8. Specific characteristics of concrete#
8.1. Keyword factor THER_HYDR#
Allows you to define the behavior associated with the hydration of concrete.
The hydration of concrete is a phenomenon that is accompanied by a release of heat depending on the temperature [R7.01.12].
\(\begin{array}{cc}\begin{array}{}\frac{d\beta }{\mathrm{dt}}+\text{div}q=Q\frac{d\xi (T)}{\mathrm{dt}}+s\\ q=-\lambda \mathrm{grad}T\end{array}\}& \text{éq 7.1-1}\\ \frac{d\xi }{\mathrm{dt}}=\mathrm{AFF}(\xi ,T)& \text{éq 7.1-2}\end{array}\)
8.1.1. Operands LAMBDA/BETA#
LAMBDA = normal
Isotropic thermal conductivity as a function of temperature.
BETA = beta
Volume enthalpy as a function of temperature. The extensions are at least linear, the volume enthalpy can be defined as the integral of the volume heat.
8.1.2. Operand AFFINITE#
AFFINITE = AFF
Depending on the degree of hydration and the temperature. In general, we use:
\(\text{AFF}(\xi ,T)=A(\xi )\mathrm{exp}(-\frac{{E}_{a}}{\mathrm{RT}})\) with \(\text{QSR\_K}=\frac{{E}_{a}}{R}\) the Arrhenius constant expressed in degrees Kelvin, and A determined by a calorimetric test of concrete (a function of the quantity HYDR).
8.1.3. Operand CHAL_HYDR#
CHAL_HYDR = Q
Heat released per hydration unit (assumed to be constant), this function depends on the type of concrete.
8.2. Keyword factor SECH_GRANGER#
Definition of the parameters characterizing the diffusion coefficient \(D\left(C,T\right)\) used in the nonlinear drying equation proposed by Granger (cf. [R7.01.12]). These characteristics are constants, while the diffusion coefficient depends on the calculation variable, i.e. the current \(C\) water concentration, (as thermal conductivity depended on temperature).
8.2.1. A/B operands/ QSR_K/TEMP_0_C#
These coefficients make it possible to express the diffusion coefficient in its form most commonly used in the literature and proposed by Granger:
\(D\left(C,T\right)=a\text{.}{e}^{\left(b\text{.}C\right)}\frac{T}{{T}_{0}}{e}^{\left[-\frac{Q}{R}\left(\frac{1}{T}-\frac{1}{{T}_{0}}\right)\right]}\)
A= a
Diffusion coefficient varying from \(0.5{10}^{-13}\) and \({2.10}^{-13}{m}^{2}/s\) for concrete.
B= b
Coefficient of the order of \(0.05\) for concrete.
QSR_K = QsR
QsR is generally \(4700.K\). (\(R\) is the ideal gas constant).
TEMP_0_C = T0
Reference temperature in the Arrhenius law. Reference temperature \(\mathit{T0}\) is in degrees Celsius, and converted to Kelvin during resolution.
8.3. Keyword factor SECH_MENSI#
Definition of the parameters characterizing the diffusion coefficient used in the nonlinear drying equation proposed by Mensi (cf. [R7.01.12]). These characteristics are constants, while the diffusion coefficient depends on the calculation variable, i.e. the current \(C\) water concentration, (as thermal conductivity depended on temperature). It is a simplified formulation of the general case, constituting Mensi’s law.
A/B operands
These coefficients make it possible to express the diffusion coefficient according to Mensi’s law:
\(\xi =\frac{{\epsilon }_{r}^{n-1}-{\epsilon }_{y}^{n-1}}{{\epsilon }_{y}^{n}-{\epsilon }_{r}^{n-1}}\)
\(D\left(C\right)=a\text{.}{e}^{\left(b\text{.}C\right)}\)
A= a
Diffusion coefficient varying from \(0.5{.10}^{-13}\) and \({2.10}^{-13}{m}^{2}/s\) for concrete.
B= b
Coefficient of the order of \(0.05\) for concrete.
8.4. Keyword factor SECH_BAZANT#
Definition of the parameters characterizing the diffusion coefficient used in the nonlinear drying equation proposed by Bazant (confer [R7.01.12]). These characteristics are constants, while the diffusion coefficient depends on the calculation variable, i.e. the current \(C\) water concentration, (as thermal conductivity depended on temperature). This formulation constitutes Bazant’s law.
8.4.1. Operands D1/ ALPHA_BAZANT /N/ FONC_DESORP#
These coefficients make it possible to express the diffusion coefficient according to Bazant’s law:
\(D(h)={d}_{1}(\alpha +\frac{1-\alpha }{1+{(\frac{1-h}{1-0\text{.}\text{75}})}^{n}})\)
where \(h\) is the degree of hydration, linked to the water concentration by the desorption curve.
D1 = d1
Diffusion coefficient which is of the order of \({3.10}^{-13}{m}^{2}/s\) for concrete.
ALPHA_BAZANT = alpha
Coefficient varying from \(0.025\) to \(0.1\) for concrete.
N = n
Exponent of the order of 6 for concrete.
FONC_DESORP = desorp
Desorption curve, allowing to go from water concentration to hydration degree \(h\).
Important note:
desorp is a function of the calculation variable, \(C\) * , the water concentration, which is assimilated for the resolution at a temperature, of type “ TEMP “ .
8.5. Keyword factor SECH_NAPPE#
The diffusion coefficient, characterizing the non-linear drying equation, is expressed using a sheet, a tabulated function of the water concentration, a calculation variable, and of the temperature, an auxiliary calculation variable, given in the form of a data structure of the evol_ther type. For the resolution of drying by operator THER_NON_LINE, the water concentration is assimilated to a temperature, of the “TEMP” type.
For the consistency of the data, the parameters of the table, i.e. the calculation variable and the auxiliary variable, cannot be of the same type. A new type of variable has been added in DEFI_NAPPE, the « type of temperature calculated prior to drying », “TSEC”, which actually corresponds to a temperature.
8.5.1. Operand FONCTION#
The diffusion coefficient is expressed using a tabulated function of the parameters \(C\) and \(T\text{.}\)
FONCTION = fund_name
Tablecloth name.
8.6. Keyword factor PINTO_MENEGOTTO#
Definitions of the coefficients of the cyclic elastoplasticity behavior relationship of steel reinforcements in reinforced concrete according to the Pinto-Menegotto model (cf. [R5.03.09]).
The initial traction curve (start of loading) is defined by:
\(\sigma =E\mathrm{.}\epsilon\) as \(\sigma \le {\sigma }_{y}\); \(E\) defined as ELAS
\(\sigma ={\sigma }_{y}\) for \(\frac{{\sigma }_{y}}{E}\le \epsilon \le {\epsilon }_{h}\)
\(\sigma \mathrm{=}{\sigma }_{u}\mathrm{-}({\sigma }_{u}\mathrm{-}{\sigma }_{y}){(\frac{{\varepsilon }_{u}\mathrm{-}\varepsilon }{{\varepsilon }_{u}\mathrm{-}{\varepsilon }_{h}})}^{4}\) for \({\epsilon }_{h}\le \epsilon <{\epsilon }_{u}\)
(\(\varepsilon\) can’t go past \({\varepsilon }_{u}\))
Curve \(s=f(e)\) in the \({n}^{\mathit{ième}}\) cycle is defined by:
\({\sigma }_{L}^{\text{*}}=b{\varepsilon }_{L}^{\text{*}}+(\frac{1-b}{{(1+{({\varepsilon }_{L}^{\text{*}})}^{R})}^{1/R}}){\varepsilon }_{L}^{\text{*}}\) with \(R={R}_{0}-\frac{{a}_{1}\xi }{{a}_{2}+\xi }\)
and \(b=\frac{{E}_{h}}{E}\), \({E}_{h}\): asymptotic work hardening slope
where \({\epsilon }^{\text{*}}\) is defined by: \({\varepsilon }^{\text{*}}=\frac{\varepsilon -{\varepsilon }_{r}^{n-1}}{{\varepsilon }_{y}^{n}-{\varepsilon }_{r}^{n-1}}\).
where \({\sigma }^{\text{*}}\) is defined by: \({\sigma }^{\text{*}}=\frac{\sigma -{\sigma }_{r}^{n-1}}{{\sigma }_{y}^{n}-{\sigma }_{r}^{n-1}}\).
Quantity \({\epsilon }_{y}^{n}\) is deduced from cycle \(n-1\) by:
\(\begin{array}{}{\varepsilon }_{y}^{n}={\varepsilon }_{r}^{n-1}+\frac{{\sigma }_{y}^{n}-{\sigma }_{r}^{n-1}}{E}\\ {\sigma }_{y}^{n}={\sigma }_{y}^{n-1}\text{.}\mathrm{sign}({\varepsilon }_{y}^{n-1}-{\varepsilon }_{r}^{n-1})+{\varepsilon }_{H}({\varepsilon }_{r}^{n-1}-{\varepsilon }_{y}^{n-1})\end{array}\)
Variable \(\xi\) is defined by:
where \({\epsilon }_{r}^{n-1}\) represents the deformation reached at the end of the \(n-1\) th half-cycle
and \({\epsilon }_{y}^{n-1},{\epsilon }_{y}^{n}\) represent the deformations at the end of linearity of the half-cycles and \(n\).
