The bowing of natural stone panels is especially known for marble slabs. The bowing of granite is mainly known from tombstones in subtropical humid climate. Field inspections in combination with laboratory investigations with respect to the thermal expansion and the bowing potential was performed on two different granitoids (Cezlak granodiorite and Flossenbürg granite) which differ in the composition and rock fabrics. In addition, to describe and explain the effect of bowing of granitoid facade panels, neutron time-of-flight diffraction was applied to determine residual macro-and microstrain. The measurements were combined with investigations of the crystallographic preferred orientation of quartz and biotite. Both samples show a significant bowing as a function of panel thickness and destination temperature. In comparison to marbles the effect of bowing is more pronounced in granitoids at temperatures of 120°C. The bowing as well as the thermal expansion of the Cezlak sample is also anisotropic with respect to the rock fabrics. A quantitative estimate was performed based on the observed textures. The effect of the locked-in stresses may also have a control on the bowing together with the thermal stresses related to the different volume expansion of the rock-forming minerals.
We study the normal distribution on the rotation group SO(3). If we take as the normal distribution on the rotation group the distribution defined by the central limit theorem in Parthasarathy (1964) rather than the distribution with density analogous to the normal distribution in Eucledian space, then its density will be different from the usual (1 / exp(-(x-m)2/2cr2) one. Nevertheless, many properties of this distribution will be analogous to the normal distribution in the Eucledian space. It is possible to obtain explicit expressions for density of normal distribution only for special cases. One of these cases is the circular normal distribution.The connection of the circular normal distribution SO(3) group with the fundamental solution of the corresponding diffusion equation is shown. It is proved that convolution of two circular normal distributions is again a distribution of the same type. Some projections of the normal distribution are obtained. These projections coincide with a wrapped normal distribution on the unit circle and with the Perrin distribution on the two-dimensional sphere. In the general case, the normal distribution on SO(3) can be found numerically. Some algorithms for numerical computations are given. These investigations were motivated by the orientation distribution function reproduction problem described in the Appendix.
Recently, several studies have focused on the crystallographic texture of bivalve mollusc shells. Unfortunately, these investigations have been limited to the local level. We demonstrate the similarities and differences between the texture measured over the whole shell and that measured over a small part of a shell. An analysis of the global crystallographic texture of bivalve mollusc shells of different genera was carried out using time-of-flight neutron diffraction. The reason for this analysis was to determine whether the crystallographic texture character was similar within the class Bivalvia. It was observed that the shells of mollusc species of the genus Mytilus consist of two phases, calcite and aragonite. Ostrea edulis shells consist of almost pure calcite, whereas Mya arenaria shells consist of almost pure aragonite. It was concluded that the character of the global textures of the different phases in the same shells is different. The advantages of characterisation of the global crystallographic texture are also discussed.
S U M M A R YDependent on the 'intrinsic' effects on the crystal lattice of the rock constituents and the diminishing 'extrinsic' effects of pores and microcracks, elastic wave velocity versus pressure trends in cracked rocks are characterized by non-linear velocity increase at low pressure. At high pressure the 'extrinsic' influence vanishes and the velocity increase becomes approximately linear. Usually, the transition between non-linear and linear behaviour, the 'crack closure pressure', is not accessible in an experiment, because actual equipment is limited to lower pressure. For this reason, several model functions for describing velocity-pressure trends were proposed in the literature to extrapolate low-pressure P-wave velocity measurements to high pressures and, in part, to evaluate the 'intrinsic' velocity-pressure trend from low-pressure data. Knowing the 'intrinsic' velocity trend is of particular importance for the quantification of the crack influence at low pressure, at high pressure, the 'intrinsic' trend describes the velocity trend as a whole sufficiently well. Checking frequently used model functions for suitability led to the conclusion that all relations are unsuitable for the extrapolation and, if applicable, the estimation of the 'intrinsic' velocity trend. However, it can be shown that the 'intrinsic' parameters determined by means of a suitable model function, the zero pressure velocity and the pressure gradient depend on maximum experimental pressure in a non-linear way. Our approach intends to obtain better estimates of particular parameters from observed non-linear behaviour. A converging exponential function is used to approximate particular trends, assuming that the point of convergence of the function represents a better estimate of the zero pressure velocity and the pressure gradient, respectively. Whether the refined 'intrinsic' velocity trend meets the 'true intrinsic' velocity trend within acceptable errors cannot be proven directly due to missing experimental data at very high pressure. We, therefore, conclude that our approach cannot ensure absolutely certain 'intrinsic' velocity trends, however, it can be shown that the optimized trends approximate the 'true intrinsic' velocity trend better as all the other relations do.
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