Experimental and Numerical Studies of Transient Shock Wave Propagation in a Geomedium

Abstract

This paper presents the results of an experimental and numerical investigation of the shock wave energy transmission and reflection through boundary layers (joints, fractures) or between inhomogeneous geomedium with different physical properties. The conclusions from these studies significantly clarify the dynamic failure mechanics due to shock wave energy transmission and reflection through different rock materials and their jointed interfaces. Thorough studies were performed in order to document the transient, two-dimensional deformation characteristics and the normal and shear stress fields for inhomogeneous geomedium (focusing on the influence of the internal geometrical shapes, relative density variations and packing of the grain boundaries).

Introduction

A distinctly different set of concepts must be considered when discussing rock breakage dynamic loading than when considering rock breakage under static loads (Kolsky and Shearman, 1949; Schardin, 1950; Rinehart and Pearson, 1954; Persson, Holmberg, Lee, 1994; Petr, 1996 and 2001). Dynamic loads generate transient stress disturbances that move through a geomedium and create highly localized inhomogeneities within a geomedium. The fracture patterns produced by dynamic loads in geomedium are exceptionally predictable, e.g. spalling, crushing, radial fractures, corner fractures. Numerous techniques have been developed by the following: Rinehart, 1962; Duvall and Atchison, 1957; Johnson, 1963. It is becoming increasingly clear through these studies that much of the fragmentation in brittle geomedium comes about as dynamic failure of tensile fracturing due to pre-existing weak planes in the geomedium. These may consist of partially open crack planes, bedding planes in sedimentary rocks or filled joints, which have a tensile strength lower than that of the surrounding solid rock material.

It is of the utmost importance, for these reasons, to develop a theoretical model validated with experimental data that provides a better understanding of how the shock wave energy, due to dynamic loading, is transmitted and reflected through inhomogeneous geomedium and through boundary layers (joints, fractures) between inhomogeneous geomedium with different physical properties. Generally speaking, a great number of geomedium can be treated as isotropic inhomogeneous, although they are not microscopic isotropic. The individual grains exhibit crystalline anisotropy and inhomogeneous characteristics. When they form a polycrystalline aggregate and are randomly oriented, the material is macroscopically isotropic (the elastic modulus is the same in all directions).

This experimental study provided data points for tensile strength and acoustic impedance of general model rock sample and information on deviatoric strain energy density per unit transmitted through jointed individual grains. The acoustic impedance is represented as a material resistance to the propagation of a transient shock wave within the geomedium (rock sample). The geomedium is represented by a model set of grains and exhibits crystalline anisotropy and inhomogeneous characteristics. Quantitative analyses of high speed imaging of the photoelastic model were employed to investigate stress wave propagation generated by a dynamic load.

Experimental photoelastic models were fabricated from similar and dissimilar physical bodies with frictional contacts. The experimental models were subjected to dynamic loading through a specifically designed attachment, which created plane shock waves in the specimen. A high-speed camera observed the behavior of the laboratory test model during the shock loading with a speed of one million frames per second. A numerical technique using nonlinear, multi-body, transient dynamic finite element (FE) simulation was employed to model the time history response during and after contact of the impactor and any subsequent body contact. The numerical data was then compared with experimental results.

Experimental Procedure

The photoelasticity method was used to obtain full-field stress information in the geomedium. Specimens were loaded dynamically using a plane-wave generator and photographs were taken of the resulting fringe patterns at various times. The design of the plane-wave generator developed for the experiment is shown in Figure 1. The experimental arrangement was loaded with a small amount of explosives, 0.3-7 grams, in a specially designed charge holder. The specimen was furnished with an electric bridge-wire exploding detonator. The electric bridge-wire exploding detonator was fixed to the surface of the plane-wave generator. The plane-wave generator is fabricated from a polycarbonate cone (Hydex 4301) with a 15-degree apex angle. This cone base was acoustically coupled over the top of the specimen with its axis normal to the surface. To improve the synchronization of the camera initiation to the event, the detonator was detonated with a 7 x 10^-6 second delay.

Dynamic fringe patterns were recorded by employing a SMD-64k1M camera operating at one million frames per second using a 256 x 256 array with the ability to store 16 frames. For each experimental setup, a total of 16 images showing the fringe patterns associated with propagation of the stress waves induced by the load source were recorded.