\(b\) represents either the value provided by the user (keyword EP_SUR_E) or, failing that:
\(b=\frac{{E}_{H}}{E}\text{avec}{E}_{H}=\frac{{\sigma }_{u}-{\sigma }_{y}}{{\varepsilon }_{u}-\frac{{\sigma }_{y}}{E}}\)
In case of buckling, (if \(L/D>5\)):
in compression we replace \(b\) by \({b}_{c}=a\left(5.0-L/D\right){e}^{\left(b{\xi }^{\text{'}}\frac{E}{{\sigma }_{y}-{\sigma }_{\mathrm{\infty }}}\right)}\)
In traction, we calculate a new slope \({E}_{r}=E\left({a}_{5}+\left(1.0-{a}_{5}\right){e}^{\left(-{a}_{6}\left({\epsilon }_{r}^{n-1}-{\epsilon }_{y}^{n-1}\right)\right)}\right)\) with \({a}_{5}=1+\frac{5-L/D}{7.5}\).
\({\xi }^{\text{'}}\) represents the biggest « plastic excursion » during loading: \({\xi }^{\text{'}}=\underset{n}{\mathit{max}}\left({\epsilon }_{r}^{n}-{\epsilon }_{y}^{n}\right)\) and \({\sigma }_{\infty }=4\frac{{\sigma }_{y}}{L/D}\)
In the case of buckling, we add to \({\sigma }_{y}^{n}\) the value \({\sigma }_{s}^{\text{*}}={\gamma }_{s}bE\frac{b-{b}_{c}}{1-{b}_{c}}\) with \({\gamma }_{s}=\frac{11-L/D}{10({e}^{\frac{\mathrm{cL}}{D}}-1)}\).
8.6.1. Operands#
SY = sign
Initial elasticity limit, noted \({\sigma }_{y}\) in the equations.
EPSI_ULTM = epsu, noted :math:`{\varepsilon }_{u}` in the equations. Ultimate deformation.
SIGM_ULTM = sigmu, noted :math:`{\sigma }_{u}` in the equations. Ultimate constraint.
◊ ELAN = L/D
Slender bar (>5: buckling).
EPSP_HARD = epsh, noted \({\varepsilon }_{h}\) in the equations.
Deformation corresponding to the end of the plastic bearing.
◊ EP_SUR_E = b
Hardening slope to Young’s modulus ratio (if no value is given, we take \(b=\frac{{E}_{H}}{E}\)).
A 1_PM = a1
Coefficient defining the traction curve of the model.
A 2_PM = a2
Coefficient defining the traction curve of the model.
A 6_PM = a6
Coefficient defining the tensile curve of the model in case of buckling.
C_PM = c used in :math:`{\gamma }_{s}`
Coefficient defining the tensile curve of the model in case of buckling.
A_PM = a
Coefficient defining the tensile curve of the model in case of buckling.
R_PM =
Coefficient \({R}_{O}\) (20. by default).
The Young’s modulus E and the thermal expansion coefficient ALPHA should be specified by the keywords ELAS or ELAS_FO.
8.9. Keyword factor BETON_DOUBLE_DP#
The 3D behavior model developed in Code_Aster is formulated in the framework of thermo-plasticity, for the description of the non-linear behavior of concrete, in tension, and in compression, taking into account the irreversible variations in the thermal and mechanical characteristics of concrete, which are particularly sensitive to high temperatures [R7.01.03].
Functions may depend on the following command variables:
“TEMP”, “INST”, “HYDR”, “SECH”.
BETON_DOUBLE_DP allows you to define all the characteristics associated with the law of behavior with the double criteria of Drücker Prager. In addition to these characteristics, the elasticity module, the Poisson’s ratio, and the thermal expansion coefficient \(\alpha\), as well as the endogenous shrinkage and desiccation shrinkage coefficients, must be defined under the keyword ELAS for the real coefficients, or ELAS_FO, for the coefficients defined by functions, or sheets. All the characteristics of the model, \(\text{(E, nu,}\alpha ,f\text{'}c,f\text{'}t,\beta ,\text{Gc,Gt)}\) of the [function] type, may depend on one or two variables including temperature, hydration and drying. When they depend on temperature, they are a function of the maximum temperature reached during the loading history \(\theta\), which is stored in memory for each Gauss point, as an internal variable. This makes it possible to take into account the irreversible variations in these characteristics at high temperature.
8.9.1. F_C /F_T/COEF_BIAX operands#
F_C= f'c
Uniaxial compression strength \(f{\text{'}}_{c}\).
F_T= f't
Uniaxial tensile strength \(f{\text{'}}_{t}\).
COEF_BIAX = beta
The ratio of biaxial compressive strength to uniaxial compressive strength \(\beta\).
8.9.2. Operands ENER_COMP_RUPT/ENER_TRAC_RUPT/COEF_ELAS_COMP#
ENER_COMP_RUPT = GC
The breaking energy in compression \({G}_{c}\) ,
ENER_TRAC_RUPT = Gt
Breakthrough energy in traction \({G}_{t}\).
COEF_ELAS_COMP = phi
The compressive elastic limit, given by a proportionality coefficient as a percentage of the strength at peak \({f}_{c}^{\text{'}}(\theta )\), is generally of the order of 30% for standard concrete. It is important to note that this parameter is a real and not a function.
8.9.3. Operands LONG_CARA#
This operand makes it possible to overload the characteristic length calculated automatically, for each mesh, according to its dimensions (from its surface in 2D, from its volume in 3D).
The characteristic length calculated automatically makes it possible, when the fineness of the mesh varies from one calculation to another, to maintain stable results while avoiding localization phenomena. This length, calculated automatically or given by the user, leads to the value of the ultimate tensile work hardening according to the formula (for linear post-peak work hardening):
\({\kappa }_{u}\left(\theta \right)=\frac{2\text{.}{G}_{t}\left(\theta \right)}{{l}_{c}\text{.}{f}_{t}^{\prime }\left(\theta \right)}\)
In the particular case of a mesh containing adjacent cells whose dimensions are very different, the ultimate work-outs of the model BETON_DOUBLE_DP calculated on the basis of the characteristic length of the cells are therefore very different, which can cause convergence problems or lead to a state of constraints that are not very physical. (This characteristic length is calculated from the volume of the current mesh). For this reason, it is proposed to give the user the possibility to define an average length that overrides the characteristic length calculated for each mesh. The default value of*Code_Aster* is the characteristic length calculated for each mesh.
Choosing an arbitrary and identical length for all cells can also cause convergence difficulties. The best solution consists in creating a mesh whose variations in the dimensions of the cells respect the direction of variation in the stress field, and to use the characteristic length calculated automatically according to the size of the cells. Overloading by LONG_CARA should be reserved for specific cases, when the user cannot freely intervene on the mesh.
In the case where the user defines the characteristic length in the material, he will choose a \(({G}_{t},\text{LONG\_CARA})\) torque such that \(\frac{2\text{.}{G}_{t}\left(\theta \right)}{{l}_{c}\text{.}{f}_{t}^{\prime }\left(\theta \right)}\) is worth the value he wants for the ultimate work hardening in traction \({\kappa }_{u}\). (The usual value of the deformation associated with the ultimate tensile work hardening of an average concrete is \(5.E\mathrm{-}4\)).
8.9.4. Operands ECRO_COMP_P_PIC/ECRO_TRAC_P_PIC#
The parameters used to define the compression and tension softening curve are optional, and have values by default.
ECRO_COMP_P_PIC =/'LINEAIRE'
/'PARABOLE'
Shape of the post-peak curve in text-type compression, which can take the values” LINEAIRE “and” PARABOLE “. The nonlinear curve is then of the parabolic type.
ECRO_TRAC_P_PIC =/'LINEAIRE'
/'EXPONENT'
Shape of the post-peak curve in text-type tension, which can take the values” LINEAIRE “and” EXPONENT “. The nonlinear curve is then of the exponential type.
8.10. Tag factor BETON_GRANGER, V_BETON_GRANGER#
Definition of material parameters for the Granger viscoelastic model, modeling the natural flow of concrete. There are 2 behavioral relationships: the first (BETON_GRANGER) does not take into account the phenomenon of aging but models the humidity effect. The second (BETON_GRANGER_V) takes into account the effects of aging and humidity (Cf. [R7.01.01]).
In 1D and under the action of a constant stress \({\sigma }_{0}\), the creep deformation as a function of time and time of loading \({t}_{c}\) is written as: \({\epsilon }_{\mathit{fl}}\left(t\right)=J\left(t,{t}_{c},\right)\cdot {\sigma }_{0}\)
\((0.5{10}^{-4}<{\epsilon }_{d0}<1.5{10}^{-4})\)
The creep function \(J\left(t,{t}_{c}\right)\) is equal to:
\(J(t,{t}_{c})=k\left(a\left({t}_{c}\right)\right)\cdot \sum _{s=0}^{n}{J}_{s}\left(1-\mathit{exp}\left(\frac{t-{t}_{c}}{{\mathrm{\tau }}_{s}}\right)\right)\)
where \(a\) is the age of the material, i.e. how long it took for the concrete to be put in place. It is an internal variable in the model.
