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 $\delta$:

$$\tau \left(x\right)={\tau }_0+\frac{{\mu }^{\prime }}{\pi }{\int }_{\mathrm{\Omega }}\frac{1}{s-x}\frac{\partial \delta }{\partial s}ds.$$

${\tau }_0$ 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:

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

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

where $\mathrm{\Delta }\tau =E:\mathrm{\Delta }\delta$ and $\mathrm{\Delta }\sigma =E:\mathrm{\Delta }ϵ$ are the changes in shear and normal stress respectively, $\mathrm{\Delta }\delta$ and $\mathrm{\Delta }ϵ$ the changes in slip and opening repectively, and the two functions $f_{\tau }$ and $f_{ϵ}$ 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 $f_{ϵ}$ 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