8. Operands for mechanical options

8.1. Stress calculation options (Operand CONTRAINTE)

The components of the fields of constraints and generalized efforts are detailed in the document [U2.01.05].

| 'EFGE_ELGA'
| 'EFGE_ELNO'
| 'EFGE_NODE'

Calculation of generalized forces (structural elements).

This is either an extraction of the forces contained in the SIEF_ELGA/STRX_ELGA field (case of beam/pipe elements or discretes), or a calculation by integrating the constraints (case of multi-fiber beam elements or of plates and shells).

Note 1

Field EFGE_ELNO is not always an extrapolation of field EFGE_ELGA; especially for a linear calculation where this field is calculated directly from the displacement. This is why some components are not calculated (set to zero) in a non-linear way.

Note 2

For eccentric plates, the forces are calculated in the « plane » of the mesh. If you want these efforts in the middle « plane » of the plate, you must use the POST_CHAMP/COQUE_EXCENT command.

| 'SIEF_ELGA'
| 'SIEF_ELNO'
| 'SIEF_NOEU'

Calculation of the stress state (generalized stresses or forces according to the modeling) from the displacements (linear elasticity), see [U2.01.05].

For incompressible formulations (INCO_UP, INCO_UPO and INCO_UPG), this tensor contains, in addition to the components SIXX, SIYY, SIZZ, SIXY, SIYZ and SIXZ in 3D, the component SIP which is the hydrostatic pressure. This component is calculated from the constraint trace (part of the law of behavior) and the unknowns PRES and GONF (for INCO_UPG) of the nodal unknowns field. The components SIXX, SIYY and SIZZ are then the purely deviatory contributions of the stress tensor

note

The field “SIEF_ELGA” is calculated natively by the non-linear resolution operators. It is always present in a result data structure such as evol_noli.

| 'SIGM_ELGA'
| 'SIGM_ELNO'
| 'SIGM_NODE'

Calculation of the stress state.

It is actually an extraction of the constraints contained in field SIEF_ELGA, see [U2.01.05].

| 'SIPO_ELNO'
| 'SIPO_NOEU'

Calculation of the stresses in the beam section broken down into the contributions of each generalized effort.

List of field components:

SN	Normal effort contribution \(N\) to \({\sigma }_{\mathit{xx}}\), \({\sigma }_{\mathrm{xx}}=\frac{N}{A}\)
SMFY	Contribution of the bending moment \(\mathit{MFY}\) to \({\sigma }_{\mathit{xx}}\), \({\sigma }_{\mathit{xx}}=z\frac{\mathit{MFY}}{{I}_{Y}}\)
SMFZ	Contribution of the bending moment \(\mathit{MFZ}\) to \({\sigma }_{\mathit{xx}}\), \({\sigma }_{\mathit{xx}}=-y\frac{\mathit{MFZ}}{{I}_{Z}}\)
SVY	Sharp effort contribution \(\mathit{VY}\) to \({\sigma }_{\mathit{xy}}\), \({\sigma }_{\mathit{xy}}=\frac{\mathit{VY}{a}_{Y}}{A}\) \({a}_{Y}\) shear coefficient in the \(y\) direction
SVZ	Sharp effort contribution \(\mathit{VZ}\) to \({\sigma }_{\mathit{xz}}\), \({\sigma }_{\mathit{xz}}=\frac{\mathit{VZ}{a}_{Z}}{A}\) \({a}_{Z}\) shear coefficient in the \(z\) direction
SMT	Contribution of the torsional moment \(\mathit{MX}\) to \({\sigma }_{\mathit{yz}}\), \({\sigma }_{\mathit{yz}}=\frac{\mathit{MX}{R}_{T}}{{J}_{x}}\)

The above constraints are expressed in the local coordinate system, i.e. the main inertia coordinate system of the straight section [R3.08.01].

The values of \({\sigma }_{\mathrm{xx}}\) due to the two bending moments are the maximum values of those calculated in \({Y}_{\mathit{min}}\), \({Y}_{\mathit{max}}\) on the one hand, and in \({Z}_{\mathit{min}}\), \({Z}_{\mathit{max}}\) on the other hand (except for a general section where it is the user who provides the location of the extremum with the keywords \(\mathit{RY}\), \(\mathit{RZ}\) and \(\mathit{RT}\) cf.). AFFE_CARA_ELEM [U4.42.01]).

