3. Operands#

3.1. Operand CHARGEMENT#

This keyword allows the user to specify the type of load processed. The load can be “UNIAXIAL”, “MULTIAXIAL”, or “QUELCONQUE”. Each load has its own method (or methods) for evaluating fatigue damage.

Note: When the load is multi-axial, simply give the history of the load over a period or a block of cous-cycles. In the event that the load is of any kind, the entire history of the load must be provided.

3.2. Operands specific to type UNIAXIAL calculation#

3.2.1. Operand HISTOIRE#

The loading history may be the evolution of a uniaxial stress or deformation value over time,

Note:

This does not mean that the load cannot be multi-axial, but only that for the calculation of the damage, the load is characterized by the evolution of a scalar component, over time (Von-Mises signed, 2nd order invariant signed,…). It is the evolution of this scalar component that the user must provide to the command POST_FATIGUE.

3.2.1.1. Operand SIGM#

◊ SIGM = histsigm,

Name of the function or formula describing the history of stress loading at a point. It is a function or a formula of the parameter INST, which gives the evolution over time of a scalar component characterizing the stress state of the structure.

This operand is mandatory for calculating damage using a WOHLER method.

3.2.1.2. Operand EPSI#

◊ EPSI = histepsi,

Name of the function or formula describing the history of deformation loading at a point. It is a function or a formula of the parameter INST, which gives the evolution over time of a scalar component characterizing the state of deformations of the structure.

This operand is mandatory for calculating the damage using the methods of MANSON_COFFIN or TAHERI_MANSON or TAHERI_MIXTE.

3.2.2. Operand COMPTAGE#

♦ COMPTAGE =

To be able to calculate the damage suffered by a structure, it is first necessary to extract the elementary cycles of the loading history. For this, numerous methods are available. In*Code_Aster*, three methods have been programmed.

/”RAINFLOW”,

Method for counting cascading ranges or method of RAINFLOW (recommendation AFNOR A03-406 of November 1993) for extracting elementary cycles from the loading history [R7.04.01].

/”RAINFLOW_MAX”, This method is similar to that of Rainflow except that the elementary cycle with a maximum amplitude is placed at the beginning of the loading story to take into account the effects of overloads.

/”RCCM”,

RCC -M [R7.04.01] method.

/”NATUREL”,

So-called natural method which consists in generating cycles in the order in which they are applied [R7.04.01].

In the special case where the load history is constant (for example, average load applied), Code_Aster will count the entire load history as a zero amplitude cycle.

3.2.3. Operand DELTA_OSCI#

◊ DELTA_OSCI = delta,

Filtering the loading history. In all cases, if the function remains constant or decreasing over more than two consecutive points, the intermediate points are removed to keep only the two extreme points. Then, the points for which the variation in the stress value is less than the delta value are removed from the loading history. By default delta is equal to zero, which is equivalent to keeping all load oscillations, even those of low amplitude.

It is noted that if the COEF_MULT and DELTA_OSCI keywords are all present, Code_Aster will apply COEF_MULT first and then DELTA_OSCI.

Example: Consider the following loading story:

Extracting the peaks from this loading history, with a value of \(\mathit{delta}\) from \(\mathrm{0,9}\), leads to the destruction of all oscillations with an amplitude of less than 0.9. Which leads to the following loading story:

We removed:

point 5 because \(\Delta \sigma \mathrm{=}\mathrm{\mid }\sigma (5)\mathrm{-}\sigma (4)\mathrm{\mid }<\mathrm{0,9}\),
point 6 because \(\Delta \sigma =\mid \sigma (6)-\sigma (4)\mid <\mathrm{0,9}\),
point 12 because \(\Delta \sigma =\mid \sigma (12)-\sigma (11)\mid <\mathrm{0,9}\),
point 13 because \(\Delta \sigma =\mid \sigma (13)-\sigma (11)\mid <\mathrm{0,9}\).

Likewise, point 22 is deleted because the loading history is increasing between points 21, 22 and 23 and therefore only the extreme points are kept.

3.2.4. Keyword COEF_MULT#

◊ COEF_MULT = _F

This keyword factor groups together the amplification coefficients of the loading history. For now, only one multiplying factor for the loading history is available: the stress concentration coefficient KT.

Stress concentration coefficient values are available in RCC_M.

3.2.4.1. KT operand#

◊ KT = kt

kt is the stress concentration coefficient that depends on the geometry of the part, on the geometry of a possible defect and on the type of load. This coefficient is used to apply a kt ratio homothetic to the « filtered » loading history.

It is noted that if the COEF_MULT and DELTA_OSCI keywords are all present, Code_Aster will apply COEF_MULT first and then DELTA_OSCI.

3.2.5. Operand CORR_KE#

◊ CORR_KE = 'RCCM',

This operand makes it possible to take into account an elasto-plastic concentration coefficient, \({K}_{e}\), which is defined by RCC -M as being the ratio between the actual deformation amplitude and the deformation amplitude determined by an elastic analysis.

\(\mathrm{\{}\begin{array}{cccccc}{K}_{e}\mathrm{=}1& \text{si}& & & \mathit{Ds}& \text{<}{\mathrm{3S}}_{m}\\ {K}_{e}\mathrm{=}1+(1\mathrm{-}n)(\frac{\Delta \sigma }{3\mathrm{.}{S}_{m}}\mathrm{-}1)\mathrm{/}(n(m\mathrm{-}1))& \text{si}& {\mathrm{3S}}_{m}& \text{<}& \mathit{Ds}& \text{<}{\mathrm{3mS}}_{m}\\ {K}_{e}\mathrm{=}1\mathrm{/}n& \text{si}& {\mathrm{3mS}}_{m}& \text{<}& \mathit{Ds}& \end{array}\)

where \({S}_{m}\) is the maximum allowable stress and \(n\) and \(m\) are two constants that depend on the material.

The values \({S}_{m}\text{,}n\) and \(m\) are provided in the operator DEFI_MATERIAU [U4.43.01] under the keyword factor FATIGUE and the operands SM_KE_RCCM, N_ KE_RCCM, and M_ KE_RCCM.

3.2.6. Operand DOMMAGE#

To calculate the damage suffered by a structure at a point, various methods are available [R7.04.01]. Methods based on uniaxial tests: Wöhler method, Manson-Coffin method, Taheri methods. What these methods have in common is that they determine a damage value from the evolution over time of a scalar component characterizing the stress state or deformation of the structure.

This does not mean that the stress state cannot be multi-axial, but only that for the calculation of the damage a uniaxial component characterizing the stress or deformation state (signed Von-Mises stress, signed Von-Mises stress, signed 2nd order invariant of the deformation tensor,…) was chosen.