The \(k\left(a\left({t}_{c}\right)\right)\) function that appears in \(J\left(t,{t}_{c}\right)\) is used to model aging because it introduces the direct dependence on the loading time. For example, we can use curve CEB which models aging due to hydration:
\(k\left(a\right)=\frac{{28}^{0.2}+0.1}{{a}^{0.2}+0.1}\) with \(a\) in days
Without aging this function is constant and is equal to \(1\), the creep function depends in this case only on the time elapsed as soon as charging \(t-{t}_{c}\) was carried out.
Hygrometry is taken into account through an equivalent constraint \(S=h\cdot \sigma\), \(h\) being the relative humidity of the material. It is therefore necessary to fill in the isothermal desorption curve \(c\) which makes it possible to go from the water content \(C\) to \(h\) : \(h={c}^{-1}\left(C\right)\). This curve is provided to Code_Aster under the keyword ELAS_FO (FONC_DESORP, see § 4.1.8).
Note:
This behavior may be associated with drying and thermohydration withdrawals defined by the operands K_DESSIC, B_ENDOGE and ALPHAsous the key word* ELAS_FO.
For law BETON_GRANGER, the parameters of the law should be entered under the keyword: BETON_GRANGER. For law BETON_GRANGER_V, you must also enter the keyword BETON_GRANGER but you must add the keyword V_BETON_GRANGER for parameters specific to the aging law.
The internal variables of the law of behavior are described in [R7.01.01].
8.10.1. Operands for clean creep#
8 material coefficients of the creep function, homogeneous at one time.
J1 = J1
...
...
J8 = J8
8 delay times of the creep function.
TAUX_1 =tau1
...
...
TAUX_8 =tau8
8.10.2. Operands for aging#
FONC_V = k (a)
Aging function.
8.11. Tag factor MAZARS, MAZARS_FO#
The Mazars behavior model is a model of damaging elastic behavior used to describe the softening behavior of concrete. It distinguishes between traction and compression behavior, but uses only one scalar damage variable (confer [R7.01.08]). The Mazars model implanted corresponds to the 2012 version, i.e. to the reformulation improving bi-compression and pure shear behavior.
The parameters can be a function of temperature, hydration and drying. In this case you should use MAZARS_FO. In the case of this law of behavior, the parameters depend on the maximum temperature reached during the entire loading history.
Functions may depend on the following command variables: “TEMP”, “HYDR”, “SECH”.
MAZARS (or MAZARS_FO) allows you to define all the characteristics associated with the Mazars behavior model. In addition to these characteristics, elastic constants should be defined under the keyword ELAS for real coefficients or ELAS_FO for temperature-dependent coefficients.
8.11.1. Operands: EPSD0/EPSC0/EPST0 /AC/ AT/ AT/ BC/ BT/ k#
♦/epsd0 = epsd0
Deformation damage threshold.
♦/ePSc0 = epsc0
epst0 = epst0
Threshold of damage in deformation, compression and traction.
If \({\mathrm{\epsilon }}_{d0}\) is given \({\mathrm{\epsilon }}_{t0}\) and \({\mathrm{\epsilon }}_{c0}\) are calculated by:
\(\begin{array}{c}{\mathrm{\epsilon }}_{t0}={\mathrm{\epsilon }}_{d0}\\ {\mathrm{\epsilon }}_{c0}=\frac{{\mathrm{\epsilon }}_{d0}}{\mathrm{\nu }\ast \sqrt{2}}\end{array}\)
♦ AC = ac
Coefficient allowing to fix the shape of the post-peak curve under compression. Introduce a horizontal asymptote which is the \(\varepsilon\) axis for \(\mathrm{Ac}=1\) and the horizontal for going through the peak for \(\mathrm{Ac}=0\) (generally).
♦ AT = at
Coefficient for fixing the shape of the post-peak curve under traction. Introduce a horizontal asymptote which is the \(\varepsilon\) axis for \(\mathrm{Ac}=1\) and the horizontal through the peak for \(\mathrm{Ac}=0\) (usually \(0.7<\mathrm{At}<1\)).
♦ bc=BC
Coefficient allowing to fix the shape of the post-peak curve under compression. Depending on its value, it may correspond to a sudden drop in stress (\(\mathrm{BC}<{10}^{4}\)) or a preliminary phase of stress increase followed by a more or less rapid decrease (generally \({10}^{3}<\mathrm{Bc}<2.{10}^{3}\)).
♦ BT=bt
Coefficient for fixing the shape of the post-peak curve under traction. Depending on its value, it may correspond to a sudden drop in stress (\(\mathrm{BC}<{10}^{4}\)) or a preliminary phase of stress increase followed by a more or less rapid decrease (generally \({10}^{\mathrm{4 }}<\mathrm{Bt}<{10}^{5}\)).
♦ k=K
Parameter introducing a horizontal asymptote in pure shear. It is between 0 and 1. Recommended value \(\mathrm{0,7}\).
8.11.2. Operand CHI#
♦ CHI =chi
As part of the coupling BETON_UMLV with the law of MAZARS. The chi parameter makes it possible to define the importance of the coupling:
\(\mathrm{CHI}=0\): no coupling,
\(\mathrm{CHI}=1\): total coupling.
Total coupling leads to premature onset of concrete damage. The recommended value is in the range: \([0.4;0.7]\).
8.11.3. Operand SIGM_LIM, EPSI_LIM#
◊ SIGM_LIM =sglim
Definition of limit stress.
◊ EPSI_LIM =eplim
Definition of limit deformation.
The sigm_ LIM and espi_ LIM operands make it possible to define the stress and deformation limits that correspond to the service and ultimate limit states, classically used during studies in civil engineering.
The sigm_ LIM and espi_ LIM operands make it possible to define the stress and deformation limits that correspond to the service and ultimate limit states, classically used during studies in civil engineering.
Note: These terminals are mandatory when using the mazars_ UNIL behavior (confer [R7.01.08] Damage model from MAZARS, [U4.42.07] DEFI_MATER_GC). In other cases they are not taken into account.
8.12. Keyword BETON_UMLV#
The creep law UMLV assumes a total decoupling between spherical and deviatory components: the deformations induced by spherical stresses are purely spherical and the deformations induced by deviatory stresses are purely deviatory [R7.01.06]. Moreover, the natural creep deformation is assumed to be proportional to the internal relative humidity:
Spherical part: \({\epsilon }^{s}=h\cdot f\left({\sigma }^{s}\right)\) and, deviatory part: \(\underline{\underline{{\epsilon }^{d}}}=h\cdot f\left(\stackrel{~}{\underline{\underline{\sigma }}}\right)\)
where \(h\) refers to internal relative humidity.
Behavioral model BETON_UMLV is a non-aging viscoelastic model developed in partnership with the University of Marne-la-Vallée to describe the natural creep of concrete. It is particularly suitable for multiaxial configurations by not assuming the value of the creep Poisson’s ratio.
Spherical stresses are at the origin of the migration of water absorbed at the interfaces between hydrates at the level of macro-porosity and absorbed within the micro-porosity in the capillary porosity. The diffusion of inter-lamellar water from the hydrate pores to the capillary porosity takes place irreversibly. The total spherical creep deformation is therefore written as the sum of a reversible part and an irreversible part:
\({\varepsilon }^{\mathrm{fs}}=\underset{\begin{array}{}\mathrm{partie}\\ \mathrm{réversible}\end{array}}{\underset{\underbrace{}}{{\varepsilon }_{r}^{\mathrm{fs}}}}+\underset{\begin{array}{}\mathrm{partie}\\ \mathrm{irréversible}\end{array}}{\underset{\underbrace{}}{{\varepsilon }_{i}^{\mathrm{fs}}}}\)
The process of spherical creep deformation is governed by the following system of coupled equations:
\(\{\begin{array}{c}{\dot{\varepsilon }}^{\mathrm{fs}}=\frac{1}{{\eta }_{r}^{s}}\cdot \left[h\cdot {\sigma }^{s}-{k}_{r}^{s}\cdot {\varepsilon }_{r}^{\mathrm{fs}}\right]-{\dot{\varepsilon }}_{i}^{\mathrm{fs}}\\ {\dot{\varepsilon }}_{i}^{\mathrm{fs}}=\frac{1}{{\eta }_{i}^{s}}{\langle \left[{k}_{r}^{s}\cdot {\varepsilon }^{\mathrm{fs}}-({k}_{r}^{s}+{k}_{i}^{s})\cdot {\varepsilon }_{i}^{\mathrm{fs}}\right]-\left[h{\sigma }^{s}-{k}_{r}^{s}\cdot {\varepsilon }_{r}^{\mathrm{fs}}\right]\rangle }^{+}\end{array}\)
where \({k}_{r}^{s}\) refers to the apparent stiffness associated with the skeleton formed by hydrate blocks at the mesoscopic scale; \({\eta }_{r}^{s}\) the apparent viscosity associated with the diffusion mechanism within the capillary porosity; \({k}_{i}^{s}\) refers to the apparent stiffness intrinsically associated with hydrates at the microscopic scale and the \({\eta }_{i}^{s}\) apparent viscosity associated with the mechanism of interfoliar diffusion.
