3. Description of the Beremin model#

In this section, the foundations of the Beremin model are first reviewed. The hypotheses related to the definition of potential cleavage sites are explained. The mechanism considered to be critical is the propagation of a cleavage crack in the adjacent grain, which is itself considered to be initiated by plasticity mechanisms. The probability of rupture of these sites, which leads to the probability of rupture of the structure, is deduced from this. The classical formulation of the Beremin model is extended to more general loads by including a plastic correction variant often used in this model [24].

The Beremin model is based on the knowledge of the local mechanical fields requiring the structure in question. We consider a structure subjected to a history of thermomechanical stresses starting from the moment \(t=0\) fixed arbitrarily. The aim is to determine the cumulative probability of failure of this structure at any time.

Hypothetically, this structure consists (at least in part) of a steel, capable of breaking by cleavage at low temperature, exhibiting an elastoviscoplastic behavior law and one of the internal variables of which corresponds to the cumulative equivalent plastic deformation: an example of a list of compatible laws is provided in paragraph § 4.

3.1. Foundations#

3.1.1. General Assumptions#

The model considers a representative elementary volume \(\mathrm{\Omega }\) of the material and is based on several fundamental assumptions summarized in [25]:

It is assumed that the initiation of microcracks at damage sites can only be done when plasticity is active (the cumulative plastic deformation rate \(\dot{p}(u)\) at time \(u\) must be positive \(\dot{p}(u)>0\)) and their number no longer increases during the history of the stresses.

Note:

Note that this condition of active plasticity which will be the one considered throughout the rest of the document is different from the condition traditionally adopted ( \(p>0\) ) . These two conditions are equivalent in the case of a monotonous loading path.

For more general loading paths, this condition of active plasticity \(\dot{p}>0\) on the other hand leads to much better results [26 ] .

It should be noted that only the maximum main stress \({\mathrm{\sigma }}_{I}\) intervenes in the propagation of the defect, the defects then being considered to be oriented perpendicular to the main direction.
The propagation of micro-cracks is controlled by a criterion type Griffith.
The rupture will start from the most important defects in terms of size, only knowledge of the distribution of these is necessary.
The structure requested can be considered as the juxtaposition of several elements of volume \({V}_{0}\) that are perfectly independent from the point of view of rupture. \({V}_{0}\) should be as small as possible to verify statistical independence, but large enough for the probability of finding a defect of sufficient size to be reasonable (in practice \({V}_{0}\) includes a few grains).
The weakest link theory is used, according to which the breakage of one of the elementary volumes \(\mathrm{\delta }V\) causes the entire structure to break.
A form of distribution of defects \(g\) is postulated for the positive constraints \(g(\mathrm{\sigma })=\mathrm{\alpha }\text{'}{\mathrm{\sigma }}^{m-1}\) and \(g(\mathrm{\sigma })=0\text{si}\mathrm{\sigma }<0\). For each of these sites, we note \(g(\mathrm{\sigma })d\mathrm{\sigma }\) the probability of having a critical cleavage constraint included in \(\left[\mathrm{\sigma };\mathrm{\sigma }+d\mathrm{\sigma }\right]\). The probability that one of the damage sites has a cleavage stress lower than an applied stress \({\mathrm{\sigma }}_{\mathit{Ic}}\) is therefore:

(3.1)#\[ \ underset {0} {\ overset {{\ mathrm {\ sigma}}} _ {\ mathit {Ic}}} {\ int}}} {\ int}} g (u)\ mathit {du}\]

3.2. Cumulative probability of structural failure#

It is assumed here to know the cumulative probability of rupture (distribution function) of each damage site of volume \({V}_{0}\), noted \({p}_{r}(\text{site})\) and considered to be identical for all sites. The probability of survival is equal to \(1-{p}_{r}(\text{site})\). We can then write the cumulative failure probability of an elementary volume \(\mathrm{\delta }V\) whose characteristic dimension is smaller than the macroscopic fluctuations of the mechanical fields (\(\mathrm{\delta }V>{V}_{0}\)) and whose stress field is assumed to be homogeneous:

\[\]

: label: eq-4

1- {p} _ {r} (delta V) =prod _ {text {site}indelta V} (1- {p} _ {r} (text {site}))

