3. Operands#

3.1. Keyword TYPE_CALCUL#

This keyword allows you to calculate

or a field of fatigue damage suffered by a structure, if TYPE_CALCUL = “CUMUL_DOMMAGE”;
be the critical plane in which the shear is maximum, if TYPE_CALCUL = “FATIGUE_MULTI”;
or the maximum vibration amplitude admissible by a structure subjected to a vibratory loading, if TYPE_CALCUL = “FATIGUE_VIBR”.

In the first two cases, we know the loading of the structure (temporal evolution of the stresses or deformations) and we are interested in the damage or the associated critical plane.

In the latter case, we know the static loading of the structure (typically the centrifugal forces for a turbine blade) but not the dynamic loading (typically the vibration of the blade). The option “FATIGUE_VIBR” then makes it possible to estimate the maximum vibration amplitude admissible by the structure in order to have unlimited endurance. The principle of the calculation is described in § 22.

3.2. Operands common to all options#

3.2.1. 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 to be subdued must contain the definition of the Wöhler curve of the material for the calculation of the damage by the methods” WOHLER “and” TAHERI_MIXTE “and the definition of the Manson-Coffin curve of the material for the calculation of the damage by the methods” MANSON_COFFIN “,” TAHERI_MANSON “and” TAHERI_MIXTE “for the calculation of the damage by the methods” “,” “and” “.

For “FATIGUE_VIBR” calculations, the material must also contain the breaking stress (operator DEFI_MATERIAU, keyword factor RCCM, operand SU).

3.2.2. Operand INFO#

◊ INFO =/1 No impression.

Printing of damage calculation parameters (number of order numbers, number of calculation points, number of calculation points, type of damage calculation (stresses, deformations), location of damage (knots or Gauss points), type of equivalent component (VMIS_SG or INVA_2SG), method of extracting cycles (or), method of extracting cycles (RAINFLOW), and method of calculating damage (WOHLER or MANSON_COFFIN or TAHERI_MANSON or TAHERI_MIXTE).

point by point of the loading history, the cycles extracted and the value of the damage.
of the damage field.

The prints are made in file MESSAGE.

3.3. Operands specific to type CUMUL_DOMMAGE calculation#

3.3.1. Keyword factor HISTOIRE#

This key word factor includes the entire phase of defining the loading history.

The loading history is the evolution of a value of stress or deformation over time.

3.3.1.1. Operand RESULTAT#

♦ RESULTAT = res

Name of the result concept containing the stress fields or deformation fields defining the loading history. More precisely, the result concept must contain one of the symbolic name fields SIEQ_ELNO, SIEQ_ELGA,, EPEQ_ELNO, EPEQ_ELGA, EPMQ_ELNO or EPMQ_ELGA depending on the desired calculation option.

3.3.1.2. Operand EQUI_GD#

♦ EQUI_GD =/'VMIS_SG',

/”INVA_2_SG”

To be able to calculate the damage suffered by a structure, by a Wöhler method, Manson-Coffin method or a Taheri method, it is necessary to have a history of loading in stresses or « uniaxial » deformations. To do this, it is necessary to transform the stress tensor or the deformation tensor into an « equivalent » uniaxial (scalar) field.

“VMIS_SG”	to calculate damage from a signed von Mises constraint loading history,
“INVA_2_SG”	to calculate the damage from a signed 2nd order invariant load history of the deformation.

3.3.2. Operand OPTION#

This factor keyword allows you to specify the type of damage to be calculated:

