Extended Finite Element Method for the Analysis of Discontinuities in Rock Masses

Abstract

The strength and deformability of rock mass primarily depend on the condition of joints and their spacing and partially on the engineering properties of rock matrix. Till today, numerical analysis of discontinuities e.g. joint, fault, shear plane and others is conducted placing an interface element in between two adjacent rock matrix elements. However, the applicability of interface elements is limited in rock mechanics problems having multiple discontinuities due to its inherent numerical difficulties often leading to non-convergent solution. Recent developments in extended finite element method (XFEM) having strong discontinuity imbedded within a regular element provide an opportunity to analyze discrete discontinuities in rock masses without any numerical difficulties. This concept is based on partition of unity principle and can be used for cohesive rock joints. This paper summarizes the mathematical frameworks for the implementation of strong discontinuities in 3 and 6 nodded triangular elements and also provides numerical examples of the application of XFEM in one and two dimensional problems with single and multiple discontinuities.

Appendix: Analytical solution for joints in a rock bar

As mentioned in Eq. 1, the displacement u(y)can be decomposed into two parts, continuous and discontinuous part as given below;

$$ \, u(y) = \hat{u}(y) + \sum\limits_{i = 1}^{n} {H_{i} \left[\kern-0.15em\left[ {u_{i} } \right]\kern-0.15em\right]} . $$

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Assuming displacement $\hat{u}$ and $\left\| {u_{i} } \right\|$ are linear functions of coordinate y, we find that $\hat{u}(y) = a + by \,$ and $\left[\kern-0.15em\left[ {u_{i} } \right]\kern-0.15em\right] = c_{i} + d_{i} \left( {y - L_{{_{i} }} } \right)$ and H _i is Heaviside function define as

$$ H_{i} = \left\{ {\begin{array}{*{20}c} {1;\quad L_{i} \le y \le L} \\ {0;\quad {\text{elsewhere }}} \\ \end{array} } \right\}\quad \forall \, i = 1,2, \ldots ,n. $$

Here a, b, c _i and d _i are constants to be determined from the geometry, material properties, loading conditions and joint properties of the rock bar. The boundary condition is u(y) = 0 at y = 0 prescribed in Fig. 7 implies that $u(y) = by + \sum\nolimits_{i = 1}^{n} {H_{i} \left\{ {c_{i} + d_{i} \left( {y - L_{i} } \right)} \right\}}$

Hence, first considering the $\varepsilon = \frac{du}{dy}$ and then σ = Eɛ, the total potential energy can be derived using Eq. 17 as

$$ \Uppi = \frac{A}{2}\left\{ {\int\limits_{0}^{L} {b^{2} Edy + \sum\limits_{i = 1}^{n} {\left( {\int\limits_{{L_{i} }}^{L} {2bd_{i} Edy + \int\limits_{{L_{i} }}^{L} {d_{i}^{2} Edy} } } \right)} } } \right\} + \frac{A}{2}\sum\limits_{i = 1}^{n} {c_{i}^{2} k_{{nn_{i} }} } - P\left\{ {bL + \sum\limits_{i = 1}^{n} {d_{i} (L - L_{i} )} } \right\} $$

$$ \Uppi = \frac{AE}{2}[b^{2} L + 2bd_{i} (L - L_{i} ) + d_{i}^{2} (L - L_{i} )] + \frac{A}{2}\sum\limits_{i = 1}^{n} {c_{i}^{2} k_{{nn_{i} }} } - P\left\{ {bL + \sum\limits_{i = 1}^{n} {d_{i} (L - L_{i} )} } \right\} $$

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where $k_{{nn_{i} }}$ is the normal stiffness of the ith joint. Now taking variations with respect to b,c _i and d _i leads linear system of equations as$Lb + (L - L_{i} )d_{i} = \frac{PL}{AE}$, $c_{i} = {\frac{P}{{Ak_{nn} }}}$ and $b + d_{i} = \frac{P}{AE}$ for $i = 1,2, \cdots n$.

Solving the above equations, we find

$$ b = \frac{P}{AE},c_{i} = {\frac{P}{{Ak_{{nn_{i} }} }}}\quad {\text{and}}\,d_{i} = 0\,{\text{for}}\,{\text{all}}\;i = 1, 2, \ldots n. $$

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Here, the displacement field for one dimensional bar having multiple joints at any locations inside the bar is given by

$$ u(y) = {\frac{Py}{\text{AE}}} + \sum\limits_{i = 1}^{n} {H_{i} \left( {{\frac{P}{{Ak_{{nn_{i} }} }}}} \right)} . $$

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