2. The Bordet model

The Bordet model is briefly presented here. For more details, we can refer to [3], [4] or [5].

Bordet’s model is based on the same foundations as Beremin’s. It defines a probability of rupture by local cleavage. However, in the Beremin model, the creation of microcracks is assumed when the plasticity threshold is reached, and these microcracks remain potentially active throughout the subsequent loading. However, in steels, the overall failure is mainly linked to newly created microcracks. It is therefore necessary to take into account the level of plastic deformation reached at each moment. This is already possible in Beremin’s model via the plastic correction option defined in doc [U4.81.22].

In the Bordet model, this is taken into account by considering that the probability of rupture by cleavage is the product of the probability of nucleation and propagation at the same time.

2.1. Probability of local rupture by cleavage of the Bordet model

The local criterion is here defined as the statistical event of simultaneously meeting the conditions for nucleation and propagation of microcracks. In the model, the nucleation and propagation of microcracks are considered to be independent events; nucleation refers to the failure of a carbide leading to the formation of a microcrack, while propagation is defined as the local cleavage instability guided only by local stresses.

The probability of local rupture by cleavage is then written as:

(2.1)\[ {P} _ {\ mathit {cliv}} = {P}} _ {\ mathit {nucl}} {P} _ {\ mathit {propa}}\]

By plastic deformation \({\varepsilon }_{p}\) we mean the equivalent plastic deformation defined by:

(2.2)\[ {\ mathrm {\ epsilon}} _ {p} =\ sqrt {\ frac {2} {3} {\ mathrm {\ epsilon}} _ {p}\ mathrm {::} {\ mathrm {\ epsilon}} _ {p}}\]

This definition is valid only for elastoplastic behavior laws based on the von Misès criterion. In code_aster, only laws of behavior of this type are therefore adapted to this model.

2.1.1. Local probability of nucleation

With the definition given above, the probability of nucleation of a microcrack during a plastic deformation increment \(d{\mathrm{\epsilon }}_{p}\) is proportional to the elastic limit \({\mathrm{\sigma }}_{\mathit{ys}}(T,{\dot{\mathrm{\epsilon }}}_{p})\) at temperature \(T\) and the plastic deformation rate \({\dot{\varepsilon }}_{p}\):

(2.3)\[ {P} _ {\ mathit {nucl}}\ propto {N}} _ {\ mathit {unc}} ({\ mathrm {\ epsilon}} _ {p}) {\ mathrm {\ sigma}}} _ {\ sigma}}} _ {\ sigma}}} _ {\ sigma}}} _ {\ sigma}}} _ {\ sigma}}} _ {\ sigma}}} _ {\ sigma}}} _ {\ sigma}}} _ {\ sigma}}} _ {\ sigma}}} _ {\ sigma}}} _ {sigma}}} _ {sigma}}} _ {sigma}}} _ {sigma}}} _ {sigma}}} _ {sigma}}} _ {sigma}}} _ {sigma}}} _ {sigma}}} _ m {\ epsilon}} _ {p}\]

This remains true as long as the number of microfissured carbides remains low compared to the number of healthy carbides \({N}_{\mathrm{unc}}({\varepsilon }_{p})\). The rate of microcracking of carbides is constant for a given plastic deformation; the number of healthy sites varies exponentially with \({\varepsilon }_{p}\). By calling \({\mathrm{\sigma }}_{\mathit{ys}\mathrm{,0}}\) the elastic limit at a reference temperature and plastic deformation rate and \({\varepsilon }_{p\mathrm{,0}}\) a reference plastic deformation, the probability \({P}_{\mathrm{nucl}}\) can be expressed as follows:

\[\]

: label: eq-4

{P} _ {mathit {nucl}}proptofrac {{mathrm {sigma}} _ {mathit {ys}} (T, {dot {mathrm {epsilon}}}}} _ {epsilon}}}} _ {mathrm {epsilon}}}} _ {mathrm {epsilon}}}} _ {mathrm {epsilon}}}} _ {mathrm {epsilon}}}} _ {mathrm {epsilon}}}} _ {mathrm {epsilon}}}} _ {mathrm {epsilon}}}} _ {p})} {{mathrm {sigma}} m {exp}left (frac {- - {mathrm {sigma}}} _ {mathit {ys}} (T, {dot {mathrm {epsilon}}}}} _ {p})}} {{mathrm {sigma}}}} {{mathrm {mathrm}}}frac {{mathrm {mathrm {sigma}}} _ {mathrm {sigma}}}frac {mathrm {mathrm}}epsilon}} _ {p}} {{mathrm {epsilon}} _ {pmathrm {,0}}}}}right)

