2024-12-04

VE0067 | Incompressible Material - Thick–Walled Vessel

Description

This verification example is a modification of VE0064 - Thick-Walled Vessel, where the only difference is that the material of the vessel is incompressible. A thick-walled vessel is loaded by inner and outer pressure. The vessel is open-ended, thus there is no axial stress. The problem is modeled as a quarter model and described by the following set of parameters. While neglecting self-weight, determine the radial deflection of the inner and outer radius u_r(r₁), u_r(r₂).

Material	Elastic Incompressible	Modulus of Elasticity	E	1.000	MPa
Material	Elastic Incompressible	Poisson´s Ratio	ν	0.499	-
Geometry		Inner Radius	r₁	200.000	mm
Geometry		Outer Radius	r₂	300.000	mm
Load		Inner Pressure	p₁	60.000	kPa
Load		Outer Pressure	p₂	0.000	kPa

Incompressible Material – Thick-Walled Vessel

The aim of this verification example is to show the phenomenon of volumetric locking of finite elements. This problem can occur in incompressible material cases when the Poisson ratio ν is approaching 0.5 (rubber materials, materials in plastic state). In this case, the finite element displacements tend to zero. The volumetric locking cannot be avoided by the mesh refinement. All fully integrated elements will lock when the incompressible material is used. The simplest way to avoid locking is using reduced integration. In this case the number of integration points is reduced. The integration scheme is one order less accurate than the standard integration scheme. In RFEM the reduced integration is used so the result will be correct, as seen bellow.

Analytical Solution

The analytical solution is the same as in the case of VE0064 - Thick-Walled Vessel. Desired radial deflection of the inner and outer radius of the open-ended vessel u_r(r₁), u_r(r₂) can be determined using the following equations:

u_{r} (r_{1}) = \frac{r_{1}}{E} [σ_{t} (r_{1}) - ν σ_{r} (r_{1})],

u_{r} (r_{2}) = \frac{r_{2}}{E} [σ_{t} (r_{2}) - ν σ_{r} (r_{2})]

Radial Deflection of Inner and Outer Radius of Open-Ended Vessel

RFEM Settings

Modeled in RFEM 5.06 and RFEM 6.06
The element size is l_FE = 2.000 mm
Isotropic linear elastic material model is used

Results

RFEM 6 Results – Total Deformation

Quantity	Analytical Solution	RFEM 6	Ratio	RFEM 5	Ratio
u_r(r₁) [mm]	29.988	29.986	1.000	29.987	1.000
u_r(r₂) [mm]	22.497	22.495	1.000	22.495	1.000

454x

Verification Example 000067 | 1

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