2. Benchmark solution#

2.1. Calculation method used for the reference solution#

For an axisymmetric crack in a cylinder of infinite length, the method of Singular Integral Equations and Asymptotic Developments [bib1] makes it possible to calculate the values of the stress intensity factors.

  1. Case 1: Traction and Torsion

Traction induces an opening in mode 1. \(\mathrm{K1}\) is given by the following formula:

\({K}_{I}=\frac{P}{\pi {a}^{2}}\sqrt{\pi a}{F}_{1}(a/b)\)

where :math:`P` is the force applied to the upper and lower faces and :math:`{F}_{1}` is a given function [:ref:`Figure 2.1-a <Figure 2.1-a>`].

Torsion induces an opening in mode 3. :math:`\mathrm{K3}` is given by the following formula:

\({K}_{\mathrm{III}}=\frac{\mathrm{2T}}{\pi {a}^{3}}\sqrt{\pi a}{F}_{3}(a/b)\)

where T is the moment applied to the upper and lower faces and :math:`{F}_{3}` is a given function [:ref:`Figure2.1‑a <Figure2.1‑a>`].

  1. Case 2: Non-contact bending

Flexion induces an opening in \(1\) mode. The value of \(\mathrm{K1}\) at the maximum opening point \(A\) is given by the following formula:

\({K}_{{I}_{A}}=\frac{\mathrm{4M}}{\pi {a}^{3}}\sqrt{\pi a}{F}_{2}(a/b)\)

where :math:`M` is the moment applied to the upper and lower faces and :math:`{F}_{2}` is a given function [:ref:`Figure2.1‑a <Figure2.1‑a>`].

  1. Case 3: Flexion with contact

There is no analytical solution to this problem. On the one hand, it is expected that \(\mathrm{K1}\) will be close to the case without contact on the part of the crack that is being opened, and on the other hand that \(\mathrm{K1}\) will be zero on the part of the crack that is being closed.

_images/100000000000040000000300F8F4D0D03F8E367A.png

Figure 2.1-a: Functions \(\mathrm{F1}\), \(\mathrm{F2}\), and \(\mathrm{F3}\)

These three functions come from [bib1].

2.2. Benchmark results#

Digital application:

Unless otherwise stated, in the rest of this document, the parameters used for \(a\) and \(b\) are:

\(a=0.4m\)

\(b=0.5m\)

Case 1: Traction and Torsion

Case 2: Flexion

\(\mathrm{K1}=5.35{\mathrm{MPa.m}}^{1/2}\) \(\mathrm{K3}=11.22{\mathrm{MPa.m}}^{1/2}\)

\({\mathrm{K1}}_{A}=11.71{\mathrm{MPa.m}}^{1/2}\)

Table 2.2-1: Reference values

2.3. Bibliographical references#

  • TADA, PARIS, IRWIN: The Stress Analysis Of Cracks Handbook, Del Research Corporation, Hellertoxn, Pennsylvania (1973).

Calculation of stress intensity factors by extrapolation of the displacement field,*Code_Aster Reference Manual, R7.02.08

  • CORNELIU: Quarter-point elements for curved crack fronts, Computers & Structures Vol. 17, No. 2, pp. 227-231, 1983