Thickness dependence of cracking resistance in thin aluminium plates

J. Mech. Phys. Solids 47 (1999) 2093±2123

Thickness dependence of cracking resistance in thin aluminium plates T. Pardoen 1, Y. Marchal 2, F. Delannay* DeÂpartement des Sciences des MateÂriaux et des ProceÂdeÂs, Universite catholique de Louvain, PCIM, Place Sainte Barbe 2, B-1348 Louvain-la-Neuve, Belgium Received 7 July 1998; received in revised form 6 January 1999

Abstract The in¯uence of thickness on the fracture toughness of aluminium 6082T0 thin plates of 1±6 mm thicknesses was investigated experimentally and numerically from tensile testing of cracked DENT specimens. The critical J-integral, Jc, critical CTOD, dCTODc, and essential work of fracture, we, are found to increase with thickness and to constitute equivalent measures of fracture toughness at small thickness. For larger thickness, Jc and dCTODc increase non-linearly with thickness and reach a maximum for 5±6 mm thickness whereas we keeps increasing linearly with thickness. This di€erence is related to a more progressive development of the necking zone in front of the crack tip when thickness increases: at large thickness, cracking initiates well before the neck has developed to its stationary value during propagation. we is more directly related to the steady-state crack growth resistance. A linear regression on the fracture toughness/thickness curve allows further separation of the two contributions of the essential work of fracture: the necking work and the fracture work spent for damaging. The maximum of the stress triaxiality ratio is shown to constitute a pertinent parameter for characterising how constraint a€ects cracking initiation in the present context where out-of-plane constraint dominates in-plane constraint. It allows justifying the shape of the Jc/thickness relationship and results in the proposal of a 3D Jc/ thickness/triaxiality fracture locus. As fracture pro®les are macroscopically ¯at with microscopic dimples and with only very small shear lips along the edges, a local criterion

* Corresponding author. Tel.: +32-1047-2426; fax: +32-1047-4028. E-mail address: [email protected] (F. Delannay) 1 Present address: Division of Engineering and Applied Sciences, Harvard University, Pierce Hall, Room 312, Cambridge, MA 02138, USA 2 Present address: SONACA SA, Parc Industriel, Route Nationale Cinq, B-6041 Gosselies, Belgium 0022-5096/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 5 0 9 6 ( 9 9 ) 0 0 0 1 1 - 3

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based on the growth and coalescence of voids has been used in order to predict fracture initiation. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: A. Fracture toughness; B. Plates; Elastic-plastic material; C. Finite elements; Mechanical testing

1. Introduction Today, a consensus seems to emerge about the measurement and transferability of fracture toughness parameters in the case of pure or nearly pure plane strain situations, even in the case of Large Scale Yielding (LSY) with low plastic constraint. In contrast, in the case of thin ductile plates, the characterisation of fracture toughness and the transferability of fracture parameters to real structures remain not suciently documented. The diculty lies in the three-dimensional character of the stress and strain ®elds at the crack tip: pure plane stress analysis is acceptable only in the case of very thin sheets. Furthermore, the mechanisms of fracture are more complex than in pure plane strain situations. Di€use plastic yielding and localised plastic yielding coexist under the form of a neck at the crack tip surrounded by a large plastic zone. The fracture surface usually shows evidence of two di€erent mechanisms: a conventional ductile fracture with coalesced voids is observed in the centre of the plate where the stress triaxiality is the highest whereas 458 shear lips dominate in the vicinity of the surfaces. Purely 458-shear fracture is frequently observed. Crack propagation analysis must account for large tunnelling e€ects and for the possible progressive twist of the originally ¯at crack into a slant fracture (i.e. the proportion of ¯at or shear fracture evolves during ductile tearing partly due to the tunneling e€ect (Knott, 1973)). Fracture toughness of very thin plates mainly results from plastic energy spent in the neck in front of the crack tip. This work of necking is often very large in comparison to the energy spent for damaging the material. As the energy spent in the neck depends on thickness, it results in the variation of toughness as a function of thickness mentioned by several authors (Bluhm, 1961; Swedlow, 1965; Knott, 1973; Broek, 1978; Barsom and Rolfe, 1987) as long as the stress state is purely plane stress, fracture toughness linearly increases with thickness. As thickness increases, the crack tip stress ®eld evolves towards plane strain and the increase of the fracture toughness slows down, reaches a maximum and decreases ®nally to the plane strain value. However, experimental results about the rising part of the fracture toughness/thickness curve are scarce. (It is worth mentioning the recent numerical work of Mathur et al. (1996) about dynamic crack growth in ductile thin plates in which a similar variation of the crack growth rates as a function of plate thickness is reported.)

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A few authors have described the development of the 3D ®elds at the crack tip during necking, the e€ect of strain-hardening and the physical mechanisms responsible for crack initiation (Hom and McMeeking, 1990; Nakamura and Parks, 1990). Several authors have addressed the problem of crack propagation in thin plate from experimental, theoretical (always steady-state crack growth) and computational points of view (Rice, 1973; Broberg, 1975, and, more recently, May and Kobayashi, 1995; Dawicke et al., 1995; Gullerud et al., 1998). Nonetheless, the literature still lacks systematic studies of the relation between the resistance to cracking initiation and plate thickness which remains insuciently understood. In this paper, the conditions for cracking initiation in 6082T0 aluminium plates are investigated from an experimental and numerical point of view using DENT (Double Edge Notched Tension) specimens of various thicknesses, t0, (1±6 mm) and various ligament lengths, l0 (see Fig. 1). These plates present a macroscopically ¯at fracture pro®le with dimples at the microscale and very small shear lips. The aim is to elucidate the e€ect of thickness t0 on cracking initiation in thin plates. Fracture toughness is ®rst discussed in terms of the critical J-integral Jc, critical crack tip opening displacement dCTODc, and essential work of fracture we. A method is proposed for separating the fracture work spent for localised necking and a more intrinsic fracture resistance of thin plates. The plastic constraint, de®ned here in terms of the stress triaxiality ®eld, is analysed in order to better understand the dependence of fracture toughness on geometry. A simple local

Fig. 1. Testing of a DENT specimen.