“DOMA_ELNO_SIGM” for calculating node damage from a stress field.
The result data structure specified under the keyword factor RESULTAT must contain the symbolic name field SIEQ_ELNO (computable by CALC_CHAMP), which among other things defines the value of the signed equivalent von Mises stress (component VMIS_SG) calculated at the nodes.
“DOMA_ELGA_SIGM” for calculating the damage to Gauss points based on a stress field.
The result data structure specified under the keyword factor RESULTAT must contain the symbolic name field SIEQ_ELGA (computable by CALC_CHAMP), which defines, among other things, the value of the signed equivalent von Mises stress (component VMIS_SG) calculated at Gauss points.
“DOMA_ELNO_EPSI” for calculating node damage from a deformation field.
The result data structure specified under the keyword factor RESULTAT must contain the symbolic name field EPEQ_ELNO, which among other things defines the value of the signed 2nd order invariant (component INVA_2SG) calculated at the nodes.
“DOMA_ELGA_EPSI” for calculating the damage at Gauss points from a deformation field.
The result data structure specified under the keyword factor RESULTAT must contain the symbolic name field EPEQ_ELGA, which among other things defines the value of the signed 2nd order invariant (component INVA_2SG) calculated at Gauss points.
“DOMA_ELNO_EPME” for calculating node damage from a field of mechanical, non-thermal deformations: \(\varepsilon \mathrm{=}B\mathrm{\cdot }u\mathrm{-}{\varepsilon }_{\mathit{th}}\).
The result data structure specified under the keyword factor RESULTAT must contain the symbolic name field EPMQ_ELNO (computable by CALC_CHAMP), which among other things defines the value of the signed 2nd order invariant (component INVA_2SG) calculated at the nodes.
“DOMA_ELGA_EPME” for calculating the damage at Gauss points from a field of mechanical, non-thermal deformations: \(\varepsilon \mathrm{=}B\mathrm{\cdot }u\mathrm{-}{\varepsilon }_{\mathit{th}}\).
The result data structure specified under the keyword factor RESULTAT must contain the symbolic name field EPMQ_ELGA, which among other things defines the value of the signed 2nd order invariant (component INVA_2SG) calculated at Gauss points.

3.3.3. Operand DOMMAGE#

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. The method available in Code_Aster for calculating damage by the Wöhler or Manson-Coffin method, is the cascading extent counting method or Rainflow method [R7.04.01].

To calculate the damage using methods TAHERI_MANSON and TAHERI_MIXTE, the so-called natural counting method is used, which consists in generating cycles in the order in which they are applied.

Once the elementary cycles have been extracted, this operand makes it possible to specify the method for calculating the damage for each elementary cycle.

♦ DOMMAGE =/'WOHLER'

For a stress-type loading history, the number of cycles at break is determined by interpolation of the Wöhler curve of the material for a given alternating stress level (to each elementary cycle there is a stress amplitude level \(\Delta \sigma \mathrm{=}∣{\sigma }_{\mathit{max}}\mathrm{-}{\sigma }_{\mathit{min}}∣\) and an alternating stress \({S}_{\mathit{alt}}\mathrm{=}1\mathrm{/}2\Delta \sigma\) corresponding to each elementary cycle).

You can only use the WOHLER method for the “DOMA_ELNO_SIGM” or “DOMA_ELGA_SIGM” options. In addition, the specified result concept must contain the symbolic name field SIEQ_ELNO or SIEQ_ELGA respectively (computable by CALC_CHAMP).

The Wöhler curve of the material must be introduced into the operator DEFI_MATERIAU [U4.43.01], in one of the three possible forms [R7.04.02]:

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

Note on fatigue curves:

For small amplitudes, the problem of extending the fatigue curve may arise: for example, for RCC -M fatigue curves beyond 106 cycles, the corresponding stress 180 MPa is considered to be an endurance limit, i.e. any stress less than 180 MPa must produce a zero use factor, or an infinite allowable number of cycles.

In Code_Aster, the endurance limit is set at 10 million 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 loading such as deformations, the number of cycles at break is determined by interpolation of the Manson-Coffin curve of the material for a given alternating deformation level (to each elementary cycle there is a deformation amplitude level \(\Delta \varepsilon =∣{\varepsilon }_{\mathrm{max}}-{\varepsilon }_{\mathrm{min}}∣\) and an alternating deformation \({E}_{\mathit{alt}}\mathrm{=}1\mathrm{/}2\Delta \varepsilon\) corresponding to each elementary cycle).