For a rectangular section:

• we calculate the value of SMFY in \(z\mathrm{=}\mathit{HZ}\mathrm{/}2\),

• we calculate the value of SMFZ in \(y\mathrm{=}\mathit{HY}\mathrm{/}2\).

For a circular section, we calculate the values of SMFY and SMFZ for \(y\) and \(z\) equal to \(R\).

| 'SIMM_ELNO'

Calculation of maximum and minimum stresses in the beam section based on generalized forces (linear elasticity).

The same remark as for SIPO_ELNO applies in the case of a general section.

| 'SIRO_ELEM'

Calculation of the stresses projected on the skin of a volume (for example on the facings of a hydraulic structure) or on the edge of a surface.

List of field components:

SIG_NX SIG_NY SIG_NZ	Components \({\sigma }_{X}\), \({\sigma }_{Y}\), \({\sigma }_{Z}\) in the global benchmark of \(\overrightarrow{\sigma }{\text{}}_{n}\)
SIG_N	Value \({\mathit{SIG}}_{N}\)
SIG_TX SIG_TY SIG_TZ	Components \({\sigma }_{X}\), \({\sigma }_{Y}\), \({\sigma }_{Z}\) in the global benchmark of \(\overrightarrow{\sigma }{\text{}}_{t}\)
SIG_T1X SIG_T1Y SIG_T1Z	Components \({\sigma }_{X}\), \({\sigma }_{Y}\), \({\sigma }_{Z}\) in the global benchmark of \(\overrightarrow{\sigma }{\text{}}_{\mathit{t1}}\)
SIG_T1	Eigenvalue \({\mathit{SIG}}_{\mathit{T1}}\)
SIG_T2X SIG_T2Y SIG_T2Z	Components \({\sigma }_{X}\), \({\sigma }_{Y}\), \({\sigma }_{Z}\) in the global benchmark of \(\overrightarrow{\sigma }{\text{}}_{\mathit{t2}}\)
SIG_T2	Eigenvalue \({\mathit{SIG}}_{T2}\)
SIG_TN	Value \({\mathit{SIG}}_{\mathit{TN}}\)

These fields are evaluated using a constraint field calculated on volume (MODELISATION =”3D” or “3D_SI”) or surface meshes (MODELISATION =” D_PLAN “or” D_PLAN_SI “). For volume meshes, the procedure is as follows:

• Identification of the volume cells corresponding to the facets of the group of surface elements. For each facet (surface mesh), the volume mesh located on the « - » side of the normal to the facet is chosen. If there is no volume mesh on the « - » side of the facet, we do not calculate SIRO_ELEM on this facet;

• Retrieving 3D constraints to assign them to the nodes of the faces;

• Average of each of the components of the stress tensor at the center of the element faces;

• We place ourselves in a coordinate system composed of the vector normal \(\overrightarrow{n}\) to the facet and the plane of the facet. We get a tensor noted \(\mathrm{[}\sigma \text{}\mathrm{]}\).

• We evaluate \(\mathrm{[}\sigma \text{}\mathrm{]}\overrightarrow{n}\mathrm{=}\overrightarrow{\sigma }{\text{}}_{n}+\overrightarrow{\sigma }{\text{}}_{t}\), \(\overrightarrow{\sigma }{\text{}}_{n}\) being a vector collinear to \(\overrightarrow{n}\). \(\overrightarrow{\sigma }{\text{}}_{t}\) is then a vector representing shear, which is negligible in the case of upstream/downstream facings of a dam. Note \(\overrightarrow{\sigma }{\text{}}_{n}\mathrm{=}{\mathit{SIG}}_{N}\overrightarrow{n}\) and \({\mathit{SIG}}_{N}\) indicate the presence of traction if it is positive and compression if it is negative.

• We assume negligible shear so \(\mathrm{[}\sigma \text{}\mathrm{]}\mathrm{=}\left[\begin{array}{cc}\sigma {\text{}}_{\mathrm{2D}}& 0\\ 0& {\mathit{SIG}}_{N}\end{array}\right]\) We are looking for the main stress vectors corresponding to \(\sigma {\text{}}_{\mathrm{2D}}\): we therefore obtain the vectors \(\vec{\mathrm{\sigma }}{\text{\%}}_{t1}\) and \(\overrightarrow{\sigma }{\text{}}_{\mathit{t2}}\) which are in the plane of the facet and the eigenvalues \({\mathit{SIG}}_{\mathit{T1}}\) and \({\mathit{SIG}}_{\mathit{T2}}\)

For area cells, the procedure for evaluating field SIRO_ELEM is the same as for volume cells, taking into account the following conventions:

• \(\vec{{\mathrm{\sigma }}_{t2}}\) collinear to the Z direction (a D_PLAN model being defined exclusively in the XY plane), \(\vec{{\mathrm{\sigma }}_{t1}}\) being therefore in the XY plane.