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criterion of fracture related to the mechanisms of ductile fracture by void growth and void coalescence is then introduced in order to predict cracking initiation in the studied specimens. Void growth rates will be estimated using the Rice and Tracey model (Rice and Tracey, 1969) and void coalescence will be considered to start at a constant critical porosity (McClintock, 1968; d'Escatha and Devaux, 1979). The discussion will focus on the comparison of the di€erent methods for characterising fracture toughness in thin plates, on the separation of fracture energies, on the size of the Fracture Process Zone (FPZ) and on the problem of transferability of failure criteria in the case of thin plates.

2. Theoretical background 2.1. Continuum approaches to fracture toughness A coordinate system is taken, oriented in such a way that the X1 axis lies in the crack plane and is normal to the crack front, the X3 axis is tangential to the crack front and the X2 axis is orthogonal to the crack plane. The J integral (Rice, 1968) related to a point s on a crack front in a nonlinear elastic material is de®ned by (Bakker, 1984)    … …  @ui @ @ui Wd1j ÿ Pji nj dG ÿ P3i dA, J…s† ˆ @ X1 @ X1 G AG @ X3

…1†

where G is a curve enclosing the crack front at the position given by s, in the plane X3=0, AG is the surface area de®ned by G, W is the deformation work per unit volume, ui is a component of the displacement vector, Pij is a component of the Piola±Kirchho€ stress tensor, Xi is a component of the position vector of a material point in the undeformed con®guration, nj is a component of a unit vector perpendicular to G. In thin plates, J varies along the crack front due to 3D e€ects (see Nakamura and Parks, 1990, for a discussion of the through-thickness variation of J in thin plates). This work will only make use of the mean value of the J integral over the crack front that will be denoted simply J. For the DENT geometry, a relation based on the load/displacement curve was derived by Rice et al. (1973): JRice ˆ

… K 2I 1 …2 P dup ÿ Pup †, ‡ l0 t0 E

…2†

where KI is the stress intensity factor, E is the Young's modulus, P is the applied load and up is the plastic displacement. KI for a DENT plate is given by (Anderson, 1995)

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r pa0 " P   2w0 a0 r 1:122 ÿ 0:561 KI ˆ a0 p w0 t0 w0 1 ÿ w0 

a0 ÿ 0:205 w0

2



a0 ‡0:471 w0

3



a0 ‡0:19 w0

4 #

,

…3†

where a0 is the initial crack length and w0 is the half-width of the plate (see Fig. 1). A valid J integral can be calculated at the crack tip in an elastoplastic solid, subject to the condition that the crack tip ®nite strain zone in which damaging develops is suciently con®ned. J validity can be proved by verifying the path independence of J. J controls cracking when there exists, outside the ®nite strain zone, a zone of so-called `J dominance', i.e. a zone where the sole J parameter suces for characterising the stress and strain ®elds. If J dominates at the initiation of cracking, its critical value Jc constitutes a transferable measure of the fracture toughness of the material. In thin plates, for dimensional reasons, Jc depends on thickness and thus Jc can be a transferable measure of the fracture toughness only at a given thickness. However, before steady-state crack growth, J does not generally dominate in thin plates even at a given thickness because the stress state at the crack tip evolves, during loading, from closer to plane strain to closer to plane stress (Nakamura and Parks, 1990). This change of the stress state can be discussed under the general framework of constraint e€ects in fracture mechanics. Constraint changes in thin plates are mainly caused by out-of-plane e€ects. They depend both on the initial thickness (which determines the initial constraint) and on the level of the loading. The most popular approaches for characterising constraint e€ects in LSY are the J±Q method of O'Dowd and Shih, 1991a, b and the J-A2 method introduced by Yang et al. (1993) and recently assessed by Chao and Zhu (1998). However, the limit of validity of these approaches for LSY and 3D situations with signi®cant out-of-plane constraint e€ects is not clearly de®ned (Dadkhah and Kobayashi, 1994; Brocks and Schmitt, 1995; May and Kobayashi, 1995). Consequently, the discussion about constraint e€ects will be addressed directly from the analysis of the distribution of the triaxiality ratio h(X1, X2, X3) which is de®ned as h…X1 ,X2 ,X3 † ˆ

sm …X1 ,X2 ,X3 † , seq …X1 ,X2 ,X3 †

…4†

where sm is the hydrostatic stress and seq is the von Mises norm of the Cauchy stress tensor. The parameter h was ®rst used by Kordish et al. (1989) to account for the e€ect of constraint on fracture resistance (see Brocks and Schmitt, 1995,

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for a discussion of various methods based on a stress triaxiality parameter for quantifying the constraint). The concept of crack tip opening displacement, dCTOD, refers to the progressive increase of the displacement at the crack tip during blunting and crack propagation. All the conditions of dominance of the J integral apply, in principle, to dCTOD. The critical dCTOD, dCTODc, can thus be used as a measure of fracture toughness when J dominance conditions are ful®lled at initiation of cracking. J and dCTOD are proportionally related by the Shih factor, dn, in the form dn ˆ

s0 dCTOD , J

…5†

where s0 is the yield stress. dn has been tabulated for power law hardening materials (Shih, 1983). When the stress state changes (due to a constraint change), J and dCTOD do not evolve proportionally (Anderson, 1989; Nakamura and Parks, 1990; Pardoen and Delannay, 1996). dn thus also constitutes a convenient parameter for characterising the stress state. The value of dn at cracking initiation is denoted dnc. Cotterell and Reddel (1977) have suggested another way for characterising the cracking resistance of a plate. The work required to fracture the ligament of a DENT specimen is partitioned in two components. The ®rst component, called the essential work of fracture (the method will be called the `EWF method'), is assumed to be proportional to the initial ligament cross section l0t0, while the second component, called the non-essential work of fracture is assumed to be proportional to l20t0. The speci®c work of fracture obtained by dividing the total work of fracture by the initial ligament area l0t0 can thus be expressed as wf ˆ we ‡ bl0 wp :

