Finite strain theories and comparisons with seismological data

All the finite strain equations that we are aware of that are worth considering in connection with the interior of the Earth are given, with the assumptions on which they are based and corresponding relationships for incompressibility and its pressure derivatives in terms of density. In several cases, equations which have been presented as new or independent are shown to be particular examples of more general equations that are already familiar. Relationships for deriving finite strain equations from atomic potential functions or vice versa are given and, in particular it is pointed out that the Birch-Murnaghan formulation implies a sum of power law potentials with even powers. All the equations that survive simple plausibility tests are fitted to the lower mantle and outer core data for the PEM earth model. For this purpose the model data are extrapolated to zero temperature, using the Mie-Grüneisen equation to subtract the thermal pressure (at fixed density) and the pressure derivative of this equation to substract the thermal component of incompressibility. Fitting of finite strain equations to such zero temperature data is less ambiguous than fitting raw earth model data and leads immediately to estimates of the low temperature zero pressure parameters of earth materials. On this basis, using the best fitting equations and constraining core temperature to give an extrapolated incompressibilityK ₀=1.6×10¹¹Pa, compatible with a plausible iron alloy, the following numerical data are obtained: Core-mantle boundary temperature 3770 K Zero pressure, zero temperature densities: lower mantle 4190 kg m^–3 outer core (solidified) 7500 kg m^–3 Zero pressure, zero temperature incompressibility of the lower mantle 2.36×10¹¹PaHowever, an inconsistency is apparent betweenP() andK() data, indicating that, even in the PEM model, in which the lower mantle is represented by a single set of parameters, it is not perfectly homogeneous with respect to composition and phase.     

Envelopes of maximum seismic response for a partially symmetric single storey building model

An analysis is made of the coupled lateral-torsional response of a partially symmetric single-storey building model to horizontal translatory earthquake excitation. Interest centres on the evaluation of realistic estimates for two equivalent static actions (a shear and a torque) which account for the worst dynamic consequences of torsional unbalance. The results substantiate the findings of previous investigations which have given rise to the belief that strong modal coupling and severely coupled lateral and torsional responses are possible even in nominally symmetric buildings. The response of the model is assumed to be linearly elastic and viscously damped. In a preliminary analysis the equations of motion are solved using the modal analysis technique and the conditions necessary for full modal coupling are ascertained. Then by employing the design spectrum concept, together with suitably conservative procedures for combining the modal maxima, dimensionless forms of the equivalent static actions are evaluated as functions of two independent parameters. The final results are furnished by modified square root of the sum of the squares (SRSS) combination functions which take account of the spacing between the translational and torsional frequencies. Examples at the end of the paper illustrate the practical significance of the work.     

A note on the numerical evaluation of Duhamel's integral

General expressions are derived for the numerical evaluation of Duhamel's integral and its derivative. The work comprises an extension (to unequal time steps) and an application (to a piecewise linear forcing function) of the numerical integration approach adopted by Cronin.¹ The application is particularly relevant to the digital computation of response spectra from strong motion earthquake records.     

Die Laufzeitfunktion in der Reflexionsseismik des quasi Einachsig inhomogenen Körpers

Zusammenfassung Es wird die Laufzeitfunktion (Reflexionsseismik) für einen quasi einachsig inhomogenen Körper berechnet und zwei einfache Beispiele und ihre Anwendungen behandelt.

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Summary The travel-time function (reflection seismic) for a quasi uniaxial inhomogeneous medium is calculated and two simple examples are given.

     

Das ebenen-Schichtenproblem der Reflexionsseismik mit konstanten Frontgeschwindigkeiten,beliebigen Neigungen und beliebiger Lage der Quelle

Zusammenfassung Es wird das ebenen-Schichtenproblem der Reflexionsseismik mit konstanten Frontgeschwindigkeiten, beliebigen Neigungen und beliebiger Lage der Quelle allgemein behandelt und durch Verallgemeinerung der tt Methode ein einfaches Aufschlussverfahren für geneigte Schichten entwickelt.

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Summary The two-dimensional problem of the reflection seismic with arbitrary inclined surfaces of discontinuity, const. wave-velocities and an arbitrary position of the source is generally treated. Finally a tt method for inclined layers is outlined.

     

Ergebnisse der Registrierungen der Zahl der Kondensationskerne in Basel und seiner nächsten Umgebung

Zusammenfassung In Fortsetzung früher mitgeteilter Ergebnisse von Registrierungen mit demVerzár'schen Kernzähler in Basel wird der Einfluss der Luftmassenwechsel, des Sonnenscheins und der Niederschläge auf den Kerngehalt untersucht. Weiterhin werden die Ergebnisse gleichzeitiger Registrierungen im Stadtinnern und am Stadtrand, sowie am Stadtrand und einem Villenvorort eingehend diskutiert.Ein weiterer Vergleich zwischen Basel und Payerne ergibt Beziehungen zwischen den Tagesmitteln der Kernzahlen dieser fast 100 Km von einander entfernten Orte.

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Summary Continuing previously communicated results of recordings with theVerzár nuclei counter at Basle the influences of changes of air masses, of sunshine and precipitation on nuclei concentration are studied. Furthermore the results of simultaneous recordings in the centre of the city and on the outskirts, as well as on the out-skirts of the city and at a near-by village are thoroughly discussed. A further comparison between Basle and Payerne shows connections between daily means of nuclei concentration of these two towns, which are about 100 Km apart.

     

Die Bestimmung einer beliebig gekrümmten Schichtgrenze aus seismischen Reflexionsmessungen

Zusammenfassung Es wird dargelegt wie man eine beliebig gekrümmte Schichtgrenze aus seismischen Reflexionsmessungen längs einer beliebig gekrümmten Messfläche berechnen kann. Dazu wird vorausgesetzt, dass die Flächez=f(x,y), die die Schichtgrenze darstellt und die Messflächez=(x,y) zusammen eine glatte geschlossene Fläche bilden und, dass das eingeschlossene Material homogen und isotrop ist.

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Summary Under the assumption that the measuring surfacez=(x,y) and the reflection horizon of a structurez=f(x,y) have an arbitrary curvature a simple method is developed for calculating the surfacef from seismic reflection data measured along the surface . In addition the general solution is discussed and some special cases are treated.