Re-examination of the shock wave’s peak pressure attenuation in soils D.Z. Yankelevsky * , Y.S. Karinski, V.R. Feldgun National Building Research Institute, Faculty of Civil & Environmental Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel article info Article history: Received 8 February 2009 Received in revised form 28 March 2011 Accepted 27 May 2011 Available online 22 July 2011 Keywords: Explosion in soil Shock wave attenuation Peak pressure Theoretical simulation Empirical formulas 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 attenua- tion 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 obser- vations 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. Ó 2011 Elsevier Ltd. All rights reserved. 1. 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 [1e5]. The shock wave propagation is accompa- nied 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 [8e10]: pðRÞ¼ A,f ,ðrCÞ, 0 B B @ R W 1 3 1 C C A k (1) where: A - a constant; f - a coupling factor depending on the scaled depth of burial (d/W 1/3 ) of the explosive; r - the undisturbed soil’s density; C - the seismic velocity of sound; r,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]: pðRÞ¼ p 0 ,R k (2) 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. * Corresponding author. National Building Research Institute, Technion City, Haifa 32000, Israel. Tel.: þ972 4 8292242; fax: þ972 4 8324534. E-mail address: davidyri@technion.ac.il (D.Z. Yankelevsky). Contents lists available at ScienceDirect International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng 0734-743X/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijimpeng.2011.05.011 International Journal of Impact Engineering 38 (2011) 864e881