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Texture and anisotropy in the Al–Mg alloy AA 5005 – Part II: Correlation of texture and anisotropic properties
Materials Science & Engineering A 618 (2014) 663–671
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Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Texture and anisotropy in the Al–Mg alloy AA 5005 – Part II: Correlation of texture and anisotropic properties Olaf Engler n, Johannes Aegerter Hydro Aluminium Rolled Products GmbH, Research & Development Bonn, PO Box 2468, D-53014 Bonn, Germany
art ic l e i nf o
a b s t r a c t
Article history: Received 5 August 2014 Received in revised form 14 August 2014 Accepted 17 August 2014 Available online 28 August 2014
Control of the plastic anisotropy during forming of a metallic sheet requires knowledge about its microstructure and crystallographic texture. In the present study the correlation of texture and anisotropy is studied in sheets of the Al–Mg alloy AA 5005. A series of samples with different textures was produced by various rolling and annealing treatments. In the 1st part of this study the microstructure and texture evolution during back-annealing of rolled sheet and during subsequent cold rolling was analysed. The present 2nd part describes the correlation of the resulting textures with anisotropic sheet properties, including Young's modulus, yield strength and r-values in uniaxial tensile tests as well as earing upon cup drawing. Texture-induced anisotropy was simulated with the help of the visco-plastic self-consistent (VPSC) model. Best agreement between simulations and experiments was obtained by using an affine formulation. For the work-hardened material the anisotropic substructure topology was also taken into consideration. With a view to a comprehensive through-process simulation of the evolution of texture and texture-related properties, in-plane anisotropy was simulated from modelled textures in place of the experimental ones. Here, larger deviations were obtained, since the errors of texture modelling and subsequent simulation of anisotropic properties tend to add up. Nonetheless, the VPSC simulations based on modelled textures were capable of reproducing the overall tendencies of anisotropy in AA 5005 sheet in a variety of tempers. & 2014 Elsevier B.V. All rights reserved.
Keywords: Texture Anisotropy r-value Earing Aluminium alloy Modelling
1. Introduction Sheets of the aluminium–magnesium alloy AA 5005 with about 1 wt% Mg find applications where more strength is required than achieved by unalloyed aluminium and where combinations of good formability, very good resistance to corrosion, good weldability, or good electrical properties are necessary. AA 5005 is typically used either in the fully soft condition (O temper) or in various semi-hard, i.e. strain-hardened (“H1x”) or back-annealed (“H2x”) tempers, depending on the required combinations of mechanical strength and formability. As many products made of AA 5005 sheets are subjected to forming operations, evolution of crystallographic texture and texture-related anisotropy is an important topic for such alloys (e.g. [1–6]). For instance, the occurrence of earing during deep drawing of a textured sheet can cause major problems in the production of deep drawn containers from aluminium sheet (e.g. [4–8]). It was the aim of the present study to correlate the anisotropic properties of AA 5005 sheets with their crystallographic texture. In the
first part of this study the evolution of microstructure and texture during processing of the sheets was analysed [1]. Especially, the texture changes upon back-annealing were simulated with an analytical softening model AlSoft, while texture evolution during subsequent cold rolling was tackled with the GIA polycrystal-plasticity code. In the present paper we address the correlation of the resulting textures with anisotropic sheet properties, including Young's modulus, yield strength and the plastic strain ratio (r-value) in uniaxial tensile tests as well as earing upon cup drawing. For simulation of textureinduced anisotropy the visco-plastic self-consistent (VPSC) formulation was employed. For the work-hardened states the impact of an anisotropic substructure topology was also taken into consideration. Furthermore, with a view to a comprehensive through-process simulation of the evolution of texture and texture-related properties, in-plane anisotropy was simulated from modelled textures in place of the experimental ones.
2. Experimental procedures n
Corresponding author. Tel.: þ 49 228 552 2792; fax: þ49 228 552 2017. E-mail address: [email protected] (O. Engler).
http://dx.doi.org/10.1016/j.msea.2014.08.040 0921-5093/& 2014 Elsevier B.V. All rights reserved.
As described in detail in part I of this work [1], hot strip of the Al–Mg alloy AA 5005 was cold rolled to 1.4 mm gauge to state H18.
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Samples taken from the cold rolled sheet were subjected to various annealing treatments, resulting in different backannealed states H2x up to the fully soft state O. Finally, soft annealed samples were cold rolled to different thicknesses ranging from 1.0 to 0.2 mm, corresponding to various strain hardened states H1x. Details about the evolution of microstructure and texture during back-annealing and subsequent strain-hardening can be found in part I of this work [1]. Tensile tests were carried out according to the international standard ISO 6892-1:2009. Standard size tensile specimens with 80 mm original gauge length and 20 mm width were machined in five different directions in the sheets, viz. parallel, under 22.51, 451, 67.51 and perpendicular to the rolling direction, RD. For each orientation a minimum of two specimens were tested in a screwdriven testing machine Zwick Z100 at a nominal strain rate of 0.00025 s 1 up to the yield strength Rp0.2 and 0.0067 s 1 up to fracture. For the determination of the modulus of elasticity the increased conditions of the new Annex G “Determination of the modulus of elasticity on metallic materials under uniaxial tensile loading” of the draft international standard ISO/DIS 6892-1:2014 were applied. Samples were loaded up to approximately 40% of the expected yield strength. The use of high resolution, double-sided, averaging extensometers of class 0.5 enables determination of the modulus of elasticity with increased accuracy and reduced measurement uncertainty. The plastic strain ratios rα, which describe the ratio of the true plastic strain in the width direction to that in the thickness direction (both in a state of uniaxial tension), were determined according to ISO 10113:2006. For that purpose, the plastic strains in the length and width directions were measured using two separate extensometers on the gauge section. The recommended regression method of the standard for materials displaying inhomogeneous yielding were applied to determine the slope mr of the true plastic width strain versus true plastic length strain through the origin. Finally, the r values were calculated from the condition of plastic incompressibility according to r¼mr/(1þ mr). The r values determined by the regression method were validated by manual measurements of width and thickness at various positions on the tensile specimen outside the necking and fracture area. For the sake of completeness it must be emphasised that the measurement uncertainty for the r-value determination is increased for the back-annealed and work hardened states H1x and H2x due to the inherently small changes in the dimensions of the test specimen. In order to determine the earing behaviour, cup drawing tests were performed on a hydraulic sheet forming machine following the guidelines of EN 1669. Blanks with a diameter of dB ¼60 mm were deformed with a punch of dP ¼33 mm, resulting in a drawing ratio of dB/dP E1.8. The tests were performed with blank holder forces ranging from 4 to 14 kN with lanolin lubrication. After the mechanical testing, the earing profiles were determined in steps of 11 with a mechanical probe using a set-up developed by Huxley Bertram Engineering, Cottenham, UK. In general, the results of minimum three cups were averaged. Fig. 1 shows an example of the earing height, h, as a function of the angle α to the rolling direction (RD) for the soft annealed sheet AA 5005-O. The profile depicts a characteristic four-ear profile with ears under 01, 901, 1801 and 2701 to the RD (see inlay). Because of the orthotropic symmetry of rolled sheets, the profiles can be symmetrised with respect to the RD and transverse direction (TD), see thick red curve in Fig. 1. For better comparison the earing profiles h(α) are normalised by the average cup height:
n
h ðαÞ ¼ hðαÞ U
αmax ∑α hðαÞΔα
ð1Þ
Fig. 1. Earing profile of soft annealed AA 5005-O, showing a characteristic four-ear profile with ears under 01, 901, 1801 and 2701 to the RD (thick red line: symmetrised profile). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
3. Correlation of texture and plastic anisotropy 3.1. Texture and plastic anisotropy during tensile testing As described in more detail in the experimental section, sheet anisotropy was assessed by performing standard uniaxial tensile tests in different in-plane direction. From these tests, the yield strength Rp0.2 and the strain ratio (r-value) were determined as a function of the in-plane angle α between the tensile axis and the original rolling direction. In order to assess the correlation of texture and plastic anisotropy in tensile tests, three samples with distinctively different textures were selected (see Figs. 2 and 3). (i) cold rolled and soft-annealed sheet in thickness 1.4 mm, i.e. AA 5005-O, with a fully recrystallised microstructure (Fig. 2(a)) and a characteristic cube recrystallisation texture (Fig. 3(a)); (ii) cold rolled and back-annealed sheet in thickness 1.4 mm, i.e. AA 5005-H24, with a well recovered, layered microstructure (Fig. 2(b)) and a typical plane-strain rolling texture (Fig. 3(b)); (iii) sheet rolled to 0.7 mm, i.e. AA 5005-H16, with a heavily elongated grain structure (Fig. 2(c)) and a mixed cube plus rolling texture (Fig. 3(c)).
