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Models of Weibull and Rice and Tracey
Date:
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Author (S):
R. MASSON, G. BARBER, G. ROUSSELIER Key
:
R7.02.06-B Page
: 1/12
Organization (S): EDF/MMC
Handbook of Référence
R7.02 booklet: Breaking process
Document: R7.02.06
Models of Weibull and Rice and Tracey
Summary
One first of all points out the bases of these two models of local approach of the rupture allowing of
to model, for one, brittle fracture (model of Beremin known as of WEIBULL), for the other ductile starting
(model of Rice and Tracey). Concerning the model of Beremin, one describes how the probability is calculated of
rupture of a structure starting from knowledge of the mechanical fields requesting it. While placing itself in
general case of a nonmonotonous way of thermomechanical loading and by supposing that parameters of
this model do not depend on the temperature, one establishes the general expression of this probability of rupture
cumulated, including the case of a correction of plastic deformation. Then, one presents the model leading to
law of growth of the cavities of Rice and Tracey as well as the ductile criterion of starting being referred to it. Lastly, of
indications concerning the implementation of these two models in Code_Aster are summarized.
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Models of Weibull and Rice and Tracey
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Count
matters
1 Introduction ............................................................................................................................................ 3
2 the model of Beremin (or Weibull) ...................................................................................................... 3
2.1 Bases .................................................................................................................................... 3
2.1.1 General assumptions ........................................................................................................... 3
2.1.2 Probability of cumulated rupture of the structure ...................................................................... 4
2.2 Expressions of the probability of cumulated rupture of the sites. ......................................................... 4
2.2.1 Parameters of cleavage independent of the temperature ....................................................... 5
2.2.2 Constraint of cleavage depend on the temperature .............................................................. 6
2.3 Correction of deformation ............................................................................................................... 7
2.4 Establishment in Code_Aster ........................................................................................................ 8
3 Modèle of Rice and Tracey ................................................................................................................. 8
3.1 Cavity insulated in an infinite plastic rigid matrix ................................................................... 8
3.2 Approximate law of the growth of the cavities ................................................................................... 9
3.3 Ductile criterion of starting ............................................................................................................ 10
3.4 Establishment in Code_Aster ...................................................................................................... 11
3.4.1 Search of the maximum value of the growth rate ................................................... 11
3.4.2 Calculation of the average value of the growth rate ........................................................... 12
4 Bibliography ........................................................................................................................................ 12
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Models of Weibull and Rice and Tracey
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1 Introduction
One is interested here in a metal structure requested thermomécaniquement. One seeks with
to determine criteria of rupture of this structure, representative of the two mechanisms met
on certain steels:
· at low temperature, certain metallic materials (as the steel of tank) can
to comprise like fragile materials while breaking brutally by cleavage,
· at higher temperature, appears the ductile tear.
In opposition to the global solution, the models of Beremin and Rice - Tracey introduced here
are based on the knowledge of the mechanical fields in the zones most requested for
to obtain a local criterion of rupture representative of the physical mechanisms brought into play (instability of
microscopic cracks of cleavage or increase then coalescence in porosity).
2
The model of Beremin (or Weibull)
The generally allowed mechanism of rupture by cleavage is as follows: the plasticization of material
conduit with the starting of microscopic cracks. Taking into account the metallurgical heterogeneity of material, these
microscopic cracks have a random size and a position. The total rupture is then obtained
when the normal constraint with one of these microscopic cracks becomes sufficiently large to return it
unstable.
The model of Beremin (cf POST_ELEM [U4.61.04]), proposed at the beginning of the years 1980 [bib1] begins again
these ideas while being based on the knowledge of the local mechanical fields requesting the structure
considered. We present here the broad outline of them by adopting the framework more general bench in
[bib2]. By abuse language, we call this model, in what follows, model of Weibull, in
reference to the law of probability to which it leads.
For that, one considers a structure subjected to a history of thermomechanical stresses with
to leave the moment t=0 fixed arbitrarily. This structure is made up (at least partly) of a steel,
likely to break by cleavage at low temperature. One seeks to determine the probability of
cumulated rupture of this structure at any moment.
