A. Tensile test

Owing to their crystal structure all metals exhibit some directionality; in particular rolled sheets exhibit direction-dependent material behavior. To determine material anisotropy for zircaloy-4 by tensile tests, we print a grid of circles of diameter 0.9 and 2 mm distance between circles on the tensile specimen (Fig. 1). Grids on metal specimens are usually printed by electro-chemical etching, but due to high corrosion resistance of zircaloy-4, silk-screen printing is used instead. The anisotropy ratio is then calculated from the residual deformation of the grid. To measure strains, marks are printed on the specimen (Fig. 1); the (initial) distances between marks are l _l = 50 mm in length direction and l _w = 20 mm in width direction. From the corresponding strains (ε_l and ε_w, respectively), we calculate the anisotropic ratio r (=ε_w/ε_t), where ε_t is the strain in thickness direction. Assuming incompressibility of plastic deformation, the plastic strain components are related to each other through ε_l + ε_w + ε_t = 0. The relative elongation up to necking is evaluated to 8% for zircaloy-4. Anisotropy is decomposed into normal and planar anisotropy. The normal anisotropy ratio r _n is calculated by averaging the anisotropic ratios r ₀, r ₄₅, and r ₉₀ where the subscript denotes the angle with respect to the rolling direction (angle to rolling direction, ARD)

henceforth, r _n is taken to be the material's anisotropy ratio. We obtain r _n = 2.67 for zircaloy-4.

FIG. 1.

Schematic of grid and tensile test specimen for anisotropic coefficient.

Material properties for zircaloy-4 (Young's modulus E, yield strength σ_o, fracture strain ε_f, definition of which will come below, and Poisson's ratio ν) are further obtained by tensile tests. Tensile specimens are manufactured according to KS B0801–5 (Korean Standards Association, Seoul, Korea) with l _l = 50 mm and three different ARD (Fig. 2). Tests are conducted on an Instron 5882 tensile testing machine (Instron, Norwood, Massachusetts) with elongation rate of 4.8 mm/min. Displacements are measured via a contactless advanced video extensometer. Young's modulus and yield strength are extracted from the load–displacement curve. Due to the sudden change of the cross sectional area, it is difficult to acquire uniaxial true stress–strain data beyond the necking point. For round specimens, Bridgman ^{Reference Bridgman23} suggested a method to transform the triaxial stress state in the necking region into an equivalent uniaxial stress state, yet, measurement of necessary parameters (radius of curvature and necking cross-section) is problematic. To correct stress–strain curves obtained with flat specimens, Ling ^{Reference Ling24} extrapolated true stress–strain (σ_t–ε_t) data after diffuse necking assuming a weighted power law function. The final σ_t–ε_t curve is established by varying the weight coefficient w (0 ≤ w ≤ 1) until the analytical engineering stress–strain (σ_e–ε_e) curve is in agreement with experimental data. This method has the advantage that a Bridgman-like transformation from a triaxial to a uniaxial stress state is not necessary. Denoting the true stress at diffuse necking by σ_d and the corresponding true strain by ε_d, Ling ^{Reference Ling24} proposed the following post-diffuse necking σ_t–ε_t relation

if w = 1, the material behavior follows a linear hardening, and if w = 0 it follows a power law function. Linear and power law hardening functions can thus be regarded as upper and lower limits of σ_t–ε_t curves. In this study, we refer to Eq. (2) as the weighted-average method (WAM).

FIG. 2.

Specimen for tensile test and FE model.

For numerical analysis the commercial software Abaqus ²⁵ is used. The tensile specimen is modeled according to above standard. Considering symmetry, a quarter model is sufficient; it consists of approximately 32,000 8-node brick elements (C3D8 in Abaqus) (Fig. 2). Assuming ARD θ = 0°, the numerical σ_e–ε_e curve from Abaqus output is fit to experimental data by varying w in Eq. (2) [Fig. 3(a)] which is Abaqus input. We take the onset of local necking as follows ^{Reference Hyun, Kim, Bang and Lee26}

where ρ is the strain ratio (ρ ≡ ε₂/ε₁). The fracture strain ε_f is the true major strain corresponding to the fracture displacement in P–δ curve. ^{Reference Hyun, Kim, Bang and Lee26} We obtain ε_f = 1.1 and w = 0.36 for zircaloy-4. Here w = 0.36 means the material does not follow the pure power law function. Corresponding σ_t–ε_t curves are shown in Fig. 3(b), where nonhardening is assumed after ε_f = 1.1; values for E, σ_o, and ε_f obtained from these curves are listed in Table I. Poisson's ratio is usually given in the literature as ν = 0.37 for zircaloy-4.

FIG. 3.

(a) σ_e–ε_e curves and (b) σ_t–ε_t curves for zircaloy-4.

TABLE I.