The true stress, σ, vs. true strain, ε, curves for AA 5005 in states O, H24 and H16 are plotted in Fig. 4 for in-planes angles α of 01, 451 and 901 (the curves at α¼22.51 and 67.51 were omitted for clarity). As expected, the tensile curves reveal an increase in strength with decreasing ductility in the sequence O-H24-H16. For the softannealed temper O there is only minor anisotropy for both yield strength and subsequent work hardening. It is noted that there is some variation in the maximum elongation and, in consequence, in ultimate tensile strength, Rm. Differences in elongation and ductility are not the topic of the present study, however. Fig. 5(a) shows the absolute values of the in-plane anisotropy in yield strength, Rp0.2, modulus of elasticity, E, and plastic strain ratio, r, as a function of α. Despite the fairly pronounced cube texture in the fully recrystallised O-temper, there is only minor inplane anisotropy in strength, with the yield strength fluctuating by 72 MPa. The elastic modulus E displays more severe in-plane anisotropy, with E varying from about 70 GPa at 01 and 901 to a maximum of 73 GPa under 451. The plastic strain ratio r, given on
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Fig. 2. Microstructure of AA 5005 after various rolling and/or back-annealing treatments, (a) AA 5005-O, (b) AA 5005-H24, (c) AA 5005-H16 (longitudinal sections, anodically oxidised, RD up).
Fig. 4. True stress, σ, vs. true strain, ε, curves at various in-planes angles α for AA 5005 in states O, H24 and H16, showing increased strength and decreasing ductility.
Fig. 3. Texture of AA 5005 after various rolling and/or back-annealing treatments, (a) AA 5005-O, (b) AA 5005-H24, (c) AA 5005-H16 (ODF φ2-sections, intensity levels 1 – 2 – 4 – 8 – 16).
the right scale, reveals the opposite behaviour, with maximum values of about 0.8 at 01 and 901 and a minimum of 0.5 around 451. In the back-annealed state H24 the in-plane anisotropy is more pronounced (Fig. 4). The yield strength steadily increases from 130 MPa along the RD to 137 MPa along the TD (Fig. 5(b)). The
r-values increase from below 0.5 in the RD to values of about 1 in the TD. The elastic modulus shows the opposite behaviour form the soft state, with maxima of about 73 GPa at 01 and 901 and a minimum below 70 GPa at 451. In the strain-hardened state H16 the maximum elongation is very low, below 2% (Fig. 4). In-plane anisotropy is slightly more pronounced than in state H24. Here the yield strength increases from 172 MPa along the RD to 182 MPa along the TD (Fig. 5(c)). From 01 to 451 the r-values stay at low values between 0.4 and 0.6, before sharply increasing to values in excess of 1.2 in the TD. The elastic modulus shows a somewhat unsystematic evolution with low in-plane variations of just over 1 GPa and a weak maximum under 451. In conclusion, all three samples show pronounced in-plane anisotropy with relatively high r-values in the TD. The two states AA 5005-H16 and H24 with prevalent rolling textures display similar evolution in r-values, with low r-values in the RD before r strongly increases to values in excess of 1 in the TD. Note that the soft AA 5005-O sheet which comprises a strong cube recrystallisation texture depicts a slight asymmetry in r values between 01 and 901. This is attributed to the remainders of the deformation texture observed in this state (Fig. 3(a)).
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Fig. 5. In-plane anisotropy in yield strength, Rp0.2, elastic modulus, E, and plastic strain ratio, r, as a function of α; (a) state O, (b), state H24 and (c) H16.
3.2. Polycrystal-plasticity simulation of plastic anisotropy during tensile testing The correlation of texture and plastic anisotropy was modelled with the help of a polycrystal-plasticity code, here, the viscoplastic self-consistent (VPSC) formulation. VPSC simulates the plastic deformation of polycrystalline aggregates subjected to external strains and stresses. It accounts for full anisotropy in properties and response of the single crystals and the aggregate. VPSC is based on the physical mechanisms of dislocation slip and accounts for grain interaction effects. In addition to providing the macroscopic stress–strain response, it accounts for hardening, reorientation and shape change of individual grains. As a consequence, it predicts the evolution of hardening and texture associated with plastic forming. The formal theory upon which the VPSC code is based is explained in the papers by Lebensohn and Tomé [9,10] and in the textbook “Texture and Anisotropy” by Kocks et al. [11]. In the VPSC model each orientation, or grain, is treated as an ellipsoidal visco-plastic inclusion embedded in and interacting with an effective visco-plastic medium, which has the average polycrystal properties. The stresses and strains in the individual grains can vary as a function of orientation, depending on the stiffness of the interaction. The properties of the HEM are not known a priori, but are adjusted ‘self-consistently’ to coincide with the average response of all orientations constituting the aggregate, i.e., its texture. The single crystal response is described by means of a rate sensitive constitutive law of the form N
dij ¼ γ_ 0 ∑ msij s¼1
mskl σ ckl τsc
n ð2Þ
Index s labels the individual slip systems (N ¼12 for fcc materials with {111}〈110〉 slip), ms is the geometric Schmid tensor, γ_ 0 is a reference shear rate, τsc is the threshold stress controlling the activation of the slip system and n is the inverse of the materials strain rate sensitivity. The strain rate dij and stress σ cij in each grain are related to the average polycrystal strain rate Dij and stress σij through an interaction equation of the form: ~ : ðσ σ c Þ ðD dÞ ¼ nef f M
ð3Þ
~ is a visco-plastic interaction compliance and neff is an The tensor M effective inverse rate sensitivity which tunes the strength of the coupling between the stress deviations and the associated strain
rate deviations. The effective stiffness of the matrix increases as neff decreases until, for neff ¼0, the Taylor condition is reproduced. In order to simulate a tensile test with the VPSC approach, the discretised texture of the sheet is subjected to the external stress and strain boundary conditions. Then, the VPSC model is employed to derive the materials response in terms of the active slip systems and resulting strains, stresses, reorientation, work hardening, etc., which can be utilised to predict the evolution of hardening and texture associated with plastic forming. For a tensile test along the former rolling direction the strain rate D consists of an elongation along direction X1 plus the resulting shape changes in transverse direction, X2, and through-thickness direction, X3. For a textured material, the exact values of the inplane strain rate D2 and the through-thickness strain rate D3 are not known, however. Therefore, the boundary conditions of uniaxial tension are described by assuming zero stresses σ2 and σ3 in directions X2 and X3, respectively (Fig. 6), see Refs. [12,13]. The ratio of the resulting in-plane to through-thickness strain rates D2/D3 then gives the r-value. Thus, under making use of a feature of the VPSC model which permits enforcing mixed displacement/ stress boundary conditions, the following load case is imposed: 0 1 0 1 1 0 0 n n n B C B C D ¼ @ 0 n 0 A U ε_ σ ¼ @ n 0 n A ð4Þ n n 0 0 0 n where ε_ is a scalar measure of the strain rate increment [s 1]; the asterisks n indicate that the corresponding value is not prescribed. The VPSC code offers modelling of the directional elastic modulus E(α) which is a measure of the elastic stiffness of the material when subjected to a tensile stress in the direction of α in the rolling plane (Fig. 6). E(α) is calculated as the ratio between the stress and the deformation measured along the sample axis (superscript α) [11,14]: EðαÞ ¼
σ α11 1 ¼ εα11 Sα1111
where
Sα1111 ¼ Rα1i Rα1j Rα1k Rα1l Sijkl
ð5Þ
Sijkl is the polycrystal compliance expressed in the reference system, which is computed with the self-consistent approach. Rα is the rotation matrix for a rotation by angle α around ND ¼X3 (Fig. 6). We have calculated E(α) for the three different textures considered here, using the polycrystal elastic constants of aluminium alloys C11 ¼108 GPa, C12 ¼61 GPa and C44 ¼28.5 GPa. Note that the constant were slightly adapted (by less than 1 GPa) to give best overall fit with the experimental data points. In all three cases
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Table 1 Voce hardening parameters at the crystal level (Eq. (6)) used at the polycrystalplasticity simulation for the various tempers.
State O State H24 State H16
Fig. 6. (a) Scheme of tensile tests under different in-plane angles α to the rolling direction, RD, to determine in-plane plastic anisotropy.
Fig. 7. Simulated in-plane anisotropy in elastic modulus, E for AA 5005 in three different states (VPSC).
the simulated curves E(α) reproduce the elastic in-plane anisotropy of the materials reasonably well (Fig. 7). In the soft-annealed state O with a characteristic cube recrystallisation texture (Fig. 3(a)) the elastic modulus E reveals minima at 01 and 901 and a pronounced maximum under 451. In the recovered material H24 with a welldefined rolling texture (Fig. 3(b)) the opposite behaviour is observed, with a minimum in the elastic modulus E at 451. The strain-hardened sheet H16 with a mixed texture (Fig. 3(c)) shows a lower in-plane elastic anisotropy with a weak maximum at 45–601 which, again, is modelled by the VPSC simulations. In order to simulate a full tensile test along direction X1, the combined stress and strain state (Eq. (4)) was imposed in sequential, incremental steps, starting with the original sheet texture. The strain rate increments ε_ were chosen in the range 0.0005 s 1 (for AA 5005-H16) over 0.002 s 1 (for AA 5005-H24) to 0.005 s 1 (for AA 5005-O) such that the relevant strain regime was covered with approximately 20–50 points (Fig. 4). After each deformation step, reorientation and shape change of the grains were determined. The VPSC scheme offers several options to solve the equation of inclusion-matrix interaction. It has repeatedly been shown that for prediction of anisotropy compliant modes – either the tangent approach or an intermediate linearization scheme with a reduced effective inverse rate sensitivity neff – gives much better results than the much stiffer secant approach [15–17]. In the present study, best results were obtained with the more recent affine formulation by Masson et al. [18].
