Curved Cantilever Beam: `curvedCantilever`

Prepared by Ivan Batistić

Tutorial Aims

Demonstrate how to perform a solid-only analysis in solids4foam;
Examine solver performance under conditions in which the shear-locking phenomena can occur.

Case Overview

The curved cantilever beam is fixed on the left side, and the right side has prescribed horizontal traction, as shown in Fig 1. The prescribed horizontal traction is $200$ Pa. The beam's inner radius is set to $a= 0.31$ m and the beam's outer radius to $b = 0.33$ m. The Young's modulus is $E= 100$ Pa and the Poisson's ratio is $\nu = 0.3$. Gravitation effects are neglected, and there are no body forces. The problem is solved as static, using one loading increment.

Figure 1: Problem geometry [1]

Expected results

Stress is dependent on the angular coordinate $\theta$;
The analytical solution for the stress field is [1]:

\[N = {a}^2 - {b}^2 + ({a}^2+{b}^2)\;\text{ln}\left(\frac{b}{a}\right),\] \[\sigma_{r} = \frac{P}{N}\left(r+\frac{a^2b^2}{r^3}-\frac{a^2+b^2}{r}\right)\sin (\theta),\] \[\sigma_{\theta} = \frac{P}{N}\left(3r-\frac{a^2b^2}{r^3}-\frac{a^2+b^2}{r}\right)\sin (\theta),\] \[\tau_{r\theta} = \tau_{\theta r} =-\frac{P}{N}\left(r+\frac{a^2b^2}{r^3}-\frac{a^2+b^2}{r}\right)\cos (\theta).\]

where $a$ is beam inner radius, $b$ is beam outer radius and $P$ is applied shear force.

The analytical solution is generated alongside solution fields using the curvedCantileverAnalyticalSolution function object located in the system/controlDict, where one needs to input geometry and material data:

  functions
  {
      analyticalSolution
      {
          type    curvedCantileverAnalyticalSolution;

          // Inner beam radius
          rInner  0.31;

          // Outer beam radius
          rOuter  0.33;

          // Applied shear force (in N/m)
          force   4;

          // Young's modulus
          E       100;

          // Poisson's ratio
          nu      0.3;
      }
  }
  

The function object writes the analytical stress tensor field analyticalStress in the time directories.

Figures 2 and 3 show $\sigma_{xx}$, $\sigma_{xy}$ and $\sigma_{yy}$ stress distributions along the line $\theta=45^{\circ}$ for computational meshes consisting of $100\times100$ and $200\times50$ cells. One can see that with mesh refinement, numerical results converge to analytical ones. The data for plots presented in Figures 2 and 3 is extracted using the sampleDict located in the system directory.

Figure 2: Stress distribution for mesh consisting of 100x10 cells

Figure 3: Stress distribution for mesh consisting of 200x50 cells

Running the Case

The tutorial case is located at solids4foam/tutorials/solids/linearElasticity/curvedCantilever. The case can be run using the included Allrun script, i.e. > ./Allrun. In this case, the Allrun script compiles the tutorial-local library in the src directory, creates the mesh using blockMesh (> ./blockMesh), runs the solids4foam solver (> ./solids4Foam), and then runs the > ./sample utility. Optionally, if gnuplot is installed, the stress distribution is plotted in the sigmaAtTheta45deg.png file.

Warning

The coupled version of this case, which uses thecoupledUnsLinearGeometryLinearElastic, can currently only be run usingsolids4foam built on foam-extend. To modify the case to run with the segregatedlinearGeometryTotalDisplacement solid model, follow the instructions intutorials/solids/linearElasticity/narrowTmember/README.md file.

References

[1] Sadd MH. Elasticity: Theory, Applications, and Numerics. Elsevier 2009.

[2] Pablo Castrillo, Alfredo Canelas, Eugenio Schillaci, Joaquim Rigola and Asensio Oliva. High-order finite volume method for linear elasticity on unstructured meshes. Computers & Structures. Volume 268, 2022,

Curved Cantilever Beam: curvedCantilever