Theory of dynamic crack branching in brittle materials

The problem of dynamic symmetric branching of an initial single brittle crack propagating at a given speed under plane loading conditions is studied within a continuum mechanics approach. Griffith's energy criterion and the principle of local symmetry are used to determine the cracks paths. The bifurcation is predicted at a given critical speed and at a specific branching angle: both correlated very well with experiments. The curvature of the subsequent branches is also studied: the sign of $T$, with $T$ being the non singular stress at the initial crack tip, separates branches paths that diverge from or converge to the initial path, a feature that may be tested in future experiments. The model rests on a scenario of crack branching with some reasonable assumptions based on general considerations and in exact dynamic results for anti-plane branching. It is argued that it is possible to use a static analysis of the crack bifurcation for plane loading as a good approximation to the dynamical case. The results are interesting since they explain within a continuum mechanics approach the main features of the branching instabilities of fast cracks in brittle materials, i.e. critical speeds, branching angle and the geometry of subsequent branches paths.Comment: 41 pages, 15 figures. Accepted to International Journal of Fractur

Determination of the polynomial moments of the seismic moment rate density distribution with positivity constraints

We present a new formulation of the inverse problem of determining the temporal and spatial power moments of the seismic moment rate density distribution, in which its positivity is enforced through a set of linear conditions. To test and demonstrate the method, we apply it to artificial data for the great 1994 deep Bolivian earthquake. We use two different kinds of faulting models to generate the artificial data. One is the Haskell-type of faulting model. The other consists of a collection of a few isolated points releasing moment on a fault, as was proposed in recent studies of this earthquake. The positions of 13 teleseismic stations for which P- and SH-wave data are actually available for this earthquake are used. The numerical experiments illustrate the importance of the positivity constraints without which incorrect solutions are obtained. We also show that the Green functions associated with the problem must be approximated with a low approximation error to obtain reliable solutions. This is achieved by using a more uniform approximation than Taylor's series. We also find that it is necessary to use relatively long-period data first to obtain the low- (0th and 1st) degree moments. Using the insight obtained into the size and duration of the process from the first-degree moments, we can decrease the integration region, substitute these low-degree moments into the problem and use higher-frequency data to find the higher-power moments, so as to obtain more reliable estimates of the spatial and temporal source dimensions. At the higher frequencies, it is necessary to divide the region in which we approximate the Green functions into small pieces and approximate the Green functions separately in each piece to achieve a low approximation error. A derivation showing that the mixed spatio-temporal moments of second degree represent the average speeds of the centroids in the corresponding direction is given

Realistic inversions to obtain gross properties of the earthquake faulting process

The problem is to find if we are able to determine correctly the average properties of the actual earthquake faulting process, which occurs in reality in the Earth at microscopic scales, from a solution of a feasible inverse problem. In order to investigate this, we use synthetic accelerograms constructed in the vicinity of a 20 km × 5 km dipping thrust fault for a discrete analog of the Haskell-type of fault model as the data and perform the inversion using much coarser spatial and temporal grids than used in constructing the data and widely used inversion methods. We show that we are essentially unable to obtain the proper (known) results when we use a least-squares (Singular Value Decomposition) method due to the fact that many negative values of moment rate, which did not exist in the forward problem, are produced though the data are well fit. Inversions using the SVD method together with smoothing constraints or exclusion of small singular values yield an improved solution but the actual values to be used must be obtained by trial-and-error and do not contribute to our insight as to how to solve the problem with real data. Finally, we show that if we enforce the condition that there are no negative moment rates, using in this case the method of linear programming, we are able to reproduce many aspects of the original solution better