Material properties for zircaloy-4.

B. LDH test

The FLD can be decomposed into three regions (Fig. 4). In Region I, the material is subject to biaxial tension. Diffuse necking occurs at maximum load and the whole sheet becomes uniformly thinner; shortly afterward the material fractures. In Region II, diffuse necking is followed by local necking, i.e., the thickness drops locally; fracture occurs soon afterward. Lastly, when ε₂ = −ε₁, the sheet is in pure shear. Theoretically, local necking does not occur and the forming limit is predicted with a ductile fracture criterion.

FIG. 4.

Schematic figure of forming limit diagram.

Experimental FLD's are usually obtained as follows: for positive ε₂ (Region I, Fig. 4), the hemispherical punch test proposed by Hecker ^{Reference Hecker27} is applied, while for negative ε₂ (Regions II and III, Fig. 4), the flat-headed punch test proposed by Marciniak ^{Reference Marciniak28} is used. However, a disadvantage is that each region of the FLD needs a different test. This problem can be mitigated by using the LDH test proposed by Nakajima workgroup. ^{Reference Hotz29} This test produces the strain states from uniaxial tension [ε₁/ε₂ = –(r + 1)/r] to biaxial tension (ε₁ = ε₂) by varying specimen width and lubrication between punch and sheet. Here r is one of r ₀, r ₄₅, and r ₉₀.

The limit dome stretch test was originally contrived by Ghosh ^{Reference Ghosh30} and Ayres et al. ^{Reference Ayres, Brazier and Sajewski31} as a test to determine formability. Ghosh subjected rectangular specimens with width of 117–138 mm to displacements. Realizing deformation modes from drawing to stretching, the LDH was expressed by a function of specimen width or minor strain around fracture. Ayres et al. proposed a LDH test-based method to evaluate formability in a relatively simple manner. They noticed that approximately 85–90% of failures occur close to plane strain mode and suggested to use the minimum failure height (LDH_o), which is obtained in plane strain, as a stamping formability criterion. Higher dome height means better formability. The Nakajima workgroup ^{Reference Hotz29} of the International Deep Drawing Research Group provided a method that is similar to that of Bragard et al. ^{Reference Bragard, Baret and Bonnarens32} A guideline for the determination of FLC with this method is presented in the other research work. ^{Reference Liebertz, Duwel, Illig, Hotz, Keller, Koehler, Kroeff, Merklein, Rauer, Staubwasser, Steinbeck and Vegter33} It describes in detail the geometry and the number of the specimens, the geometry of the tool and testing conditions such as punch speed, binder force, and lubricant. Major/minor strain pairs are obtained from the grid deformation in the vicinity of local necking. Based on the 3D measurement, some systems have been developed by the commercial company to determine the FLC automatically. Among them, GOM Company has developed the software called ARGUS™ (optical forming analysis) for the determination of FLC. The methodology follows the recommendation of Nakajima workgroup. After the LDH test, the specimen is scanned by ATOS™ (3D digitizing and full-field scanning equipment) and then ARGUS™ calculates the major/minor strains from the image of deformed grids. The strains near crack are then plotted along the 5 path-lines normal to the crack. This approach of using both ATOS™ and ARGUS™ is robust, and shows high repeatability.

C. Experimental FLS determination

Despite the drawbacks raised by the Nakajima workgroup—namely sensitivity of LDH test to not only specimen shape and ARD but also to grid size and pattern and to the strain measurement method—we follow the ISO 12004-2 (Ref. 34) procedure here. Test conditions are listed in Table II; equipment and notation regarding the LDH test are shown in Fig. 5. To measure the fracture limit for several deformation modes, the specimen length is fixed at l _LDH = 200 mm while specimen widths are set to w _LDH = 25, 50, 75, 100, 125, 150, 170, 200 mm. ARD is θ = 0° and thickness is t = 0.67 ± 0.01 mm. For the FLD sector of positive minor strain (in biaxial tension), the variation in major/minor strains with specimen width is too small and therefore it is hard to vary the degree of biaxiality by only changing the specimen width. To establish the state of (equi-) biaxial tension and to make the fracture occur close to the dome apex of full-width (square) specimens, lubrication is thus added. For (nearly) full-width specimens as w _LDH = 150, 200 mm, we tallow the specimen surface and put a polyethylene foil between dome punch and the tallowed specimen surface to minimize the friction. The same grid as above in Fig. 1 is printed on the specimen. After test, the specimen is scanned by ATOS™ and major/minor strains are calculated from the deformed grid imaged by ARGUS™ (Fig. 6). Figure 7 shows the major strain distributions of zircaloy-4 specimens with width w _LDH = 200 mm, from ATOS™, for (i) nonlubrication (ii) lubrication. Also shown in Fig. 7(c) are corresponding LDH test P–δ curves. We state that low friction makes fracture occur near the dome pole while the limiting dome height and the corresponding peak load increase.

