Ahmed Allal

C Guerret-Piécourt

S Tonon

A Lavernhe-Gerbier

F Flores

Christelle Guerret
email: christelle.guerret@univ-pau.fr

Electrical charging during the sharkskin instability of a metallocene melt

Keywords: Metallocene polyethylene, Flow instabilities, Sharkskin instability, Electrical charging

Electrical charging during the sharkskin instability of a metallocene melt.

Introduction and bibliography

On the one hand, flow instabilities occur during extrusion upon reaching a critical shear rate or stress. Due to their economical and theoretical interest, these phenomena have been extensively studied since their first report in the late 50's [START_REF] Dennison | Flow instability in polymer melts: a review[END_REF][START_REF] Denn | Issue in viscoelastic fluid mechanics[END_REF]. On the other hand, the electrification of polymers during their manufacturing process is also a well-known problem in industry, often resolved by the addition of antistatic agents. However few studies have been performed to correlate these both phenomena. Indeed among the hundreds of publications concerning the flow instabilities, only a tens of them are focused on the electrification of the polymer melt during their stable or instable flow [START_REF] Vinogradov | Viscoelastic properties and flow of narrow distribution polybutadienes and polyisoprenes[END_REF][START_REF] Taylor | The electrokinetic charging of polymers during capillary extrusion[END_REF][START_REF] Dreval | Linear flexible-chain polymer electrification conditions and pattern at T>TG[END_REF][START_REF] Dreval | The rheology and static electrication of linear flexible chain polymers[END_REF][START_REF] Pérez-González | Exploration of the slip phenomenon in the capillary flow of linear lowdensity polyethylene via electrical measurements[END_REF].

The first study was performed by Vinogradov et al. on polybutadienes (PB) by using a cylindrical capacitor for the charge measurements under spurt and overspurt conditions. They have shown that a measurable electrification was only observed for pressure greater than the critical for spurt [START_REF] Vinogradov | Viscoelastic properties and flow of narrow distribution polybutadienes and polyisoprenes[END_REF].

In the case of stable capillary extrusion of polyethylene (PE), polystyrene (PS) and polycarbonate (PC), Taylor et al. measured -by collecting the charges in a Faraday cup-an electrification attributed to the double layer formation, but no presence of flow instabilities were noted [START_REF] Taylor | The electrokinetic charging of polymers during capillary extrusion[END_REF].

Ten years later the results of Dreval et al., -who has fulfilled the study begun by Vinogradov-on linear flexible chain polymers (PB, PI, fluorine rubber) did not confirm the electrification during the stable regime, indeed they only measured electrical charges during the spurt regime.

Very recently, the evidencing of electrification during the capillary extrusion of polymer has known a renewed interest, with the studies of Pérez-González et al. [START_REF] Pérez-González | Exploration of the slip phenomenon in the capillary flow of linear lowdensity polyethylene via electrical measurements[END_REF][START_REF] Perez-González | Flow enhancement in the continuous extrusion of linear low-density polyethylene[END_REF]. He used an experimental set-up where the charges are collected by swiping the probe of an electrometer on the molten extrudate. In these last papers no electrification during the sharkskin regime has been measured, since the appearance of the static electrification was linked to the conditions of strong slip.

All these previous studies have evidenced the influence of various parameters on the electrification of polymer melts such as the material of the die [START_REF] Taylor | The electrokinetic charging of polymers during capillary extrusion[END_REF][START_REF] Pérez-González | Exploration of the slip phenomenon in the capillary flow of linear lowdensity polyethylene via electrical measurements[END_REF], the molecular characteristics of the polymer for instance the molecular weight [START_REF] Vinogradov | Viscoelastic properties and flow of narrow distribution polybutadienes and polyisoprenes[END_REF], the molecular weight distribution, the segmental mobility [START_REF] Dreval | Linear flexible-chain polymer electrification conditions and pattern at T>TG[END_REF], and the temperature [START_REF] Taylor | The electrokinetic charging of polymers during capillary extrusion[END_REF]. Two mechanisms have been proposed for the electrification of the polymer during the extrusion, the first one is a tribological effect, the second one is a double layer effect.

In this paper, we focus on the electrification of a metallocene polyethylene (mPE) during the sharkskin defect occurrence. First, we will show that contrary to the studies of Vinogradov, Dreval and Pérez-González, electrical charges can be measured simultaneously with the presence of this defect. The fact that the wall slip appears at the same time as the electrical charging will be also underlined. Second, the effect of the die geometry on the electrification will be quantified, leading to some conclusions about the electrification mechanism. Third, the results at various temperatures will be presented and the timetemperature superposition principle will be applied to these curves. The whole study will show that rheological properties and electrical charging are strongly correlated.