τ0 [MPa]
τ1 [MPa]
θ0 [MPa]
θ1 [MPa]
16 48 65
21.5 4.0 2.5
325 160 2500
29 39 55
The above stress/strain state (Eq. (4)) applies for uniaxial tensile deformation along direction X1, which generally relates to the former sheet rolling direction, RD. In order to determine the variation of the property under consideration in the sheet plane, the stress/strain state has to be rotated by angle Δα about the common normal direction, ND¼X3 (Fig. 6). For the sake of simplicity, we rather rotate the sheet texture and grain (ellipsoid) axes by –Δα about the ND, while keeping the load state constant. As described already in part I [1], work hardening of the material during tensile testing was simulated by means of the generalised Voce law [19] to calculate the increase of the critical shear stress τc with shear strain Γ for all grains (see Eq. (2)): τc ðΓ Þ ¼ τ0 þ ðτ1 þ θ1 UΓ Þ U 1 exp Γθ0 =τ1 ð6Þ The single crystal hardening parameters τ0, θ0, τ1 and θ1 were determined by fitting the experimental tensile test data for alloy AA 5005 in the various tempers (Fig. 4) and are summarised in Table 1. Note that the hardening parameters of the soft material differ considerably from the Voce parameters obtained for cold rolling with the GIA model (Table 1 in [1]). This is due to the different deformation mode, rolling vs. uniaxial tension, and different polycrystal-plasticity model, GIA vs. VPSC. It has been described in the literature that grain shape also affects in-plane anisotropy, though less pronounced than texture [17,20]. In VPSC, grain shape effects modify the grain-matrix interaction through the dependence of the Eshelby tensor upon the ellipsoid aspect ratios of non-spherical inclusions. The soft annealed material in temper O had a rather equiaxed microstructure (Fig. 2(a)); accordingly, here the assumption of a spherical grain structure with an aspect ratio (RD:TD:ND) 1.0:1.0:1.0 is reasonable. Both the cold rolled sheet H16 and, especially, the back-annealed state H24 revealed a strongly elongated microstructure, in contrast (Fig. 2(b), (c)). However, it does not appear likely that these deformed grains per se will have a noticeable impact on anisotropy [21]. Rather, it is anticipated that other microstructural features, most probably the as-deformed substructure, exert a significant effect in altering the in-plane anisotropy compared to the ubiquitous influence of crystallographic texture [21–24]. Thus, in agreement with the TEM micrographs shown in part I [1], for the well-recovered material in state H24 again a spherical (sub)grain structure was assumed, i.e. 1.0:1.0:1.0. For the cold rolled state H16 a non-spherical substructure with an aspect ratio of 4:1 was assumed, more precisely, (RD:TD:ND) 2.0:1.0:0.5. Fig. 8 shows the resulting in-plane variations in yield strength and r values for alloy AA 5005 in the three different tempers. As for the strength anisotropy the stress after the onset of yielding was taken as a measure of the yield strength Rp0.2 (Fig. 8(a)). For the determination of the r-values the strain rates in throughthickness direction, D3, and transverse direction, D2, were integrated over the simulated tensile test in order to yield average r-values similarly as in the experiments (Fig. 8(b)). In the fully recrystallised O-temper the weak in-plane anisotropy in strength is reproduced by the VPSC model, though the W-shape of the experimental data points is not matched. The plastic strain ratio r showed much stronger anisotropy, with a distinct minimum at 451. The polycrystal-plasticity simulations predict the overall
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Fig. 8. Correlation of texture and in-plane anisotropy in (a) yield strength Rp0.2 and (b) r-values.
Fig. 9. Correlation of texture and in-plane anisotropy, based on modelled textures rather than experimental ones (a) yield strength Rp0.2 and (b) r-values (see text for details).
shape and the individual data points from 01 to 67.51 with reasonable accuracy, yet the r-value at 901 is heavily overpredicted. For the other two tempers similar effects are obtained. In the back-annealed state H24 the increase in yield strength at α¼901 as well as the general evolution of r(α) is simulated, but the simulated curve r(α) appears to be less undulated than its experimental counterpart (Fig. 8(b)). In the strain-hardened state H16 the increases in yield strength is very nicely matched (Fig. 8(a)). As in state O, the r-values are satisfactorily modelled up to α¼67.51, but are strongly overpredicted at α¼901. With a view to a comprehensive through-process simulation of texture and texture-related properties it appeared worthwhile to use simulated textures rather than experimental ones as an input for the VPSC earing model. Fig. 9 shows the same examples of anisotropy as Fig. 8, but based on simulated textures rather than experimental ones. It has been demonstrated in part I that both rolling textures and recrystallisation textures can be modelled with great accuracy [1]. Accordingly, the predictions of yield strength anisotropy and r-values based on modelled textures in
general resemble the simulations based on experimental textures (Fig. 8), but the deviations between simulation and experiments are larger (Fig. 9). In particular this affects predictions of the rvalues (Fig. 9(b)), which illustrates how sensitive r-values are on texture. Presumably, the errors of texture modelling and subsequent simulation of anisotropic properties tend to add up, leading to increasing discrepancies from the experimental results. However, despite the enlarged discrepancies, the VPSC simulations based on modelled textures are obviously capable of reproducing the overall tendencies of in-plane anisotropy in yield strength and r-values.
3.3. Texture and earing The earing behaviour was analysed by means of deep drawing of small cups from circular blanks which were taken from the various sheets with different crystallographic texture. The characteristic earing profiles of deep drawn cups form because the
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sheet texture gives rise to different radial elongations in different directions of the blank. In most soft annealed Al alloys four ears are obtained, which are situated under angles α of 01 and 901 to the rolling direction (Fig. 1). In heavily cold rolled sheet often fourfold earing prevails as well, but at the four positions 7451 from the rolling direction. In many cases of industrial relevance mixed textures will form, by contrast, which are associated with the formation of more complex earing profiles with six ears (at 01/1801 and 451) or eight ears (at 01/901 and 451) (e.g. [4–8]). The evolution of earing profiles in the present alloy AA 5005 is displayed in Fig. 10(a) for the various back annealing treatments H18-O and in Fig. 10(b) for subsequent cold rolling of recrystallised AA 5005-O to temper H19. In the initial, cold rolled state H18 a characteristic mixed eight-ear profile is obtained, showing maxima under 01 (and 1801) and under 451 (and 1351/2251/ 3151). The ear at 901 (and 2701) is not very pronounced and sometimes not discernible at all, which is then the typical appearance of a six-ear profile [25]. Recovery upon back annealing (H18-H24-H22) does not lead to significant changes in texture [1] and, in consequence, in the earing profiles. Partial recrystallisation in the state H21 entails a significant modification of the anisotropy, in contrast, with the 01 and 901 ears being fortified at the expenses of the 451 ears. Accordingly, AA 5005-H21 shows a textbook-example of a smooth eight-ear profile. In principle, this texture would be well suited to produce Al sheet with low earing properties, yet it is hard to control partial recrystallisation with sufficient accuracy in industrial practice (e.g. [26]). Further progressing recrystallisation then gives rise to the complete disappearance of the 451 ears, leading to the typical four-ear profiles with ears at 01/901 of a fully recrystallised sheet (see Fig. 1). Subsequent cold rolling of the fully soft sheet leads to the opposite effects. With increasing strain the 451 ears intensify to the disadvantage of the 01 ears and, even more rapidly, the 901 ears. Accordingly, at high strains (0.3 mm, i.e. H19) a six-ear profile is obtained, which resembles the profile of the initially cold rolled material (H18). Note that larger strains will eventually lead to a four-ear profile with ears under 7 451 only (0.2 mm, H19). At intermediate strains, here for samples H16 and H18, mixed eightear profiles with low overall earing level are obtained. This practice of inter-annealing and medium final rolling to designated temper is utilised in industry to produce low-earing grade Al sheet (e.g. [27,28]). It should be emphasised here that the two
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low-earing qualities H16 and H18 both displayed rather pronounced anisotropy in strain ratios, i.e. r-values (e.g. Fig. 8(b)). This disproves the often quoted direct relationship between r-values and earing; rather, the anisotropy in yield strength (Fig. 8(a)) must be taken into consideration as well [29]. 3.4. Polycrystal-plasticity simulation of earing from texture The evolution of earing during cup deep drawing was simulated by means of the visco-plastic self-consistent (VPSC) scheme which was already used to simulated anisotropy in the tensile tests (Section 3.2.). Polycrystal-plasticity simulation of earing is slightly more complicated, however, which is caused by the more complex behaviour of stresses and strains operating during cup deep drawing. The present polycrystal-plasticity earing model, described in detail in Ref. [30], relies on the fact that plastic flow and, therefore, the formation of ears and troughs is largely concentrated in the flange of the blank under the blank-holder. In the flange a plane stress state prevails, in that the material is exposed to a tensile stress σr in the radial direction and a compressive stress σt in the tangential direction (Fig. 11). The stress in the through-thickness direction σz as well as the offdiagonal shear stresses σrt are almost zero. This stress state generates a positive radial strain rate Dr and a negative tangential strain rate Dt, while the through-thickness strain rate Dz is not necessarily zero. Thus, the following mixed load case is imposed: 0 1 0 1 σr n n n 0 0 B B C n C ð7Þ Dij ¼ @ 0 1 0 A U ε_ ; σ ij ¼ @ n n A n n σ n 0 0 z ¼0 (_ε strain rate increment; the asterisks * indicate that the corresponding value is not prescribed.) It is seen that the stress components σr and σz are prescribed rather than the corresponding strain rate components Dr and Dz. All simulations in this study were performed with a non-zero tangential stress of σr ¼–0.25 σt (see Ref. [31]). Since in the first time step the stress values are not yet known, this is accomplished iteratively within a few VPSC steps until stable values for σr and σt are achieved. Eq. (7) is expressed in the frame defined by radial direction, r, tangential direction, t, and through-thickness direction, z. This frame is related to the standard rolling frame {RD, TD, ND} through
Fig. 10. Evolution of earing profiles h*(α) in AA 5005, (a) after various back annealing treatments H18-O, and (b) after cold rolling of recrystallised AA 5005-O to temper H19 (normalised, symmetrised).
