9. Formula criteria

9.1. For periodic charging

*For periodic loading, damage is calculated only over the first full cycle. The first part of the loading story corresponding to monotonic loading is not taken into account because it aims to impose a non-zero average loading. For elastic behavior, the calculation is carried out between the maximum value and the minimum value of the cycle in question. For the elasto-plastic behavior, the calculation is made**between the first discharge and the second discharge. *

The list of available sizes can be found in the following table:

TYPE_CHARGE= 'PERIODIQUE', CRITERE = 'FORMULE_CRITERE'

The available quantities are: “DTAUMA “:half-amplitude of shear at maximum stress (\(\Delta \tau (\text{n*})/2\))” PHYDRM “:hydrostatic pressure (\(P\)) “NORMAX “:maximum normal stress on the plane of normal (\({N}_{\mathit{max}}(\text{n*})\))” NORMOY “:mean normal stress on the critical plane (\({N}_{\mathit{moy}}(\text{n*})\)) 'EPNMAX':déformation normale maximale sur le plan critique(:math:`{\epsilon }_{\mathit{Nmax}}(\text{n*})`) 'EPNMOY':déformation normale moyenne sur le plan critique(:math:`{\epsilon }_{\mathit{Nmoy}}(\text{n*})`) 'DEPSPE':demi-amplitude de la déformation plastique équivalente (:math:`\Delta {\epsilon }_{\mathit{eq}}^{p}/2`) :math:`\Delta {\epsilon }_{\mathit{eq}}^{p}/2=\frac{1}{2}{\mathit{max}}_{\mathit{t1}}{\mathit{max}}_{\mathit{t2}}\sqrt{\frac{2}{3}({ϵ}^{p}({t}_{1})-{ϵ}^{p}({t}_{2}))\mathrm{:}({ϵ}^{p}({t}_{1})-{ϵ}^{p}({t}_{2}))}` 'EPSPR1':demi-amplitude de la première déformation principale (avec la prise en compte du signe) :math:`\frac{{ϵ}_{\mathit{max}}^{1}-{ϵ}_{\mathit{min}}^{1}}{2}` 'SIGNM1':contrainte normale maximale sur le plan associé avec :math:`{\epsilon }_{1}` :math:`{\mathit{max}}_{t}(\sigma (t)\mathrm{.}{n}_{1}(t)\mathrm{.}{n}_{1}(t))` où :math:`{n}_{1}(t)`est le vecteur normal du plan associé avec :math:`{\epsilon }_{1}`. 'DENDIS':densité d'énergie dissipée (:math:`{W}_{\mathit{cy}}`) :math:`{W}_{\mathit{cy}}=\underset{\mathit{cylce}}{\int }\sigma \mathrm{:}\dot{{ϵ}^{p}}\mathit{dt}` où :math:`\dot{{ϵ}^{p}}`représente le taux de la déformation plastique. 'DENDIE':densité d'énergie des distorsions élastiques (:math:`{W}_{e}`) :math:`{W}_{e}\mathrm{=}\underset{\mathit{cylce}}{\mathrm{\int }}\langle s\mathrm{:}\dot{{e}^{e}}\rangle \mathit{dt}` où :math:`s`représente la partie déviatorique de la contrainte :math:`\sigma`, :math:`{e}^{e}`représente la partie déviatorique de la contrainte :math:`{ϵ}^{e}`et :math:`\langle x\langle`donne :math:`x`si :math:`x\ge 0`et donne 0 si :math:`x<0`. 'APHYDR':demi-amplitude de la pression hydrostatique (:math:`{P}_{a}`) :math:`{P}_{a}=\frac{{P}_{\mathit{max}}-{P}_{\mathit{min}}}{2}` 'MPHYDR':pression hydrostatique moyenne (:math:`{P}_{m}`) :math:`{P}_{m}=\frac{{P}_{\mathit{max}}-{P}_{\mathit{min}}}{2}` 'DSIGEQ':demi-amplitude de la contrainte équivalente (:math:`\Delta {\sigma }_{\mathit{eq}}/2`) :math:`\frac{\Delta {\sigma }_{\mathit{eq}}}{2}=\frac{1}{2}{\mathit{max}}_{\mathit{t1}}{\mathit{max}}_{\mathit{t2}}\sqrt{\frac{3}{2}(s({t}_{1})-s({t}_{2}))\mathrm{:}(s({t}_{1})-s({t}_{2}))}` 'SIGPR1':demi-amplitude de la première contrainte principale (avec la prise en du signe) :math:`\frac{{\sigma }_{\mathit{max}}^{1}-{\sigma }_{\mathit{min}}^{1}}{2}` 'EPSNM1':déformation maximale normale sur le plan associé avec :math:`{\sigma }_{1}` :math:`{\mathit{max}}_{t}(ϵ(t)\mathrm{.