10. Thermo-hydro-mechanical and soil behaviors

10.1. Simple keyword COMP_THM

Allows the coupling law THM to be selected as soon as the material is defined. The table below specifies the mandatory keywords according to the coupling law chosen.

O	Mandatory keyword
T	Mandatory thermal keyword
	Unnecessary keyword for this type of coupling law

10.2. Keyword factor THM_INIT

For all Thermo-Hydro-Mechanical behaviors, it makes it possible to describe the initial state of the structure (cf. [R7.01.11] and [R7.01.14]).

To fully understand this data, we need to distinguish between the unknowns at the nodes, which we call \({\left\{u\right\}}^{\mathit{ddl}}\), and the values defined under the THM_INIT keyword, which we call \({p}^{\mathit{ref}}\) and \({T}^{\mathit{ref}}\).

\({\left\{u\right\}}^{\mathrm{ddl}}=\left\{\begin{array}{}{u}_{x}\\ {u}_{y}\\ {u}_{z}\\ {\mathrm{PRE1}}^{\mathrm{ddl}}\\ {\mathrm{PRE2}}^{\mathrm{ddl}}\end{array}\right\}\)

The meaning of the unknowns \(\mathrm{PRE1}\) and \(\mathrm{PRE2}\) varies between models. Noting \({p}_{w}\) the pressure of water, \({p}_{\mathrm{ad}}\) the pressure of dissolved air, the pressure of dissolved air, \({p}_{l}\) the pressure of liquid, the pressure of liquid \({p}_{l}={p}_{w}+{p}_{\mathrm{ad}}\), \({p}_{\mathrm{as}}\) the pressure of dry air, the pressure of dry air, \({p}_{g}={p}_{\mathrm{as}}+{p}_{\mathrm{vp}}\) the total gas pressure and \({p}_{c}={p}_{g}-{p}_{l}\) the capillary pressure (also called suction), we have the following meanings of the unknowns \({p}_{\mathrm{vp}}\) \(\mathrm{PRE1}\) and \(\mathrm{PRE2}\):

Behavior KIT	LIQU_SATU	LIQU_VAPE	LIQU_GAZ_ATM	GAZ	LIQU_VAPE_GAZ	LIQU_GAZ	LIQU_AD_GAZ_VAPE	LIQU_AD_GAZ
\(\mathrm{PRE1}\)	\({p}_{l}\)	\({p}_{l}\)	\(-{p}_{l}\)	\({p}_{g}\)	\({p}_{c}={p}_{g}-{p}_{l}\)	\({p}_{c}={p}_{g}-{p}_{l}\)	\({p}_{c}={p}_{g}-{p}_{l}\)	\({p}_{c}={p}_{g}-{p}_{l}\)
\(\mathrm{PRE2}\)					\({p}_{g}\)	\({p}_{g}\)	\({p}_{g}\)	\({p}_{g}\)

You can refer to the [§4.4.3] of the documentation [U4.51.11].

The « total » pressures and temperature are then defined by:

\(p={p}^{\mathrm{ddl}}+{p}^{\mathrm{ref}}\text{;}T={T}^{\mathrm{ddl}}+{T}^{\mathrm{ref}}\)

The values written by IMPR_RESU are the \({p}^{\mathrm{ddl}}\text{et}{T}^{\mathrm{ddl}}\) nodal unknowns. Likewise, the boundary conditions must be expressed in relation to the nodal unknowns.

On the other hand, it is the total pressures and the temperature that are used in the laws of behavior \(\frac{P}{\rho }=\frac{R}{M}T\) for ideal gases, \(\frac{d{\rho }_{l}}{{\rho }_{l}}=\frac{{\mathrm{dp}}_{l}}{{K}_{l}}-3{\alpha }_{l}\mathrm{dT}\) for the liquid and in the saturation/capillary pressure relationship.

Note that nodal values can be initialized by the ETAT_INIT keyword from the STAT_NON_LINE command or by a combination of the two (this can allow easy conversion from Kelvin to degrees for example).

10.2.1. Operand TEMP

Reference temperature \({T}^{\mathit{ref}}\) expressed in Kelvin. By default it is taken equal to zero. Attention the value of the initial \(T={T}^{\text{ddl}}+{T}^{\text{ref}}\) temperature must be strictly greater than zero.

The reference temperature value entered behind the keyword **** VALE_REFde the command**** AFFE_VARC is ignored. **

10.2.2. Operand PRE1

By default it is taken equal to zero.

For behaviors: LIQU_SATU, LIQU_VAPE reference liquid pressure.

For the behavior: GAZpression of reference gas. In this case the initial gas pressure :math:`p={p}^{\text{ddl}}+{p}^{\text{ref}}` must be non-zero.

For behavior: LIQU_GAZ_ATMpression of liquid reference changed sign.

For behaviors: LIQU_VAPE_GAZ, LIQU_AD_GAZ_VAPE, LIQU_AD_GAZet LIQU_GAZpression reference capillary.

10.2.3. Operand PRE2

By default it is taken equal to zero. For the behaviors: LIQU_VAPE_GAZ, LIQU_AD_GAZ_VAPE, LIQU_AD_GAZ and LIQU_GAZpression of reference gas. The initial pressure of gas \(p={p}^{\text{ddl}}+{p}^{\text{ref}}\) must be non-zero.

10.2.4. Operand PORO/PRES_VAPE/DEGR_SATU

PORO = porn

Initial porosity \({\Phi }_{0}\).

note

For models AXIS_JHMS and PLAN_JHMS, this value does not make sense and is not used during the calculation.

PRES_VAPE = pvap

For behaviors: LIQU_VAPE_GAZ, LIQU_VAPE, LIQU_AD_GAZ_VAPE and LIQU_GAZ initial vapor pressure.

DEGR_SATU = ds

For all unsaturated behaviors: initial degree of saturation (now useless, given as an indication).

10.3. Keyword factor THM_LIQU

This key word concerns all THM behaviors involving liquid (cf. [R7.01.11]).

10.3.1. Operand RHO

Liquid density for the pressure defined under the keyword PRE1 of the keyword factor THM_INIT.

10.3.2. Operand UN_SUR_K

Inverse of the \({K}_{l}\) compressibility of liquid. A zero value means an incompressible liquid.

10.3.3. Operand ALPHA

Coefficient of thermal expansion \({\alpha }_{l}\) of the liquid.

If \({p}_{l}\) refers to the pressure of the liquid, \({\rho }_{l}\) its density and \(T\) the temperature, the behavior relationship of the liquid is: \(\frac{d{\rho }_{l}}{{\rho }_{l}}=\frac{{\mathrm{dp}}_{l}}{{K}_{l}}-3{\alpha }_{l}\mathrm{dT}\).

10.3.4. CP operand

Specific heat at constant liquid pressure.

10.3.5. Operands VISC/D_VISC_TEMP

VISC = life

Dynamic viscosity of the liquid (reminder of the units: [pressure]). [time]). Temperature function.

D_VISC_TEMP = dvi

Derived from the viscosity of the liquid with respect to temperature. Temperature function. The user must ensure consistency with the function associated with VISC.

10.4. Keyword factor THM_GAZ

This key word factor concerns all THM behaviors involving gas (cf. [R7.01.11]). For behaviors involving both a liquid and a gas, and when liquid evaporation is taken into account, the coefficients given here relate to dry gas. The properties of steam are given under the keyword THM_VAPE_GAZ.

10.4.1. Operand MASS_MOL

Molar mass of dry gas \({M}_{\mathrm{gs}}\mathrm{.}\)

If \({p}_{\mathrm{gs}}\) refers to the pressure of dry gas, \({\rho }_{\mathrm{gs}}\) its density, its density, \(R\) the ideal gas constant, and \(T\) the temperature, the behavior of dry gas is: \(\frac{{p}_{\mathrm{gs}}}{{\rho }_{\mathrm{gs}}}=\frac{\mathrm{RT}}{{M}_{\mathrm{gs}}}\).

10.4.2. CP operand

Specific heat at constant dry gas pressure.

10.4.3. Operand VISC

Viscosity of dry gas. Temperature function.

10.4.4. Operand D_VISC_TEMP

Derived from the viscosity of dry gas with respect to temperature. A function of temperature.

The user must ensure consistency with the function associated with VISC.