2.1 Bases
2.1.1 Assumptions
general
Let us consider first of all an elementary volume representative Vrep of material considered. One
suppose that the microstructural heterogeneity of material led to the existence of sites
of damage (microdéfauts) appearing with plasticity. One notes V0 the volume of each
Vrep
site, so that in a volume plasticized Vrep, the number of sites of damage is
.
V0
For each one of these sites, one notes G () D the probability of having a critical stress of cleavage
included/understood in [; +
D]. Probability that one of the sites of damage has one
forced cleavage lower than I (c) is thus:
Ic
G (
) D
0
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In practice, one postulates a form of G for the positive constraints
:
G
m
() = '- 1 (G () = 0 if < 0).
If, one considers now the structure, we suppose that 3 V
3
V 0, where V is a volume
elementary whose characteristic dimension is lower than the macroscopic fluctuations of
mechanical fields.
Lastly, the events of rupture of the sites of damage are supposed to be independent the ones
others, rupture of one of the sites involving the rupture of the whole of the structure (assumption of
the weakest link).
2.1.2 Probability of cumulated rupture of the structure
One supposes here to know the probability of cumulated rupture of each site, noted Pr (sit)
E. One can
then successively to write the probability of cumulated rupture of an elementary volume, then of
complete structure. The stress field being homogeneous in V, the first is worth:
1 - p (V
) =
1
(- p
R
R (site)) ,
site V
that is to say:
V
p (V
) = 1 - (1 - p
R
R (site)) V0.
The probability so that at the end of the loading, our structure (volume) is not broken raises
then with:
V
V
1 - P
V
R =
1
(- p (V
R)) =
1
0
(- Pr (if)
you)
= exp L (
N 1 - Pr (if)
you)
,
V
V
V
0
V
Knowing that p (site) remains small, in front of the unit, the preceding expression can be simplified for
R
to give finally:
V
V
P 1 - exp (- p (sit)
E
) = 1 - exp- p (site)
R
R
R
.
V
V
V
0
0
That is to say:
V
P
1 ex (
p
X
R = -
-) with X = p
R (site)
V0
2.2
Expressions of the probability of cumulated rupture of the sites.
At any moment, the evolution of the mechanical fields in each element V is supposed to be radial and
not necessarily monotonous. This evolution is characterized in any point by a history of
principal constraint maximum I (U) 0ut.
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2.2.1 Parameters of cleavage independent of the temperature
The loading being radial, the direction of maximum principal constraint is supposed to be constant.
When there is plasticization, the sites of damage appear. We suppose that one
condition necessary of rupture of a site of damage is that plasticity is active. So that
this volume did not break at the moment T, it is necessary and it is enough that:
Ic I (U), U
< T,
such as p& (U
) > 0,
&p (U) being the rate of plastic deformation cumulated at the moment U.
Note:
Let us stress that this condition of active plasticity is different from the condition classically
adopted (p > 0, to see document [R7.02.06] Indice A Version 5.0). It is clear that these two
conditions are equivalent in the case of a monotonous way of loading.
For more general ways of loading, this condition of active plasticity leads in
revenge with much better results [bib7].
Only times U are considered for which plasticity is active, since the rupture is not
possible that at these moments there. One notes {U < T, p
& (U) >}
the 0 whole of these moments for the element V
considered. The preceding condition is thus written:
Ic
max
()
{
I U.
u<t, p& (U) >}
0
Its probability of rupture being equal, as in the preceding section, with the probability so that it
have a critical stress of cleavage lower or equal to the member of straight line of the inequality
the preceding one, it is thus written:
{max
(U
I
)
m
u<t, p
& (U) >}
0
max
(U)
{
I
u<t, p& (U) >}
0
p (site) =
G () D
R
=
,
U
0
1
m
m
U =
indicating the constraint of cleavage (forced for which probability of rupture
'
cumulated potential sites of cleavage is worth 1).