}{n}_{1}(t)\mathrm{.}{n}_{1}(t))` où :math:`{n}_{1}(t)`est le vecteur normal du plan associé avec :math:`{\sigma }_{1}`. 'INVA2S':demi-amplitude du deuxième invariant de la déformation (:math:`{J}_{2}(\Delta ϵ)`) :math:`{J}_{2}(\Delta ϵ)=\frac{1}{2}{\mathit{max}}_{\mathit{t1}}{\mathit{max}}_{\mathit{t2}}\sqrt{\frac{2}{3}(e({t}_{1})-e({t}_{2}))\mathrm{:}(e({t}_{1})-e({t}_{2}))}` 'DSITRE':demi-amplitude de la demi-contrainte Tresca (:math:`({\sigma }_{\mathit{max}}^{\mathit{Tresca}}-{\sigma }_{\mathit{min}}^{\mathit{Tresca}})/4`) 'DEPTRE': demi-amplitude de la demi-déformation Tresca (:math:`({ϵ}_{\mathit{max}}^{\mathit{Tresca}}-{ϵ}_{\mathit{min}}^{\mathit{Tresca}})/4`) 'EPSPAC': déformation plastique accumulé :math:`p` 'RAYSPH': le rayon de la plus petite sphère circonscrite au trajet de chargement dans l'espace des déviateurs des contraintes:math:`R`. Voir le document [:external:ref:`R7.04.01 <R7.04.01>`] pour la définition de ce paramètre. 'AMPCIS': amplitude de cission (:math:`{\tau }_{a}\mathrm{=}\frac{1}{2}\underset{0\mathrm{\le }{t}_{0}\mathrm{\le }T}{\text{Max}}\underset{0\mathrm{\le }{t}_{1}\mathrm{\le }T}{\text{Max}}\mathrm{\parallel }{\sigma }_{({t}_{1})}^{D}\mathrm{-}{\sigma }_{({t}_{0})}^{D}\mathrm{\parallel }`) 'DEPSEE': demi-amplitude de la déformation élastique équivalente (:math:`\Delta {\epsilon }_{e}^{p}/2`) Il existe des grandeurs dépendant de l'orientation du plan qui passes au travers un point de matériel. Pour ces grandeurs, on définit des critères du type de plan critique. Le plan critique est le plan qui rend la valeur maximale d'une formule critique maximum. “DTAUCR”: half magnitude of shear stress on the plane of normal**n**(\(\Delta \tau (\text{n})/2\)) “DGAMCR”: half magnitude of deformation (engineering) shear on the plane of normal**n**(\(\Delta \gamma (\text{n})/2\)) “DSINCR”: half magnitude of normal stress on the plane of normal**n**( \(\Delta N(\text{n})/2\)) “DEPNCR”: half-amplitude of normal deformation on the plane of normal**n**(\(\Delta {ϵ}_{n}(\text{n})/2\)) “()” MTAUCR “: maximum shear stress on the plane of normal**n**(\({\tau }_{\mathit{max}}(\text{n})\))” MGAMCR “: deformation (engineering) shear maximum on the plane of normal**n**()” “: maximum normal stress on the plane of normal**n**(\({\gamma }_{\mathit{max}}(\text{n})\))” MSINCR “: maximum normal stress on the plane of normal**n**(\({N}_{\mathit{max}}(\text{n})\)) “MEPNCR “:maximum normal deformation on the plane of normal**n**:math:{epsilon }_{mathit{nmax}}(text{n}) “ DGAMPC “: half magnitude of plastic deformation (engineering) shear on the plane of normal**n**(\(\Delta {\gamma }^{p}/2\))” () “DEPNPC “:half magnitude of normal plastic deformation on the plane of normal**n**(\(\Delta {\epsilon }_{e}^{p}/2\) ) “MGAMPC”: plastic deformation (engineering) maximum shear on the plane of normal**n**(\({\gamma }_{\mathit{max}}^{p}(\text{n})\)) “MEPNPC”: maximum normal plastic deformation on the plane of normal**n** \({\epsilon }_{\mathit{nmax}}^{p}(\text{n})\) Note that there are two types of shear deformation measurement: shear distortions \({\gamma }_{\mathit{ij}}\) (\(i\ne j\)) and deformations of shear \({ϵ}_{\mathit{ij}}\) (\(i\ne j\)). Note that \({\gamma }_{\mathit{ij}}=2{ϵ}_{\mathit{ij}}\). For “DGAMCR”, “MGAMCR”, “MGAMPC”, shear distortions \({\gamma }_{\mathit{ij}}\) were used.