Experimental section

Material

The studied polymer is a metallocene polyethylene (mPE) from Repsol. Its molecular characteristics are a weight average molecular weight of 118 000 g/mol and a polydispersity of 2.15. As shown by RMN study, the studied mPE is not branched, that is confirmed by the absence of strain hardening. Because of the polymerisation catalyst system, some metallic impurities are still present in the polymer melt. According to the safety data sheet of the mPE, their amounts were around 3.6 ppm for zirconium and 60 times more for aluminium.

The flow curve of this mPE is shown on figure 1 (a). It has been established in a capillary rheometer at a temperature of 190°C using a die of length over diameter L/D=20, D=2mm and a 180° entry angle. The sharkskin defect is the first occurring defect when the apparent shear rate • γ app reaches a value around 50 s -1 . It is replaced by the spurt flow defect at about 110 s -1 , and the last one is the gross-melt fracture (g.m.f) defect. In this paper, we will focus on the shear rate domain of the sharkskin defect.

To characterize the polymer, its elastic modulus G' and its viscous modulus G" have been studied on an imposed strain rheometer at different temperatures. The viscoelastic curves could be shifted horizontally and vertically according to the time-temperature superposition principle (see Fig. 1-b). The corresponding activation energy was 27.06 kJmol. -1 K -1 .

From the rheological master curve, the values of the Williams-Landel-Ferry (WLF) shift factors, that will be used later for charging phenomena, are a T = 1.753, and 1.363, for 155°C and 170°C respectively, with a reference temperature of 190°C and b T ≈1 in both cases.

One can note also on figure 1.b, that the cross-over appears for ω=80 rad/s, that corresponds to an average relaxation time for this polymer around 0.08 s at 150°C, or 0,05 s at 190°C.

Experimental set-up

The experimental set-up for measuring the electrokinetic charging of melt during the capillary extrusion is mainly composed of two coupled apparatus as shown on figure 2: (i) The rheometer is a capillary extruder, where the polymer is first melt, then pushed by a piston at constant velocity through a die. The dies are made of tungsten carbide with an entry angle of 180° and the study is performed for dies of different length -L-over diameter -Dratios -L/D-. Note that the minimum time spent by the polymer in the die, in the sharkskin zone, is about 1s at 150°C and 0,4s at 190°C.

(ii) The set-up for measuring the electrical charging is based on the measurement of the electric field induced by the charges present on the polymer extrudate. The mPE melt is collected, at the die exit, in a Faraday cup. The charges present on the extrudate induce the appearance of influence charge on the outside of the pail (equal and opposite to the nett quantity of charge placed into the cup). The electric field generated by these charges is then measured by a field-meter mill placed under the Faraday pail. The Faraday pail is a sensitive instrument for precise and reliable measurement of net electrostatic charge. This experimental set-up has a very good sensitivity that allows charges to be measured with a resolution down to 1pC -corresponding to surface charge down to 10 -9 C/m 2 -. The sensitivity is comparable to the one of the previous experimental set-up used by Taylor et al., that is based on the same principle of collecting the electrical charges [START_REF] Taylor | The electrokinetic charging of polymers during capillary extrusion[END_REF].

Results on the electrification during extrusion

Typical electrification curve

One typical curve of electrical charging versus wall shear stress τ w is presented on figure 3. Note that the charges remain zero until the extrudate is smooth. Then the density of charges increases regularly, with a charge density per unit area σ in the range of 10 -8 -10 -7 C/m 2 . The end of the sharkskin defect, that corresponds to the beginning of the spurt defect also coincides to a sudden increase of the electrical charging.

It is interesting to recall that the onset of wall slippage has been observed for such mPE before the onset of spurt [START_REF] Hatzikiriakos | Interfacial phenomena in the capillary extrusion of metallocene polyethylenes[END_REF][START_REF] Pérez-González | Temperaturedependent instabilities in the capillary flow of a metallocene linear low-density polyethylene melt[END_REF]. This wall slip might be correlated to the appearance of the electrical charges.

But it has been impossible to obtain the velocity curve versus wall shear stress for the studied mPE by the classical Mooney method. Indeed the use of different die diameters has evidenced the variation of the flow curve with the die diameter (figure 3 (b)), that is generally considered as a proof of wall slippage. But, for the studied mPE, the measurements for three different diameters leads to some inconsistencies in the order of the flow curve versus the length diameter, so that the Mooney method is not reliable for determining the slip values, as previously noted by Robert et al. [START_REF] Robert | Instabilité oscillante de polyéthylènes linéaires: observations et interprétations[END_REF] . In this case, the slip velocity is probably not simply related to the wall shear stress, or the values of the slip are too small. The fact that wall slip is probably lower than the usual values observed for other mPE is consistent with the fact that this metallocene polymer, on the contrary to the one studied by Hatziriakos, is not branched [START_REF] Hatzikiriakos | Interfacial phenomena in the capillary extrusion of metallocene polyethylenes[END_REF][START_REF] Pérez-González | Temperaturedependent instabilities in the capillary flow of a metallocene linear low-density polyethylene melt[END_REF]. Indeed Hatziriakos has explained the slip in terms of chains detachment from the interface, this detachment being favored by the branched feature of the mPE.

