4. Dezember 2024

VE0067 | Inkompressibles Material - Dickwandiger Behälter

Beschreibung

Dieses Verifikationsbeispiel ist eine Modifikation des VE0064 - Dickwandige Behälter, wo der einzige Unterschied ist dass das Material des Behälters inkompressibel ist. 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	Elastizitätsmodul	E	1.000	MPa
Material	Elastic Incompressible	Poisson´s Ratio	ν	0.499	-
Geometrie		Inner Radius	r₁	200.000	mm
Geometrie		Outer Radius	r₂	300.000	mm
Last		Inner Pressure	p₁	60.000	kPa
Last		Outer Pressure	p₂	0.000	kPa

Inkompressibles Material - Dickwandiger Behälter

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.

Analytische Lösung

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})]

Radiale Durchbiegung des Innen- und Außenradius eines offenen Behälters

RFEM-Einstellungen

Modeled in RFEM 5.06 and RFEM 6.06
The element size is l_FE = 2.000 mm
Es wird ein isotropes linear-elastisches Materialmodell vorausgesetzt.

Ergebnisse

RFEM 6 Ergebnisse – Gesamte Verformung

Anzahl	Analytische Lösung	RFEM 6	Verhältnis	RFEM 5	Verhältnis
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

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Verifikationsbeispiel 000067 | 1

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