(The square brackets:math: {⟨⟩}} ^ {text {+}}} refer to the Mac Cauley operator: :math: {⟨x⟩}} ^ {text {+}}} =frac {1} {1} {2}left (x+|x|right)} =frac {1} {1} {2}left (x+|x|right))
Deviatory stresses are at the origin of a sliding mechanism (or mechanism of virtual dislocation) of sheets of CSH in nano-porosity. Under deviatoric stress, creep takes place at constant volume. Moreover, creep law UMLV assumes the isotropy of deviatoric creep. Phenomenologically, the sliding mechanism includes a reversible viscoelastic contribution of water strongly adsorbed to the sheets of CSH and an irreversible viscous contribution of free water:
\(\underset{\begin{array}{c}\mathit{déformation}\\ \mathit{déviatorique}\\ \mathit{totale}\end{array}}{\underset{\underbrace{}}{{\underline{\underline{\varepsilon }}}^{\mathit{fd}}}}\mathrm{=}\underset{\begin{array}{c}\mathit{contribution}\\ \mathit{eau}\\ \mathit{absorbée}\end{array}}{\underset{\underbrace{}}{{\underline{\underline{\varepsilon }}}_{\text{}r}^{\mathit{fd}}}}+\underset{\begin{array}{c}\mathit{contribution}\\ \mathit{eau}\\ \mathit{libre}\end{array}}{\underset{\underbrace{}}{{\underline{\underline{\varepsilon }}}_{\text{}i}^{\mathit{fd}}}}\)
The*th* main component of total deviatoric deformation is governed by the following system of equations:
\({\tilde{\dot{\sigma }}}^{j}(1+\frac{{\eta }_{r}^{d}}{{\eta }_{i}^{d}})+\frac{{k}_{r}^{d}}{{\eta }_{i}^{d}}{\tilde{\sigma }}^{j}\mathrm{=}{\eta }_{r}^{d}{\ddot{\varepsilon }}^{d,j}+{k}_{r}^{d}{\dot{\varepsilon }}^{d,j}\)
where \({k}_{r}^{d}\) refers to the stiffness associated with the capacity of the absorbed water to transmit charges (load bearing water); \({\eta }_{r}^{d}\) the viscosity associated with the water adsorbed by the hydrate sheets and \({\eta }_{i}^{d}\) refers to the viscosity associated with free water.
8.12.1. Operand#
K_RS = K_RS
\({k}_{r}^{s}\) apparent stiffness associated with the skeleton formed by hydrate blocks at the mesoscopic scale.
K_IS = K_IS
:math:`{k}_{i}^{s}` apparent stiffness intrinsically associated with hydrates at the microscopic scale.
K_RD = K_RD
:math:`{k}_{r}^{d}` stiffness associated with the capacity of adsorbed water to transmit loads (load bearing water).
ETA_RS = ETA_RS
:math:`{\eta }_{r}^{s}` apparent viscosity associated with the diffusion mechanism within the capillary porosity.
ETA_IS = ETA_IS
:math:`{\eta }_{i}^{s}` apparent viscosity associated with the interlamellar diffusion mechanism.
ETA_RD = ETA_RD
:math:`{\eta }_{r}^{d}` viscosity associated with water absorbed by hydrate sheets.
ETA_FD = ETA_FD
makes it possible to take into account desiccation creep according to Bazant’s law.
Note:
The desorption curve giving the hygrometry \(h\) as a function of the water concentration \(C\) must be entered under the keyword ELAS_FO.
8.13. Keyword factor BETON_ECRO_LINE#
Definition of a linear work-hardening curve taking into account confinement in the specific case of concrete. In order to improve compression behavior, a reversibility threshold is defined ([R7.01.04] model ENDO_ISOT_BETON).
8.13.1. Operands#
D_SIGM_EPSI = dsde (AND)
Slope of the traction curve.
SYT = sigt
Maximum stress in simple traction.
SYC = SIGC
Maximum stress in simple compression (it does not exist for a Poisson’s ratio \(\nu =0\), in this case \(\mathrm{SYC}\) is not specified)
Young’s module \(E\) should be specified by the keywords ELAS or ELAS_FO.
8.14. Keyword factor ENDO_ORTH_BETON#
Definition of the parameters of the law of behavior ENDO_ORTH_BETON, allowing to describe the anisotropy induced by concrete damage, as well as the unilateral effects [R7.01.09]. Refer to documents [R7.01.09] and [V6.04.176] for the precise meaning of the parameters and the identification procedure.
8.14.1. Operand ALPHA#
Coupling constant between the evolution of tensile damage and that of compression damage. It should be taken between \(0\) and \(1\), rather close to \(1\). The value by default is \(0.9\).
8.14.2. Operands K0/K1/K2#
K0 = k0
A constant part of the threshold function. Allows you to calibrate the height of the peak under traction.
K1 = k1
Parameter of the threshold function used to increase the compression threshold.
K2 = k2
Parameter for controlling the shape of the rupture envelope for biaxial tests. The value by default is \({7.10}^{-4}\).
8.14.3. Operands ECROB/ECROD#
ECROB = ecrob
Term for blocked energy (equivalent to work hardening energy) relating to the evolution of tensile damage. It allows you to control the shape of the peak under traction.
ECROD = record
Term for blocked energy (equivalent to work hardening energy) relating to the evolution of compression damage. It makes it possible to control the shape of the peak under compression.
The Young’s modulus \(E\) and the Poisson’s ratio \(\nu\) are to be specified by the keywords ELAS or ELAS_FO.
8.15. Keywords factor ENDO_SCALAIRE/ENDO_SCALAIRE_FO#
Definition of the parameters of the law of behavior ENDO_SCALAIRE [R5.03.25], which describes the brittle elastic rupture of a homogeneous isotropic material. This law is only available for damage gradient modeling GRAD_VARI.
8.15.1. K, P, M operand#
These are the internal parameters of the model that define work hardening, see [R5.03.25]: \(k\) refers to an energy density \(\mathrm{Pa}\), \(k\) and \(m\) are dimensionless parameters. \(k\) and \(m\) can be adjusted using the non-local scale \(D\) (approximately the half-width of the location bandwidth) and the following macroscopic parameters: \(E\) the Young’s modulus, \({G}_{f}\) the cracking energy and \({f}_{t}\) the value of the peak stress in simple tension. The readjustment relationships are then written as:
\(k=\frac{3{G}_{f}}{4D};\phantom{\rule{4em}{0ex}}m=\frac{3E{G}_{f}}{2{f}_{t}^{2}D};\phantom{\rule{4em}{0ex}}c=\frac{3}{8}D{G}_{f}\)
where \(c\) is the parameter specified by NON_LOCAL = _F (C_GRAD_VARI = c), which also depends on the macroscopic response. As for the parameter \(p\), greater than 1, it controls the curvature of the post-peak response.
8.15.2. Operands C_COMP, C_VOLU#
These are the internal, dimensionless parameters of the model that define the shape of the load surface (with one exception), see [R5.03.25]. The default values make it possible to find the energy model (symmetric) for which the load surface corresponds to a level line of the elastic energy density (ellipsoid of rotation around the \((\mathrm{1,1}\mathrm{,1})\) axis which is centered at the beginning of coordinates).
In the more general case, the non-centered ellipsoidal load surface (always the \(\mathrm{1,1}\mathrm{,1}\) axis) can be defined by three parameters that are more accessible to measurement: \({f}_{t}\) the value of the stress at the peak in simple tension, \({f}_{c}\) the value of the stress at the peak in simple compression, and \(\tau\) the value of the stress at the peak in pure shear. The adjustment relationships are as follows:
\({c}_{\mathit{comp}}=\frac{1+\nu }{1-2\nu }\frac{({f}_{c}-{f}_{t})\tau \sqrt{3}}{2{f}_{t}{f}_{c}};\phantom{\rule{6em}{0ex}}{c}_{\mathit{volu}}=\frac{2(1+\nu )}{1-2\nu }\left[{\left(\frac{({f}_{c}+{f}_{t})\tau \sqrt{3}}{2{f}_{t}{f}_{c}}\right)}^{2}-1\right]\)
8.15.3. Operands COEF_RIGI_MINI#
COEF_RIGI_MINI
This is the parameter for regularizing the matrix tangent to the break, to avoid zero pivots if the cracking were to cut the part into several pieces not maintained by the boundary conditions. It does not depend on control variables.
The Young’s modulus \(E\) and the Poisson’s ratio \(\nu\) are to be specified by the keywords ELAS or ELAS_FO.