…6†

If the product bwp is constant, the speci®c essential work of fracture we can be obtained by carrying out a linear regression on the values of the speci®c work of fracture wf measured for a range of DENT specimens with di€erent ligament lengths. However, for such a linearity to apply, the regression must be performed for ligament lengths large enough as to warrant that all these specimens present globally the same stress state across the ligament. More details about this condition of validity have been reported by Marchal et al. (1998). Many studies have shown that we adequately characterises the fracture toughness of the material for a given sheet thickness (Mai and Cotterell, 1980; Wnuk and Read, 1986; Chan and Williams, 1994; Levita et al., 1996). Moreover, theoretical and experimental work has aimed at demonstrating that we is equal or very close to the critical J integral (Mai and Powell, 1991). In a similar manner, a linear regression on the values of the ®nal displacement, uf , measured at completion of cracking for zeroligament length yields a displacement dc which can be interpreted as the critical crack tip opening displacement dCTODc of the plate (Cotterell and Reddel, 1977). The ratio s0dc/we, denoted here d nc, can be compared to dnc.

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2.2. Local approaches to fracture toughness At room temperature, ductile fracture of metal alloys usually ensues from the succession of nucleation, growth and coalescence of small internal voids. Many sophisticated models have been developed in the recent literature in order to account for the mechanisms governing these three stages (for a review paper, see Tvergaard, 1990) and implemented in the crack tip region (Xia et al., 1995). Only a simple model is used in this article. The voids are considered to be present from the beginning of straining and the void growth rate is estimated using the Rice and Tracey (1969) model. This model, adapted by Beremin (1981) to account for strain-hardening, evaluates the growth of an initially spherical void in an in®nite plastic material subjected to a uniform remote strain ®eld. With the assumption of spherical void growth, the average growth rate is approximately given by   dR 3sm ˆ a exp depeq , 2seq R

…7†

where R is the radius of the cavity, sm is the hydrostatic stress, equal to sii/3, e peq is the equivalent plastic strain, and a is a constant. The ratio sm/seq expresses the stress triaxiality. A value a=0.283 was predicted by Rice and Tracey (1969), but Huang (1991) re-evaluated the model and obtained a=0.427 and a=0.427(sm/ seq)0.25 for sm/seq>1 and for sm/seq < 1, respectively. A major approximation in the model is that no account is taken of the coupling between damage and the stress state. The onset of void coalescence is predicted assuming the attainment of a critical void size, Rc/R0 (McClintock, 1968; d'Escatha and Devaux, 1979). This criterion is evaluated from the void growth relation (7). The critical void growth must be independently determined on another geometry: in practice, notched round bars are commonly used. In this work, the critical void growth rate is determined from a tensile test on a cylindrical, notched round bar for which the average stress triaxiality is nearly the same as at the tip of a crack in the DENT specimens. When a coalescence model or, more generally, a local fracture criterion is applied at a crack tip, a characteristic volume, representative of the physical process of damage is introduced. This characteristic volume is called the Fracture Process Zone (FPZ). The Fracture Process Zone (FPZ) is the zone in front of the crack tip in which all the fracture phenomena localise during loading (Broberg, 1971). The FPZ is de®ned in the present modelling as the zone in which R/ R0>Rc/R0. The size of this zone at cracking initiation is sometimes assumed to be a constant in the local approach methodology. This constant is related to the void spacing, which is also frequently related to the critical CTOD dc (Rice and Johnson, 1970; Ritchie and Thompson, 1985). However, some authors (Mai and Cotterell, 1980) have clearly mentioned the thickness-dependence of the FPZ size in thin plates.

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3. Experimental and numerical procedures 3.1. Material The material consists of 6 mm-thick plates of aluminium alloy 6082 in the T0 state. Alloy 6082 is a heat-treatable alloy containing Mg and Si. The T0 state is obtained merely after the casting, heating and hot rolling processes. Three kinds of particles are present. The larger particles (about 5 mm size) are the primaries (AlFeMnSi-type), which form during the solidi®cation after casting. The secondaries or dispersoids contain the same elements, but are 10 times smaller and form from a supersaturated solid solution during the preheating stage preceding hot rolling. The third type of particles consist of Mg2Si and Si precipitates, which are spherical and have sizes of about 1 mm. These microstructural characteristics yield low values of hardness (35 HBN) and yield stress (s0=50 MPa), which are well suited for the purpose of this study. Indeed, high ductility results in a large range of thicknesses for which 3D e€ects (out-of-plane constraint e€ects) signi®cantly a€ect fracture toughness. The uniaxial stress-strain curve was measured by tension testing of cylindrical specimens of initial radius d0=5 mm machined from the 6 mm-thick plates. The Young's modulus is equal to 70 GPa, and the Poisson's ratio is equal to 0.34. Necking starts at eu=0.17 and su=130 MPa. In order to extend the strain range of the curve beyond necking, use was made of Bridgman's correction (Bridgman, 1952). The ¯ow rule is well represented by a power law curve s ˆ 202:1e0:247 :

…8†

3.2. Testing procedures In order to obtain a large range of thicknesses without change of the microstructure, the 6 mm-thick plates were thinned by mechanical milling. Microstructure was found homogeneous along thickness. Six plate thicknesses were obtained: 1, 2, 3, 4, 5 and 6 mm. The other dimensions of the DENT specimens, length and width, were 150  60 mm. Notches were made by cutting the plate ®rst with a saw and subsequently with a fresh razor blade in such a way as to obtain sharp initial notch tips with a radius of approximately 25 mm. The validity of this precracking method is justi®ed by the large values of dCTODc (see Section 4.3.3). The ligament lengths l0 were measured before testing by use of a travelling microscope. In order to simplify the notations, the di€erent specimens will be referred to as Al(t0 ÿ l0) with t0 and l0 expressed in mm. All DENT specimens were strained along the rolling direction until completion of cracking using a screw driven universal testing machine. The crosshead speed was 0.5 mm/min. The displacements were measured by means of an extensometer with an initial gauge length of 45 mm. Load/displacement curves were recorded for all specimens. Fracture initiation, de®ned by the appearance of thumb nails on

