FLAC3D Theory and Background • Constitutive Models

Swell Model*

Note

*This model is available in FLAC3D only.

This model is based on the Mohr-Coulomb model with nonassociated shear and associated tension flow rules. The difference is that the wetting-induced deformations are taken into account by means of coupling wetting strains with the model state prior to wetting. For model application and modeling procedures, see Noorany et al. (1999), Pathak et al. (2003), and Pathak et al (2010). The yield and potential functions, plastic flow rules, and stress corrections are identical to those of the Mohr-Coulomb model.

Wetting Strain and Stress Corrections

The wetting-induced strains can be expressed by the following logarithmic or linear function of total compressive stress, $$\sigma_{z'z'}$$, in the principal swelling direction normalized by the atmospheric pressure, $$p_a$$:

• logarithmic function
(1)$\begin{split} \begin{matrix} \varepsilon_{z'z'} = c_1 \log [a_1 (-\sigma_{z'z'}/p_a)] \mbox{ , } \sigma_{z'z'} < 0 \\ \\ \varepsilon_{x'x'} = \varepsilon_{y'y'} = c_3 \log [a_3 (-\sigma_{z'z'}/p_a)] \mbox{ , } \sigma_{z'z'} < 0 \end{matrix}\end{split}$
• linear function
(2)$\begin{split} \begin{matrix} \varepsilon_{z'z'} = c_1 - a_1 \sigma_{z'z'}/p_a \mbox{ , } \sigma_{z'z'} < 0 \\ \\ \varepsilon_{x'x'} = \varepsilon_{y'y'} = c_3 - a_3 \sigma_{z'z'}/p_a \mbox{ , } \sigma_{z'z'} < 0 \end{matrix}\end{split}$

In the above equations, $$a_1$$, $$c_1$$, $$a_3$$, and $$c_3$$ are non-negative welling properties determined from laboratory tests, and, by convention, compressive stresses and strains are negative. Note that the wetting strain is assumed to be isotropic in the lateral directions ($$\varepsilon_{x'x'} = \varepsilon_{y'y'}$$). Stress $$\sigma_{z'z'}$$ is the total vertical stress component in the local axes obtained at the equilibrium state prior to wetting and is not modified in Equations (1) and (2) during calculations.

Corresponding wetting stresses in the principal swelling directions (defined by $$x', y', z'$$) are then calculated based on the incremental form of Hooke’s Law:

(3)$\begin{split} \begin{matrix} \varepsilon_{x'x'} = \alpha_1 \varepsilon_{x'x'} + \alpha_2 (\varepsilon_{y'y'} + \varepsilon_{z'z'}) \\ \\ \varepsilon_{y'y'} = \alpha_1 \varepsilon_{y'y'} + \alpha_2 (\varepsilon_{z'z'} + \varepsilon_{x'x'}) \\ \\ \varepsilon_{z'z'} = \alpha_1 \varepsilon_{z'z'} + \alpha_2 (\varepsilon_{x'x'} + \varepsilon_{y'y'}) \end{matrix}\end{split}$

where $$\alpha_1 = K + 4G/3$$, $$\alpha_2 = K - 2G/3$$, and $$K$$ and $$G$$ are bulk and shear modulus, respectively.

Finally, the global swelling stresses are obtained through resolution of the local stresses in the principal swelling direction into the global axes. Global swelling stresses are then added (in increments) to the global stress components over a specified number of steps, $$N_s$$. At each step, the resulting stress components are examined for failure based on the Mohr-Coulomb failure criteria. Note that the model should be cycled to equilibrium after $$N_s$$ steps are taken.

Implementation Procedure

In the implementation of the Swell model, stresses corresponding to the elastic guess for the step are first analyzed as described in the Mohr-Coulomb model. The corresponding swelling stresses in the local axis (principal swelling directions) are calculated using Equation (3) and then resolved into the global axes. The swelling stresses in the global axes are added over a specified number of steps, $$N_s$$. The resulting stresses are then examined for failure based on the Mohr-Coulomb failure criteria.

Upper bounds are defined for the swelling strains in and vertical to the swelling plane based on the swelling measured under zero vertical stress. The code checks the wetting strains at each step and sets values to the upper bounds once they exceed the limits.

References

Pathak, Y., Alfaro, M., & Detournay, C. Wetting-induced deformation of geosynthetic reinforced slopes with expansive soils. In 56th Canadian Geotechnical Conference. Winnipeg, Manitoba. 266-273(2003).

Pathak, Y.P., and M.C. Alfaro. Wetting-drying behaviour of geogrid-reinforced clay under working load conditions. Geosynthetics International 17(3) 144-156 (2010).

Noorany, I., Frydman, S., & Detournay, C. Prediction of soil slope deformation due to wetting. in FLAC and numerical modeling in geomechanics. Balkema, Rotterdam, 101-107 (1999).

swell Model Properties

Use the following keywords with the zone property command to set these properties of the swell model.

swell
bulk f

elastic bulk modulus, $$K$$

cohesion f

cohesion, $$c$$

constant-a-1 f

swelling property, $$a_1$$

constant-a-3 f

swelling property, $$a_3$$

constant-c-1 f

swelling property, $$c_1$$

constant-c-3 f

swelling property, $$c_3$$

constant-m-1 f

swelling property, $$m_1$$

constant-m-3 f

swelling property, $$m_3$$

dilation f

dilation angle, $$\psi$$. The default is 0.0.

dip f

dip angle [degrees] of the local swelling plane

dip-direction f

dip direction [degrees] of the local swelling plane

friction f

internal angle of friction, $$\phi$$

normal v

normal direction of the local swelling plane, ($$n_x$$, $$n_y$$, $$n_z$$)

normal-x f

$$x$$-component of the normal direction to the local swelling plane, $$n_x$$

normal-y f

$$y$$-component of the normal direction to the local swelling plane, $$n_y$$

normal-z f

$$z$$-component of the normal direction to the local swelling plane, $$n_z$$

poisson f

Poisson’s ratio, $$\nu$$

pressure-reference f

atmospheric pressure, $$p_a$$

shear f

elastic shear modulus, $$G$$

tension f

tension limit, $$\sigma^t$$. The default is 0.0.

young f

Young’s modulus, $$E$$

flag-swell i (a)

flag of swelling function, (1 - logarithmic; 2 - linear). The default is 1.

flag-brittle b (a)

If true, the tension limit is set to 0 in the event of tensile failure. The default is false.

number-start i (a)

number of steps over which swelling strains are introduced, $$N_s$$. The default is 1.

count-swell i (r)

count of step number after swelling starts, must be reset to zero to start a new swelling episode

stress-local-vertical f (r)

local total stress vertical stress when swelling starts; must be reset to zero when soil swelling properties are changed

stress-swell-xx f (r)

xx swelling stress component

stress-swell-yy f (r)

yy swelling stress component

stress-swell-zz f (r)

zz swelling stress component

stress-swell-xy f (r)

xy swelling stress component

stress-swell-xz f (r)

xz swelling stress component

stress-swell-yz f (r)

yz swelling stress component

Key

• Only one of the two options is required to define the elasticity: bulk modulus $$K$$ and shear modulus $$G$$, or Young’s modulus $$E$$ and Poisson’s ratio $$\nu$$. When choosing the latter, Young’s modulus $$E$$ must be assigned in advance of Poisson’s ratio $$\nu$$.
• The tension cut-off is $${\sigma}^t = min({\sigma}^t, c/\tan \phi)$$.