) and the linearly decreasing cohesive stress. Furthermore, the problem for a more general cohesive stress distribution
is analyzed based on the fracture energy concept. The analytical solutions are confirmed by comparing them with the results
of numerical computations performed using the body force method. The analytical solution suggests a substantial increase in
the apparent fracture toughness due to increased confining pressures, even if the size of the fracture process zone is small. 相似文献
The Lesser Qinling carbonatite dykes are mainly composed of calcites. They are characterized by unusually high heavy rare earth element concentrations (HREE; e.g. Yb > 30 ppm) and flat to weakly light rare earth element (LREE) enriched chondrite-normalized patterns (La/Ybn = 1.0–5.5), which is in marked contrast with all other published carbonatite data. The trace element contents of calcite crystals were measured in situ by laser ablation inductively coupled plasma mass spectrometry (LA-ICPMS). Some crystals show reduced LREE from core to rim, whereas their HREE compositions are relatively constant. The total REE contents and chondrite-normalized REE patterns from the cores of carbonate crystals are similar to those of the whole rock. The carbon and oxygen isotopic compositions of calcites fall within the range of primary, mantle-derived carbonatites. The initial Sr isotopic compositions (0.70480–0.70557) of calcites are consistent with an EM1 source or mixing between HIMU and EM1 mantle sources. However these sources cannot produce carbonatite parental magmas with a flat or slightly LREE enrichment pattern by low degrees of partial melting. Analyses of carbonates from other carbonatites show that carbonates have nearly flat REE pattern if they crystallize from a LREE enriched carbonatite melt. This implies that when carbonates crystallize from a carbonatite melt the calcite/melt partition coefficients (D) for HREE are much greater than the D for the LREE. The nearly flat REE patterns of the Lesser Qinling carbonatites can be explained if they are carbonate cumulates that contain little trapped carbonatite melt. Strong enrichment of HREE in the carbonatites may require their derivation by small degrees of melting from a garnet-poor source. 相似文献
Combining Lu–Hf garnet geochronology with in situ trace element analyses in garnet allowed us to gain new insight into the metamorphic evolution of UHP–UHT rocks in the Stary Gierałtów region, in the Polish Sudetes. Prograde garnet growth recorded by Rayleigh-type heavy REE (HREE) zoning in the felsic granulites indicates that the obtained 386.6 ± 4.9 Ma Lu–Hf age represents the time of garnet crystallization on a prograde UHP metamorphic path. The surrounding rocks were metamorphosed at the same time as indicated by 381.2 ± 6.7 Ma Sm–Nd garnet age obtained for the mid-crustal metapelites. The second metamorphic episode, which affected most of the lower crust in the Orlica–Śnieżnik Massif (OSM) occurred at ca. 340 Ma as determined by U–Pb zircon and Sm–Nd garnet dating of granulites in this and previous studies is interpreted as a high temperature event, which took place on a retrograde path.
Trace element distribution in garnets from the layered granulites showed significant differences in distribution of medium and HREE in garnets from mafic and felsic protoliths over the course of the metamorphic evolution. This had strong impact on the isotopic dating results and led to “decoupling” of the Sm–Nd and Lu–Hf clocks, which recorded timing of the two different metamorphic episodes separated by as much as 40 Ma. Moreover, the preservation of the HREE growth zonation profile in garnets from the felsic granulites whose minimum metamorphic temperature was established at 900 °C implies that the Lu–Hf system under relatively dry conditions does not undergo significant diffusional re-equilibration even at such extreme temperatures and therefore it sill provides the age of prograde garnet growth. Under hydrous conditions, at least some resetting will take place, as documented by the partially relaxed HREE zonation profile in the amphibolitised mafic granulite, which yielded a 10 Ma younger age. The HREE distribution study appeared to be a particularly valuable and essential tool, which allowed us to distinguish garnet growth from post-growth complexities and hence, provide improved age interpretation. Medium REE, on the other hand, did not show any obvious correlation with the isotopic signature of garnet.
Two distinct metamorphic episodes recorded in the Stary Gierałtów region show that buoyancy-driven uplift of UHP rocks can be arrested at the base of a continental crust if not supported by any additional force. In our case study, the UHP rocks would have never reached the surface if their uplift had not been resumed after a long pause under a different tectonic regime. The multistage, discontinuous uplift revealed by the UHP rocks of the OSM provides a new scenario for the exhumation of continental crust from mantle depths distinct from the fast-track exhumation histories recognized in UHP terranes elsewhere. 相似文献