The Manson-Coffin and Taheri methods use the deformations generated by the load.

The Wöhler method uses the constraints generated by loading.

◊ DOMMAGE = 'WOHLER',

For a history of stresses associated with uniaxial loading, the number of cycles at break is determined using the Wöhler curve of material \(({N}_{\text{rupt}}=\text{WOHLER}(\frac{\Delta \sigma }{2}))\).

The Wöhler curve of the material must be introduced into the operator DEFI_MATERIAU [U4.43.01] in one of the three possible mathematical forms [R7.04.01]:

discretized function point by point (keyword factor FATIGUE, operand WOHLER),
Basquin analytical form (keyword factor FATIGUE, operands A_ BASQUIN and BETA_BASQUIN),
« current zone » form (keyword factor FATIGUE, operands E_ REFE, A0, A1, A2, A3 and SL and keyword factor ELAS operand E).

Note on fatigue curves:

For small stress amplitudes, the problem of extending the fatigue curve may arise: for example, for RCC -M fatigue curves beyond \({10}^{6}\) cycles, the corresponding stress, \(180\mathit{MPa}\) is considered as an endurance limit, i.e. any stress less than \(180\mathit{MPa}\) must produce a zero use factor or an infinite allowable number of cycles.

The method adopted here corresponds to this concept of endurance limit: if the stress amplitude is less than the first x-axis of the fatigue curve, then a zero use factor is taken, that is to say an infinite number of admissible cycles.

◊ DOMMAGE = 'MANSON_COFFIN',

For a history of uniaxial loading such as deformations, the number of cycles at break is determined using the Manson-Coffin curve of material \(({N}_{\text{rupt}}=\text{MANSON\_COFFIN}(\frac{\Delta \varepsilon }{2}))\).

The Manson-Coffin curve of the material must be introduced into the operator DEFI_MATERIAU [U4.43.01] (keyword factor FATIGUE, operand MANSON_COFFIN).

◊ DOMMAGE = 'TAHERI_MANSON',

This damage calculation method only applies to deformed loads.

Let \(n\) elementary cycles (extracted by a counting method) have a half-amplitude \(\frac{\Delta {\varepsilon }_{1}}{2},\cdots ,\frac{\Delta {\varepsilon }_{n}}{2}\).

The value of the elementary damage of the first cycle is determined by interpolation on the Manson-Coffin curve of the material.

The calculation of the elementary damage of the following cycles is carried out by the algorithm described below:

If \(\frac{\Delta {\varepsilon }_{i+1}}{2}\ge \frac{\Delta {\varepsilon }_{i}}{2}\)

the calculation of the elementary damage of cycle \((i+1)\) is determined by interpolation on the Manson-Coffin curve of the material,

If \(\frac{\Delta {\varepsilon }_{i+1}}{2}<\frac{\Delta {\varepsilon }_{i}}{2}\)

we determine:

\(\begin{array}{}\frac{\Delta {\sigma }_{i+1}}{2}=\text{Fnappe}(\frac{\Delta {\varepsilon }_{i+1}}{2},\underset{j<i}{\text{Max}}(\frac{\Delta {\varepsilon }_{j}}{2}))\\ \frac{\Delta {\varepsilon }_{i+1}^{\text{*}}}{2}=\text{Ffonc}(\frac{\Delta {\sigma }_{i+1}}{2})\end{array}\)

where:

Fis a tablecloth introduced under the operand TAHERI_NAPPE,

Ffunc is a function introduced under operand TAHERI_FONC.

The damage value for cycle \((i+1)\) is obtained by interpolation of \(\frac{\Delta {\varepsilon }_{i+1}^{\text{*}}}{2}\) on the Manson-Coffin curve of the material.

\({N}_{{\text{rupt}}_{i+1}}\) is the number of cycles at the break of cycle \((i+1)\)

\({N}_{{\text{rupt}}_{i+1}}=\text{MANSON\_COFFIN}(\frac{{\Delta \varepsilon }_{i+1}^{\text{*}}}{2})\)

and \({\text{Dom}}_{i+1}\) is the damage of the \((i+1)=\frac{1}{{N}_{{\text{rupt}}_{i+1}}}\) cycle.

The Manson-Coffin curve of the material must be introduced into the operator DEFI_MATERIAU [U4.43.01] (keyword factor FATIGUE, operand MANSON_COFFIN).

◊ DOMMAGE = 'TAHERI_MIXTE',

This damage calculation method only applies to deformed loads.

Let \(n\) elementary cycles (extracted by a counting method) have a half-amplitude \(\frac{\Delta {\varepsilon }_{1}}{2},\cdots ,\frac{\Delta {\varepsilon }_{n}}{2}\).

The value of the elementary damage of the first cycle is determined by interpolation on the Manson-Coffin curve of the material.

The calculation of the elementary damage for the following cycles is carried out by the algorithm described below:

If \(\frac{\Delta {\varepsilon }_{i+1}}{2}\ge \frac{\Delta {\varepsilon }_{i}}{2}\)

the calculation of the elementary damage of cycle \((i+1)\) is determined by interpolation on the Manson-Coffin curve.

If \(\frac{\Delta {\varepsilon }_{i+1}}{2}<\frac{\Delta {\varepsilon }_{i}}{2}\)

we determine:

\(\begin{array}{}\frac{\Delta {\sigma }_{i+1}}{2}=\text{Fnappe}(\frac{\Delta {\varepsilon }_{i+1}}{2},\underset{j<i}{\text{Max}}(\frac{\Delta {\varepsilon }_{j}}{2}))\end{array}\)

where Fis a tablecloth introduced under the operand of TAHERI_NAPPE.

The damage value for cycle \((i+1)\) is obtained by interpolating \(\frac{\Delta {\sigma }_{i+1}}{2}\) onto the material’s Wöhler curve.

\({N}_{{\text{rupt}}_{i+1}}\) is the number of cycles at the break of cycle \((i+1)\)

\({N}_{{\text{rupt}}_{i+1}}=\text{WOHLER}(\frac{{\Delta \sigma }_{i+1}}{2})\)

and \({\text{Dom}}_{i+1}\) is the damage of the \((i+1)=\frac{1}{{N}_{{\text{rupt}}_{i+1}}}\) cycle.

This method requires the data from the Wöhler curve and the Manson-Coffin curve of the material that must be entered into the operator DEFI_MATERIAU [U4.43.01] (keyword factor FATIGUE).

3.2.7. Operand MATER#

♦ MATER = subdue,

Allows you to specify the name of the material to be subdued created by DEFI_MATERIAU [U4.43.01].

