6. Formalization of endogenous withdrawal and desiccation#
6.1. Withdrawal in Code_Aster#
In the context of formalizing shrinkage in terms of deformation, the total deformation increment can be broken down into the sum of a thermal component, a component representing endogenous shrinkage, and a component representing desiccation shrinkage, added to the mechanical component (elasticity, creep,…).
The easiest way to represent desiccation shrinkage is to put it in the form:
\(\mathrm{\Delta }{\mathrm{\epsilon }}_{\text{dessiccation}}=\left[-\mathrm{\kappa }\left({C}_{0}-{C}^{+}\right)+\mathrm{\kappa }\left({C}_{0}-{C}^{-}\right)\phantom{\rule{0.5em}{0ex}}\right]\text{.}\phantom{\rule{0.5em}{0ex}}{I}^{d}=-\mathrm{\kappa }\mathrm{\Delta }C\text{.}\phantom{\rule{0.5em}{0ex}}{I}^{d}\) eq 6.1-1
where \(C\) is the water concentration, \({C}_{0}\) the initial water concentration.
and \(\kappa\) a coefficient characterizing the shrinkage, assumed to be constant.
Endogenous withdrawal can be modelled as:
\(\Delta {\varepsilon }_{\text{endogène}}=-\beta \text{.}\Delta \xi \text{.}{I}^{d}\) eq 6.1-2
where \(\xi\) is hydration, and \(\beta\) is a coefficient characteristic of the material whose dependencies are poorly known.
Desiccation and endogenous withdrawals can therefore intervene by a law of behavior by replacing the usual terms \(\Delta \varepsilon -\Delta {\varepsilon }_{\text{thermique}}\) by \(\Delta \varepsilon -\Delta {\varepsilon }_{\text{thermique}}-\Delta {\varepsilon }_{\text{dessiccation}}-\Delta {\varepsilon }_{\text{endogène}}\). In Code_Aster, these terms are taken into account for elastic and elastoplastic behaviors of the Von Mises type and as well as several concrete models [3] _ (BETON_UMLV, BETON_BURGER, BETON_DOUBLE_DP, ENDO_ISOT_BETON, MAZARS). For example, in 1D elasticity, we then have:
\(\Delta \varepsilon =\frac{1}{E(\xi )}\Delta \sigma +(\Delta {\varepsilon }_{\text{thermique}}+\Delta {\varepsilon }_{\text{endogène}}+\text{.}\Delta {\varepsilon }_{\text{dessiccation}})\) eq 6.1-3
The mechanical parameter \(E\) (Young’s modulus) depends mainly on the hydration variable \(\xi\).
This formulation of desiccation withdrawal and endogenous withdrawal has the advantage of directly using the \(C\) water content, which can be linked to weight loss by simple integration into the volume. If relative humidity \(h\) was used, it would have to be translated into terms of water content through the desorption isotherm of each of the different concretes.
For the*Code_Aster*, these parameters can be defined in a relatively general framework, as functions of the various calculation variables and auxiliary variables (temperature, hydration, water concentration, or constants) to give the user the choice to freely define the dependencies of the parameters. It remains up to the user to use the Code_Aster functions to reproduce the Young’s modulus expression given in equation [éq 6.1-3].
For more detail on these formulations, and on the means of calculating the coefficients \(\kappa\) and \(\beta\), reference will be made to L. Granger’s thesis, [bib2], on pages 99 and following, and on pages 210 and following.
For mechanical calculation the variables \(\xi\) (hydration) and \(C\) (water concentration) are data, as is the temperature during a thermomechanical calculation.
6.2. Integration of shrinkage into mechanical laws of behavior#
Thermal and drying are decoupled from mechanical resolution, just as drying is an operation linked to thermal and hydration. This decoupling makes it possible to integrate shrinkage into the nonlinear mechanics resolution operator, without adding new phenomena, behaviors, types of elements and calculation options. In addition, it makes it possible to introduce withdrawal in a simple way into all non-linear laws of behavior. The syntax of the mechanics operators STAT_NON_LINE and MECA_STATIQUE is not changed.
In the current version of the nonlinear mechanics operator, shrinkage has been integrated into elastic behavior (ELAS), elastoplastic Von Mises behavior (VMIS_ISOT_ ) and models specific to concrete: MAZARS, ENDO_ISOT_BETON, BETON_DOUBLE_DP,,, GRANGER, BETON_UMLV_FP. It consists in removing the terms of shrinkage from the total deformation, before solving the equilibrium equations at the Gauss points, in the same way that thermal expansion is taken into account.
The coefficients \(\kappa\) and \(\beta\) characterizing endogenous and desiccation withdrawals are defined under the keyword “ELAS_FO” (operands K_ DESSIC and B_ ENDOGE respectively), as constants. The other mechanical characteristics, Poisson’s ratio and Young’s modulus can also be defined as functions of the new variables HYDR and SECH, which have been added to the catalogs of the two operators DEFI_FONCTION and DEFI_NAPPE.
The results, of the [evol_ther] type, resulting from a non-linear thermal calculation, or thermohydration, and from a drying calculation, and corresponding respectively to the thermo-hydraulic fields of type “TEMP/HYDR”, or to the drying field of type “TEMP”, or to the drying field of type “”, are transmitted to the mechanical calculation by means of the operator AFFE_MATERIAU (keyword AFFE_VARC). Mechanical calculations combining these fields allow:
to calculate endogenous and desiccation withdrawals, in the case where the associated material characteristics have been defined previously in DEFI_MATERIAU,
to interpolate the Young’s modulus and the Poisson’s ratio when these are functions of the hydration or drying variables.
Note 1:
In the presence of a drying field, it is necessary to enter the keyword VALE_REFdans the command AFFE_MATERIAU. This value defines the value of SECHpour where the desiccation shrinkage is zero.
You must therefore be careful to be consistent with the values SECHutilisés (especially at the initial moment!).
Note 2:
Drying and hydration are not taken into account for structural elements. For this type of element, it will be necessary to define coefficients \(\kappa\) and \(\beta\) zero, otherwise the calculation will stop in a fatal error.
6.3. Tangent matrix#
The calculation of the tangent matrices of the various laws of nonlinear behavior is not affected by the addition of endogenous shrinkage and desiccation shrinkage, because derivatives with respect to the hydration and drying variables, terms of the equilibrium equations, as well as the derivatives with respect to temperature of these same terms are usually neglected. These derivatives occur in the second order.