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}

This article was written by Xing-Yuan Miao. If you are missing something or you find an error please let us know.
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