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the crack front at the centre of the plates (El Soudani, 1990), was detected in each test by means of a camera equipped with a zoom. On several polished sections (i.e. at di€erent locations in the thickness) of unloaded specimens, dCTOD was measured as the displacement of the crack ¯anks at the intersection with a 908 vertex centered at the crack tip. Elastic displacement can be neglected owing to the very large plastic part of the opening. The measurement of we for the six sheet thicknesses ranging from 1 to 6 mm involved the testing of a large number of DENT specimens with di€erent ligament lengths. For each specimen, the total work of fracture was calculated as … uf P du, …9† Wf ˆ 0

where uf is the displacement at the completion of cracking along the ligament. The total speci®c work of fracture wf was obtained by dividing Wf by the initial ligament area. In order to evaluate the critical damage ratio Rc/R0 at coalescence, cylindrical specimens of 5 mm diameter with axisymmetrical round notches of 1 mm radius and 1 mm depth were machined from the 6 mm-thick A6082T0 plates. The tensile tests were performed at a loading rate of 0.5 mm/min on a universal testing machine. The diameter reduction was measured using a radial extensometer. The average equivalent strain is calculated as   D0 , …10† eeqm ˆ 2 ln D

Fig. 2. Finite element mesh, modelling one eighth of an Al3-10 DENT panel.

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where D0 is the initial diameter of the minimum section and D is the current diameter. 3.3. Numerical procedure The tensile tests were simulated until cracking initiation for DENT specimens with thickness t0 equal to 1±6 mm and ligament lengths l0 equal to 5, 10, 15, and 20 mm. The width of the specimens was equal to 60 mm in each case. Threedimensional ®nite element modelling was based on a ®nite strain formulation of the J2 ¯ow theory with isotropic hardening and performed using the generalpurpose program ABAQUS (Hibbit, Karlsson & Sorensen, Inc.). The ®nite element meshes were made of eight-node bricks. Owing to symmetry, only one eighth of each DENT specimen was modelled. Several mesh designs were used in order to test convergence. Fig. 2 presents a typical mesh for specimen Al3-10. The initial radius of the crack tip was always taken lower than 0.1dCTODc. The meshes from which the results of this paper are derived consisted in 6-7-8-9-10-10 bricks in the half-thickness, with thicknesses of 1-2-3-4-5-6 mm, respectively. The length of the ®rst element close to the crack tip was chosen equal to half the initial radius of the crack tip in the more re®ned meshes (from which all the results presented hereafter are derived). A mixed formulation was used in order to handle near incompressibility resulting from large plastic strains. The uniaxial ¯ow behaviour is given by relation (8). The evaluation method of the pointwise values of the J integral (relation (1)) over the crack front is based on the 3D domain integration technique proposed by Shih et al. (1986). J(s ) was computed from various contours in order to check the path independence. A mean value for the J-integral was obtained by averaging the local values on the entire thickness. Whatever the result of the path independence analysis, the values of J discussed hereafter are taken from the largest contour. Values of JRice were also estimated numerically using relation (2). The ratio J/JRice is denoted l, with lc the value of l at cracking initiation. The numerical parameter l will be used in order to correct the values of JRice derived from the experimental load/displacement curves. The numerical dCTOD was measured as the displacement at the intersection of a 908 vertex centered at the crack tip with the crack ¯anks (i.e. an identical de®nition as for the experimental measurement of dCTOD). The tensile test on the notched round bars was simulated numerically in ®nite strains using the program ABAQUS. Axisymmetrical 8-noded subintegrated quadrilaterals were used. A number of 30 elements was chosen on the diameter of the minimum section. A J2 elastoplastic rate independent formulation was used. Void growth rates (relation (7)) in the minimum section of the notched specimen and in the crack plane of all DENT plates were computed by post-processing. Void nucleation is assumed to start at the beginning of straining.

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4. Experimental and numerical results 4.1. Fracture pro®les As shown on the SEM micrograph in Fig. 3, the fracture pro®le of broken DENT specimens of alloy A6082T0 present shear lips restricted only to the very surface of the plates. The fracture pro®le is predominantly ¯at along the whole ligament length (on a macroscopic point of view). A classical pro®le with dimples is observed at the microscale. The phenomenon responsible for cracking initiation is thus void growth and coalescence in the centre of the plate. Shear localisation is not the dominant phenomenon. In contrast, in an alloy A6082 thermally treated to the T6 state, the whole fracture pro®le consists of 458 shear lips (slant fracture). The very much higher yield stress in the T6 state (320 MPa) and the lower strainhardening exponent favour plastic localisation in shear bands where ductile damage concentrates. Slant fracture in plates (see Knott, 1973) does not enter within the scope of the present paper. A sketch of the ligament shape at cracking initiation is presented in Fig. 4(a). A transient zone of length ltrans exists before the ligament in which crack propagates reaches a constant thickness reduction. The thickness at the crack tip is denoted ttip and the thickness in the region of constant thickness is denoted tf . ltrans, ttip, and tf were measured on several fracture pro®les for each initial thickness. ttip was also deduced from the numerical simulation at the step corresponding to cracking initiation. A very good agreement was found between experimental and numerical values of ttip. Fig. 4(b) shows the variation of the ratios t0 ÿ ttip/t0, t0 ÿ tf /t0, and ltrans/t0 as a function of t0. The degree of necking in the constant thickness zone is

Fig. 3. Fracture pro®le of a broken Aluminium 6082T0 DENT specimens.

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Fig. 4. Fracture pro®le analysis: (a) de®nition of the parameters; (b) variation of the relative thickness reductions as a function of the initial thickness of the plate and variation of the transient distance as a function of the initial thickness.

similar in all tested specimens whatever the initial thickness whereas the degree of necking at cracking initiation decreases with increasing thickness. This justi®es why ltrans/t0 markedly increases with t0. 4.2. Load-displacement curves As shown in Fig. 5 for specimens Al3-5, Al3-10 and Al3-20, a good agreement

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Fig. 5. Comparison of experimental and numerical load-displacement curves for Al3-5, Al3-10 and Al320.

was observed between the experimental and numerical load-displacement curves. Numerical and experimental curves diverge after cracking initiation due to the fact that modelling does not account for damage and crack propagation. Cracking initiation is always detected at the midplane and after the maximum of the load/ displacement curve.