You can only use the MANSON_COFFIN method for the options” DOMA_ELNO_EPSI “or” “or” DOMA_ELGA_EPSI “,” DOMA_ELNO_EPME “, or” DOMA_ELGA_EPME “. In addition, the specified result concept must contain the symbolic name field EPEQ_ELNO, EPEQ_ELGA, EPMQ_ELNO, or EPMQ_ELGA respectively (computable by CALC_CHAMP).

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

This damage calculation method only applies to deformation-type loads, i.e. for options” DOMA_ELNO_EPSI “,” DOMA_ELGA_EPSI “,” DOMA_ELNO_EPME “, or” DOMA_ELGA_EPME “. In addition, the specified result concept must contain the symbolic name field EPEQ_ELNO, EPEQ_ELGA, EPMQ_ELNO, or EPMQ_ELGA respectively (computable by CALC_CHAMP).

Let’s say \(n\) elementary cycles of half amplitude \(\frac{\Delta {\varepsilon }_{1}}{\mathrm{2,}}\mathrm{...},\frac{\Delta {\varepsilon }_{n}}{2}\).

The calculation 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 determined by the algorithm described below:

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

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:

\(\frac{\Delta {\sigma }_{i+1}}{2}={F}_{\mathrm{NAPPE}}(\frac{\Delta {\varepsilon }_{i+1}}{2},\underset{j<i}{\mathrm{max}}(\frac{\Delta {\varepsilon }_{j}}{2}))\)

\(\frac{\Delta {\varepsilon }_{i+1}^{\text{*}}}{2}={F}_{\mathrm{FONC}}(\frac{\Delta {\sigma }_{i+1}}{2})\)

where \({F}_{\mathrm{NAPPE}}\) is a tablecloth introduced under the operand TAHERI_NAPPE.

\({F}_{\mathrm{FONC}}\) is a function introduced under the TAHERI_FONC operand.

The damage value of cycle \((i+1)\) is obtained by interpolation of \(\frac{\Delta {\varepsilon }_{i+1}^{\text{*}}}{2}\) on the Manson-Coffin curve of the material (\({\mathrm{Nrupt}}_{i+1}\) = number of cycles at break for cycle \((i+1)=\text{MANSON\_COFFIN}(\frac{\Delta {\varepsilon }_{i+1}^{\text{*}}}{2})\) and \({\mathrm{Dom}}_{i+1}\) = damage from cycle \((i+1)=\)).

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

Notes:

The sheet or the formula introduced under the operand * TAHERI_NAPPE is in fact the cyclic work hardening curve with pre-stressing the material.
The function or formula introduced under the operand * TAHERI_FONC is in fact the cyclic work hardening curve of the material.
The sheet or the formula introduced under the operand TAHERI_NAPPE, must have “X’and “ EPSI “as parameters.
The function or formula introduced under the operand TAHERI_FONC, must have the parameter “ SIGM “ .

♦ DOMMAGE =/'TAHERI_MIXTE'

Let’s say \(n\) elementary cycles of half amplitude \(\frac{\Delta {\varepsilon }_{1}}{\mathrm{2,}}\mathrm{...},\frac{\Delta {\varepsilon }_{n}}{2}\).

The calculation 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 determined 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:

\(\frac{\Delta {\sigma }_{i+1}}{2}={F}_{\mathrm{NAPPE}}(\frac{\Delta {\varepsilon }_{i+1}}{2},\underset{j<i}{\mathrm{max}}(\frac{\Delta {\varepsilon }_{j}}{2}))\)

where \({F}_{\mathrm{NAPPE}}\) is a tablecloth introduced under the operand TAHERI_NAPPE.

The damage value for cycle \((i+1)\) is obtained by interpolation of \(\frac{\Delta {\sigma }_{i+1}}{2}\) on the material’s Wöhler curve (\({\mathrm{Nrupt}}_{i+1}\) = number of cycles at break for cycle \((i+1)=\mathrm{WOHLER}(\frac{\Delta {\sigma }_{i+1}}{2})\) and \({\mathrm{Dom}}_{i+1}\) = damage from cycle \((i+1)=\)).