The material must contain the values of all the material data necessary for the calculation of the damage.

3.2.8. Operand CORR_SIGM_MOYE#

◊ CORR_SIGM_MOYE =/'GOODMAN',

/”GERBER”,

This operand is only used when calculating damage using the WOHLER method.

If the part is not subjected to pure or symmetric alternating stresses, that is, if the mean stress of the cycle is not zero, the Wöhler curve can be weighted to calculate the number of effective cycles at break using the Haigh diagram [R7.04.01].

From a cycle \(({S}_{\mathrm{alt}},{\sigma }_{m})\) identified in the signal, the value of the corrected alternating stress \({S}_{\mathrm{alt}}^{\text{'}}\) is calculated.

If we use the Goodman line

\({S}_{\mathit{alt}}^{\text{'}}\mathrm{=}\frac{{S}_{\mathit{alt}}}{1\mathrm{-}\frac{{\sigma }_{m}}{{S}_{u}}}\)

If we use Gerber’s parable

\({S}_{\mathrm{alt}}^{\text{'}}=\frac{{S}_{\mathrm{alt}}}{1-{(\frac{{\sigma }_{m}}{{S}_{u}})}^{2}}\)

The value of the breaking limit of material \({S}_{u}\) must be entered into the operator DEFI_MATERIAU [U4.43.01] (keyword factor RCCM, operand SU).

3.2.9. Operand TAHERI_NAPPE#

◊ TAHERI_NAPPE = tablecloth,

This operand allows you to specify the name of a tablecloth.

\(\text{Fnappe}=(\frac{\Delta \varepsilon }{2},{\varepsilon }_{\mathrm{max}})\) required to calculate the damage using the TAHERI_MANSON and TAHERI_MIXTE methods.

The tablecloth must have \(X\) and \(\mathit{EPSI}\) parameters. Parameter \(X\) corresponds to the maximum deformation reached during possible pre-work hardening.

The layer introduced under operand TAHERI_NAPPE is the cyclic work hardening curve with pre-work hardening of the material.

The cyclic work hardening curve without pre-work hardening, given under the keyword TAHERI_FONC, must be one of the curves defining the sheet. This curve should be given for \(X\mathrm{=}0\).

3.2.10. Operand TAHERI_FONC#

◊ TAHERI_FONC = so,

This operand allows you to specify the name of a \(\text{Ffonc}\mathrm{=}(\frac{\Delta \sigma }{2})\) function required to calculate the damage using the TAHERI_MANSON method.

The parameter for this function should be SIGM.

This function is the cyclic work hardening curve of the material.

3.2.11. Operand CUMUL#

◊ CUMUL = 'LINEAIRE',

The methods in WOHLER, MANSON_COFFIN, and TAHERI calculate a damage value for each elementary cycle extracted from the uniaxial load introduced by the user.

The CUMUL operand allows you to request the calculation of the total damage suffered by the structure at a point.

The only rule available is Miner’s rule, which consists of summing all elemental damage \(D\mathrm{=}\mathrm{\sum }_{i}{D}_{i}\).

3.3. Operands specific to type MULTIAXIAL calculation#

3.3.1. Operand TYPE_CHARGE#

This operand allows you to specify the type of load applied to the structure:

PERIODIQUE, charging is periodic;
NON_PERIODIQUE, charging is not periodic.

3.3.2. Operand HISTOIRE#

This keyword covers the entire phase of defining the loading history. The loading history can be the evolution of the stress tensor, total deformation, and plastic deformation over time.

Note that at least one type of load (stress, total deformation, plastic deformation) must be provided. For one type of tensor, all six components must be provided.

In this operator, elastic deformation = total deformation - plastic deformation. For the criterion that requires elastic deformation, the request for total deformation is mandatory. If the plastic deformation is not entered, the value zero will be taken.

3.3.2.1. Operands SIGM_XX/SIGM_YY/SIGM_ZZ/SIGM_XY//SIGM_XZ/SIGM_YZ#

Names of functions or formulas that describe the history of each component of the stress tensor over time. Each function or formula depends on the INST parameter. All functions or formulas must be defined for the same time points in time.

3.3.2.2. Operands EPS_XX/EPS_YY/EPS_ZZ/EPS_XY//EPS_XZ/EPS_YZ#

Names of functions or formulas that describe the history of each component of the total strain tensor over time. Each function or formula depends on the INST parameter. All functions or formulas must be defined for the same time points in time.

3.3.2.3. Operands EPSP_XX/EPSP_YY/EPSP_ZZ/EPSP_XY//EPSP_XZ/EPSP_YZ#

3.3.3. Operand CHAM_MATER#

◊ CHAM_MATER = cham_mater

Allows you to specify the name of the field of the cham_master material created by AFFE_MATERIAU [U4.43.03].

The mater material defined with the DEFI_MATERIAU command and used to assign the material to the mesh with the AFFE_MATERIAU command must contain the definition of the Wöhler curve as well as the information necessary to implement the criterion, see the keywords factors FATIGUE and CISA_PLAN_CRIT of the command DEFI_MATERIAU [U4.43.01].

The CHAM_MATER keyword is not mandatory when using a damage formula.

3.3.4. Operand COEF_PREECROU#

◊ COEF_PREECROU =/coef_pre,

/1.0,

This coefficient is used to take into account the effect of possible pre-work hardening.

3.3.5. Operand COEF_CORR#

◊ COEF_CORR = heart,

The Crossland and Dang Van-Papadopoulos criteria allow for periodic loading to calculate a value \({R}_{\mathrm{crit}}\) which indicates whether or not there is damage for the number of cycles associated with the endurance limits \({\tau }_{0}\) and \({d}_{0}\).

These criteria do not give a value of the damage, which can however be interesting.

To do this, it is proposed to use the value of the criterion and the Wöhler curve of the material, by defining an equivalent stress:

\({\sigma }^{\text{*}}=({R}_{\mathrm{crit}}+b)\times \text{corr}\)

Most Wöhler curves are obtained with alternating pure traction-compression tests. However, the Dang-Van—Papadopoulos criterion is a shear criterion. Therefore, it is necessary to « correct » the equivalent stress \({\sigma }^{\text{*}}\) before applying it to a Wöhler curve obtained with tension-compression tests; this is the role of the COEF_CORR operand* . *

The damage value is obtained by applying \({\sigma }^{\text{*}}\) to the material’s Wöhler curve.

