STATIC AND DYNAMIC 3D SPATIAL DEFECTS
© Copyright 1988-2015 John Bruce Davies All Rights Reserved
An explosion is an injection of energy, which, if of sufficient magnitude, can create a hole or gap in the material continuum. Such holes, along with cracks, faults, breaks and discontinuities are defects in the continuum. They are governed by the Bianchi Identities, which are conservation laws for distributions of such defects. In general, these defect distributions are combinations of two interlinked components, namely, dislocations and disclinations. These fields manifest topologically as curvature representing disclination density, and torsion representing dislocation density, together with the associated metric tensor. In general, the partial differential equations governing these fields are the highly nonlinear Bianchi Identities. We shall develop and use the simpler linearized conservation laws of this self-consistent model incorporating nonsymmetric torsion and curvature in a generalized continuum.
These linear laws are applied to 3-dimensional cases in order to define the geometries of static defect distributions. The static equations are used to comprehend tensile and shear cracks and explain the defect patterns observed. In the simplest linear case of a long narrow crack, the body of the crack is due to translational displacements which are represented by the dislocation field. At the crack ends, gradients in the dislocation field are mathematically shown to yield a disclination field which represents the crack tip and is due to rotational displacements. Explosive sources are equivalent to creation of disclinations (the cavity) and associated dislocations (the surrounding radial fracturing) and their disclination fields at the fracture tips, from which both compressional and shear waves are emitted. The far-field dilatation, deviatoric strain and rotation potentials are due to both dislocation (torsion) gradients and disclinations (curvature). The geometric constraints of the particular distribution of defects govern the shape of the resultant explosive structure and the specific types of strain potentials produced. Non-linear conservation laws apply close to the explosive source resulting in crack bifurcation and inhomogeneous defect distributions.
In order to understand the largest explosion in and of the Universe, the Big Bang, we note the similarities to the physics of a small explosion. The Big Bang was the injection of energy into a uniform flat spacetime vacuum where no previous structure or form existed. A small explosion occurs in a homogeneous 3-dimensional continuum devoid of any pre-existing forms. The characteristic fields and structures that are produced by this smaller explosion have analogies and similar physical and mathematical properties when transposed to 4-dimensional spacetime.
The conservation laws governing defects must be applied in order to determine self-consistent defect density distributions, deWit (1973), and their motions, Kossecka and deWit (1977a,b), in 3-dimensional elastic continua. Defect distributions are composed of the usual dislocations, due to translational displacements, and also must incorporate disclinations, due to rotational displacements, which make defect theory self-consistent. We shall show that the geometry of a linear crack is due to translational displacement (dislocations), which end in crack tips which are rotational displacements (disclinations).
Using the mathematical relations governing closure failure in parallel transport, Lardner (1973), we obtain the continuity and conservation laws of defect distributions in static continua, Edelen (1980), Davies (1988). This continuum distribution approach is applied to static crack geometries and predicts the effects of finite cracks in simple media and the important geometrical conditions that exist at the crack ends. The fundamental line defect in a continuum is a dipole consisting of disclinations of opposite signs separated and joined by the dislocation field. This model of cracks is applied to the two main types of sources, namely, laminar and spherical defect distributions. The far-field elastic radiation from these sources is obtained from the linearized conservation laws where only disclinations and dislocation gradients are shown to contribute.
The usual model of a compressional pulse on the inner wall of a cavity in an elastic medium is insufficient to explain certain strain and crack phenomena observed in explosions, Masse (1981). This is especially so in the case of high-frequency shear waves which are observed to be produced by and inherent to the explosive source, Kisslinger et al. (1961). The explosion cavity and its surrounding cracked zone, formed by the explosive point source, are shown to be a particular disclination and associated dislocation distribution. Such a self-consistent defect distribution is found to emit transverse shear waves as well as the usual and expected compressional waves. Thus, a particular defect distribution yields a specific crack geometry and governs the types of near and far field strains so produced. Non-linear conservation laws derived from the full Bianchi identities govern the elastoplastic zone next to the crack tip, where the self-interaction of the defect densities leads to bifurcation of the crack and the inhomogeneous defect distributions found in many cracked systems.
Defect Fields in 3-Dimensions
The basic tensor densities in a 3-dimensional continuum and their relationships are derived from the fundamental mathematical technique of computing the closure failure and change in the amplitude and direction of a vector transported around the defect, Lardner (1973). Let us introduce the following field definitions for a 3-dimensional (ijk) continuum.