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

Notes:

The sheet or the formula introduced under the operand * TAHERI_NAPPE is in fact the cyclic work hardening curve with pre-stressing the material.
The sheet or the formula introduced under the operand TAHERI_NAPPE, must have “X’and “ EPSI “as parameters.

3.3.4. Operand TAHERI_NAPPE#

This operand allows you to specify the name of a \({F}_{\mathit{NAPPE}}(\frac{\Delta \varepsilon }{2},{\varepsilon }_{\mathit{MAX}})\) sheet required to calculate the damage using the “TAHERI_MANSON” and “TAHERI_MIXTE” methods.

The tablecloth should have “X” and “EPSI” as parameters.

Note:

This sheet is in fact the cyclic work hardening curve with pre-stressing the material.

3.3.5. Operand TAHERI_FONC#

This operand allows you to specify the name of a \({F}_{\mathrm{FONC}}(\frac{\Delta \sigma }{2})\) function required to calculate the damage using the “TAHERI_MANSON” method.

The parameter for this function should be “SIGM”.

Note:

This function is in fact the cyclical work-hardening curve of the material.

3.4. Operands specific to type FATIGUE_MULTI calculation#

3.4.1. 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.

3.4.2. Operand OPTION#

This operand allows you to specify the place where the post-processing will be done:

DOMA_ELGA, post-processing is done at the Gauss points of the mesh;
DOMA_NOEUD, post-processing is done at the nodes of the mesh or part of the mesh, cf. operands: GROUP_MAetGROUP_NO.

3.4.3. Operand RESULTAT#

♦ RESULTAT = res

Name of the result concept containing the stress and deformation fields defining the loading history. More specifically, the result concept should contain the symbolic name field

SIEF_ELGA, EPSI_ELGA, EPSP_ELGA are the stress, total deformation, and plastic deformation fields, respectively, for the fatigue calculation at the fields at the elements
SIGM_NOEU/SIEF_NOEU, EPSI_NOEU, EPSP_NOEU are the stress, total deformation, and plastic deformation fields, respectively, for the calculation of fatigue at the fields at the elements

The criterion is first analyzed. Depending on the criteria parameters, the fields above are requested.

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.

3.4.4. 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].

3.4.5. Operand CRITERE#

♦ CRITERE =/'MATAKE_MODI_AC',

/'DANG_VAN_MODI_AC',

/'MATAKE_MODI_AV',

/'DANG_VAN_MODI_AV',

/'FATESOCI_MODI_AV',

/'FORMULE_CRITERE',

/'VMIS_TRESCA',

Note:

*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 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”	“MATAKE_MODI_AV”, “DANG_VAN_MODI_AV” “” FATESOCI_MODI_AV “ “ FORMULE_CRITERE “

For constant amplitude loading, the operand CRITERE 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 for 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 \({\sigma }_{\mathrm{eq}}({\text{n}}^{\text{*}})\):

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

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]:

\(\frac{\Delta \tau }{2}({\text{n}}^{\text{*}})+aP\le b\) 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:

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

with \(\Delta {\sigma }_{1}\) the loading amplitude for the alternating case \(({\sigma }_{m}=0)\) \(\Delta {\sigma }_{2}\) 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 \({\sigma }_{\mathrm{eq}}({\text{n}}^{\text{*}})\):

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

where \(c/t\) 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 the \({\tau }_{\mathrm{p2}}^{i}(\text{n})\) s values of the projected shear stresses of the subcycle \(i\) and \({N}_{1}^{i}(\text{n})\) and the \({N}_{2}^{i}(\text{n})\) s 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:

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

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 \({N}_{\mathrm{max}}\):

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

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:

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

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 \({\gamma }_{p}\mathrm{=}f(t)\). After extracting the elementary subcycles with method RAINFLOW we define an elementary equivalent deformation for any elementary subcycle \(i\):

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

with \(a=\frac{k}{{\sigma }_{y}}\), \(\text{n}\) the normal to the current plane, \({\gamma }_{\mathrm{p1}}^{i}(\text{n})\) 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 \({\varepsilon }_{\mathit{eq}}^{i}(\text{n})\) 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:

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

Criterion FORMULE_CRITERE

This type of criterion allows the user to build a criterion as a formula for predefined 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_COFFIN” defined previously in DEFI_MATERIAU.