10.5. Keyword factor THM_VAPE_GAZ

This key word factor concerns all behaviors THM involving both a liquid and a gas, and taking into account the evaporation of the liquid (confer [R7.01.11]). The coefficients given here refer to steam.

10.5.1. Operand MASS_MOL

MASS_MOL = m

Molar mass of steam \({M}_{\mathrm{vp}}\).

If \({M}_{\mathrm{vp}}\) refers to the pressure of the vapor, \({\rho }_{\text{vp}}\) its density, the constant \(R\) for ideal gases and \(T\) the temperature, the behavior of the vapor is: \(\frac{{p}_{\mathrm{vp}}}{{\rho }_{\mathrm{vp}}}=\frac{\mathrm{RT}}{{M}_{\mathrm{vp}}}\mathrm{.}\)

10.5.2. CP operand

CP = cp

Specific heat at constant vapour pressure.

10.5.3. Operand VISC

VISC = v

Viscosity of the steam. Temperature function.

10.5.4. Operand D_VISC_TEMP

D_VISC_TEMP = dvi

Derived from the viscosity of the vapor in relation to temperature. Temperature function.

The user must ensure consistency with the function associated with VISC.

10.6. Keyword factor THM_AIR_DISS

This key word factor concerns behavior THM THM_AD_GAZ_VAPE taking into account the dissolution of air in liquid (cf. [R7.01.11]). The coefficients given here refer to dissolved air.

10.6.1. CP operand

CP = cp

Specific heat at constant pressure of dissolved air.

10.6.2. Operand COEF_HENRY

COEF_HENRY = h

Henry’s constant \({K}_{H}\), a function of temperature, making it possible to relate the molar concentration of dissolved air \({C}_{\mathrm{ad}}^{\mathrm{ol}}\) (\(\mathrm{moles}/{m}^{3}\)) to the pressure of dry air:

\({C}_{\mathrm{ad}}^{\mathrm{ol}}=\frac{{p}_{\mathrm{as}}}{{K}_{H}}\)

10.7. Keyword factor THM_DIFFU

Mandatory for all THM behaviors (cf. [R7.01.11]). The user must ensure the consistency of the functions and their derivatives.

10.7.1. Operands R_GAZ/RHO /CP/ BIOT_COEF

R_GAZ = gas

Ideal gas constant.

RHO = rho

For hydromechanical behaviors, homogenized density. In static cases, it is useless to specify the density in the elastic behavior (see section 4.1.3).

CP = cp

For thermal behaviors: specific heat at constant stress of the solid alone.

BIOT_COEF = organic

Biot coefficient. Must be less than or equal to \(1\).

10.7.2. Operands BIOT_L/BIOT_T/BIOT_N

Replace BIOT_COEF in the anisotropic case. In the case of transverse isotropy (3D), BIOT_L and BIOT_N are respectively the Biot coefficients in the L and N directions (perpendicular to the isotropy plane). In the orthotropic case in 2D, we will define the Biot coefficients in the three directions L, T, N: BIOT_L, BIOT_T and BIOT_N.

Note: if you choose this anisotropic version of the Biot coefficients, you must be careful to choose the anisotropic version of the elasticity in the same way. Otherwise you will get an error message.

10.7.3. Operands SATU_PRES/D_SATU_PRES

For unsaturated material behaviors (LIQU_VAPE_GAZ, LIQU_VAPE,,, LIQU_AD_GAZ,,,, LIQU_AD_GAZ_VAPE, LIQU_GAZ, LIQU_GAZ_ATM).

SATU_PRES = sp

Saturation isotherm as a function of capillary pressure.

D_SATU_PRES = dsp

Derived from saturation with respect to pressure.

10.7.4. Operands PESA_X/PESA_Y/PESA_Z/PESA_MULT

PESA_X = px, PESA_y = py, PESA_z = pz,

Gravity according to \(x\), \(y\) or \(z\), used only if the modeling chosen in AFFE_MODELE includes 1 or 2 pressure variables.

PESA_MULT = fpesa

Time function as a factor of the components of gravity PESA_X, PESA_Y and PESA_Z. Optional, it is by default constant and equal to 1.

10.7.5. Operand PERM_IN

Intrinsic permeability \({K}_{\text{int}}\), whose dimension is that of a surface: a function of porosity \(\varphi\) (in the isotropic case). In studies, the user can express the dependence of intrinsic permeability on \(\varphi\) by the following usual cubic law, using a previously defined function:

\(\frac{{K}_{\text{int}}(\varphi )}{{K}_{{\text{int}}_{0}}}=\begin{array}{c}\mathrm{si}\varphi -{\varphi }_{0}<0:1\\ \mathrm{si}0<\varphi -{\varphi }_{0}<{10}^{-2}:1+\chi {(\varphi -{\varphi }_{0})}^{3}\\ \mathrm{si}{10}^{-2}<\varphi -{\varphi }_{0}:1+\chi \ast {10}^{-6}\end{array}\)

Other laws are of course possible.

Permeability in the usual sense \(K\), whose dimension is that of a speed, is calculated as follows:

\(K=\frac{{K}_{\text{int}}{K}_{\mathrm{rel}}}{\mu }{\rho }_{l}g\) where \({K}_{\text{int}}\) is the intrinsic permeability, \({K}_{\mathrm{rel}}\) the relative permeability of the liquid (in dimension), \(\mu\) the dynamic viscosity of the liquid, \({\rho }_{l}\) the density of the liquid and \(g\) the acceleration of gravity. \({K}_{\text{int}}\) is in fact a diagonal tensor, in the isotropic case its three components are equal to the value entered.

10.7.6. Operands PERMIN_L/PERMIN_T/PERMIN_N

For the definition of isotropy plans, reference will be made to Error: Reference source not found and Error: Reference source not found. In the case of transverse isotropy (3D), PERMIN_L and PERMIN_T are respectively the intrinsic permeabilities in the L and N directions (perpendicular to the isotropy plane). In the orthotropic case in 2D, we will define the permeabilities in the L and T planes: PERMIN_L and PERMIN_T.

10.7.7. Operands PERM_LIQU/D_PERM_LIQU_SATU

For unsaturated material behaviors (LIQU_VAPE_GAZ, LIQU_VAPE,,, LIQU_AD_GAZ,,,, LIQU_AD_GAZ_VAPE, LIQU_GAZ, LIQU_GAZ_ATM).

Relative permeability to liquid (dimensionless): function of saturation and its derivative with respect to saturation.

10.7.8. Operands PERM_GAZ/D_PERM_SATU_GAZ/D_PERM_PRES_GAZ

For unsaturated material behaviors (LIQU_VAPE_GAZ, LIQU_VAPE,,,, LIQU_AD_GAZ, LIQU_AD_GAZ_VAPE, LIQU_GAZ).

Relative gas permeability (dimensionless): a function of gas saturation and pressure and its derivatives with respect to gas saturation and pressure.

10.7.9. Operands VG_N/VG_PR/VG_SR

For the behaviors of unsaturated materials liquid gas with two components and two unknowns (LIQU_VAPE_GAZ, LIQU_AD_GAZ, LIQU_AD_GAZ_VAPE, LIQU_GAZ,) and in the case where the hydraulic law is HYDR_VGM or HYDR_VGC (see doc. U4.51.11), respectively designate the parameters \(N\), \(\mathit{Pr}\), and \(\mathit{Sr}\) of the Mualem Van-Genuchten law used to define the capillary pressure and the permeabilities relating to water and gas.

10.7.10. Operands VG_SMAX/VG_SATUR

For the behaviors of unsaturated materials liquid gas with two components and two unknowns (LIQU_VAPE_GAZ, LIQU_AD_GAZ, LIQU_AD_GAZ_VAPE, LIQU_GAZ,) and in the case where the hydraulic law is HYDR_VGM or HYDR_VGC (see document [U4.51.11]).

VG_SMAX = smax

refers to the maximum saturation for which Mualem Van-Genuchten’s law is applied. Beyond this saturation, the Mualem-Van Genuchten curves are interpolated (see document [R7.01.11]). This value should be very close to 1.

VG_SATUR = stur

Beyond the saturation defined by VG_SMAX, the saturation is multiplied by this corrective factor. This value must be very close to 1 (see document [R7.01.11]).