(1) lkij = 1/2( δejk/δxi + δeik/δxj + δeij/δxk )
where e is the metric strain tensor, and the quantities lkij are the Christoffel symbols defined as above from the strain tensor, and where the connection Ljlm is defined as usual in terms of the Christoffel symbols. Defining:
(2) Tjlm = 1/2(Ljlm - Ljml)
where Tjlm is the torsion tensor associated with the connection Ljlm . On rotation of indices and combining these equations, we can obtain, Lardner (1973), to first order, the usual linearized relation between these tensors:
(3) Lkij = 2lkij + Tkij + Tijk - Tjki
Let C be a closed circuit in the body, starting and finishing at the point x, and construct a continuous local reference configuration (triad) for the points of C. If, upon completing the circuit, the orientation of the reference configuration of the triad N(x) is rotated from its original orientation then C encloses angular defects represented by the disclination density tensor. Also, the curve C in its local reference configuration will not in general be a closed curve. This closure failure is represented by the dislocation density tensor to leading order in the limit as C shrinks to zero. However, for general curves C, the closure failure may be caused both by dislocations and disclinations.
By using the curve C as an infinitesimal parallelogram, we can relate, Lardner (1974), through the definition of the Burger's vector, the torsion to the dislocation density, A , where the usual Einstein convention applies:
(4) Tpkm = 1/2 Apj εjkm
where ε is the usual alternating tensor. Thus, to first order, the closure failure can be considered to be caused by a dislocation density within the curve which manifests itself as torsion in the manifold.
Analogously, we examine the change in orientation of a vector around a closed path C. The rotation of this vector is obtained in terms of the components of the curvature tensor, Lardner (1974):
(5) Llnim = δLlnm/δxi - δLlni/δxm + Llji Ljnm - Lljm Ljni
In the weak field limit, assume that the tensor densities are small and keep only terms of first order in these quantities, so that the torsion tensor is also small and so are the connection coefficients. To first order, we may ignore the last two terms in equation (5) and so Llnim is antisymmetric between the pair of indices (ln) as well as (im). Thus it suffices to consider the related tensor, Lpq , where:
(6) Lpq = -1/4 εpln εqim Llnim
We note that for small strains this tensor is essentially equivalent to the density of disclinations. Combining these equations yields the fundamental conservation law:
(7) εpkr εqim δ2erm/δxkδxi = -Lpq + εqim δAmp/δxi -1/2 εpqi δAmm/δxi
This is the linearized First Bianchi identity that yields the metric strain gradient fields produced by both the disclination density and dislocation density gradient fields. One other continuity relation will be used in the following examination of these fields. The tensor of curvature Lpq obeys the conservation law, the Second Bianchi Identity:
(8) Lpq,q = 0
Geometry of Static Distributions
We now use the 3-dimensional conservation laws to extend the previous analysis of the geometric structure of self-consistent static cracks, Davies (1980). Intuitively, a dislocation can be crudely envisaged as a translational defect while a disclination can be considered as a rotational defect. From the definition of the dislocation density tensor Apj the displacement is in direction p while the line sense of the dislocation field is in direction j. The diagonal and off-diagonal components of Apj represent the screw and edge component of the dislocation density respectively. Similarly, the diagonal and off-diagonal components of the equivalent disclination density tensor Lpq represent the wedge and twist components respectively, where p is the direction of the rotation vector and q is the line sense, c.f. Figure 1.1.
The dislocation density tensor gradient, after operating on equation (7) with the alternating tensor, is:
(9) Apj,j = εpqr Lqr
This conservation law implies that dislocation gradients yield dislocations, i.e. curvature fields. Thus, dislocations can only end on disclinations, or conversely, if the disclination density is asymmetric, dislocations must emerge from it, deWit (1973). However, we also see from equation (9) that the direction of the rotation vector of the disclination density is orthogonal to the line sense and translational displacement of the dislocation density.
In a tensile crack, at each end a dislocation gradient exists and a disclination density will be produced at the crack tip. The disclination is composed of a rotational displacement, the axis of which is orthogonal to the crack body, a cross-section of such a 3-dimensional crack illustrated in Fig.1.2a. This same approach is immediately applicable to the antiplane shear crack in a three-dimensional continuum with the rotation vector at the ends of the crack being orthogonal to the line sense and translational displacement. Once again equation (9) governs this asymmetric dislocation field with Fig. 1.2a applicable to the different directions.
Now, the symmetric static disclination field is governed by the symmetric component of equation (7), which implies that when gradients of dislocation density exist they can also yield an associated incompatibility of strain, together with symmetric disclination components. This situation is realized, for example, in long lenticular cracks with dislocation gradients orthogonal to the line sense 'i', the disclinations so produced being the i'th on-diagonal component.