By using at least one of the first six quantities, we will implicitly build the « critical plan » criterion. In this case, we will find two different planes on which the shear is maximum.

Note that the names of the quantities are identical to those used in programming. The operators used in the formula should conform to Python syntax as shown in note [U4.31.05].

Note that the equivalent quantity output for periodic loading is under the name “SIG1” in the result.

9.2. For non-periodic charging

The list of available sizes can be found in the following table:

TYPE_CHARGE= 'NON-PERIODIQUE', CRITERE = 'FORMULE_CRITERE'

The available quantities are: “TAUPR_1”: projected shear stress from the first vertex of the subcycle (\({\tau }_{\mathit{p1}}(\text{n})\)) “TAUPR_2”: projected shear stress from the second vertex of the subcycle (\({\tau }_{\mathit{p2}}(\text{n})\)) “SIGN_1”: normal stress from the first vertex of the subcycle (\({N}_{1}(\text{n})\)) “SIGN_2”: normal stress of second peak of the sub-cycle (\({N}_{2}(\text{n})\)) “PHYDR_1”: hydrostatic pressure of the first top of the sub-cycle “PHYDR_2”: hydrostatic pressure of the second top of the sub-cycle “EPSPR_1”: projected deformation shear from the first top of the sub-cycle (\({\gamma }_{\mathit{p1}}(\text{n})\)) “EPSPR_2”: projected deformation shear from the second top of the sub-cycle (\({\gamma }_{\mathit{p2}}^{i}(\text{n})\)) 'SIPR1_1':première contrainte principale du premier sommet du sous-cycle (:math:`{\sigma }_{1}(1)`) 'SIPR1_2':première contrainte principale du deuxième sommet du sous-cycle (:math:`{\sigma }_{1}(2)`) 'EPSN1_1':déformation normale sur le plan associé avec :math:`{\sigma }_{1}(1)`du premier sommet du sous-cycle 'EPSN1_2':déformation normale sur le plan associé avec :math:`{\sigma }_{1}(2)`du deuxième sommet du sous-cycle 'ETPR1_1':première déformation totale principale du premier sommet du sous-cycle (:math:`{ϵ}_{1}^{\mathit{tot}}(1)`) 'ETPR1_2':première déformation totale principale du deuxième sommet du sous-cycle (:math:`{ϵ}_{1}^{\mathit{tot}}(2)`) 'SITN1_1':contrainte normale sur le plan associé avec :math:`{ϵ}_{1}^{\mathit{tot}}(1)`du premier sommet du sous-cycle 'SITN1_2':contrainte normale sur le plan associé avec :math:`{ϵ}_{1}^{\mathit{tot}}(2)`du deuxième sommet du sous-cycle 'EPPR1_1':première déformation plastique principale du premier sommet du sous-cycle (:math:`{\epsilon }_{1}^{p}(1)`) 'EPPR1_2':première déformation plastique principale du deuxième sommet du sous-cycle (:math:`{ϵ}_{1}^{p}(2)`) 'SIPN1_1':contrainte normale sur le plan associé avec :math:`{ϵ}_{1}^{p}(1)`du premier sommet du sous-cycle 'SIPN1_2':contrainte normale sur le plan associé avec :math:`{ϵ}_{1}^{p}(2)`du deuxième sommet du sous-cycle 'SIGEQ_1':contrainte équivalente du premier sommet du sous-cycle (:math:`{\sigma }_{\mathit{eq}}(1)`) 'SIGEQ_2':contrainte équivalente du deuxième sommet du sous-cycle (:math:`{\sigma }_{\mathit{eq}}(2)`) 'ETEQ_1':déformation totale équivalente du premier sommet du sous-cycle (:math:`{ϵ}_{\mathit{eq}}^{\mathit{tot}}(1)`) 'ETEQ_2':déformation totale équivalente du deuxième sommet du sous-cycle (:math:`{ϵ}_{\mathit{eq}}^{\mathit{tot}}(2)`)

For non-periodic loading, after extracting the elementary subcycles with method RAINFLOW, we calculate an elementary equivalent quantity using the criterion formula for any elementary subcycle. It should be noted that the sub-cycle is represented by two states of stress or deformation, noted by the first and the second vertices of the sub-cycle.

Using the criterion in the formula, we will implicitly build the criterion to determine the maximum damage plan with a linear accumulation of the damage.

It should be noted that the use of the quantities” TAUPR_1 “and” TAUPR_2 “excludes that of” EPSPR_1 “and” EPSPR_2 “because it is possible to project either the shear stress or the shear in deformation. It is not possible to project all of these two parameters simultaneously.

Note that the names of the quantities are identical to those used in programming. The operators used in the formula must respect Python syntax as shown in note U4.31.05.