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a rotation by angle α about the common direction z¼ ND (Fig. 11). Thus, in order to derive the earing tendency under a given angle α, the strain in radial direction is determined for the texture in this particular position of the blank. For this purpose, the sheet texture is rotated by Δα in steps of typically 51 about z. The earing profile along the rim of a deep-drawn cup is then given by the radial strain or, more precisely, by the normalised strain rate ratio q¼ Dr/Dt as a function of α. The above model was applied to the textures of AA 5005 in the various states H18-O and O-H19; Fig. 12(a) shows the normalised, symmetrised earing profiles for a number of characteristic states. Comparison with the experimental profiles, shown in Fig. 10, reveals the good accuracy of the VPSC earing model mentioned above. Recrystallisation of the initial, cold rolled state H18 gave rise to an increase in 01 and 901 ears to the disadvantage of the 451 ears, leading to the typical four-ear profiles of a fully recrystallised sheet with ears at 01 and 901 (see Fig. 1). On the other hand, recovery reactions occurring during the early stages of back-annealing (H18-H22) led to an appreciable sharpening of the rolling texture [1]. Since the texture type is not affected, however, this texture sharpening did not alter the earing profiles.
Subsequent cold rolling of the fully recrystallised sheet is accompanied by an intensification of 451 earing at the expenses of the 01 ears and, even more rapidly, the 901 ears. That is to say, the four-ear profile with ears at 01 and 901 is transformed via mixed eight-ear profiles at intermediate strains (e.g. H16) towards a six-ear profile (H19), which resembles the profile of the initially cold rolled material (H18). Again, it appears that the transitional states (H21 and H16) are distinguished by mixed earing profiles with low overall earing grade. Similarly as described for the simulations of in-plane anisotropy, the VPSC earing model was also applied to simulate earing from simulated textures rather than experimental ones. Fig. 12(b) shows the same examples of earing profiles as Fig. 12(a), but based on simulated textures in place of their experimental counterparts. The curves nicely reproduce the transition from 451 earing towards 01/901 earing upon back-annealing and the re-appearance of 451 earing during subsequent cold rolling O-H19. Compared with the profiles simulated from experimental textures the deviations with regard to the experimental earing profiles are larger, however. Nevertheless, the main earing features are clearly reproduced, illustrating the potential of a comprehensive through-process simulation of texture and texture-related properties by combining models for texture evolution with polycrystal-plasticity models to simulated texture-related properties.
4. Discussion and conclusions
Fig. 11. Geometry of cup deep drawing used in the present study.
It is widely accepted that crystallographic texture of Al alloy sheet is the main cause for the occurrence of anisotropic sheet properties (e.g. [1–8]). However, plastic anisotropy may also be influenced by other microstructural parameters, including the average grain shape [17,20], the topology of second-phase particles [32] or the substructure topology, e.g. arrangement of dislocations etc. [21–24]. In the course of the present study we have explored the correlation of texture and anisotropy in sheets of the Al–Mg alloy AA 5005 in a number of differently rolled and annealed states (see Ref. [1]). Clearly, the experiments substantiate a relationship between texture and anisotropic properties, especially for in-plane variation in modulus of elasticity (Young's modulus) E, yield strength Rp0.2 and r-values (Figs. 4,5) as well as earing during cup drawing (Fig. 10).
Fig. 12. Correlation of texture and earing in AA 5005, (a) simulation based on experimental textures, (b) simulation based on modelled textures (see text for details).
O. Engler, J. Aegerter / Materials Science & Engineering A 618 (2014) 663–671
Evolution of texture and texture-induced anisotropy can be modelled with the help of polycrystal-plasticity codes like the familiar Taylor model. In the present study, softening during backannealing was modelled with an analytical softening model named AlSoft treating both recovery and recrystallisation including the resulting texture changes. Texture evolution during subsequent cold rolling was tackled with the GIA polycrystal-plasticity code considering grain-interaction [1]. Then, the visco-plastic selfconsistent (VPSC) model was employed to simulate the resulting texture-induced anisotropy [9–11]. For the simulation of tensile tests in different in-plane directions, the load case (i.e. strains and stresses) of uniaxial tensile deformation under an angle α to the original sheet rolling direction is imposed on the sheet texture (Eq. (4)). The load case upon cup drawing is approximated by a plane-stress state with a constant ratio of radial and tangential stresses in the flange (Eq. (7)). VPSC then computes the crystal response in terms of reorientation and shape change of the individual grains as well as the resulting macroscopic stresses and strains. Best results were obtained with an affine formulation to solve the interaction between inclusion and matrix in the VPSC code [18]. For the work-hardened material in state H16 a slight improvement of the simulations could be achieved by taking the anisotropic substructure topology into consideration [21]. By contrast, the well-recovered state H24 comprised an equiaxed subgrain structure [1]. Hence, despite its strongly layered microstructure (Fig. 2(b)), best simulation results were achieved by assuming a spherical substructure. For the soft-annealed material in state O likewise a spherical grain structure (Fig. 2(a)) was utilised. Comparison of the simulated anisotropy in materials properties with the experimental results showed that the VPSC model is able to predict the in-plane anisotropy obtained in tensile tests and the earing behaviour observed in cup drawing tests with reasonable accuracy (Figs. 8, 12(a)). In some cases, most notably for r-values at α¼ 901, quite large discrepancies were obtained, however. Despite these discrepancies we presume that the accuracy obtained will be sufficient to utilise texture-based anisotropy data as an input for subsequent forming simulations (e.g. [33–35]). With a view to a comprehensive through-process simulation of texture and texture-related properties we have used simulated textures in place of the experimental ones as an input for the VPSC model. For both the in-plane anisotropy obtained in tensile tests and for the earing behaviour in cup drawing tests the predictions based on modelled textures largely resemble the simulations based on experimental textures (see Figs. 8 and 9 as well as Fig. 12(a) and (b)); however, the deviations between simulation and experiments are generally larger. Presumably, the errors of texture modelling and subsequent simulation of anisotropic properties tend to add up, leading to increasing discrepancies from the experimental results. Nonetheless, the VPSC simulations based on modelled textures are capable of reproducing the overall tendencies of anisotropic properties in AA 5005 alloy sheet in a variety of tempers and textures. Accordingly, a combined application of the present models yields valuable information on the evolution of
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texture and earing as a function of process parameters that can be used to improve the quality of aluminium sheet alloys.
Acknowledgements The VPSC software was provided by Dr. C.N. Tomé (Los Alamos Nat. Lab., USA). The authors would like to thank their colleagues Dr. H. Aretz and Prof. J. Hirsch for helpful discussions.