If the plastic deformation is small, so if the cracking of carbides is quite limited, i.e. \(\frac{{\mathrm{\sigma }}_{\mathit{ys}\mathrm{,0}}{\mathrm{\epsilon }}_{p\mathrm{,0}}}{{\mathrm{\sigma }}_{\mathit{ys}}(T,{\dot{\mathrm{\epsilon }}}_{p})}\gg {\mathrm{\epsilon }}_{p}\), then the probability of nucleation is reduced to \({P}_{\mathit{nucl}}\propto \frac{{\mathrm{\sigma }}_{\mathit{ys}}(T,{\dot{\mathrm{\epsilon }}}_{p})}{{\mathrm{\sigma }}_{\mathit{ys}\mathrm{,0}}}\).

2.1.2. Local probability of spread

As in the Beremin model [R4.02.04], the size density of carbide microcracks is assumed to be distributed according to an inverse power law (with \(\alpha\) and \(\beta\) material parameters independent of temperature and deformation rates and \(l\) the size of the microcrack):

\[\]

: label: eq-5

f (l) =frac {alpha} {{alpha}} {{l} ^ {beta}}

The cracking of a carbide will only propagate in the ferrite (due to the inertia of propagation alone, by dynamic effect) only under the condition of the presence of a sufficiently high local stress. An upper limit for the size of nucleated ferritic microcrack \({l}_{\mathit{max}}=l\left({\mathrm{\sigma }}_{\mathit{th}}\right)\) is therefore introduced; it makes it possible to define the minimum local stress necessary for the propagation of the largest ferritic microcrack possible. The probability of propagation is therefore written as:

(2.4)\[ {P} _ {\ mathit {propa}}} ({\ mathrm {\ sigma}}} _ {I}) = {\ int} _ {l} _ {c} ({\ mathrm {\ sigma}}} _ {I}} _ {I})}} ^ {{l}} _ {\ mathit {max}}}} f (l)\ mathit {dl}} (l)\ mathit {dl}}\]

With \({\mathrm{\sigma }}_{I}\) the maximum principal stress, \({\sigma }_{\mathrm{th}}\) the threshold below which propagation cannot take place and \({l}_{c}\) the critical ferritic microcrack size, obeying the Griffith relationship for an elliptical microcrack:

(2.5)\[ {l} _ {c}\ left (\ mathrm {\ sigma}\ right) =\ frac {\ mathrm {\ pi} E {\ mathrm {\ gamma}} _ {p}} {2}} {2\ left (1- {\ mathrm {\ sigma}}} {2}\ right) {\ mathrm {\ sigma}} {p}}} {2}} {2}} {2}} {2}}\]

With \(E\) the Young’s modulus, \(\mathrm{\nu }\) the Poisson’s ratio and \({\mathrm{\gamma }}_{p}\) the surface energy density. We finally get:

(2.6)\[\begin{split} \ {\ begin {array} {ccc} {P} _ {\ mathit {propa}}} ({\ mathrm {\ sigma}} _ {I}) &\ text {=} 0& {\ mathrm {\ sigma}} {\ sigma}} _ {\ mathit {sigma}} _ {\ mathit {th}}}\\ {P} 0& {\ mathrm {\ sigma}} _ {\ mathit {propa}} _ {\ mathit {propa}} _ {\ mathit {propa}}} ({\ mathrm {\ sigma}} _ {I}}) &\ text {\ I}) &\ text {=}) &\ text {=}\ left (\ frac {{\ mathrm {\ sigma}}} _ {\ sigma}}} _ {sigma}}} _ {\ sigma}} _ {\ mathit {th}}} {{\ mathrm {\ sigma}}} _ {u}}\ right)} ^ {m}\ right) & {\ mathrm {\ sigma}} _ {I} {\ mathrm {\ sigma}}} _ {\ mathit {th}}}\ end {array}} _ {\ mathit {th}}}\ end {array}\end{split}\]

With \(m\) parameter independent of temperature, like \(\alpha\) and \(\beta\), like and, and \({\sigma }_{u}\) which can depend on it (if Young’s modulus depends on it).