Numerical Analysis

A combined technique, experimental and numerical, was developed to obtain the complete stress field distribution and the normal loads in geomedium simulated by cemented circular disks. In order to understand the shock wave energy transmission within geomedium, it is essential to correctly identify and analyze the state of stress involved. By definition, the state of stress generated by a planar, normal shock wave (particle velocity is perpendicular to the shock front) is a uniaxial strain state (Wasley, 1973). This state of stress should be decomposed into a deviator and hydrostatic component. The uniaxial strain state can be represented as:

(1.1)

 

 

This strain can be decomposed into its hydrostatic and its deviatoric components as follows:

(1.2)

 

 

The corresponding stresses (using Eq. 2.75 and 2.76 from Dieter, 1976) for an isotropic material are:

(1.3)

 

 

It should be emphasized that Eqs. 1.1-1.3 assume elastic behavior of the material. It is assumed here that the material is responding elastically to the passage of the wave. The pressure is the hydrostatic component:

(1.4)

 

dP is an elastic hydrostatic stress. The original dimensions of the specimen would be recovered, once dP was released.

The second term of importance in Equation 1.3 is the deviatoric stress. The maximum shear can be found from the deviatoric stresses and occurs at 45° to the shock front.

(1.5)

 

Substituting Equation 1.4 into Equation 1.5:

(1.6)

 

Integration yields:

(1.7)

 

G, the modulus of rigidity, and K, the bulk modulus, can also be used to express equation 1.7 where :

(1.8)

 

From G and K, it is possible to determine the maximum shear stress. Using for granite, we have:

Thus a pressure of 20 GPa, common for such experiments, will give shear stresses on the order of 9 GPa. Such stresses are required for creating plastic deformation or fracture.

Energy dissipation is of the form of strain, micro and macro friction and fracture (crack) surface creation (de-bonding). It is recognized that there are two classes of important parameters when understanding the effects of shock waves on the structure and properties of geomedium: (1) shock-wave parameters: pressure, pulse duration, rarefaction rate, attenuation rate, arrangement and normality of wave; and (2) physical material parameters: grain size, grain boundary and substructure changes (such as point-deformation, dislocation, twins and grain orientation, flows and fractures).

Coupling these equations to the experimental data was accomplished through the use of the governing stress-optic law. The resulting equation is:

(1.9)

 

where is the photoelastic material stress optic coefficient, N is the fringe order and h is the photoelastic material thickness. The maximum shear stress, , can be computed from the photoelastic data. A number of data points obtained from the photoelastic image gave a shock stress wave, which was used to compute the appropriate shock wave velocity.

Review of Deviatoric Strain Energy Density Distribution from Computer Modeling

The deviatoric strain energy density is expended by the action of external forces in deforming an elastic geomedium. All work performed during elastic deformation is stored as elastic energy and this energy is recovered on the release of the applied forces. Energy is equal to a force multiplied by the distance over which it acts. In the deformation of an elastic geomedium, the force and deformation increased linearly from initial values of zero so that the average energy is equal to one-half of their product. The ideal impact load would be one in which all energy of impact is absorbed by the target. Actually this ideal is never realized, since some of the impact energy is always lost through a) External aspects: friction, volumetric deformation, heat, sound and b) Internal aspects: grain deformation, friction between the grain and opening pores, flaws or micro fractures within the medium.

In the real test situation, it was impossible to obtain a truly accurate measure of the strain energy absorbed by a specimen. The definition of critical deviatoric strain energy density (SED) is the maximum deviatoric strain energy density that is capable of withstanding fracture. and determines the deviatoric strain energy density at uniaxial loading.

Thus at fracturing, the deviatoric strain energy density for brittle materials can be calculated as follows:

(1.10)

 

Where is Poisson’s ratio and E is the modulus of elasticity of the material. For fracturing (yielding), a general state of stress is equal to the value defined by the von Mises fracture criterion for a material under uniaxial state of stress. Therefore, the deviatoric strain energy density is given by:

(1.11)

 

Results and Discussion

The results of the current experiments are presented in the following manner. Initially, the evaluation between selected experimental and selected numerical results will be covered. This will be followed by calculations of deviatoric strain energy density for different boundary conditions. Concluding with the presentation of data, interpretation is presented. All of the systems shown in Figure 2 have the ability to show the front shock wave propagation within experimental tests and the FE models. An experienced observer will notice that the relatively plane shock wave propagates through the homogeneous material in frames 5 through 11, and the reflected waves from frames 12 through 16. Unfortunately, the combination of a very sharp fringe gradient, high fringe velocity and minimum exposure time yielded poorly resolved fringe patterns for the first few frames.

