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A basic, sinusoidal solution to the linearized equations of equilibrium for compressible, elastic materials provides solutions to several problems of folding of multilayers. Theoretical wavelengths are comparable to those predicted by Ramberg, using viscosity theory, and to those predicted by elementary folding theory. The linearized analysis of buckling of a single, stiff, elastic layer, either isolated or within a soft medium, suggests that wavelengths computed with elementary beam theory are remarkably similar to those computed with the linearized theory for wavelength-to-thickness ratios greater than about five. This is half the limit of ten normally assumed for use of the elementary theory.The theory and experiments with deep beams of rubber or gelatin indicate that thick, homogeneous layers folded with short wavelengths assume internal forms strikingly similar to those of the ideal concentric fold. Thus, mechanical layering clearly is not required to produce concentric-like forms.Further, the theory suggests that “arc and cusp” structure, or “pinches”, at edges of deep beams as well as chevron-like forms in single or multiple stiff layers are a result of a peculiar, plastic-like behavior of elastic materials subjected to high normal stresses parallel to layering. In a sense, the elastic material “yields” to form the hinge of the chevron fold, although the strain vanishes if the stresses are released. Accordingly, it may be impossible to distinguish chevron forms produced in elastic-plastic materials, such as cardboard or aluminum and perhaps some rock, from chevron forms produced in purely elastic materials, such as rubber.Analysis of the theory shows that, just as high axial stresses make straight, shortened multilayers the unstable form and sinusoidal waves the stable form, stresses induced by sinusoidal displacements of the multilayer make the sinusoidal waveform unstable and concentric-like waves the stable form. Thus, concentric-like folds appear to be typical of folded multilayers according to our analysis. Further, where the layers have short wavelengths in the cores of the concentric-like folds, the stiff layers “yield” elastically at hinges and straighten in limbs. Thus the concentric-like pattern is replaced by chevron folds as the multilayer is shortened. In this way we can understand the sequence of events from uniform shortening, to sinusoidal folding, to concentric-like folding, to chevron folding in multilayers composed of elastic materials. 相似文献
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This part concerns folding of elastic multilayers subjected to principal initial stresses parallel or normal to layering and to confinement by stiff or rigid boundaries. Both sinusoidal and reverse-kink folds can be produced in multilayers subjected to these conditions, depending primarily upon the conditions of contacts between layers. The initial fold pattern is always sinusoidal under these ideal conditions, but subsequent growth of the initial folds can change the pattern. For example, if contacts between layers cannot resist shear stress or if soft elastic interbeds provide uniform resistance to shear between stiff layers, sinusoidal folds of the Biot wavelength grow most rapidly with increased shortening. Further, the Biot waves become unstable as the folds grow and are transformed into concentric-like folds and finally into chevron folds. Comparison of results of the elementary and the linearized theories of elastic folding indicates that the elementary theory can accurately predict the Biot wavelength if the multilayers contain at least ten layers and if either the soft interbeds are at most about one-fifth as stiff as the stiff layers, or there is zero contact shear strength between layers.Multilayers subjected to the same conditions of loading and confinement as discussed above, can develop kink folds also. The kink fold can be explained in terms of a theory based on three assumptions: each stiff layer folds into the same form; kinking is a buckling phenomenon, and shear stress is required to overcome contact shear strength between layers and to produce slippage locally. The theory indicates that kink forms will tend to develop in multilayers with low but finite contact shear strength relative to the average shear modulus of the multilayer. Also, the larger the initial slopes and number of layers with contact shear strength, the more is the tendency for kink folds rather than sinusoidal folds to develop. The theoretical displacement form of a layer in a kink band is the superposition of a full sine wave, with a wavelength equal to the width of the kink band, and of a linear displacement profile. The resultant form resembles a one-half sine curve but it is significantly different from this curve. The width of the kink band may be greater or less than the Biot wavelength of sinusoidal folding in the multilayer, depending upon the magnitude of the contact shear strength relative to the average shear modulus. For example, in multilayers of homogeneous layers with contact strength, the Biot wavelength is zero so that the width of the kink band in such materials is always greater than the Biot wavelength. In general, the higher the contact strength, the narrower the kink band; for simple frictional contacts, the widths of kink bands decrease with increasing confinement normal to layers. Widths of kink bands theoretically depend upon a host of parameters — initial amplitude of Biot waves, number of layers, shear strength of contacts between layers, and thickness and modulus ratios of stiff-to-soft layers — therefore, widths of kink bands probably cannot be used readily to estimate properties of rocks containing kink bands. All these theoretical predictions are consistent with observations of natural and experimental kink folds of the reverse variety.Chevron folding and kink folding can be distinctly different phenomena according to the theory. Chevron folds typically form at cores of concentric-like folds; they rarely form at intersections of kink bands. In either case, they are similar folds that develop at a late stage in the folding process. Kink folds are more nearly akin to concentric-like folds than to chevron folds because kink folds form early, commonly before the sinusoidal folds are visible. Whereas concentric-like folds develop in response to higher-order effects near boundaries of a multilayer, kink folds typically initiate in response to higher-order shear, as at inflection points near mid-depth in low-amplitude, sinusoidal fold patterns. Chevron folding and kink folding are similar in elastic multilayers in that elastic “yielding” at hinges can produce rather sharp, angular forms. 相似文献
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The system Ag-Sb-S was studied between 600°C and 200°C in evacuated silica glass tubes. Results from lower temperature runs require shifts in the stable tie-line configuration found by Barstad at 400°C. It is proposed that the configuration changes near 300°C, and that at 200°C the equilibrium assemblages correspond to those usually reported for minerals in ores. Most of the minerals of the system were synthesized. In addition, the synthetic phase Ag7SbS6 (antimony analogue of the arsenic mineral billingsleyite) is characterized, and the ease of its synthesis in the composition area bounded by argentite-pyrargyrite-sulfur suggests the probable existence of a mineral of this composition. The relatively common mineral stephanite (Ag5SbS4) was not formed as a synthetic product in the temperature range of this study. Combined DTA and X-ray data show that at 197±5°C stephanite decomposes in the absence of sulfur to form pyrargyrite plus argentite, whereas with excess sulfur the products are Sb-billingsleyite plus pyrargyrite. Pyrostilpnite (Ag3SbS3), the low temperature dimorph of pyrargyrite, is unstable above 192±5°C.
Das ternäre System Silber-Antimon-Schwefel wurde zwischen 600° und 200°C untersucht und versucht, die Gleichgewichtszustände aller stabilen Phasen zu analogen natürlichen Mineralien in Beziehung zu setzen. Neben den Elementen wurden an binären Phasen Allargentum, Dyskrasit, Antimonit, Argentit bzw. Akanthit gefunden oder bestätigt. Auf dem pseudo-binären Schnitt Ag2S-Sb2S3 liegen Pyrargyrit und Miargyrit, während eine als Mineral unbekannte ternäre Phase Ag7SbS6 (entsprechend dem natürlichen As-Analogon Billingsleyit) nur bei höherem Schwefelangebot beständig ist. Hier nicht synthetisch dargestellte Silber-Antimon-Sulfosalze liegen vermutlich unterhalb der 200°C-Grenze. So ließ sich mittels Differential-Thermo-Analyse und röntgenographischer Bestimmungsmethoden der inkongruente Zerfall von Stephanit in Argentit und Pyrargyrit bei 197±5°C bestimmen. Pyrostilpnit (Ag3SbS3) ist nur unterhalb 192±5°C beständig.相似文献
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