• SIG_NZ = SIG_TZ = 0. and SIG_T1Z = SIGT2X = SIGT2Y = 0.

• \({\mathit{SIG}}_{\mathit{TN}}\) is the value of the shear stress in the XY plane.

Note 1

In the case of facets/edges immersed in the volume/surface, the user can use the command MODI_MAILLAGE/ORIE_PEAU_3D/GROUP_MA_VOLUou MODI_MAILLAGE/ORIE_PEAU_2D/GROUP_MA_SURF to reorient this normal as he wishes. He can thus choose the volume mesh that will be used for the calculation.

If on the « - » side, we find a « joint » mesh (which is volumic/surface), the calculation of SIRO_ELEM is impossible because the stresses stored in the joint elements do not allow the calculation detailed above.

Note 2

If you enter TOUT =” OUI “, the list of stitches is filtered to keep only the skin/edge meshes.

8.2. Options for calculating deformations (Operand DEFORMATION)

The components of the deformation fields are detailed in document [U2.01.05].

| 'DEGE_ELGA'

Calculation of generalized deformations from displacements. This option only makes sense for plate and pipe structural elements, not for beams

The generalized deformations are obtained in the local coordinate system of the element.

| 'DEGE_ELNO'
| 'DEGE_NODE'

Calculation of generalized deformations from displacements. This option only makes sense for beam, plate, and pipe structural elements.

The generalized deformations are obtained in the local coordinate system of the element.

| 'EPFD_ELGA'
| 'EPFD_ELNO'
| 'EPFD_NODE'

Calculation of desiccation creep deformations, for models BETON_UMLV_FP and BETON_BURGER_FP.

| 'EPFP_ELGA'
| 'EPFP_ELNO'
| 'EPFP_NODE'

Calculation of the natural creep deformations associated with model GRANGER_FP, model BETON_UMLV_FP or model BETON_BURGER_FP.

| 'EPME_ELGA'
| 'EPME_ELNO'
| 'EPME_NODE'

Calculation of « mechanical » deformations based on displacements. This calculation is done in theory of « small displacements ». The calculated deformations are equal to the total deformations minus the thermal deformations. Drying and hydration deformations are also subtracted as well as fluid pressure deformations and anelastic deformations. On the other hand, creep deformations are not subtracted.

\({\varepsilon }_{\mathit{ij}}^{m}(u)\mathrm{=}\frac{1}{2}({u}_{i,j}+{u}_{j,i})\mathrm{-}{\varepsilon }^{\mathit{th}}\)

| 'EPMG_ELGA'
| 'EPMG_ELNO'
| 'EPMG_NODE'

Calculation of « mechanical » deformations based on displacements. This calculation is done in theory of « major displacements ». The calculated deformations are equal to the total deformations minus the thermal deformations.

\({E}_{\mathit{ij}}^{m}(u)\mathrm{=}\frac{1}{2}({u}_{i,j}+{u}_{j,i}+{u}_{k,i}{u}_{k,j})\mathrm{-}{\varepsilon }^{\mathit{th}}\)

| 'EPSG_ELGA'
| 'EPSG_ELNO'
| 'EPSG_NODE'

Calculation of Green-Lagrange deformations.

\({E}_{\mathrm{ij}}(u)=\frac{1}{2}({u}_{i,j}+{u}_{j,i}+{u}_{k,i}{u}_{k,j})\)

| 'EPSI_ELGA'
| 'EPSI_ELNO'
| 'EPSI_NODE'

Calculation of deformations based on displacements.