Criterion VMIS_TRESCA

Criterion VMIS_TRESCA is not strictly speaking a fatigue criterion since it does not allow damage to be calculated. It determines the maximum amplitude variation of the stress tensor over time. Concretely, we apply the Von Mises and Tresca criteria to tensors that result from the difference in the stress tensor taken at two distinct moments. By varying these moments we can calculate the maximum values of the Von Mises and Tresca criteria [R7.04.04].

3.4.6. 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 (\(\Delta \tau (\text{n*})\mathrm{/}2\))” 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:`{\varepsilon }_{\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 {\varepsilon }_{\mathit{eq}}^{p}\mathrm{/}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}}\mathrm{-}{\sigma }_{\mathit{min}}^{\mathit{Tresca}})\mathrm{/}4`)

'DEPTRE':demi-amplitude de la demi-déformation Tresca (:math:`({\epsilon }_{\mathit{max}}^{\mathit{Tresca}}\mathrm{-}{\epsilon }_{\mathit{min}}^{\mathit{Tresca}})\mathrm{/}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 \({\gamma }_{\mathit{max}}(\text{n})\) MSINCR 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 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'

Available quantities:

“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 “:projected deformation shear from the first top of the sub-cycle (\({\gamma }_{\mathit{p1}}(\text{n})\)) “:hydrostatic pressure from the first top of the sub-cycle ()” EPSPR_2 “:projected deformation shear of 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:`{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 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., \({ϵ}^{\mathit{tot}}={ϵ}^{e}+{ϵ}^{p}\) .

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

7) For periodic loading, the equivalent quantity evaluation is output under the name “SIGEQ1”.

3.4.7. Operand COURBE_GRD_VIE#

♦ COURBE_GRD_VIE =/”WOHLER”,

/”MANSON_COFFIN”, /”FORM_VIE”

Allows you to provide a curve relating the equivalent quantity to the number of cycles at breakage.

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

If COURBE_GRD_VIE = “MANSON_COFFIN”, we will take the Manson_Coffin curve (\({N}_{f}\mathrm{=}f(\mathit{EPSN})\)) defined in AFFE_MATERIAU.

If COURBE_GRD_VIE = “FORM_VIE”, we will provide a function defining the life curve, see 3.4.8.

3.4.8. 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:

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

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

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

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

3.4.9. 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.

3.4.10. 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].

3.4.11. Operand INST_INIT_CYCL#

* INST_INIT_CYCL = /inst_ini_cyc

Allows you to specify the initial time of the part of the cyclic loading. If this operand is not specified or inst_ini_cyc is not part of the calculated times, the initial value stored in the result is taken as the initial moment of the cycle. This operand also allows users to apply a non-zero mean loading.

3.4.12. Operand INST_CRIT#

* INST_CRIT = /” RELATIF “

/”ABSOLU”

Allows you to specify the criteria to search for the initial moment INST_INIT_CYCL

3.4.13. Operand PRECISION#

◊ PRECISION =/prec [R]

/1.E-6,

Allows you to specify the precision of the initial moment INST_INIT_CYCL

3.4.14. 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.

3.4.15. 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.

3.4.16. Operands GROUP_MA GROUP_NO#

◊ GROUP_MA = lgma,

The options are calculated based on the mesh groups contained in the lgma list.

◊ GROUP_NO = lgno,

The options are calculated based on the node groups contained in the lgno list.

3.4.17. Operand COEF_PREECROU#

◊ COEF_PREECROU =/coef_pre,

/1.0,

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

3.4.18. Operand MAILLAGE#

♦ MAILLAGE = mesh,

Allows you to specify the mesh name given by the user.