Either:

\[\]

: label: eq-5

{p} _ {r} (delta V) =1- {(1- {p} _ {r} (text {site})))} ^ {frac {delta V} {{V}} _ {0}}}

The probability of survival of the structure (volume \(\mathrm{\Omega }\)) at the end of the loading is equal to the product of the survival probabilities of each of the elementary volumes \(\mathrm{\delta }V\) (statistical independence). The probability of survival of the structure then amounts to:

(3.2)#\[ 1- {P} _ {r} =\ prod _ {\ delta V\ in\ Omega} (1- {p} _ {r} (\ delta V)) =\ prod _ {\ delta V\ in\ Omega} (1- {p}\ omega} (1- {p} _ {\ omega}} (1- {p} _ {\ omega}} (1- {p} _ {r}}} (\ text {site})} (\ text {site}) {)} ^ {\ frac {\ delta V} {{V} _ {Omega}} (1- {p} _ {0}}} =\ prod _ {\ delta V\ in\ Omega}\ text {exp}\ text {exp} (\ text {exp}} (\ text {site}))\ frac {\ delta V} {\ delta V} {{V}} _ {0}})\]

Knowing that \({p}_{r}(\text{site})\) remains small in front of unity, the previous expression can be simplified to finally give:

(3.3)#\[ {P} _ {r}\ approx 1-\ prod _ {\ mathrm {\ delta} V\ in\ mathrm {\ Omega}}\ text {exp} (- {p} _ {r} (\ text {site}) (\ text {site})\ frac {\ site})\ frac {\ site})\ frac {\ mathrm {\ delta} V\ in\ mathrm {\ delta} V} {{V} _ {0}}) =1-\ text {exp}\ r} (\ text {site}) (\ text {site})\ frac {\ site})\ frac {\ mathrm {\ delta} V} {\ mathrm {\ delta} V} {{V} _ {0}}) =1-\ text {\ underset {\ mathrm {\ Omega}} {\ int} {\ int} {p} {p} _ {r} (\ text {site})\ frac {\ mathrm {\ delta} V}} {{V}} {V} _ {0}}\ right)\]

Either:

(3.4)#\[ \begin{align}\begin{aligned} {P} _ {r} =1-\ text {exp}\ left (-x\ right)\\ \ textrm {with}\\ x=\ underset {\ mathrm {\ Omega}} {\ int} {\ int} {p} _ {r} (\ text {site})\ frac {\ mathrm {\ delta} V}} {{V} _ {0}}\end{aligned}\end{align} \]

3.3. Cumulative probability of site breakage#

We consider here the case of the radial and not necessarily monotonic loading path, the evolution of mechanical fields in each element \(\mathrm{\delta }V\) is characterized at all points by a history of maximum principal stress \({\mathrm{\sigma }}_{I}(u{)}_{0\le u\le t}\).

3.3.1. Case where the critical cleavage stress is independent of temperature#

Since the load is radial, the direction of maximum principal stress is assumed to be constant. We only consider past times \(u\) for which plasticity is active (\(\dot{p}(u)>0\)), since breakage is only possible at these times (hypothesis 1). We note \(\left\{u<t,\dot{p}(u)>0\right\}\) all of these moments for the element \(\mathrm{\delta }V\) under consideration.

According to hypothesis (2), only the maximum main constraint \({\mathrm{\sigma }}_{I}\) intervenes in the propagation of the defect. For volume \(\mathrm{\delta }V\) to be broken at time \(t\), you need to:

(3.5)#\[ {\ mathrm {\ sigma}} _ {\ text {Ic}}\ the\ underset {\ left\ {u0 \<t,\ dot {p} (u) > right\}} {\ text {max}} {\ mathrm {\ sigma}}} {\ mathrm {\ sigma}}} _ {I} (u)\]

Taking into account hypothesis (7), the probability of rupture of this volume \(\mathrm{\delta }V\) can therefore be written as:

(3.6)#\[ {p} _ {r} (\ text {site}) =\ underset {0} {\ underset {0} {\ underset {\ left\ {u0<t,\ dot {p} (u) >\ right\}} {\ text {max}} {\ mathrm {\ sigma}} {\ sigma}} {\ sigma}} {\ sigma}} {\ sigma}} {\ left (\ frac {\ underset {\ left\ {u0 \<t,\ dot {p} (u) > right\}}} {\ text {max}} {\ mathrm {\ sigma}} _ {I} (u)}} {{\ mathrm {\ sigma}}} {\ sigma}}} _ {u}}\ right)} ^ {m}}\]

By defining \({\mathrm{\sigma }}_{u}={\left(\frac{m}{\mathrm{\alpha }\text{'}}\right)}^{\frac{1}{m}}\) as the cleavage stress (i.e. the stress for which the cumulative probability of failure of the potential cleavage sites is \(1\)), the probability of breaking the structure can be written from () as:

(3.7)#\[ {P} _ {r} =1-\ text {exp}\ left (- {\ left (\ frac {{\ mathrm {\ sigma}} _ {W}}} {{\ mathrm {\ sigma}}}} _ {u}}\ right)} ^ {m}\ right)} ^ {m}\ right)\]

with \({\mathrm{\sigma }}_{W}\) being the Weibull constraint [23] at time \(t\) defined by:

(3.8)#\[ \begin{align}\begin{aligned} {\ mathrm {\ sigma}} _ {W} (t) = {\ left [\ underset {\ mathrm {\ omega}} {\ int} {\ mathrm {\ sigma}} _ {I} ^ {m} {m}\ m}\ frac {\ m}\ m}\ frac {\ m}\ frac {1} {m}\ frac {1} {m}\ right]} ^ {\ frac {1} {m}}\\ \ textrm {with}\\ {\ mathrm {\ sigma}} _ {I} =\ underset {\ left\ {u0 \<t,\ dot {p} (u) > right\}} {\ text {max}} {\ mathrm {\ sigma}}} _ {I} (u)\end{aligned}\end{align} \]

In this expression, the Weibull module \(m\), gives an idea of the dispersion of the size of defects likely to initiate brittle failure. \({\mathrm{\sigma }}_{u}\) represents an average breaking stress for a sample of volume \({V}_{0}\).

In the case, where, at any moment, the evolution of the mechanical fields in each element \(\mathrm{\delta }V\) is assumed to be radial and monotonic **** (:math:`dot{p}>0` ), the previous expression of the Weibull constraint is reduced to:

(3.9)#\[ {\ mathrm {\ sigma}} _ {W} = {\ left [\ left [\ underset {\ mathrm {\ omega}} {\ int} {\ mathrm {\ sigma}} _ {I} ^ {m}}\ frac {\ mathrm {\ mathrm {\ delta} V} {\ delta} V} {{V}}\ right]} ^ {\ frac {1} {m}}}\]

3.3.2. Case where the critical cleavage stress is dependent on temperature#

Hitherto, temperature-independent parameters of the cleavage model have been considered. Applying the local approach to real study transients (for example, simulations of hot preloading tests on 16 MND5 steel with the Beremin model) requires the modification of the initial formulation to be able to take into account loading paths involving mechanical and thermal discharges [24, 26, 27] and also to be able to take into account the dependence of the stress of temperature cleavage.

For this, we assume that the evolution of mechanical fields in each element \(\mathrm{\delta }V\) is radial and not monotonic. This evolution is characterized by a history of maximum principal stress \({\mathrm{\sigma }}_{I}(u{)}_{0\le u\le t}\) as well as by a history of temperature \(\mathrm{\theta }{\left(u\right)}_{0\le u\le t}\).

For all time \(u\), we assume that in the vicinity of each damage site, the normal « microscopic » stress verifies:

(3.10)#\[ {\ mathrm {\ sigma}} _ {I\ left (\ text {micro}\ right)}\ left (u\ right) =f {\ mathrm {\ sigma}}} _ {I}\ left (u\ right)\]

\(f\) being a location parameter that only depends on the mean temperature \(\mathrm{\theta }\left(u\right)\) in \(\mathrm{\delta }V\). So that the damage site did not break, it is therefore necessary that:

(3.11)#\[ \begin{align}\begin{aligned} {\ mathrm {\ sigma}} _ {\ mathit {Ic}}\ ge {\ mathrm {\ sigma}} _ {I\ left (\ mathit {micro}\ right)} (u)\\ \ textrm {such as}\\ \ dot {p}\ left (u\ right) >0\end{aligned}\end{align} \]

