Journal of Materials Processing Technology

Transcription

1 Journal of Materials Processing Technology 234 (216) Contents lists available at ScienceDirect Journal of Materials Processing Technology jo ur nal ho me page: Fracture toughness and failure limits in sheet metal forming M.B. Silva a, K. Isik b, A.E. Tekkaya b, A.G. Atkins c, P.A.F. Martins a, a IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, Lisboa, Portugal b Institute of Forming Technology and Lightweight Construction, Technical University of Dortmund, Baroper Str. 33, D Dortmund, Germany c Department of Engineering, University of Reading Box 225, Reading RG6 6AY, UK a r t i c l e i n f o Article history: Received 3 April 215 Received in revised form 1 September 215 Accepted 31 March 216 Available online 1 April 216 Keywords: Sheet metal forming Fracture loci Fracture toughness Crack opening mode a b s t r a c t This paper proposes a link between plastic flow, void coalescence and growth, ductile damage, crack opening modes and fracture toughness in sheet metal forming. This new integrated view is based on an analytical framework that allows estimating the location of the fracture loci in the principal strain space directly from material stress strain response and from fracture toughness and thickness at fracture obtained from double-notched test specimens loaded in tension and plane torsion (in-plane shear). Experiments in AA15-H111 aluminium sheets with 1 mm thickness give support to the proposed analytical framework. 216 Elsevier B.V. All rights reserved. 1. Introduction Until recently, the formability limits by fracture have not been of interest to sheet metal forming because once a neck appears and spreads sideways under subsequent deformation, thinning will progress very fast under decreasing loads or pressures until the sheet cracks. As a result of this, research has been focused on the formability limits at the onset of plastic instability (also known as the forming limit curves, FLC s). Nowadays, the experimental methods and procedures for determining the FLC s of metal sheets at room temperature are well established in the international standard ISO (ISO, 28) and involve carrying out Nakazima and Marciniak sheet formability tests. The widespread utilization of finite element analysis in sheet metal forming relaunched the discussion on the utilization of ductile damage mechanics for predicting the onset of failure by fracture and on the experimental methods and procedures for determining the fracture loci in the principal strain space and in the space of effective strain vs. stress triaxiality. Some authors combine data retrieved from sheet and bulk formability tests (Wierzbicki et al., 25) while others consider that the differences in plastic flow resulting from the plane stress conditions of sheet metal forming and the three dimensional stress conditions of bulk metal forming that are commonly used as a rationale to classify metal forming processes into two-different groups must be treated differently, Corresponding author. address: pmartins@ist.utl.pt (P.A.F. Martins). in order to distinguish the circumstances under which different processes fail by fracture (Isik et al., 214). The commonly accepted fact that FLC s are not material properties and that its determination is greatly influenced by strain loading paths, by combination of in-plane loading and bending effects and by difficulties in measuring the onset of necking (Centeno et al., 214) further contribute to the growing interest in the formability limits by fracture. The utilization of ductile damage mechanics for predicting the onset of failure by fracture has a long pedigree and can be systematized as a function of the associated theoretical background into two different categories: (i) uncoupled procedures based on the utilization of simple ductile damage criteria that are weighted integrations of the effective plastic strain (Atkins, 1996) and (ii) coupled procedures based on micro-based damage mechanics (Tvergaard and Needleman, 1984) built upon the macroscopic yield surface for porous materials (Gurson, 1977), or based on continuum damage mechanics (Lemaitre, 1985). Uncoupled procedures based on the utilization of ductile damage criteria due to Cockcroft Latham, McClintock and Oyane, among others, do not account for the progressive deterioration of the material during loading and unloading but are easier to implement and to calibrate than fully coupled procedures. Recent developments by Bai and Wierzbicki, (21) and Li et al. (21) allowed determining the fracture loci in both the principal strain space and the space of effective strain vs. stress triaxiality by means of new uncoupled ductile damage procedures that combine experimentation in bulk and sheet metal formability tests and finite element inverse calibration procedures / 216 Elsevier B.V. All rights reserved.