Fig. 6. Critical J-integral as a function of the sheet thickness and ligament length (Aluminium 6082T0)Ðcomparison with results by Swedlow (1965) for an Aluminium 7075T6.

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4.3. Fracture toughness characterisation 4.3.1. J integral measurement The l factor (=J/JRice) was computed from the ®nite element results for all the specimens (see Section 3.3). It was always found between 0.92 and 1, except for specimen Al5-5 for which it was equal to 0.88. The formula of Rice et al. (1973) is thus accurate in the LSY range even for l0/t0 ratios close to 1. The values of J at cracking initiation, Jc, were derived from the experimental load/displacement curve using Rice's analytical formula (2) corrected with the numerical factor lc. Path independence at cracking initiation was observed for all geometries except for specimens Al5-5, Al4-5, Al3-5, and Al2-5 i.e. for specimens with l0/t0 R 2.5 and a small ligament length (l0 R 5 mm) for which the whole ligament undergoes ®nite strains at cracking initiation. Strictly speaking, the Jc values of these specimens are thus not valid. Fig. 6 presents the variation of Jc with thickness. Jc increases with increasing sheet thickness. However, the rate of increase of Jc as a function of thickness decreases with increasing thickness and Jc reaches a maximum for t0=5±6 mm. The decrease of Jc at larger thickness is expected. Moreover, Jc does not depend much on the ligament length. It is worth mentioning that the non-valid Jc values (values corresponding the largest contour in specimens Al5-5, Al4-5, Al3-5, and Al2-5) do not diverge from the other values. The reason is that loss of path independence is only observed at high loads. The relative variations of the local value of J computed along the crack front were found to remain small except close to the surface (in agreement with the results of

Fig. 7. Determination of the essential work of fracture for di€erent sheet thicknesses. wf is the total speci®c work of fracture (after completion of cracking).

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Nakamura and Parks, 1990). In Fig. 6, a curve has also been added presented by Swedlow (1965), giving the toughness/thickness variation of a 7075T6 aluminium alloy. This aluminium is less ductile than the material studied in the present work, resulting in a smaller range of thickness for which fracture toughness is an increasing function of thickness. It is worth mentioning the similarity between the values of maximum toughness of both alloys. 4.3.2. Essential work of fracture Fig. 7 shows how the essential work of fracture we was obtained for each thickness by linear regression on the speci®c total work of fracture data. The scatter of the data increases with sheet thickness. Fig. 8 gathers the Jc and we data. As the ligament length has little e€ect on Jc (see Fig. 6), only the mean values of Jc are compared to we. Fig. 8 emphasises the fact that Jc and we have similar values, except for the largest thicknesses for which we is signi®cantly higher than Jc. Whereas we keeps increasing as sheet thickness increases, Jc reaches a maximum for t0=5±6 mm. Linear regressions on the variations of these two measures of fracture toughness as a function of t0 yield a similar value for the constant term: 24 kJ/m2 and about 29 kJ/m2 for Jc and we, respectively. The slope of the regression line is 29 MJ/m3 for Jc and 33 kJ/m2 for we. 4.3.3. Crack tip opening displacement Variations of dCTOD along the crack front were found very small in both experimental and numerical data. As an excellent agreement was found between experimental and numerical critical values dCTODc, only the numerical values are

Fig. 8. Comparison of the essential work of fracture and critical J integral (average value on the valid values of J for the di€erent ligament lengths) for di€erent sheet thicknesses.

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presented. As can be seen in Fig. 9, dCTODc increases with thickness in a similar way as Jc. Once again, the dCTODc displays no ligament length dependence. dc was obtained by linear regression on the uf (l0) data (similar method as for we in Fig. 7). The values of dc were also plotted in Fig. 9 which shows that the values of dc are very similar to the average dCTODc, except at the largest thicknesses investigated (5±6 mm). 4.3.4. Shih factors The variations of the Shih factors at cracking initiation dnc (relation (6)) as a function of thickness and ligament length are presented in Fig. 10. Comparison is made with the theoretical values of dn tabulated by Shih (1983), which are equal to 0.46 and 0.3 in plane stress and in plane strain, respectively. Fig. 10 shows that increasing the thickness brings about a decrease of dn which means that the stress states are progressively departing from the pure plane stress. A decrease of ligament length induces a lower value of dnc for the largest thicknesses. The dnc values corresponding to specimens Al5-5, Al4-5, Al3-5, Al2-5 are not meaningful because of the non-validity of J. The ratios s0dc/we=d nc are also plotted on Fig. 11. They match the dnc values at small thicknesses and decrease more markedly at high thicknesses. 4.4. Stress triaxiality variations Fig. 11 shows the variation of h(X1, 0, 0), i.e. at the midplane, in the crack plane, as a function of rs0/J for di€erent loading conditions, whereas Fig. 12 presents the variation of h(X1, 0, 0) for various DENT geometries at cracking

Fig. 9. dCTODc as a function of the sheet thickness and ligament length; comparison with dc.

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Fig. 10. Variation of the critical value of the Shih factor dnc as a function of the sheet thickness and ligament length.

initiation. The plane stress HRR value h = 0.62, is also indicated. Stress triaxiality decreases with increasing loading and decreasing thickness. h is always close to 0.62 in the middle of the ligament except in specimens Al5-5, Al4-5, Al3-5, Al2-5, Al4-10, and Al5-10 (i.e. for small l0/t0 ratios) for which h>0.62. Thus, for the major part of the specimens, one can consider that the 3D zone is embedded in a 2D plane stress zone. Even though there exists a zone where the sole J parameter suces for characterising the stress and strain ®elds (in the plane stress zone), the stress and strains in the 3D zone are far from evolving proportionally to J. Strictly speaking, J is thus not dominant in the sense that it does not characterise alone fracture toughness but in another sense it dominates in the plane stress zone. This problem does not appear in plane strain. It is a matter of terminology (that will not be solved here) to de®ne what is J dominance: (i) the fact that J univocally characterises fracture toughness; (ii) or, only, the existence of a zone where stress and strains are dictated by J. The evolution of hmax with loading (monitored by J/Jc) is plotted in Fig. 13 for various DENT geometries. hmax decreases when load increases. Two opposite e€ects can be observed on Fig. 13. First, for a given load, hmax appears to be independent of the ligament length but to increase considerably with sheet thickness. Second, hmax decreases less with loading for larger thicknesses. This conclusion is supported in Fig. 13 by also showing the result of the computation of the stress triaxiality variations in a DENT plate Al10-20. (In order to plot the result for the specimen Al10-20, a

