DDMFrictionalSlip

DDMFrictionalSlip is a julia implementation of the Displacement Discontinuity Method (DDM) for two-dimensional domains (one-dimensional fracture). Main features:

• Choice of Piecewise Constant (PWC), Piecewise Linear Collocation (PWLC), and Piecewise Quadratic Collocation (PWQ) shape functions
• Multithreaded assembly and solve
• Flexible problem formulation
• Non-equally sized elements

This package discretize the quasi-static changes in stress (normal or shear) $\tau$ expressed as a integral of the displacement discontinuity $\delta$:

$$ \tau\left(x\right) = \tau_{0} + \frac{\mu^{\prime}}{\pi} \int_{\Omega} \frac{1}{s - x} \frac{\partial \delta}{\partial s} ds. $$

$\tau_{0}$ is here the initial stress and $\mu^{\prime}$ the effective shear modulus. The previous expression is discretized into:

$$ \tau_{i} = \tau_{0} + E_{ij} : \delta_{j}, $$

where $E_{ij}$ is the elastic collocation matrix (dense matrix).

This package can be used to solve for systems of coupled equations which can be expressed in the following way:

$$ R_{\tau} = \Delta \tau\left(\Delta \delta\right) - f_{\tau}\left(\Delta \epsilon, \Delta \delta\right) = 0 $$

$$ R_{\epsilon} = \Delta \sigma\left(\Delta \epsilon\right) - f_{\epsilon}\left(\Delta \epsilon, \Delta \delta\right) = 0 $$

where $\Delta \tau = E: \Delta \delta$ and $\Delta \sigma = E : \Delta \epsilon$ are the changes in shear and normal stress respectively, $\Delta \delta$ and $\Delta \epsilon$ the changes in slip and opening repectively, and the two functions $f_{\tau}$ and $f_{\epsilon}$ can be defined to account for applied stress, frictional constraints, and/or fluid pressure coupling.

The user needs to specify the two functions $f_{\tau}$ and $f_{\epsilon}$ together with their derivatives with respect to the displacement discontinuity variables to properly calculate the jacobian matrix of the problem. Please see the test suite in test/ for examples of formulations.

Author: Dr. Antoine B. Jacquey