The first system is the elastic bar model (a). When shock waves impinge against a cross-section boundary between two different materials they have a tendency for the incident shock wave to be partially reflected, refracted and transmitted with different velocities as shown in model (b). Similarly, the BCC model is loaded by dynamic impact (c). The material has the ability to produce two shock fronts. The investigation describes the behavior when the shock wave propagates through a denser material and the second part is when it propagates through less dense material.

Each system is represented by four images ((a) through (d)). The selection was made to represent all three different models comparing the experimental test results with the numerical simulations. A numerical technique using nonlinear, multi-body, transient dynamic finite element (FE) simulation was employed in this section. Comparison is provided between the results of experimental high-speed images and extracted numerical data.

The FE technique provides a good match with experimental data as shown in Figure 3. Unfortunately, the experimental data did not provide consistent shock wave velocities. The clarity of the image observation was distorted by the reflection and different optical index of the glass plates and the photoelastic material, as well as, the glue between it. This resulted in an image that was extremely difficult to determine the exact position of the front of the shock wave position at different times. This is why the calculated shock wave velocity shows significant discrepancy from 4445 m/s to 4843 m/s as can be seen in Figure 3. The computer simulation, which can be seen on the right side of Figure 3, shows a consistent shock wave front velocity of 3876 m/s. In the numerical simulation, the velocity was determined by using a specific monitoring feature of the program. Applying the monitor, we can study the velocity at two nodes, separated by a known distance. It is from this information that the velocity of the shock wave front was calculated. We note that in this simulation, the nodal velocity is equal to the particle wave velocity.

In Figure 4, the shock wave velocity observed during experimentation was compared with the FE simulation for a specimen made of two different materials. The same difficulties in determining a velocity were experienced. The computer model consistently shows 5 to 20 percent discrepancy in the shock wave front velocities. It is interesting to note that the computer modeling consistently resulted in a lower shock wave velocity than the experimental observation.

In Figure 5, the comparison between the experimental observation and the numerical simulation can be seen for a Body-Centered Cubic Arrangement (BCC). This comparison is somewhat better than in the elastic bar presented in Figure 3. Interestingly, the computer code resulted in a consistent prediction of higher shock wave velocities.

The inspection and the drawing of a qualitative conclusion from the photoelastic images required a very experienced observer. The experimental study and setup requires additional development in order to make a more reliable comparison between the experimental data and computer model. It was interesting to note that the fringe patterns around the contact points in the BCC model were symmetric on either sides of the contact points and were similar to the fringes obtained with the numerical simulation. Both the numerical and experimental method indicated that the wave propagates slower around the contact zone than in the cemented discs, as was expected.

For summary in Table 1, the von Mises stresses and the deviatoric strain energy densities are provided for the different models. Figure 6 represents a comparison of an approximation of deviatoric strain energy density and shock wave in a solid medium with a Poisson’s ratio of 0.35.

These results indicate the primary role that a local microstructure plays in determining the wave propagation behavior. Both inclusions and voids produce local wave velocity scattering through various reflection mechanisms and the results seem to indicate that deviatoric strain energy density produces more local attenuation. Table 1 shows the transmission results of the shock wave propagation within simulation for different models.

It is apparent that the vertical z-component of load transmission is predominant for these samples. Although it is not evident from the results, the wave speed, as determined by the arrival time of these average transmission profiles, is different for the two propagation directions, with a vertical wave speed approximately three times that of the horizontal motion. These results correlate with physical properties and that physical property is related to the transmission of waves in different mediums.

Similar results occur for moderately anisotropic models. However, in these cases, the total energy transmitted through the medium due to dynamic impact is highly influenced by boundaries and the interfaces between two dissimilar material properties.

When waves propagate through a geomedium, the local micro dynamic loads are transferred through chains of grains linked through contact. Between these chains, grains may carry little or no load. It would be expected that such load-carrying paths, which are relatively stronger, would act as a better propagator of wave energy, than would highly weaker grains. Thus, it appears that the local microstructure of load carrying paths within a geomedium are also frequency- or wavelength-dependent.

Acknowledgment

This work was supported by the Mining Engineering Department at Colorado School of Mines. The authors would like to express our thanks to Erick Howard from OPSCI for providing the high-speed camera for this research project. The authors also wish to give a very special thank you to Frank W. Davies from K-tech Corp., Larry Brown, and Bob Lynch from Applied Research Associates, Inc. for their constructive discussion and assistance during this research.

References

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