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Fig. 11. E€ect of loading on the variation of the stress triaxiality h as a function of the distance to the crack tip.

value of Jc equal to that of specimen Al6-10 has been assumed. Indeed, from Fig. 6 it can be expected that Jc 10-20E Jc 6-10.) 4.5. Void growth modelling at the crack tip 4.5.1. Parameters identi®cation Fig. 14 presents the tensile curves of ®ve notched round A6082T0 bar: the average stress, sm, i.e. the load divided by the current area of the minimum section, is plotted as a function of the average equivalent strain, eeqm. Cracking initiation is detected by a sudden drop in the curve. Fig. 14 also shows the good agreement between simulation and experiment. Void growth computed by relation (7) is maximum in the centre of the minimum section of the bar. Void growth variation at this location is added on Fig. 14. The critical void growth, Rc/R0, obtained at the mean experimental coalescence strain, 0.64, is equal to 2.6. The average stress triaxiality ratio in the central element of the minimum section is relatively constant and equal to about 1. 4.5.2. Growth and coalescence of voids in the FPZ Fig. 15 shows the evolution of the void growth rate, R/R0, with the distance r

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Fig. 12. E€ect of geometry on the variation of the stress triaxiality h as a function of the distance to the crack tip, at J=Jc.

to the crack tip at the midplane of a DENT specimen 3-10 for di€erent loading levels. The loading is characterised by the value of J normalised by Jc. The R/R0 values above Rc/R0 do not carry much physical meaning. Indeed, when coalescence develops, void growth follows an evolution law which is very di€erent from that predicted by the Rice and Tracey model. The distance from the crack tip at which the coalescence criterion is ful®lled (i.e. Rc/R0=2.6) is denoted X(X3/ t0). In Fig. 15, X3/t0=0. The FPZ in the crack plane corresponds to the area between the crack front and the line of the points located at the distance Xc(X3/ t0), where Xc(X3/t0) is the X value at the loading corresponding to the experimentally observed crack initiation, i.e. when J=Jc. Fig. 16 gives an example of the 2-D FPZ size and shape (plain line) in the crack plane of the DENT specimen 3-10. The dashed lines represent the evolution of the X(X3/t0) locus with loading. The FPZ length is maximum at the midplane where the stress triaxiality is the highest. Fig. 17 compares the FPZ lengths Xc(0), for all DENT specimens investigated. The results are plotted as a function of thickness t0 for the four di€erent ligament sizes. The FPZ is strongly dependent on thickness. The increase of the FPZ length with increasing thickness is fairly linear as long as t0 R 3 mm. Comparison of

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Fig. 13. Variation of the maximum of the stress triaxiality hmax as a function of loading (described in terms of J/Jc) for various DENT specimens.

Figs. 9 and 17 shows that the ratio Xc(0)/dCTODc remains close to 0.5±0.6 except for specimens Al5-5 and Al4-5.

5. Discussion 5.1. Global approach Detection of cracking initiation is a requisite in fracture mechanics tests. The fracture toughness measurements shown in Figs. 6 and 9 present a dispersion which mainly results from the inaccuracy in the detection of cracking initiation. Figs. 8 and 9 compare Jc or dCTODc to the essential work of fracture we and to dc, respectively. They demonstrate that these parameters provide equivalent measures of fracture toughness for small sheet thicknesses. This point is of major practical importance: measurement of we and dc does not require determination of cracking initiation because we and dc are computed from the total load-displacement curve. At large thickness, the change of stress state induces a di€erence between the fracture toughness predicted by Jc and by Cotterell's method. A criterion for predicting whether we and dc can be identi®ed with Jc and dCTODc (without detecting cracking initiation) may be tentatively proposed from Fig. 10: the ratio s0dc/we=d nc has to remain close to the plane stress Shih factor of the material.

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Fig. 14. Comparison of experimental and numerical average stressÐaverage strain curves of Al6082T0 notched round specimens (left y-axis). Variation of the void growth in the centre of the minimum section of the bar as a function of the strain (right y-axis).

For larger thicknesses, constraint e€ects modify the way these two approaches account for the work spent in the FPZ. Let us now further discuss the discrepancies between Jc and we, especially at large thickness. First of all, at large thickness, the domain of valid application of the EWF method, i.e. the domain within which all specimens present globally the same stress state across the ligament, becomes restricted and the linearity hypothesis underneath the EWF method becomes less valid (for example, the ligament must be suciently large in order to avoid interactions between the fracture process zones). This leads to a large uncertainty on the linear regression values, as con®rmed by the larger scatter of the speci®c work of fracture data. Second, Fig. 8 shows that we remains proportional to thickness with a slope equal to about 33 MJ/m3. This slope corresponds to the supplement of work required for the necking of the specimen when thickness increases. Fig. 4(b) shows that the relative thickness reduction after completion of cracking is indeed a constant. At small thickness, the necking zone is localised close to the crack tip and the neck presents a width ttip slightly larger that the ®nal width tf . Jc is thus only slightly lower than we. As thickness increases, the degree of necking at the crack tip becomes much lower than the steady-state degree of necking. The transition

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Fig. 15. Variation of the void growth rate with the distance to the crack tip at the midplane of a DENT specimen 3-10 for di€erent loading levels.