For there to be coherence between the criterion and the Wöhler curve, it is necessary that:

\(\left\{\begin{array}{cc}{\sigma }^{\text{*}}\le {\tau }_{0}& \text{pas de dommage}\\ {\sigma }^{\text{*}}>{\tau }_{0}& \text{dommage}\end{array}\right\}\)

for a Wöhler curve defined in shear and that:

\(\left\{\begin{array}{cc}{\sigma }^{\text{*}}\le {d}_{0}& \text{pas de dommage}\\ {\sigma }^{\text{*}}>{d}_{0}& \text{dommage}\end{array}\right\}\)

for a Wöhler curve defined in traction-compression.

The user can therefore specify a corr value, taking into account the type of Wöhler curve he has. The value taken by default for COEF_CORR is \(\frac{{d}_{0}}{{\tau }_{0}}\), consistent with Wöhler curves in traction-compression.

Note:

In the case \({R}_{\mathrm{crit}}<0\) , if the left-hand extension of the Wöhler curve is linear (in DEFI_FONCTION (… PROL_GAUCHE = “LINEAIRE”…)), the user will get damage that is different from zero. To get zero damage when \({R}_{\mathrm{crit}}<0\) , the left extension must be equal to “ EXCLU “or “ CONSTANT “ .

3.3.6. Operand CRITERE#

♦ CRITERE =/'MATAKE_MODI_AC',

/'DANG_VAN_MODI_AC',

/'MATAKE_MODI_AV',

/'DANG_VAN_MODI_AV',

/'FATESOCI_MODI_AV',

/'FORMULE_CRITERE',

/”CROSSLAND”,

/”PAPADOPOULOS”,

The user enters the values for each component of the stress tensor at various times \(({t}_{0},\dots ,{t}_{N})\), and it is assumed that \(\mathrm{[}{t}_{\mathrm{0,}}{t}_{N}\mathrm{]}\) is a loading period.

Loads can be stresses, total deformations, plastic deformations, or combinations of these parameters.

The following table lists the boot criteria that are available for two load types.

TYPE_CHARGE= 'PERIODIQUE'	TYPE_CHARGE = “NON_PERIODIQUE”

“MATAKE_MODI_AC” “DANG_VAN_MODI_AC” “FORMULE_CRITERE” “ “ CROSSLAND “ “ PAPADOPOULOS “	“MATAKE_MODI_AV”, “DANG_VAN_MODI_AV” “” FATESOCI_MODI_AV “ “ FORMULE_CRITERE “

For constant amplitude loading, the CRITERE operand makes it possible to specify the criterion that the maximum shear half-amplitude must satisfy. For variable amplitude loading, the CRITERE operand makes it possible to specify the criterion that the maximum damage must satisfy.

The boot criteria in Code_Aster can be called by a name for well-established criteria. It is also possible for the user to build a priming criterion by himself as a formula of predefined quantities.

Rating:

\({\text{n}}^{\text{*}}\): normal to the plane in which the shear amplitude is maximum;

\(\Delta \tau (\text{n})\): magnitude of stress shear in a plane with normal \(\text{n}\);

\(\Delta \gamma (\text{n})\): amplitude of shear deformation in a plane of normal \(\text{n}\);

\({N}_{\mathrm{max}}(\text{n})\): normal maximum stress on the plane of normal \(\text{n}\);

\({\tau }_{0}\): endurance limit in pure alternating shear;

\({d}_{0}\): endurance limit in pure alternating traction-compression;

\(P\): hydrostatic pressure;

\({c}_{p}\): coefficient used to take into account possible pre-work hardening;

\({\sigma }_{y}\): elastic limit.

Criterion MATAKE_MODI_AC

The initial criterion for MATAKE is defined by the inequality [éq.3.12-1]:

\(\frac{\Delta \tau }{2}({\text{n}}^{\text{*}})+a{N}_{\mathit{max}}({\text{n}}^{\text{*}})\mathrm{\le }b\) eq 3.12-1

where \(a\) and \(b\) are two constants given by the user under the keywords MATAKE_A and MATAKE_B of the key word factor CISA_PLAN_CRIT of DEFI_MATERIAU, they depend on the material characteristics and are equal to:

\(a=({\tau }_{0}-\frac{{d}_{0}}{2})/\frac{{d}_{0}}{2}\) \(b={\tau }_{0}\)

If the user has the results of two tensile compression tests, one alternating and the other not, the constants \(a\) and \(b\) are given by:

\(a=\frac{\Delta {\sigma }_{2}-\Delta {\sigma }_{1}}{(\Delta {\sigma }_{1}-\Delta {\sigma }_{2})-2{\sigma }_{m}}\),

\(b=\frac{{\sigma }_{m}}{(\Delta {\sigma }_{2}-\Delta {\sigma }_{1})+2{\sigma }_{m}}\times \frac{\Delta {\sigma }_{1}}{2}\),

with \(\Delta {\sigma }_{1}\) the loading amplitude for the alternating case \(({\sigma }_{m}=0)\) and \(\Delta {\sigma }_{2}\) the loading amplitude for the case where the mean stress is non-zero \(({\sigma }_{m}\ne 0)\).

We are modifying the initial criterion of MATAKE by introducing the definition of an equivalent constraint, noted \({\sigma }_{\mathrm{eq}}({\text{n}}^{\text{*}})\):

\({\sigma }_{\mathit{eq}}({\text{n}}^{\text{*}})\mathrm{=}({c}_{p}\frac{\Delta \tau }{2}({\text{n}}^{\text{*}})+a{N}_{\mathit{max}}({\text{n}}^{\text{*}}))\frac{f}{t}\),

where \(f/t\) represents the ratio of the alternating flexure and torsional endurance limits, and must be entered under the keyword COEF_FLEX_TORS of the keyword factor CISA_PLAN_CRIT of DEFI_MATERIAU.

Criterion DANG_VAN_MODI_AC

The initial criterion for DANG VAN is defined by the inequality [éq 3.12-2]:

\(\frac{\Delta \tau }{2}({\text{n}}^{\text{*}})+aP\mathrm{\le }b\) eq 3.12-2

where \(a\) and \(b\) are two constants given by the user under the keywords D_ VAN_A and D_ VAN_B of the keyword factor CISA_PLAN_CRIT of DEFI_MATERIAU, they depend on the material characteristics. In the case where the user has two tensile compression tests, one alternates the other without the constants \(a\) and \(b\) equal:

\(a\mathrm{=}\frac{3}{2}\mathrm{\times }\frac{\Delta {\sigma }_{2}\mathrm{-}\Delta {\sigma }_{1}}{(\Delta {\sigma }_{1}\mathrm{-}\Delta {\sigma }_{2})\mathrm{-}2{\sigma }_{m}}\) \(b\mathrm{=}\frac{{\sigma }_{m}}{(\Delta {\sigma }_{2}\mathrm{-}\Delta {\sigma }_{1})+2{\sigma }_{m}}\mathrm{\times }\frac{\Delta {\sigma }_{1}}{2}\)

with \(\Delta {\sigma }_{1}\) the loading amplitude for the alternating case \(({\sigma }_{m}=0)\) \(\Delta {\sigma }_{2}\) and for the case where the mean stress is non-zero \(({\sigma }_{m}\ne 0)\).