2 25 M.B. Silva et al. / Journal of Materials Processing Technology 234 (216) Fig. 1. Failure by fracture in tension. (a) Schematic representation of void growth and (b) fracture forming limit line (FFL) in the principal strain space. This paper is built upon uncoupled ductile damage procedures but contrary to other publications in the field it is aimed at establishing a link between void coalescence and growth, ductile damage, crack opening modes and fracture toughness. The work draws from two recently published works by Isik et al. (214) and Martins et al. (214) on the circumstances under which each crack opening mode will occur in terms of plastic flow and microstructural ductile damage and it is aimed at proposing an analytical framework to characterize fracture loci under plane stress conditions directly from material stress-strain response and from fracture toughness and thickness at fracture obtained from experiments with double-notched test specimens loaded in tension and plane torsion, which fail by fracture in crack opening modes I and II of fracture mechanics. The fracture loci determined by means of the new proposed analytical framework are checked against experimental strain pairs at failure that were previously obtained by the authors in plastic deformation scenarios that are distinct from the fracture mechanics calibration test cases. 2. Theoretical background 2.1. Ductile damage, void coalescence and growth Crack propagation by void coalescence and growth in tension or shear stress fields may always be viewed as a process of continuous re-initiation along the path of cracking. In tension, Atkins and Mai (1985) worked on McClintock s (1968) continuum mechanics of void growth to established a relation between the inter-hole l (inter-particle/inclusion) spacing, the diameter d of the hole (particle/inclusion) and the stress triaxiality m / (defined as the ratio of the average and the effective stress) at the onset of cracking (Fig. 1a), ln ( l ) = d m d (1) Fig. 2. Failure by fracture in shear. (a) Schematic representation of void growth and (b) in-plane shear fracture forming limit line (SFFL) in the principal strain space.

3 M.B. Silva et al. / Journal of Materials Processing Technology 234 (216) Fig. 3. Double notched test specimen under tension. (a) Schematic representation of the volumes of the necked down region at the crack tip in which the cracks run during the applied tension. (b) Schematic representation of the strain loading path in the principal strain space. where the right hand side term is a simplified version of McClintock (1968) ductile damage criterion. Eq. (1) allows concluding that the critical value of damage D I crit calculated from of McClintock s (1968) criterion at the onset of fracture is related to crack opening by mode I of fracture mechanics because stress triaxiality is related to dilatation changes in voids rather than distortional changes, D I crit = m d (2) In case of pure shear m / =, McClintock et al. (1966) proposed the following relation between the inter-hole l (inter-particle/inclusion) spacing, the diameter d of the hole (particle/inclusion) and the shear strain at the onset of cracking (Fig. 2a), ( l ) ln = (3) d Contrary to Eq. (1) that is presented in integral form and, therefore, can be readily taken as a measure of the accumulated void growth damage in tension, Eq. (3) is already the result of integration. This implies that Eq. (3) cannot be directly related to damage in shear. However, a connection between l/d and ductile damage in shear can be achieved if the term in the right hand side of Eq. (3) is approximated by /3 for the typical working range of shear strains < 3. Under these circumstances, the following integral form of Eq. (3) can be built upon application of the Levy-Mises constitutive equations in pure shear d = 3(/)d, ln ( l ) d 1 3 d = d (4) The above integral form of Eq. (3) allows concluding that a ductile damage criterion based on the ratio of the shear stress to the effective stress can be utilized to model the distortional changes in voids, D II crit = d (5) The symbol D II crit denotes the critical value of damage at the onset of fracture by in-plane shear (crack opening mode II of fracture mechanics). The derivation of Eq. (5) provides theoretical support to the empirical ductile fracture criterion that was utilized by Isik et al. (214) and Martins et al. (214) to characterize the fracture locus by in-plane shear in the principal strain space and in the space of effective strain vs. stress triaxiality Ductile damage and fracture loci Martins et al. (214) recently showed that by using the constitutive equations associated to Hill (1948) anisotropic yield criterion and assuming plane stress loading conditions and rotational symmetry anisotropy r = r = r, where r is the normal anisotropy, it is possible to rewrite Eqs. (2) and (5) as a function of the major and minor in-plane strains ( 1f, 2f ) at the onset of fracture, D I crit = D II crit = m d = d = 1f 1f (1 + r) 3 ( 1 (1 + r) ˇ 1 2 (1 + 2r) ˇ ( ) ˇ + 1 ˇ ) d 1 = (1 + r) 3 d 1 = 1 (1 + r) ) ( 1f 2f 2 (1 + 2r) ( 1f + 2f ) (6) The integrands in Eqs. (6) and (7) have the form (A + B/ˇ) implying that the damage functions for a constant strain ratio ˇ = d 1 /d 2, are independent of the loading path history. In other words, the integrated values of Eqs. (6) and (7) are only dependent on the initial and final values of strain at the onset of fracture. This occurrence is comprehensively discussed by Atkins and Mai (1985) (7)

4 252 M.B. Silva et al. / Journal of Materials Processing Technology 234 (216) loaded in tension (Fig. 3a), for example, the necked down process region is located in-between the notches and is assumed to have a height h < d, which should be similar to the thickness t of the specimens (Hill, 1952). Under these circumstances, the incremental work RdA dissipated within an incremental volume hda during crack nucleation (or propagation) corresponds to the plastic work per unit of volume at the onset of fracture (please refer to the volumes that are schematically shown in the detail of Fig. 3a), RdA hda = R t = d (1) In a strain loading path that consists of two components (before and after necking) the above equation may be written as, Fig. 4. Forming limits by necking and fracture in the principal strain space of the aluminium AA15-H111 sheets with 1 mm thickness. and justifies the reason why strain loading paths in Figs. 1b and 2b were assumed as linear. Moreover, it also follows from Eq. (6) that the critical value of damage D I crit associated with stress triaxiality and dilatational changes in voids defines a straight line with slope 1 falling from left to right in close agreement with the FFL (fracture forming line, Fig. 1b) and the ( condition ) of critical reduction of thickness at fracture 3f = ln 1 R f, where R f given by (t t f )/t with t and t f being the initial thickness and the thickness at fracture. Conversely, Eq. (7) allows concluding that the critical value of damage D II crit associated with in-plane shear and distortional changes in voids defines a straight line rising from left to right with a slope equal to +1 in agreement with the condition of critical distortion f along the SFFL (shear fracture forming line, Fig. 2b). In connection to what was said above about the fracture loci, it is worth noting that if the lower limits of the integrals in Eqs. (6) and (7) are equal to rather than zero, corresponding to situations where there is a threshold strain below which damage is not accumulated, the FFL and the SFFL deviate from straight lines and present upward curvatures as it is schematically represented by the dashed solid curves in Figs. 1b and 2b, D I crit = D II crit = m d = (1 + r) 3 ( ( d = 1 (1 + r) 2 (1 + 2r) 1f + 2f 1f 2f 2.3. Fracture toughness and fracture loci ( ) ) ˇ + 1 ˇ ( ) ) ˇ 1 The link between fracture toughness R and fracture loci is based on the assumption that there should be a connection between the specific essential work of fracture, which characterizes the ability of a sheet to resist crack initiation, and the work required locally to nucleate or propagate (under continuous nucleation) a crack in tension or shear stress loading conditions. As discussed by Atkins and Mai (1985), the specific essential work of fracture R may be converted to a local work done per unit of volume at the crack tip by dividing R by the height h of the necked down process region. In case of the double notched test specimen ˇ (8) (9) neck R t = d + neck d (11) where neck and f are the effective strains at neck formation (FLC) and fracture (refer to points A and B of Fig. 3b). In case of a material with a stress-strain behaviour = K n, the right hand side terms of Eqs. (1) and (11) corresponding to the strain loading paths OB and OAB provide the same result, R t = K n+1 f (12) n + 1 The path independence result of Eq. (12) is understandable because the integrand of the critical damage D I associated with crack crit opening mode I (Eq. (6)) has the form (A + B/ˇ) and, therefore, it is also independent from the loading path history. The above analytical procedure and the resulting Eq. (12) can also be applied to double notched circular test specimens loaded in plane torsion because the critical damage D II crit associated with crack opening mode II (Eq. (7)) has also