As with the Beremin model, the effects of the orientation of the microcracks in relation to the direction of the maximum principal stress are only taken into account via the parameter \({\sigma }_{u}\).

2.1.3. Local probability of cleavage

As specified above, the local probability of cleavage is assumed to be the product of the probability of nucleation and the probability of propagation; for this purpose, it is considered that during an infinitesimal plastic deformation increment, the active stress is constant. Where, if \({\mathrm{\sigma }}_{I}⩾{\mathrm{\sigma }}_{\mathit{th}}\):

(2.7)\[\begin{split} \ begin {array} {ccc} {P} _ {\ mathit {cliv}}}\ left ({\ mathrm {\ sigma}} _ {I}, d {\ mathrm {\ epsilon}}} _ {p}\ right) _ {p}\ right) &\ propto & {P}}\ right) &\ propto & {P}} _ {\ mathit {nucl}}\ left (d {\ mathrm {\ epsilon}}} _ {p}\ right)\ mathrm {.} {P} _ {\ mathit {propa}}}\ left ({\ mathrm {\ sigma}} _ {I}\ right)\\\ text {} &\ propto &\ frac {{\ mathrm {\ sigma}}}\ left (T, {\ dot {\ mathrm {\ epsilon}}}\ frac {{\ mathrm {\ sigma}}} _ {\ sigma}}} _ {\ sigma}} _ {\ sigma}}} _ {p}\ right)} {{\ mathrm {\ sigma}} _ {\ mathit {ys}}\ mathrm {,0}}}\ mathrm {exp}\ left (\ frac {- {\ mathrm {\ sigma}}} _ {\ mathit {\ sigma}}} _ {\ sigma}} _ {\ sigma}}} _ {\ sigma}} _ {\ sigma}} _ {\ sigma}} _ {\ sigma}} _ {\ sigma}} _ {\ sigma}} _ {\ sigma}} _ {\ sigma}} _ {\ sigma}}} _ {\ sigma}} _ {\ sigma}}} _ {\ sigma}}} _ {\ sigma}}} _ {\ sigma}}} _ {\ sigma}}} _ {\ sigma}}} _ {\ sigma}}}\ mathrm {\ sigma}} _ {\ mathit {ys}\ mathrm {ys}}\ mathrm {,0}}}\ frac {{\ mathrm {\ epsilon}} _ {\ mathrm {\ epsilon}}} _ {p\ mathrm {\ epsilon}}} _ {p\ mathrm {\ epsilon}} _ {\ mathrm {\ epsilon}} _ {\ mathrm {\ epsilon}} _ {\ mathrm {\ epsilon}} _ {\ mathrm {\ epsilon}} _ {\ mathrm {\ epsilon}} _ {\ mathrm {\ epsilon}} _ {\ mathrm {\ epsilon}} _ {\ mathrm {} _ {I}\ left ({\ mathrm {\ epsilon}}} _ {p}\ right)} {{\ mathrm {\ sigma}} _ {u}}\ right)}} ^ {m} - {m} - {\ m} - {\ left (\ frac {\ mathrm {\ mathrm {\ right)}}} ^ {m}} - {m} - {m} - {\ left (\ frac {{\ mathrm {\ sigma}}} _ {u}}\ right)} ^ {m}}\ right sigma}} _ {\ mathit {th}}}}} {{\ mathrm {\ sigma}}} _ {u}}\ right)} ^ {m}\ right) d {\ mathrm {\ epsilon}}} _ {p}\ end {array}\end{split}\]

This equation does not indicate that the nucleated microcracks remain active during the plastic deformation increment, but that the propagation condition for each of these microcracks is determined by the value of the local stress field at the time of creation.

