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This validation case belongs to solid mechanics. In this case, an infinite cylinder is subjected to a voluminal load. The parameter being validated is:

- Volume load

SimScale simulation results were compared to the analytical solution presented in [SSLA100]\(^1\).

The geometry for this project consists of a slice of an annulus between two cylinders, as seen in Figure 1:

The dimensions of the geometry are given in Table 1:

Geometry parameters | Dimension \([m]\) |

Outer radius \((R)\) | 1.4 |

Inner radius \((r)\) | 1 |

Thickness of the slice \((t)\) | 0.5 |

**Tool Type**: Code_Aster

**Analysis Type**: Linear static

**Mesh and Element Types**: The mesh used in this case is a second-order mesh created in SimScale with the standard algorithm.

Table 2 contains details to the resulting mesh:

Mesh Type | Nodes | Element Type |

Second-order standard | 991445 | Standard |

Figure 2 shows the standard mesh used for this case:

**Material**:

:*Material behavior**Linear elastic*- Young’s modulus
*\(E\) = 10 \(Pa\)* - Poisson’s ratio
*\(\nu\) = 0.3* - Density
*\(\rho\) = 1 \(kg/m³\)*

**Boundary Conditions**:

The boundary conditions will be defined based on the following nomenclature:

- Constraints
- \(d_z\) = 0 on both sides
*Isotropic elastic support*applied to the inner wall, with a spring constant \(k\) = 0.001 \(N/m\). The purpose of this boundary condition is to avoid rigid body motion.

- Surface loads
*Pressure*of 1 \(Pa\) applied to the*Volume load*applied in the radial direction, with a magnitude of \(\alpha r^2\ (\frac {N}{m^3})\), where \(\alpha = 1\ \frac {N}{m^5}\) and \(r\) is any given radius, in meters.

Note

Since the volume load is in cylindrical coordinates\(^1\), it was necessary to convert it to Cartesian coordinates using trigonometric functions. The resulting input formulae are:

$$Load_x = (x^2+y^2)cos(atan2(y,x)) \tag {1}$$

$$Load_y = (x^2+y^2)sin(atan2(y,x)) \tag {2}$$

$$Load_z = 0 \tag {3}$$

In equations 1 and 2, the \((x^2+y^2)\) term represents radius squared. The second part of the equations, consisting of trigonometric functions, is responsible for the conversion from cylindrical to Cartesian coordinates. As a result, the loads will be applied in a radial direction.

The analytical solution for the displacements of the inner and outer wall are presented in [SSLA100]\(^1\). Find below a comparison between SimScale results and the analytical solution:

Radius \([m]\) | Analytical displacement\(^1\) \([m]\) | Displacement – SimScale \([m]\) | Error [%] |

1 | 0.521309 | 0.521259 | -0.0096 |

1.4 | 0.442031 | 0.441986 | -0.0102 |

In Figure 5, we can see the resulting displacement contours. SimScale results show great agreement with the analytical solution.

Last updated: September 4th, 2023

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