between cracking initiation and steady-state cracking is more progressive (see Fig. 4). Due to the higher stress triaxiality in thick specimens, the crack starts propagating before the neck has completely formed (see Fig. 13). The ligaments used for the regression for computing we are suciently long for avoiding the in¯uence of this transient e€ect. In conclusion, we seems related to the steady-state fracture resistance whereas Jc is the usual measure of the resistance to cracking initiation. At small thickness, as necking is nearly plainly developed at cracking initiation, we and Jc have similar values. There is then little increase of fracture resistance during propagation. At larger thickness, the work of fracture increases during crack propagation due to the larger development of the neck. This constitutes a true increase of toughness in contrast with the apparent increase of toughness observed in JR-curves of plane strain specimens (see also Cotterell and Atkins, 1996). Figs. 8 and 9 show that linear regression on the fracture toughness data yields a constant term with a value of 24 kJ/m2 (29 kJ/m2 for we) or about 0.25 mm for dCTODc (0.36 mm for dc). It is proposed that this regression thus allows separation of the fracture work spent for localised necking and of the fracture work for damaging. This fracture work more intrinsically characterises fracture resistance of thin plates because it does not depend on thickness and it is more directly related to the mechanisms occurring in the FPZ. The fact must be stressed that this more `intrinsic' work of fracture does not correspond to the work of fracture of very

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Fig. 16. Evolution with loading of the 2-D X(X3/t0) locus in the crack plane of the DENT specimen 310. The plain line represents the Xc(X3/t0) locus at fracture initiation, which corresponds thus to the FPZ frontier.

Fig. 17. Variations of the FPZ lengths with the thickness and ligament size of the DENT specimens.

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thin plates or sheets (t0 typically R 100 mm), for which the fracture mechanisms are di€erent. In other words, the linear regression cannot be assimilated to an extrapolation of the fracture toughness values at t0=0. Fig. 13 suggests that Jc reaches a maximum at thickness equal to 5±6 mm because of the increasing stress triaxiality (and of the smaller decrease of h with loading). The crack initiates for a lower degree of necking at larger thickness because void growth is favoured by a higher stress triaxiality. Fracture toughness is expected to decrease for larger thickness and to reach the plane strain (and thus minimum) value for very thick specimens. Fig. 10 shows that dnc globally expresses the stress state change with increasing thickness. However, this parameter is not as sensitive as hmax. The Q parameter (O'Dowd and Shih, 1991a,b) has been computed as a function of loading for the di€erent specimens but no consistent results were derived: for example, Q was found not to depend on plate thickness, or not pertinent as it varied with the distance to the crack tip. It is worth also mentioning the approach of Nakamura and Parks (1990) who de®ne the `dominance parameter' r as rˆ

ksij ÿ sHRRpl:strain k ij k ksHRRpl:strain ij

,

…11†

where k.k is the spectral matrix norm: ksijk=vs1±s3v, where s1 and s3 are the maximum and minimum principal stresses respectively. This dominance parameter is well adapted for analyzing the departure from plane strain which exists in thin plates very close to the crack tip when the distance from the crack tip and/or the loading increase (Nakamura and Parks, 1990). The use of h is preferred because stress triaxiality directly in¯uences the physical process of ductile fracture by void growth and coalescence. Furthermore, it is possible to consider a particular value of h, namely its maximum hmax, which avoids thus introducing a characteristic length, as required in the r approach or in the J±Q approach when wishing to quantify constraint changes. The use of hmax (also discussed by Brocks and Schmitt, 1995, in the context of plane strain fracture) is also justi®ed by the fact that hmax is located inside the FPZ where damage develops but at a position not too close to the crack tip where numerical solutions can be questionable due to the large distortion of the elements. (The maximum of the stress triaxiality was always found located in elements presenting a reasonable aspect ratio.) 5.2. Local approach The local approach of fracture through the modelling of void growth appears physically pertinent because of the macroscopically ¯at fracture pro®le with dimples at the micro-scale. The length of the FPZ in the mid-plane, Xc, is proportional to dCTODc (Xc 1 0.55 dCTODc) except for specimens Al5-5 and Al4-5 for which the interaction between the fracture process zones at the tip of opposite cracks brings about an increase of the FPZ lengths. The FPZ length which is also

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scaled by the size of the localised necking zone (see also Broberg, 1975) is geometry dependent, as already mentioned by several authors (Mai and Cotterell, 1980). As supported by the previous discussion, linear regression on the Xc values would give, for the constant term, a distance related only to the damage mechanisms (about 120 mm). In the Al6082T0 plates, the inclusions responsible for the initiation of voids are very small (<1±5 mm), with interdistances of the order of a few micrometers. The crack propagation mechanism does not result from the interaction of the crack tip with only one single void or one single cluster of voids, as frequently described in the literature (Rice and Johnson, 1970; Ritchie and Thompson, 1985; Hom and McMeeking, 1989). The number of voids inside the FPZ is very large especially for thicknesses corresponding to the maximum of fracture toughness. In order to predict cracking initiation, the following fracture criterion could be proposed: cracking initiates when the critical void growth corresponding to the onset of void coalescence is attained in the entire FPZ; Fig. 17 can be used as a means for determining independently the size of the FPZ as a function of thickness (FPZ length=0.12+0.15t0 (in mm)). However, no reason exists for claiming that the FPZ size must remain the same, for a given thickness, when the loading con®guration or the geometry is di€erent. A more physical approach is to use 3D computational cells (as Gao et al., 1998) based on the Gurson±Tvergaard constitutive relationships (Gurson, 1977; Tvergaard, 1981). The cell size is directly related to the void spacing which in the meantime constitutes a way to solve the problem of mesh sensitivity. Some attempts have already been made in that direction. At this time, with the damage model as it stands, the fracture toughness results presented in this work can be reproduced only by adjusting the cell size with respect to the thickness. The authors feel that extensions of the existing damage models are required in order to allow consistent application of computational cells in thin plates where the low stress triaxiality and the strain distributions are very di€erent relative to the plane strain situation. Examples of extensions of the damage models which are thought to constitute a way for solving that problem and to allow the use in thin plates of computational cells with initial dimension related to the void spacing would be: 1. to incorporate void shape e€ects, important at low stress triaxiality and which can signi®cantly a€ect the damage evolution (see Gologanu et al., 1995); 2. to use a more physical criterion for the onset of void coalescence (Thomason, 1990) accounting, amongst others, for the void shape, the void spacing, and the stress state in the intervoid ligament; 3. to model more adequately the void coalescence phase. For example, very small strain biaxiality ratios were found to develop in the crack tip ®nite strain zone of the studied thin plates. A small strain biaxiality ratio may signi®cantly hinder the void coalescence process (Faleskog and Shih, 1997) and result in a large delay between the onset of void coalescence somewhere in front of the crack tip and the initiation of the cracking of the blunted crack tip.

