2. Benchmark solution#

2.1. Calculation method#

2.1.1. Dirichlet boundary conditions#

The analytical stress field corresponding to the problem of the infinite cracked plate subjected to mode I is written [1]:

\({\sigma }_{\mathit{XX}}=\frac{1}{\sqrt{2\pi r}}{K}_{I}\mathrm{cos}\frac{\theta }{2}(1-\mathrm{sin}\frac{\theta }{2}\mathrm{sin}\frac{3\theta }{2})\), \({\sigma }_{\mathit{YY}}=\frac{1}{\sqrt{2\pi r}}{K}_{I}\mathrm{cos}\frac{\theta }{2}(1+\mathrm{sin}\frac{\theta }{2}\mathrm{sin}\frac{3\theta }{2})\), and \({\sigma }_{\mathit{XY}}=\frac{1}{\sqrt{2\pi r}}{K}_{I}\mathrm{sin}\frac{\theta }{2}\mathrm{cos}\frac{\theta }{2}\mathrm{cos}\frac{3\theta }{2}\).

With \({K}_{I}=1.0\), we have:

\({\sigma }_{\mathit{XX}}=\frac{1}{\sqrt{2\pi r}}\mathrm{cos}\frac{\theta }{2}(1-\mathrm{sin}\frac{\theta }{2}\mathrm{sin}\frac{3\theta }{2})\), \({\sigma }_{\mathit{YY}}=\frac{1}{\sqrt{2\pi r}}\mathrm{cos}\frac{\theta }{2}(1+\mathrm{sin}\frac{\theta }{2}\mathrm{sin}\frac{3\theta }{2})\), and \({\sigma }_{\mathit{XY}}=\frac{1}{\sqrt{2\pi r}}\mathrm{sin}\frac{\theta }{2}\mathrm{cos}\frac{\theta }{2}\mathrm{cos}\frac{3\theta }{2}\).

2.1.2. Neumann boundary conditions#

Knowing that \(\nu =0.0\) and that the pressure imposed is constant, the stress field simply corresponds to the pressure imposed:

\({\sigma }_{\mathit{XX}}=-P=20\mathit{MPa}\), \({\sigma }_{\mathit{YY}}=0\mathit{MPa}\), and \({\sigma }_{\mathit{XY}}=0\mathit{MPa}\).

2.2. Reference quantities and results#

The constraint field SIGM_ELNO calculated at the nodes by the operator CALC_ERREUR is compared and it is verified by non-regressions that they correspond to the field SIEF_ELNO calculated by the operator CALC_CHAMP.

We also test the constraint field calculated at the nodes of the X- FEM sub-elements.

Finally, we test the field for estimating the residual error [2].

2.2.1. Dirichlet boundary conditions#

Knowing that the refinement of the mesh will be insufficient to capture the asymptotic stress variation at the bottom of the crack, the calculation of the error indicator based on these numerical variations is not in a position to make a good estimate of the error. It is therefore decided to apply non-regression tests.

2.2.2. Neumann boundary conditions#

The stresses are constant in the plate. Since the solution is P0, the numerical error must be zero. The estimated error should also be zero.

2.3. Bibliographical references#

    1. IRWIN, « Analysis of stresses and strains near the end of a crack crossing a plate, » Journal of Applied Mechanics 24, 361-364, 1957.

  1. Document R4.10.02, Residual Error Estimator, Code_Aster Reference Manual.