Perforated Plate: perforatedPlate
Prepared by Philip Cardiff and Ivan Batistić
Tutorial Aims
- Demonstrate a small-strain elastoplastic analysis of a perforated plate.
Case Overview
This case considers a 2-D plate with a circular hole subjected to a time-varying uniaxial traction. The full plate has dimensions of \(36 \times 20\) mm and contains a central circular hole with a diameter of \(10\) mm. Owing to the two symmetry planes, only one quarter of the plate is modelled, as shown in Figure 1. The problem is represented as a plane-strain case.
Figure 1 - Perforated plate geometry [1]
The traction is applied on the right boundary in the \(x\) direction. It increases linearly from zero to \(133.65\) MPa during the first \(10\) s and then unloads linearly to zero by \(20\) s. The hole and upper boundaries are traction free, while symmetry conditions are applied on the left and lower boundaries. Gravitational and inertial effects are neglected, and the case is solved using time steps of \(0.1\) s.
The default mesh, shown in Figure 2, consists of 300 cells. The total displacement solid model is selected in constant/solidProperties using linearGeometryTotalDisplacement.
Figure 2 - Computational mesh
The aluminium plate is modelled using the linearElasticMisesPlastic mechanical law with isotropic hardening. The mechanical properties are given in Table 1, and the tabulated hardening curve is specified in constant/plasticStrainVsYieldStress.
Table 1 - Mechanical properties
| Parameter | Symbol | Value |
|---|---|---|
| Young's modulus | \(E\) | \(70\) GPa |
| Poisson's ratio | \(\nu\) | \(0.3\) |
| Yield stress | \(\sigma_Y\) | \(243\) MPa |
| Density | \(\rho\) | \(3500\) kg/m\(^3\) |
Expected Results
Figure 3 compares the predicted \(y\)-displacement at point A on the upper rim of the hole with Abaqus finite-element results. The mesh-refinement study shows that the finite-volume predictions become mesh independent and agree well with the finite-element reference solution [1].
Figure 3 also shows the residual \(\sigma_{xx}\) stress component along the path A-B after unloading at \(t = 20\) s. Material near the hole yields during loading and returns to a compressive residual stress state after unloading. Additional comparisons are available in Cardiff et al. [1] and in the [2] where vertex-centred discretisation is employed.
Figure 3 - Displacement and residual stress predictions [1]
The solidForcesDisplacements function object records the force and displacement on the right patch. The solidPointDisplacement function object records the displacement at point A, located at (0 0.005 0).
Running the Case
The tutorial case is located at solids4foam/tutorials/solids/elastoplasticity/perforatedPlate. The case can be run using the included Allrun script, i.e. > ./Allrun.
The Allrun script first creates the mesh using blockMesh and runs the solids4Foam solver.