Modeling A ============== Characteristics of modeling ----------------------------------- This modeling is carried out on a cube that is free to deform. Charging is a temperature and drying that vary over time. **# Common list for** interpolation **of temperature and drying** L_evol **= DEFI_LIST_REEL (VALE =(** 0.0, 2.0, 2.0, 2.0, 2.0, 10.0, 10.0, 200.0, **))** **# Temperature Field** TFONC **= DEFI_FONCTION (NOM_PARA =' INST '**, **PROL_DROITE =' CONSTANT '**, **PROL_GAUCHE =' CONSTANT '**, **VALE =(** 0.0, 10.0, 2.0, 10.0, 10.0, 40.0, 40.0, 50.0, 100.0, 20.0, 200.0, 50.0, **)**, **)** **# Drying Field** SFONC **= DEFI_FONCTION (NOM_PARA =' INST '**, **PROL_DROITE =' CONSTANT '**, **PROL_GAUCHE =' CONSTANT '**, **VALE =(** 0.0, 0.30, 2.0, 0.30, 50.0, 0.60, 80.0, 0.30, 100.0, 0.70, 110.0, 0.70, **)**, **)** The figure shows the evolution of loading over time. .. image:: images/1000020100000849000005A96D8C02F88684F931.png :width: 6.2992in :height: 4.7083in .. _RefImage_1000020100000849000005A96D8C02F88684F931.png: Figure 2.1-a: Evolution of temperature loading and drying. Benchmark solution --------------------- The equations are taken from [:ref:`R7.01.26 `]. Chemical advancement ~~~~~~~~~~~~~~~~~~~~~ Chemical advancement follows the following law: :math:`\frac{\partial A}{\partial t}\phantom{\rule{2em}{0ex}}=\phantom{\rule{2em}{0ex}}{a}_{0}.\mathrm{exp}\left[\frac{{E}_{a}}{R}\left(\frac{1}{{T}_{\mathit{ref}}}-\frac{1}{T}\right)\right]\frac{{⟨{S}_{r}-{S}_{r}^{0}⟩}_{\text{+}}}{\left(1-{S}_{r}^{0}\right)}{⟨{S}_{r}-A⟩}_{\text{+}}` [:ref:`éq2.2.1-1 <éq2.2.1-1>`] This equation depends on temperature and drying. It can therefore be verified without knowing the state of deformation and stress of the sample. Water pressure ~~~~~~~~~~~~~~~ Capillary pressure is related to the degree of saturation, by the relationship: :math:`\mathit{Pc}=-a.{S}_{r}.{({\mathit{Sr}}^{-b}-1)}^{1-1/b}` [:ref:`éq2.2.2-1 <éq2.2.2-1>`] This equation only depends on drying. It can therefore be verified without knowing the state of deformation and stress of the sample. Gel pressure ~~~~~~~~~~~~~~~~ The effective freeze pressure follows the following law: :math:`{P}_{\mathit{gel}}={b}_{g}.{M}_{g}{⟨A.{V}_{g}-{⟨{A}_{0}.{V}_{g}+{b}_{g}\mathit{tr}({\mathrm{\epsilon }}^{\mathit{total}})⟩}_{\text{+}}⟩}_{\text{+}}` [:ref:`éq2.2.3-1 <éq2.2.3-1>`] This equation depends on chemical progress and on the trace of deformation. It can therefore only be verified by knowing the state of deformation of the sample. Mechanical condition of the sample ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ To solve the preceding equations it is necessary to determine the deformations. The following equations govern the problem: :math:`{\mathrm{\epsilon }}^{\text{e}}=\frac{1+{\mathrm{\nu }}^{0}}{{E}^{0}}.{\mathrm{\sigma }}^{\text{e}}-\frac{{\mathrm{\nu }}^{0}}{{E}^{0}}\text{tr}({\mathrm{\sigma }}^{\text{e}}).\mathit{Id}` with: :math:`{\mathrm{\epsilon }}^{\text{total}}={\mathrm{\epsilon }}^{\text{e}}+{\mathrm{\epsilon }}^{\text{fluage}}+{\mathrm{\epsilon }}^{\text{vdt}}+{\mathrm{\epsilon }}^{\text{ther}}` :math:`{\mathrm{\sigma }}^{\mathit{total}}={\mathrm{\sigma }}^{\text{e}}-{P}_{\mathit{gel}}.\mathit{Id}-{P}_{c}.\mathit{Id}` We are in the case where :math:`{\mathrm{\sigma }}^{\mathit{total}}=0` due to boundary conditions. The material parameters are chosen so that :math:`{\mathrm{\epsilon }}^{\text{fluage}}=0`, :math:`{\mathrm{\epsilon }}^{\text{vdt}}=0`, :math:`{\mathrm{\epsilon }}^{\text{ther}}=0`. So we have :math:`\mathit{tr}({\mathrm{\epsilon }}^{\text{total}})=\frac{3.