Rigid vs. Deformable

An important consideration when doing a discontinuum analysis is whether to use rigid or deformable blocks to represent the behavior of intact material. The considerations for rigid versus deformable blocks are discussed in this section. If a deformable block analysis is required, there are several different models available to simulate block deformability; these are discussed in the section Choice of Constitutive Model.

As mentioned in Theory and Background, early distinct element codes assumed that blocks were rigid. However, the importance of including block deformability has become recognized, particularly for stability analyses of underground openings and studies of seismic response of buried structures. One of the most obvious reasons to include block deformability in a distinct element analysis is the requirement to represent the “Poisson’s ratio effect” of a confined rock mass.

Poisson’s Effect

Rock mechanics problems are usually very sensitive to the Poisson’s ratio chosen for a rock mass. This is because joints and intact rock are pressure-sensitive; their failure criteria are functions of the confining stress (e.g., the Mohr-Coulomb criterion). Capturing the true Poisson behavior of a jointed rock mass is critical for meaningful numerical modeling.

The effective Poisson’s ratio of a rock mass comprises two parts: a component due to the jointing; and a component due to the elastic properties of the intact rock. Except at shallow depths or low confining stress levels, the compressibility of the intact rock makes a large contribution to the compressibility of a rock mass as a whole. Thus, the Poisson’s ratio of the intact rock has a significant effect on the Poisson’s ratio of a jointed rock mass.

Strictly speaking, a single Poisson’s ratio, \(\nu\), is defined only for isotropic elastic materials. However, there are only a few jointing patterns that lead to isotropic elastic properties for a rock mass. Therefore, it is convenient to define a “Poisson effect” that can be used for discussion of anisotropic materials.

Note

The following discussion assumes 2D plain strain conditions with \(y\) as the vertical direction.

The Poisson effect will be defined as the ratio of horizontal-to-vertical stress when a load is applied in the vertical direction and no strain is allowed in the horizontal direction; plane-strain conditions are assumed. As an example, the Poisson effect for an isotropic elastic material is

(1)\[\frac{\sigma_{xx}}{\sigma_{yy}} = \frac{\nu}{1-\nu}\]

Consider the Poisson effect produced by the vertical jointing pattern shown in the figure below. If this jointing were modeled with rigid blocks, applying a vertical stress would produce no horizontal stress at all. This is clearly unrealistic, because the horizontal stress produced by the Poisson’s ratio of the intact rock is ignored.

Figure 1: Model for Poisson’s effect in rock with vertical and horizontal jointing

The joints and intact rock act in series. In other words, the stresses acting on the joints and on the rock are identical. The total strain of the jointed rock mass is the sum of the strain due to the jointing and the strain due to the compressibility of the rock. The elastic properties of the rock mass as a whole can be derived by adding the compliances of the jointing and the intact rock:

(2)\[\begin{split}\begin{bmatrix}\epsilon_{xx}\\\epsilon_{yy}\end{bmatrix} = \biggl(C^{\mathrm{rock}}+ C^{\mathrm{jointing}}\biggr)\begin{bmatrix}\sigma_{xx}\\\sigma_{yy}\end{bmatrix}\end{split}\]

If the intact rock were modeled as an isotropic elastic material, its compliance matrix would be

(3)\[\begin{split}C^{\mathrm{rock}} = \frac{1+\nu}{E}\begin{bmatrix}1-\nu & -\nu\\-\nu & 1-\nu\end{bmatrix}\end{split}\]

The compliance matrix due to the jointing is

(4)\[\begin{split}C^{\mathrm{jointing}} = \begin{bmatrix}\frac{1}{Sk_n}&0\\0&\frac{1}{Sk_n}\end{bmatrix}\end{split}\]

where \(S\) is the joint spacing, and \(k_n\) is the normal stiffness of the joints.

If \(\epsilon_{xx}\) = 0 in Equation (2) then

(5)\[\frac{\sigma_{xx}}{\sigma_{yy}}=-\frac{C^{(total)}_{12}}{C^{(total)}_{11}}\]

where \(C^{(total)} = C^{(rock)} + C^{(jointing)}\).

