STRESS RATE INTEGRAL EQUATIONS OF ELASTOPLASTICITY

The stress rate integral equations of elastoplasticity are deduced based on Ref. [1] by consistent methods. The point at which the stresses and/or displacements are calculated can be in the body or on the boundary, and in the plastic region or elastic one. The existence of the principal value integral in the plastic region is demonstrated strictly, and the theoretical basis is presented for the paticular solution method by unit initial stress fields. In the present method, programming is easy and general, and the numerical results are excellent.

THE EXISTENCE OF LIMIT CYCLES FOR THE SYSTEM X=Q(x,y),Y=P(x)

In [1], by a transformation on the Liemrd equation system suchihai the trajectories of (1)on both left and right half -planes change into thoseintegral curves of the new equation system merely on the right half-plane,A.F.Hilippov shows that under some certain conditions the stable limit cycles of system (1)must exist.Applying the Filippov’s method on the more generalized systemthis paper provides a sufficient condition for the existence of the stable limit cycles oftvstem (2).

Constraining Shear Wave Velocity and Density Contrast at the Inner Core Boundary with PKiKP/P Amplitude Ratio

Shear velocity and density contrast across the inner core boundary are essential for stud- ying deep earth dynamics, geodynamo and geomagnetic evolution. In previous studies, amplitude ratio of PKiKP/PcP at short distances and PKiKP/P at larger distances are used to constrain the shear veloc- ity and density contrast, and shear velocity in the top inner core is found to be substantially smaller than the PREM prediction. Here we present a large dataset of PKiKP/P amplitude ratio measured on 420 seismic records at ILAR array in Alaska for the distance range of 800-90~, where the amplitude ra- tio is sensitive to shear velocity and density contrast. At high frequency （up to 6 Hz）, mantle attenuation is found to have substantial effects on PKiKP/P. After the attenuation effects are taken into account, we find that the density contrast is about 0.2-1.0 g/cm3, and shear velocity of inner core is 3.2-4.0 km/s, close to the PREM （Preliminary Reference Earth Model） prediction （0.6 g/cm3 and 3.5 kin/s, respec- tively）. The relatively high shear velocity in inner core does not require large quantities of defects or melts as proposed in previous studies.

Research on the scatter features of the PKiKP/PcP amplitude ratio and the inner core boundary density contrasts beneath Northeast Asia

Quantifying the density contrasts of the Earth’s inner core boundary(ICB)is crucial to understand core-mantle coupling and the generation of the geodynamo.The PKiKP/PcP amplitude ratio is commonly used to obtain the density contrast at the ICB,but its applications are limited by scattered observed data.In this study,we selected the PKiKP and PcP phases reflected at the same region of inner-core and core-mantle boundaries beneath Northeast Asia from different earthquakes for the first time,and the observations suggested that the PKiKP/PcP amplitude ratio is widely scattered.We also compared the PKiKP and PcP amplitudes,which demonstrated that the scatter cannot be attributed only to ICB anomalies but might also arise from raypath differences and heterogeneities throughout the crust and mantle.By fitting the observed PKiKP/PcP amplitude ratio,we obtained a density contrast of approximately 0.65 g cm^(-3) and a compressional velocity contrast of approximately 0.87 km s^(-1) at the ICB beneath Northeast Asia.The larger contrast values indicate the possible occurrence of local crystallization occurring at the inner core surface.