\({\varepsilon }_{\mathit{ij}}(u)\mathrm{=}\frac{1}{2}({u}_{i,j}+{u}_{j,i})\)

For structural elements, these deformations are obtained in the local coordinate system of the element.

| 'EPSP_ELGA'
| 'EPSP_ELNO'
| 'EPSP_NODE'

Calculation of anelastic deformations using the displacement field \(u\), the stresses, the stresses \(\sigma\), the temperature \(T\), any anelastic deformations \({\varepsilon }^{a}\), and the internal variables,

\({\varepsilon }^{p}=\varepsilon (u)-{A}^{-1}\sigma -{\varepsilon }^{\mathrm{th}}(T)-{\varepsilon }^{a}-{\varepsilon }^{\mathrm{fl}}\)

where \({\varepsilon }^{\mathrm{fl}}\) is Granger’s own creep deformation.

| 'EPVC_ELGA'
| 'EPVC_ELNO'
| 'EPVC_NODE'

Calculation of deformations related to control variables. For the time being only the following components are defined:

• thermal deformations: EPTHER_L, EPTHER_T, EPTHER_N such as: \({\varepsilon }_{i}^{\mathrm{th}}={\alpha }_{i}(T-{T}_{\mathrm{ref}});i\in \text{{}L,T,N\text{}}\) (if the material is isotropic, the 3 components are equal), \(T\) being the temperature and \({\alpha }_{i}\) the thermal expansion coefficient;

• drying shrinkage EPSECH (used for laws describing the behavior of concrete) \({\varepsilon }^{\mathrm{sech}}=-{K}_{\mathrm{dessic}}({S}_{\mathrm{ref}}-S)\),, \(S\) being the drying control variable and \({K}_{\mathrm{dessic}}\) being the desiccation shrinkage coefficient;

• hydration shrinkage EPHYDR (used for laws describing the behavior of concrete) \({\varepsilon }^{\mathrm{hydr}}=-{B}_{\mathrm{endog}}h\),, \(h\) being the hydration control variable, and \({B}_{\mathrm{endog}}\) being the endogenous shrinkage coefficient.

• Deformation related to fluid pressure (for thermo-hydro-mechanics with a resolution by chaining): EPPTOT such as: \({\varepsilon }^{\mathrm{ptot}}=\frac{b}{\mathrm{3K}}{p}_{\mathrm{tot}}\), \({p}_{\mathrm{tot}}\) is the total fluid pressure control variable, \(b\) is the Biot coefficient, \(K\) is the elasticity module.

| 'EPSL_ELGA'
| 'EPSL_ELNO'
| 'EPSL_NODE'

Calculation of logarithmic deformations.

\({E}_{l}(u)=\frac{1}{2}\left(\mathrm{ln}({F}^{T}F)\right)\)

8.3. Options for extracting internal variables (Operand VARI_INTERNE)

| 'VARC_ELGA'
| 'VARC_ELNO'
| 'VARC_NODE'

Calculation of control variables used for mechanical calculation.

List of field components:

TEMP	See command documentation AFFE_MATERIAU [U4.43.03] for the definition of each component.
HYDR
SECH
CORR
IRRA
PTOT
DIVU
NEUT1
NEUT2

note

All components are systematically calculated. Variables that have not been defined are initialized to the value R 8VIDE () (very large real number of the order of 1.0E+308). If you want to manipulate this field then in post-processing (and in post-processing only), it is therefore necessary to set the undefined components to zero before under penalty of numerical error.

| 'VARI_ELNO'
| 'VARI_NODE'

Calculation of internal variables.

List of field components:

V1	Internal Variable 1
…
Vi	Internal variable i
…
Vn	Internal variable n

The number and type of these internal variables are specific to each behavior model (cf. [U4.51.11]).

note

The field “VARI_ELGA” is calculated natively by non-linear resolution operators. It is always present in an evol_noli result SD.

8.4. Energy calculation options (Operand ENERGIE)

| 'DISS_ELEM'

Calculation of the energy dissipated by the damage. The resulting field has only one component with the name “ENDO”.

List of field components:

ENDO	Energy dissipated by damage

note

Valid only for elements DKTG and law GLRC_DM. His expression is given in [R7.01.32].

| 'DISS_ELGA'
| 'DISS_ELNO'
| 'DISS_NODE'

Calculation of the density of energy dissipated by the damage. The resulting field has only one component with the name “ENDO”.

List of field components:

ENDO	Energy dissipated by damage

note

Valid only for elements DKTG and law GLRC_DM. His expression is given in [R7.01.32].

| 'ECIN_ELEM'

Calculation of kinetic energy.

