2. Benchmark solution#

2.1. Calculation method used for the reference solution#

2.1.1. Modeling A#

The reference solution comes from WILSON and YU [bib1]:

_images/Object_1.svg

In plane deformations, the IRWIN formula gives:

_images/Object_2.svg

or numerically:

_images/Object_3.svg

2.1.2. B modeling#

The reference solution for the stress intensity factor for mode I \({K}_{I}\) is expressed as follows [bib3]:

\({K}_{I}=\mathrm{\sigma }\sqrt{\mathrm{\pi }a}F\left(\frac{a}{W}\right)\)

with \(F\left(\frac{a}{W}\right)=\mathrm{1,122}-\mathrm{0,231}\left(\frac{a}{W}\right)+\mathrm{10,550}{\left(\frac{a}{W}\right)}^{2}-\mathrm{21,710}{\left(\frac{a}{W}\right)}^{3}+\mathrm{30,382}{\left(\frac{a}{W}\right)}^{4}\)

The \(G\) energy return rate is obtained thanks to Irwin’s formula: \(G=\frac{\left(1-{\mathrm{\nu }}^{2}\right)}{E}{K}_{I}^{2}\).

2.2. Benchmark results#

2.2.1. Modeling A#

The reference results are those from the reference solution from WILSON and YU [bib1]:

_images/Object_4.svg

2.2.2. B modeling#

The reference results are those from the reference solution from [bib3]: \(G=\mathrm{572,05}J\mathrm{.}{m}^{-2}\)

2.3. Bibliographical references#

  1. The Use of J-Integrals in thermal stress crack problems - International Journal of Fracture (1979) WILSON and YU.

  2. Complementary qualification of codes INCA/MAYA in linear thermo-elasticity. Technical note DRE/STRE/LMA 84/598

    1. Tada, P. Paris, G. Irwin, G. Irwin, The Stress Analysis of Cracks Handbook, 3rd edition, 2000