So the numerical link between v s and σ could not have been established for this specific polymer but further work is in progress to evaluate the value of slippage for this mPE. .

Influence of the dies geometry

The second point that has drawn our attention is the influence of the die geometry on the electrification. Figure 4 (a,b) shows the two curves of surface charge density σ versus wall shear stress τ w for two die diameters D=1mm (Fig. 4a), and D=2mm (Fig. 4b). They sum up very well the observed trends : charging differs with both dies diameter and length. For example in the case of D=1mm, when increasing the length L of the die, the charge density σ seems to increase. On the contrary, for D=2 mm, increasing the length of the die seems to decrease σ, suggesting that the relationship between electrical charging and geometry of the die is not simple. But the fact that the charging rate is not linear versus dies length, at imposed diameter, is an important fact that will be discuss later.

A particular feature appears for the die with D=1mm and L=30mm where there is a sudden decrease that could be attributed to dielectric breakdown. Some dielectric breakdowns were also reported by Dreval et al. in spurt conditions, they attributed these breakdowns to the increase of the charge density with L, limited by the electric discharge processes between the polymer and the die wall at σ >4/6 10 -5 C/m 2 [START_REF] Dreval | The rheology and static electrication of linear flexible chain polymers[END_REF].

Influence of the temperature

The electrical charging measurements were performed for three temperatures of the mPE melt: T=155°C, T=170°C and T=190°C. On figure 5(a), for a die with L=30mm and D=2mm, one can observe that for these three temperatures, the variations of the charge density σ with the shear rate Analysis of the data from the figure 5(a) indicates that the thermo-dependence of the electrical charging is probably governed by an Arrhénius law like for all the linear viscoelastic functions. We assume that the activation energy for the electrical charging is the same as the one determined for the rheological G'-G" master curve building (cf. § 2.1), so that the same reduction parameters a T has been used to built the figure 5 (b). The resulting master curve for the electrical charging is correct, and this leads to an important conclusion: the timetemperature principle is applicable to the electrification process during polymer melt extrusion, that implies a contribution of the segmental mobility of the polymer in this electrification mechanism. By using the same horizontal shift factor, it has been also possible to built a master curve for L=20mm. An additional vertical factor would be necessary to obtain a good continuity of the master curve for L=10 mm.

Discussion

Comparison with previous studies

The comparison between these results and the previous studies evidences four main points :

First, for sharkskin instability, the charge density values are in the same range as the ones Second, it is the first time -to our knowledge-that electrical charges are observed during sharkskin defect. For Taylor study, they used a LDPE which did not exhibit sharkskin, and the other authors have noticed the presence of charge only for strong slip velocities under spurt and overspurt conditions. One explanation could be the better resolution of the experimental set up with a Faraday collector compared to the measurement by swiping the probe of the electrometer on the extrudate surface [START_REF] Pérez-González | Exploration of the slip phenomenon in the capillary flow of linear lowdensity polyethylene via electrical measurements[END_REF], or by the variation of a cylindrical capacitance [START_REF] Vinogradov | Viscoelastic properties and flow of narrow distribution polybutadienes and polyisoprenes[END_REF].

If the correlation between electrical charges generation and slippage could be demonstrated, another explanation could be the fact that mPE generally exhibits higher slip values in the sharkskin domain as the other polymers [START_REF] Hatzikiriakos | Interfacial phenomena in the capillary extrusion of metallocene polyethylenes[END_REF].

Third this study has confirmed the influence of experimental parameters such as the length and the diameter of the die, but no clear variations of the charge density versus the surface area of the die could be evidenced.

Fourth, the influence of the temperature on the electrical charging has shown that the electrification is related to the molecular properties of the polymer such as the segmental mobility, since the time-temperature superposition is applicable.

About possible mechanisms

Two tentative explanations of the electrical charging can be proposed:

The first one is to imagine a mechanism like in solid-solid contact [START_REF] Lowell | Contact electrification[END_REF], named further triboelectrification mechanism. In the case of solids, the mechanism is well-known for the contact between two metals. Charges that are exchanged are electrons and the transferred quantity is proportional to the difference between the two work functions, that corresponds to the equality of the Fermi levels. Even in solid-solid contact, the charge transfer between a metal and an insulator is more complicated, because it depends on a lot of parameters: duration of the contact, number of contacts, area of contact... Even if no certain mechanism has been ascertained, most of the studies conclude that the exchanged charges are electrons and the quantity of transferred charges is governed by the difference between the metal Fermi level and some energy level characteristic of the insulator. In solid case, the transferred charges are usually in the range 10 -5 to 10 -3 C/m 2 . It is also well-known that, for polymers, the determination of the real contact area is fundamental in the determination of the transferred charge density because of the increase of charge with time due to viscoelastic increase in the area of contact. For contact with sliding, it has been found that the charge deposited could be independent of the speed of sliding, or not, depending on the time of transfer and on the backflow of charges [START_REF] Lowell | Contact electrification[END_REF]. The transfer of charge to a polymer has been supposed to be assisted by the stirring of molecules near to the surface, or by the number of polymer segments in contact with the metallic surface, and so by the molecular motion [START_REF] Ohara | Contribution of molecular motion of polymers to frictional electrification[END_REF].