the form (A + B/ˇ). The only important difference that needs to be taken into consideration is the utilization of different values of fracture toughness R for tension and shear loading conditions because the energy to nucleate and propagate cracks should vary as a function of the crack opening mode. In connection to what was said above, it is worth noting that the necked down region can also be formed as a result of changes in the relative ease of plastic flow in width and thickness directions rather than on classical mechanisms based on the development of unstable plastic deformation (Isik et al., 215). Now, by taking into consideration that fracture toughness is a material property, which is not influenced by strain loading paths and by combination of in-plane loading and bending effects, it follows that Eq. (12) can be utilized to determine the fracture loci. In this way, the effective strains at fracture f for strain loading paths giving rise to fracture by crack opening modes I and II can be estimated from, ( Rmode ) (n + 1) 1 n+1 f = (13) tk where R mode is the fracture toughness R in crack opening modes I or II and t = t f is the thickness at fracture. The effective strain at fracture f may be written as a function of the anisotropy r, the strain ratio ˇ and the major strain at fracture 1f, 1 + r f = 1 + 2r 1 (1 + 2r) (1 + r) ˇ + 1ˇ2 1f (14)

5 M.B. Silva et al. / Journal of Materials Processing Technology 234 (216) F t Force W w T T = tl l n l d l 1 l 2 R I w W T1 F l 1 l 2 l n Displacement (a) (b) (c) l Fig. 5. Method and procedure for determining fracture toughness R I in crack opening mode I. (a) Schematic representation of a double-notched test specimen; (b) Schematic evolution of the force with displacement for test specimens with different ligaments; (c) Determination of fracture toughness from extrapolation of the specific total energy. A first glimpse into Eq. (13) allows concluding that fracture loci derived from fracture toughness R may be seen as an effective strain based criterion where the ellipses of constant effective strain at fracture = k f in the principal strain space (refer to Figs. 1b and 2b) are modified in order to include a dependency on fracture toughness R I or R II, sheet thickness t and stress-strain response of the material by means of constant K and strain hardening exponent n (in case of a material following = K n ). The comparison between the fracture loci derived from Eq. (13) and from experimental procedures based on the determination of gauge length strains at fracture will be provided in Section 4 and will be utilized to validate the proposed link between fracture toughness and fracture loci. 3. Experimentation The investigation was carried out in aluminium AA15-H111 sheets with 1 mm thickness and the experimental tests that are needed to validate the proposed analytical framework to characterize fracture loci under plane stress conditions directly from fracture toughness in crack opening modes I and II of fracture mechanics were retrieved from previous research works performed by the authors. Consequently, this section will only present a brief summary of the methodologies and results associated with the mechanical characterization and determination of the formability limits and fracture toughness of the material that are relevant for the aims and objective of this paper. Further details and experimental testing conditions are given in Madeira et al. (215) and Isik et al. (214, 215) Mechanical characterization The mechanical characterization of the aluminium AA15- H111 sheets at room temperature was carried out by means of tensile tests in specimens cut out from the supplied sheets at, 45 and 9 degrees with respect to the rolling direction. The methodology followed the ASTM standard E8/E8 M (ASTM E8/E8 M, 213) and the results are summarized in Table 1. The stress strain curve was approximated by the following Ludwik Hollomon s equation, = 14.4 (MPa) (15) The normal r anisotropy coefficient included in Table 1 was determined from the anisotropy coefficients r of the tensile tests performed in specimens cut out from the supplied sheets at, 45 and 9 degrees with respect to the rolling direction, r = r + 2r 45 + r Formability limits (16) The formability limit by necking (FLC) was determined by means of sheet formability tests (tensile, Nakazima, bulge and hemispherical dome tests) that covered strain paths from uniaxial to biaxial stretching conditions. The procedure utilized for determining the in-plane strains ( 1, 2 ) at the onset of necking involved electrochemical etching of a grid of overlapping circles with 2 mm initial diameter on the surface of the test specimens