\({E}_{c}=\frac{1}{2}m{v}^{2}\)

List of field components:

TOTALE	Kinetic energy
Additional components for plates and shells:
MEMBRANE FLEXION	Contributions to kinetic energy (cf. [])
Additional components for curved beams:
PLAN_XY PLAN_XZ	Contributions to kinetic energy (cf. [])
Additional components for discretes:
DX DY DZ DRX DRY DRZ	Contributions to kinetic energy

| 'ENEL_ELEM'

Calculation of elastic energy.

\({E}_{p}=\frac{1}{2}\sigma {A}^{-1}\sigma\)

List of field components:

TOTALE	Elastic energy
Additional components for plates and shells: Contributions to elastic energy (cf. [])
MEMBRANE FLEXION CISAILLE COUPL_MF	Elastic energy in membrane Elastic energy in flexure Elastic energy in shear Elastic energy in membrane-flexure coupling

note

In non-linear (STAT_NON_LINE, DYNA_NON_LINE,…) the components CISAILLE and COUPL_MF are zero.

| 'ENEL_ELGA'
| 'ENEL_ELNO'
| 'ENEL_NOEU'

Calculation of elastic energy density.

List of field components:

TOTALE	Elastic energy
Additional components for plates and shells: Contributions to elastic energy (cf. [])
MEMBRANE FLEXION CISAILLE COUPL_MF	Elastic energy in membrane Elastic energy in flexure Elastic energy in shear Elastic energy in membrane-flexure coupling

note

In non-linear (STAT_NON_LINE, DYNA_NON_LINE,…) the components CISAILLE and COUPL_MF are zero.

| 'ENTR_ELEM'

Calculation of the modified elastic tensile energy. In fracture mechanics, it may be necessary to estimate a so-called tensile elastic energy, so the idea consists in calculating a modified elastic energy, making it possible to annihilate the participation of spherical compression and compression in each specific direction of deformation. Thus elastic energy becomes:

\({E}_{\mathit{el}}^{\mathit{traction}}=\frac{\lambda }{2}H(\mathit{tr}(ϵ))\mathit{tr}{(ϵ)}^{2}+\mu \sum _{i=1}^{3}H({ϵ}_{i}){ϵ}_{i}^{2}\)

where \(H\) represents the Heaviside function,

\(ϵ\) represents the elastic deformation tensor,

\({ϵ}_{i}\) represents the main elastic deformations.

List of field components:

TOTALE

Modified elastic tensile energy

note

For now, only valid in small deformations (DEFORMATION =” PETIT “or DEFORMATION =” PETIT_REAC”).

| 'EPOT_ELEM'

Calculation of the potential deformation energy, using the displacement field \(u\) and the temperature field \(T\):

List of field components:

TOTALE	Potential energy
Additional components for plates and shells:
MEMBRANE FLEXION	Contributions to potential energy (cf. [])
Additional components for straight beams:
TRAC_COM TORSION FLEX_Y FLEX_Z	Contributions to potential energy (cf. [])
Additional components for curved beams:
PLAN_XY PLAN_XZ	Contributions to potential energy (cf. [])
Additional components for discretes:
DX DY DZ DRX DRY DRZ	Contributions to potential energy

• for 2D and 3D continuous media elements:

\({E}_{\mathrm{pot}}=\frac{1}{2}\underset{\text{element}}{\int }\varepsilon (U)\mathrm{.}A\mathrm{.}\varepsilon (U)\mathrm{dv}-\underset{\text{element}}{\int }\varepsilon (U)\mathrm{.}A\mathrm{.}{\varepsilon }^{\mathrm{th}}(U)\mathrm{dv}+\frac{1}{2}\underset{\text{element}}{\int }{\varepsilon }^{\mathrm{th}}(U)\mathrm{.}A\mathrm{.}{\varepsilon }^{\mathrm{th}}(U)\mathrm{dv}\)

• for beam elements:

\({E}_{\mathrm{pot}}=\frac{1}{2}{u}^{\mathrm{T.}}{K}_{e}\mathrm{.}u-{u}^{\mathrm{T.}}{B}^{T}\mathrm{.}A\mathrm{.}{\varepsilon }^{\mathrm{th}}+\frac{1}{2}{\varepsilon }^{\mathrm{th.}}A\mathrm{.}{\varepsilon }^{\mathrm{th}}\)

• for plate and shell elements:

\({E}_{\mathrm{pot}}=\frac{1}{2}{u}^{T}\mathrm{.}{K}_{e}\mathrm{.}u-{u}^{T}\mathrm{.}{B}^{T}\mathrm{.}A\mathrm{.}{\varepsilon }^{\mathrm{th}}\)

| 'ETOT_ELEM'

Calculation of the total deformation energy increment between the current instant and the previous instant.