References [1] O. Engler, Mater. Sci. Eng. A, (2014) (in press) http://dx.doi.org/10.1016/j.msea. 2014.08.037. [2] P.S. Bate, Y.G. An, Scr. Mater. 51 (2004) 973–977. [3] O. Engler, Y. An, Solid State Phenom. 105 (2005) 277–284. [4] M.A. Zaidi, T. Sheppard, Mater. Sci. Technol. 1 (1985) 593–599. [5] S.E. Naess, B. Andersson, in: H.D. Merchant, J.G. Morris (Eds.), Texture in NonFerrous Metals and Alloys, TMS-AIME, Warrendale, PA, 1985, pp. 61–78. [6] C. Liu, L. Zhuang, M.R. van der Winden, T.J. Hurd, B. Charot, Mater. Sci. Forum 331–337 (2000) 787–792. [7] P.M.B. Rodrigues, P.S. Bate, in: H.D. Merchant, J.G. Morris (Eds.), Texture in Non-Ferrous Metals and Alloys, The Metall. Soc., Warrendale, PA, 1985, pp. 173–187. [8] A. Oscarsson, W.B. Hutchinson, H.-E. Ekström, Mater. Sci. Technol. 7 (1991) 554–564. [9] R.A. Lebensohn, C.N. Tomé, Acta Metall. Mater. 41 (1993) 2611–2624. [10] R.A. Lebensohn, C.N. Tomé, Mater. Sci. Eng. A 175 (1994) 71–82. [11] U.F. Kocks, C.N. Tomé, H.R. Wenk, Texture and Anisotropy: Preferred Orientations and their Effect on Materials Properties, Cambridge Univ. Press, Cambridge, UK, 1998. [12] S.-H. Choi, F. Barlat, Scr. Mater. 41 (1999) 981–987. [13] O. Engler, Aluminium 80 (2004) 719–723. [14] S.-H. Choi, K.-G. Chin, J. Mater. Process. Technol. 171 (2006) 385–392. [15] C.N. Tomé, R.A. Lebensohn, C.T. Necker, Metall. Mater. Trans. 33A (2002) 2635–2648. [16] M.A. Iadicola, L. Hu, A.D. Rollett, T. Foecke, Int. J. Solids Struct. 49 (2012) 3507–3516. [17] Q. Xie, P. Eyckens, H. Vegter, J. Moerman, A. Van Bael, P. Van Houtte, Mater. Sci. Eng. A 581 (2013) 66–72. [18] R. Masson, M. Bornert, P. Suquet, A. Zaoui, J. Mech. Phys. Solids 48 (2000) 1203–1227. [19] C. Tomé, G.R. Canova, U.F. Kocks, N. Christodoulou, J.J. Jonas, Acta Metall. 32 (1984) 1637–1653. [20] S.-H. Choi, J.C. Brem, F. Barlat, K.H. Oh, Acta Mater. 48 (2000) 1853–1863. [21] O. Engler, Mater. Sci. Technol. (2014) (MST11659, Submitted for publication). [22] B. Peeters, M. Seefeldt, S.R. Kalidindi, P. Van Houtte, E. Aernoudt, Mater. Sci. Eng. A 319–321 (2001) 188–191. [23] S. Mahesh, C.N. Tomé, R.J. McCabe, G.C. Kaschner, I.J. Beyerlein, A. Misra, Metall. Mater. Trans. 35A (2004) 3763–3774. [24] Z.J. Li, G. Winther, N. Hansen, Acta Mater. 54 (2006) 401–410. [25] O. Engler, J. Hirsch, Mater. Sci. Eng. A 452–453 (2007) 640–651. [26] J. Hirsch, O. Engler, in: G. Gottstein, D.A. Molodov (Eds.), Proceedings of the First Joint International Conference on Recrystallization and Grain Growth, Springer, Berlin, 2001, pp. 731–740. [27] S.E. Naess, Z. Metallkd. 82 (1991) 259–264. [28] O. Engler, Mater. Sci. Eng. A 538 (2012) 69–80. [29] S.C. Soare, F. Barlat, Eur. J. Mech. A-Solids 30 (2011) 807–819. [30] O. Engler, S. Kalz, Mater. Sci. Eng. A 373 (2004) 350–362. [31] P. Van Houtte, G. Cauwenberg, E. Aernoudt, Mater. Sci. Eng. 95 (1987) 115–124. [32] P. Bate, W.T. Roberts, D.V. Wilson, Acta Metall. 29 (1981) 1797–1814. [33] B. Bacroix, P. Gilormini, Model. Simul. Mater. Sci. Eng. 3 (1995) 1–21. [34] S. Li, E. Hoferlin, A. Van Bael, P. Van Houtte, Adv. Eng. Mater. 3 (2001) 990–994. [35] O.-G. Lademo, O. Engler, S. Keller, T. Berstad, K.O. Pedersen, O.S. Hopperstad, Mater. Des. 30 (2009) 3005–3019.
Contents lists available at ScienceDirect
Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Texture and anisotropy in the Al–Mg alloy AA 5005 – Part II: Correlation of texture and anisotropic properties Olaf Engler n, Johannes Aegerter Hydro Aluminium Rolled Products GmbH, Research & Development Bonn, PO Box 2468, D-53014 Bonn, Germany
art ic l e i nf o
a b s t r a c t
Article history: Received 5 August 2014 Received in revised form 14 August 2014 Accepted 17 August 2014 Available online 28 August 2014
Control of the plastic anisotropy during forming of a metallic sheet requires knowledge about its microstructure and crystallographic texture. In the present study the correlation of texture and anisotropy is studied in sheets of the Al–Mg alloy AA 5005. A series of samples with different textures was produced by various rolling and annealing treatments. In the 1st part of this study the microstructure and texture evolution during back-annealing of rolled sheet and during subsequent cold rolling was analysed. The present 2nd part describes the correlation of the resulting textures with anisotropic sheet properties, including Young's modulus, yield strength and r-values in uniaxial tensile tests as well as earing upon cup drawing. Texture-induced anisotropy was simulated with the help of the visco-plastic self-consistent (VPSC) model. Best agreement between simulations and experiments was obtained by using an affine formulation. For the work-hardened material the anisotropic substructure topology was also taken into consideration. With a view to a comprehensive through-process simulation of the evolution of texture and texture-related properties, in-plane anisotropy was simulated from modelled textures in place of the experimental ones. Here, larger deviations were obtained, since the errors of texture modelling and subsequent simulation of anisotropic properties tend to add up. Nonetheless, the VPSC simulations based on modelled textures were capable of reproducing the overall tendencies of anisotropy in AA 5005 sheet in a variety of tempers. & 2014 Elsevier B.V. All rights reserved.
Keywords: Texture Anisotropy r-value Earing Aluminium alloy Modelling
1. Introduction Sheets of the aluminium–magnesium alloy AA 5005 with about 1 wt% Mg find applications where more strength is required than achieved by unalloyed aluminium and where combinations of good formability, very good resistance to corrosion, good weldability, or good electrical properties are necessary. AA 5005 is typically used either in the fully soft condition (O temper) or in various semi-hard, i.e. strain-hardened (“H1x”) or back-annealed (“H2x”) tempers, depending on the required combinations of mechanical strength and formability. As many products made of AA 5005 sheets are subjected to forming operations, evolution of crystallographic texture and texture-related anisotropy is an important topic for such alloys (e.g. [1–6]). For instance, the occurrence of earing during deep drawing of a textured sheet can cause major problems in the production of deep drawn containers from aluminium sheet (e.g. [4–8]). It was the aim of the present study to correlate the anisotropic properties of AA 5005 sheets with their crystallographic texture. In the
first part of this study the evolution of microstructure and texture during processing of the sheets was analysed [1]. Especially, the texture changes upon back-annealing were simulated with an analytical softening model AlSoft, while texture evolution during subsequent cold rolling was tackled with the GIA polycrystal-plasticity code. In the present paper we address the correlation of the resulting textures with anisotropic sheet properties, including Young's modulus, yield strength and the plastic strain ratio (r-value) in uniaxial tensile tests as well as earing upon cup drawing. For simulation of textureinduced anisotropy the visco-plastic self-consistent (VPSC) formulation was employed. For the work-hardened states the impact of an anisotropic substructure topology was also taken into consideration. Furthermore, with a view to a comprehensive through-process simulation of the evolution of texture and texture-related properties, in-plane anisotropy was simulated from modelled textures in place of the experimental ones.
2. Experimental procedures n
Corresponding author. Tel.: þ 49 228 552 2792; fax: þ49 228 552 2017. E-mail address: [email protected] (O. Engler).
http://dx.doi.org/10.1016/j.msea.2014.08.040 0921-5093/& 2014 Elsevier B.V. All rights reserved.
As described in detail in part I of this work [1], hot strip of the Al–Mg alloy AA 5005 was cold rolled to 1.4 mm gauge to state H18.