The non-location parameter is entered under the keyword C_GRAD_VARI behind the keyword factor NON_LOCAL. It is linked to the macroscopic parameters by:
8.16. Keyword factor ENDO_FISS_EXP/ENDO_FISS_EXP_FO#
Definition of the parameters of the law of behavior ENDO_FISS_EXP [R5.03.27], which describes the brittle elastic rupture of a homogeneous isotropic material. This law is only available for damage gradient modeling GRAD_VARI.
8.16.1. Operand K, M, P, Q#
These are the internal parameters of the model that define work hardening, see [R5.03.27]. Their identification is supported by the DEFI_MATER_GC [U4.42.07] command, using quantities that are experimentally accessible.
8.16.2. Operands TAU, SIG0, BETA#
These are the internal parameters of the model that define the shape of the load surface (with one exception), see [R5.03.27]. It is based on the Von Mises stress and the exponential stress tensor and compares well to experimental results on concrete under biaxial loading. The parameter BETA is more numerical in nature and only has the advantage of making the elasticity domain limited, including for hydrostatic compressions; the default value fulfills this function well, without affecting the shape of the domain in the areas of interest.
Again, the DEFI_MATER_GC [U4.42.07] command makes it possible to identify these parameters from quantities that are experimentally accessible (limits in tension and compression).
8.16.3. Operand REST_RIGIDITE#
Stiffness restoration is active for compression deformation directions. To avoid an abrupt change in regime during the transition from traction to compression, a function S regulates the jump in stiffness, see [R5.03.27]. The parameter REST_RIGIDITE, which is positive, controls this regularization; it corresponds to the coefficient gamma of the function S”. A value of 0 leads to not restoring stiffness (i.e. the model is without restoring stiffness) while a very large value amounts to virtually eliminating regularization. The DEFI_MATER_GC [U4.42.07] command makes it easy to quantify this parameter by stipulating what proportion of the stiffness is restored for a deformation level corresponding to the initial compression threshold.
8.16.4. Operands COEF_RIGI_MINI#
This is the parameter for regularizing the matrix tangent to the break, to avoid zero pivots if the cracking were to cut the part into several pieces not maintained by the boundary conditions. It does not depend on control variables.
The Young’s modulus \(E\) and the Poisson’s ratio \(\nu\) are to be specified by the keywords ELAS or ELAS_FO.
The non-locality parameter is entered under the keyword C_GRAD_VARI behind the keyword factor NON_LOCAL; the command DEFI_MATER_GC [U4.42.07] makes it possible to identify it from quantities that are experimentally accessible.
8.17. Keyword factor GLRC_DM#
This keyword factor allows you to define the parameters of the GLRC_DM law of behavior. It is a model of global damage to a reinforced concrete slab formulated in terms of generalized deformation/stress relationships (membrane extension, membrane flexure and force, bending moment), cf. [R7.01,32].
8.17.1. Operands#
NYT = Not
Membrane force of the damage threshold under simple traction of a reinforced concrete slab (unit of force per length).
NYC = Nc
Membrane force of the « damage » threshold (end of linearity of the compression curve) in simple compression of a reinforced concrete slab (unit of force per length).
MYF = Mf
Flexing moment of the damage threshold in simple bending of a reinforced concrete slab (unit of force).
GAMMA_T = GMT
Relative damaging slope compared to the elastic slope under simple traction (\(0<{\gamma }_{\mathrm{MT}}<1\)).
GAMMA_C = GMC
Relative damaging slope compared to the elastic slope in simple compression (\(0<{\gamma }_{\mathit{MC}}<1\)).
GAMMA_F = GMF
Relative damaging slope compared to the elastic slope in simple bending (\(0<{\gamma }_{F}<1\)).
ALPHA_C = Alfc
Parameter for modulating the compression damage function to introduce a decoupling of the traction and compression thresholds and inducing a curvature of the compression curve. The membrane damage function is written as:
:math:`{\xi }_{m}(x,{d}_{\mathrm{1,}}{d}_{2})=\frac{1}{2}((\frac{1+{\gamma }_{\mathrm{mt}}{d}_{1}}{1+{d}_{1}}+\frac{1+{\gamma }_{\mathrm{mt}}{d}_{2}}{1+{d}_{2}})H(x)+(\frac{{\alpha }_{c}+{\gamma }_{\mathrm{mc}}{d}_{1}}{{\alpha }_{c}+{d}_{1}}+\frac{{\alpha }_{c}+{\gamma }_{\mathrm{mc}}{d}_{2}}{{\alpha }_{c}+{d}_{2}})H(-x))`
We can refer to the reference documentation [:ref:`R7.01.32 <R7.01.32>`] section §3.2.4 where a summary of the identification of model parameters is presented.
8.18. Keyword factor DHRC#
This factor keyword makes it possible to define the parameters of the DHRC law of behavior. It is a global damage model of a reinforced concrete slab formulated using a homogenization method, in terms of generalized deformation/stress relationships (membrane extension, membrane flexure and force, flexing moment) and including internal state variables of damage and sliding at the steel-concrete interface, see [R7.01.36].
The \(258\) parameters of the law to be identified, by homogenization and by the method of least squares on different damage values, correspond to:
to the parameters controlling the components of the tensors of damageable elastic stiffness \(A\), of coupling of generalized deformations and slips \(B\) and of stored energy in sliding \(C\), for which no analytical expression is available;
to macroscopic threshold parameters which are linked to microscopic threshold parameters.
8.18.1. Operands#
NYD = nyd
List of the two damage thresholds \({G}^{\zeta ,\mathit{crit}}\) in simple traction of reinforced concrete
SCRIT = script
List of the four sliding thresholds \({\Sigma }_{\alpha }^{\zeta ,\mathit{crit}}\) equivalent steel-concrete
AA_C = Alpha_AC
Parameters (\(42\)) \({\alpha }^{\mathrm{Ac}}\) of the dependencies in component damage variables (\(21\) supra-diagonal terms) of the tensor of order \(4\) symmetric \(A\) membrane-flexure of the plate, in the compression domain, in the reinforcement coordinate system \((x,y)\), in Voigt notations, in Voigt notations, identified by homogenization and by the method of least squares on various values of \({D}_{\rho }\), in the upper zone (\(1\)) then in the lower zone (\(2\)):
\({A}_{\beta \delta \tau \upsilon }^{\rho }({D}_{\rho })={A}_{\beta \delta \tau \upsilon }^{0}\frac{{\alpha }_{\beta \delta \tau \upsilon }^{\mathrm{Ac}\rho }+{\gamma }_{\beta \delta \tau \upsilon }^{\mathrm{Ac}\rho }{D}_{\rho }}{{\alpha }_{\beta \delta \tau \upsilon }^{\mathrm{Ac}\rho }+{D}_{\rho }}\)
Note:
In current practice, due to the isotropy of concrete and the orientation of the steels along the axis \((x,y)\) , we will have:
AAC131 = 1., AAC161 = 1., AAC231 = 1., = 1., = 1., = 1., = 1. AAC261 AAC341 AAC351 ,
AAC461 = 1., AAC561 = 1.; AAC132 = 1.; = 1.; = 1.; = 1.; = 1. AAC162 AAC232 AAC262 ,
AAC342 = 1., AAC352 = 1., AAC462 = 1., AAC562 = 1..
AA_T = Alpha_AT
Parameters (\(42\)) \({\alpha }^{\mathrm{At}}\) of the dependencies in component damage variables (\(21\) supra-diagonal terms) of the tensor of order \(4\) symmetric \(A\) membrane-flexure of the plate, in the tensile domain, in the frame coordinate system \((x,y)\), in Voigt notations, in Voigt notations, identified by homogenization and by the method of least squares on various values of \({D}_{\rho }\), in the upper zone (\(1\)) then in the lower zone (\(2\)):
\({A}_{\beta \delta \tau \upsilon }^{\rho }({D}_{\rho })={A}_{\beta \delta \tau \upsilon }^{0}\frac{{\alpha }_{\beta \delta \tau \upsilon }^{\mathrm{At}\rho }+{\gamma }_{\beta \delta \tau \upsilon }^{\mathrm{At}\rho }{D}_{\rho }}{{\alpha }_{\beta \delta \tau \upsilon }^{\mathrm{At}\rho }+{D}_{\rho }}\)
Note:
In current practice, because of the isotropy of concrete and the orientation of the steels along the axis \((x,y)\) , we will take:
AAT131 = 1., AAT161 = 1., AAT231 = 1., = 1., = 1., = 1., = 1. AAT261 AAT341 AAT351 ,
AAT461 = 1., AAT561 = 1.; AAT132 = 1.; = 1.; = 1.; = 1.; = 1. AAT162 AAT232 AAT262 ,
AAT342 = 1., AAT352 = 1., AAT462 = 1., AAT562 = 1..