Probably the most interesting geometric result of this self-consistent approach is the case of a static shear crack with screw dislocations along the line. Here, we have dislocations with translational displacement and line sense in the same direction. For a dislocation gradient in the i’th direction, equation (7) shows that this produces an extra dislocation as well as the disclination Lpq at the ends of the crack where the gradient is greatest. Here p and q directions are orthogonal to each other as well as to the direction of the dislocation gradient (i) thus yielding a disclination at the end, with the whole system manifesting as an echelon crack pattern. This is sketched in Fig. 1.2b in cross-section. At each end, the conservation of curvature shows that on either side of the tip a disclination will be formed of opposite sign. Depending on the sign of the disclination, at a free surface these end disclinations will produce either an apparent 'pulled apart' zone or the opposite being a 'local upwarped' zone. When more than one such crack is close but discontinuous to another, these effects manifest themselves on the surface of the Earth as topographic basins or domes, Segal and Pollard (1980).
The other important linear Bianchi Identity is the conservation law of disclination density equation (8) which is readily understood through the observation that a linear crack has equal and opposite tips. Alternatively, this conservation law can be interpreted as an argument for closed loops of disclinations in a three-dimensional continuum. This can be seen in a lenticular-shaped crack in such a continuum where the crack tip forms a closed circular boundary around the body of the crack.
Integral expressions for the strain, stress, dilatation and rotation potentials, have been derived in terms of the static dislocation and disclination density tensors convolved with the Green's tensor and its derivatives, deWit (1973). These Green's integral expressions can be used in the modeling of various sources by their appropriate self-consistent defect distributions. The primary interest is in comparing compressional and shear wave radiation from various sources. Thus, let us examine the deviatoric strain, dilatation and rotation potentials due to dislocation and disclination sources in a homogeneous, isotropic infinite medium. These are based on partial integrations, which assume, essentially, no cracks at infinity.
From equation (9) and the Green’s integrals, deWit (1973), it can be shown that only anti-symmetric dislocation and on-diagonal disclination components contribute to the dilatation. Similarly, in the case of the rotation term, contributions are made both by dislocation components and by general disclination components.
Incorporating the gradients of the Green's function, at a distance R, it can be shown that the dislocation components are multiplied by terms of order R-3 or worse, while the disclination components are only multiplied by terms of order R-1 or worse. Thus the dilatation and deviatoric strains have far-field components due solely to the disclinations while the dislocations contribute only to the near-fields. This agrees with the argument, Nabarro (1967), that the sources of long-range strains are the disclinations. However, the dislocation gradients can also be a producer of far-field strains. A similar conclusion can be made for the rotation potentials.
Most natural sources are laminar and much thinner in one direction than in the others. Consider the simple crack with only the component Axy , i.e. displacement in x direction and line sense in the y direction. Gradients in the y direction produce, via equation (9), only the antisymmetric disclination density terms Lyz . However, a dislocation gradient in the z direction produces, from equation (7), an on-diagonal disclination component Lyy and an associated incompatibility of strain. For such cracks, it can be shown that the far-field dilatation and deviatoric strain are produced only by this on-diagonal disclination component and its associated dislocation gradient. However, the rotational potentials in the far-field will, in general, be due to both on- and off-diagonal disclinations. A similar result holds for other off-diagonal dislocation density distributions.
In the other case of the shear crack, Ayy , where the screw dislocation displacement and line sense are in the same direction, equation (7) yields for a gradient in the i direction an antisymmetric disclination orthogonal to this direction together with associated dislocation gradients. If the line sense is also in the direction of the gradient, the resultant antisymmetrical disclination will not yield any net far-field dilatation and deviatoric strain. However, when the I direction is orthogonal to the line sense of this strike-slip crack, the resultant symmetric disclination will produce dilatation and strain contributions in the far-field. Again, the rotational potentials can expect contributions from both on- and off-diagonal disclinations. For all these natural sources the far-field stresses and strains are controlled by the dislocation gradients and associated disclinations.
The artificial defect distribution of interest is the explosive or cavity-forming device in natural media. Fig. 1.3 depicts the usual idealized model of such a crack system consisting of radial zones. The innermost zone consists of a cavity, formed at highest pressures, outside of which is a hydrodynamic zone where the material behaves as a fluid, then a crushed solid zone where the particles are broken into a sand or powder, then a fractured zone ranging from a high density of interacting cracks, then to a radial cracked region where linear crack theory is assumable and, finally, into the elastic zone where strains are reversible. The idealized model in Fig. 1.3 is not to scale and the cracked zone is approximately 10 to 20 times larger in radius than the inner zones and it is this cracked zone, which our linearized defect distributions are able to model.
