Understanding Non-Linearity in Barton's Criteria for Rock Mechanics

[2685866]

Homework Statement

I have been given the task of approximating Barton's Criteria of failure with rock mechanics to a linear function or proving that it cannot be done. Through reading some of Barton's papers - some which are 40 years old I have come to understand that the empirical relationship is not linear, but do not know why/how.

All variables are independent and can be used with positive rational numbers.

Any and all assistance is appreciated.

Homework Equations

\tau = \sigma r * \tan(\phi c + JRC * \log (JCS/\sigma r))

The Attempt at a Solution

I spent some time considering disproving this through the subspace theorem considering I know that all variables are independent and that obviously it has to be within the rational numbers.

 

[HallsofIvy]

First, while this function is certainly not linear, it is differentiable and any differentiable function can be approximated, locally, by a linear function.
It is not clear to me which of r, c, ϕ, J, R, C, S are intended to be variables and which are constants. If they all are variables, then the linear approximation, about the point (r₀, c₀, ϕ₀, J₀, R₀, C₀, S₀, would be
\tau= \frac{\partial \tau}{\partial r}(r- r_0)+ \frac{\partial \tau}{\partial c}(c- c_0)+ \frac{\partial \tau}{\partial \phi}(\phi- \phi_0)+ \frac{\partial \tau}{\partial J}(J- J_0)+ \frac{\partial \tau}{\partial C}(C- C_0)+ \frac{\partial \tau}{\partial S}(S- S_0)

 

[BvU]

Hello $\approx$3M, welcome to PF !

Good thing I could find some explanation of the symbols http://www.rocscience.com/products/4/feature/172 . You really do want to make it a little easier for potential helpers (see guidelines)
because you can't expect all helpers to go out and reseaarch what on Earth it is you are talking about. So:

The Barton-Bandis failure criterion is an empirical relationship widely used to model the shear strength of rock discontinuities (e.g. joints). It is useful for fitting a strength envelope to field or laboratory shear test data of discontinuities. The Barton-Bandis criterion is non-linear, and relates shear strength to normal stress using the following equation: ...

where $phi_r$ is the residual friction angle of the failure surface, JRC is the joint roughness coefficient, and JCS is the joint wall compressive strength.

You have a $\phi_c$ (unless you mean $\phi \times c$ ?? and "The original Barton equation. has a $\phi_b$ , the basic friction angle of the failure surface. On the basis of direct shear test results for 130 samples of variably weathered rock joints, Barton and Choubey revised this to Eqn.2 ... (same), but with a $phi_r$.
​

So your question is about linearizing wrt $\sigma_n$ ? See link mentioned above. Acceptable or not depends on the criteria you apply.

 

[Wxfsa]

Those symbols don't mean anything to me, so I do not know which one is the input.
Even so, I can say that it is non linear because there is nothing in that expression for which there could be a linear relationship.
It would be a good idea to actually understand what "linearity" means whether it is necessary for this particular question or not.

The way you see if something is linear is:
If the input of the expression is f(x) and the output is some function S[f(x)], you would consider what would happen if the input was f(x)+g(x). If you have S[f(x)+g(x)]=S[f(x)] + S[g(x)] then it is linear
tan(x)f(x) is linear
f(x)+1 is not linear
x^2 f(x) ,