List of field components:

TOTALE

Total strain energy increment

| 'ETOT_ELGA'
| 'ETOT_ELNO'
| 'ANDOT_NODE'

Calculation of the total deformation energy density increment between the current instant and the previous instant.

List of field components:

TOTALE

Total strain energy increment

8.5. Criteria calculation options (Operand CRITERES)

| 'DERA_ELGA'
| 'DERA_ELNO'
| 'DERA_NOEU'

Calculation of the local discharge indicator and radiality loss indicator [R4.20.01].

List of field components:

DCHA_V	Discharge indicator calculated on the stress deviating tensor
DCHA_T	Discharge indicator calculated on the total stress tensor
IND_DCHA	Indicator to know if the discharge would remain elastic or if there would be a risk of plasticization if pure kinematic work hardening was used
VAL_DCHA	Indicates the proportion of criterion output in the case of abusive discharge
X11 X22 X33 X12 X13 X23	Kinematic tensor components used to calculate IND_DCHA
RADI_V	Indicator of the variation in the direction of the constraints between the times \(t\) and \(t+\Delta t\)
ERR_RADI	Error \(\eta\) due to time discretization, directly related to the rotation of the normal to the load surface

DCHA_V and DCHA_T express, in both cases, the relative variation of the stress norm in the sense of Von Misès: \({I}_{1}=\frac{\parallel \sigma (M,t+\Delta t)\parallel -\parallel \sigma (M,t)\parallel }{\parallel \sigma (M,t+\Delta t)\parallel }\), the norm being a function of the behavior (isotropic work hardening or linear kinematics)

IND_DCHA can take the following values:

• 0: initial value without constraint;

• 1: if elastic load;

• 2: if plastic filler;

• -1: if lawful elastic discharge (regardless of the type of work hardening);

• -2: if abusive discharge (we would have plasticized with kinematic work hardening).

RADI_V is given by the following relationship:

\({I}_{2}=1-\frac{\mid \sigma (M,t)\text{.}\Delta \sigma \mid }{\parallel \sigma (M,t)\parallel \parallel \Delta \sigma \parallel }\)

This quantity is zero when the radiality is maintained during the time increment.

ERR_RADI is the angle between \({n}^{-}\), the normal to the plasticity criterion at the start of the time step (instant \({t}^{-}\)), and \({n}^{+}\), the normal to the plasticity criterion calculated at the end of the time step (instant \({t}^{+}\)) as follows:

\(\eta =\frac{1}{2}∣∣\Delta n∣∣=\frac{1}{2}∣∣{n}^{\text{+}}-{n}^{\text{-}}∣∣=∣\mathrm{sin}(\frac{\alpha }{2})∣\)

This provides a measure of the error (also used to refine the time step) [U4.51.11]. This criterion is operational for Von Mises elastoplastic behaviors with isotropic, linear and mixed kinematics work hardening: VMIS_ISOT_LINE, VMIS_ISOT_TRAC, VMIS_ISOT_PUIS, VMIS_CINE_LINE, VMIS_ECMI_LINE, VMIS_ECMI_TRAC, and for the elasto-visco-plastic behaviors of Chaboche: VMIS_CIN1_CHAB, VMIS_CIN2_CHAB, VMIS_CIN2_MEMO, VISC_CIN1_CHAB, VISC_CIN2_CHAB, VISC_CIN2_MEMO.

note

The calculation of these options requires comparing the constraint fields at times \({t}_{i}\) and \({t}_{i+1}\). The result is sorted by the order number associated with moment \({t}_{i}\).

The discharge indicator is calculated by: \(\mathrm{ID}=\frac{\parallel {\sigma }_{i+1}\parallel -\parallel {\sigma }_{i}\parallel }{\parallel {\sigma }_{i+1}\parallel }\).

By default, the calculation is done for order numbers \(1\) to \(n–1\). But if we specify the list of moments (with « holes » possibly), the calculation will only concern the moments requested but it will always compare the instant \({t}_{i}\) with the instant \({t}_{i+1}\) in the list of the moments that were used to do the non-linear calculation.

| 'ENDO_ELGA'
| 'ENDO_ELNO'
| 'ENDO_NODE'

Calculation of damage \(d\) using the stress tensor and the cumulative plastic deformation \(p\).