GA_C = gamma_AC
Parameters (\(42\)) \({\gamma }^{\mathrm{Ac}}\) of the dependencies in component damage variables (\(21\) supra-diagonal terms) of the tensor of order \(4\) symmetric \(A\) membrane-flexure of the plate, in the compression domain, in the reinforcement coordinate system \((x,y)\), in Voigt notations, in Voigt notations, identified by homogenization and by the method of least squares on various values of \({D}_{\rho }\), in the upper zone (\(1\)) then in the lower zone (\(2\)).
Note:
In current practice, because of the isotropy of concrete and the orientation of the steels along the axis \((x,y)\) , we will take:
GAC131 = 1., GAC161 = 1., GAC231 = 1., = 1., = 1., = 1., = 1. GAC261 GAC341 GAC351 ,
GAC461 = 1., GAC561 = 1.; GAC132 = 1.; = 1.; = 1.; = 1.; = 1. GAC162 GAC232 GAC262 ,
GAC342 = 1., GAC352 = 1., GAC462 = 1., GAC562 = 1..
GA_T = gamma_AT
Parameters (\(42\)) \({\gamma }^{\mathrm{At}}\) of the dependencies in damage variables of the components (\(21\) supra-diagonal terms) of the tensor of order \(4\) symmetric \(A\) membrane-flexure of the plate, in the tensile domain, in the frame coordinate system \((x,y)\), for sliding in the upper grid (\(1\)) or lower ( \(2\)), in Voigt notations, identified by homogenization and by the least squares method on various values of \({D}_{\rho }\), in the upper zone (\(1\)) then the lower zone (\(2\)).
Note:
In current practice, because of the isotropy of concrete and the orientation of the steels along the axis \((x,y)\) , we will take:
GAT131 = 1., GAT161 = 1., GAT231 = 1., = 1., = 1., = 1., = 1. GAT261 GAT341 GAT351 ,
GAT461 = 1., GAT561 = 1.; GAT132 = 1.; = 1.; = 1.; = 1.; = 1. GAT162 GAT232 GAT262 ,
GAT342 = 1., GAT352 = 1., GAT462 = 1., GAT562 = 1..
AB = alpha_B
Parameters (\(24\)) \({\alpha }^{B}\) of the dependencies in component damage variables (\(24\) supra-diagonal terms) of the symmetric 3rd-order tensor \(B\) of membrane-flexion-sliding coupling of the plate, in the frame coordinate system \((x,y)\), for sliding in the upper grid (\(1\)) then lower grid () then lower (\(2\)) grid, in Voigt notations , identified by homogenization and by the least squares method on various values of \({D}_{\rho }\):
with: \({B}_{\beta \delta \zeta }^{m\rho \pi }({D}_{\rho })=\frac{{\gamma }_{\beta \delta \zeta }^{Bm\rho \pi }{D}_{\rho }}{{\alpha }_{\beta \delta \zeta }^{Bm\rho \pi }+{D}_{\rho }}\); \({B}_{\beta \delta \zeta }^{f\rho \pi }({D}_{\rho })=\frac{{\gamma }_{\beta \delta \zeta }^{Bf\rho \pi }{D}_{\rho }}{{\alpha }_{\beta \delta \zeta }^{Bf\rho \pi }+{D}_{\rho }}\)
Note:
In current practice, because of the isotropy of concrete and the orientation of the steels along the axis \((x,y)\) , we will take:
AB311 = 1., AB321 = 1., AB611 = 1., AB621 = 1. ,
AB312 = 1., AB322 = 1., AB612 = 1., AB622 = 1.
GB = gamma_B
Parameters (24) \({\gamma }^{B}\) of the dependencies in component damage variables (\(24\) supra-diagonal terms) of the tensor of order \(3\) symmetric \(B\) of membrane-flexion-sliding coupling of the plate, in the frame coordinate system \((x,y)\), for sliding in the upper grid (\(1\)) then lower () then lower (\(2\)) grid sliding, in Voigt notations , identified by homogenization and by the least squares method on various values of \({D}_{\rho }\).
Note:
In current practice, because of the isotropy of concrete and the orientation of the steels along the axis \((x,y)\) , we will take:
GB311 = 0, GB321 = 0, GB611 = 0, = 0, GB621 = 0,
GB312 = 0, GB322 = 0, GB612 = 0, GB622 = 0.
C0 = C0
Components (\(6\) non-zero supra-diagonal terms) of the \(2\) order symmetric tensor of the free energy of free sliding steel-concrete \({C}^{0}\) of the plate before damage, according to the directions of the slides considered, in the frame coordinate system \((x,y)\), in the frame coordinate system, in an upper grid (\(1\)) or lower grid () or lower (\(2\)) grid, in Voigt notations, identified by homogenization:
\((\begin{array}{cccc}{C}_{\mathrm{xx}}^{01}& {C}_{\mathrm{yx}}^{01}& 0& 0\\ \text{}& {C}_{\mathrm{yy}}^{01}& 0& 0\\ \text{}& \text{}& {C}_{\mathrm{xx}}^{02}& {C}_{\mathrm{yx}}^{02}\\ \text{}& \text{}& \text{}& {C}_{\mathrm{yy}}^{02}\end{array})\)
Note:
In current practice, because of the isotropy of concrete and the orientation of the steels along the axis \((x,y)\) , we will have: C0211= 0, C0212= 0.
AC = alpha_C
Parameters (\(6\)) \({\alpha }^{C}\) of the dependencies in damage variables of the components (\(6\) supra-diagonal terms) of the symmetric tensor \(C\) identified by homogenization and by the least squares method on different values of \({D}_{\rho }\):
\({C}_{\beta \delta }^{\rho }({D}_{\rho })={C}_{\beta \delta }^{0\rho }\frac{{\alpha }_{\beta \delta }^{C\rho }+{\gamma }_{\beta \delta }^{C\rho }{D}_{\rho }}{{\alpha }_{\beta \delta }^{C\rho }+{D}_{\rho }}\)
Note:
In current practice, because of the isotropy of concrete and the orientation of the steels along the \((x,y)\) axes, such as: C0211= 0, C0212= 0, we will take: AC211 = 1., AC212 = 1., = 1..
GC = gamma_C
Parameters (\(6\)) \({\gamma }^{C}\) of the dependencies in damage variables of the components (\(6\) supra-diagonal terms) of the symmetric tensor \(C\) identified by homogenization and by the least squares method on different values of \({D}_{\rho }\).
Note:
In current practice, because of the isotropy of concrete and the orientation of the steels along the \((x,y)\) axes, such as: C0211= 0, C0212= 0, we will take: GC211 = 1., GC212 = 1., = 1..
Refer to the reference documentation [R7.01.36] where a summary of the identification of model parameters is presented.
8.19. Keyword factor BETON_REGLE_PR#
This keyword is used to define the material parameters used by behavior BETON_REGLE_PR (« Parabole-Rectangle » rule). This behavior can only be used in 2D (plane stresses or plane deformations) or in shells (models DKT, COQUE_3D) (see for example the ssnop129a test). It is reduced to a one-dimensional behavior, which is written, in each of the main directions of the 2D deformation tensor:
In traction: \(\{\begin{array}{ccc}\sigma \text{=}E\varepsilon & \text{si}& 0<\epsilon <\frac{{\sigma }_{y}^{t}}{E}\\ \sigma \text{=}{\sigma }_{y}^{t}+{E}_{T}(\varepsilon -\frac{{\sigma }_{y}^{t}}{E})& \text{si}& \frac{{\sigma }_{y}^{t}}{E}<\varepsilon <\frac{{\sigma }_{y}^{t}}{E}(1-\frac{E}{{E}_{T}})\\ \sigma \text{=}0& \text{sinon}& \end{array}\)
In compression: \(\{\begin{array}{ccc}\sigma ={\sigma }_{y}^{c}\left[1-{(1-\frac{\varepsilon }{{\varepsilon }_{c}})}^{n}\right]& \text{si}& \varepsilon >{\varepsilon }_{c}\\ \sigma ={\sigma }_{y}^{c}& \text{sinon}& \end{array}\)
8.19.1. Operands#
DSIGM_EPSI = And
Post-peak tangent module in traction \({E}_{t}\) (negative).
SYT = Syt
Ultimate tensile stress \({\sigma }_{y}^{t}\).
SYC = Syc
Ultimate compression stress \({\sigma }_{y}^{c}\). It must be given positive.
EPSC = Epsc
Ultimate compression deformation \({\varepsilon }_{c}\). It must be given positive.
N = n
Exponent of the law of work hardening in compression.
8.20. Keyword JOINT_BA#
These parameters define the nonlinear behavior model of the steel-concrete bond used for the fine calculation of reinforced concrete structures where the prediction of cracks and the redistribution of stresses in concrete are very important. Available for analyses under the effect of monotonic and cyclic loads, the model is written in the framework of the thermodynamic formulation of irreversible processes. It makes it possible to take into account the damage to the shear interface, in combination with the effects of crack friction, as well as irreversible deformations. The [R7.01.21] document describes the corresponding details.
This model should be used with 2D « joint » elements [R3.06.09]. Steel reinforcements can be modelled with plane (QUAD4) or one-dimensional (BARRE) elements.
Note:
Taking into account the effect of thermal loading is not possible for the moment.