before forming and measuring the major and minor axes of the ellipses that resulted from plastic deformation. The characterization of the formability limits by fracture (FFL and SFFL) included additional results from double-notched test specimens loaded in tension and in plane torsion. The latter were needed to characterize the SFFL because all the remaining sheet formability tests failed by fracture under crack opening mode I. The determination of the strains at fracture involved measuring the thickness of the specimens before and after fracture at several locations along the crack in order to obtain the gauge length strains. The gauge length strains were subsequently fitted by two straight lines; (i) one straight line (FFL) falling from left to right with a slope.68 in agreement with Eq. (7) of Section 2.2 and (ii) another straight line (SFFL) rising from left to right with a slope equal to in agreement with Eq. (8) of Section 2.2. The formability limits by necking and fracture of the aluminium AA15-H111 sheets with 1 mm thickness are shown in Fig. 4. Table 1 Summary of the mechanical properties of aluminium AA15-H111 sheets with 1 mm thickness. Modulus of elasticity (GPa) Yield strength (MPa) Ultimate tensile strength (MPa) Elongation at break (%) Anisotropy coefficient RD RD RD Average r =.84

6 254 M.B. Silva et al. / Journal of Materials Processing Technology 234 (216) Fig. 6. Schematic link between plastic flow, void coalescence and growth, ductile damage, crack opening modes, and fracture toughness in sheet metal forming Fracture toughness The determination of fracture toughness in plane stress in crack opening modes I and II made use of double-notched test specimens loaded in tension and plane torsion. The methodology followed the original developments of Cotterell and Reddel (1977) for crack opening mode I and the extension for crack opening mode II that was recently proposed by Isik et al. (215). Characterization of fracture toughness R I (mode I) using doublenotched test specimens loaded in tension is summarized in Fig. 5 and involved three main procedures; (i) determination of the total energy W T directly from the force-displacement evolution as a function of the starting ligament length l of each test specimen, (ii) determination of the specific total energy w T = W T /tl as a function of the starting ligament length l of each test specimen and (iii) determination of the specific energy of fracture (fracture toughness in crack opening mode I) R I by extrapolating the specific total energy w T to the limiting conditions in which the starting ligament length l approaches zero. A similar procedure was employed for determining fracture toughness in crack opening mode II (R II ) from double-notched test specimens loaded in plane torsion in which the force F and the displacement were replaced by the torque T and the degree of rotation (Isik et al., 215). Table 2 summarizes the results obtained in both test cases. To conclude, it is worth mentioning that fracture initiation points are located in the middle of the notches because the specimens were slightly indented with a razor blade in the opposite sides of the notched tips before testing in order to localize crack opening. 4. Results and discussion 4.1. Comparison of theoretical models for predicting fracture loci Fig. 6 provides a schematic comparison of the theoretical estimate of the fracture loci (FFL and SFFL) obtained from ductile damage and void coalescence/growth models associated to crack opening modes I and II (Sections 2.1 and 2.2) with the theoretical estimates provided by fracture toughness based models with values determined from double-notched test specimens loaded in tension and plane torsion (Section 2.3). As seen in Fig. 6, the fracture loci obtained from fracture toughness tests in crack opening modes I and II (Eqs. (13) and (14)) are represented in the principal strain space as two different ellipses ( = f (R I ) and = f (R II )) with major axis along pure shear 1 = 2 and minor axis along equal biaxial stretching 1 = 2. This allows concluding that the slope +1 of the in-plane shear fracture forming line (SFFL) associated to crack opening mode II and to the ductile damage criterion based on the ratio of the shear stress to the effective stress (Eq. (7)) exactly matches the slope of the derivative to Fig. 7. Finite element distribution of effective strain for double-notched test specimens loaded in tension and plane torsion. (a) Double-notched test specimen with a ligament l = 15 mm loaded in tension at 1 mm vertical displacement; (b) Double-notched test specimens with a ligament l = 6.5 mm loaded in plane torsion at 2.5 of rotation.