The expansion of a spherical cavity in a plastic-elastic medium has been analyzed using plasticity theory, Hill (1950). When such a cavity is expanded from zero radius in an infinite medium, the stresses are found to be a function of the ratio (r/a) only while the ratio (c/a) remains constant; here a is the radius of the inner cavity and c is the radius where the plastic and elastic regions separate. It was found that (c/a) is approximately 0.7(E/Y)1/3 where E is Young's modulus and Y is the yield strength of the medium. For most rock materials, this is of order 100 giving an approximate (c/a) of 3, which closely agrees with the field observations of cracked zones around explosions.
Thus our model consists of an effective cavity, due to explosive expansion, surrounded by a cracked zone. A hole is a disclination of unit strength, Nabarro (1970), so that formation of a hole is equivalent to the introduction of a disclination density into the material. The inner spherical cavity can be considered as a symmetrical disclination density with the associated defect gradients corresponding to the surrounding radial cracking, which will end in crack tips that are the equivalent and opposite disclinations to the central source. In spherical coordinates, the dislocation distribution is described by Aθψ as, for the simple spherically symmetric cracking, only radial gradients exist. This implies that components of disclination forming the cavity are Lθθ and Lψψ. In Cartesian coordinates, this will convert to off-diagonal dislocation density and symmetrical disclination density components.
Depending on the particular material properties, the disclination density will vary as a function of radius, which will thereby affect the strains and their properties. Using the Green's functions, it can be shown that the disclination density produces dilatation, which has a far-field term, and a deviatoric strain, which has both near and far-field components. As there are both dislocation gradients and disclinations produced, far-field rotational potentials can be expected. Thus shear waves can be expected in the far-field of both SH and Love type as well as the usual P and SV waveforms. In a series of controlled experiments, it was observed that SH and Love shear wave patterns are produced, Kisslinger et al. (1961), from small dynamite explosions in soil and that the SH wave motion was due to the radial fracturing formed by the explosion.
Non-linear Defect Distributions
A mathematical theory of defect dynamics and kinematics has been developed for a linearly elastic homogeneous continuum. The static relations for the strain potentials have been extended to the dynamic case with introduction of the concept of the plastic velocity. This approach has a foundation in the defect conservation laws, which have been derived through a linearization assumption that the defect and strain fields are sufficiently small. This approximation is valid in regions where linear elasticity governs the stress-strain relation. When the field and defect densities are larger, this linearization is not valid and elasto-plasticity is the governing constitutive relation. Especially near the crack tips, i.e. the disclinations, the defect densities and strains can be so large that non-linear effects must be accounted for.
The non-linear conservation laws, known as the full Bianchi Identities, govern torsion and curvature and the metric deformation, Davies (1988), Schouten (1954). Their main additional characteristics are that quadratic terms in dislocation and disclination densities are present. Thus the laws depend on the product of the defect densities so that self-interaction of the defects can lead to non-linear behavior. This manifests in such phenomena as bifurcation of the crack tip in areas of intense field strengths, e.g. closer to the explosion center and inward of the linear cracking zone in Figure 2. Also, in this same non-linear region we get inhomogeneous distributions of defects with large crack-free zones between the bifurcated cracks.
The non-linear Bianchi Identities possess solutions of non-linear character due to the presence of Klein-Gordon operators, Edelen and Lagoudas (1988). Similar non-linear behavior for defects is predicted from examining their physics at the atomic level. Dislocations in crystals, subject to sinusoidal atomic potentials, obey the non-linear Sine-Gordon wave equation. Solutions manifest behaviour, Whitham (1974), such as wave superposition and splitting (bifurcation). This is related to the amplitude dependence of the applicable dispersion relation. Other interesting phenomena include the production of solitons of constant shape and energy representing kinks and antikinks. Other solitary waves include ‘Breathers’ where temporal oscillations, usually periodic, are experienced in a localised region for the co-moving coordinate system.
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The 3 fundamental types of dislocations and disclinations are depicted.
Illustrates cross-sections of 3D cracks with the location and type of dislocation and disclination. Figure 1.2a depicts the off-diagonal dislocation density where the displacement vector and line sense are orthogonal. Figure 1.2b depicts the on-diagonal dislocation density where displacement and line sense are parallel. The solid-line disclination is due to a defect (absence) while the dashed-line disclination is due to an infect (excess).
Illustrates (not to scale) the usual model of an explosion-produced cavity and associated cracking.