Benchmark solution ===================== Calculation method ----------------- A, C, E, G modeling ~~~~~~~~~~~~~~~~~~~~~~~~~~ The stress intensity factor :math:`{K}_{I}` is tested in relation to the analytical value without contact. For a specimen in bending and a right crack with an abscissa :math:`[-a\phantom{\rule{2em}{0ex}},\phantom{\rule{2em}{0ex}}a]`, the analytical value of :math:`{K}_{I}` without closure of the crack is determined by the Bui formula [:ref:`7 `]: :math:`{K}_{I}\mathrm{=}\frac{1}{\sqrt{\pi a}}\underset{\mathrm{-}a}{\overset{a}{\mathrm{\int }}}\mathrm{-}t\sqrt{\frac{a\mathrm{-}t}{t+a}}\mathit{dt}` where :math:`a` refers to the half-length of the crack. From the integral above, we calculate the following simplified expression for :math:`{K}_{I}`: :math:`{K}_{I}\mathrm{=}\frac{{a}^{\frac{3}{2}}\sqrt{\pi }}{2}` Indeed, in the integral, several changes of variables are made: :math:`t\to \mathit{at}` we have :math:`{K}_{I}\mathrm{=}\frac{1}{\sqrt{\pi a}}{a}^{2}\underset{\mathrm{-}1}{\overset{1}{\mathrm{\int }}}\mathrm{-}t\sqrt{\frac{1\mathrm{-}t}{t+1}}\mathit{dt}` :math:`t\to t\mathrm{-}1` we have :math:`{K}_{I}\mathrm{=}\frac{1}{\sqrt{\pi a}}{a}^{2}\underset{0}{\overset{2}{\mathrm{\int }}}\mathrm{-}(t\mathrm{-}1)\sqrt{\frac{2\mathrm{-}t}{t}}\mathit{dt}` :math:`t\to \mathrm{2t}` we have :math:`{K}_{I}\mathrm{=}\frac{1}{\sqrt{\pi a}}{\mathrm{2a}}^{2}\underset{0}{\overset{1}{\mathrm{\int }}}\mathrm{-}(\mathrm{2t}\mathrm{-}1)\sqrt{\frac{1\mathrm{-}t}{t}}\mathit{dt}` :math:`t\to {t}^{2}` we have :math:`{K}_{I}\mathrm{=}\frac{1}{\sqrt{\pi a}}{\mathrm{2a}}^{2}\underset{0}{\overset{1}{\mathrm{\int }}}\mathrm{-}2({\mathrm{2t}}^{2}\mathrm{-}1)\sqrt{1\mathrm{-}{t}^{2}}\mathit{dt}` :math:`t\to \mathrm{sin}t` we have :math:`{K}_{I}\mathrm{=}\frac{1}{\sqrt{\pi a}}{\mathrm{2a}}^{2}\underset{0}{\overset{\frac{\pi }{2}}{\mathrm{\int }}}\mathrm{2cos}\mathrm{2t}{\mathrm{cos}}^{2}t\mathit{dt}` After product development by :math:`\mathrm{cos}`: :math:`{K}_{I}\mathrm{=}\frac{1}{\sqrt{\pi a}}{\mathrm{2a}}^{2}\underset{0}{\overset{\frac{\pi }{2}}{\mathrm{\int }}}(\frac{1}{2}+\mathrm{cos}\mathrm{2t}+\frac{1}{2}\mathrm{cos}\mathrm{4t})\mathit{dt}` Hence the final expression (after integration): :math:`{K}_{I}\mathrm{=}\frac{{a}^{\frac{3}{2}}\sqrt{\pi }}{2}` The half-length being unitary (:math:`a=1`), we therefore test the stress intensity factor: :math:`{K}_{I}=\frac{\sqrt{\mathrm{\pi }}}{2}\approx \mathrm{0,88629}` By construction, there is no mode II. So we test: :math:`{K}_{\mathit{II}}=0` On the other hand, the rate of return of analytical energy is equal to: :math:`G=\frac{1-{\mathrm{\nu }}^{2}}{E}({K}_{I}^{2}+{K}_{\mathit{II}}^{2})\approx \mathrm{7,853982}E-7` B, D, F modeling ~~~~~~~~~~~~~~~~~~~~~~ For a specimen in bending and a right crack with abscissa :math:`[-a\phantom{\rule{2em}{0ex}},\phantom{\rule{2em}{0ex}}a]`, the analytical value of :math:`{K}_{I}` with crack closure is determined by the Bui formula [:ref:`7 `]: :math:`{K}_{I}=\sqrt{\frac{2}{\pi (a-c)}}\underset{c}{\overset{a}{\int }}t\sqrt{\frac{t-c}{a-t}}\mathit{dt}` where :math:`c` is the abscissa of the point where the crack starts to open (see []) and :math:`a` refers to the half-length of the crack. We show analytically [:ref:`7 `] that the abscissa :math:`c\mathrm{=}\mathrm{-}\frac{a}{3}` So, :math:`{K}_{I}=\sqrt{\frac{3}{2\pi a}}\underset{-a/3}{\overset{a}{\int }}t\sqrt{\frac{t+a/3}{a-t}}\mathit{dt}` From the integral above, we calculate the following simplified expression for :math:`{K}_{I}`: :math:`{K}_{I}={(\frac{2}{3}a)}^{\frac{3}{2}}\sqrt{(\pi )}` Indeed, in the integral, several changes of variables are made: :math:`t\to \mathit{at}` we have :math:`{K}_{I}=\sqrt{\frac{3}{2\pi a}}{a}^{2}\underset{-1/3}{\overset{1}{\int }}t\sqrt{\frac{t+1/3}{1-t}}\mathit{dt}` :math:`t\to t-\frac{1}{3}` we have :math:`{K}_{I}=\sqrt{\frac{3}{2\pi a}}{a}^{2}\underset{0}{\overset{4/3}{\int }}(t-\frac{1}{3})\sqrt{\frac{t}{4/3-t}}\mathit{dt}` :math:`t\to \frac{4}{3}t` we have :math:`{K}_{I}=\sqrt{\frac{3}{2\pi a}}{a}^{2}\frac{4}{9}\underset{0}{\overset{1}{\int }}(\mathrm{4t}-1)\sqrt{\frac{t}{1-t}}\mathit{dt}` :math:`t\to {t}^{2}` we have :math:`{K}_{I}=\sqrt{\frac{3}{2\pi a}}{a}^{2}\frac{4}{9}\underset{0}{\overset{1}{\int }}({\mathrm{4t}}^{2}-1)\frac{{\mathrm{2t}}^{2}}{\sqrt{1-{t}^{2}}}\mathit{dt}` :math:`t\to \mathrm{sin}t` we have :math:`{K}_{I}=\sqrt{\frac{3}{2\pi a}}{a}^{2}\frac{4}{9}\underset{0}{\overset{\pi /2}{\int }}({\mathrm{4sin}}^{2}t-1)({\mathrm{2sin}}^{2}t)\mathit{dt}` After developing the powers of :math:`\mathrm{sin}t`: :math:`{K}_{I}=\sqrt{\frac{3}{2\pi a}}{a}^{2}\frac{4}{9}\underset{0}{\overset{\pi /2}{\int }}(2-\mathrm{3cos}\mathrm{2t}+\mathrm{cos}\mathrm{4t})\mathit{dt}` Hence the final expression (after integration): :math:`{K}_{I}\mathrm{=}{(\frac{2}{3}a)}^{\frac{3}{2}}\sqrt{(\pi )}` The half-length being unitary (:math:`a=1`), we therefore test the stress intensity factor: :math:`{K}_{I}={(\frac{2}{3})}^{\frac{3}{2}}\sqrt{(\mathrm{\pi })}\approx \mathrm{0,9648017}` By construction, there is no mode II. So we test: :math:`{K}_{\mathit{II}}=0` On the other hand, the rate of return of analytical energy is equal to: :math:`G=\frac{1-{\mathrm{\nu }}^{2}}{E}({K}_{I}^{2}+{K}_{\mathit{II}}^{2})\approx \mathrm{9,308423}E-7` Reference quantities and results ----------------------------------- We're testing :math:`{K}_{I}`, :math:`{K}_{\mathit{II}}`, and :math:`G`. A, C, E, F, G models ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ :math:`{K}_{I}=\frac{\sqrt{\mathrm{\pi }}}{2}\approx \mathrm{0,88629}` :math:`{K}_{\mathit{II}}\mathrm{=}0` :math:`G\mathrm{=}\frac{1\mathrm{-}{\nu }^{2}}{E}({K}_{I}^{2}+{K}_{\mathit{II}}^{2})\mathrm{\approx }\mathrm{7,853982}E\mathrm{-}7` B and D modeling ~~~~~~~~~~~~~~~~~~~~~ :math:`{K}_{I}={(\frac{2}{3})}^{\frac{3}{2}}\sqrt{(\mathrm{\pi })}\approx \mathrm{0,9648017}` :math:`{K}_{\mathit{II}}\mathrm{=}0` :math:`G=\frac{1-{\mathrm{\nu }}^{2}}{E}({K}_{I}^{2}+{K}_{\mathit{II}}^{2})\approx \mathrm{9,308423}E-7` Uncertainty about the solution --------------------------- Weak, semi-analytical solution. .. _RefNumPara__949_821110035: Bibliographical references --------------------------- .. _Ref112833260: .. _Ref112833703: 1. GENIAUT S., MASSIN P.: eXtended Finite Element Method, *Code_Aster* Reference Manual, [:ref:`R7.02.12 `] 2. GENIAUT S.: XFEM approach for contact cracking of industrial structures, PhD thesis GeM Laboratory, 2006. 3. H.D. Bui: Mechanics of brittle fracture, Ed. Masson, 1978. 4. THRESHER R.W., SMITH F.W.: The Partially Closed Griffith Crack, *International Journal of Fracture*, vol. 9, no. 1, pp. 33-41, 1973 5. Rice, J.R. (1968), "A path independent integral and the approximate analysis of strain concentration by notches and cracks," *Journal of Applied Mechanics* **35**: 379—386