2. Benchmark solution#
2.1. Calculation method used for the reference solution#
The reference solution comes from THAU and LU [bib1] and L.B. book FREUND [bib2]. The following figure represents the infinite medium and not the geometry of the test.
\({K}_{I}^{D}(t)\mathrm{=}\mathrm{2P}H(t)\frac{\sqrt{1\mathrm{-}2\nu }}{(1\mathrm{-}\nu )}\sqrt{\frac{{c}_{d}t}{\pi }},0<t<\mathrm{2a}\mathrm{/}{c}_{d}\)
The expression for the energy recovery rate is as follows:
\(G(t)\mathrm{=}\frac{{K}_{I}^{{D}^{2}}(t)}{E}(1\mathrm{-}{\nu }^{2})\mathrm{=}{\mathrm{4P}}^{2}H(t)\frac{(1\mathrm{-}2\nu )(1+\nu )}{(1\mathrm{-}\nu )E}\frac{{c}_{d}t}{\pi },0<t<\mathrm{2a}\mathrm{/}{c}_{d}\)
\(G\mathrm{=}\frac{(1\mathrm{-}{\nu }^{2})}{E}{K}_{1}^{2}\) with \({K}_{1}\mathrm{=}\frac{\alpha E}{P(1\mathrm{-}\nu )}{T}_{0}\sqrt{\mathit{Pa}}\)
That is: \(G\mathrm{=}\frac{(1\mathrm{-}{\nu }^{2})}{P{(1\mathrm{-}\nu )}^{2}}{\alpha }^{2}{\mathit{ET}}_{0}^{2}a\)
2.2. Benchmark result#
So the reference result is: \(G\mathrm{=}5.9115{10}^{\mathrm{-}7}N\mathrm{/}\mathit{mm}\)
\(t\) |
|
2e-3 |
5.677e5 |
4e-3 |
1.135e6 |
6e-3 |
1.703e6 |
2.3. Bibliographical reference#
Transient stress intensity factors for a finite crack in an elastic solid caused by a dilatational wave, International Journal of Solids and Structures 7, THAU and LU (1971)
Dynamic Fracture Mechanics L.B FREUND.