10. Dynamic Rock Fragmentation
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DYNAMIC ROCK F RACME NTATl0N
D. E. Grady and M. E. Kipp
Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
10.1. INTRODUCTION
The response of a single crack to both static and impulsive loading has received considerable attention over the past several decades and is reasonably well understood. The mechanics of a system of cracks under impulsive or stress-wave loading, and how the cooperative response of such a system relates to the transient strength and ultimate failure of a solid body is less well understood, and has been a subject of study over the past several years. Experimental studies of fracture under high-rate loading have revealed unusual features associated with the phenomenon, such as greatly enhanced material strength and fracture stress dependence on loading conditions. Although such observations have led to the postulation of rate-dependent material properties, most of these features can be understood through fundamentai fracture concepts when considered in terms of a system of interacting cracks. In events involving dynamic fracture, there are a number of features for which a predictive capability is needed. Perhaps the first, and most fundaFracture Mechanics of Rock ISBN: 0-12-066265-5 Copyright 01987 by Academic Press Tnc (London) Ltd. All rights of reproduction in any form reserved
429
430
D. E. CRADY AND M. E. KlPP
mental, is the transient strength, or ability to support an impulsive load, either without sustaining fracture damage, or sustaining fracture damage within some tolerable level without permitting total failure. In partially fractured bodies, the spacing of the cracks may be important, along with the void volume and extent of intersection, which relate to the permeability of the crack system. In completely failed bodies the degree of fragmentation is of interest in many applications. The size and velocity of ejected fragments is also of concern, as is the distribution in fragment sizes and how this relates to the conditions of loading. This chapter will focus on several of the features of dynamic fracture and fragmentation which have come to light over the past several years. In the second section, the concept of dynamic fracture strength is examined, and related to both inherent flaw concepts and the strength properties of a single crack subjected to stress-wave loading. The latter has been found to correlate well with the behaviour of a system of cracks within a body as a whole. In the third section, the properties of the material and the conditions of loading which determine the number of fractures participating in the fracture process, and the number and size of fragments resultingin the failed body are considered. Two related concepts are important here. First is the inherent distribution of flaws or sites of weakness in the body which constitute the points of fracture activation. The second relates to the energy or the rate of energy application required to sustain the system of growing cracks. Although the inherent flaw distribution, a material property, has been used to determine fracture number and fragment size in most previous work, it has been found that in numerous applications a kinematic energy condition appears to govern the fracture fabric. The following section is concerned with the mechanical and statistical conditions which determine the distributions in fragment size resulting from catastrophic fracture events. This is a diverse and extremely complex topic which is not well understood. There is evidence that both the mode and the multiplicity of the dynamic fracture event is important in determining the shape of the distribution. Statistical concepts and theoretical approaches which appear important to the dynamic fragmentation problem are discussed in this section. In the final section, the various approaches to continuum modelling of dynamic fracture and fragmentation currently under consideration are presented. Such modelling represents a necessary final goal in that the complexities of stress loading, geometry, and the interaction of stress and relief waves necessitate the use of wave-propagation codes to address realistic problems.
10. DYNAMIC ROCK FRAGMENTATION
43 1
10.2. DYNAMIC FRACTURE STRENGTH
In an investigation of the underlying causes of the time-dependent tensile fracture strength of rock and rock-like materials, two issues are immediately recognized as important to the phenomena. The first is the inherent distribution of flaws, or sites of weakness. These are the points of fracture activation under tensile loading which clearly must play an important role in the failure process. Extensive microstructural studies of flaw features relevant to dynamic fracture have been performed (Curran et al., 1977). In continuum modelling of dynamic fracture, inherent flaws are frequently characterized through familiar analytic representations, such as a Weibull distribution (Grady and Kipp, 1980). The second issue involves the response of isolated cracks subjected to dynamic tensile loading. Illuminating studies on the strength of a body due to stress concentrating effects of cracks under static loading have emerged from the concepts initiated by Griffith (1920). The response of an elastic solid containing a crack and subjected to abrupt tensile loading normal to the crack surface has also been well characterized and is extensively discussed by Chen and Sih (1977). Response of a solid to more general loading functions applied to the crack has been considered by Freund (1973). Application of these methods to constant tensile strain-rate loading have been explored in detail by Kipp et al. (1980). These studies provide an understanding of many of the features observed in the dynamic fracture of rock. The concepts of both inherent flaws and the dynamic response of isolated cracks can independently lead to a theoretical description of the dynamic fracture strength of rock. Each approach illuminates different physical features of the transient fracture process. A theoretical reconciliation of the two methods has not yet been clearly established. An understanding of the interdependence is beginning to emerge, however, and is discussed in the present and following sections (in particular, see Section 10.3.5).