The probability that a ferritic microcrack is created and propagated over a plastic deformation interval \(\left[\mathrm{0,}{\mathrm{\epsilon }}_{p,u}\right]\) is then:

(2.8)\[ {P} _ {\ mathit {cliv}}\ left ({\ mathrm {\ sigma}} _ {I}, {\ mathrm {\ epsilon}} _ {p, u}\ right)\ propto {\ int}}\ right)\ propto {\ int} _ {\ int} _ {\ int} _ {\ mathrm {\ mathrm {\ epsilon}} _ {p, u}\ right)\ propto {\ int} _ {\ mathrm} _ {\ mathrm {\ mathrm {\ epsilon}} _ {p, u}\ right)\ propto {\ int} _ {\ mathrm {\ mathrm {\ mathrm {\ epsilon}}} _ {p, u}\ right)\ propto sigma}} _ {\ mathit {ys}}}\ left (T, {\ dot {\ mathrm {\ epsilon}}}} _ {p}\ right)} {{\ mathrm {\ sigma}}} _ {\ mathit {ys}}\ left (\ mathrm {\ sigma}}}}\ mathrm {\ exp}\ left (\ frac {-\ sigma}}} _ {\ mathrm {\ sigma}}} _ {\ mathrm {\ sigma}} _ {\ mathit {ys}}\ left (T, {\ dot {\ mathrm {\ epsilon}}}} _ {p}\ right)} {{\ mathrm {\ sigma}}} _ {\ mathit {ys}}\ mathrm {ys}}\ mathrm {,0}}}}\ frac {{\ mathrm {\ epsilon}}} _ {\ mathrm {\ epsilon}} _ {p}}} _ {\ mathrm {ys}}\ mathrm {ys}}\ mathrm {,0}}}}\ frac {{\ mathrm {\ epsilon}}} _ {\ mathrm {\ epsilon}} _ {\ mathrm {\ sigma}}} _ {\ mathrm {epsilon}} _ {p\ mathrm {,0}}}}\ right)}\ right)\ left ({\ left (\ frac {{\ mathrm {\ sigma}} _ {I}\ left ({\ mathrm {\ epsilon}}}} _ {p}\ right)} {{\ mathrm { \ sigma}} _ {u}}\ right)}}} ^ {m}} - {\ m} - {\ left (\ frac {{\ mathrm {\ sigma}}}}} {{\ mathrm {\ sigma}}}} _ {\ sigma}}}} _ {\ sigma}}} _ {p}} _ {p}}\]

This probability is reduced to \(0\) if the constraint remains less than \({\sigma }_{\mathrm{th}}\) throughout the loading path.

2.2. Overall probability of rupture by Bordet cleavage

The preceding paragraph made it possible to determine the local probability of rupture by cleavage. Following the weak-link principle, the overall probability of breakage by cleavage at \({n}_{c}\) potential initiation sites is written as:

(2.9)\[ {P} _ {\ mathit {Bordet}}}\ text {=}}\ text {=} 1-\ mathrm {exp}\ left (-\ sum _ {=} 1} ^ {{n} _ {c}}} {P}} {P}} {P}} {P} _ {\ mathit {cliv}}}\ left ({\ mathrm {\ sigma}} _ {I, i}, {\ mathrm {\ epam}} _ {i, i}, {\ c}}} {P}} {P}} {P} _ {\ mathrm {\ ep. silon}} _ {p, u, i}\ right)\ right)\]

This equation can be expressed as a function of the volume of the process zone by introducing an infinitesimal volume \(\mathrm{dV}\) on which the deformations and the stresses are constant (in the numerical case, the Gauss point). In order to simply compare the Beremin and Bordet probabilities on a given example, we can define a Bordet constraint of the same type as that of Weibull:

(2.10)\[ {\ mathrm {\ sigma}} _ {\ mathit {Bordet}}\ text {=} {\ left\ {{\ int} _ {v} _ {p}}\ left ({\ int} _ {0} _ {0}} ^ {{\ mathrm {\ Bordet}}} ^ {{\ mathrm {\ Bordet}}} ^ {{\ mathrm {\ Bordet}}}\ text {=}}\ text {=}} {\ mathrm {\ epsilon}}}\ text {=}} {\ mathrm {\ epsilon}}} _ {\ mathit {ys}}\ left (T, {\ dot {\ mathrm {\ epsilon}}}} _ {p}\ right)} {{\ mathrm {\ sigma}} _ {\ mathit {ys}}\ mathrm {,0}}}\ mathrm {,0}}}}}\ left ({\ left ({\ mathrm {\ sigma}}} _ {I}\ left ({\ mathrm {\ epsilon}} _ {I}\ left ({\ mathrm {\ epsilon}}}} _ {p},\ mathit {dV}\ right)\ right)\ right)}\ right)}}} ^ {m} - {\ left ({\ mathrm {\ sigma}}}\ right)} ^ {m}\ right)} ^ {m}\ right)}\ right)\ right)\ right)\ mathrm {exp}\ right)}\ right)\ mathrm {exp}\ left (\ frac {- - {\ mathrm {\ sigma}}} _ {\ mathit {ys}}} _ {\ mathit {ys}}\ left (T, {\ dot {\ mathrm {\ epsilon}}}}} _ {p}\ right)} {{\ mathrm {\ sigma}} _ {\ mathit {ys}\ mathrm {,0}}}}\ mathrm {,0}}}}\ frac {{\ mathrm {\ epsilon }} _ {p}} {{\ mathrm {\ epsilon}}} _ {p\ mathrm {0}}}\ right) d {\ mathrm {\ epsilon}} _ {p}\ right)\ frac {\ mathit {dv}}} {\ mathit {dv}}} {{v}}}\ right\}} ^ {\ frac {1}\ right}}\ right)\ frac {1}\ right)\ frac {1} {m}}}\]

and the probability of global breakup by cleavage is written with a Weibull distribution:

(2.11)\[ {P} _ {\ mathrm {Bordet}}}\ text {=}}\ text {=}} 1-\ mathrm {exp} (- {\ frac {{\ sigma}} _ {\ mathrm {Bordet}}}}}} {{\ sigma}}}} {\ sigma}}} {\ sigma}}} {\ sigma}}} {\ sigma}}} {\ sigma}}} {\ sigma} _ {u}})} ^ {m})\]

If the number of microfissured carbides is low compared to the number of healthy carbides, the exponential term is close to \(1\) and the Bordet constraint is thereby simplified.

2.3. Discussion

2.3.1. Bordet or Beremin?

The Bordet model is slightly more complex and finer than the Beremin model.

One of its advantages is to consider the maximum principal stress at all times, and not the maximum principal stress during loading; therefore nothing prevents Bordet’s probability from decreasing, unlike that of Beremin.

In addition, the Bordet model accounts for the fact that a region with lower stress but greater plastic deformation may be more critical than an area in which the stresses are higher but the level of plastic deformation is lower.

However, the Bordet model requires the knowledge of additional material parameters, as will be described in the following paragraph.

2.3.2. Material parameters

The usual Beremin model requires the knowledge of three material parameters: the two shape parameters of the Weibull law, \(m\) and \({\sigma }_{u}\), as well as the elementary volume of the plastic zone \({V}_{0}\); only the parameter \({\sigma }_{u}\) depends on the temperature. To these three parameters, it is possible to add the threshold plastic deformation making it possible to define the plastic zone on which the integration is carried out.

The first three parameters are formally retained by the Bordet model.

Note: they are only formally preserved, they may be different and require a calibration of the same type as that done for Beremin and shown in [R7.02.09].

Other parameters are added.

The plasticity threshold \({\mathrm{\sigma }}_{\mathit{ys}}(T,{\dot{\mathrm{\epsilon }}}_{p})\) is an a minima function of temperature and potentially of the plastic deformation rate and its reference value \({\mathrm{\sigma }}_{\mathit{ys}\mathrm{,0}}\).

The critical stress below which the propagation of ferritic microcracks cannot take place, \({\sigma }_{\mathrm{th}}\), independent of loading conditions.

In the full version only (with the term exponential taken into account), a plastic deformation of reference \({\varepsilon }_{p\mathrm{,0}}\) whose identification method is not given and which seems quite delicate. It should be noted that the author himself (cf. [8], [9]) seems to use for certain studies and validation the version of the model in which this parameter does not intervene (for the code_aster user, all that is needed is to specify PROBA_NUCL =” NON “)

2.4. Implemented in code_aster

The calculation of the probability and the Bordet constraint is performed by the operator POST_BORDET. It requires having carried out an elastoplastic thermomechanical calculation via the operator STAT_NON_LINE. Tips for using this model are given in the documentation [U2.05.08].

In the currently implemented version, the temperature of the medium on which the calculation is carried out must be uniform (but may change over time); this limitation does not seem prohibitive insofar as the plastic zone at the crack point is generally quite small.

It is possible to perform the calculation with or without the term in exponential; if this term is requested, the material parameter \({\varepsilon }_{p\mathrm{,0}}\) must be entered.

