1. Reference problem#

The test is test case LA6 of a benchmark from the European Brite EurAM project BE97 -4547 « LISA » studied by three organizations:

  • EDF (old Norton-Hoff method with three values of the regularization parameter: the Norton-Hoff coefficient \(n\));

LTAS in Liège (regularized kinematic method of the dedicated university software « * ELSA ** »);

ForschungZentrum de Jülich (static method approximated by finite elements in displacement, reduced representation of self-constraint fields and algorithm for optimizing the**Permas* code).

An axisymmetric tank with a torispherical bottom subjected to internal pressure is considered.

The same network is used by the three participating organizations. It contains two Q8 elements in thickness, and in total there are 34 elements, and 141 knots. The constituent material is homogeneous and meets the von Mises criterion with the elastic limit (\({\sigma }_{y}\)) of \(100\mathrm{MPa}\) as a threshold.

+——————————————————————————————–++ | || +——————————————————————————————–++ |Results EDF LISA: initial and deformed mesh obtained for the Norton-Hoff coefficient n =31 | +——————————————————————————————–++

The calculation of the limit load by the Norton-Hoff-Friaâ method is detailed in the reference document [R7.07.01].

Let us recall briefly that the aim is to find the limit load \({\lambda }_{\text{lim}}\), for which the structure, made of a material of the perfect elastoplastic type with a von Mises threshold, can support surface loads \({\lambda }_{\text{lim}}F\) and volume loads \({\lambda }_{\text{lim}}f\), to which we can optionally add constant loads (\({F}_{0}\) and \({f}_{0}\)). Code_aster makes it possible to calculate, for each moment of the calculation, that is to say for increasingly weaker regularizations, two parameters:

  • An estimate of an upper bound of the limit load:

(1.1)#\[ {\ widehat {\ mathrm {\ lambda}}}} _ {m}} =\ underset {\ mathrm {\ omega}} {\ int} {\ mathrm {\ sigma}} _ {y}\ sqrt {\ frac {2} {m}}\ sqrt {\ frac {2} {m}}\ sqrt {\ frac {2} {3} {3}\ mathrm {\ epsilon} ({u} _ {m})\ cdot\ mathrm {\ epsilon})\ cdot\ mathrm {\ epsilon} ({u} _ {m}})\ cdot\ mathrm {\ epsilon} ({u} _ {m}})\ cdot\ mathrm {\ epsilon} Silon} ({u} _ {m})} d\ mathrm {\ omega} - {L} _ {0} ({u} _ {m})\]

  • and, in the absence of constant loading, an estimate of a lower bound \({\underline{\lambda }}_{m}\):

(1.2)#\[ {\ underline {\ lambda}} _ {m} =\ underset {\ omega} {\ omega} {\ omega} {\ int}\ frac {A (m)} {m}}\ cdot {(\ sqrt {\ varepsilon ({u} _ {m}) =\ underset {\ omega} _ {m})}\ cdot {m} (\ underset {x\ in\ Omega} {\ text {Sup}} {\ text {Sup}} (\ sqrt {\ frac {3} {3} {2} {\ sigma}} ^ {m})\ cdot {\ sigma} {m})\ cdot {\ sigma} {{m}})} {\ sigma}} ({u} _ {m})} {\ sigma}} ({u} _ {m}))} ^ {-1}\ {m}))} ^ {-1}\ the {\ stackrel {} {\ lambda}} _ {m}\]

If constant loading is present, we have the power of constant loading in the speed field to solve the problem.

It can be seen that the deviatoric stress tensor verifies the Norton-Hoff behavior relationship:

(1.3)#\[ {\ sigma} ^ {D} (u) =A (m)\ text {.} {(\ sqrt {{\ epsilon} ^ {D} (u)\ text {.} {\ epsilon} ^ {D} (u)})}} ^ {m-2}\ text {.} {\ epsilon} ^ {D} (u)\]

with \(\text{tr}\varepsilon (u)=0\), and \(A(m)={k}^{1-m}{\left(\frac{2}{3}\right)}^{m/2}{{\mathrm{\sigma }}_{y}}^{m}\).

1.1. Data for modeling#

Geometry

The axisymmetric tank with a torispherical bottom has the following characteristics: * internal radius of the cylindrical part: \(49\mathrm{mm}\); * thickness: \(2\mathrm{mm}\); * radius of the spherical part at the apex: \(98\mathrm{mm}\); * radius of the connection core: \(20\mathrm{mm}\).

Material Properties

Elastic limit: \({\sigma }_{y}=100\mathrm{MPa}\)

Boundary conditions

Zero axial displacement \(\mathrm{DY}\) on the \({\mathrm{BORD}}_{\mathrm{INF}}\) end of the cylindrical part (symmetry conditions)

Loads

Internal pressure of \(1\mathrm{MPa}\) applied to the inner wall \({B}_{D}\)

1.2. Benchmark results#

For this test case, we do not have analytical results but only numerical values from the calculations carried out as part of the benchmark LISA and recalled Tableau 1.2-1.

Modeling

Estimated upper value

Estimated lower value

EDF

\(m=1,\text{0476}\)

\(n=21\)

\(t=\mathrm{2,3222}\)

\(\mathrm{3,9514}\)

\(\mathrm{3,6049}\)

\(m=1,\text{0322}\)

\(n=31\)

\(t=\mathrm{2,4914}\)

\(\mathrm{3,9456}\)

\(\mathrm{3,7090}\)

\(m=1,\text{0141}\)

\(n=71\)

\(t=\mathrm{2,8513}\)

\(\mathrm{3,9404}\)

\(\mathrm{3,8372}\)

\(m=1,\text{0099}\)

\(n=101\)

\(t=\mathrm{3,0043}\)

\(\mathrm{3,9396}\)

\(\mathrm{3,8673}\)

Univ. of Liège/ LTAS

\(\mathrm{3,931}\)

nought

Research Center FZJ

nought

\(\mathrm{3,997}\)

Table 1.2-1: Benchmark results LISA