Similarly, in the case of a polymer melt going through a die, one can imagine that electrons are transferred between the polymer melt and the material of the die, and that the wall shear stress τ w is mainly responsible of the number of contact between the polymer and the die wall, and so of the transferred charges . However in that case the quantity of transferred charge should be proportional to the length of the die at imposed diameter. But it has been shown previously on the experimental curves that the charging is not linearly linked to the die length . This contradiction induces two conclusions. Either the true contact area is not directly given by the surface of the die, which is likely for a viscoelastic material, or the hypothesis of the charging by electron transfer is not valid. Indeed, for an imposed die diameter D, one can suppose that the surface charge σ 0 exchanged between a surface unit of the die wall (located at the height x) and an infinitesimal disk of polymer (diameter D) is constant along the die ( from the entrance of the die (x=0) to the exit of the die (x=L),) σ 0 (x, D, T, τ wx )=σ 0 (0,D,T, τ w0 )

This assumption σ 0 (x, τ wx )=σ 0 (0, τ w0 ) is based on the fact that the wall shear stress τ w is mainly responsible of the transferred quantity of charges, and that in the present study, we can consider that τ wx = τ w (x) = τ w (0)= τ w0 . Indeed, (i) the pressures used in this study have sufficient low values (P<500 bar) to consider that the influence of the viscosity is negligible on τ w , and (ii) the residence time of the melt in the die is much higher than the mean relaxation time of the mPE (see § 2.1 and 2.2), so that the flow can be considered to be established very quickly inside the die.

In that case of an exchange of charge constant along the die length, at imposed diameter and wall shear rate, the total surface charging of the infinitesimal disk of polymer might be directly proportional to the length of the die. So that the surface density of charges measured at the exit of the die might be linearly related to the die length.

If the charging by electron transfer is not valid, an alternative hypothesis might be the stripping of the well-known double layer. Indeed the tribocharging in liquids can be due to the relative motion between the fluid and the solid interface [START_REF] Taylor | The electrokinetic charging of polymers during capillary extrusion[END_REF][START_REF] Harvey | Effect of oil quality on electrostatic charge generation and transport[END_REF][START_REF] Touchard | Flow electrification of liquids[END_REF]. The charged metal surface attracts opposite charges and repels like charges. That leads to the formation of a double layer structure composed of an innerlayer formed of the adsorbed ions and of a diffuse layer (thickness λ) that compensates the innerlayer charge [START_REF] Israelachvili | Intermolecular and Surface Forces[END_REF]. The properties of double-layers depend on the temperature, on the properties of the liquid and solid and also on the concentration of charged species in the liquid.

Under forced convection, two other layers can be defined, a stagnant layer (thickness δ d )

near the interface and a flow layer further. For insulating liquids the thickness of the diffuse layer is greater than the thickness of the stagnant layer, λ>δ d , so that the electroneutrality is not recovered within the stagnant layer [START_REF] Touchard | Flow electrification of liquids[END_REF]. In that case, charges can be entrained in the flowing liquid. This theory, that has been initially developed for the electrostatic charging of oil during pipeflow, could be applied to the polymer extrusion [START_REF] Taylor | The electrokinetic charging of polymers during capillary extrusion[END_REF][START_REF] Taylor | Outlets effects in the measurements of streaming currents[END_REF]. Indeed charges are present in the polymer melt due for instance to metallic impurities in the case of the mPE (cf. § 2.1). The amount of ionic impurities necessary to built up an electrical double layer is extremely low (some ppm) [START_REF] Harvey | Effect of oil quality on electrostatic charge generation and transport[END_REF]. This mechanisms fits well with the experimental facts observed for the tribocharging during sharkskin instability, i.e. the facts that the charging is linked to the molecular properties of the polymer, or to experimental parameters (length and diameter of the die, temperature...). Indeed parameters such as the diffusion coefficient of the liquid, the temperature, the flow velocity, die geometry etc, are directly linked to the tribocharging in this theory. Note also that, for newtonian fluids, δ d is proportional to the reciprocal of the wall slip velocity, leading to an increase of the charging when the wall slip increases [START_REF] Harvey | Effect of oil quality on electrostatic charge generation and transport[END_REF]. This point is consistent with our observation of a link between wall slippage and electrical charging. However in the case of mPE, further work is necessary to determine the nature of the ions present in the double layer, the influence of the slip velocity, and to adapt the published models to non-newtonian fluid...

Perhaps the both mechanisms are suitable successively. One can assume that the double layer stripping mechanism is dominant in the case of stable flow, and that the triboelectrification mechanism becomes dominant for unstable flow with strong wall slip.