10.7.11. Operands FICKV_T/FICKV_S/FICKV_PG/FICKV_PV

For behaviors LIQU_VAPE_GAZ and LIQU_AD_GAZ_VAPE, Fick coefficient as a function of temperature for the diffusion of steam in the gas mixture. Since the Fick coefficient can be a function of saturation, temperature, gas pressure and vapor pressure, it is defined as a product of 4 functions: FICKV_T, FICKV_S, FICKV_PG, FICKV_VP. In the case of LIQU_VAPE_GAZ and LIQU_AD_GAZ_VAPE, only FICKV_T is mandatory.

10.7.12. Operands D_FV_T/D_FV_PG

For LIQU_VAPE_GAZ and LIQU_AD_GAZ_VAPE behaviors.

Derived from the FICKV_T coefficient with respect to temperature.

Derived from the coefficient FICKV_PG with respect to gas pressure.

10.7.13. Operands FICKA_T/FICKA_S/FICKA_PA/FICKA_P

For behavior LIQU_AD_GAZ_VAPE, Fick coefficient as a function of temperature for the diffusion of dissolved air in the liquid mixture. Since the Fick coefficient can be a function of saturation, temperature, dissolved air pressure and liquid pressure, it is defined as a product of 4 functions: FICKA_T, FICKA_S, FICKV_PA, FICKV_PL. In the case of LIQU_AD_GAZ_VAPE, only FICKA_T is mandatory.

10.7.14. Operand D_FA_T

For behavior LIQU_AD_GAZ_VAPE, derived from the FICKA_T coefficient with respect to temperature.

10.7.15. Operands LAMB_T/LAMB_S/LAMB_PHI/LAMB_CT

LAMB_T = lambt

Multiplicative part of the thermal conductivity of the mixture depending on the temperature (cf. [R7.01.11]). This operand is mandatory in the case of thermal modeling.

LAMB_S = lambs, LAMB_PHI = lambp

Multiplicative part (equal to 1 by default) of the thermal conductivity of the mixture depending respectively on saturation and porosity.

LAMB_CT = Lambct

Part of the thermal conductivity of the mixture that is constant and additive (confer [R7.01.11]). This constant is zero by default.

10.7.16. Operands LAMB_TL/LAMB_TN/LAMB_TT

Replace LAMB_T in the anisotropic case. In the case of transverse isotropy (3D), LAMB_TL and LAMB_TT are respectively the conductivities in the L and N directions (perpendicular to the isotropy plane). In the orthotropic case in 2D, we will define the conductivities in the L and T planes: LAMB_TL and LAMB_TN.

10.7.17. Operands LAMB_C_L/LAMB_C_N/LAMB_C_T

Replace LAMB_CT in the anisotropic case. In the case of transverse isotropy (3D), LAMB_C_L and LAMB_C_T are respectively the conductivities in the L and N directions (perpendicular to the isotropy plane). In the orthotropic case in 2D, we will define the conductivities in the L and T planes: LAMB_C_L and LAMB_C_N.

10.7.18. Operands D_LB_T/D_LB_S/D_LB_PHI

D_LB_T = flashing

Derived from the part of the thermal conductivity of the mixture that is dependent on temperature in relation to temperature.

D_LB_S = dlambs, D_LB_PHI = dlambp

Derived from that part of the thermal conductivity of the mixture depending respectively on saturation and porosity.

10.7.19. Operands D_LB_TL/D_LB_TN/D_LB_TT

In the anisotropic case, derived with respect to the temperature of LAMB_TL, LAMB_TN and LAMB_TT respectively.

10.7.20. Operand EMMAG

Storage coefficient. This coefficient is only taken into account in the case of models without mechanics. It relates the change in porosity to the change in liquid pressure.

10.7.21. Operand PERM_END

Permeability a function of damage, used by mechanical behaviors with damage.

10.7.22. Operands EPAI /A0/ SHUTTLE//S_BJH/W_BJH

For the behaviors of unsaturated materials liquid gas with two components and two unknowns (LIQU_VAPE_GAZ, LIQU_AD_GAZ, LIQU_AD_GAZ_VAPE, LIQU_GAZ, LIQU_GAZ_ATM) and in the case where the hydraulic law is HYDR_TABBAL (see doc. OR 2. 04. 05). These parameters are those of the poromechanical model for strong drying defined in the thesis of G. El Tabbal. They are used to define respectively the thickness of the adsorbed layer EPAI, the specific surface area A0 of the material, the Shuttleworth parameter SHUTTLE and the functions output from the calculation BJH: S_BJH and W_BJH.

10.8. Keyword MOHR_COULOMB

The Mohr-Coulomb model is an elasto-plastic model used in soil mechanics and is specially adapted to sandy materials. The document [:ref:` R7.01.28 < R7.01.28 >`] describes the corresponding equations. This model can be used regardless of THM behaviors. Elastic characteristics should be defined under the keyword ELAS.

10.8.1. Operands PHI/ANGDIL/COHESION

PHI = phi

Friction angle (in degrees). The value should be between 0 and 60 degrees.

ANGDIL = angel

Expansion angle (in degrees). The value should be between 0 and 60 degrees.

COHESION = bumps

Cohesion of the material (in Pascal — if SI unit).

10.9. Keyword CAM_CLAY

The Cam-Clay model is an elastoplastic model used in soil mechanics and is specially adapted to clay materials. The model shown here is called modified Cam-Clay. The document [R7.01.14] describes the corresponding equations. This model can be used regardless of THM behaviors. Elastic characteristics should be defined under the keyword ELAS.

10.9.1. MU/ LAMBDA/KAPA operands

MU = mu

Elastic shear modulus.

LAMBDA = normal

Compressibility coefficient (plastic slope in a hydrostatic compression test).

KAPA = kapa

Elastic swelling coefficient (elastic slope in a hydrostatic compression test).

10.9.2. M operand

Slope of the critical state line.

10.9.3. Operand PORO

Initial porosity. If CAM_CLAY is used under RELATION_KIT, the key word PORO entered under CAM_CLAY and under THM_INIT must be the same.

10.9.4. Operands PRES_CRIT/KCAM

PRES_CRIT = press

The critical pressure is equal to half the consolidation pressure.

KCAM = kcam

Initial pressure corresponding to the initial porosity generally equal to atmospheric pressure. This parameter must be positive (\(\mathrm{kcam}>0.\)).

10.9.5. Operand PTRAC

Quantity of hydrostatic tensile stress allowed or shift of the ellipse to the left on the hydrostatic stress axis. This parameter must be negative (ptrac < 0.).

10.10. Keyword factor CJS

The law (Cambou, Jaffani, Sidoroff) is a law of behavior for soils. It comprises three mechanisms, one corresponds to non-linear elasticity, another corresponds to plasticization for isotropic stress states, and the third mechanism corresponds to plasticization linked to a deviatory stress state. The document [R7.01.13] accurately describes the corresponding equations.

Elastic characteristics should be defined under the keyword ELAS.

Law CJS covers three possible forms (CJS1, CJS2 and CJS3), depending on whether or not the activation of nonlinear mechanisms is authorized.

The table below shows the mechanisms activated for the three levels CJS1, CJS2 and CJS3:

	Elastic mechanism	Isotropic plastic mechanism	Deviatory plastic mechanism
CJS1	linear	not activated	activated, perfect plasticity
CJS2	nonlinear	activated	activated, isotropic work hardening
CJS3	nonlinear	activated	activated, kinematic work hardening

note

By adopting the correspondence of the parameters for the limit states, it is possible to use behavior CJS1 to model a Mohr Coulomb law in soil mechanics.

The various coefficients may or may not be filled in depending on the level you want to use, in accordance with the table below (F for optional, O for mandatory and nothing for not applicable).

Symbol	\(b\)	\({R}_{m}\)		\(\mu\)		\({p}_{\mathrm{co}}\)	\(c\)	\({P}_{a}\)
Keyword	B_CJS	RM	M_CJS		PCO	C_CJS	PA
CJS1		O				O	O
CJS2		O				O	O
CJS3	O	O	O	O	O	O	O

We draw the user’s attention to the fact that, for the same material, the same coefficient may take on different values depending on the level used. The level used is never entered; it is indicated by the fact that certain coefficients are entered or not.

In addition, the keyword ELAS must be entered when using law CJS (under one of its three levels). The definition of the Young’s modulus and the Poisson’s ratio make it possible to calculate the coefficients \({K}_{o}^{e}\) and \({G}_{o}\).