7 M.B. Silva et al. / Journal of Materials Processing Technology 234 (216) Table 2 Summary of the experimental tests to determine fracture toughness in aluminium AA15-H111 sheets with 1 mm thickness. Test Geometry (mm) Fracture toughness R (kj/m 2 ) Thickness at fracture t f (mm) t = w = 5 d = 3 l = 5 25 t = r = 4 r i = 21 d = 1 l = the ellipse = f (R II ) at the fracture point 1f = 2f corresponding to pure shear loading conditions. In case of the fracture forming line (FFL) associated to crack opening mode I and to the ductile damage criterion based on the stress triaxiality ratio (Eq. (6)), there is no exact match between the slope -1 of the FFL and the slope of the derivative to the ellipse = f (R I ) at the fracture point 1f = 3f that is typical of near plane strain loading conditions of the double notched test specimens loaded in tension. However, not only the ellipse = f (R I ) presents little curvature in the segment from equal biaxial stretching (where the derivative has a slope of 1 ) to plane strain, as the observed increase in the slope of the derivative is compatible with the upward curvature of the FFL when a threshold strain below which damage is not accumulated, is assumed to exist (refer to Eq. (8) and to the dashed solid curve > in Fig. 6). To conclude the comparison of the theoretical methods for predicting fracture loci in sheet metal forming it is worth noting that the linear loci with slope 1 (FFL) and +1 (SFFL) as well as the corresponding elliptical loci obtained from the fracture toughness based model were built upon the assumption that the throughthickness strain at fracture is constant for each crack opening mode and that fracture occurs without previous necking. The influence of these two assumptions in the overall agreement between theory and experimentation will be discussed in the next section Assessment of theoretical and experimental fracture loci Fig. 7 shows the finite element predicted distribution of effective strain for two double-notched test specimens taken from Table 2. As seen, the plastic deformation region has a slightly elliptical shape and the necked down zone where material experiences high values of effective strain is limited to a small volume located in-between the notches with an average height h < d. This result corroborates the assumption h = t that was previously made in the analytical development of Section 2.2. Fig. 8a shows the experimental strain loading paths in the principal strain space for the double-notched test specimens loaded in tension and plane torsion. The open markers correspond to values that were obtained from in-plane strain measurements with the Aramis commercial system whereas the solid markers correspond to gauge length strains at fracture that were obtained from through-thickness measurements along the cracks. The strain loading paths given by the two lines (please notice that one of these lines is coincident with the vertical axis) correspond to the finite element estimates provided by the test specimens that are shown in Fig. 7. The agreement between experimental and numerically predicted strain loading paths is good up to the transition of experimental data from necking to fracture, which is plotted in a simplified manner as a change into vertical direction (refer to the vertical dashed lines in Fig. 8a). In fact, finite elements were not able to model the change of the strain loading

8 256 M.B. Silva et al. / Journal of Materials Processing Technology 234 (216) Fig. 8. Forming limits by fracture of the aluminium AA15-H111 sheets with 1 mm thickness. (a) Comparison between the experimental fracture loci determined from the gauge length strains at fracture and that calculated from fracture toughness by means of Eqs. (13) and (14). Comparison against experimental values of fracture determined from tensile tests and from single point incremental forming of truncated conical, lobe conical and pyramidal shapes with varying drawing angles. path into vertical direction in case of the double-notched test specimen loaded in plane torsion and the reason for this is attributed to the modelling conditions of uncoupled damage that were utilized in order to cope with the analytical framework for estimating the fracture loci in the principal strain space. By taking the stress-strain curve of the material (Eq. (1)) and the experimental values of fracture toughness and thickness at fracture from the double-notched test specimens loaded in tension and plane torsion (Table 2), and by then applying Eqs. (13) and (14) it is possible to obtain the two different ellipses = f (R I ) and