Conclusion

It has been shown that the electrification of the melt is present simultaneously with the flow instabilities -in particular with the sharkskin-, its magnitude varying parallel to the nature of the instabilities.

The electrical charging behaviour is the same as the slip velocity behaviour according to the wall shear stress, that implies that the measurement of electrical charges could lead to an evaluation of the slip velocity -particularly for low slip velocity values-.

The study of the influence of experimental parameters leads to the conclusions that the electrification is not proportional to the apparent contact area between the melt and the die, and that the charging is related to the molecular properties of the polymer.

Finally two tentative explanations have been recalled for this electrical charge generation during extrusion: either it is due to an electron transfer between the polymer and the die like in solid-solid contact (named triboelectrification mechanism), or it is due to the stripping of

•γ 1 )

 1 are the same and that the ranges of the charge density are also similar. 8 Note also, for experiment at T=170°C, that the initial charging occurs for charging (corresponding to the beginning of spurt) for 1 -i.e. a range of Δ • γ = 90 s -1 -. So the electrical charging seems to follow the variations of the range of appearance of the sharkskin defect as shown on the cartography of the surface morphology Fig.5(c) established by Fernandez et al for the same mPE[START_REF] Santamaria | Postponing sharkskin of metallocene polyethylenes at low temperatures: the effect of molecular parameters[END_REF]. Indeed as the temperature decreases from 190°C to 170°C, the sharkskin defect arrives for smaller shear rate and the range of the sharkskin presence is reduced.For electrical charging, in the case of T=155°C, the value of σ at low shear rate ( is under-evaluate because, due to the low temperature of the melt, the extrudate became hard too quickly and could not be correctly stocked in the Faraday pail. But the same tendencies are still observed, the electrical charging occurrence corresponding to the sharkskin defect occurrence.

  measured for laminar flow by Taylor. Indeed σ varies from 10 -9 C/m 2 to 1.6 10 -7 C/m 2 as shown on figure 4, and Taylor measured charges between 10 -7 C/m 2 to 10 -5 C/m 2 for various polymers (PS, PC, LDPE...), the charges being the smallest for LDPE [4]. However this density of electrical charges is smaller than the one measured in spurt or overspurt conditions by Vinogradov et al., Dreval et al. and Pérez-González et al. Indeed they found charges around 10 -5 C/m 2 .

Figure captions Figure 1 :

 captions1 Figure captions

Figure 2 :

 2 Figure 2: Schematic diagram of the apparatus

Figure 4 :Figure 5 :Figure 1

 451 Figure 4: Surface charge density σ versus wall shear stress τ w , at T=190°C, for various

Figure 2

 2 Figure 2 S. Tonon et al., Electrical charging during the flow instabilities of a metallocene melt.

Figure 3

 3 Figure 3 S. Tonon et al., Electrical charging during the flow instabilities of a metallocene melt.

Figure 4 Figure 5

 45 Figure 4 S. Tonon et al., Electrical charging during the flow instabilities of a metal.locene melt.

Acknowledgments

The authors acknowledge M. Fernández and A. Santamaría for the permission of using their results concerning temperature-shear map. This work was financially supported by the Conseil Régional d'Aquitaine (CTP).

the double layer due to the forced convection and the presence of ionic impurities in the polymer melt. Further work is necessary to choose between these two models or to determine the relative contribution of each model.


J Besson

A Pineau

G Rousselier

Tvergaard , S. Pommier etc. A Needleman

Classes of material behaviour : relevant variables Linear elastic behaviour: linear elastic fracture mechanics (K)

The fracture mechanics concepts were still unknown Causes of fracture:

• Welded Structure rather than bolted, offering a substantial assembly time gain but with a continuous path offered for cracks to propagate through the structure.

• Low quality of the welds (presence of cracks and internal stresses) • Low quality steel, ductile/brittle transition around 0°C

Foundations of fracture mechanics : The Liberty Ships

One basic assumption :

The structure contains a singularity (ususally a geometric discontinuity, for example: a crack)

Two main questions :

What are the relevant variables to characterize the risk of fracture and to be used in fracture criteria ?

What are the suitable criteria to determine if the crack may propagate or remain arrested, the crack growth rate and the crack path ?

Classes of fracture mechanisms : criteria • Gc is a material constant (single mechanism, surfacic mechanism only)

• What if non isothermal conditions are considered ?

