Benchmark solution ===================== Benchmark results ---------------------- Calculation of stress intensity factors ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The reference solution for an opening crack of depth *a* in a plate of thickness :math:`W` stressed in pure :math:`I` mode (force :math:`\sigma` along the axis :math:`y`) is as follows: :math:`{K}_{I}\mathrm{=}Y\sigma \sqrt{a}`, with :math:`Y\mathrm{=}1.99\mathrm{-}0.41\frac{a}{W}+18.7{(\frac{a}{W})}^{2}\mathrm{-}38.48{(\frac{a}{W})}^{3}+53.85{(\frac{a}{W})}^{4}` *Numerical application:* for a crack :math:`15\mathit{mm}` deep (:math:`W\mathrm{=}72\mathit{mm}`) and the load :math:`{f}^{0}`, we get :math:`{K}_{I}\mathrm{=}\mathrm{2,874}\mathit{MPa.}\sqrt{\mathit{mm}}`. .. _Ref164581161: Calculation of the stresses around the crack bottom ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The analytical solution for the stress on a circle with radius :math:`r` around the crack bottom, obtained for a crack in an infinite environment, is as follows: :math:`{\sigma }_{\theta \theta }\mathrm{=}\frac{{K}_{I}}{4\sqrt{2\pi r}}(3\mathrm{cos}\frac{\theta }{2}+\mathrm{cos}\frac{3\theta }{2})` *Digital application:* for loading :math:`{f}^{0}` and for radius :math:`r\mathrm{=}{R}_{\mathit{AMORC}}\mathrm{=}\mathrm{0,046}\mathit{mm}`, we get: :math:`\theta \mathrm{=}0`: :math:`{\sigma }_{\theta \theta }\mathrm{=}53.5\mathit{MPa}`; :math:`\theta \mathrm{=}\pi \mathrm{/}2`: :math:`{\sigma }_{\theta \theta }\mathrm{=}18.9\mathit{MPa}` Calculation of the priming factor ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The priming factor associated with the geometric singularity in :math:`P` is calculated analytically using the procedure prescribed in RCC -M [:ref:`3 <3>`]. All the moments provided are considered to be local extremes and are therefore retained in the calculation. We note :math:`{s}^{0}` the value of :math:`{\sigma }_{\theta \theta }` taken at a radius :math:`r` from the bottom of the crack, at an angle :math:`\theta` given for loading :math:`{f}^{0}`. We can then build the table of the amplitudes of variation of :math:`{\sigma }_{\theta \theta }` for all the possible combinations: +----------------------------------------+-------------------------------------------------+----------------------------------------+-----------+-------------+--------------------------------------------+-------------+------------+ |Transitional 1 |Transitional 2 |:math: `10s°` |:math: `110s°` |:math: `50s°` | +----------------------------------------+-------------------------------------------------+----------------------------------------+-----------+-------------+ + |:math:`{\mathrm{Nb}}_{\mathrm{occur}}=i` |:math:`{\mathrm{Nb}}_{\mathrm{occur}}=j` | | +----------------------------------------+-------------------------------------------------+----------------------------------------+-----------+-------------+ + |:math:`t=0` |:math:`t=1` |:math:`t=0` |:math:`t=1`|:math:`t=2` | | +----------------------------------------+-------------------------------------------------+----------------------------------------+-----------+-------------+ + |Transitional 1 |:math:`{\mathit{Nb}}_{\mathit{occur}}\mathrm{=}i`|:math:`t=0` | |:math:`100s°`| | + + +----------------------------------------+-----------+-------------+--------------------------------------------+-------------+------------+ | | |:math:`t=1` | | |:math:`90s°` |:math:`10s°` |:math:`50s°`| +----------------------------------------+-------------------------------------------------+----------------------------------------+-----------+-------------+--------------------------------------------+-------------+------------+ |Transitional 2 |:math:`{\mathrm{Nb}}_{\mathrm{occur}}=j` |:math:`t=0` | | | |:math:`100s°`|:math:`40s°`| + + +----------------------------------------+-----------+-------------+--------------------------------------------+-------------+------------+ | | |:math:`t=1` | | | | |:math:`60s°`| + + +----------------------------------------+-----------+-------------+--------------------------------------------+-------------+------------+ | | |:math:`t=2` | | | | | | +----------------------------------------+-------------------------------------------------+----------------------------------------+-----------+-------------+--------------------------------------------+-------------+------------+ Taking into account the load ratio :math:`R` of each combination makes it possible to calculate the amplitude of variation of the effective stresses :math:`\Delta {\sigma }_{\mathit{eff}}`: +----------------------------------------+----------------------------------------+----------------------------------------+-----------+-------------+--------------------------------------------+------------------------+-----------------------+ |Transitional 1 |Transitional 2 |:math: `10s°` |:math: `110s°` |:math: `50s°` | +----------------------------------------+----------------------------------------+----------------------------------------+-----------+-------------+ + |:math:`{\mathrm{Nb}}_{\mathrm{occur}}=i` |:math:`{\mathrm{Nb}}_{\mathrm{occur}}=j` | | +----------------------------------------+----------------------------------------+----------------------------------------+-----------+-------------+ + |:math:`t=0` |:math:`t=1` |:math:`t=0` |:math:`t=1`|:math:`t=2` | | +----------------------------------------+----------------------------------------+----------------------------------------+-----------+-------------+ + |Transitional 1 |:math:`{\mathrm{Nb}}_{\mathrm{occur}}=i`|:math:`t=0` | |:math:`100s°`| | + + +----------------------------------------+-----------+-------------+--------------------------------------------+------------------------+-----------------------+ | | |:math:`t=1` | | |:math:`\mathrm{95,6}s°` |:math:`\mathrm{22,6}s°` |:math:`\mathrm{72,1}s°`| +----------------------------------------+----------------------------------------+----------------------------------------+-----------+-------------+--------------------------------------------+------------------------+-----------------------+ |Transitional 2 |:math:`{\mathrm{Nb}}_{\mathrm{occur}}=j`|:math:`t=0` | | | |:math:`\mathrm{105,9}s°`|:math:`\mathrm{45,6}s°`| + + +----------------------------------------+-----------+-------------+--------------------------------------------+------------------------+-----------------------+ | | |:math:`t=1` | | | | |:math:`\mathrm{83,6}s°`| + + +----------------------------------------+-----------+-------------+--------------------------------------------+------------------------+-----------------------+ | | |:math:`t=2` | | | | | | +----------------------------------------+----------------------------------------+----------------------------------------+-----------+-------------+--------------------------------------------+------------------------+-----------------------+ The calculation of the priming factor :math:`\mathrm{FA}` is carried out according to an iterative process: 1. identification of the transient number of non-zero occurrences leading to the maximum of .. image:: images/Object_17.svg :width: 37 :height: 25 .. _RefImage_Object_17.svg: ; 2. calculation of the associated elementary priming factor, using the law of fatigue; 3. updating the number of occurrences of the processed combination. Two cases are studied in succession with the numbers of occurrences of the following two transients: :math:`i=j=1` and :math:`i=\mathrm{1 };j=2`. We denote :math:`{N}_{\mathrm{kl}-\mathrm{mn}}` the number of cycles admissible for the combination of the :math:`l` -th time step of the -th time step, the :math:`k` -th transitory and the :math:`n` -th time step of the :math:`m` -th transient (calculated from .. image:: images/Object_18.svg :width: 37 :height: 25 .. _RefImage_Object_18.svg: of this combination and of the law of fatigue). **1st case:** .. image:: images/Object_19.svg :width: 37 :height: 25 .. _RefImage_Object_19.svg: **2nd case:** .. image:: images/Object_20.svg :width: 37 :height: 25 .. _RefImage_Object_20.svg: *Numerical application:* for modeling A of this test case, we assume that :math:`s°=1`. So we have: :math:`{\mathrm{FA}}_{1}=\mathrm{2,005}{.10}^{-\mathrm{10 }}`; F :math:`{A}_{2}=\mathrm{2,710}{.10}^{-10}` For B modeling, the value of :math:`s°` depends on the angle .. image:: images/Object_21.svg :width: 37 :height: 25 .. _RefImage_Object_21.svg: , and can be calculated using the formula in paragraph :ref:`2.1.2 `. .. image:: images/Object_22.svg :width: 37 :height: 25 .. _RefImage_Object_22.svg: : .. image:: images/Object_23.svg :width: 37 :height: 25 .. _RefImage_Object_23.svg: , :math:`{\mathrm{FA}}_{1}=\mathrm{0,607}`; :math:`{\mathrm{FA}}_{2}=\mathrm{0,819}` .. image:: images/Object_24.svg :width: 37 :height: 25 .. _RefImage_Object_24.svg: : .. image:: images/Object_25.svg :width: 37 :height: 25 .. _RefImage_Object_25.svg: , :math:`{\mathrm{FA}}_{1}=\mathrm{2,01}{.10}^{-3}`; :math:`{\mathrm{FA}}_{2}=\mathrm{2,72}{.10}^{-3}` Bibliographical references --------------------------- 1. W.F. BROWN, J.E. STRAWLEY: "Plane Strain Crack Toughness Testing of High Strength Material", American Society of Testing and Materials, STP 410 2. J.B. LEBLOND: "Mechanics of fragile and ductile fracture", Lavoisier, Paris, 2003. 3. RCC -M: "Mechanics of fragile and ductile fracture", Lavoisier, Paris, 2003.