10.10.1. Operands BETA_CJS /RM

For levels CJS1, CJS2 CJS3.

BETA_CJS = beta

Parameter \(\beta\). Control the change in plastic volume in the diverting mechanism.

RM = RM

Maximum value for the opening of the deviatory reversibility domain.

10.10.2. Operands N_CJS /KP/RC

For levels CJS2 and CJS3.

N_CJS = n

Check the dependence of the elastic modulus on the mean stress:

\(K={K}_{o}^{e}{(\frac{{I}_{1}+{Q}_{\mathrm{init}}}{{\mathrm{3P}}_{a}})}^{n}\text{}G={G}_{o}{(\frac{{I}_{1}+{Q}_{\mathrm{init}}}{{\mathrm{3P}}_{a}})}^{n}\)

KP = kp

Plastic compressibility module:

\({\dot{Q}}_{\mathrm{iso}}={K}^{p}\dot{q}={K}_{o}^{p}{(\frac{{Q}_{\mathrm{iso}}}{{P}_{a}})}^{n}\dot{q}\)

RC = rc

Critical value for variable \(R\):

\({\dot{e}}_{v}^{\mathrm{dp}}=-\beta (\frac{{s}_{\mathrm{II}}}{{s}_{\mathrm{II}}^{c}}-1)\frac{∣{s}_{\mathrm{ij}}{\dot{e}}_{\mathrm{ij}}^{\mathrm{dp}}∣}{{s}_{\mathrm{II}}}\text{}{s}_{\mathrm{II}}^{c}=-\frac{{R}_{c}{I}_{1}}{h({\theta }_{s})}\)

10.10.3. Operands A_CJS/R_INIT

For levels CJS2.

A_CJS = a

Controls the isotropic work hardening of the diverting mechanism;

\(R=\frac{{\mathrm{AR}}_{m}r}{{R}_{m}+\mathrm{Ar}}\)

R_INIT = r

Initial value for variable \(R\). At the first calculation time, if the initial value of \(R\) is zero, either because the initial state of the internal variables has not been defined by the keyword ETAT_INIT of STAT_NON_LINE, or if this initial state is zero, the initial value will be taken as the initial value that defined by the keyword R_INIT of DEFI_MATERIAU.

10.10.4. Operands B_CJS/C_CJS/PCO/MU_CJS

For levels CJS3.

B_CJS = b

Check the kinematic work hardening of the diverting mechanism:

\({\dot{X}}_{\mathrm{ij}}=-\frac{1}{b}{\dot{\lambda }}^{d}\left[\mathrm{dev}(\frac{\partial {f}^{d}}{\partial {X}_{\mathrm{ij}}})-{I}_{1}f{X}_{\mathrm{ij}}\right]{(\frac{{I}_{1}}{3{P}_{a}})}^{-1.5}\)

C_CJS = c

Control the evolution of critical pressure: \({p}_{c}={p}_{\mathrm{co}}\mathrm{exp}(-c{\varepsilon }_{v})\).

PCO = PCO

Initial critical pressure: \({p}_{c}={p}_{\mathrm{co}}\mathrm{exp}(-c{\varepsilon }_{v})\).

MU_CJS = mu

Check the break value for the variable \(R\): \({R}_{r}={R}_{c}+m\mathrm{ln}(\frac{3{p}_{c}}{{I}_{1}})\).

10.10.5. Operands GAMMA_CJS /PA/ Q_INIT

For levels CJS1, CJS2 and CJS3.

GAMMA_CJS = g

Check the shape of the criterion: \(h({\theta }_{s})={(1+\gamma \mathrm{cos}(3{\theta }_{s}))}^{1/6}={(1+\gamma \sqrt{54}\frac{\mathrm{det}(\underline{\underline{s}})}{{s}_{\mathrm{II}}^{3}})}^{1/6}\)

PA = pa

Atmospheric pressure. Should be given negative.

Q_INIT = q

Numerical parameter that makes it possible to make a zero stress state admissible. Can also be used to define cohesion, at least for level CJS1. We will use the formula: \({Q}_{\mathit{init}}=-3c\mathit{cotan}(\phi )\)

10.11. Keyword factor LAIGLE

The law of LAIGLE [R7.01.15] is a model of rheological behavior for modeling rocks. These are characterized by the following three parameters:

• \(a\) which defines the influence of the dilatance component in behavior at large deformations. This parameter depends on the level of alteration of the rock,

• \(s\) which defines the cohesion of the environment. It is therefore representative of the damage to the rock,

• \(m\) is a function of the mineralogical nature of the rock, and is associated with significant feedback.

Elastic characteristics should be defined under the keyword ELAS.

10.11.1. Operands GAMMA_ULT/GAMMA_E

GAMMA_ULT = gamma_ult

Parameter \({\gamma }_{\mathrm{ult}}\): Plastic deviatory deformation corresponding to the bearing.

GAMMA_E = gamma_e

Parameter \({\gamma }_{e}\): Plastic deviatory deformation corresponding to the complete disappearance of cohesion.

10.11.2. Operand M_ULT /M_E/A_E/ M_PIC

M_ULT = m_ult

Parameter \({m}_{\mathrm{ult}}\): Value of \(m\) of the ultimate criterion reached \({\gamma }_{\mathrm{ult}}\).

M_E = m_e

Parameter \({m}_{e}\): Value of \(m\) of the intermediate criterion reached in \({\gamma }_{e}\).

A_E = a_e

Parameter \({a}_{e}\): Value of \(a\) of the intermediate criterion reached in \({\gamma }_{e}\).

M_PIC = m_pic

Parameter \({m}_{\mathrm{pic}}\): Value of \(m\) of the peak criterion reached at the peak of stress.

10.11.3. Operands A_PIC/ETA/SIGMA_C

A_PIC = a_pic

Parameter \({a}_{\mathrm{pic}}\): Value of the exponent \(a\) at the peak of stress.

ETA = eta

Parameter \(\eta\): Exponent that regulates work hardening.

SIGMA_C = sigma_c

Parameter \({s}_{c}\): Simple compression strength.

10.11.4. Operands GAMMA/KSI

GAMMA = gamma, KSI = ksi

Parameters \(\gamma\) and \(\xi\): Parameters that control the dilatance.

One condition that must be met is that the \(\gamma /\xi\) ratio remains below \(1\). In the case of very resistant hard rocks, subject to relatively low confinement stresses, the variation in dilatance \(\mathrm{sin}\psi\) (depending on the state of the stresses - see [R7.01.15]) may tend towards \(\gamma /\xi\), which justifies this condition.

10.11.5. Operand GAMMA_CJS

Parameter \({\gamma }_{\mathit{cjs}}\): parameter for the shape of the load surface in the deflection plane.

10.11.6. Operand SIGMA_P1

Parameter \({\sigma }_{\mathit{pl}}\): intersection of the intermediate criterion and the peak criterion.

10.11.7. PA operand

Atmospheric pressure. Should be given positive.

Note:

The parameters M_E, A_E, A_E, * A_PIC, * SIGMA_P1, SIGMA_Cet MPICsont dependent on each other by the relationship: \({m}_{e}=\frac{{\sigma }_{c}}{{\sigma }_{\mathrm{pl}}}{({m}_{\mathrm{pic}}\frac{{\sigma }_{\mathrm{pl}}}{{\sigma }_{c}}+1)}^{\frac{{a}_{\mathrm{pic}}}{{a}_{e}}}\text{.}\) This dependency is checked within the code.

10.12. Keyword factor LETK

The L&K rheological model (Lagle and Kleine) is a law of elasto-visco-plastic behavior called here LETK [R7.01.24]. It is based on concepts of elastoplasticity and viscoplasticity. Elastoplasticity is characterized by positive pre-peak work hardening and post-peak negative work hardening.

Among the parameters, we find:

• parameters that are involved in the work hardening functions relating to the various elastoplastic or viscous thresholds, such as \(a\), \(s\) and \(m\),

• parameters related to viscous criteria,

• parameters related to dilatance,

• parameters related to the compressive and tensile strength of the material.

Elastic characteristics should be defined under the keyword ELAS.

10.12.1. Operands PA/ NELAS/SIGMA_C /H 0_EXT

PA = pa S_0 = s0

Parameter \(\mathrm{Pa}\): atmospheric pressure.