3.5. Operands specific to type FATIGUE_VIBR calculation#

3.5.1. Principle of calculation#

This option does not aim to calculate the damage associated with a known loading, but conversely to estimate the maximum vibratory load associated with unlimited endurance to the structure under study. The structures in question are typically the blades, stressed by a known static loading (centrifugal force linked to the rotation of the machine) and by an unknown or poorly known dynamic loading (vibrations induced by the flow of the fluid).

A fundamental hypothesis of this option is to consider a uniaxial fatigue criterion (Wöhler method). In other words, it is assumed that the main directions of static loading and dynamic loading are the same. This hypothesis seems legitimate for the usual structures referred to (fins, pipe lines,…); it induces a conservatism that is undoubtedly excessive in the general case.

The approach for a study with this option is as follows:

Calculation of static load stress \({\sigma }_{\mathrm{stat}}\) with MECA_STATIQUE or STAT_NON_LINE;
Calculation of the constraints associated with the \(N\) eigenmodes considered \({\sigma }_{\mathrm{mod}}^{i}\) with CALC_MODES;
Fatigue calculation with CALC_FATIGUE/TYPE_CALCUL = “FATIGUE_VIBR”
Introduction of a hypothesis on the relative weight of the various eigenmodes considered \({({\beta }_{i})}_{1\le i\le N}\) (corresponds to the FACT_PARTICI operand):
- - - \({\sigma }_{\mathrm{total}}(t)={\sigma }_{\mathrm{stat}}+\alpha \sum _{i=1}^{N}{\beta }_{i}{\sigma }_{\mathrm{mod}}^{i}\mathrm{cos}({\omega }_{i}t+{\phi }_{i})\),

where \({\omega }_{i}\) and \({\phi }_{i}\) are respectively the pulsation (known) and the phase shift (unknown) of mode i. The coefficient \(\alpha\) is the parameter we are trying to calculate;

Retrieval of material parameters and choice of the damage calculation criterion (operands CORR_SIGM_MOYE and MATER, cf. § 24). We note \(f\) the criterion that must be verified by the maximum amplitude of variation of the stress \({S}_{\mathrm{alt}}^{\mathrm{max}}\). \(f\) depends on the endurance limit \({S}_{l}\) and the breaking limit \({S}_{u}\) of the material:
\({S}_{\mathrm{alt}}^{\text{max}}=f({\sigma }_{\mathrm{stat}},{S}_{l},{S}_{u})\)
On all nodes or Gauss points in the mesh (depending on the choice in OPTION):
Calculation of the amplitude of variation of the stresses: \({S}_{\mathrm{alt}}=\alpha \sum _{i=1}^{N}{\beta }_{i}{\sigma }_{\mathrm{mod}}^{i}\) (note that, not knowing the phase differences between the modes, the amplitude is defined conservatively as the sum of the amplitude of each of the modes);
Calculation of the \(\alpha\) coefficient corresponding to unlimited endurance: \(\alpha \mathrm{=}\frac{f({\sigma }_{\mathit{stat}},{S}_{l},{S}_{u})}{\mathrm{\sum }_{i\mathrm{=}1}^{N}{\beta }_{i}{\sigma }_{\mathit{mod}}^{i}}\)
Interpretation and use of the result of CALC_FATIGUE: the operator provides the field (at the nodes or at the Gauss point) of the admissible values of \(\alpha\): the minimum value of \(\alpha\) on the mesh makes it possible to calculate the maximum allowable amplitude of vibration of the structure (the minimum value is displayed in the message file; it can also be found by post-processing or viewing the result field); the field makes it possible to locate the zones that limit the vibration of the structure (the minimum value is displayed in the message file; it can also be found by post-processing or viewing the result field); the field allows you to locate the zones that limit the life of the structure.