TABLE II.

Experimental conditions for LDH test.

FIG. 5.

Schematic figure of LDH test for test tool and drawbead detail.

FIG. 6.

Strain measuring equipment (ARGUS).

FIG. 7.

Major strain distributions of zircaloy-4 specimens with w _LDH = 200 mm for (a) non-lubrication (b) lubrication and (c) P–δ curves for LDH test depending on lubrication; peak load increases with lubrication.

Figure 8 shows the experimental FLS for various specimen widths; theoretical FLC will be explained in detail in Sec. III. The experimental FLS moves in upper left direction with decreasing w. Here, the FLS for w _LDH = 25 mm is lower than the one for w _LDH = 50 mm, but which seems to be a material characteristic. From Fig. 8 we find that the specimen with w _LDH = 150 mm produces a plane-strain state (ε₂ = 0) and the lubricated square specimen with w _LDH = 200 mm produces an equi-biaxial state (ε₁ = ε₂). The minor strain increases with specimen width. When using specimens with w _LDH > 150 mm, the FLS obtained are nearly the same with the FLS of w _LDH = 150 mm. The variation of major/minor strains with the variation of specimen width and lubrication state is very small. Hence, it is hard to obtain experimental data between plane strain and biaxial tension. One method was proposed by Hasek, ^{Reference Hasek35} who introduced arced cutouts into circular LDH test specimens. The arced cutouts cause fracture to occur closer to the center of the specimen and thus closer to the desired combination of principal strains.

FIG. 8.

Comparison of theoretical FLCs with experimental data.

According to ISO 12004-2, FLSs are obtained as follows. After major strains have been determined [Fig. 9(a)], minor and major strains are plotted against the section length measured along the 5 paths [0–4, Fig. 9(b)] passing through the crack [Fig. 9(c)]. The corresponding 5 curve pairs obtained by Gaussian regression are compared, and the curve with the smallest major strain peak from path no. 2 is considered as FLS [Fig. 9(d)]. If there are multiple smallest peaks of major strain, the one with the smallest minor strain value is taken as the critical FLS. When ISO 12004-2 is applied, the evaluated FLS is rather small compared with other methods, for example the one suggested by American Society for Testing and Materials (ASTM). The ISO method has the advantage that results are relatively unaffected by micro necking or fracture length. However, the FLS obtained by ISO 12004-2 is a presumed strain lying somewhere between local necking and ductile fracture and is obtained by Gaussian regression of the strains measured on the fractured specimen. To determine a more appropriate FLS, we will develop a method based on strains measured at the onset of local necking between local thinning and fracture.

FIG. 9.

Procedure of obtaining FLS (e.g., w _LDH = 150 mm) (a) major strain distribution (b) selection for 5 lines in the vicinity of fracture line (c) distribution of major and minor strain with different sections and (d) bell-shaped curve fitting sectional strain.

A. Negative minor strain sector (ε₂ < 0; between uniaxial tension and plane strain)

After determining the strain hardening exponent n by regression of stress–strain data to the PPL [Eq. (5)], the theoretical FLC is established. For direct comparison with the FLSs from LDH tests, we obtain in this section numerical FLSs by FE analysis of the LDH test according to ISO 12004-2. Punch, die, and blank holder are fixed; for sheet width w _LDH = 100 mm, the FE model of the specimen consists of 15,750 C3D8 elements (Fig. 13). Draw bead is used to hold the sheet, and the punch head is moved toward the upper side. The friction coefficient between the sheet and tools (punch, die and blank holder) is considered with Coulomb friction coefficient of µ = 0.2. Note again that Eq. (2) for zircaloy-4 can also be approximated with Eq. (5) as depicted in Fig. 10. Since the numerical P–δ curves from LDH test simulation using both WAM of Eq. (2) and PPL of Eq. (5) show a good agreement with experimental data (maximum deviation of 3%), we judge Eq. (5) also to be appropriate for zircaloy-4 (Fig. 14).

FIG. 13.

FE mesh for LDH test (e.g., w _LDH = 100 mm).

FIG. 14.

Comparison of P–δ curves for two different regressed σ_t–ε_t curves.