9 M.B. Silva et al. / Journal of Materials Processing Technology 234 (216) = f (R II ) corresponding to crack opening modes I and II that are plotted in Fig. 8a and b. The theoretically predicted fracture locus is constructed from the solid curved segments taken from these two ellipses, as it is schematically illustrated in Fig. 8a and b. As seen in Fig. 8a, the upper ellipse resulting from fracture toughness experiments with double-notched test specimens loaded in tension agrees well with the FFL that had been previously determined by the authors (Isik et al. (214), refer to Section 3.2). This may be attributed to the fact that the slope (.68 ) of the experimental FFL is closer to that of a secant between the fracture strain pairs ( 1f, 2f ) at plane strain and equal biaxial stretching (where the derivative has a slope equal to 1 ). The dashed segment of the upper ellipse does not account for the fracture loci because the strain loading paths in this region of the tension-compression quadrant do not fail by cracking in opening mode I. However, Fig. 8a also discloses a poorer agreement between the lower ellipse and the SFFL in the region of the tension-compression quadrant located at the vicinity of pure shear, 1 = 2 (refer to the solid segment of the lower ellipse). This is attributed to the following three main reasons. Firstly, the experimental SFFL has a slope larger than that of the theoretical SFFL. Secondly, the doublenotched test specimens loaded in plane torsion fail by fracture with previous necking. Thirdly, the scattered measurements of the in-plane strains of fracture in the necked down region of the ligament by the Aramis commercial system gives rise to a certain degree of uncertainty in the determination of the fracture strain pairs ( 1f, 2f ). The first of these reasons is justified by the fact that the slope ( ) of the experimental SFFL is larger than the slope ( +1 ) of the theoretical SFFL and of the derivative to the ellipse at the fracture point 1f = 2f corresponding to pure shear. This difference moves the SFFL away from the ellipse. The second reason allows explaining a better agreement between the SFFL and the lower ellipse if fracture toughness R II (mode II) were larger than that given in Table 2 as a consequence of hypothetical double-notched test specimens loaded in plane torsion that would fail by cracking without previous necking. The third reason is also exclusive of double-notched test specimens loaded in plane torsion because the fracture strain pairs ( 1f, 2f ) are constructed from the experimental values of the in-plane minor strains at fracture 2f obtained from the Aramis commercial system and the values of in-plane major strains at fracture 1f = 3f that are obtained from through-thickness measurements at several locations along the cracks by using a microscope. The reason why this problem does not affect the double-notched test specimens loaded in tension is because their strain loading paths experience near plane strain conditions 2f =. Fig. 8b provides additional comparison of the theoretically predicted fracture locus constructed from the solid curved segments taken from the two abovementioned ellipses against experimental fracture strain pairs obtained by the authors in earlier investigations on truncated conical and pyramidal shapes (Isik et al., 214) and truncated lobe conical shapes (Soeiro et al., 215) with varying drawing angles produced by single point incremental forming. The agreement is good and demonstrates the predictive capability of the new proposed vision and associated analytical framework for loading conditions that are distinct from the calibration test cases. Fracture strain pairs retrieved from tensile tests are also included for reference purposes. To conclude it is worth noting that the differences between the ellipses and the fracture loci at the transition between the FFL and the SFFL (refer to the grey schematic quadrilateral in Fig. 8a) are easy to explain because Isik et al. (214) assumed no crack opening by mixed modes in their analytical framework to characterize failure by fracture in sheet metal forming. In other words, they assumed the FFL and the SFFL to meet in a point. However, a transition loci between the FFL and the SFFL is not only likely to exist inside the grey quadrilateral of Fig. 8 as its occurrence makes sense due to the existence of infinite ellipses located in-between the upper and lower ellipses corresponding to crack opening modes I and II. The above conclusion may justify future interest in extending the overall procedure based on the utilization of double notched test specimens loaded in tension and plane torsion to include the double notched staggered test specimens loaded in tension that were originally proposed by Cotterell et al. (1982) to produce fracture in mixed modes I and II along the inclined direction, which forms at an angle to the tips of the notches. 