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Samples taken from the cold rolled sheet were subjected to various annealing treatments, resulting in different backannealed states H2x up to the fully soft state O. Finally, soft annealed samples were cold rolled to different thicknesses ranging from 1.0 to 0.2 mm, corresponding to various strain hardened states H1x. Details about the evolution of microstructure and texture during back-annealing and subsequent strain-hardening can be found in part I of this work [1]. Tensile tests were carried out according to the international standard ISO 6892-1:2009. Standard size tensile specimens with 80 mm original gauge length and 20 mm width were machined in five different directions in the sheets, viz. parallel, under 22.51, 451, 67.51 and perpendicular to the rolling direction, RD. For each orientation a minimum of two specimens were tested in a screwdriven testing machine Zwick Z100 at a nominal strain rate of 0.00025 s 1 up to the yield strength Rp0.2 and 0.0067 s 1 up to fracture. For the determination of the modulus of elasticity the increased conditions of the new Annex G “Determination of the modulus of elasticity on metallic materials under uniaxial tensile loading” of the draft international standard ISO/DIS 6892-1:2014 were applied. Samples were loaded up to approximately 40% of the expected yield strength. The use of high resolution, double-sided, averaging extensometers of class 0.5 enables determination of the modulus of elasticity with increased accuracy and reduced measurement uncertainty. The plastic strain ratios rα, which describe the ratio of the true plastic strain in the width direction to that in the thickness direction (both in a state of uniaxial tension), were determined according to ISO 10113:2006. For that purpose, the plastic strains in the length and width directions were measured using two separate extensometers on the gauge section. The recommended regression method of the standard for materials displaying inhomogeneous yielding were applied to determine the slope mr of the true plastic width strain versus true plastic length strain through the origin. Finally, the r values were calculated from the condition of plastic incompressibility according to r¼mr/(1þ mr). The r values determined by the regression method were validated by manual measurements of width and thickness at various positions on the tensile specimen outside the necking and fracture area. For the sake of completeness it must be emphasised that the measurement uncertainty for the r-value determination is increased for the back-annealed and work hardened states H1x and H2x due to the inherently small changes in the dimensions of the test specimen. In order to determine the earing behaviour, cup drawing tests were performed on a hydraulic sheet forming machine following the guidelines of EN 1669. Blanks with a diameter of dB ¼60 mm were deformed with a punch of dP ¼33 mm, resulting in a drawing ratio of dB/dP E1.8. The tests were performed with blank holder forces ranging from 4 to 14 kN with lanolin lubrication. After the mechanical testing, the earing profiles were determined in steps of 11 with a mechanical probe using a set-up developed by Huxley Bertram Engineering, Cottenham, UK. In general, the results of minimum three cups were averaged. Fig. 1 shows an example of the earing height, h, as a function of the angle α to the rolling direction (RD) for the soft annealed sheet AA 5005-O. The profile depicts a characteristic four-ear profile with ears under 01, 901, 1801 and 2701 to the RD (see inlay). Because of the orthotropic symmetry of rolled sheets, the profiles can be symmetrised with respect to the RD and transverse direction (TD), see thick red curve in Fig. 1. For better comparison the earing profiles h(α) are normalised by the average cup height:
n
h ðαÞ ¼ hðαÞ U
αmax ∑α hðαÞΔα
ð1Þ
Fig. 1. Earing profile of soft annealed AA 5005-O, showing a characteristic four-ear profile with ears under 01, 901, 1801 and 2701 to the RD (thick red line: symmetrised profile). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
3. Correlation of texture and plastic anisotropy 3.1. Texture and plastic anisotropy during tensile testing As described in more detail in the experimental section, sheet anisotropy was assessed by performing standard uniaxial tensile tests in different in-plane direction. From these tests, the yield strength Rp0.2 and the strain ratio (r-value) were determined as a function of the in-plane angle α between the tensile axis and the original rolling direction. In order to assess the correlation of texture and plastic anisotropy in tensile tests, three samples with distinctively different textures were selected (see Figs. 2 and 3). (i) cold rolled and soft-annealed sheet in thickness 1.4 mm, i.e. AA 5005-O, with a fully recrystallised microstructure (Fig. 2(a)) and a characteristic cube recrystallisation texture (Fig. 3(a)); (ii) cold rolled and back-annealed sheet in thickness 1.4 mm, i.e. AA 5005-H24, with a well recovered, layered microstructure (Fig. 2(b)) and a typical plane-strain rolling texture (Fig. 3(b)); (iii) sheet rolled to 0.7 mm, i.e. AA 5005-H16, with a heavily elongated grain structure (Fig. 2(c)) and a mixed cube plus rolling texture (Fig. 3(c)).
The true stress, σ, vs. true strain, ε, curves for AA 5005 in states O, H24 and H16 are plotted in Fig. 4 for in-planes angles α of 01, 451 and 901 (the curves at α¼22.51 and 67.51 were omitted for clarity). As expected, the tensile curves reveal an increase in strength with decreasing ductility in the sequence O-H24-H16. For the softannealed temper O there is only minor anisotropy for both yield strength and subsequent work hardening. It is noted that there is some variation in the maximum elongation and, in consequence, in ultimate tensile strength, Rm. Differences in elongation and ductility are not the topic of the present study, however. Fig. 5(a) shows the absolute values of the in-plane anisotropy in yield strength, Rp0.2, modulus of elasticity, E, and plastic strain ratio, r, as a function of α. Despite the fairly pronounced cube texture in the fully recrystallised O-temper, there is only minor inplane anisotropy in strength, with the yield strength fluctuating by 72 MPa. The elastic modulus E displays more severe in-plane anisotropy, with E varying from about 70 GPa at 01 and 901 to a maximum of 73 GPa under 451. The plastic strain ratio r, given on
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Fig. 2. Microstructure of AA 5005 after various rolling and/or back-annealing treatments, (a) AA 5005-O, (b) AA 5005-H24, (c) AA 5005-H16 (longitudinal sections, anodically oxidised, RD up).
Fig. 4. True stress, σ, vs. true strain, ε, curves at various in-planes angles α for AA 5005 in states O, H24 and H16, showing increased strength and decreasing ductility.
Fig. 3. Texture of AA 5005 after various rolling and/or back-annealing treatments, (a) AA 5005-O, (b) AA 5005-H24, (c) AA 5005-H16 (ODF φ2-sections, intensity levels 1 – 2 – 4 – 8 – 16).
the right scale, reveals the opposite behaviour, with maximum values of about 0.8 at 01 and 901 and a minimum of 0.5 around 451. In the back-annealed state H24 the in-plane anisotropy is more pronounced (Fig. 4). The yield strength steadily increases from 130 MPa along the RD to 137 MPa along the TD (Fig. 5(b)). The
r-values increase from below 0.5 in the RD to values of about 1 in the TD. The elastic modulus shows the opposite behaviour form the soft state, with maxima of about 73 GPa at 01 and 901 and a minimum below 70 GPa at 451. In the strain-hardened state H16 the maximum elongation is very low, below 2% (Fig. 4). In-plane anisotropy is slightly more pronounced than in state H24. Here the yield strength increases from 172 MPa along the RD to 182 MPa along the TD (Fig. 5(c)). From 01 to 451 the r-values stay at low values between 0.4 and 0.6, before sharply increasing to values in excess of 1.2 in the TD. The elastic modulus shows a somewhat unsystematic evolution with low in-plane variations of just over 1 GPa and a weak maximum under 451. In conclusion, all three samples show pronounced in-plane anisotropy with relatively high r-values in the TD. The two states AA 5005-H16 and H24 with prevalent rolling textures display similar evolution in r-values, with low r-values in the RD before r strongly increases to values in excess of 1 in the TD. Note that the soft AA 5005-O sheet which comprises a strong cube recrystallisation texture depicts a slight asymmetry in r values between 01 and 901. This is attributed to the remainders of the deformation texture observed in this state (Fig. 3(a)).
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Fig. 5. In-plane anisotropy in yield strength, Rp0.2, elastic modulus, E, and plastic strain ratio, r, as a function of α; (a) state O, (b), state H24 and (c) H16.
