Modeling A ============== In this modeling, the extended finite element method (:math:`\text{X-FEM}`) is used. A radius of geometric enrichment is defined with a number of element layers equal to 3. Characteristics of the mesh ---------------------------- The structure is modelled by a regular mesh composed of :math:`100\times 100` QUAD4, respectively along the :math:`x,y` axes. The crack is not meshed. .. image:: images/10000000000001860000033A3B2D7B4CE7FC8440.png :width: 2.1811in :height: 3.8756in .. _RefImage_10000000000001860000033A3B2D7B4CE7FC8440.png: **Figure** 2.1-1 **: mesh of the cracked plate** Tested sizes and results ------------------------------ For each value of the angle :math:`\theta`, we test the value of the stress intensity factors :math:`{K}_{I}` and :math:`{K}_{\mathrm{II}}` given by CALC_G (for the two crack backgrounds) as well as those given by K1 and K2 from POST_K1_K2_K3 (for the two crack bottoms). For method :math:`G-\mathrm{thêta}` (command CALC_G), two choices of theta field crowns are tested: * :math:`\mathrm{C1}`: :math:`{R}_{\mathrm{inf}}=\mathrm{0,1}a` and :math:`{R}_{\text{sup}}=\mathrm{0,3}a`; * :math:`\mathrm{C2}`: :math:`{R}_{\mathrm{inf}}=h` and :math:`{R}_{\text{sup}}=\mathrm{3h}`; where :math:`h` is the characteristic mesh size: :math:`h=\sqrt{{(\frac{\mathrm{LX}}{\mathrm{NX}})}^{2}+{(\frac{\mathrm{LY}}{\mathrm{NY}})}^{2}}` For the method by extrapolation of movement jumps (POST_K1_K2_K3), the maximum curvilinear abscissa is equal to :math:`\mathrm{0,3}a`. Results for thêta= 0° ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. csv-table:: "**Identification**", "**Reference**", "**Tolerance**" "CALC_G ", "", "" "C1 + background1: K1", "2.5725 105"," 2.0%" "C1 + background2: K1", "2.5725 105"," 2.0%" "C1 + background1: K2", "0", "257" "C1 + background2: K2", "0", "257" "C1 + background1: G", "0.29"," 2.0%" "C1 + background2: G", "0.29"," 2.0%" "C2 + background1: K1", "2.5725 105"," 2.0%" "C2 + background2: K1", "2.5725 105"," 2.0%" "C2 + background1: K2", "0", "257" "C2 + background2: K2", "0", "257" "C2 + background1: G", "0.29"," 2.0%" "C2 + background2: G", "0.29"," 2.0%" "POST_K1_K2_K3 ", "", "" "background 1: K1", "2.5725 105"," 2.00%" "background 2: K1", "2.5725 105"," 2.00%" "background 1: K2", "0", "257" "background 2: K2", "0", "257" The zero values of :math:`{K}_{2}` are tested in absolute terms with a tolerance equal to :math:`{K}_{1}^{\mathit{ref}}\mathrm{/}1000`. Results for thêta= 15° ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. csv-table:: "**Identification**", "**Reference**" "CALC_G ", "" "C1 + background1: K1", "2.4001 105" "C1 + background2: K1", "2.4001 105" "C1 + background1: K2", "6.4313 104" "C1 + background2: K2", "6.4313 104" "C2 + background1: K1", "2.4001 105" "C2 + background2: K1", "2.4001 105" "C2 + background1: K2", "6.4313 104" "C2 + background2: K2", "6.4313 104" "POST_K1_K2_K3 ", "" "background1: K1", "2.4001.105" "background 2: K1", "2 400 1 105" "background 1: K2", "6, 4313 104" "background 2: K2", "6, 4313 104" Results for thêta= 30° ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. csv-table:: "**Identification**", "**Reference**" "CALC_G ", "" "C1 + background1: K1", "1.9294 105" "C1 + background2: K1", "1.9294 105" "C1 + background1: K2", "1,139 105" "C1 + background2: K2", "1,139 105" "C2 + background1: K1", "1.9294 105" "C2 + background2: K1", "1.9294 105" "C2 + background1: K2", "1,139 105" "C2 + background2: K2", "1,139 105" "POST_K1_K2_K3 ", "" "background1: K1", "1.9294 105" "background 2: K1", "1,9294 105" "background 1: K2", "1,139 105" "background 2: K2", "1,139 105" Results for thêta= 45° ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. csv-table:: "**Identification**", "**Reference**" "CALC_G ", "" "C1 + background1: K1", "1.2863 105" "C1 + background2: K1", "1.2863 105" "C1 + background1: K2", "1.2863 105" "C1 + background2: K2", "1.2863 105" "C2 + background1: K1", "1.2863 105" "C2 + background2: K1", "1.2863 105" "C2 + background1: K2", "1.2863 105" "C2 + background2: K2", "1.2863 105" "POST_K1_K2_K3 ", "" "background 1: K1", "1, 2 863 105" "background 2: K1", "1, 2 863 105" "background 1: K2", "1, 2863 105" "background 2: K2", "1, 2863 105" Results for thêta= 60° ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. csv-table:: "**Identification**", "**Reference**" "CALC_G ", "" "C1 + background1: K1", "6.4313 104" "C1 + background2: K1", "6.4313 104" "C1 + background1: K2", "1,140 105" "C1 + background2: K2", "1,140 105" "C2 + background1: K1", "6.4313 104" "C2 + background2: K1", "6.4313 104" "C2 + background1: K2", "1,140 105" "C2 + background2: K2", "1,140 105" "POST_K1_K2_K3 ", "" "background 1: K1", "6, 43 13 104" "background 2: K1", "6, 43 13 104" "background 1: K2", "1,140 105" "background 2: K2", "1,140 105" Additional results ------------------------- Other :math:`\theta` angle values have been tested elsewhere, without being part of this test. They are carried over to :ref:`Figure 2.3-1 `. .. image:: images/1000000000000CE4000009F6E3F578DD6FCBE50E.png :width: 6.3575in :height: 4.139in .. _RefImage_1000000000000CE4000009F6E3F578DD6FCBE50E.png: .. _Ref193799714: **Figure** 2.3-1 **: stress intensity factors normalized by** :math:`{K}_{I}` **(theta= 0°) obtained by** CALC_G **for the crown** :math:`\mathrm{C1}` As an illustration, :ref:`Figure 2.3-2 ` shows a view of the deformation of the plate for an angle :math:`\theta =45°`. .. image:: images/10000000000004680000040CD96680AB2EF0C302.png :width: 3.9827in :height: 6.3236in .. _RefImage_10000000000004680000040CD96680AB2EF0C302.png: .. _Ref193799375: **Figure** 2.3-2 **: deformed for theta= 45°**