We know that we can transform the traditional Mohr-Coulomb system to the p-q system by using the equations
Stated formally, this means that, for every set of principal stresses, there is a unique pair of p and q values.
But did you know you can go the other way, if you need to? Let’s start by putting these equations into matrix format, which yields
Inverting the matrix and premultiplying the right hand side yields
The inversion is the key step. The fact that the matrix is invertible, square and of the same rank as the vectors means that the transformation is linear, one-to-one and onto. We can also say that, for every set of p and q values, there is a unique set of principal stresses.
Explosions may cause a whole series of rapid mechanical processes in soils: appearance of an explosion gas chamber within a rather short interval of time (sometimes a few thousandths of a second), which exerts an enormous pressure (of the order of a few hundred thousand atmospheres), causes the formation and propagation of explosion waves which change the stressed state of a soil mass and cause its particles to move with velocities varying from a few thousand metres per second to zero.
Explosion impulses are characterized by the maximum pressure the rise time during which this pressure is formed, the fall time during which the pressure drops from the maximum to zero, and the total time of explosion action .
As seen from experiments of Prof. G. M. Lyakhov*, the gas chambers formed in soil through explosion of deep concentrated charges of explosives are almost spherical in shape. With time, a gas chamber (the cavity in soil) is destroyed, but the time period of its destruction may be very different , from a few minutes (in sands) to several months (in dense clays).
As has been shown by the experiments, the radius of an explosion gas chamber , after it has been formed completely, is determined by the following relationship:
weight of explosive charge, kg
proportionality factor depending on the properties of the soil
According to G. M. Lyakhov, numerical values of this factor are:
for saturated sands
for loams (according to G. I. Pokrovsky)
for loess soils
for clayey soils
Explosion of a concentrated charge in a soil results in the formation of normal (radial) pressures , lateral (tangential) pressures , and the motion of particles with a velocity .
For non-saturated soils and rocks, all these three parameters are determined in calculations as functions of time, i.e.,
For saturated soils and liquid media, it is sufficient to investigate only two of these parameters, for instance,
The parameters of stress waves in soils caused by explosions and the parameters of velocities of their propagation are determined by special field tests. Using the results of such tests, empirical formulae are established for determination of the design parameters of explosion, waves in soils depending on the weight of charge, the distance from explosion centre, etc.
* Lyakhov G. M. Osnovy dinamiki vzryva v gruntakh i zhidkikh sredakh (Fundamentals of Dynamics of Explosion in Soils and Liquid Media), Moscow, Nedra Publishers, 1964.
Although geotechnical engineering has always had a military application (witness the prominence of documents such as NAVFAC DM 7 and the many others offered on this site,) this is the only elementary level soil mechanics text where I can recall seeing such a presentation.
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