Re-examination of the shock wave’s peak pressure attenuation in soils

Abstract

The paper investigates the problem of a charge exploding in soil and focuses on the characteristics of the shock wave’s peak pressure attenuation. Analysis of existing empirical data observes different attenuation factors for apparently similar certain types of soils whereas for other types of soils there is no significant difference. It was also observed that prediction of the shock wave’s peak pressure with existing power law empirical formulas yields a large discrepancy in comparison to test data. The discrepancy is significant even in case where the specific tested soil parameters are used. These observations among others motivated this study. The power law relationship has been investigated through numerical simulations of the shock wave propagation in different soils. The soil is modeled as a bulk irreversible compressible elastic plastic medium, including full bulk locking and dependence of the current deviatoric yield stress on the pressure. The Lagrange approach and the modified variational difference methods are used to simulate the process. The study shows that the shock wave’s peak pressure attenuation for certain types of soils may be well presented by a power law with a constant exponent, whereas other types of soils may be presented by a power law for a limited distance range and their behavior for a wide distance range is poorly described by a linear relationship on a logarithmic scale but is well represented by a bi-linear or a tri-linear realtionship. These findings explain some of the above mentioned observations.

Introduction

The problem of an underground explosion in soil is of great complexity and the prediction of its shock wave front parameters is of much interest [1], [2], [3], [4], [5]. The shock wave propagation is accompanied by rather large soil deformations with irreversible bulk compaction [6], [7] and it is strongly attenuated as a result. Some aspects of the shock wave’s attenuation in soils are still obscure and require further research. Therefore, the prediction of the shock wave parameters is commonly relied upon empirical expressions. These expressions are commonly based on fitting of test data to empirical expressions.

The common expression that describes the magnitude of the peak pressure at the shock front as function of the distance is the following power law [8], [9], [10]:

$$p\left(R\right)=A\mathrm{\cdot }f\mathrm{\cdot }\left(\rho C\right)\mathrm{\cdot }{\left(\frac{R}{W^{\frac{1}{3}}}\right)}^{-k}$$

where:

A - a constant;

f - a coupling factor depending on the scaled depth of burial (d/W^1/3) of the explosive;

ρ - the undisturbed soil’s density;

C - the seismic velocity of sound;

$\rho \mathrm{\cdot }C$ - the acoustic impedance of the soil medium;

R - the distance measured from the charge centre;

W - the explosive weight;

k - a constant attenuation factor.

Sometimes a similar expression to Eq. (1) is used without explicit expression of the acoustic impedance [4], [32], [38]:

$$p\left(R\right)=p_0\mathrm{\cdot }{R}^{-k}$$

The same general expression of this power laws appears in different references [11], [12], in different unit systems, and refers to different shapes of the explosive material but mostly to spherical charges. All the references generally describe the type of soil, and provide the recommended values for the attenuation factor that characterizes this type of soil. Calculation of the resulting pressure is then straightforward.

A common graphical description of the power law is a straight line on a logarithmic scale, where the slope equals to the attenuation factor.

A well-known reference that is commonly used for evaluation of the peak pressure is TM5-855-1 [8], or the programmed version CONWEP [13]. Sometimes caution is required in performing the calculations, when a mixed unit system is used [8], [12] (e.g. p(R) [psi]; $\rho \mathrm{\cdot }C$ [psi/fps]; R [ft]; W [lb]). Other references use metric unit systems as well [14], [15], [16], [17], [18], [19], [20]. Table 1 summarizes the parameters for several types of soils described in [8], [12]. At a first glance Eq. (1) and the accompanied Table 1 seems to enable the peak pressure prediction at distance R in a given type of soil. A closer examination of the table raises some questions with regard to the clarity, precision and uniqueness of the definitions of the types of soils under consideration, in light of the absence of important information such as the specific composition of the soil, the degree of saturation and the size of particles and their distribution.

Nevertheless, if a certain definition of a soil type is sufficiently close to a given soil under consideration, the pressure variation with distance may be calculated.

Extending the review to other references shows that different references provide rather similar attenuation factors for sandy soils, however some of them provide somewhat different attenuation factors with regard to a spherical charge explosion in saturated sandy clays, as shown in Table 2. We realize that reference [21] mentions an attenuation factor k = 1.5 for a spherical explosion in saturated sandy clays whereas another reference [8] recommends an attenuation factor of k = 2.5 for the same soil definition. These two suggested attenuation factors for the same type of soil differ considerably.

Two major questions then arise on that regard:

• Which is the more appropriate attenuation factor for that type of soil? Is it k = 2.5 or whether is it k = 1.5?

• Why different references recommend similar attenuation factors for sands while there appear different attenuation factors for saturated sandy clays?

Definitely, these questions evoke some thoughts.

This confusing data for the same type of soil appears once again for another type of soil. The literature survey identifies a similar observation for spherical explosions in loamy soils (Table 3). Most references [4], [8], [22] recommend an attenuation factor of k = 2.7–2.8 for a spherical explosion in this type of soil, whereas another reference [19] recommends k = 1.6.

This observation is similar to the one above regarding the saturated sandy clay hence it raises up similar questions as above.

All the above specified attenuation factors refer to the explosion of a spherical TNT charge in soil. Only limited data is available for cylindrical charges. Barlas [14] suggests an attenuation factor of k = 2.4 for “sandy loam” and Luchko [15] suggests k = 2.17 for “loamy soils” when the cylindrical charge is vertical.

As expected, the attenuation factors in the case of a cylindrical charge (k = 2.17–2.4) are somewhat smaller than the values for a spherical charge (k = 2.7–2.81) in the same type of loamy soil.