To pass from the coefficient \(\alpha\) to the admissible vibration amplitude at a given point \(\partial \tilde{u}\) (corresponding, for example, to the position of a sensor), an additional operation must be performed. We note \({\tilde{u}}_{\mathrm{mod}}^{i}\) the displacement at the point of interest associated with the \(i\) mode; the admissible vibration amplitude at this point is then:

\(\partial \tilde{u}=\mathrm{min}(\alpha )\sum _{i=1}^{N}{\beta }_{i}{\tilde{u}}_{\mathrm{mod}}^{i}\)

Note:

If the static stress exceeds the breaking stress of the material by one node, the allowable vibration amplitude is zero. In this case, an alarm message is sent and the calculation continues on the other nodes.

3.5.2. Keyword factor HISTOIRE#

This factor keyword includes the loading definition phase: static constraint (operand RESULTAT); modal constraints (MODE_MECA); number of the mode (s) to be considered (NUME_MODE); relative weight of each of its modes (FACT_PARTICI).

3.5.2.1. Operand RESULTAT#

♦ RESULTAT = res

Name of the result concept containing the constraints field associated with the static loading of the structure (a single time step). More precisely, the result concept must contain one of the symbolic name fields SIEQ_ELNO or SIEQ_ELGA depending on the desired calculation option.

3.5.2.2. Operand MODE_MECA#

♦ MODE_MECA = fashion

Name of the mode_meca concept, containing the constraint fields for the structure’s own modes.

More precisely, the result concept must contain one of the symbolic name fields SIEQ_ELNO or SIEQ_ELGA depending on the desired calculation option. These fields are calculated with the CALC_CHAMP operator, in post-processing an eigenmode calculation with CALC_MODES.

3.5.2.3. Operand NUME_MODE#

♦ NUME_MODE = list_I

Number of the method (s) to be considered for calculating the damage.

3.5.2.4. Operand FACT_PARTICI#

♦ FACT_PARTICI = list_R

Relative weight of each of the modes to be considered. The length of the list must be the same as the length of the one entered under the NUME_MODE operand.

Only the relationship between the various factors provided is important. If we want to go from the parameter calculated by CALC_FATIGUE to a maximum amplitude of movement at a given node, it is however necessary to take the same coefficients into account (see § 22).

3.5.3. Operand OPTION#

This keyword factor allows you to specify the place where the damage was calculated:

“DOMA_ELNO_SIGM” for calculating node damage from a stress field.

The static and modal results (operands RESULTAT and MODE_MECA) must contain the symbolic name field SIEQ_ELNO (computable by CALC_CHAMP), which defines, among other things, the value of the signed equivalent von Mises constraint (component VMIS_SG) calculated at the nodes.

“DOMA_ELGA_SIGM” for calculating the damage to Gauss points based on a stress field.

The static and modal results (operands RESULTAT and MODE_MECA) must contain the symbolic name field SIEQ_ELGA (computable by CALC_CHAMP), which defines, among other things, the value of the signed equivalent von Mises stress (component VMIS_SG) calculated at Gauss points.

3.5.4. Operand CORR_SIGM_MOYENNE#

♦ CORR_SIGM_MOYE =/'GOODMAN',

/”GERBER”,

The structure is subjected to a loading with a non-zero mean stress, the mean stress corresponding to the static stress.

The average stress \({\sigma }_{m}\) can be taken into account in the Wöhler fatigue curve using the Haigh diagram [R7.04.01]. Two corrections are available to calculate the permissible alternating stress \({S}_{\mathrm{alt}}^{\text{max}}\) as a function of the endurance limit \({S}_{l}\) and the breaking limit \({S}_{u}\) of the material:

Goodman’s right:

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

Gerber’s parable:

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

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). The endurance limit \({S}_{l}\) corresponds to the first point of the Wöhler curve (operator DEFI_MATERIAU, keyword FATIGUE, operand WOHLER).

3.5.5. Operand DOMMAGE#

♦ DOMMAGE =/'WOHLER'

For the moment, only the Wöhler method is available for vibration fatigue calculations. This method is based on calculating the amplitude of stress variation and comparing the material to the Wöhler fatigue curve.

The Wöhler curve of the material must be introduced into operator DEFI_MATERIAU (keyword FATIGUE, operand WOHLER). Only the endurance limit \({S}_{l}\) (i.e. the first point on the curve) is actually used in the calculation.