• Unsteady crack growth criteria, non applicable to steady crack propagation,

•

Order of this singularity

Linear elasticity:

0 1 1 0 2 2 1 0 2 2 2 2 2 0 2 2 max 0                                 r A E r elast For a crack : =-0.5   elastic n n o o 1                   n r Br r 0           2 1 2 2 2 1 max 0 0 1 1 0 0 0 max max                                   n r C E

Fracture modes

Various fractures in compression

Fracture modes

Various fractures in torsion

                           z y x z y x z y x zz yz xz yz yy xy xz xy xx          Linear isotropic elasticity : E, n     xy xy xx yy yy yy xx xx E E E  n  n  n  n  n          1 1 1 2 2   1 1  n  n  Tr E E    Compatibility equations                     x u y u y u x u y x xy y yy x xx 2 1    y x u y x xx 2 3 2 2        x y u x y yy 2 3 2 2        y x u x y u y x x y xy 2 3 2 3 2 2             y x y x xx yy xy 2 2 2 2 2 2             Combination = 3 Equations, 3 unknowns

Balance equations

Compatibility

Linear elasticity

y x y x xx yy xy 2 2 2 2 2 2                 xy xy xx yy yy yy xx xx E E E  n  n  n  n  n          1 1 1 2 2 y x y x xx yy xy 2 2 2 2 2 2             + 0 0             y x y x yy xy xy xx     + Airy function F(x,y) -1862- 1 equation, 1 unknow F(x,y) Balance equation Compatibility Assuming 0             y x y x yy xy xy xx     y x y x xx yy xy 2 2 2 2 2 2             0 2 4 4 2 2 4 4 4           y F y x F x F y x F x F y F xy yy xx           
0 ,       xy yy xx S      4 3 2 2 2 2 a y a x a x y S F         4 2 2 2 a x y S F     & y x F x F y F xy yy xx            2 2 2 2 2    Construction of Z(z) Relation   4 2 2 2 a x y S F     4 2 2 a z S Z                              z Z yI Z R y Z yR Z R F m e e e y x F x F y F xy yy xx            2 2 2 2 2                                                            3 3 3 3 2 2 3 3 2 2 z Z yR z Z yI z Z R z Z yI z Z R e xy
                                                       3 3 3 3 2 2 3 3 2 2 z Z yR z Z yI z Z R z Z yI z Z R
+ + Exact solution a z  1 a z  1 Sz z Z a z S Z        4 2 2   2 1 2 2 a z S z Z                                                             
  2 1 2 2 2 2 a z Sz z Z       2 3 2 2 2 3 3 a z Sa z Z       i re a z     2 2 1 2 2 2 2   i i e r a S are Sa z Z        2 3 2 3 2 3 3 2 1 2   i i e r a S r are Sa z Z        Exact Solution   2 1 2 2 a z S z Z     
Westergaard's stress function :

2 2 2 2  i e r a S z Z     2 3 3 3 2 1  i e r a S r z Z                             2 3 cos 2 sin 2 cos 2 2 3 sin 2 sin 1 2 cos 2 2 3 sin 2 sin 1 2 cos 2                   r a S r a S r a S xy yy xx                                                        

Similitude principle

(geometry locally planar, with a straigth crack front, self-similar, singularity)

Same KI & T → Same local field 2 3 cos 2 sin 2 cos 2 2 3 sin 2 sin 1 2 cos 2 2 3 sin 2 sin 1 2 cos 2                r K r K T r K I xy I yy I xx                  
                             cos 2 sin 2 2 cos 2 cos 2 2 2 3 cos 2 sin 2 cos 2 2 3 sin 2 sin 1 2 cos 2 2 3 sin 2 sin 1 2 cos 2                       r K u r K u r K r K T r K I y I x I xy I yy I xx                              cos 2 2 cos 2 2 cos 2 2 sin 2 2 2 3 sin 2 sin 1 2 cos 2 2 3 cos 2 cos 2 sin 2 2 3 cos 2 cos 2 2 sin 2                         r K u r K u r K r K r K II y II x II xy II yy II xx Mode II 2 sin 2 2 4 2 cos 2 2 sin 2          r K u r K r K III z III yz III xz   G    Mode III   ) 4 3 ( n    n      

Williams expansion

A self-similar solution in the form is sought directly as follows :

x r  y 0 2 4 4 4 2 2 4 4 4             F y F y x F x F        g r r F 2 ,                                                                                         4 4 2 2 2 2 2 2 2 4 4 4 2 2 2 2 2 2 2 2 2 2 2 2 4 2 2 2 2 2 2 2 2 2 2 2 2 1 1                                 g g g r F g r g r g r g r F g r g r F r r F r r r F x r  y             0 2 2 4 4 2 2 2 2 2 2 2 4                                g g g r F             0 2 2 2 2 2 2 2 2 4 4                g d g d d g d

Dans ce cas g() doit vérifier

Williams expansion

A self-similar solution in the form is sought directly as follows :

       g r r F 2 ,   67
The solution is sought as follows :

x r  y     ip Ae g                2 2 0 2 2 2 2 2 2 2 2 2 2 2 4                         p p p p p p             0 2 2 2 2 2 2 2 2 4 4                g d g d d g d Williams expansion x r  y                   2 2 2 Re ,          i i i i De Ce Be Ae r r F
Boundary conditions are defined along the crack faces which are defined as free surface (fluid pressure & friction between faces are excluded)