The probability of rupture of the structure is written then according to [§2.1.2]:
m
P
W
R = 1 - exp -
U
where the constraint of Weibull at the moment T is given by:
1
m
V
~ m
~
W (T) = I
with =
max
I
()
V
I U
0
{u<t, &p (U) >}
0
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Let us notice that in the case of a monotonous way of loading, the preceding expression of
constraint of Weibull is reduced to:
1
m
~
if the element is plasticized
m
V
I
W = I
with ~
=
.
V
I
0
0
if not
2.2.2 Constraint of cleavage depend on the temperature
That is to say (U) 0ut evolution of temperature in V.
For any moment (U), we suppose that in the vicinity of each site of damage,
normal constraint “microscopic” checks:
=
I (micro) (U)
F. I (U),
F being a parameter of localization depending only on the average temperature (U) in V.
So that the site of damage did not break, it is necessary thus that:
Ic
, <, such as &
>
I (micro)
U T
(
p U)
.
0
that is to say:
Ic F
.
, <, such as &
>
I (U)
U T
(
p U)
,
0
so that the probability of cumulated rupture of a site rises with:
m
.
I
R (site)
(U) F ((U))
p
= max
[
,
u<t, p& (U)
>0]
U
or:
m
U
p
I
R (site)
()
=
max
,
[
u<t, p& (U)
>0] U ((U))
with: (
U
U) =
,
a function of the temperature. The introduction of the parameter of localization
F ()
F thus leads to an apparent dependence of the constraint of cleavage.
In a general way, the probability of cumulated rupture of the structure rises with:
m
I (U)
FD
p
R = 1 - exp -
max
{
u<t, p& (U)
>}
0 U ((U))
V
0
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The constraint of Weibull has nothing any more but the following conventional smell then: while noting or a value
chosen arbitrarily, one can write:
O m
p
1 exp
R = -
-
,
O
U
O being defined by:
1
O
~
°
U I (U)
O
= ~°m V m
I
with
=
max
V
I
u<t, &p U >0
U (U)
0
[
()]
2.3
Correction of deformation
A great deformation of the sites resulting in ' decreasing harmfulness (relative contraction by it of
microscopic cracks in the transverse plan with the axis of traction), the constraint criticizes cleavage at one moment
U increases under the effect of this deformation (U) according to:
1
(U) = (U =)
0 exp ((U)) with (U) = N. (U) .n
I
Ic
Ic
I
2
where N (U)
indicate the direction associated with the maximum principal constraint at the moment U.
The probability of rupture of a site at one moment U given is written now:
m
I (U)
1
p (site) =
max
.exp - U
R
.
{
I
u<t, p
& (U) >}
0
2
U ((U))
()
For a monotonous way of loading (constant temperature and uniform), the preceding relation
conduit with the traditional expression [bib2]:
m
m
p
I
R (site) =
exp - I.
2
U
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2.4 Establishment
in
Code_Aster
Let us consider a field C of the studied structure which can be the whole of the studied grid, one
group of mesh or a mesh. Following an elastoplastic thermomechanical calculation, one knows
evolution of the plastic deformation and deformation, stress fields cumulated in it
field and one wish to determine his probability of cumulated rupture.
Let us stress that for calculation with correction of deformation (option CORR_PLAST: “OUI”), one
preliminary calculation (simple postprocessing) of the field of deformation of Green - Lagrange from
field of displacement on the zone of the studied structure (cf CALC_ELEM [U4.61.02]) is necessary.
In the contrary case, postprocessing stops.
Corresponding numerical integration in Code_Aster is carried out in two times:
· one calculates in each point of Gauss ~
I if the rate of plastic deformation cumulated in it
not is strictly positive,
· by quadrature on each mesh then simple summation on the field C concerned, one in
deduced the constraint from Weibull as well as the probability of associated rupture. The summation is
balanced by a multiplicative coefficient which takes account of possible symmetries and the type
of modeling selected (axi,…). One will take care well to define this coefficient (COEF_MULT)
in accordance with the indications given in [U4.61.04].