In all cases, quantities such as the maximum principal stress and the equivalent plastic deformation are calculated at all times. The value of the parameters \({\sigma }_{u}(T)\) and \({\mathrm{\sigma }}_{\mathit{ys}}(T,\dot{{\mathrm{\epsilon }}_{p}})\) is determined according to the material data provided by the user, then the global Bordet stress is calculated by summing the elements of the group of elements of the group of elements indicated by the user on the gauss points, then finally the overall probability of Bordet failure at all times up to that requested. The calculation carried out is finally written as follows:

(2.12)\[\begin{split} {\ sigma} _ {\ mathrm {Bordet}}} = {\ left\ {\ sum}} = {\ left\ {\ sum _ {\ mathrm {inst}}}}\ sum _ {\ mathrm {el}}\ sum _ {\ mathrm {el}}\ sum _ {\ mathrm {el}}}\ sum _ {\ mathrm {el}}}\ sum _ {\ mathrm {el}}\ sum _ {\ mathrm {el}}\ sum _ {\ mathrm {el}}\ sum _ {\ mathrm {el}}\ sum _ {\ mathrm {el}}\ sum _ {\ mathrm {el}}\ sum _ {\ mathrm {el}}\ sum _ {\ mathrm {} =1} ^ {{n} _ {\ mathrm {pg}}}}} (\ begin {array}} {}\ frac {{\ sigma} _ {\ mathrm {ys}} (T (o), {\ dot {\ stackrel {\ stackrel {}}} {\ stackrel {}}} (o)))} {{\ sigma} _ {\ mathrm {ys}\ mathrm {ys}\ mathrm {0,0}}}}\\\ mathrm {Max} ({\ stackrel {} {\ sigma}}} _ {\ mathrm {1,}\ mathrm {ys}}\ mathrm {el}}}}}\\ mathrm {el}}}}\\ mathrm {max} (o) - {\ sigma}} _ {\ mathrm {1,}}\ mathrm {el},},\ mathrm {el},},\ mathrm {el},},\ mathrm {el}},\ mathrm {el}},\ mathrm {el}},\ mathrm {el}}}} ^ {m}\ mathrm {,0})\\\ mathrm {exp})\\ mathrm {exp} (\ frac {- {\ sigma} _ {\ mathrm {ys}} (T (o), {\ dot {\ stackrel {}})\\ stackrel {}})\\ mathrm {}})\\ mathrm {}} {\ stackrel {}} {\ stackrel {}} {\ varepsilon}} {\ varepsilon}}} _ {p,\ mathrm {el}},\ mathrm {pg}} (o))} {{\ sigma} _ {\ mathrm {ys}\ mathrm {,0}}}}\ frac {{\ stackrel {} {\ varepsilon}} _ {p,\ mathrm {el},\ mathrm {el},\ mathrm {pg}}} (o)} {{\ varepsilon}}} {{\ varepsilon} _ {p\ mathrm {,0}}})\\\ {Delta}}})\\\ {delta\ varepsilon}})\\\ {Delta}}})\\\ {delta\ varepsilon psilon} _ {p,\ mathrm {el},\ mathrm {el}},\ mathrm {pg}},\ mathrm {el}}} {{V}}} {{V} _ {0}} (o)\ frac {{V}}}\ frac {{V}} _ {{V}} _ {{V}} _ {\ mathrm {el}},\ mathrm {pg}}}}}} {{V}}}} {{V}}}} {{V}}\end{split}\]

With \({n}_{\mathrm{inst}}\) the number of moments over which the calculation is done, \({n}_{\mathrm{elem}}\) the number of elements contained in the group of elements requested by the user and \({n}_{\mathrm{pg}}\) the number of gauss points for each of these elements. Also, for any scalar \(a\), \(\stackrel{ˉ}{a}(o)\text{=}\frac{a(o)+a(o-1)}{2}\),, \(\dot{a}(o)\text{=}\frac{a(o)-a(o-1)}{t(o)-t(o-1)}\), and \(\Delta a(o)\text{=}a(o)-a(o-1)\).

The result is a table containing the overall values of the Bordet stress and the probability of Bordet failure.

For the calculation to be correct, the user must enter the COEF_MULT keyword as recommended in the POST_BORDET user documentation.