Crack beam under tension

Problem description

We solve a homogeneous beam model under a given displacement loading. The length of the beam is 2,mm. Detailed model description can refer this PDF. ## Results and evaluation

Results show crack Phase-Field and displacement field distributions through the length of the beam:

For highlighting the deviation between the analytical and numerical solution, we provide the absolute error of the analytical solution and numerical simulation as follows:

The analytical solution is: \[ \begin{equation} d (x) = 1 - {\mathrm{e}}^{\frac{- |x|}{2 \varepsilon}} \end{equation} \] \[ \begin{equation} u (x) = \dfrac{\sigma}{E} \int_0^x \dfrac{1}{d (x)^2 + k} \mathrm{d}x \end{equation} \] with \[ \begin{equation} \sigma = \dfrac{E u_0}{I (\varepsilon, k)} \end{equation} \] \[\begin{equation} I (\varepsilon, k) = \int_0^1 \dfrac{1}{d (x)^2 + k} \mathrm{d}x \end{equation}\] $$