Either:

(3.12)#\[ \begin{align}\begin{aligned} {\ mathrm {\ sigma}} _ {\ text {Ic}}\ ge f {\ mathrm {\ sigma}} _ {I} (u)\\ \ textrm {such as}\\ \ dot {p}\ left (u\ right) >0\end{aligned}\end{align} \]

So that the cumulative probability of failure of a site () is:

(3.13)#\[ {p} _ {r}\ left (\ text {site}\ right) = {\ left [\ underset {\ left [u0 \<t,\ dot {p}\ left (u\ right) > right]} {\ text {max}}\ left\ {\ frac {{\ frac {{\ mathrm {\ mathrm {\ sigma}}}} _ {I}\ left (u\ right)\ text {.} f\ left (\ left (\ left) right)\ right)} {{\ mathrm {\ sigma}} _ {u}}\ right\}\ right]}} ^ {m}\]

Or even:

(3.14)#\[ {p} _ {r}\ left (\ text {site}\ right) = {\ left [\ underset {\ left [u0 \<t,\ dot {p}\ left (u\ right) > right]} {\ text {max}}\ left\ {\ frac {{\ frac {{\ mathrm {\ mathrm {\ sigma}}} _ {\ mathrm {\ sigma}} _ {u}\ mathrm {\ mathrm {\ sigma}} _ {u}\ mathrm {\ mathrm {\ sigma}} _ {u}\ mathrm {\ sigma}} _ {u}\ mathrm {\ sigma}} _ {u}\ mathrm {\ sigma}} {\ theta}\ left (u\ right)\ right)\ right)}\ right\}\ right]} ^ {m}\]

With \({\mathrm{\sigma }}_{u}\left(\mathrm{\theta }\right)\) a temperature function such as:

(3.15)#\[ {\ mathrm {\ sigma}} _ {u}\ left (\ mathrm {\ theta}\ right) =\ frac {{\ mathrm {\ sigma}}} _ {u}} {u}} {f\ left (\ mathrm {\ theta}\ right)}\]

The introduction of the location parameter \(f\) therefore leads to an apparent dependence on the cleavage constraint. In general, the cumulative probability of failure of the structure is expressed starting from () by:

(3.16)#\[ {P} _ {r} =1-\ text {exp}\ left (- {\ int} _ {\ mathrm {\ Omega}} {\ left [\ underset {\ left\ {u0 \<t,\ dot {p}\ left (u\ right) > right\}}} {\ text {exp}}} {\ text {max}}}\ left {max}}}\ left (u\ max}}} {\ mathrm {\ sigma}} _ {I}\ left (u\ right)} {\ mathrm {\ sigma}} _ {I}\ left (u\ right)} {\ mathrm {\ sigma}} thrm {\ sigma}} _ {u}\ left (\ mathrm {\ theta}\ left (u\ right)\ right)}\ right\}\ right]} ^ {m}\ frac {\ mathrm {\ delta}\ left (\ mathrm {\ delta}\ delta}\ delta}\ text {V}} {{V} _ {0}}\ right)\]

It is this last formulation implemented in code_aster that is used to simulate hot preload tests and more generally load with discharge paths. By noting \({\mathrm{\sigma }}_{u}^{o}\) a value chosen arbitrarily, we can write:

\[\]

: label: eq-21

{P} _ {r} =1-text {exp}left (- {left (frac {{mathrm {sigma}} _ {W} ^ {o}} {{mathrm {sigma}}} {{mathrm {sigma}}}} _ {u} ^ {o}}right)} ^ {m}right)} ^ {m}right)

The Weibull \({\sigma }_{w}^{o}\) constraint then becomes:

(3.17)#\[ \begin{align}\begin{aligned} {\ mathrm {\ sigma}} _ {W}} ^ {o} = {\ left [{\ int}} _ {\ mathrm {\ omega}} {(\ stackrel {~} {\ mathrm {\ sigma}}}}}} _ {\ sigma}}}}} _ {\ sigma}}}}} _ {\ sigma}}}} _ {0}} _ {0}} _ {0}} _ {0}}\ right]} ^ {\ frac {1} {m}}\\ \ textrm {with}\\ {\ stackrel {~} {\ mathrm {\ sigma}}}}} _ {I} ^ {o} =\ underset {\ left [u0 \<t,\ dot {p}\ left (u\ right) > right]} {\ text {max}}\ left\ {\ frac {\ frac {{\ frac {{\ frac {{\ frac {{\ frac {{\ frac {{\ frac {{\ frac {{\ frac {{\ frac {{\ frac {{\ frac {{\ mathrm {\ sigma}}} _ {\ mathrm {\ sigma}}} _ {I}\ frac {{\ frac {{\ frac {{\ frac {{\ frac {{\ mathrm {\ sigma}}} _ {\ frac {{\ frac {{} {{\ mathrm {\ sigma}}} _ {u}\ left (\ mathrm {\ theta}\ left (u\ right)\ right)}\ right)}\ right\}\end{aligned}\end{align} \]

3.4. Plastic deformation correction#

Several studies on breakage by cleavage show the beneficial effect of plastic deformation on cleavage resistance. Several reasons are mentioned by the authors and in particular the apparent reduction of microdefects in the direction perpendicular to traction with strong deformations. It is also likely that pre-deformation at temperatures higher than that of cleavage may cause the blunting of micro-defects already present in the material.

A correction to the largest principal constraint \({\mathrm{\sigma }}_{I}\) that is involved in the calculation of the Weibull constraint was proposed in the Beremin model. The main stress is corrected by a factor depending on the plastic deformation in the main direction \({\mathrm{\epsilon }}_{I}^{p}\):

(3.18)#\[ \begin{align}\begin{aligned} {\ mathrm {\ sigma}} _ {I}} ^ {\ ast} = {\ mathrm {\ sigma}} _ {I}\ text {exp}\ left (-\ frac {{\ mathrm {\ mathrm {\ epsilon}}} _ {I} ^ {p}} {k}\ right)\\ \ textrm {with}\\ 2≤k≤4\end{aligned}\end{align} \]

The new expression for the Weibull constraint is:

(3.19)#\[ {\ mathrm {\ sigma}} _ {W} = {\ left [\ left [\ underset {\ mathrm {\ omega}} {\ int} {({\ mathrm {\ sigma}} _ {I})}} ^ {I})}} ^ {I}))}} ^ {I}})} ^ {I} {I}})}} ^ {I}})} ^ {I}))} ^ {I}))} ^ {I}))} ^ {I})}\ text {exp}\ left (-\ frac {{\ mathrm {\ sigma}} _ {I}} {I}} {I}})} {I} {I}})} {I} {I}})} {I} {I}})} k}\ right)\ frac {\ mathrm {\ delta} V} {{V} _ {0}}\ right]}} ^ {\ frac {1} {1} {m}}\]

The probability of a site breaking at a given moment \(t\) is now written:

(3.20)#\[ {p} _ {r} (\ text {site}) =\ underset {\ left\ {u0 \<t,\ dot {p} (u) > right\}} {\ text {max}}\ {\ frac {{\ mathrm {\ sigma}}} _ {I}\ left (u\ right)} {{\ left (u\ right)}} {{\ mathrm {\ sigma}}} _ {u}\ left (\ mathrm {\ sigma}}} _ {u}\ left (\ mathrm {\ sigma}}} _ {u}\ left (\ mathrm {\ sigma}}} u\ right)\ right)}\ text {.} \ text {exp} {\ left (-\ frac {1} {2} {2} {\ mathrm {\ epsilon}} _ {I} ^ {p}\ left (u\ right)\ right)\ right)\}}} ^ {m}\]

For a monotonous loading path (constant and uniform temperature), the previous relationship leads to the classical expression [24]:

(3.21)#\[ {p} _ {r} (\ text {site}) = {\ left [\ frac {{\ mathrm {\ sigma}} _ {I}} {{\ mathrm {\ sigma}}} _ {u}}\ right]}\ right]}}\ right]}}}\ right]}} ^ {m}}} ^ {m}}\ right]} ^ {m}}\ right]} ^ {m}\ right]} ^ {m}\ right]} ^ {m}\ left (-\ frac {m} {2} {\ mathrm {\ sigma}}} _ {I} ^ {p}\ right)\]