1. Reference problem#

1.1. Geometry#

Linear relationships

1.2. Material properties#

The parameters of the laws of behavior are as follows:

For mechanical characteristics in linear elasticity (ELAS) :

Young’s module:	\(E=31000\text{MPa}\)
Poisson’s ratio:	\(\nu \mathrm{=}0.2\)
Coefficient of thermal expansion:	\(\alpha \mathrm{=}{10}^{\mathrm{-}5}\)
Desiccation shrinkage coefficient:	\(\kappa \mathrm{=}{10}^{\mathrm{-}5}\)

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:*|\(\beta \mathrm{=}1.16\) ||| +——————————————————————-+——————————————————————————————————————+++ |*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:*|\(33.33\text{\%}\) ||| +——————————————————————-+——————————————————————————————————————+++

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:*|\(\text{D\_sigm\_epsi}\mathrm{=}0.1N\mathrm{/}\mathit{mm²}\) || +———————–+—————————————————————-+++

For the mechanical characteristics of the creep model of GRANGER ****: **

Coefficient \({J}_{1}\) :	\({J}_{1}\mathrm{=}0.2{\mathit{MPa}}^{\mathrm{-}1}\)
Coefficient \({\tau }_{1}\) :	\({\tau }_{1}\mathrm{=}4320000s\)
Coefficient \(Q\mathrm{/}R\) :	\(\text{QsR\_K}\mathrm{=}0.\text{K}\)
The desorption curve is equal to 1 for all values of humidity, to simplify the analytical solution.

1.3. Boundary conditions and mechanical loads#

For calculations in \(\mathrm{3D}\):

\(x=0\) face of the first cube (sa):	stuck following \(\mathit{ox}\),

Face knots in \(y=0\):	stuck following \(\mathrm{oy}\),

Face knots in \(z\mathrm{=}0\):	stuck following \(\mathrm{oz}\),

Linear relationship (LIAISON_DDL) between the end nodes of the merged faces of the adjacent linear and quadratic elements (nodes \(\mathrm{c1}\), \(\mathrm{c2}\), \(\mathrm{c3}\), \(\mathrm{c4}\) linked with the nodes \(\mathrm{d1}\), \(\mathrm{d2}\), \(\mathrm{d3}\), \(\mathrm{d4}\)),

Linear relationship (LIAISON_UNIF) on the \(\mathrm{sd}\) face to link the displacements along \(\mathrm{ox}\) of the quadratic nodes of this face to those of the vertex nodes,

Face in \(x={x}_{\mathrm{max}}\) of the last cube (\(\mathrm{sf}\)):	Traction exerted following \(\mathit{ox}\).

For calculations in \(\mathrm{2D}\):

Line in \(x=0\) of the first square (the):	stuck following \(\mathrm{ox}\),

Knots of the lines in \(y=0\):	stuck following \(\mathrm{oy}\),

Linear relationship (LIAISON_DDL) between the end nodes of the merged lines of the adjacent linear and quadratic elements (nodes \(\mathrm{c1}\), \(\mathrm{c2}\) linked with the nodes \(\mathrm{d1}\), \(\mathrm{d2}\)),

Linear relationship (LIAISON_UNIF) on line \(\mathrm{ld}\) to link the displacements along \(x\) of the quadratic nodes of this line to those of the vertex nodes,

Line in \(x={x}_{\mathrm{max}}\) of the last square (\(\mathrm{lf}\)):	Traction exerted following \(\mathrm{ox}\).

The temperature field is either constant (first calculation) or increasing from \(0°C\) to \(20°C\) 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.*