3.2. Polycrystal-plasticity simulation of plastic anisotropy during tensile testing The correlation of texture and plastic anisotropy was modelled with the help of a polycrystal-plasticity code, here, the viscoplastic self-consistent (VPSC) formulation. VPSC simulates the plastic deformation of polycrystalline aggregates subjected to external strains and stresses. It accounts for full anisotropy in properties and response of the single crystals and the aggregate. VPSC is based on the physical mechanisms of dislocation slip and accounts for grain interaction effects. In addition to providing the macroscopic stress–strain response, it accounts for hardening, reorientation and shape change of individual grains. As a consequence, it predicts the evolution of hardening and texture associated with plastic forming. The formal theory upon which the VPSC code is based is explained in the papers by Lebensohn and Tomé [9,10] and in the textbook “Texture and Anisotropy” by Kocks et al. [11]. In the VPSC model each orientation, or grain, is treated as an ellipsoidal visco-plastic inclusion embedded in and interacting with an effective visco-plastic medium, which has the average polycrystal properties. The stresses and strains in the individual grains can vary as a function of orientation, depending on the stiffness of the interaction. The properties of the HEM are not known a priori, but are adjusted ‘self-consistently’ to coincide with the average response of all orientations constituting the aggregate, i.e., its texture. The single crystal response is described by means of a rate sensitive constitutive law of the form N
dij ¼ γ_ 0 ∑ msij s¼1
mskl σ ckl τsc
n ð2Þ
Index s labels the individual slip systems (N ¼12 for fcc materials with {111}〈110〉 slip), ms is the geometric Schmid tensor, γ_ 0 is a reference shear rate, τsc is the threshold stress controlling the activation of the slip system and n is the inverse of the materials strain rate sensitivity. The strain rate dij and stress σ cij in each grain are related to the average polycrystal strain rate Dij and stress σij through an interaction equation of the form: ~ : ðσ σ c Þ ðD dÞ ¼ nef f M
ð3Þ
~ is a visco-plastic interaction compliance and neff is an The tensor M effective inverse rate sensitivity which tunes the strength of the coupling between the stress deviations and the associated strain
rate deviations. The effective stiffness of the matrix increases as neff decreases until, for neff ¼0, the Taylor condition is reproduced. In order to simulate a tensile test with the VPSC approach, the discretised texture of the sheet is subjected to the external stress and strain boundary conditions. Then, the VPSC model is employed to derive the materials response in terms of the active slip systems and resulting strains, stresses, reorientation, work hardening, etc., which can be utilised to predict the evolution of hardening and texture associated with plastic forming. For a tensile test along the former rolling direction the strain rate D consists of an elongation along direction X1 plus the resulting shape changes in transverse direction, X2, and through-thickness direction, X3. For a textured material, the exact values of the inplane strain rate D2 and the through-thickness strain rate D3 are not known, however. Therefore, the boundary conditions of uniaxial tension are described by assuming zero stresses σ2 and σ3 in directions X2 and X3, respectively (Fig. 6), see Refs. [12,13]. The ratio of the resulting in-plane to through-thickness strain rates D2/D3 then gives the r-value. Thus, under making use of a feature of the VPSC model which permits enforcing mixed displacement/ stress boundary conditions, the following load case is imposed: 0 1 0 1 1 0 0 n n n B C B C D ¼ @ 0 n 0 A U ε_ σ ¼ @ n 0 n A ð4Þ n n 0 0 0 n where ε_ is a scalar measure of the strain rate increment [s 1]; the asterisks n indicate that the corresponding value is not prescribed. The VPSC code offers modelling of the directional elastic modulus E(α) which is a measure of the elastic stiffness of the material when subjected to a tensile stress in the direction of α in the rolling plane (Fig. 6). E(α) is calculated as the ratio between the stress and the deformation measured along the sample axis (superscript α) [11,14]: EðαÞ ¼
σ α11 1 ¼ εα11 Sα1111
where
Sα1111 ¼ Rα1i Rα1j Rα1k Rα1l Sijkl
ð5Þ
Sijkl is the polycrystal compliance expressed in the reference system, which is computed with the self-consistent approach. Rα is the rotation matrix for a rotation by angle α around ND ¼X3 (Fig. 6). We have calculated E(α) for the three different textures considered here, using the polycrystal elastic constants of aluminium alloys C11 ¼108 GPa, C12 ¼61 GPa and C44 ¼28.5 GPa. Note that the constant were slightly adapted (by less than 1 GPa) to give best overall fit with the experimental data points. In all three cases
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Table 1 Voce hardening parameters at the crystal level (Eq. (6)) used at the polycrystalplasticity simulation for the various tempers.
State O State H24 State H16
Fig. 6. (a) Scheme of tensile tests under different in-plane angles α to the rolling direction, RD, to determine in-plane plastic anisotropy.
Fig. 7. Simulated in-plane anisotropy in elastic modulus, E for AA 5005 in three different states (VPSC).
the simulated curves E(α) reproduce the elastic in-plane anisotropy of the materials reasonably well (Fig. 7). In the soft-annealed state O with a characteristic cube recrystallisation texture (Fig. 3(a)) the elastic modulus E reveals minima at 01 and 901 and a pronounced maximum under 451. In the recovered material H24 with a welldefined rolling texture (Fig. 3(b)) the opposite behaviour is observed, with a minimum in the elastic modulus E at 451. The strain-hardened sheet H16 with a mixed texture (Fig. 3(c)) shows a lower in-plane elastic anisotropy with a weak maximum at 45–601 which, again, is modelled by the VPSC simulations. In order to simulate a full tensile test along direction X1, the combined stress and strain state (Eq. (4)) was imposed in sequential, incremental steps, starting with the original sheet texture. The strain rate increments ε_ were chosen in the range 0.0005 s 1 (for AA 5005-H16) over 0.002 s 1 (for AA 5005-H24) to 0.005 s 1 (for AA 5005-O) such that the relevant strain regime was covered with approximately 20–50 points (Fig. 4). After each deformation step, reorientation and shape change of the grains were determined. The VPSC scheme offers several options to solve the equation of inclusion-matrix interaction. It has repeatedly been shown that for prediction of anisotropy compliant modes – either the tangent approach or an intermediate linearization scheme with a reduced effective inverse rate sensitivity neff – gives much better results than the much stiffer secant approach [15–17]. In the present study, best results were obtained with the more recent affine formulation by Masson et al. [18].
τ0 [MPa]
τ1 [MPa]
θ0 [MPa]
θ1 [MPa]
16 48 65
21.5 4.0 2.5
325 160 2500
29 39 55
The above stress/strain state (Eq. (4)) applies for uniaxial tensile deformation along direction X1, which generally relates to the former sheet rolling direction, RD. In order to determine the variation of the property under consideration in the sheet plane, the stress/strain state has to be rotated by angle Δα about the common normal direction, ND¼X3 (Fig. 6). For the sake of simplicity, we rather rotate the sheet texture and grain (ellipsoid) axes by –Δα about the ND, while keeping the load state constant. As described already in part I [1], work hardening of the material during tensile testing was simulated by means of the generalised Voce law [19] to calculate the increase of the critical shear stress τc with shear strain Γ for all grains (see Eq. (2)): τc ðΓ Þ ¼ τ0 þ ðτ1 þ θ1 UΓ Þ U 1 exp Γθ0 =τ1 ð6Þ The single crystal hardening parameters τ0, θ0, τ1 and θ1 were determined by fitting the experimental tensile test data for alloy AA 5005 in the various tempers (Fig. 4) and are summarised in Table 1. Note that the hardening parameters of the soft material differ considerably from the Voce parameters obtained for cold rolling with the GIA model (Table 1 in [1]). This is due to the different deformation mode, rolling vs. uniaxial tension, and different polycrystal-plasticity model, GIA vs. VPSC. It has been described in the literature that grain shape also affects in-plane anisotropy, though less pronounced than texture [17,20]. In VPSC, grain shape effects modify the grain-matrix interaction through the dependence of the Eshelby tensor upon the ellipsoid aspect ratios of non-spherical inclusions. The soft annealed material in temper O had a rather equiaxed microstructure (Fig. 2(a)); accordingly, here the assumption of a spherical grain structure with an aspect ratio (RD:TD:ND) 1.0:1.0:1.0 is reasonable. Both the cold rolled sheet H16 and, especially, the back-annealed state H24 revealed a strongly elongated microstructure, in contrast (Fig. 2(b), (c)). However, it does not appear likely that these deformed grains per se will have a noticeable impact on anisotropy [21]. Rather, it is anticipated that other microstructural features, most probably the as-deformed substructure, exert a significant effect in altering the in-plane anisotropy compared to the ubiquitous influence of crystallographic texture [21–24]. Thus, in agreement with the TEM micrographs shown in part I [1], for the well-recovered material in state H24 again a spherical (sub)grain structure was assumed, i.e. 1.0:1.0:1.0. For the cold rolled state H16 a non-spherical substructure with an aspect ratio of 4:1 was assumed, more precisely, (RD:TD:ND) 2.0:1.0:0.5. Fig. 8 shows the resulting in-plane variations in yield strength and r values for alloy AA 5005 in the three different tempers. As for the strength anisotropy the stress after the onset of yielding was taken as a measure of the yield strength Rp0.2 (Fig. 8(a)). For the determination of the r-values the strain rates in throughthickness direction, D3, and transverse direction, D2, were integrated over the simulated tensile test in order to yield average r-values similarly as in the experiments (Fig. 8(b)). In the fully recrystallised O-temper the weak in-plane anisotropy in strength is reproduced by the VPSC model, though the W-shape of the experimental data points is not matched. The plastic strain ratio r showed much stronger anisotropy, with a distinct minimum at 451. The polycrystal-plasticity simulations predict the overall
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Fig. 8. Correlation of texture and in-plane anisotropy in (a) yield strength Rp0.2 and (b) r-values.
Fig. 9. Correlation of texture and in-plane anisotropy, based on modelled textures rather than experimental ones (a) yield strength Rp0.2 and (b) r-values (see text for details).
shape and the individual data points from 01 to 67.51 with reasonable accuracy, yet the r-value at 901 is heavily overpredicted. For the other two tempers similar effects are obtained. In the back-annealed state H24 the increase in yield strength at α¼901 as well as the general evolution of r(α) is simulated, but the simulated curve r(α) appears to be less undulated than its experimental counterpart (Fig. 8(b)). In the strain-hardened state H16 the increases in yield strength is very nicely matched (Fig. 8(a)). As in state O, the r-values are satisfactorily modelled up to α¼67.51, but are strongly overpredicted at α¼901. With a view to a comprehensive through-process simulation of texture and texture-related properties it appeared worthwhile to use simulated textures rather than experimental ones as an input for the VPSC earing model. Fig. 9 shows the same examples of anisotropy as Fig. 8, but based on simulated textures rather than experimental ones. It has been demonstrated in part I that both rolling textures and recrystallisation textures can be modelled with great accuracy [1]. Accordingly, the predictions of yield strength anisotropy and r-values based on modelled textures in
general resemble the simulations based on experimental textures (Fig. 8), but the deviations between simulation and experiments are larger (Fig. 9). In particular this affects predictions of the rvalues (Fig. 9(b)), which illustrates how sensitive r-values are on texture. Presumably, the errors of texture modelling and subsequent simulation of anisotropic properties tend to add up, leading to increasing discrepancies from the experimental results. However, despite the enlarged discrepancies, the VPSC simulations based on modelled textures are obviously capable of reproducing the overall tendencies of in-plane anisotropy in yield strength and r-values.