NELAS = Nelas

Parameter \({n}_{\mathrm{elas}}\): exponent of the law of variation of the elastic modules \(K\) and \(G\).

SIGMA_C = SIGC

Parameter \({\sigma }_{c}\): resistance in simple compression (the unit of a stress)..

H 0_EXT = h0

Parameter \({H}_{\mathrm{0ext}}\): parameter controlling the tensile strength

10.12.2. Operand GAMMA_CJS/XAMS

GAMMA_CJS = gcjs
XAMS = spam

Parameter \({\gamma }_{\text{cjs}}\): parameter for the shape of the criterion in the deviatory plane (between 0 and 1).

Parameter \({x}_{\text{ams}}\): non-zero parameter involved in pre-peak work hardening laws.

10.12.3. Operand ETA /A_0/A_E/ A_PIC

ETA = eta

Parameter \(h\): non-zero parameter involved in post-peak work hardening laws.

A_0 = a0

Parameter \({a}_{0}\): value of \(a\) on the damage threshold.

A_E = ae

Parameter \({a}_{e}\): value of \(a\) on the intermediate threshold.

A_PIC = ap

Parameter \({a}_{\mathit{pic}}\): value of \(a\) on the peak threshold.

10.12.4. Operands S_0/S_E/M_0/M_E/ M_PIC/M_ULT

S_0 = s0

Parameter \({s}_{0}\): value of \(s\) on the damage threshold.

S_E = se

Parameter \({s}_{e}\): value of \(s\) on the intermediate threshold.

M_0 = m0

Parameter \({m}_{0}\): value of \(m\) on the damage threshold.

M_E = me

Parameter \({m}_{e}\): value of \(m\) on the intermediate threshold.

M_PIC = mp

Parameter \({m}_{\mathrm{pic}}\): value of \(m\) on the peak threshold.

M_ULT = multiple

Parameter \({m}_{\mathrm{ult}}\): value of \(m\) on the residual threshold.

10.12.5. Operands XI_E/XI_PIC/MV_MAX/XIV_MAX

XI_E = Xie

Parameter \({\xi }_{e}\): level of work hardening on the intermediate threshold.

XI_PIC = zip

Parameter \({\xi }_{\mathrm{pic}}\): level of work hardening on the peak threshold.

MV_MAX = mvmx

Parameter \({m}_{v-\mathit{max}}\): value of \(m\) on the viscoplasticity threshold.

XIV_MAX = xivmx

Parameter \({\xi }_{v-\mathrm{max}}\): level of work hardening to reach the maximum viscoplastic threshold.

10.12.6. A/N operands

A = A

Parameter \(A\): parameter characterizing the amplitude of the creep speed (in \({s}^{-1}\) or \({\mathrm{jour}}^{-1}\)).

N = n

Parameter \(n\): exponent used in the formula controlling the creep kinetics.

10.12.7. Operand SIGMA_P1

SIGMA_P1 = sp1

Parameter \({\sigma }_{\mathit{P1}}\): corresponds to the abscissa of the intersection point of the cleavage limit and the peak threshold.

10.12.8. Operands MU0_V and XI0_V

MU0_V = mu0v, XI0_V = xi0v

Parameters \({\mu }_{\mathrm{0v}}\) and \({\xi }_{\mathrm{0v}}\): parameters regulating the dilatance of pre-peak and viscoplastic mechanisms

The conditions to be respected on these parameters are:

\({\mu }_{\mathrm{0v}}<{\xi }_{\mathrm{0v}}\) or \(\{\begin{array}{c}{\mu }_{0v}>{\xi }_{0v}\hfill \\ \frac{{s}_{\mathit{pic}}^{{a}_{\mathit{pic}}}}{{s}_{0}^{{a}_{0}}}\le \frac{1+{\mu }_{0v}}{{\mu }_{0v}-{\xi }_{0v}}\end{array}\) with \({s}^{\mathit{pic}}=1\)

10.12.9. Operands MU1 and XI1

MU1 = mu1, XI1 = xi1

Parameters \({\mu }_{1}\) and \({\xi }_{1}\): parameters that regulate the dilatance of post-peak mechanisms. A condition to be met is that the \({\mu }_{1}\mathrm{/}{\xi }_{1}\) ratio remains less than or equal to 1.

10.13. Keyword factor DRUCK_PRAGER

The criterion of DRUCK_PRAGER [R7.01.16] is a behavior model for soil mechanics, it is defined by the relationship:

\({\mathrm{\sigma }}_{\mathit{eq}}+\mathrm{\alpha }{I}_{1}-R(p)\le 0\)

where:

\({\sigma }_{\mathrm{eq}}\) is a function of the effective constraints deviator \({\sigma }^{\text{'}}\),

\({I}_{1}=\mathrm{Tr}({\sigma }^{\text{'}})\) is the trace of the effective constraints,

\(\alpha\) is a pressure dependence coefficient,

\(R(p)\) is a function of cumulative plastic deformation.

In the linear case (associated law DRUCK_PRAGER and non-associated law DRUCK_PRAG_N_A), the function \(R\) is given by: \(\begin{array}{cc}0<p<{p}_{\mathit{ult}}& R(p)=\mathit{hp}+{\mathrm{\sigma }}_{y}\hfill \\ p\ge {p}_{\mathit{ult}}& R(p)=h{p}_{\mathit{ult}}+{\mathrm{\sigma }}_{y}\hfill \end{array}\).

In the parabolic case (associated law DRUCK_PRAGER), \(R(p)={\sigma }_{y}f(p)\) where the function \(f(p)\) is given by:

\(\begin{array}{cc}0<p<{p}_{\mathit{ult}}& f\left(p\right)={\left(1-\left(1-\sqrt{\frac{{\mathrm{\sigma }}_{y\mathit{ult}}}{{\mathrm{\sigma }}_{y}}}\right)\frac{p}{{p}_{\mathit{ult}}}\right)}^{2}\hfill \\ p\ge {p}_{\mathit{ult}}& f\left(p\right)=\frac{{\mathrm{\sigma }}_{y\mathit{ult}}}{{\mathrm{\sigma }}_{y}}\hfill \end{array}\)

In the parabolic case (non-associated law DRUCK_PRAG_N_A), the function \(R\) is given by:

\(\begin{array}{cc}0<p<{p}_{\mathit{ult}}& R(p)=({\mathrm{\sigma }}_{Y}-{\mathrm{\sigma }}_{\mathit{ult}}){(\frac{p}{{p}_{\mathit{ult}}})}^{2}+2({\mathrm{\sigma }}_{Y}-{\mathrm{\sigma }}_{\mathit{ult}})\frac{p}{{p}_{\mathit{ult}}}+{\mathrm{\sigma }}_{Y}\hfill \\ p\ge {p}_{\mathit{ult}}& R(p)={\mathrm{\sigma }}_{\mathit{ult}}\hfill \end{array}\)

In the exponential case (non-associated law DRUCK_PRAG_N_A), the function \(R\) is given by:

\(R(p)=({\mathrm{\sigma }}_{Y}-{\mathrm{\sigma }}_{\mathit{ult}})\mathrm{exp}(\frac{-p}{{p}_{c}})+{\mathrm{\sigma }}_{\mathit{ult}}\)

In the case of an unassociated law of behavior DRUCK_PRAG_N_A, it is also necessary to define a flow potential (in addition to the criterion defined above). This flow potential is written as:

\(G(\mathrm{\sigma },p)={\mathrm{\sigma }}_{\mathit{eq}}+\mathrm{\beta }(p){I}_{1}\)

where \(\mathrm{\beta }(p)\) is a function of the cumulative plastic deformation

In the case of linear and parabolic work hardening, this function \(\mathrm{\beta }(p)\) is written:

\(\begin{array}{cc}0<p<{p}_{\mathit{ult}}& \mathrm{\beta }(p)={\mathrm{\beta }}_{0}(1-\frac{p}{{p}_{\mathit{ult}}})\hfill \\ p\ge {p}_{\mathit{ult}}& \mathrm{\beta }(p)=0\hfill \end{array}\)

In the case of exponential work hardening, this function \(\mathrm{\beta }(p)\) is written:

\(\mathrm{\beta }(p)=({\mathrm{\beta }}_{0}-{\mathrm{\beta }}_{\mathit{ult}})\mathrm{exp}(\frac{-p}{{p}_{c}})+{\mathrm{\beta }}_{\mathit{ult}}\)

10.13.1. Operand ECROUISSAGE

ECROUISSAGE =/” LINEAIRE “, /” PARABOLIQUE”, /” EXPONENTIEL “

Allows you to define the type of work hardening required.