List of field components:

TRIAX	Triaxiality rate
SI_ENDO	Lemaître-Sermage equivalent damage stress
COENDO	Normalized Lemaître-Sermage damage stress
DOM_LEM	Lemaître-Sermage damage

The triaxiality rate \(\alpha\) is given by the following relationship:

\(\alpha =\frac{{\sigma }_{h}}{{\sigma }_{\mathrm{eq}}}\)

and the equivalent damage stress \(\sigma \text{*}\):

\(\sigma \text{*}={\sigma }_{\mathrm{eq}}\sqrt{\frac{2}{3}(1+\nu )+3(1-2\nu ){\alpha }^{2}}\)

with: \(\begin{array}{ccc}s& \text{=}& \sigma -\frac{1}{3}\mathrm{tr}(\sigma )\cdot \mathrm{Id}\\ {\sigma }_{\mathrm{eq}}& \text{=}& \sqrt{\frac{3}{2}s:s}\\ {\sigma }_{h}& \text{=}& \frac{1}{3}\mathrm{tr}(\sigma )\end{array}\)

The damage kinetics is given by the Lemaître-Sermage law:

\(\dot{d}={\left[\frac{Y}{S}\right]}^{s}\dot{p}\) if \(p\ge {p}_{\mathrm{seuil}}\) with \(Y=\frac{{\sigma \text{*}}^{2}}{2E{(1-D)}^{2}}\)

where \(S\) and \(s\) are characteristic coefficients of the material and \({p}_{\mathrm{seuil}}\) the damage threshold linked to the energy stored in the material (if \(s=1\) we obtain the classical Lemaître law).

| 'EPEQ_ELGA'
| 'EPEQ_ELNO'
| 'EPEQ_NOEU'
| 'EPMQ_ELGA'
| 'EPMQ_ELNO'
| 'EPMQ_NODE'
| 'EPGQ_ELGA'
| 'EPGQ_ELNO'
| 'EPGQ_NODE'

Calculation of the equivalent deformations:

Fields EPEQ_x are calculated from fields EPSI_x (deformations in small displacement), fields EPGQ_x are calculated from fields EPSG_x (Green-Lagrange deformations) and fields EPMQ_x are calculated from fields EPME_x (mechanical deformations).

List of field components:

INVA_2	Equivalent Von Mises deformation
PRIN_1 PRIN_2 PRIN_3	Main deformations, arranged in ascending order
INVA_2SG	Equivalent Von Mises deformation signed by the \(\epsilon\) trace
VECT_1_X VECT_1_Y VECT_1_Z VECT_2_X VECT_2_Y VECT_2_Z VECT_3_X VECT_3_Y VECT_3_Z	Main directions

The equivalent Von Mises deformation is given by the following expression:

\(\mathrm{INVA}\text{\_}2=\sqrt{\frac{2}{3}\mathit{dev}{(\mathrm{\epsilon })}_{\mathit{ij}}\mathit{dev}{(\mathrm{\epsilon })}_{\mathit{ji}}}\) with \(\mathrm{dev}{(\varepsilon )}_{\mathrm{ij}}={\varepsilon }_{\mathrm{ij}}-\frac{1}{3}\mathrm{tr}(\varepsilon ){\delta }_{\mathrm{ij}}\)

This expression is equivalent to that of the \(\overline{\mathrm{\epsilon }}\) standard in the RCC -CW code.

\(\overline{\mathrm{\epsilon }}=\frac{\sqrt{2}}{3}\sqrt{{({\mathrm{\epsilon }}_{11}-{\mathrm{\epsilon }}_{22})}^{2}+{({\mathrm{\epsilon }}_{22}-{\mathrm{\epsilon }}_{33})}^{2}+{({\mathrm{\epsilon }}_{33}-{\mathrm{\epsilon }}_{11})}^{2}+6({\mathrm{\epsilon }}_{12}^{2}+{\mathrm{\epsilon }}_{23}^{2}+{\mathrm{\epsilon }}_{31}^{2})}\)

note

It should be noted that the equivalent deformations obtained from EPSI and EPME are identical. In fact, the difference between the two tensors is a spherical tensor (thermal deformation). As the equivalent deformation is obtained from the second invariant of the deviator, the spherical tensor « disappears » when the deviator is taken.

| 'INDL_ELGA'

Calculation of the location indicator, based on the acoustic tensor (criterion of RICE).