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5.3. Transferability of cracking initiation criteria in thin plates A model for predicting fracture initiation in a structure made of Al 6082T0 thin plates does not straightforwardly emerge from sections 5.1 and 5.2. Three approaches can be envisioned in order to solve the transferability problem in thin plates. Firstly, the `intrinsic' fracture work obtained by the linear regression on the experimental fracture toughness data could be considered as the area under the stress/opening curve in a cohesive zone model as used by Needleman, 1990, or Tvergaard and Hutchinson, 1992. The maximum opening would then be equal to the constant term in the regression on dCTODc data. Assuming a nearly rectangular crack closure stress pro®le, this results, for the AlT0 material, in a peak stress equal to approximately 2s0. Necking in front of the crack tip would then be accounted for by the elastoplastic FE modelling. Crack propagation would be simulated while accounting for the transient stage just after cracking initiation.

Fig. 18. A proposal for a Jc±t0 ±hmax locus for thin aluminium 6082T0 plates.

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Nevertheless, such a method would be pertinent as long as fracture remains ¯at. Furthermore, it is not clear whether or not reasonable predictions will be obtained with this approach applied to thin plates without making the cohesive zone to depend on other parameters as stress triaxiality (Siegmund and Brocks, 1998) or strain (Tvergaard and Hutchinson, 1996). Secondly, in analogy with the JcQ locus proposed by O'Dowd and Shih (1991b), it can tentatively be proposed to combine the results of Figs. 6 and 13 in such a way as to draw a 3D Jc±t0±hmax locus for ¯at fracture in thin plates. These experimental results correspond to a line in this space. Fig. 18 gives an approximate representation of this locus. A similar idea can be found or derived from the works of Rice (1974), Chao (1993), Heerens et al. (1996) and Yuan and Brocks (1998). Full characterisation of this locus would require measurement of fracture toughness in other geometries and loading con®gurations. At very large thickness, out-of-plane e€ects are expected to disappear, which implies that intersections of the locus with planes perpendicular to the t0-axis will remain similar (a 2D representation, similar to the Jc±Q locus then becomes sucient). One can speculate that the intersection of the surface with the plane perpendicular to the hmax-axis at hmax=hHRRPl.stress would closely correspond to the variation of we plotted in Fig. 8. Indeed, it was justi®ed in section 5.1, that we characterises the steady-state fracture toughness corresponding to the complete development of the neck and that we is representative of the plane stress fracture resistance. The last solution for the transferability problem would be through the development of a proper modelling of the mechanisms of fracture along the lines drawn at the end of section 5.2. This approach is appealing because it is the most physical and also because of the diculty of constructing a three parameters fracture locus. 6. Conclusions The characterisation and prediction of cracking initiation in thin plates with thicknesses such that fracture toughness is an increasing function of thickness is a complex issue due to the 3D stress state at the crack tip. It is an important topic of research for material science when the fracture properties of materials processed under the form of plates must be characterised and for structural mechanics when the integrity of structures made of thin plates must be assessed (for example, in safety assessment of aging aircraft). The present work contributes to a better understanding of this issue from an application on ductile aluminium plates which fail by ¯at ductile fracture.= 1. The parameters Jc±dCTODc and we±dc were found equivalent in pure plane stress situations. Indeed, for small thicknesses, the resistances to fracture initiation and to fracture propagation are very similar because necking has already nearly fully developed at cracking initiation. In this range of thicknesses, fracture toughness evolves linearly with thickness because the necking work is

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3.

4.

5.

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proportional to thickness. we constitutes a convenient way for characterising fracture toughness because it does not require the determination of cracking initiation nor any analysis of the fracture surface. The interest of parameters such as we and Jc lies in the separation made between the non-essential work of di€use plasticity and the essential work for deforming, damaging and fracturing the FPZ. It is suggested that a linear regression on the fracture toughness/thickness curve allows further separation of the two contributions of the essential work of fracture: the necking work and the fracture work spent for damaging the FPZ. This last term is a more intrinsic measure of the fracture resistance of thin plates because it does not depend on specimen dimensions. At larger thicknesses, the combination of a higher constraint and a smaller decrease of the constraint with loading brings about cracking initiation well before necking attains steady-state. This results in a true increase of the fracture resistance during propagation. The maximum of the stress triaxiality was found to constitute a pertinent parameter for characterising constraint in this situation of dominant out-of-plane constraint. Local approach modelling allows quanti®cation of the FPZ sizes for the various thicknesses. The marked dependence of the FPZ on thickness complicates the prediction of cracking initiation in thin plates using local approaches. Three approaches were suggested in order to solve the problem of transferability of fracture criteria in thin plates: (i) a cohesive model based on the experimental separation of fracture energy; (ii) a 3D Jc/thickness/triaxiality fracture locus; and (iii) a local approach based on 3D computational cells, requiring proper modelling of the void coalescence stage.

Acknowledgements T. Pardoen acknowledges a fellowship of FNRS, Belgium. The authors are grateful to the Referees for their detailed reviewing and constructive suggestions. The authors are also grateful to R. Ghislain, M. Sinnaeve and E. Clembos for technical assistance. This work was carried out in the framework of program PAI41 supported by SSTC Belgium. The aluminium plates were kindly supplied by Hoogovens Aluminium N.V. Belgium. References Anderson, T.L., 1989. Crack tip parameters for large scale yielding and low constraint con®gurations. Int. J. Fract. 41, 79±104. Anderson, T.L., 1995. Fracture MechanicsÐFundamentals and Applications. CRC Press, Boca Raton. Bakker, A., 1984. On the numerical evaluation of J in three dimensions. In: Sih, G.C., Sommer, E.,

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