# Crack beam under tension

Project file on GitHub

## 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: $$$d (x) = 1 - {\mathrm{e}}^{\frac{- |x|}{2 \varepsilon}}$$$ $$$u (x) = \dfrac{\sigma}{E} \int_0^x \dfrac{1}{d (x)^2 + k} \mathrm{d}x$$$ with $$$\sigma = \dfrac{E u_0}{I (\varepsilon, k)}$$$ $$$I (\varepsilon, k) = \int_0^1 \dfrac{1}{d (x)^2 + k} \mathrm{d}x$$$ 

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