Granular Materials in Civil Engineering: Recent Advances in the Physics of Their Mechanical Behavior and Applications to Engineering Works - Multiscale Geomechanics: From Soil to Engineering Projects - page 54

This basic description of grain breakage raises several questions:
− What is the relation between the microcrack size and the grain size?

− What is the relation between local stresses within the grains and the
macroscopic stresses applied on the granular assembly?
− Are there relations with the grain size distribution?

The answers to these questions, often found through simplifying assumptions,
have led to the description of the macroscopic effects of grain breakage.

3.2.1.2. Statistical representations
The influence of grain size was investigated long ago [MAR 72], the results
showing that the average crushing force, F cr , of gravels or rock fragments is a power
function of the average grain diameter (see Figure 3.9c):

λ

F = η . d

[3.22]

cr

This expression can be connected to Weibull's theory, which gives the
probability of survival within a population of brittle objects subjected to stress
condition near failure:

m



[3.23]

V σ

σ

PV () exp

s

= −

 

V

0 0

As the volume, V , of a grain is proportional to the cube of its diameter, Weibull's
approach for a given value of the probability of survival, P s , also leads to an average
crushing force proportional to a power function of the grain diameter:

-3/m

σ ∝ d

[3.24]

cr

The average crushing force being proportional to the crushing stress multiplied
by the average grain section, the comparison of the two approaches gives a simple
means to fit a Weibull distribution for a given material from a set of crushing tests
on grains of different sizes:

3 3

[3.25]

λ = 2 - o r m =

m 2 − λ