8.20.1. Operands#
HPEN = HPEN
Parameter of penetration between surfaces by crushing concrete.
We check that :math:`\mathrm{HPEN}>0.`.
GTT = GTT
Link stiffness module.
We check that :math:`{\text{G}}_{\mathit{beton}}\mathrm{\le }\text{GTT}\mathrm{\le }{\text{G}}_{\mathit{acier}}\mathrm{.}`
GAMD0 = Gam0
Perfect adherence threshold or elastic deformation limit.
We check that :math:`1.E-4<\mathrm{Gam0}<1.E-2`.
AD1 = ad1
Parameter for the evolution of damage in region 1 (transition from small deformations to large landslides).
We check that :math:`1.E-1<\mathrm{AD1}<1.E+1`.
BD1 = bd1
Power parameter describing the evolution of the damage variable in region 1 (transition from small deformations to large slides).
We check that :math:`\mathrm{BD1}<1.E-1`.
GAMD2 = Gam2
Threshold of major landslides.
We check that :math:`1.E-4<\mathrm{Gam2}<1.E+0`.
AD2 = ad2
Parameter for the evolution of damage in region 2 (maximum bond strength and friction degradation).
We check that :math:`\mathrm{AD2}<1.E-6`.
BD2 = bd2
Power parameter describing the evolution of the damage variable in region 2 (maximum bond strength and friction degradation).
We check that :math:`\mathrm{BD2}<1.E-1`.
VIFROT = vifrot
Material parameter describing the influence of crack friction.
We check that :math:`\mathrm{VIFROT}<0.0E+0`.
FA = alpha
Material parameter related to kinematic work hardening by friction of cracks.
We check that :math:`\mathrm{FA}<0.0E+0`.
FC = c
Parameter describing the influence of confinement on bond strength.
We check that :math:`\mathrm{FC}<0.0E+0`.
EPSTR0 = EPSN
Threshold of elastic deformation in the normal direction before rupture. We check that \(1.E-4<\mathrm{EPSN}<1.E+0\).
ADN = DNA
Parameter of damage in the normal direction by opening the crack.
We check that :math:`\mathrm{ADN}<1.E-10`.
BDN = good
Power parameter describing the evolution of the damage variable in the normal direction.
We check that :math:`\mathrm{BDN}<1.E-1`.
8.21. Keyword BETON_RAG#
This model is used to estimate the long-term behavior of structures affected by the alkali-granulate reaction [R7.01.26]. It makes it possible to assess deformations and anisotropic damage (cracking) of affected structures. It includes a Rankine criterion in tension and a Drücker-Prager criterion in compression. Both criteria are associated with a law of evolution leading to softening behavior.
8.21.1. Operands#
8.21.1.1. Operands linked to the damage model#
♦ ENDO_MC
Parameter of fragility of concrete under compression.
♦ ENDO_MT
Parameter of fragility of concrete under tension.
♦ ENDO_SIGUC
Equivalent stress of concrete in compression. The unit of this parameter is consistent with a constraint.
♦ ENDO_SIGUT
Equivalent tensile stress of concrete. The unit of this parameter is consistent with a constraint.
♦ ENDO_DRUPRA
This term is a characteristic of the compression criterion. It corresponds to the angle in radians of the Drucker Prager criterion.
8.22. Keyword BETON_BURGER#
The creep model BETON_BURGER assumes a decomposition between spherical and deviatory components: the deformations induced by spherical stresses are purely spherical and the deformations induced by deviatory stresses are purely deviatory [R7.01.35]. Moreover, the natural creep deformation is assumed to be proportional to the internal relative humidity:
Spherical part: \({\epsilon }^{s}=h\cdot f\left({\sigma }^{s}\right)\) and, deviatory part: \(\underline{\underline{{\epsilon }^{d}}}=h\cdot f\left(\stackrel{~}{\underline{\underline{\sigma }}}\right)\)
where \(h\) refers to internal relative humidity.
Behaviour model BETON_BURGER is a model based on the BETON_UMLV [R7.01.06] model for describing the natural creep of concrete. It is particularly suitable for multiaxial configurations by not assuming the value of the creep Poisson’s ratio. The changes made focus on taking into account a consolidation of creep translated by a non-linear term on the long-term behavior of the model. In addition, the parts spherical and deviatory are now constructed identically, leaving the possibility of controlling the apparent Poisson’s ratio of creep.
The spherical and deviatory parts are described by equivalent rheological chains, the so-called Bürger chain. This model was initially built using a Kelvin Voigt stage (reversible part) coupled in series to a Maxwell body (irreversible part).
The model also makes it possible to take into account the effect of temperature on creep deformations via an Arrhenius law.
The elastic mechanical characteristics E and NU must be defined in parallel under the keyword ELAS. Values under the ELAS keywords will be compared to the values entered under the BETON_BURGER keyword. If they are different, a fatal error will be issued.
If the ELASn keyword is not entered, code_aster will do it automatically by taking the elastic characteristics of the BETON_BURGER keyword.
8.22.1. Operands#
YoungModulus = E
\(E\) Young’s module. This operand is mandatory due to the use of Mfront
FishRatio = NU
\(\mathrm{\nu }\) Poisson’s ratio. This operand is mandatory due to the use of Mfront
K_RS = K_RS
\({k}_{r}^{s}\) apparent stiffness associated with the reversible spherical part of the creep deformations
K_RD = K_RD
:math:`{k}_{r}^{d}` apparent stiffness associated with the reversible deviatoric portion of creep deformations
ETA_RS = ETA_RS
:math:`{\eta }_{r}^{s}` apparent viscosity associated with reversible spherical deformations
ETA_IS = ETA_IS
:math:`{\mathrm{\eta }}_{i}^{s}` apparent viscosity associated with irreversible spherical deformations
ETA_RD = ETA_RD
:math:`{\eta }_{r}^{d}` viscosity at reversible deviatory deformations
ETA_ID = ETA_ID
:math:`{\eta }_{i}^{d}` viscosity with irreversible deviatory deformations
KAPPA = KAPPA
:math:`\kappa` term affecting the long-term viscosity (:math:`{\eta }_{i}^{s}` and :math:`{\eta }_{i}^{d}`) of the material
QSR_K = EAC/R
\({E}_{\mathit{ac}}/R\) is generally equivalent to \(4700.K\). (\(R\) is the ideal gas constant).
TEMP_0_C = T0
Reference temperature in the Arrhenius law. Reference temperature \({T}_{0}\) is in degrees Celsius, and converted to Kelvin during resolution.
ETA_FD = ETA_FD
makes it possible to take into account desiccation creep according to Bazant’s law.
Note:
The desorption curve giving the hygrometry \(h\) as a function of the water concentration \(C\) must be entered under the keyword ELAS_FO.
8.23. Keyword FLUA_PORO_BETON#
This model is used to estimate the behavior of structures subjected to delayed creep deformations [R7.01.30]. Deferred deformations of the solid skeleton are called natural creep deformations. They can be permanent or reversible, depending on the underlying phenomena. In the model, the constitutive equations of natural creep are always used coupled with a plastic deformation model that ensures the compatibility of the stress field with the resistance criteria. In practice, the proper creep model involves two usual rheological modules: a Kelvin-Voigt stage for reversible visco-elastic creep and a plastic Maxwell stage for permanent creep. The effects of intraporeous pressure are considered through the poromechanical framework.
The units of the material parameters are given in R7.01.30.
8.23.1. Operands#
8.23.1.2. Operands related to the creep model#
♦ TREF
Reference temperature for creep.
♦ TAUK
Characteristic time of the Kelvin module.
♦ YKSY
Ratio: Kelvin Stiffness/Young’s Modulus
♦ TAUM
Characteristic time of the Maxwell module.
♦ EKFL
Characteristic deformation of the creep potential.
♦ XFLU
Non-linearity coefficient.
♦ NRJM
Creep activation energy.
♦ DFMX
Creep damage.
8.24. Keyword ENDO_PORO_BETON#
This model is used to estimate the behavior of structures subjected to mechanical (in tension, compression or shear) or thermal damage [R7.01.30]. Concrete cracking is described using a non-linear model combining plastic criteria and damages, all anisotropic. These damages form the link between total stress and effective stress in the sense of damage theory. In traction, the Rankine criteria make it possible to reproduce localized structural macro-cracks. In compression, the Drucker-Prager criterion makes it possible to establish shear damage, in connection with the expansion resulting from the non-associated flow based on the Drucker-Prager criterion.
The units of the material parameters are given in R7.01.30.
8.24.1. Operands#
8.24.1.1. Operands related to mechanical behavior#
♦ AND
Tensile strength.
♦ EPT
Deformation at peak traction.
♦ RC
Compressive strength.
♦ EPC
Deformation at the peak of compression.
♦ DELT
Coefficient for taking into account confinement.
♦ BETA
Dilatance for the non-associated plastic Drucker-Prager flow.
♦ REF
Crack closure stress.
♦ EKDC
Characteristic plastic deformation of Drucker-Prager damage.