The first stage makes it possible to introduce an alternative (key word SIGM_ELMOY instead of SIGM_ELGA)
leading to appreciably different results in the case of a fissured structure (presence of
gradient): in each mesh, ~
I is given starting from the average on this mesh of the field
of constraint (and, possibly, field of deformation of Green - Lagrange). It is nonnull if
the rate of plastic deformation cumulated at the moment considered is strictly positive in a point of
Gauss at least.
3
Modèle of Rice and Tracey
One is interested now in the case of ductile starting. By considering an element of volume
initially healthy, ductile this element tears it results from the following elementary mechanisms:
· nucleation of cavities caused by the decoherence of inclusions present in material,
· growth then coalescence of these cavities.
3.1
Cavity insulated in an infinite plastic rigid matrix
In an analytical step of comprehension, Rice and Tracey [bib3] studied the behavior
of a cavity, initially spherical (Sv surface), insulated in an infinite isotropic medium (volume V), of
behavior of plastic rigid Von Mises (elastic limit 0), incompressible, ad infinitum subjected to
a speed of deformation & unspecified (noted constraint with L `infinite).
They show that the field rate of travel, solution of the posed mechanical problem,
minimize the functional calculus:
Q (u&) = [S (&) - S] & FD -
N U
& dS
ij
ij
ij
ij
I J
V
Sv
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3.2
Approximate law of the growth of the cavities
While managing to minimize this functional calculus in various situations, Rice and Tracey have then
m
shown the dominating influence of the rate of triaxiality (=
, m: trace and equivalent
eq
eq
(von Mises) of the constraint imposed on the element of volume considered) on the growth rate of
cavities.
They exhibent even a law of growth of the cavities, certainly approached, but very near to the results
preceding model. Thus, in each principal direction (K) associated at the speed of
deformation &, the rate of elongation of a cavity rises with:
&R = {& + &
} DR.
K
K
eq
K
(R
K: radius of the cavity in the direction (K), &
K, &eq: principal value in the direction (K) and
equivalent (von Mises) of the speed of deformation imposed ad infinitum), relation in which
coefficients and D depend on the situation considered:
5
·
= for a linear matrix of work hardening or a perfectly plastic matrix with weak
3
rate of triaxiality or = 2 in the case of a perfectly plastic matrix atstrong rate of
triaxiality,
3
·
D
m
= exp
for a perfectly plastic matrix or D
m
=
for one
2
0
4 eq
stamp linear work hardening. = 0 283
,
is the value given by Rice and Tracey whereas
more precise calculations (cf [bib4]) showed that this coefficient is higher (= 1 2
, 8).
Mudry [bib6] then proposed to apply these theoretical results to the case of the steel of tank, i.e.:
· intermediate behavior enters the extreme cases of behavior studied by Rice and
Tracey with a reasonable work hardening not no one but,
· fissured structures (high rate of triaxiality).
It deduced from it the approximate law following, valid for sufficiently high rates of triaxiality
(superiors with 0,5):
3
&
R = & p
exp
m R
eq
,
2 eq
expression in which:
·
&
p
eq was substituted by &
eq (equivalent (von Mises) of the plastic part the speed of
deformation) in order to extend the law of Rice and Tracey to the elastoplastic case,
· elastic limit
0 were substituted by eq in order to take account of the hardening of
stamp around the cavity.
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Experimental measurements of growth of porosity for various rates of triaxiality allowed
to validate this expression (cf Figure following). These results show that, when the proportion of air voids
initial weak remainder, exponential character of the relation between the radius of the cavities and the rate of
triaxiality is well confirmed. On the other hand, the coefficient depends on material considered as well as
initial fraction of porosity.
Experimental results of measurement of the growth of cavities in various metallic materials
(figure extracted the ref. 6, eq indicating the equivalent of the plastic deformation noted p
eq in
body text) according to the rate of triaxiality
m
eq
3.3
Ductile criterion of starting
R0 and R (T) indicating the initial radius of the cavities and at the moment T considered, the criterion of starting
ductile adoptee here is:
R (T)
R
=
,
R
R
0
0 C
expression in which the first member results from the integration of the law of growth,
in accordance with the indications of the preceding paragraph.