DYNAMIC ROCK F RACME NTATl0N
D. E. Grady and M. E. Kipp
Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
10.1. INTRODUCTION
The response of a single crack to both static and impulsive loading has received considerable attention over the past several decades and is reasonably well understood. The mechanics of a system of cracks under impulsive or stress-wave loading, and how the cooperative response of such a system relates to the transient strength and ultimate failure of a solid body is less well understood, and has been a subject of study over the past several years. Experimental studies of fracture under high-rate loading have revealed unusual features associated with the phenomenon, such as greatly enhanced material strength and fracture stress dependence on loading conditions. Although such observations have led to the postulation of rate-dependent material properties, most of these features can be understood through fundamentai fracture concepts when considered in terms of a system of interacting cracks. In events involving dynamic fracture, there are a number of features for which a predictive capability is needed. Perhaps the first, and most fundaFracture Mechanics of Rock ISBN: 0-12-066265-5 Copyright 01987 by Academic Press Tnc (London) Ltd. All rights of reproduction in any form reserved
429
430
D. E. CRADY AND M. E. KlPP
mental, is the transient strength, or ability to support an impulsive load, either without sustaining fracture damage, or sustaining fracture damage within some tolerable level without permitting total failure. In partially fractured bodies, the spacing of the cracks may be important, along with the void volume and extent of intersection, which relate to the permeability of the crack system. In completely failed bodies the degree of fragmentation is of interest in many applications. The size and velocity of ejected fragments is also of concern, as is the distribution in fragment sizes and how this relates to the conditions of loading. This chapter will focus on several of the features of dynamic fracture and fragmentation which have come to light over the past several years. In the second section, the concept of dynamic fracture strength is examined, and related to both inherent flaw concepts and the strength properties of a single crack subjected to stress-wave loading. The latter has been found to correlate well with the behaviour of a system of cracks within a body as a whole. In the third section, the properties of the material and the conditions of loading which determine the number of fractures participating in the fracture process, and the number and size of fragments resultingin the failed body are considered. Two related concepts are important here. First is the inherent distribution of flaws or sites of weakness in the body which constitute the points of fracture activation. The second relates to the energy or the rate of energy application required to sustain the system of growing cracks. Although the inherent flaw distribution, a material property, has been used to determine fracture number and fragment size in most previous work, it has been found that in numerous applications a kinematic energy condition appears to govern the fracture fabric. The following section is concerned with the mechanical and statistical conditions which determine the distributions in fragment size resulting from catastrophic fracture events. This is a diverse and extremely complex topic which is not well understood. There is evidence that both the mode and the multiplicity of the dynamic fracture event is important in determining the shape of the distribution. Statistical concepts and theoretical approaches which appear important to the dynamic fragmentation problem are discussed in this section. In the final section, the various approaches to continuum modelling of dynamic fracture and fragmentation currently under consideration are presented. Such modelling represents a necessary final goal in that the complexities of stress loading, geometry, and the interaction of stress and relief waves necessitate the use of wave-propagation codes to address realistic problems.
10. DYNAMIC ROCK FRAGMENTATION
43 1
10.2. DYNAMIC FRACTURE STRENGTH
In an investigation of the underlying causes of the time-dependent tensile fracture strength of rock and rock-like materials, two issues are immediately recognized as important to the phenomena. The first is the inherent distribution of flaws, or sites of weakness. These are the points of fracture activation under tensile loading which clearly must play an important role in the failure process. Extensive microstructural studies of flaw features relevant to dynamic fracture have been performed (Curran et al., 1977). In continuum modelling of dynamic fracture, inherent flaws are frequently characterized through familiar analytic representations, such as a Weibull distribution (Grady and Kipp, 1980). The second issue involves the response of isolated cracks subjected to dynamic tensile loading. Illuminating studies on the strength of a body due to stress concentrating effects of cracks under static loading have emerged from the concepts initiated by Griffith (1920). The response of an elastic solid containing a crack and subjected to abrupt tensile loading normal to the crack surface has also been well characterized and is extensively discussed by Chen and Sih (1977). Response of a solid to more general loading functions applied to the crack has been considered by Freund (1973). Application of these methods to constant tensile strain-rate loading have been explored in detail by Kipp et al. (1980). These studies provide an understanding of many of the features observed in the dynamic fracture of rock. The concepts of both inherent flaws and the dynamic response of isolated cracks can independently lead to a theoretical description of the dynamic fracture strength of rock. Each approach illuminates different physical features of the transient fracture process. A theoretical reconciliation of the two methods has not yet been clearly established. An understanding of the interdependence is beginning to emerge, however, and is discussed in the present and following sections (in particular, see Section 10.3.5).