Reference problem ===================== Geometry --------- .. image:: images/Shape11.gif .. _RefSchema_Shape11.gif: .. image:: images/Shape12.gif .. _RefSchema_Shape12.gif: .. image:: images/Shape13.gif .. _RefSchema_Shape13.gif: F1 D1 C1 B1 E1 A1 .. image:: images/Shape14.gif .. _RefSchema_Shape14.gif: .. image:: images/Shape15.gif .. _RefSchema_Shape15.gif: Ux Linear relationships Material properties ----------------------- The parameters of the laws of behavior are as follows: **For mechanical characteristics in linear elasticity (ELAS) :** .. csv-table:: "Young's module:", ":math:`E=31000\text{MPa}`" "*Poisson's ratio:*", ":math:`\nu \mathrm{=}0.2`" "*Coefficient of thermal expansion:*", ":math:`\alpha \mathrm{=}{10}^{\mathrm{-}5}`" "*Desiccation shrinkage coefficient:*", ":math:`\kappa \mathrm{=}{10}^{\mathrm{-}5}`" **For the non-linear mechanical characteristics of the model** **** BETON_DOUBLE_DP **** **:** +-------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+++ |*Uniaxial compression strength:* |:math:`f\text{'}c\mathrm{=}40N\mathrm{/}{\mathit{mm}}^{2}` ||| +-------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+++ |*Uniaxial tensile strength:* |:math:`f\text{'}t\mathrm{=}4N\mathrm{/}{\mathit{mm}}^{2}` ||| +-------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+++ |*Ratio of resistances in biaxial compression/uniaxial compression:*|:math:`\beta \mathrm{=}1.16` ||| +-------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+++ |*Breakdown energy in compression:* |:math:`\mathit{Gc}\mathrm{=}10\mathit{Nmm}\mathrm{/}{\mathit{mm}}^{2}` ||| +-------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+++ |*Tensile breaking energy:* |:math:`\text{Gt}\mathrm{=}10000\mathit{Nmm}\mathrm{/}{\mathit{mm}}^{2}` ***to simulate almost zero work hardening* | +-------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+++ |*Ratio of the yield strength to the uniaxial compression strength:*|:math:`33.33\text{\%}` ||| +-------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+++ **For the mechanical characteristics of the linear work hardening model** VMIS_ISOT_LINE **:** +-----------------------+----------------------------------------------------------------+++ |*Elastic limit:* |:math:`\mathit{Sy}\mathrm{=}4N\mathrm{/}\mathit{mm²}` ||| +-----------------------+----------------------------------------------------------------+++ |*Work hardening slope:*|:math:`\text{D\_sigm\_epsi}\mathrm{=}0.1N\mathrm{/}\mathit{mm²}` || +-----------------------+----------------------------------------------------------------+++ **For the mechanical characteristics of the creep model of** **GRANGER** ****: ** +-----------------------------------------------------------------------------------------------------+--------------------------------------------------------+ |*Coefficient* :math:`{J}_{1}` *:* |:math:`{J}_{1}\mathrm{=}0.2{\mathit{MPa}}^{\mathrm{-}1}`| +-----------------------------------------------------------------------------------------------------+--------------------------------------------------------+ |*Coefficient* :math:`{\tau }_{1}` *:* |:math:`{\tau }_{1}\mathrm{=}4320000s` | +-----------------------------------------------------------------------------------------------------+--------------------------------------------------------+ |*Coefficient* :math:`Q\mathrm{/}R` *:* |:math:`\text{QsR\_K}\mathrm{=}0.\text{K}` | +-----------------------------------------------------------------------------------------------------+--------------------------------------------------------+ |*The desorption curve is equal to 1 for all values of humidity, to simplify the analytical solution.* | +-----------------------------------------------------------------------------------------------------+--------------------------------------------------------+ Boundary conditions and mechanical loads ------------------------------------------------ For calculations in :math:`\mathrm{3D}`: +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------+ | - :math:`x=0` face of the first cube (sa): |stuck following :math:`\mathit{ox}`, | + + + | | | +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------+ | - Face knots in :math:`y=0`: |stuck following :math:`\mathrm{oy}`, | + + + | | | +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------+ | - Face knots in :math:`z\mathrm{=}0`: |stuck following :math:`\mathrm{oz}`, | + + + | | | +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------+ | - Linear relationship (LIAISON_DDL) between the end nodes of the merged faces of the adjacent linear and quadratic elements (nodes :math:`\mathrm{c1}`, :math:`\mathrm{c2}`, :math:`\mathrm{c3}`, :math:`\mathrm{c4}` linked with the nodes :math:`\mathrm{d1}`, :math:`\mathrm{d2}`, :math:`\mathrm{d3}`, :math:`\mathrm{d4}`), | + + | | +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------+ | - Linear relationship (LIAISON_UNIF) on the :math:`\mathrm{sd}` face to link the displacements along :math:`\mathrm{ox}` of the quadratic nodes of this face to those of the vertex nodes, | + + | | +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------+ | - Face in :math:`x={x}_{\mathrm{max}}` of the last cube (:math:`\mathrm{sf}`): |Traction exerted following :math:`\mathit{ox}`.| + + + | | | +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------+ For calculations in :math:`\mathrm{2D}`: +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------+ | - Line in :math:`x=0` of the first square (the): |stuck following :math:`\mathrm{ox}`, | + + + | | | +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------+ | - Knots of the lines in :math:`y=0`: |stuck following :math:`\mathrm{oy}`, | + + + | | | +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------+ | - Linear relationship (LIAISON_DDL) between the end nodes of the merged lines of the adjacent linear and quadratic elements (nodes :math:`\mathrm{c1}`, :math:`\mathrm{c2}` linked with the nodes :math:`\mathrm{d1}`, :math:`\mathrm{d2}`), | + + | | +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------+ | - Linear relationship (LIAISON_UNIF) on line :math:`\mathrm{ld}` to link the displacements along :math:`x` of the quadratic nodes of this line to those of the vertex nodes, | + + | | +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------+ | - Line in :math:`x={x}_{\mathrm{max}}` of the last square (:math:`\mathrm{lf}`): |Traction exerted following :math:`\mathrm{ox}`.| + + + | | | +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------+ The temperature field is either constant (first calculation) or increasing from :math:`0°C` to :math:`20°C` for all other calculations. In the case where the temperature varies, it is assumed that the drying field varies from 1 to 0. The material characteristics are constant. In addition, a non-zero desiccation shrinkage coefficient is applied, in such a way that the desiccation shrinkage compensates for thermal expansion, to verify that these 2 phenomena are properly taken into account. **Note. The temperature variation impacts the calculation of withdrawals but not the law of mechanical behavior, which does not depend on temperature.*