        0 , , 0 , , 2 2                                       r r F r r r r F r r Williams expansion x r  y                   2 2 2 Re ,          i i i i De Ce Be Ae r r F                                     0 2 2 Re 0 2 2 Re 0 , 0 Re 0 Re 0 , 2 2 2 2 2 2 2 2                                                                                     i i i i i i i i r i i i i i i i i De Ce Be Ae De Ce Be Ae r De Ce Be Ae De Ce Be Ae r

Williams expansion

A sery of eligible solutions is obtained :

x r  y                                                                                1 2 sin 1 2 sin 1 2 cos 1 2 cos 1 2 n C n A g n D n B g n n even n odd       g r r F n 1 2 ,  
La solution en contrainte s'exprime alors à partir des dérivées d'ordre 2 de F, toutes les valeurs de n sont possibles, tous les modes apparaissent • History effects in mixed mode

Williams expansion

Williams versus Westergaard

        0 0 , , 2 2 0 , , 2 0 , 0 ,            n        r r K r r K r r xy I zz I yy xx     n      1 3 2 2 0 , r K r p I H     r K r I eq  n   2 2 1 0 ,    Yield criterion :   Y Y eq r      0 ,
 n   2 2 1 2 max 0 0 max   2 2 2 2 1 2 Y I Y pm K r r   n    Monotonic plastic zone Cyclic plastic zone   2 2 max 2 2 1 Y I mpz K r   n     2 2 2 4 2 1 Y I cpz K r   n D          a S K yy I         
• Observations

•

  2700 Liberty Ships were built between 1942 and the end of WWII • The production rate was of 70 ships / day • duration of construction: 5 days • 30% of ships built in 1941 have suffered catastrophic failures

  grain boundary corrosion, creep, oxydation, persistent slip band in fatigue etc… ) • Plastic zone scale or damaged zone (material hardening or softening, continuum damage, ductile damage.problem, planar crack with a straight crack) Curved cracks, branched cracks, merging cracks (3D problem, nonplanar cracks, curved crack fronts) Short cracks (3D problem, influence of free surfaces, scale and gradients effects) energy release rate G Criteria : An unsteady crack growth occurs if the cohesion energy released by the structure because of the creation of new cracked surfaces reaches the energy required to create these new cracked surfaces the elastic energy of the structure : variation of the surface energy of the structure (Rice's integral if q is coplanar) q vector: the crack front motion If the crack faces are free surfaces (no friction, no fluid pressure …), If volume forces can be neglected (inertia, electric field...)Then the J integral is shown to be independent of the choice of the selected integration contour 𝐺 = 𝐽= Γ 𝜑 𝑓𝑟𝑒𝑒 𝑒𝑛𝑒𝑟𝑔𝑦 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑑𝑦 -𝜎𝑛. faces must be free surfaces (no friction, no fluid pressure)

  The surfacic energy 2 may be negligible compared with the energy dissipated in plastic work or continuum damage / localization process Linear Elastic Fracture Mechanics (LEFM) Characterize the state of the structure where useful (near the crack front where damage occurs) for a linear elastic behavior of the material Stress concentration factor Kt of an elliptical hole, With a length 2a and a curvature radius r

F=F

  the plane is defined by a complex number z = x + i y Z a function of z : Z(z)=F(x,y) y=0 x=+a & y=0 x=-a Symmetric with respect to y=0 & x=0 2D problem, plane (x,y) : S zz =n(S xx +S yy ) 6 boundary or symmetry conditions 2 singularities, 0 boundary conditions along the crack faces Exact solution Taylor's development with respect to the distance to the crack front Separated variables Similitude principle Boundary conditions & Symmetries symmetries

  for any 2D problem, with symmetries along the planes y=0 & x=0, and biaxial BCs



  for the 3 modes, determined for one specific geometry -Taylor development, 1 st order → asymptotic solution generalized to any other cracks -First order -Solution expressed with separate variables f (r) g () and f (r) self-similar -Solution : f (r) a power function, r  , with  = -1/2 -Higher Orders -A unique stress intensity factor for all terms -The exponent of (r/a) increasing with the order of the Taylor's development -Boundary conditions -Singularity along the crack front, symmetries, planar crack and straight front -no prescribed BCs along the crack faces, -Boundary conditions defined at infinity 6 independent components of the stress tensor at infinity → 6 degrees of freedoms in MLER: KI, KII, KIII and T, Tz, and G

-

  The boundary conditions are free surface conditions along the crack faces (apply on 3 components of the stress tensor), no boundary condition at infinity → absence of T, Tz, and G -Super Singular terms → missing BCs -The first singular term of the Williams expansion is identical to the first term of the Taylor expansion of the exact solution of Westergaard -The stress intensity factors of the higher order terms are not forced to be the same as the one of the first term, -advantage, leaves some flexibility to ensure the compatibility of the solution with a distant, non-uniform field -drawbacks, it replaces the absence of boundary conditions at infinity by condition of free surface on the crack, and it lacks 3 BCs, it is obliged to add constraints T, Tz, and G

