Resolving the Issue of Jean-Louis Briaud’s “Pet Peeve” (or at least clarifying the problem)

Three years ago I posted Jean-Louis Briaud’s “Pet Peeve” on the Analysis of Consolidation Settlement Results. Since that time he has been elected President of the American Society of Civil Engineers and I am in the process of retiring from full-time teaching, so our trajectories are a little different. (He’ll catch up, don’t worry.)

Nevertheless his Presidency would go unfinished if some explanation of the pet peeve wasn’t given. To remind my readers it is as follows:

The consolidation e versus log p’ curve is a stress-strain curve. Typically, stress-strain curves are plotted as stress on the vertical axis and strain on the horizontal axis. Both axes are on normal scales, not log scales. It’s my view that consolidation curves should be plotted in a similar fashion: effective vertical stresses on the vertical axis in arithmetic scale, and normal strain on the horizontal axis in arithmetic scale. When doing so, the steel ring confining the test specimen influences the the measurements and skews the stiffness data. Indeed the stress-strain curve, which usually has a downward curvature, has an upward curvature in such a plot.

This post won’t be very rigourous or mathematically detailed, but more of a qualitative statement of the problem. Perhaps a proper solution will solve this dilemma; I think it certainly needs it.

To start, let’s pick up where we left off, with the E vs. $\epsilon$ plot below:

It was noted at the time that the apparent elastic modulus increased more or less (that’s about as good as it gets with most geotechnical phenomena) linearly with strain.

From this, it can be noted that the shear modulus can be estimated for a soil (excluding strain-softening effects) as follows:

$\frac{G_{0}}{p_{atm}}=SF\left(e\right)\left(\frac{\sigma_{0}}{p_{atm}}\right)^{\bar{n}}$

where the notation is shown in the source. Let’s make some assumptions:

Poisson’s ratio remains constant, thus the relationship between strain and elastic modulus is constant.
$S$ is constant for a given soil type.
Effective stress $\sigma_o$ for a given sample (that’s just about a given for triaxial tests in any event)
Other constants, such as $p_{atm}$ and $\bar{n}$ also remain constant.

That leaves the variable $F(e)$ to change. The shear (and by extension the elastic) modulus of a material is a function of the void ratio. From the same source,

$F\left(e\right)=\left(1+e\right)^{-3}$

We can convert this to strain by noting the following relationship, which is written so that compressive strain is positive:

$\epsilon={\frac {{\it e_0}-{\it e_1}}{1+{\it e_0}}}$

Substituting that into the equation before it yields

$F(e) = -{\frac {1}{\left (1+{\it e0}\right )^{3}\left (-1+\epsilon\right )^{3}}}$

Let’s consider the case of $e_0 = 1$ . Substituting that into the previous equation and plotting it yields the following result:

It’s not perfect, but it’s close to a linear relationship, at least in the strains under consideration. And, of course, it shows an increasing shear modulus with increasing strain (or decreasing void ratio.)

Verruijt makes two important observations that should be noted. The first is his commentary on the image below, in the caption.

When performing the test, it is observed, as expected, that the increase of vertical stress caused by a loading from say 10 kPa to 20 kPa leads to a larger deformation than a loading from 20 kPa to 30 kPa. The sample becomes gradually stiffer, when the load increases. Often it is observed that an increase from 20 kPa to 40 kPa leads to the same incremental deformation as an increase from 10 kPa to 20 kPa. And increasing the load from 40 kPa to 80 kPa gives the same additional deformation. Each doubling of the load has about the same effect. This suggests to plot the data on a semi-logarithmic scale. In this figure log(σ/σ0 ) has been plotted against ε, where σ0 denotes the initial stress. The test results appear to form a straight line, approximately, on this scale. The logarithmic relation between vertical stress and strain has been found first by Terzaghi, around 1930.

The second is his comment on the use of strain vs. void ratio:

It is of course unfortunate that different coefficients are being used to describe the same phenomenon. This can only be explained by the historical developments in different parts of the world. It is especially inconvenient that in both formulas the constant is denoted by the character C, but in one form it appears in the numerator, and in the other one in the denominator.

The need to treat compression due to settlement completely differently than that of elastic (or elasto-plastic) settlement is one of the anomalies of geotechnical engineering. The observation that the elastic modulus decreases with void ratio (or increases with strain) is a start in putting the two together and presenting a more or less unified theory of soil deformation. Coupled with agreement on using strain in consolidation tests, this would bring us a long way to solving the dilemma of Jean-Louis Briaud’s–and some of the rest of our–pet peeves.