Simulations

Three-point bending

Fields	Equations	BCs
\(\boldsymbol{u}(\boldsymbol{x},t)\)	\(\operatorname{div}(\underline{\boldsymbol{\sigma}})+\bar{\boldsymbol{b}}=\boldsymbol{0}\)	\(\begin{aligned}\boldsymbol{u}&=\bar{\boldsymbol{u}} &&\text{on }\partial\mathcal{B}_{u}\times[0,T]\\ \underline{\boldsymbol{\sigma}}\boldsymbol{n}&=\bar{\boldsymbol{t}} &&\text{on }\partial\mathcal{B}_{t}\times[0,T]\end{aligned}\)
\(d(\boldsymbol{x},t)\)	\(\operatorname{div}\boldsymbol{q}_{d}-\underline{\mathcal{H}}_{d}+f_{d}=0\)	\(\boldsymbol{q}_{d}\cdot\boldsymbol{n}_{\mathcal{L}}=0\quad\text{on }\partial\mathcal{L}\times[0,T]\)

Crack propagation from a notch, modeled with a phase-field formulation.

Damage field \(d\) on the deformed mesh. Displacements are magnified 15× for visibility. The inset shows the notch and propagating crack. Playback follows 100 saved simulation steps.

Fluid–solid interaction

Interaction between fluid flow, heat transfer, mechanical equilibrium and fracture propagation in a porous medium.

An illustration from my doctoral thesis, showing the coupled processes involved in hydraulic fracturing of porous media. Drawing by Muhittin Babaoglu.