POD based post treatment Elastic domain : generalized Von Mises Criterion

  Same values of K max , K min , DK for each mode Fatigue crack growth experiments Solution of an elastic FE analyses with 𝑲 𝒊 ∞ =1MPa.m 1/2 for each mode 𝒗 𝒓é𝒔𝒊𝒅𝒖_𝒊 𝑷, 𝒕 = 𝒗 𝑬𝑭_𝒊 𝑷, 𝒕 -𝝂 𝒊 𝒆 𝑷, 𝒕 𝑷𝑶𝑫𝟏 → 𝒗 𝒓é𝒔𝒊𝒅𝒖_𝒊 𝑷, 𝒕 ≈ 𝝆 𝒊 𝒕 . 𝒖 𝒊 𝒄 (𝑷)

	Growth criteria in mixed mode conditions ? POD based post treatment Model
	𝑑𝑁 𝑢 𝑖 𝑒 (𝑃) = 𝐶∆𝐾 𝑒𝑞 𝑚 𝑓 𝑌 = 𝐾 𝐼 ∞ -𝐾 𝐼 𝑋 𝐾 𝐼 𝑌 2 + 𝐾 𝐼𝐼 ∞ -𝐾 𝐼𝐼 𝑋 𝐾 𝐼𝐼 𝑌 𝒇 = 𝑲 𝑰 ∞ -𝑲 𝑰 𝑿 𝟐 𝑲 𝑰 𝒀 𝟐 + 𝑲 𝑰𝑰 ∞ -𝑲 𝑰𝑰 𝑿 𝟐 𝑲 𝑰𝑰 𝒀 𝟐 + 𝑑𝑎 Yield criterion	2 𝑲 𝑰𝑰𝑰 -1 𝑲 𝑰𝑰𝑰 𝒀 𝟐 ∞ -𝑲 𝑰𝑰𝑰 𝑿 𝟐	-𝟏
	𝛥𝐾 𝑒𝑞 = ∆𝐾 𝐼 𝒇 𝑮 𝑰 , 𝑮 𝑰𝑰 , 𝑮 𝑰𝑰𝑰 = 𝑮 𝑰 𝑮 𝑰 𝒀 + 𝑛 + 𝛽∆𝐾 𝐼𝐼 𝑮 𝑰𝑰 𝑮 𝑰𝑰 𝒀 + 𝑮 𝑰𝑰𝑰 𝑮 𝑰𝑰𝑰 𝒀 -𝟏	𝑛 + 𝛾∆𝐾 𝐼𝐼𝐼	𝑛 1	𝑛
	• Crack growth rate • Crack path • Simulation • Modelling Crack growth rate Crack path 𝑲 𝒊 𝒕 = 𝝂 𝒊 𝒆 𝑷, 𝒕 = 𝑲 𝒊 𝒕 𝒖 𝒊 𝒆 (𝑷) 𝒗 𝒓é𝒔𝒊𝒅𝒖_𝒊 𝑷, 𝒕 = 𝒗 𝑬𝑭_𝒊 𝑷, 𝒕 -𝝂 𝒊 𝑷𝝐𝑫 𝒗 𝑬𝑭_𝒊 𝑷, 𝒕 . 𝒖 𝒊 𝒆 (𝑷) 𝑷𝝐𝑫 𝒆 (𝑷). 𝒖 𝒊 𝒆 (𝑷) 𝒆 𝑷, 𝒕 𝑃𝑂𝐷2 → 𝑢 𝑖 𝑐 (𝑃) ≈ f 𝑟 𝑔 𝑖 𝑐 (𝜃) 𝑔 𝐼𝑦 𝑐 𝜃 = 𝜋 = 𝐼𝑦 1 𝑓 𝑌 = 𝐺 𝐼 𝐺 𝐼 𝑌 + 𝐺 𝐼𝐼 Flow rule 𝑌 -1 𝐺 𝐼𝐼 𝒔𝒊𝒈𝒏𝒆 𝑮 𝒊 𝝆 𝒊 = 𝝀 𝒀 𝑮 𝒊 𝑐 𝜃 = -𝜋 = 2 lim 𝑟→0 𝑓 𝑟 =1 𝐺 𝑖 = 𝑠𝑖𝑔𝑛 𝐾 𝑖 ∞ -𝐾 𝑖 𝑋 𝐾 𝑖 ∞ -𝐾 𝑖 Evolution equation 𝑋 2 𝐸 * 𝑲 𝑿 = 𝑪 𝝆 -𝚪 𝑲 𝑿𝒆𝒒 𝑴-𝟏 𝟏 + 𝚪 𝑲 𝑿𝒆𝒒 𝑲 𝑿 𝑴-𝟏 𝒅 𝝆 𝒅 𝒘𝒉𝒆𝒓𝒆 𝒅 = 𝑲 𝒆𝒒 𝑿
					131 159

Ideally elastic-plastic material  Y =600 MPa, E=200 GPa, n=0.3 plane strain, along the plane =0