List of field components:

INDICE	Location indicator 0si \(\mathrm{det}(\mathrm{N.H.N})>0\) 1 otherwise, which corresponds to the initiation of the localization
DIR1	First normal to location zone
DIR2	Second normal to location zone
DIR3	Third normal to the location zone
DIR4	Fourth normal to the location zone

This indicator defines a state at which the local problem of integrating behavior loses its uniqueness. It is defined by: \(\mathrm{det}(\mathrm{N.}H\mathrm{.}N)\le 0\), where \(H\) refers to the tangent operator and \(N\) the normal to the location directions.

note

The method is only developed in case \(\mathrm{2D}\) and for the laws of behavior of the DRUCK_PRAGER and HUJEUX types.

| 'PDIL_ELGA'

Calculation of the modulus of rigidity of micro-expansion.

List of field components:

A 1_LC2

Micro-expansion stiffness module

In the context of environments with a second expansion gradient, option PDIL_ELGA provides the value of the module A 1_LC2, making it possible to control the periodicity of the non-trivial solution of the initially homogeneous problem [R5.04.03].

The calculation of A 1_LC2 is obtained via the evaluation of a function dependent on the geometric orientation of the hardware band in question. The angular discretization currently imposed is equal to \(0.1°\).

note

The method is only developed for behavior laws such as DRUCK_PRAGER and HUJEUX.

| 'SIEQ_ELGA'
| 'SIEQ_ELNO'
| 'SIEQ_NOEU'

Calculate equivalent constraints calculated from the constraint fields.

List of field components:

VMIS	Von Mises equivalent stress
TRESCA	Tresca constraint
PRIN_1 PRIN_2 PRIN_3	Main constraints, arranged in ascending order
VMIS_SG	Equivalent Von Mises stress signed by the trace of \(\sigma\)
VECT_1_X VECT_1_Y VECT_1_Z VECT_2_X VECT_2_Y VECT_2_Z VECT_3_X VECT_3_Y VECT_3_Z	Main directions
TRSIG	Trace of \(\sigma\)
TRIAX	Triaxiality rate

The equivalent Von Mises stress is given by the following expression:

\(\mathrm{VMIS}=\sqrt{\frac{3}{2}{s}_{\mathrm{ij}}{s}_{\mathrm{ji}}}\) with \({s}_{\mathrm{ij}}={\sigma }_{\mathrm{ij}}-\frac{1}{3}\mathrm{tr}(\sigma ){\delta }_{\mathrm{ij}}\)

The triaxiality rate is given by the following expression:

\(\mathrm{TRIAX}=\frac{\mathrm{TRSIG}}{3\times \mathrm{VMIS}}\)

8.6. Hydraulic flow calculation option (Operand HYDRAULIQUE)

| 'FLHN_ELGA'

Calculation of hydraulic flows in THM \({\Phi }_{\mathit{ij}}\mathrm{=}{M}_{\mathit{ij}}\mathrm{.}\nu\) on the edge elements (2D or 3D) from the flow vector at the nodes.

\({M}_{\mathit{ij}}\) is the hydraulic flow vector for component \(\mathit{ij}\).

List of field components:

• FH11

• FH22

• FH12

• FH21

8.7. Options related to calculation parameters (Operand PROPRIETES)

| 'MATE_ELGA'
| 'MATE_ELEM'

Recovery of the values of the elastic material parameters E, NU, RHO,,, ALPHAet thermal LAMBDA, RHO_CPavec taking into account the dependence on the control variables and on the calculation time. A zero value is assigned to the components of the field if the associated parameters have not been filled in DEFI_MATERIAU. The coordinates of the integration points are also present in the field.

These options are only available for ELAS, ELAS_FO,, THER, THER_FO, THER_NL behaviors.

For field MATE_ELGA, we calculate the values at each Gauss point, for field MATE_ELEM, we calculate the values at the barycenter of the element (family FPG1).

For the field MATE_ELGA and the absorbing border elements (3D_ABSO, D_PLAN_ABSO), the spatial coordinates taken into account are those of the barycenter of the element (as in the other calculation options). The values at each Gauss point are therefore always equal to each other and also to the values in field MATE_ELEM.