In addition, we define an equivalent constraint in the sense of DANG VAN, denoted by \({\sigma }_{\mathrm{eq}}({\text{n}}^{\text{*}})\):

\({\sigma }_{\mathit{eq}}({\text{n}}^{\text{*}})\mathrm{=}({c}_{p}\frac{\Delta \tau }{2}({\text{n}}^{\text{*}})+aP)\frac{c}{t}\)

where \(c\mathrm{/}t\) represents the ratio of the alternating shear and tensile endurance limits, and must be entered under the keyword COEF_CISA_TRAC of the keyword factor CISA_PLAN_CRIT of DEFI_MATERIAU.

For more information, consult the document [R7.04.04].

Criterion MATAKE_MODI_AV

Criterion MATAKE_MODI_AV is an evolution of the MATAKE criterion. Unlike the previous two criteria, this criterion selects the critical plan based on the damage calculated in each design. It is the plan in which the damage is maximum that is retained. This criterion is adapted to non-periodic loads, which implies the use of a cycle counting method in order to calculate elementary damages. To count the cycles, we use the RAINFLOW method.

Once known, the elementary damage is accumulated linearly to determine the damage.

To calculate the elementary damages we project the history of shear stresses on one or two axes in order to reduce this one to a one-dimensional function of the \({\tau }_{p}=f(t)\) time. After extracting the elementary subcycles of \({\tau }_{p}\) with the RAINFLOW method we define an elementary equivalent constraint for any elementary subcycle \(i\):

\({\sigma }_{\mathrm{eq}}^{i}(\text{n})=\alpha ({c}_{p}\frac{\mathrm{Max}({\tau }_{\mathrm{p1}}^{i}(\text{n}),{\tau }_{\mathrm{p2}}^{i}(\text{n}))-\mathrm{Min}({\tau }_{\mathrm{p1}}^{i}(\text{n}),{\tau }_{\mathrm{p2}}^{i}(\text{n}))}{2}+a\mathrm{Max}({N}_{1}^{i}(\text{n}),{N}_{2}^{i}(\text{n}),0))\) eq 3.12-3

with \(\text{n}\) the normal of the current plane, \({\tau }_{\mathrm{p1}}^{i}(\text{n})\) and \({\tau }_{\mathit{p2}}^{i}(\text{n})\) the values of the projected shear stresses of the subcycle \(i\) and \({N}_{1}^{i}(\text{n})\) and \({N}_{2}^{i}(\text{n})\) the normal stresses of the subcycle \(i\). From \({\sigma }_{\mathrm{eq}}^{i}(\text{n})\) and a fatigue curve we determine the number of cycles at elementary failure \({N}^{i}(\text{n})\) and the corresponding damage \({D}^{i}(\text{n})=1/{N}^{i}(\text{n})\). In [éq3.12‑3] \(\alpha\) is a corrective term that allows the use of a tension-compression fatigue curve. The constants \(a\) and \(\alpha\) must be entered under the keywords MATAKE_A and COEF_FLEX_TORS of the key word factor CISA_PLAN_CRIT of DEFI_MATERIAU.

We use a linear accumulation of damage. Let \(k\) be the number of elementary subcycles, for a fixed normal \(\text{n}\), the cumulative damage is equal to:

\(D(\text{n})=\sum _{i=1}^{k}{D}^{i}(\text{n})\) eq 3.12-4

To determine the normal vector \({\text{n}}^{\text{*}}\) corresponding to the maximum cumulative damage we vary \(\text{n}\), the normal vector \({\text{n}}^{\text{*}}\) corresponding to the maximum cumulative damage is then given by:

\(D({\text{n}}^{\text{*}})=\underset{\text{n}}{\mathrm{Max}}(D(\text{n}))\)

Criterion DANG_VAN_MODI_AV

The approach and techniques used to calculate this criterion are identical to those used for criterion MATAKE_MODI_AV. The only difference is in the definition of the elementary equivalent stress where the hydrostatic pressure \(P\) replaces the maximum normal stress \({N}_{\mathrm{max}}\):

\({\sigma }_{\mathit{eq}}^{i}(\text{n})=\alpha \left({c}_{p}\frac{\mathit{Max}\left({\tau }_{\mathit{p1}}^{i}(\text{n}),{\tau }_{\mathit{p2}}^{i}(\text{n})\right)-\mathit{Min}\left({\tau }_{\mathit{p1}}^{i}(\text{n}),{\tau }_{\mathit{p2}}^{i}(\text{n})\right)}{2}+a\mathit{Max}\left({P}_{1}^{i}(\text{n}),{P}_{2}^{i}(\text{n}),0\right)\right)\)

The constants \(a\) and \(\alpha\) are to be filled in by the user under the keywords D_ VAN_A and COEF_CISA_TRAC of the keyword factor CISA_PLAN_CRIT of DEFI_MATERIAU.

For more information consult the document [R7.04.04].

Criterion FATESOCI_MODI_AV

The criteria for FATEMI and SOCIE are defined by the relationship:

\({\epsilon }_{\mathit{eq}}(n)=\frac{\Delta \gamma (n)}{2}\left(1+k\frac{{N}_{\mathit{max}}(n)}{{\sigma }_{y}}\right)\)

where \(k\) is a constant that depends on material characteristics. Unlike the other criteria, it uses deformation shear instead of stress shear. In addition, the various quantities that contribute to the criterion are multiplied and not added together. The criterion of FATEMI and SOCIE can be used after an elastic or elastoplastic calculation. This criterion selects the critical plan based on the damage calculated in each plan. It is the plan in which the damage is maximum that is retained.

This criterion is adapted to non-periodic loads, which leads us to use the cycle counting method RAINFLOW to calculate elementary damage. The elementary damage is then accumulated linearly to determine the damage.