10.13.2. Operand ALPHA

ALPHA = alpha

Refers to the pressure dependence coefficient. We recall that the operand ALPHA is linked to the angle of friction \(\mathrm{\phi }\) by the relationship: \(\mathrm{\alpha }=\frac{2\mathrm{.}\mathrm{sin}(\mathrm{\phi })}{3-\mathrm{sin}(\mathrm{\phi })}\).

10.13.3. Operand P_ULTM

P_ULTM = p_ult

Refers to the ultimate cumulative plastic deformation for linear and parabolic work hardening (operand ECROUISSAGE =” LINEAIRE “, /” PARABOLIQUE”).

10.13.4. SY operand

SY = sy

Refers to plastic stress. This operand is linked to the combination of the cohesion coefficient \(C\) with the friction angle \(\varphi\) in the following way: \(S=\frac{6C\mathrm{cos}(\varphi )}{3-\mathrm{sin}(\varphi )}\).

10.13.5. H operand

H = h

Refers to the work hardening module, \(h<0\) if the law is softening. This operand is mandatory for linear type work hardening (operand ECROUISSAGE =” LINEAIRE “).

10.13.6. Operand SY_ULTM

SY_ULTM = sy_ult Refers to the ultimate constraint. This operand is mandatory for parabolic work hardening (operand ECROUISSAGE =” PARABOLIQUE “and” EXPONENTIEL “).

10.13.7. Operand DILAT

DILAT = beta_0

Refers to the initial dilatance (for the unassociated law DRUCK_PRAG_N_A). We recall that operand DILAT is linked to the angle of dilatance \({\mathrm{\psi }}_{0}\) by the relationship: \({\mathrm{\beta }}_{0}=\frac{2\mathrm{.}\mathrm{sin}({\mathrm{\psi }}_{0})}{3-\mathrm{sin}({\mathrm{\psi }}_{0})}\)

10.13.8. Operand DILAT_ULTM

DILAT_ULTM = beta_ult

Refers to the final dilatance (for the unassociated law DRUCK_PRAG_N_A and ECROUISSAGE =” EXPONENTIEL “).

10.14. Keyword factor VISC_DRUC_PRAG

Rheological model VISC_DRUC_PRAG is a law of elasto-visco-plastic behavior [R7.01.22]. It is characterized by a viscoplastic mechanism that collapses between three thresholds: elastic, peak and ultimate. Elastoplasticity is of the Drucker Prager type with positive pre-peak work hardening and post-peak negative work hardening and viscoplasticity is a Perzyna power law.

Among the parameters, we find:

• parameters that are involved in the work hardening functions relating to the various elastic, peak and ultimate thresholds « \(\alpha\) », « \(R\) » and « \(\beta\) »,

• parameters related to the creep law « \(A\) » and « \(n\) »,

• the cumulative viscoplastic deformations corresponding to each of the thresholds \({p}_{\mathit{pic}}\) and \({p}_{\mathit{ult}}\);

• a reference pressure « \({P}_{\mathit{ref}}\) »

Elastic characteristics should be defined under the keyword ELAS.

10.14.1. Operands PREF /N/A/ P_PIC/P_ULT

PREF = pref

Parameter \({P}_{\mathrm{ref}}\): reference pressure (unit of a constraint)

N = n

Parameter \(n\): exponent of the creep law

A = a

Parameter \(A\): viscoplastic parameter (in \({s}^{-1}\) or \({\mathrm{jour}}^{-1}\))

P_PIC = peak

Parameter \({p}_{\mathrm{pic}}\): cumulative viscoplastic deformation at the peak threshold

P_ULT = pult

Parameter \({p}_{\mathrm{ult}}\): cumulative viscoplastic deformation at the ultimate threshold level

10.14.2. Operands ALPHA_0/ALPHA_PIC/ALPHA_ULT

ALPHA_0 = alpha0

Parameter \({\alpha }_{0}\): parameter of the cohesion function \(\alpha (p)\) at the level of the elastic threshold

ALPHA_PIC = alphapic

Parameter \({\alpha }_{\mathrm{pic}}\): parameter of the cohesion function \(\alpha (p)\) at the peak threshold level

ALPHA_ULT = Alphault

Parameter \({\alpha }_{\mathrm{ult}}\): parameter of the cohesion function \(\alpha (p)\) at the level of the ultimate threshold

10.14.3. Operands R_0/ R_PIC/R_ULT

R_0 = r0

Parameter \({R}_{0}\): parameter of the work hardening function \(R(p)\) at the level of the elastic threshold (in stress units \(\mathit{Pa}\) or \(\mathit{MPa}\))

R_PIC = rpic

Parameter \({R}_{\mathrm{pic}}\): parameter of the work hardening function \(R(p)\) at the peak threshold level (in stress units \(\mathit{Pa}\) or \(\mathit{MPa}\))

R_ULT = Rult

Parameter \({R}_{\mathrm{ult}}\): parameter of the work hardening function \(R(p)\) at the level of the ultimate threshold (in stress units \(\mathit{Pa}\) or \(\mathit{MPa}\))

10.14.4. Operands BETA_0/BETA_PIC/BETA_ULT

BETA_0 = beta0

Parameter \({\beta }_{0}\): parameter of the dilatance function \(\beta (p)\) at the level of the elastic threshold

BETA_PIC = betapic

Parameter \({\beta }_{\mathrm{pic}}\): parameter of the dilatance function \(\beta (p)\) at the peak threshold

BETA_ULT = Betault

Parameter \({\beta }_{\mathrm{ult}}\): parameter of the dilatance function \(\beta (p)\) at the level of the ultimate threshold

10.15. Keyword factor BARCELONE

The Barcelona model describes the elasto-plastic behavior of unsaturated soils coupled with hydraulic behavior (Cf. [R7.01.17] for more details). This model is reduced to the Cam-Clay model in the saturated case. Two criteria are involved: a mechanical plasticity criterion (that of Cam-Clay) and a water criterion controlled by suction (or capillary pressure). It can only be used in the context of THHM and HHM behaviors. The characteristics required for the model should be given under this keyword and under the keywords CAM_CLAY and ELAS.

It is therefore mandatory to fill in the parameters of the keywords CAM_CLAY and ELAS.

10.15.1. Operands MU/ PORO/LAMBDA//KAPA /M

MU = mu

Elastic shear modulus.

PORO = porn

Porosity associated with initial pressure and linked to the initial void index: \(n=\frac{{e}_{0}}{1+{e}_{0}}\).

LAMBDA = normal

Compressibility coefficient (plastic slope in a hydrostatic compression test).

KAPA = kapa

Elastic swelling coefficient (elastic slope in a hydrostatic compression test).

M = m

Slope of the critical state line.

10.15.2. Operands PRES_CRIT and PA

PRES_CRIT = pc, PA = pa

Critical pressure equal to half of the consolidation pressure and atmospheric pressure.

10.15.3. R/ BETA /MC operands

R = r, BETA = beta

Dimensional coefficients used in expression: \(\lambda ({p}_{c})=\lambda (0)\left[(1-r)\mathit{exp}(-\beta {p}_{c})+r\right]\)

KC= kc

An adimensional parameter controlling the increase in cohesion with suction (capillary pressure).

10.15.4. Operands PCO_INIT/KAPAS/LAMBDAS/ALPHAB

PC0_INIT = Pc0 (0)

Initial capillary pressure threshold (homogeneous to stresses).

KAPAS = Kappas

Dimensional stiffness coefficient associated with the change in suction in the elastic domain.

LAMBDAS = Lambdas

Compressibility coefficient linked to a variation in suction in the plastic field. (dimensionless).

ALPHAB = alphabet

Correction coefficient for the normality of plastic flow, cf. [R7.01.17].

Optional and dimensionless corrective term to better take into account experimental results. By default, it is calculated based on the slope of the critical state line, the swelling coefficient, and the compressibility coefficient.

