Fracture around Mining Excavations
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The high stresses in the rock around deep mining excavations induce extensive fractures which can cause rockfalls or rockburst instabilities like minor earthquakes. To design safer excavations, a fundamental understanding of the fracturing and post-fracture behaviour of brittle rock is essential. My work in this area has involved both numerical and analytic studies of rock fracture processes. The analytic work gives a complete characterization of the types of fracture instabilities observed in the vicinity of deep underground excavations in the post-failure regime and the effect of micro-scale displacements that occur in the elastic plastic models. In 1998 I organized a workshop on microstructural models of rock fracture which was sponsored by the Pacific Institute for the Mathematical Sciences (PIMS). I have also been involved in development of a number of numerical algorithms for modeling rock fracture interactions which are currently being used in the mining industry to design safer and more stable tabular excavations. |
Effect of Microstructure on Localization in Elasto-Plastic Models
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Abstract:At certain critical values of the hardening modulus, the governing equations of elasto-piastic flow may lose their hyperbolicity and exhibit two modes of ill-posedness: shear-band and flutter ill-posedness. These modes of ill-posedness are characterized by the uncontrolled growth of modes at infinitely fine scales, which ultimately violates the continuum assumption. In previous work [L. An and A. Peirce, SIAM J. Appl. Math., 54(1994), pp. 708-730], a continuum model accounting for microscale deformations was built. Linear analysis demonstrated the regularizing effect of the microstructure and provided a relationship between the width of the localized instabilities and the microlength scale. In this paper a weakly nonlinear analysis is used to explore the immediate post-critical behavior of the solutions. For both one-dimensional and anti-plane shear models, post-critical deformations in the plastic regions are shown to be governed by the Boussinesq equation (one of the completely integrable PDEs having soliton solutions), which describes the essential coupling between the focusing effect of the nonlinearity and the dispersive effect of the microstructure terms. The soliton solution in the plastic region is patched to the solution in the elastic regions to provide a special solution to the weakly nonlinear system. This solution is used to derive a relation between the width of the shear band and the length scale of the microstructure. A multiple scale analysis of the constant displacement solution is used to reduce the perturbed problem to a nonlinear Schrodinger equation in the amplitude functions-which turn out to be unstable for large time scales. Stability analyses of more complicated special solutions show that the low wave number solutions are unstable even on the fast time scales while the high wave numbers are damped by the dispersive microstructure terms. These theoretical results are corroborated by numerical evidence. This pervasive instability in the strain-softening regime immediately after failure, indicates that the material will rapidly move to a lower residual stress state with well-defined shear bands. |
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Abstract:For large deformations, the governing equations of elastic-plastic flow may lose their hyperbolicity and become ill posed at some critical values of the hardening modulus. This ill-posedness is characterized by uncontrolled growth of the amplitude of plane wave solutions in certain directions. To capture post-critical behavior, microstructure is built into the constitutive relations. Two types of microstructure are included: one accounts for intergranular rotation via Cosserat theory, and the other accounts for the formation of voids at the microscale by means of a new pressure term related to the gradient of the dilational deformation. Using both a linearized analysis and integral estimates, it is shown that the microstructure terms provide regularizing mechanisms that inhibit the occurrence of both shear band ill-posedness and flutter ill-posedness. Moreover, a local analysis shows that the problem can be reduced to two turning point singular Schrodinger equations in the neighborhood of points where the equations reach the critical value of the hardening modulus. Using matched asymptotics and Wentzel-Kramers-Brillouin (WKB) theory, a relation is derived between the thickness of the localization (internal layer) and the internal length scale of the material introduced by the microstructure terms. |
Explicit Modeling of Large-Scale Fracture Interactions
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Abstract:In
this paper we introduce a method to reduce the solution cost for Boundary
Element (BE) models from 0(N^3) operations to 0(N^2logN) operations (where
N is the number of elements in the model). Previous attempts to achieve
such an improvement in efficiency have been restricted in their applicability
to problems with regular geometries defined on a uniform mesh. We have developed
the Spectral Multipole Method (SMM) which can be used not only for problems
with arbitrary geometries but also with a variety of element types. The memory necessary to store the required influence coefficients for the spectral multipole method is 0{N) whereas the memory required for the traditional Boundary Element method is 0{N^2). We demonstrate the savings in computational speed and fast memory requirements in some numerical examples. We have established that the break-even point for the method can be as low as 500 elements, which implies that the method is not only suitable for extremely large-scale problems, but that it also provides a useful bridge between the small-scale and large-scale problems. We also demonstrate the performance of the multipole algorithm on the solution of large-scale granular assembly models. The large-scale BE capacity provided by this algorithm will not only prove to be useful in large macroscopic models but it will also make it possible to model microscopic damage processes that form the fundamental mechanisms in plastic flow and brittle fracture. |
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Abstract:
This paper describes a novel spectral method based on the FFT for solving
boundary integral equations incorporating the effect of non-linear material
behaviour. The mathematical properties of this method are developed and
illustrated by means of a simple model problem. The spectral boundary element
technique is shown to provide a new framework for volumetric modelling with
easy access to a number of interesting features not available to spatially
implemented algorithms. A fundamental set of point Fourier kernels is introduced
from which, a variety of approximation schemes (including standard piecewise
polynomial approximations) can be constructed in the frequency domain by
' introducing high frequency filters. The frequency domain implementation
of these approximation schemes avoids the tedious integrations associated
with spatial discretizations of the integral equations and provides considerable
flexibility for general-purpose user-defined approximation schemes. Techniques
are described to overcome the periodicity constraint imposed by the FFT
so that general non-repeating geometries can be modelled. It is shown how
the same periodicity can also be exploited to model repeating geometries.