(1-2.{\mathrm{\nu }}^{0})}{{E}^{0}}.({P}_{\mathit{gel}}+{P}_{c})` [:ref:`éq2.2.4-1 <éq2.2.4-1>`] Tested sizes and results ------------------------------ For chemical advancement, the corresponding internal variable is BR_AVCHI. The figure represents its evolution over time. .. csv-table:: "**Instant**", "**Reference Value**", "**Precision**" "**10.00**", "4.981207357356948E-03", "1.0E-03" "**40.00**", "5.361979299970E-01", "1.0E-03", "5.361979299970E-01" "**50.00**", "5.924644217241E-01", "1.0E-03" "**80.00**", "5.938195769343E-01", "1.0E-03", "5.938195769343E-01" "**100.00**", "5.938216310035E-01", "1.0E-03" "**110.00**", "5.9383380868086705E-01", "1.0E-03" "**200.00**", "7.000000000002E-01", "1.0E-03" .. image:: images/1000020100000804000005BE39CC600AB4B83752.png :width: 5.9055in :height: 4.2283in .. _RefImage_1000020100000804000005BE39CC600AB4B83752.png: Figure 2.3-a: Chemical progress, comparison between the theoretical solution and that obtained by code_aster. For gel pressure, the corresponding internal variable is BR_PRGEL. The figure represents its evolution over time. .. csv-table:: "**Instant**", "**Reference Value**", "**Precision**" "**40.00**", "2.101217954854E+06", "1.0E-03" "**50.00**", "2.448598919982E+06", "1.0E-03", "2.448598919982E+06" "**80.00**", "2.460563938661E+06", "1.0E-03", "2.460563938661E+06" "**100.00**", "2.455008850947E+06", "1.0E-03" "**110.00**", "2.455084251465E+06", "1.0E-03" "**200.00**", "3.112433957654E+06", "1.0E-03" .. image:: images/100002010000080D000005A3339714A6F5E1DC12.png :width: 5.9055in :height: 4.1256in .. _RefImage_100002010000080D000005A3339714A6F5E1DC12.png: Figure 2.3-b: Gel pressure, comparison between the theoretical solution and that obtained by code_aster. For capillary pressure, the corresponding internal variable is BR_PRCAP. The figure represents its evolution over time. .. csv-table:: "**Instant**", "**Reference Value**", "**Precision**" "**10.00**", "-2.810249099279E+06", "1.0E-03" "**40.00**", "-2.529791246328E+06", "1.0E-03" "**50.00**", "-2.400000000000E+06", "1.0E-03" "**80.00**", "-2.861817604251E+06", "1.0E-03" "**100.00**", "-2.142428528528563E+06", "1.0E-03" "**110.00**", "-2.142428528528563E+06", "1.0E-03" "**200.00**", "-2.142428528528563E+06", "1.0E-03" .. image:: images/1000020100000813000005A6CD7E36F7C3EF36B9.png :width: 5.9055in :height: 4.1256in .. _RefImage_1000020100000813000005A6CD7E36F7C3EF36B9.png: **Figure** 2.3-c **:**: ** Gel pressure, comparison between the theoretical solution and that obtained by code_aster. The quantities tested are the deformations. The names of the components in field EPSI_ELGA are EPXX, EPYY, EPZZ. The figure shows the evolution of the deformation trace over time. Given the symmetry of the problem the 3 deformations are identical, so we have: :math:`\mathit{EPXX}=\mathit{EPYY}=\mathit{EPZZ}=\mathit{Trace}(\mathit{Epsi})/3` .. csv-table:: "**Instant**", "**Reference Value**", "**Precision**" "**10.00**", "-4.391014214217624E-05", "1.0E-03" "**40.00**", "-6.696457679679281E-06", "2.1E-03" "**50.00**", "7.59358121247263E-07", "1.6E-02" "**80.00**", "-6.269588588524836E-06", "1.0E-03" "**100.00**", "4.88406757537256E-06", "3.2E-03" "**110.00**", "4.885245670348E-06", "1.0E-03" "**200.00**", "1.515633483482955E-05", "1.0E-03" .. image:: images/1000020100000831000005A35727822822715084.png :width: 5.9055in :height: 4.0634in .. _RefImage_1000020100000831000005A35727822822715084.png: Figure 2.3-d: Trace of deformations, comparison between the theoretical solution and that obtained by code_aster.