5. Conclusions This paper enlarges a recently proposed vision for the fracture loci in sheet metal forming that makes use of fundamental concepts of plastic flow, ductile damage and void coalescence and growth to include fracture toughness in two different crack opening modes. An analytical framework is proposed to estimate the location of the fracture loci in the principal strain space directly from the experimental measurements of fracture toughness and thickness at fracture in double notched test specimens loaded in tension and plane torsion. The comparison between the fracture loci of AA15-H111 aluminium sheets with 1 mm thickness obtained by means of the new proposed analytical framework and that retrieved from experiments with different sheet formability tests shows a good agreement in case of failure by tension and a fair agreement in case of failure by in-plane shear. The investigation also allows inferring the existence of a transition fracture loci associated to crack opening by mixed modes. Acknowledgements K. Isik and A. E. Tekkaya gratefully acknowledge funding by the German Research Foundation (DFG) within the scope of the Transregional Collaborative Research Centre on sheet-bulk metal forming (SFB/TR 73) in the subproject C4 Analysis of load history dependent evolution of damage and microstructure for the numerical design of sheet-bulk metal forming processes. M.B. Silva and P.A.F. Martins would like to acknowledge Fundaç ão para a Ciência e a Tecnologia of Portugal and IDMEC under LAETA UID/EMS/522/213 and PDTC/EMS-TEC/626/214. References ASTM E8/E8 M, 213. Standard Test Methods for Tension Testing of Metallic Materials. ASTM International, West Conshohocken, USA. Atkins, A.G., Mai, Y.W., Elastic & Plastic Fracture, Chichester, Ellis Horwood. Atkins, A.G., Fracture in forming. J. Mater. Process. Technol. 56, Bai, Y., Wierzbicki, T., 21. Application of extended Mohr Coulomb criterion to ductile fracture. Int. J. Fract. 161, 1 2. Centeno, G., Bagudanch, I., Martínez-Donaire, A.J., García-Romeu, M.L., Vallellano, C., 214. Critical analysis of necking and fracture limit strains and forming forces in single-point incremental forming. Mater. Des. 63, Cotterell, B., Reddel, J.K., The essential work of plane stress fracture. Int. J. Fract. 13, Cotterell, B., Lee, E., Mai, Y.W., Mixed mode plane stress ductile fracture. Int. J. Fract. 2, Gurson, A., Continuum theory of ductile rupture by void nucleation and growth. I: Yield criteria and flow rules for porous ductile media. J. Eng. Mater. Technolo. Trans. ASME 99, Hill, R., A theory of yielding and plastic flow of anisotropic metals. Proc. R. Soc. Lond. Ser. A 193, Hill, R., On discontinuous plastic states, with special reference to localized necking in thin sheets. J. Mech. Phys. Solids 1, Isik, K., Silva, M.B., Tekkaya, A.E., Martins, P.A.F., 214. Formability limits by fracture in sheet metal forming. J. Mater. Process. Technol. 214,

10 258 M.B. Silva et al. / Journal of Materials Processing Technology 234 (216) Isik, K., Silva, M.B., Atkins, A.G., Tekkaya, A.E., Martins, P.A.F., 215. A new test for determining fracture toughness in plane stress in mode II. J. Strain Anal. 5, Lemaitre, J., A continuous damage mechanics model for ductile fracture. J. Eng. Mater. Technol. 17, Li, Y., Luo, M., Gerlach, J., Wierzbicki, T., 21. Prediction of shear-induced fracture in sheet metal forming. J. Mater. Process. Technol. 21, Madeira, T., Silva, C.M.A., Silva, M.B., Martins, P.A.F., 215. Failure in single point incremental forming. Int. J. Adv. Manuf. Technol. 8, Martins, P.A.F., Bay, N., Tekkaya, A.E., Atkins, A.G., 214. Characterization of fracture loci in metal forming. Int. J. Mech. Sci. 83, McClintock, F.A., Kaplan, S.M., Berg, C.A., Ductile fracture by hole growth in shear bands. Int. J. Fract.Mech. 2, McClintock, F.A., A criterion for ductile fracture by the growth of holes. J. Appl. Mech. Trans. ASME 35, Soeiro, J.M.C., Silva, C.M.A., Silva, M.B., Martins, P.A.F., 215. Revisiting the formability limits by fracture in sheet metal forming. J. Mater. Process. Technol. 217, Tvergaard, V., Needleman, A., Analysis of the cup-cone fracture in a round tensile bar. Acta Metall. 32, Wierzbicki, T., Bao, Y., Lee, Y.W., Bai, Y., 25. Calibration and evaluation of seven fracture models. Int. J. Mech. Sci. 47,