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One can object several arguments of principle against the direct use of this law of growth
cavities of Rice and Tracey like ductile criterion of starting. As follows:
· inclusions, and thus the cavities, are not actually insulated. Worse, they are often
gathered in cluster,
· the coalescence of the cavities undoubtedly results from interactions which, they also, are not
described in the established model,
· in a fissured structure, the presence of gradients in bottom of fissure returns less
directly applicable the preceding analysis which relates to an infinite medium subjected to
boundary conditions homogeneous.
Nevertheless, by using the preceding criterion, one hopes that this law remains realistic, on average, even
in clusters or zones of strong gradients (average on an element of dimensions
comparable with that of the model of Beremin). In addition, one makes the assumption that the critical size
reserve, in general readjusted on geometries given (test-tube CT, for example), translated
coalescence, which amounts supposing that coalescence does not depend too much on nature on
mechanical stresses imposed on the element of volume (triaxiality, shearing,…).
Let us notice to finish that the model of Rice and Tracey is only one approached law, valid for
important rates of triaxiality (i.e higher than 0,5).
3.4 Establishment
in
Code_Aster
Let us consider a field C of the studied structure which can be the whole of the studied grid, one
group of mesh or a mesh. Following an elastoplastic thermomechanical calculation, one knows
evolution of the plastic deformation and deformation, stress fields in this field and
one wishes to determine the space and temporal variations growth rate of the cavities in
this field.
In each point of Gauss of the field C, one assimilates the constraints and speeds of deformation
calculated at every moment with the quantities applied to the infinite medium considered previously. The law
of growth of Rice and Tracey is thus integrated step by step using the approximate formula
following:
R (T)
R (T
)
(T)
(T)
Log
N
= Log
N 1
- +, 0283 sign m N
Exp
m
N
p
, 1 .5
(T
p
) - (T
)
R
R
(T)
(T) (eq N
eq
N 1
-)
0
0
eq
N
eq
N
R
The values of the report/ratio are thus at every moment obtained
in each point of Gauss of the field
R0
C, the sign of the rate of triaxiality allowing the taking into account of evolutions as well in traction
that in compression. Two functionalities are then offered in Code_Aster:
3.4.1 Search of the maximum value of the growth rate
At every moment, one seeks on the whole of the field C the point of Gauss (and the volume of
R
associated under-mesh) maximizing
.
R0
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3.4.2 Calculation of the average value of the growth rate
By quadrature on each mesh then moyennation on the field C concerned, one deduces with each
R
moment the average value of
on
R
C
0
As in the case of the model of Weibull, an alternative is introduced: temporal integration
the preceding one is then carried out starting from the constraint and of the average plastic deformation by
net.
4 Bibliography
[1]
F. BEREMIN: “A local criterion for cleavage fracture off has nuclear presses vessel steel”,
Metall. Trans. 14A, p 2277-2287, 1981.
[2]
R. MASSON and Al: “Definition of the local approach (into fragile) in the case of a way of
unspecified loading “, Note EDF HT-26/00/021.
[3]
J.R. RICE and D.M. TRACEY: “One the ductile enlargement off voids in triaxial stress fields”,
J. Mech. Phys. Solids, Vol. 17, pp. 202-217, 1969.
[4]
Y. HUANG: “Accurate Dilatation Rates for Spherical Voids in Triaxial Stress Fields”,
Transactions off the ASME J. a. Mech., vol. 58, n°4, p 1084-86, 1991.
[5]
F. MUDRY: “Study of Rupture Ductile and Rupture by Clivage d' Aciers Faiblement
Combined “, Thèse of State, Université de Technologie of Compiegne, 1982.
[6]
B. MARINI, F. MUDRY, A. PINEAU: “Experimental study off cavity growth in ductile
rupture “, Engineering Fracture Mechanics Vol 22, No 6, pp. 989-996, 1985.
[7]
W. LEFEVRE, R. MASSON, G. BARBER: “Digital simulations using the model
improved of Beremin of tests of preloading hot carried out with steel 18MND5 “.
Note EDF HT-26/00/028.
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