Figure 15(a) shows the Mises stress distribution in the vicinity of the fracture region for w _LDH = 50 mm, for example. Point A is the point where fracture occurs. Point B and C are the neighboring nodes below A in y-direction. Figure 15(b) shows the ρ–(δ/δ_max) curves from nodes in the vicinity of the fracture region; again ρ ≡ ε₂/ε₁, and δ_max denotes the punch displacement at maximum load. The gray symbol is the loading-point where curve C departs from curves A and B. We may take this loading-point as the initiation of local-necking. The ρ–(δ/δ_max) curves at the point A are further plotted for different values of w _LDH [Fig. 16(b)]. Following the punch's movement, ρ increases, decreases and increases again; the length of the increasing or decreasing section varies with w _LDH. When we observe the region where ρ increases again (after its decrease), the slope first increases gently and then changes abruptly. The intersection of two regression curves for the region of gentle increase and the region of abrupt change may be taken as a numerical FLS acquisition point. Note that the gray symbols for w _LDH = 50 mm in Figs. 15(b) and 16(a) provide the same δ/δ_max = 0.922. The numerical FLS point can be regarded as the starting point of local thinning. Denoting the displacement at the point of intersection by δ_c, the ε₁ corresponding to δ_c is taken as FLS. However, the ρ–(δ/δ_max) curves for biaxial tension (i.e., for specimens of w _LDH = 150, 170, 200 mm) shows no such sudden change in ρ [Fig. 16(b)]. Here, the LDH test P–δ curves overlap (Fig. 17) and experimental FLSs are difficult to determine. We therefore need to find an alternative to obtain FLSs for the sector of ε₂ ≥ 0.

FIG. 15.

(a) Distribution of σ₁ at δ_max and (b) ρ versus (δ/δ_max) curves for w _LDH = 50 mm.

FIG. 16.

ρ versus (δ/δ_max) curves for (a) ε₂ < 0 and (b) ε₂ ≥ 0.

FIG. 17.

P–δ curves for LDH with w _LDH = 150, 170, 200 mm.

Both numerical and experimental FLSs for ε₂ < 0 are depicted in Fig. 18. The numerical ε₁ values are somewhat smaller than the experimental ones, but in general, the numerical FLSs are in a good agreement with experimental data. Numerical values are smaller because the experimental critical major strain is a presumed strain (obtained between local necking and ductile fracture), while the numerical critical major strain is obtained at the onset of local thinning.

FIG. 18.

Comparison of numerical FLSs with experimental FLSs (ε₂ < 0).

B. Positive minor strain sector (ε₂ ≥ 0; between plane strain and equi-biaxial tension)

Theoretically, local necking does not occur in the region where ε₂ ≥ 0. Therefore, to obtain the FLSs for the biaxial region, we propose a biaxial tensile model and apply diverse displacement ratios ρ_d (≡δ₂/δ₁). The FE model of the flat specimen for biaxial tension consists of 35,536 C3D8 elements [Fig. 19(a)]. Length and width of the specimen are 200 mm; thickness is 0.67 mm. To realize necking, we taper the specimen in its center by 0.01% the specimen's thickness [Fig. 19(b)]. Maximum load values obtained using the models with/without taper differ by less than 0.1%, and major and minor strains are equal at maximum load. If the thickness-to-length ratio is sufficiently large, the taper ratio does not affect the P–δ curve or the strain results. For biaxial tension, ρ_d is set to values between 0 and 1.0 with the interval of 0.2. Invoking that at diffuse necking dP = 0, for uniaxial tension, work W and load P are related by ^{Reference Hill19}

FIG. 19.

(a) Biaxial FE mesh for obtaining a FLS [region I (ε₂ ≥ 0), Fig. 4] and (b) Schematic diagram of biaxial FE model.

Since ρ_d = constant, for biaxial tension Eq. (6) can be written as

where P ₁ is a transverse (major) load and P ₂ is a longitudinal (minor) load [see Fig. 19(a)]. For ρ_d ≠ 1, loads and corresponding displacements in major and minor directions are different (Fig. 20). To apply the diffuse necking model, we introduce the relation between work and load, and determine the diffuse necking point. The (effective) load P is then $P \,\equiv\, {P_1} + {\rho _{\rm{d}}}{P_2}$ . Consequently, diffuse necking occurs when the following equation is satisfied

FIG. 20.

P–δ curves with definition of P (e.g., ρ_d = 0.8).

The numerical FLSs are the major/minor strain pairs satisfying Eq. (8). The gap between the FLSs and the theoretical FLCs are below 5% (Fig. 21).

FIG. 21.

Comparison of numerical FLSs with experimental FLSs (ε₂ ≥ 0).

Figure 22 reveals a good agreement between the numerical and experimental FLSs for the whole range of ε₂. Note that as stated in Sec. II. C, it is hard to obtain the distributed data of experimental FLSs in biaxial tension mode (ε₂ > 0). Applying our numerical approach to the materials obeying the power law, we further confirmed that our FE approach is valid for general materials. However, those data are omitted here, as this paper focuses only on zirconium alloys. In the near future, the matter will be reported in detail. Therefore, our FE approach can both substitute and supplement experimental LDH tests which are the expensive and time-demanding.

FIG. 22.

Comparison of numerical FLSs with experimental FLSs for all regions of ε₂.