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# Validation Case: Infinite Cylinder Subjected to a Volume Load

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

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

## Geometry

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:

## Analysis Type and Mesh

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:

Figure 2 shows the standard mesh used for this case:

## Simulation Setup

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.
• Pressure of 1 $$Pa$$ applied to the inner wall
• 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.

## Result Comparison

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:

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

Last updated: June 14th, 2021