10.16. Keyword factor HUJEUX

Law of elastoplastic behavior in soil mechanics (granular geo-materials: sandy clays, normally consolidated or over-consolidated, gravel…). This model is a multi-criteria model that includes a non-linear elastic mechanism, 3 deviatory plastic mechanisms and an isotropic plastic mechanism (see [R7.01.23]).

The elastic mechanical characteristics E, NU, and ALPHA must be defined in parallel under the keyword ELAS. Since Hujeux’s law exhibits non-linear elastic behavior, the values of these parameters are associated with the reference pressure PREF of Hujeux’s law.

10.16.1. Operands N/ BETA /B/D/ PHI

N = n

Value of the characteristic parameter of the non-linear elastic power law, between 0 and 1.

BETA = beta

Value of the coefficient of volume plastic compressibility or critical state law, (positive).

B= b

Value of the parameter influencing the load function in plane \((P',Q)\), between 0 (Mohr-Coulomb) and 1 (Cam-Clay).

D = d

Value of the parameter characterizing the distance between the critical state line and the isotropic consolidation line, (positive).

PHI = phi

Value of the parameter characterizing the angle of internal friction, in degrees.

10.16.2. Operands ANGDIL/PCO/PREF

ANGDIL = angel

Value of the parameter characterizing the angle of dilatance, in degrees.

PCO = PCO

Initial reference critical pressure value, (negative).

PREF = pref

Reference confinement pressure value, (negative).

Operands ACYC/AMON/CCYC/CMON

ACYC = acyc, AMON = amon, CCYC = ccyc, CMON = cmon

Values of the hardening parameters of deviatory plastic mechanisms, in cyclic and monotonic, and of plastic mechanisms of consolidation, in cyclic and monotonic, respectively.

10.16.3. Operands RD_ELA/RI_ELA

RD_ELA = rdela, RI_ELA = riela,

Values of the initial radii of the thresholds of the monotonic deviatory and monotonic consolidation mechanisms, respectively, between 0 and 1.

RD_ELA = rdela, RI_ELA = riela,

Values of the initial radii of the thresholds of the monotonic deviatory and monotonic consolidation mechanisms, respectively, between 0 and 1.

10.16.4. Operands RD_CYC/RI_CYC

RD_CYC = rdcyc, RI_CYC = ricyc

Values of the initial radii of the thresholds of the cyclic deviation and cyclic consolidation mechanisms, respectively, between 0 and 1.

10.16.5. Operands RHYS/RMOB /XM/ DILA/PTRAC

RHYS = rhys

Value of the parameter defining the size of the hysteretic domain.

RMOB = rmob

Value of the parameter defining the size of the mobilized domain.

XM = xm

Value of the control parameter in the hysteretic domain.

DILA = Dila

Value of the expansion coefficient, between 0 and 1.

PTRAC = fart

cohesion of the material, homogeneous to a stress (positive or zero value). Allows you to shift the load surface towards \(p>0\) in order to take into account a slight pull in the material.

10.17. Keyword factor HOEK_BROWN

Law of behavior in rock mechanics such as the law of HOEK - BROWN modified (Cf. [R7.01.18]

The elastic mechanical characteristics E, NU, and ALPHA must be defined in parallel under the keyword ELAS.

10.17.1. Operands GAMMA_RUP/GAMMA_RES

GAMMA_RUP = group

Value of the work-hardening parameter at material breakage.

GAMMA_RES = sandstone

Value of the work-hardening parameter at the beginning of the residual resistance.

10.17.2. Operands S_END/S_RUP/M_END/M_RUP

S_END = send

Product value S*SIGMA_c**2 attained upon initiation of damage.

S_RUP = srup

Product value S*SIGMA_c**2 reached in GAMMA_RUP.

M_END = Mend

Value of the product M* SIGMA_c attained at the onset of damage.

M_RUP = mrup

Product value M* SIGMA_c reached in GAMMA_RUP.

10.17.3. Operand BETA/ALPHAB

BETA = beta

Parameter characterizing the post-rupture behavior of the material.

ALPHAHB = alpha b

Parameter characterizing the post-rupture behavior of the material.

10.17.4. Operand PHI_RUP/PHI_RES/PHI_END

PHI_RUP = prup

Value of the friction angle reached in GAMMA_RUP.

PHI_RES = near

Value of the friction angle reached in GAMMA_RES.

PHI_END = Phiend

Value of the angle of friction at the start of damage (taken as zero by default).

10.18. Keyword: Factor: GonfElas

Behavioral law in rock mechanics - written under MFront - used to describe the behavior of « swelling clay » materials (bentonite). This model was developed at LAEGO. It is a non-linear elastic model relating net stress to inflation pressure, which itself depends on suction (or capillary pressure). It can only be used in the context of THHM and HHM behaviors.

The elastic mechanical characteristics E, NU, and ALPHA must be defined in parallel under the keyword ELAS.

The GonfElas law is a behavior model for swelling clays (bentonite type), it is defined by the relationship:

\(d\tilde{\sigma }={K}_{0}d{\varepsilon }_{V}+b(1+\frac{s}{A}){e}^{-{\beta }_{m}{(\frac{s}{A})}^{2}}\mathrm{ds}\)

with \(\tilde{\sigma }\): net constraint (trace) \(\sigma =\tilde{\sigma }-{p}_{g}\).

In the saturated field:	\(d\tilde{\sigma }={K}_{0}d{\varepsilon }_{V}-{\mathrm{bdp}}_{w}+{\mathrm{dp}}_{g}\)
Or again:	\(d\tilde{\sigma }={K}_{0}d{\varepsilon }_{V}-{\mathrm{bdp}}_{c}+(1-b){\mathrm{dp}}_{g}\)

\({K}_{0}\) is the compressibility module of the material

\(b\) is the Biot coefficient

\(A\) is a parameter that is homogeneous at one pressure

\({\beta }_{m}\) is a dimensionless parameter

\(s\) suction (or capillary pressure)

From there, identification is done by looking for the inflation pressure.

Let \({P}_{\mathrm{gf}}\) be the expected inflation pressure and let \({P}_{\mathrm{gf}}({s}_{0})\) be the inflation pressure found by the model when we re-saturate a sample in a blocked deformation test and starting from a \({s}_{0}\) suction.

It’s easy to see that: \(\frac{{P}_{\mathit{gf}}({s}_{0})}{A}=\frac{\sqrt{\pi }}{2\sqrt{{\beta }_{m}}}\mathit{Erf}\left(\frac{{s}_{0}}{A}\sqrt{{\beta }_{m}}\right)+\frac{1}{2{\beta }_{m}}\left(\begin{array}{c}1-{e}^{-{\beta }_{m}{\left(\frac{{s}_{0}}{A}\right)}^{2}}\\ \phantom{\rule{2em}{0ex}}\end{array}\right)\)

We have to have \({P}_{\mathrm{gf}}={P}_{\mathrm{gf}}^{\infty }\). We know that \(\mathrm{Erf}(\infty )=1\) and therefore: \(\frac{{P}_{\mathit{gf}}({s}_{0})}{A}=\frac{\sqrt{\pi }}{2\sqrt{{\beta }_{m}}}+\frac{1}{2{\beta }_{m}}\)

For more details, refer to [R7.01.41].

10.18.1. YoungModulus and PoissonRatio operand

Classic elasticity parameters (Young’s modulus and Poisson’s ratio).

10.18.2. BioTCoef operand

Biot coefficient. Be careful to take the same one as the one entered in module THM.

10.18.3. BetAM operand

Dimensionless material parameter corresponding to \({\beta }_{m}\) of the law above. Identification is done by looking for the inflation pressure.

10.18.4. ReferencePressure operand

Homogeneous parameter at a pressure corresponding to \(A\) of the law above.

10.19. Keyword factor JOINT_BANDIS

Law of behavior of a hydraulic joint in rock mechanics. In the direction normal to the joint, the behavior is given by

\(d\sigma {\text{'}}_{n}=-{K}_{\text{ni}}\frac{\mathrm{dU}}{{(1-\frac{U}{{U}_{\mathrm{max}}})}^{\gamma }}\)

\(\sigma {\text{'}}_{n}\) is the normal effective stress.

\({K}_{\text{ni}}\) is the normal initial stiffness.

\(U\) is crack closure (opening at zero loading minus common opening).

\({U}_{\text{max}}\) is the asymptotic closure of the crack (with infinite stress).

\(\gamma\) is a material parameter.