In order to calculate the elementary damages we project the history of deformation shear on one or two axes in order to reduce this one to a one-dimensional function of time \({\gamma }_{p}=f(t)\). After extracting the elementary subcycles with method RAINFLOW we define an elementary equivalent deformation for any elementary subcycle \(i\):

\({\epsilon }_{\mathit{eq}}^{i}(\text{n})=\alpha {c}_{p}\left(\frac{\mathit{Max}\left({\gamma }_{\mathit{p1}}^{i}(\text{n}),{\gamma }_{\mathit{p2}}^{i}(\text{n})\right)-\mathit{Min}\left({\gamma }_{\mathit{p1}}^{i}(\text{n}),{\gamma }_{\mathit{p2}}^{i}(\text{n})\right)}{2}\right)\left(1+a\mathit{Max}\left({N}_{1}^{i}(\text{n}),{N}_{2}^{i}(\text{n}),0\right)\right)\) eq 3.12-5

with \(a=\frac{k}{{\sigma }_{y}}\), \(\text{n}\) the normal to the current plane, \({\gamma }_{\mathit{p1}}^{i}(\text{n})\) and \({\gamma }_{\mathrm{p2}}^{i}(\text{n})\) the values of the projected deformation shear values of the sub-cycle \(i\), \({N}_{1}^{i}(\text{n})\) and \({N}_{2}^{i}(\text{n})\) being the two values of the normal stress of the sub-cycle \(i\). From \({\varepsilon }_{\mathrm{eq}}^{i}(\text{n})\) and a Manson-Coffin curve we determine the number of cycles at elementary failure and \({N}^{i}(\text{n})\) the corresponding damage \({D}^{i}(\text{n})=1/{N}^{i}(\text{n})\).

Note that the shear deformations used in the criteria of FATEMI and SOCIE are distortions \({\gamma }_{\mathit{ij}}\) (\(i\ne j\)) *. If you use tensor-type shear deformations*\({ϵ}_{\mathit{ij}}\) (\(i\ne j\))**, you have to multiply them by a factor of 2 because**:math:`{gamma }_{mathit{ij}}=2{ϵ}_{mathit{ij}}`** . **

In equation [éq 3.12-5], \(\alpha\) is a corrective term that uses a Manson-Coffin curve obtained in traction-compression. \({c}_{p}\) is a coefficient that makes it possible to take into account possible pre-work hardening.

The constants \(a\) and \(\alpha\) must be entered under the keywords FATSOC_A and COEF_CISA_TRAC of the key word factor CISA_PLAN_CRIT in the DEFI_MATERIAU command.

It is noted that a rigorous approach is to use the Manson-Coffin curve obtained directly in torsion (which is not always available). The use of the Manson-Coffin curve obtained in traction-compression with the corrective term \(\alpha\) (which is the relationship between two endurance limits), as programmed in Code_Aster, is therefore an approximation.

As we use a linear accumulation of damage, if \(m\) is the number of elementary subcycles, then for a fixed \(\text{n}\) normal, the cumulative damage is equal to:

\(D(\text{n})=\sum _{i=1}^{m}{D}^{i}(\text{n})\)

To find the normal vector \({\text{n}}^{\text{*}}\) corresponding to the maximum cumulative damage we vary \(\text{n}\). The normal vector \({\text{n}}^{\text{*}}\) associated with the maximum cumulative damage is then given by:

\(D({\text{n}}^{\text{*}})=\underset{\text{n}}{\mathrm{Max}}(D(\text{n}))\)

Criterion FORMULE_CRITERE

This type of criterion allows the user to build a criterion as a formula for pre-defined quantities. This criterion is based on a general relationship:

« Equivalent quantity » = « Life curve »

where the « Equivalent quantity » is a formula provided under operand FORMULE_GRDEQ (see 3.4.6) and the « Life curve » is provided under operand COURBE_GRD_VIE (see 3.4.7) either by a function (table or formula, under the operand of “FORMULE_VIE”, see 3.4.8), or by a curve name “WOHLER” or “MANSON_C” defined previously in DEFI_MATERIAU.

Crossland criterion

The criterion is written as:

\({R}_{\mathit{crit}}\mathrm{=}{\tau }_{a}+a\mathrm{.}{P}_{\mathit{max}}\mathrm{-}b\)

where

\({\tau }_{a}\mathrm{=}\frac{1}{2}\underset{0\mathrm{\le }{t}_{0}\mathrm{\le }T}{\text{Max}}\underset{0\mathrm{\le }{t}_{1}\mathrm{\le }T}{\text{Max}}\mathrm{\parallel }\tilde{S}({t}_{1})\mathrm{-}\tilde{S}({t}_{0})\mathrm{\parallel }\) is the amplitude of cission

with \(\tilde{S}\) stress tensor deviator \(\sigma\)

\({P}_{\mathit{max}}\mathrm{=}\underset{0\mathrm{\le }t\mathrm{\le }T}{\text{Max}}(\frac{1}{3}\text{trace}\sigma )\) is the maximum hydrostatic pressure

\(a\mathrm{=}\frac{({\tau }_{0}\mathrm{-}\frac{{d}_{0}}{\sqrt{3}})}{\frac{{d}_{0}}{3}}\) and \(b\mathrm{=}{\tau }_{0}\)

with \({\tau }_{0}\) the endurance limit in pure alternating shear

and \({d}_{0}\) the endurance limit in pure alternating traction-compression

Dang Van-Papadopoulos criterion

The criterion is written as:

\({R}_{\mathit{crit}}\mathrm{=}{k}^{\text{*}}+a\mathrm{.}{P}_{\mathit{max}}\mathrm{-}b\)

where

\({k}^{\text{*}}\mathrm{=}R\)

\(R\) radius of the smallest sphere circumscribed to the loading path in the space of the constraint deviators \(\tilde{S}\)

\(R\mathrm{=}\underset{0\mathrm{\le }t\mathrm{\le }T}{\text{Max}}\sqrt{\frac{1}{2}\mathrm{.}(\tilde{S}(t)\mathrm{-}{C}^{\text{*}})\mathrm{:}(\tilde{S}(t)\mathrm{-}{C}^{\text{*}})}\)

\({C}^{\text{*}}\mathrm{=}\text{Min}\text{Max}\sqrt{(\tilde{S}(t)\mathrm{-}C)\mathrm{:}(\tilde{S}(t)\mathrm{-}C)}\) is the center of the hypersphere

\({P}_{\mathit{max}}\mathrm{=}\underset{0\mathrm{\le }t\mathrm{\le }T}{\text{Max}}(\frac{1}{3}\text{trace}\sigma )\) is the maximum hydrostatic pressure

\(a\mathrm{=}\frac{({\tau }_{0}\mathrm{-}\frac{{d}_{0}}{\sqrt{3}})}{\frac{{d}_{0}}{3}}\) and \(b\mathrm{=}{\tau }_{0}\)

with \({\tau }_{0}\) the endurance limit in pure alternating shear

and \({d}_{0}\) the endurance limit in pure alternating traction-compression

Notes:

The original purpose of these criteria Crossland and Dang Van-Papadopoulos*is not a step to determine a damage value, but a criterion value* \({R}_{\mathrm{crit}}\) such as:

\(\mathrm{\{}\begin{array}{cc}{R}_{\mathit{crit}}\mathrm{\le }0& \text{pas de dommage}\\ {R}_{\mathit{crit}}>0& \text{dommage possible}\end{array}\)

However, it is also possible to determine a damage value.