Thus, the Poisson effect for the rock mass as a whole is

(6)\[\frac{\sigma_{xx}}{\sigma_{yy}}=\frac{\nu(1+\nu)}{E/(Sk_n)+(1+\nu)(1-\nu)}\]

Equation (6) is graphed as a function of the ratio \(E/(Sk_n)\) in the next figure. Also graphed are the results of several two-dimensional UDEC simulations run to verify the formula. The ratio \(E/(Sk_n)\) is a measure of the stiffness of the intact rock in relation to the stiffness of the joints. For low values of \(E/(Sk_n)\), the Poisson effect for the rock mass is dominated by the elastic properties of the intact rock. For high values of \(E/(Sk_n)\), the Poisson effect is dominated by the jointing.

Now consider the Poisson effect produced by joints dipping at various angles. The Poisson effect is a function of the orientation and elastic properties of the joints. Consider the special case shown in Figure 3. A rock mass contains two sets of equally spaced joints dipping at an angle, \(θ\), from the horizontal. The elastic properties of the joints consist of a normal stiffness, \(k_n\), and a shear stiffness, \(k_s\). The blocks of intact rock are assumed to be completely rigid.

Figure 2: Poisson’s effect for vertically jointed rock (\(ν\) = 0.3 for intact rock)

where \(S\) is the joint spacing, and \(k_n\) is the normal stiffness of the joints.

Figure 3: Model for Poisson’s effect in rock with joints dipping at angle \(θ\) from the horizontal and with spacing \(S\)

The Poisson effect for this jointing pattern is

(7)\[\frac{\sigma_{xx}}{\sigma_{yy}}=\frac{\cos^2\theta[(k_n\, / \,k_s)-1]}{\sin^2\theta+cos^2\theta(k_n\, / \,k_s)}\]

This formula is illustrated graphically for several values of \(θ\) in the next figure. Also shown are the results of numerical simulations using UDEC. The UDEC simulations agree closely with Equation (7).

Figure 4: Poisson’s effect for jointed rock at various joint angles (blocks are rigid)

Equation (7) demonstrates the importance of using realistic values for joint shear stiffness in numerical models. The ratio of shear stiffness to normal stiffness dramatically affects the Poisson response of a rock mass. If shear stiffness is equal to normal stiffness, the Poisson effect is zero. For more reasonable values of \(k_n/k_s\) , from 2.0 to 10.0, the Poisson effect is quite high, up to 0.9.

Next, the contribution of the elastic properties of the intact rock will be examined for the case of θ = 45º. Following the analysis for the vertical jointing case, the intact rock will be treated as an isotropic elastic material. The elastic properties of the rock mass as a whole will be derived by adding the compliances of the jointing and the intact rock.

The compliance matrix due to the two equally spaced sets of joints dipping at 45º is

(8)\[\begin{split}C^{(jointing)}=\frac{1}{2S\:k_nk_s}\begin{bmatrix}k_s+k_n&k_s-k_n\\k_s-k_n&k_s+k_n\end{bmatrix}\end{split}\]

Thus, the Poisson effect for the rock mass as a whole is

(9)\[\frac{\sigma_{xx}}{\sigma_{yy}}=\frac{\nu(1+\nu)\,/E+(k_n-k_s)\,/\,(2S\,k_nk_s)}{[(1+\nu)(1_-\nu)]\,/E+\,(k_n+k_s)\,/\,(2S\,k_nk_s)}\]

Equation (9) is graphed for several values of the ratio \(E/(Sk_n)\) for the case of \(ν\) = 0.2 (see the next figure). Also plotted are the results of UDEC simulations. For low values of \(E/(Sk_n)\), the Poisson effect of a rock mass is dominated by the elastic properties of the intact rock. For high values of \(E/(Sk_n)\), the Poisson effect is dominated by the jointing.

Figure 5: Poisson’s effect for rock with two equally spaced joint sets with θ = 45◦ (blocks are deformable with ν = 0.2)