Two novel iterative methods are described to solve the discretized BE equations
efficiently. The first method uses the information provided by the FFT to
construct an approximate inverse extremely efficiently for use in a preconditioned
conjugate gradient algorithm. This method can reduce the operation count
for solution of the discretized problem to 0(N log N) operations. The second
method is an adaptation of Jacobi iteration which can roughly double the convergence rate for linear problems andean help to inhibit undesirable simultaneous failure of neighbouring elements when modelling brittle rock fracture. Two appendices containing expressions for the boundary element kernels and their Fourier transforms are provided. |
6. Efficient
multigrid solution of boundary element models of cracks and faults
by A. Peirce
Int. J. Num. and Anal. Meth. in Geomechanics, Vol. 15, pp. 549-572,
1991.
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Abstract:
This paper describes a novel spectral method based on the FFT for solving
boundary integral equations incorporating the effect of non-linear material
behaviour. The mathematical properties of this method are developed and
illustrated by means of a simple model problem. The spectral boundary element
technique is shown to provide a new framework for volumetric modelling with
easy access to a number of interesting features not available to spatially
implemented algorithms. A fundamental set of point Fourier kernels is introduced
from which, a variety of approximation schemes (including standard piecewise
polynomial approximations) can be constructed in the frequency domain by
' introducing high frequency filters. The frequency domain implementation
of these approximation schemes avoids the tedious integrations associated
with spatial discretizations of the integral equations and provides considerable
flexibility for general-purpose user-defined approximation schemes. Techniques
are described to overcome the periodicity constraint imposed by the FFT
so that general non-repeating geometries can be modelled. It is shown how
the same periodicity can also be exploited to model repeating geometries.
Two novel iterative methods are described to solve the discretized BE equations
efficiently. The first method uses the information provided by the FFT to
construct an approximate inverse extremely efficiently for use in a preconditioned
conjugate gradient algorithm. This method can reduce the operation count
for solution of the discretized problem to 0(N log N) operations. The second
method is an adaptation of Jacobi iteration which can roughly double the convergence rate for linear problems andean help to inhibit undesirable simultaneous failure of neighbouring elements when modelling brittle rock fracture. Two appendices containing expressions for the boundary element kernels and their Fourier transforms are provided. |
Instabilities in Elastodynamic Boundary Integral Models
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Abstract: Evidence of numerical instabilities in two-dimensional time domain direct boundary element methods is presented. The e!ects of numerical versus analytical integration of spatial integrals on stability are shown,and two new time-stepping algorithms are introduced and compared to existing formulations. The so-called new 'direct half-step' scheme and the 'epsilon' scheme are shown to improve the numerical stability of direct boundary element methods. |
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Abstract: Time domain elastodynamic boundary element methods are prone to numerical instabilities. Under suitable conditions, these instabilities can swamp the transient response of a system. We show evidence of these instabilities in both the direct and indirect boundary element methods. We summarize the literature on the evidence and causes of these instabilities, and refer to improved algorithms and alternative formulations which are less prone to numerical instabilities. Finally, we make suggestions as to where research should concentrate so that these methods can reach their full potential. |
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Abstract: There is growing evidence of intermittent numerical instabilities in boundary integral elastodynamic models (Andrews, 1994; Mack, 1991; Siebrits, 1992). We have used the term "intermittent instabilities" because of the way in which the instabilities appear and disappear as the time step is changed. As an example of this type of instability, consider a fixed spatial discretization of a given elastodynamic problem, and allow the time step to change. The time domain boundary element model can be unstable for a certain time step and berome stable if the time step is increased If the time step is increased further, th^ element model may become unstable again. In addition, these instabilities may occur for certain problems and not for others depending on the specific geometry of die problem, Le. the particular combination of spatial modes that are active in the problem. This intermittent instability is clearly unacceptable, as one cannot provide coherent guidelines about the appropriate choice of meshing parameters. In this article, we will restrict ourselves to the space-time formulation of the boundary element equations. We briefly outline die various boundary element formulations. We show how the indirect boundary element methods can be obtained from the direct boundary element formulation. We also outline the various temporal and spatial discretization procedures as well as a strategy for time marching the resulting system of algebraic equations. In order to illustrate the nature of the instabilities and how pervasive they are, we provide examples of instabilities for the direct formulation, and both the fictitious stress and displacement discontinuity indirect formulations. |
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Abstract: In the literature there is growing evidence of instabilities in standard time-stepping schemes to solve boundary integral elastodynamic models. However, there has been no theory to support scientists and engineers in assessing the stability of their boundary element algorithms or to help them with the design of new, more stable algorithms. In this paper we present a general framework for the analysis of the stability of any time-domain boundary element model. We illustrate how the stability theory can be used to assess the stability of existing boundary element models and how the insight gained from this analysis can be used to design more stable time-stepping schemes. In particular, we describe a new time-stepping procedure that we have developed, which has substantially enhanced stability characteristics and greater accuracy for the same computational effort. The new scheme, which we have called `the half-step scheme', is shown to have substantially improved performance for the displacement discontinuity boundary element method commonly used to model dynamic fracture interaction and propagation. |
1. Stability
analysis of model problems for elastodynamic boundary element discretizations
by A. Peirce and E. Siebrits.
Numerical Methods for Partial Differential Equations, Vol. 12, pp
585-613, 1996.
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Abstract:
In the literature there is growing evidence of instabilities in standard
time-stepping schemes to solve boundary integral elastodynamic models. In
this article we use three distinct model problems to investigate the stability
properties of various discretizations that are commonly used to solve elastodynamic
boundary integral equations. Using the model problems, the stability properties
of a large variety of discretization schemes are assessed. The features
of the discretization procedures that are likely to cause instabilities
can be established by means of the analysis. This new insight makes it possible
to design new time-stepping schemes that are shown to be more stable. |