3.3.7. Operand FORMULE_GRDEQ#

♦ FORMULE_GRDEQ = for_grd, [formula]

Allows you to provide the criterion formula as a function of the available quantities. The lists of sizes available for each type of load can be found in the following table:

TYPE_CHARGE= 'PERIODIQUE', CRITERE = 'FORMULE_CRITERE'

The available quantities are:

“DTAUMA “:half-magnitude of maximum shear stress (\(\Delta \tau (\text{n*})/2\))” PHYDRM “:hydrostatic pressure (\(P\)) “NORMAX “:maximum normal stress on the critical plane (\({N}_{\mathit{max}}(\text{n*})\))” 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:`{\varepsilon }_{\mathit{Nmoy}}(\text{n*})`)

'DEPSPE':demi-amplitude de la déformation plastique équivalente (:math:`\Delta {\epsilon }_{\mathit{eq}}^{p}/2`)

'EPSPR1':demi-amplitude de la première déformation principale (avec la prise en compte du signe)

'SIGNM1':contrainte normale maximale sur le plan associé avec :math:`{\varepsilon }_{1}`

'DENDIS':densité d'énergie dissipée (:math:`{W}_{\mathit{cy}}`)

'DENDIE':densité d'énergie des distorsions élastiques (:math:`{W}_{e}`)

'APHYDR':demi-amplitude de la pression hydrostatique (:math:`{P}_{a}`)

'MPHYDR':pression hydrostatique moyenne (:math:`{P}_{m}`)

'DSIGEQ':demi-amplitude de la contrainte équivalente (:math:`\Delta {\sigma }_{\mathit{eq}}\mathrm{/}2`)

'SIGPR1':demi-amplitude de la première contrainte principale (avec la prise en compte du signe)

'EPSNM1':déformation normale maximale sur le plan associé avec :math:`{\sigma }_{1}`

'INVA2S':demi-amplitude du deuxième invariant de la déformation :math:`{J}_{2}(\epsilon )`

'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`

'AMPCIS':amplitude de cission:math:`{\tau }_{a}`

'DEPSEE':demi-amplitude de la déformation élastique équivalente (:math:`\Delta {\epsilon }_{e}^{p}/2`)
There are quantities depending on the orientation of the plane that pass through a material point. For these quantities, criteria of the critical plane type are defined. The critical plan is the plan that maximizes a critical formula (see).

“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 magnitude 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 \({N}_{\mathit{max}}(\text{n})\)) “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 shear deformations \({ϵ}_{\mathit{ij}}\) (\(i\ne j\)). Note that \({\gamma }_{\mathit{ij}}=2{ϵ}_{\mathit{ij}}\). For “DGAMCR”, “MGAMCR”, “”, “MGAMPC”, shear distortions \({\gamma }_{\mathit{ij}}\) were used.

TYPE_CHARGE = “NON - PERIODIQUE”, CRITERE = “FORMULE_CRITERE”

The available quantities are:

“TAUPR_1 “:projected shear stresses from the first vertex of the subcycle (\({\tau }_{\mathit{p1}}(\text{n})\))” TAUPR_2 “:projected shear stresses 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 “:shear in deformation (of the tensorial type \({ϵ}_{\mathit{ij}}\) (i #j)) projected from the first vertex of the sub-cycle (\({\gamma }_{\mathit{p1}}(\text{n})\)) “EPSPR_2 “:shear in deformation (of the tensorial type (i))) projected from the first summit of the sub-cycle ()” “:shear in deformation (of the tensorial type (i))) projected from the first summit of the subcycle () “”:shear in deformation (of the tensorial type (i \({ϵ}_{\mathit{ij}}\) (i #j)) projected from the second summit 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:`{\epsilon }_{1}^{\mathit{tot}}(1)`)

'ETPR1_2':première déformation totale principale du deuxième sommet du sous-cycle (:math:`{\epsilon }_{1}^{\mathit{tot}}(2)`)

'SITN1_1':contrainte normale sur le plan associé avec :math:`{\epsilon }_{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:`{ϵ}_{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)`)

Notes:

1) For periodic loading, the criterion formula is used to determine the maximum shear plane if the parameter “ DTAUMA “is introduced into the formula.

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.

3) The input parameters for the command FORMULE must be among those listed in the table above.

Expressions of certain sizes can be found in the document [R7.04.04].

5) It is emphasized that thermal deformation was not taken into account, i.e., \({\varepsilon }^{\mathit{tot}}\mathrm{=}{\varepsilon }^{e}+{\varepsilon }^{p}\) .

The operators used in the formula should conform to Python syntax as shown in note [U4.31.05].

3.3.8. Operand FORMULE_CRITIQUE#

◊ FORMULE_CRITIQUE = for_grd, [formula]

This keyword makes it possible to define a critical quantity that the critical plan maximizes. This formula must contain at least one parameter depending on the orientation of the plane.

3.3.9. Operand DOMMAGE#

♦ DOMMAGE =/”WOHLER”,

/”MANSON_C”, /”FORM_VIE”

This keyword makes it possible to provide a curve relating the quantity equivalent to the number of cycles at break.

In Code_Aster, the endurance limit is set at 10 million cycles. If the calculated equivalent quantity is less than the endurance limit, the calculated damage is 0.

If DOMMAGE = “WOHLER”, we will take the Wohler curve (\({N}_{f}\mathrm{=}f(\mathit{SIGM})\)) defined in AFFE_MATERIAU.

If DOMMAGE = “MANSON_C”, we will take the Manson-Coffin curve (\({N}_{f}\mathrm{=}f(\mathit{EPSN})\)) defined in AFFE_MATERIAU.

If DOMMAGE = “FORM_VIE”, we will provide a function defining the life curve.

3.3.9.1. Operand FORMULE_VIE#

/[function]

Allows you to specify the curve relating the equivalent quantity and the lifetime.

If for_vie is provided by a tabulated function, it should be in the form:

\({N}_{f}\mathrm{=}f\) (equivalent_size).