In the tangential direction, the behavior is linear elastic:

\(\sigma {\text{'}}_{t}={K}_{t}[[{u}_{t}]]\)

10.19.1. K operand

Normal stiffness at zero load \({K}_{\text{ni}}\) (stress per unit length).

10.19.2. Operand DMAX

Asymptotic closure \({D}_{\text{max}}\) (length).

10.19.3. Operand GAMA

Dimensionless \(\gamma\) material parameter.

10.19.4. KT operand

Tangential stiffness \({K}_{t}\) (stress per unit length).

10.20. Keyword factor THM_RUPT

Behavioral law for cracks with hydro-mechanical coupling (see [R7.02.15]).

When the masses surrounding the crack are impermeable, the flow is no longer well defined on the unopened joint elements. In this case, the displacement jump is replaced by a fictional crack opening \({\varepsilon }_{\text{fict}}\) which makes it possible to regulate the flow and to transfer to the crack tip the boundary condition written at the end of the crack path.

It is also possible to define a Biot module \(N\) for the cohesive zone.

10.20.1. Operand OUV_FICT

Fictional crack opening \({\varepsilon }_{\text{fict}}\) (length).

10.20.2. Operand UN_SUR_N

Inverse of the Biot module for crack \(N\) (stress per unit length).

10.21. Keyword factor: Iwan

Law of elastoplastic behavior in soil mechanics adapted for cyclic deviatoric behavior. Iwan’s law of behavior [R7.01.38] makes it possible to reproduce shear modulus degradation curves. The law is calibrated based on the parameters of a hyperbolic model of the form:

\(\frac{\text{G}}{{\text{G}}_{\mathit{max}}}=\frac{1}{1+{\left(\frac{\text{γ}}{{\text{γ}}_{\mathit{ref}}}\right)}^{n}}\)

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 Iwan keyword. If they are different, a fatal error will be issued.

If the ELAS keyword is not entered, it will be done automatically using the elastic characteristics of the Iwan keyword.

10.21.1. YoungModulus operand

Young’s module.

10.21.2. Operand Poisson Ratio

Poisson’s ratio.

10.21.3. HypDistortion operand

Shear strain value \({\text{γ}}_{\mathit{ref}}=\text{2}{\text{ε}}_{\text{i}\text{j}}\) from the hyperbolic law. This is the value for which \(\text{G}={\text{G}}_{\mathit{max}}/2\) in the behavior curve.

10.21.4. HypExponent operand

Exponent of hyperbolic law.

10.22. Keyword factor LKR

The LKR model (Lagle, Kleine and Raude) is a law of thermo-elasto (visco) plastic behavior [R7.01.40]. It is based on concepts from the theories of elastoplasticity and viscoplasticity. The plastic mechanism is characterized by positive work hardening in the pre-peak regime and negative in the post-peak regime. Temperature influences plastic and viscoplastic workings.

Elastic characteristics should be defined under the keyword ELAS.

The reader will refer to the reference documentation [R7.01.40] for the meaning, definition intervals, and by default values (if optional parameter) of each parameter specific to the keyword factor LKR, whose syntax is detailed at the beginning of the document.

10.23. Keyword factor Mohr-Coulombas

Law of elastoplastic behavior in soil mechanics (granular geo-materials) adapted for monotonic loads, written under MFront. This elastoplastic behavior law with a smoothed Mohr-Coulomb load surface [R7.01.43] makes it possible, for example, to analyze the load-bearing capacity of a geotechnical structure. The proposed smoothing of the Mohr-Coulomb criterion, compared to the law of origin [R7.01.28], aims to increase the robustness of the implicit numerical integration. The data required for the material field also include the smoothing parameters. The elastic mechanical characteristics E and NU must be defined in parallel under the keyword ELAS. Values under the ELAS keyword will be compared to those entered under the MohrCoulombas keyword. If they are different, a fatal error will be issued.

If the ELAS keyword is not entered, it will be done automatically by taking the elastic characteristics of the MohrCoulombas keyword.

10.23.1. Operands

10.23.1.1. YoungModulus operand

Young’s module.

10.23.1.2. Operand Poisson Ratio

Poisson’s ratio.

10.23.1.3. Cohesion operand

Cohesion (positive scalar).

10.23.1.4. FrictionAngle operand

Friction angle \({\varphi }_{\mathit{pp}}\) (supplied in°).

10.23.1.5. DilatancyAngle operand

Expansion angle \(\psi\) (provided in°) which must be less than or equal to the internal friction angle \({\varphi }_{\mathit{pp}}\). When \(\psi ={\varphi }_{\mathit{pp}}\), the law of plastic flow becomes associated.

10.23.1.6. TransitionAngle operand

Parameter associated with smoothing: transition Lode angle \({\theta }_{T}\) (provided in°) .It must be strictly less than 30°. A typical value can be taken to be equal to 25°.

10.23.1.7. Cutoff tension operand

Positive parameter associated with smoothing: traction truncation \({a}_{\mathit{tt}}\). If \({a}_{\mathit{tt}}=0\), the criterion returns to the top of the Mohr-Coulomb pyramid, so it is no longer smoothed.

10.23.1.8. HardeningCoef operand

Positive or zero parameter describing the work hardening acting on cohesion as a function of the equivalent plastic deformation. If it is strictly positive, we will generally take a low value if the aim is to reduce the unstable response regime, cf. [R7.01.43].

10.24. Keyword factor NLH_CSRM

The law of behavior NLH_CSRM makes it possible to model the elasto-visco-plastic behavior of coherent geomaterials. It will eventually replace laws LETK and LKR.

The scope of use of model NLH_CSRMest for the time being limited to isotropic mechanical behavior. Taking into account structural anisotropy and the dependence of work hardenings on temperature — effective in the latest version of model LKR — will be the subject of future renderings; as will the dependence of the criteria on the third invariant of the stress deflector (Lode angle).

The NLH_CSRM model is part of the thermodynamic framework of generalized standard materials with two irreversible mechanisms,

• Plastic mechanism —irreversible instantaneous behavior: hardening and/or softening workings, contracting or expanding volume behavior, influence of the average stress on the fragile nature of the compression response,…;

• Viscoplastic mechanism — irreversible delayed behavior (Perzyna type): primary, secondary, and tertiary creep (ruin).

A coupling between these two mechanisms makes it possible — among other things — to represent the dependence of the maximum resistance of certain geomaterials on the stress rate.

This law was integrated implicitly via the Mfront tool. Its reference material has the key r7.01.46 Elasto-visco-plastic law NLH_CSRM for geomaterials.

YoungModulus and PoissonRatio should also be entered in the keyword factor ELAS.

10.25. Keyword factor MCC

Behavioral model MCC makes it possible to qualitatively represent the phenomenology of granular soils under monotonic loads, taking into account the effects of dilatance/contraction, softening/hardening, as well as the critical state. The MCC model is part of the generalized standard materials. It updates the formulation and the numerical resolution method of law CAM_CLAY [R7.01.14], while remaining faithful to the elements of phenomenology predicted by the law, in order, ultimately, to replace it.

The parameters in model MCC have a total of seven. Three define isotropic nonlinear elastic behavior; three represent the initial reversibility domain. Combined kinematic-isotropic work hardening carried out by volume plastic deformation is controlled by a last parameter.

Behavioral equations are solved using an implicit time direct integration method, implemented in MFront.

The reference documentation for model MCC has the key [R7.01.48].

10.26. Keyword factor CSSM

Behavioral model CSSM (« Critical State Soil Model ») is a combination of two models aimed at representing the monotonic and cyclical drained behavior of soils, respectively. The first component is based on the modified Cam-Clay model, which describes the phenomena of dilatance or contraction as well as confinement effects under monotonic and drained loads. The second component is an Iwan multi-mechanism model, designed to represent the cyclical nonlinear behavior of soils at low levels of deformation. The model adheres to the framework of generalized standard materials.

Behavior model CSSM has a total of eleven parameters: two to describe the reversible (elastic) behavior, five for the irreversible behavior related to the modified Cam-Clay component, and three for the irreversible behavior related to the Iwan component. A final parameter weighs the contributions of the two components on the shear behavior of model CSSM.

Behavioral equations are solved using an implicit time direct integration method, implemented in MFront.

The reference documentation for model CSSM has the key [R7.01.49].