♦ GFT
Tensile cracking energy.
♦ GFR
Energy to close tensile cracks.
♦ DIM3
Size of the element in the 3rd direction (for cases AXIS or 2D).
8.24.1.2. Operands related to thermal damage#
♦ DT80
Damage at 80°C.
♦ TSTH
Threshold temperature for thermal damage.
8.24.1.3. Hydration-related operands#
♦ HYDR
Advancement of hydration.
♦ HYDS
Threshold for solidification of progress.
8.25. Keyword FLUA_ENDO_PORO#
This model is used to estimate the behavior of structures subjected to delayed creep deformations and mechanical (in tension, compression or shear) or thermal damage in a coupled manner [R7.01.30]. This law takes up the laws of behavior FLUA_PORO_BETON and ENDO_PORO_BETON by adding couplings between physical phenomena.
The units of the material parameters are given in R7.01.30.
8.25.1. Operands#
8.25.1.1. Operands related to mechanical behavior#
♦ AND
Tensile strength.
♦ EPT
Deformation at peak traction.
♦ RC
Compressive strength.
♦ EPC
Deformation at the peak of compression.
♦ DELT
Coefficient for taking into account confinement.
♦ BETA
Dilatance for the non-associated plastic Drucker-Prager flow.
♦ REF
Crack closure stress.
♦ EKDC
Characteristic plastic deformation of Drucker-Prager damage.
♦ GFT
Tensile cracking energy.
♦ GFR
Energy to close tensile cracks.
♦ DIM3
Size of the element in the 3rd direction (for cases AXIS or 2D).
8.25.1.2. Operands related to the creep model#
♦ TREF
Reference temperature for creep.
♦ TAUK
Characteristic time of the Kelvin module.
♦ YKSY
Ratio: Kelvin Stiffness/Young’s Modulus
♦ TAUM
Characteristic time of the Maxwell module.
♦ EKFL
Characteristic deformation of the creep potential.
♦ XFLU
Non-linearity coefficient.
♦ NRJM
Creep activation energy.
♦ DFMX
Creep damage.
8.25.1.3. Operands related to shrinkage/creep in desiccation#
♦ PORO
Porosity of the material.
◊ MSHR
Van Genuchten module.
◊ MVGN
Van Genuchten exhibitor.
◊ BSHR
Biot coefficient of capillary pressure.
◊ SFLD
Characteristic desiccation stress.
8.25.1.4. Operands related to thermal damage#
♦ DT80
Damage at 80°C.
♦ TSTH
Threshold temperature for thermal damage.
8.25.1.5. Hydration-related operands#
♦ HYDR
Advancement of hydration.
♦ HYDS
Threshold for solidification of progress.
8.26. Keyword RGI_BETON#
This model is used to estimate the behavior of structures subjected to internal swelling reactions (RGI) whether they come from the alkali-aggregate reaction (RAG) or from the DEF (Deferred Ettringite Formation) resulting from the RSI (Internal Sulfatic Reaction) [R7.01.30]. The volume of expansive product created and calculated by chemical models, allows the calculation of intraperous pressure which, combined with external loading, makes it possible to evaluate anisotropic diffuse cracking using anisotropic cracking criteria and work hardening laws. It should be noted that all the phenomena and equations of law FLUA_ENDO_BETON and therefore a fortiori of FLUA_PORO_BETON and ENDO_PORO_BETON are reused in RGI_BETON. In fact, the correct consideration of swelling phenomena requires that creep and damage be taken into account in a coupled manner.
The units of the material parameters are given in R7.01.30.
8.26.1. Operands#
8.26.1.4. Operands related to mechanical behavior#
♦ AND
Tensile strength.
♦ EPT
Deformation at peak traction.
♦ RC
Compressive strength.
♦ EPC
Deformation at the peak of compression.
♦ DELT
Coefficient for taking into account confinement.
♦ BETA
Dilatance for the non-associated plastic Drucker-Prager flow.
♦ REF
Crack closure stress.
♦ EKDC
Characteristic plastic deformation of Drucker-Prager damage.
♦ GFT
Tensile cracking energy.
♦ GFR
Energy to close tensile cracks.
♦ DIM3
Size of the element in the 3rd direction (for cases AXIS or 2D).
8.26.1.5. Operands related to the creep model#
♦ TREF
Reference temperature for creep.
♦ TAUK
Characteristic time of the Kelvin module.
♦ YKSY
Ratio: Kelvin Stiffness/Young’s Modulus
♦ TAUM
Characteristic time of the Maxwell module.
♦ EKFL
Characteristic deformation of the creep potential.
♦ XFLU
Non-linearity coefficient.
♦ NRJM
Creep activation energy.
♦ DFMX
Creep damage.
8.26.1.6. Operands related to shrinkage/creep in desiccation#
♦ PORO
Porosity of the material.
◊ MSHR
Van Genuchten module.
◊ MVGN
Van Genuchten exhibitor.
◊ BSHR
Biot coefficient of capillary pressure.
◊ SFLD
Characteristic desiccation stress.
8.26.1.7. Operands related to thermal damage#
♦ DT80
Damage at 80°C.
♦ TSTH
Threshold temperature for thermal damage.
8.26.1.8. Hydration-related operands#
♦ HYDR
Advancement of hydration.
♦ HYDS
Threshold for solidification of progress.
8.27. Keyword RGI_BETON_BA#
This model is an extension of law RGI_BETON integrating reinforcements to model the behavior of reinforced concrete. It uses the same parameters as well as additional parameters for defining the reinforcements. The definitions of the parameters common to RGI_BETON are not given again.
8.27.1. Operands#
8.27.1.1. Global operands#
♦ NREN
Number of frames (between 1 and 5)
♦ YOUM
Young’s modulus of concrete alone.
♦ NUM
Poisson’s ratio of concrete alone.
8.27.1.2. Operands linked to a given frame#
The following operands exist for i from 1 to 5. Parameters with no by default values are required if NREN is greater than or equal to i.
◊ ROAi
Density of the frame i.
◊ Ei
Young’s modulus of the frame i.
◊ SYi
Limit of elasticity of the frame i.
◊ TYRi
Stress at the steel/concrete interface of the reinforcement i.
◊ VRi1
Coordinate x of the direction of the frame i.
◊ VRi2
Coordinate y of the reinforcement direction i.
◊ VRi3
The z coordinate of the reinforcement direction i.
◊ D_SIGM_EPSI i
Armature work hardening module i.
◊ TAUKi
Reversible characteristic time for the Kelvin creep of the frame i.
◊ TAUMi
Irreversible characteristic time for Maxwell creep of the frame i.
◊ EKRi
Characteristic deformation for the relaxation of the frame i.
◊ SKRi
Characteristic constraint for setting EKRi. (The value by default is in MPa)
◊ ATRi
Reference value for the thermal activation coefficient
◊ CTMi
Coupling coefficient for the impact of stresses on activation energy
◊ XFLi
Exhibitor for thermal activation of relaxation
◊ PREi
Initial prestress (imposed as initial stress at the first step)
◊ TTRi
Reference temperature for setting TAUMi (°C)
◊ XNRi
Nonlinear relaxation coefficient
◊ MUSi
Threshold beyond which thermal activation becomes a function of the charge rate
◊ YKYi
Reversible relaxation reduction coefficient rated at TTRi
8.28. Keyword factor ENDO_LOCA_EXP/ENDO_LOCA_EXP_FO#
Definition of the parameters of the law of behavior ENDO_LOCA_EXP [R7.01.42], which describes the homogeneous cracking of a concrete structure via an isotropic local quasibrittle damage model.
8.28.1. Operands KAPPA and P#
These are the internal parameters of the model that define work hardening, see [R7.01.42]. Their identification is supported by the DEFI_MATER_GC [U4.42.07] command, using quantities that are more experimentally accessible.
Operands SIGC, SIG0, BETA0
These are the internal parameters of the model that define the shape of the load surface (with one exception), see [R7.01.42]. It is based on the Von Mises stress and the exponential stress tensor and compares well to experimental results on concrete under biaxial loading. The parameter BETA0 is more numerical in nature and only has the advantage of making the elasticity domain limited, including for hydrostatic compressions; the default value fulfills this function well, without affecting the shape of the domain in the areas of interest.
Again, the DEFI_MATER_GC [U4.42.07] command makes it possible to identify these parameters from quantities that are experimentally accessible (limits in tension and compression).
8.28.2. Operand REST_RIGIDITE#
Stiffness restoration is active for compression deformation directions. To avoid an abrupt change in regime during the transition from traction to compression, a function S regulates the jump in stiffness, see [R7.01.42]. The parameter REST_RIGIDITE, which is positive, controls this regularization; it corresponds to the coefficient gamma of the function S”. A value of 0 leads to not restoring stiffness (i.e. the model is without restoring stiffness) while a very large value amounts to virtually eliminating regularization. The DEFI_MATER_GC [U4.42.07] command makes it easy to quantify this parameter by stipulating what proportion of the stiffness is restored for a deformation level corresponding to the initial compression threshold.