If for_vie is provided by a formula, it should be in the form:

\(\text{grandeur équivalente}\mathrm{=}f({N}_{f})\).

In this case, the input parameter for command FORMULEdoit will be “NBRUPT” (i.e., \({N}_{f}\)).

3.3.10. Operand METHODE#

♦ METHODE = 'CERCLE_EXACT'

Allows you to specify the name of the method that will be used to calculate the maximum half-shear amplitude.

The “CERCLE_EXACT” method is used to determine the circumscribed circle at points in shear planes. This method is based on the process of obtaining the circle that passes through three points, cf. document [R7.04.04].

3.3.11. Operand PROJECTION#

♦ PROJECTION =/”UN_AXE”,

/”DEUX_AXES”,

In the case where the loading is non-periodic, it is necessary to project the shear history on one or two axes, see document [R7.04.04].

UN_AXE, the history of shear is projected onto an axis;
DEUX_AXES, the history of shear is projected on two axes.

3.3.12. Operand DELTA_OSCI#

◊ DELTA_OSCI =/delta,

/0.0,

Filtering the history of the load. In all cases, if the function remains constant or decreasing over more than two consecutive points, the intermediate points are removed to keep only the two extreme points. Then, the points for which the variation in the stress value is less than the delta value are removed from the loading history. By default delta is equal to zero, which is equivalent to keeping all load oscillations, even those of low amplitude. For more information see the documentation for command POST_FATIGUE, [U4.83.01], same operand.

3.4. Operands specific to type QUELCONQUE calculation#

The loading history can be the evolution of the stress tensor, the cumulative plastic deformation, and the temperature over time.

3.4.1. Operand EPSP#

◊ EPSP = p,

Name of the function describing the history of cumulative plastic deformation over time, only for the calculation of the damage of LEMAITRE.

This function or formula depends on the parameter INST and must be defined for the same times as the functions or formulas describing the history of the components of the stress tensor.

Operand EPSP should be used in conjunction with operands SIGM_XX,…

3.4.2. Operand TEMP#

◊ TEMP = time,

Name of the function or formula describing the history of temperature over time, only for calculating the damage of LEMAITRE. In this case, it is used to determine the value of the mechanical characteristics (Young’s modulus \(E\), Poisson’s ratio \(\nu\) and material parameter \(S\)) at the time of calculation of the damage.

This function or formula depends on the parameter INST and must be defined for the same times as the functions or formulas describing the history of the components of the stress tensor.

Operand TEMP should be used in conjunction with operands EPSP, SIGM_XX,…

3.4.3. Lemaître and Lemaître-Sermage methods#

These two methods make it possible to calculate the damage \(D(t)\) from the data of the stress tensor \(\sigma (t)\) and the cumulative plastic deformation \(p(t)\).

They therefore apply to any load and are only used in post-processing a plastic or viscoplastic law with \(p\) as a variable.

The evolution of \(D\) is defined by:

\(\begin{array}{c}\mathrm{\{}\begin{array}{cc}\dot{D}\mathrm{=}\frac{1}{{(1\mathrm{-}D)}^{\mathrm{2s}}}{(\frac{1}{\mathrm{3ES}}\mathrm{.}(1+\nu ){\sigma }_{\mathit{eq}}^{2}+\frac{3}{\mathrm{2ES}}(1\mathrm{-}2\nu )\mathrm{.}{\sigma }_{H}^{2})}^{s}\dot{p}& \mathit{si}p>{p}_{d}\\ D\mathrm{=}0& \mathit{sinon}\end{array}\end{array}\)

where \(E\): Young’s modulus, \(\nu\): Poisson’s ratio, \(S\) and \(s\): material parameters, \({\sigma }_{\mathrm{eq}}\): equivalent von Mises stress, \({\sigma }_{H}\): hydrostatic pressure, \(p\): cumulative plastic deformation and \({p}_{d}\): damage threshold.

◊ DOMMAGE = 'LEMAITRE',

Allows you to calculate the Lemaître or Lemaître-Sermage \(D(t)\) damage from the data of the stress tensor \(\sigma (t)\) and the cumulative plastic deformation \(p(t)\). Note that Lemaître damage is obtained by assigning the value \(1.0\) to the exponent \(s\) (\(s=1\)).

3.5. Operand INFO#

◊ INFO =/1,

Printing:

elementary cycles determined by the counting method chosen by the user,
the elementary damage associated with each cycle for methods WOHLER, MANSON_COFFIN and TAHERI,
damage of LEMAITRE at each calculation point,
of the total damage (if the user requested its calculation).

◊ INFO =/2,

Printing:

the loading story entered by the user under the operands SIGM and EPSI,
peaks extracted from the loading history (introduced under the operands SIGM and EPSI),
elementary cycles determined by the counting method chosen by the user,
the elementary damage associated with each cycle for methods WOHLER, MANSON_COFFIN and TAHERI,
damage of LEMAITRE at each calculation point,
of the total damage (if the user requested its calculation).

The prints are made in the message file.

3.6. Operand TITRE#

◊ TITRE = title

Title associated with the table.

3.7. Table produced#

The POST_FATIGUE operator creates a table that is different depending on the post-processing calculations performed:

Uniaxial load (Wöhler, Manson-Coffin and Taheri methods).

The table includes five parameters:

NB_CYCL	:	number of elementary cycles extracted by the counting method,

VALE_MIN	:	values of the minimum stresses or deformations for each elementary cycle,

VALE_MAX	:	values of the maximum stresses or deformations for each elementary cycle,

DOMMAGE	:	damage values for each elementary cycle,

DOMM_CUMU	:	value of the total damage after accumulation over all elementary cycles.

Multiaxial load

The table includes all the parameters that make up the criteria used.

In addition, for all criteria, the table includes:

CRITERE	:	name of the criterion

VALE_CRITERE	:	value of the criterion (equivalent quantity)

NBRUP	:	Number of the cycle at break (associated with a cycle or a block of sub-cycles)

DOMMAGE	:	Wöhler damage value (if requested by the user).

For the Crossland and Dang Van-Papadopoulos criteria:

AMPLI_CISSION	:	Amplitude of the cission

PRES_HYDRO_MAX	:	value of the maximum hydrostatic pressure,

RAYON_SPHERE	:	the radius of the smallest sphere circumscribed to the load path in the space of the constraint deviators:math:

Any load (damage from Lemaître and Lemaître-Sermage).

The table includes two parameters:

DOMMAGE

damage value at each discretization point of the load,

The command IMPR_TABLE [U4.91.03] allows you to print the table produced.