3.3. Texture and earing The earing behaviour was analysed by means of deep drawing of small cups from circular blanks which were taken from the various sheets with different crystallographic texture. The characteristic earing profiles of deep drawn cups form because the
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sheet texture gives rise to different radial elongations in different directions of the blank. In most soft annealed Al alloys four ears are obtained, which are situated under angles α of 01 and 901 to the rolling direction (Fig. 1). In heavily cold rolled sheet often fourfold earing prevails as well, but at the four positions 7451 from the rolling direction. In many cases of industrial relevance mixed textures will form, by contrast, which are associated with the formation of more complex earing profiles with six ears (at 01/1801 and 451) or eight ears (at 01/901 and 451) (e.g. [4–8]). The evolution of earing profiles in the present alloy AA 5005 is displayed in Fig. 10(a) for the various back annealing treatments H18-O and in Fig. 10(b) for subsequent cold rolling of recrystallised AA 5005-O to temper H19. In the initial, cold rolled state H18 a characteristic mixed eight-ear profile is obtained, showing maxima under 01 (and 1801) and under 451 (and 1351/2251/ 3151). The ear at 901 (and 2701) is not very pronounced and sometimes not discernible at all, which is then the typical appearance of a six-ear profile [25]. Recovery upon back annealing (H18-H24-H22) does not lead to significant changes in texture [1] and, in consequence, in the earing profiles. Partial recrystallisation in the state H21 entails a significant modification of the anisotropy, in contrast, with the 01 and 901 ears being fortified at the expenses of the 451 ears. Accordingly, AA 5005-H21 shows a textbook-example of a smooth eight-ear profile. In principle, this texture would be well suited to produce Al sheet with low earing properties, yet it is hard to control partial recrystallisation with sufficient accuracy in industrial practice (e.g. [26]). Further progressing recrystallisation then gives rise to the complete disappearance of the 451 ears, leading to the typical four-ear profiles with ears at 01/901 of a fully recrystallised sheet (see Fig. 1). Subsequent cold rolling of the fully soft sheet leads to the opposite effects. With increasing strain the 451 ears intensify to the disadvantage of the 01 ears and, even more rapidly, the 901 ears. Accordingly, at high strains (0.3 mm, i.e. H19) a six-ear profile is obtained, which resembles the profile of the initially cold rolled material (H18). Note that larger strains will eventually lead to a four-ear profile with ears under 7 451 only (0.2 mm, H19). At intermediate strains, here for samples H16 and H18, mixed eightear profiles with low overall earing level are obtained. This practice of inter-annealing and medium final rolling to designated temper is utilised in industry to produce low-earing grade Al sheet (e.g. [27,28]). It should be emphasised here that the two
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low-earing qualities H16 and H18 both displayed rather pronounced anisotropy in strain ratios, i.e. r-values (e.g. Fig. 8(b)). This disproves the often quoted direct relationship between r-values and earing; rather, the anisotropy in yield strength (Fig. 8(a)) must be taken into consideration as well [29]. 3.4. Polycrystal-plasticity simulation of earing from texture The evolution of earing during cup deep drawing was simulated by means of the visco-plastic self-consistent (VPSC) scheme which was already used to simulated anisotropy in the tensile tests (Section 3.2.). Polycrystal-plasticity simulation of earing is slightly more complicated, however, which is caused by the more complex behaviour of stresses and strains operating during cup deep drawing. The present polycrystal-plasticity earing model, described in detail in Ref. [30], relies on the fact that plastic flow and, therefore, the formation of ears and troughs is largely concentrated in the flange of the blank under the blank-holder. In the flange a plane stress state prevails, in that the material is exposed to a tensile stress σr in the radial direction and a compressive stress σt in the tangential direction (Fig. 11). The stress in the through-thickness direction σz as well as the offdiagonal shear stresses σrt are almost zero. This stress state generates a positive radial strain rate Dr and a negative tangential strain rate Dt, while the through-thickness strain rate Dz is not necessarily zero. Thus, the following mixed load case is imposed: 0 1 0 1 σr n n n 0 0 B B C n C ð7Þ Dij ¼ @ 0 1 0 A U ε_ ; σ ij ¼ @ n n A n n σ n 0 0 z ¼0 (_ε strain rate increment; the asterisks * indicate that the corresponding value is not prescribed.) It is seen that the stress components σr and σz are prescribed rather than the corresponding strain rate components Dr and Dz. All simulations in this study were performed with a non-zero tangential stress of σr ¼–0.25 σt (see Ref. [31]). Since in the first time step the stress values are not yet known, this is accomplished iteratively within a few VPSC steps until stable values for σr and σt are achieved. Eq. (7) is expressed in the frame defined by radial direction, r, tangential direction, t, and through-thickness direction, z. This frame is related to the standard rolling frame {RD, TD, ND} through
Fig. 10. Evolution of earing profiles h*(α) in AA 5005, (a) after various back annealing treatments H18-O, and (b) after cold rolling of recrystallised AA 5005-O to temper H19 (normalised, symmetrised).
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a rotation by angle α about the common direction z¼ ND (Fig. 11). Thus, in order to derive the earing tendency under a given angle α, the strain in radial direction is determined for the texture in this particular position of the blank. For this purpose, the sheet texture is rotated by Δα in steps of typically 51 about z. The earing profile along the rim of a deep-drawn cup is then given by the radial strain or, more precisely, by the normalised strain rate ratio q¼ Dr/Dt as a function of α. The above model was applied to the textures of AA 5005 in the various states H18-O and O-H19; Fig. 12(a) shows the normalised, symmetrised earing profiles for a number of characteristic states. Comparison with the experimental profiles, shown in Fig. 10, reveals the good accuracy of the VPSC earing model mentioned above. Recrystallisation of the initial, cold rolled state H18 gave rise to an increase in 01 and 901 ears to the disadvantage of the 451 ears, leading to the typical four-ear profiles of a fully recrystallised sheet with ears at 01 and 901 (see Fig. 1). On the other hand, recovery reactions occurring during the early stages of back-annealing (H18-H22) led to an appreciable sharpening of the rolling texture [1]. Since the texture type is not affected, however, this texture sharpening did not alter the earing profiles.
Subsequent cold rolling of the fully recrystallised sheet is accompanied by an intensification of 451 earing at the expenses of the 01 ears and, even more rapidly, the 901 ears. That is to say, the four-ear profile with ears at 01 and 901 is transformed via mixed eight-ear profiles at intermediate strains (e.g. H16) towards a six-ear profile (H19), which resembles the profile of the initially cold rolled material (H18). Again, it appears that the transitional states (H21 and H16) are distinguished by mixed earing profiles with low overall earing grade. Similarly as described for the simulations of in-plane anisotropy, the VPSC earing model was also applied to simulate earing from simulated textures rather than experimental ones. Fig. 12(b) shows the same examples of earing profiles as Fig. 12(a), but based on simulated textures in place of their experimental counterparts. The curves nicely reproduce the transition from 451 earing towards 01/901 earing upon back-annealing and the re-appearance of 451 earing during subsequent cold rolling O-H19. Compared with the profiles simulated from experimental textures the deviations with regard to the experimental earing profiles are larger, however. Nevertheless, the main earing features are clearly reproduced, illustrating the potential of a comprehensive through-process simulation of texture and texture-related properties by combining models for texture evolution with polycrystal-plasticity models to simulated texture-related properties.
4. Discussion and conclusions
Fig. 11. Geometry of cup deep drawing used in the present study.
It is widely accepted that crystallographic texture of Al alloy sheet is the main cause for the occurrence of anisotropic sheet properties (e.g. [1–8]). However, plastic anisotropy may also be influenced by other microstructural parameters, including the average grain shape [17,20], the topology of second-phase particles [32] or the substructure topology, e.g. arrangement of dislocations etc. [21–24]. In the course of the present study we have explored the correlation of texture and anisotropy in sheets of the Al–Mg alloy AA 5005 in a number of differently rolled and annealed states (see Ref. [1]). Clearly, the experiments substantiate a relationship between texture and anisotropic properties, especially for in-plane variation in modulus of elasticity (Young's modulus) E, yield strength Rp0.2 and r-values (Figs. 4,5) as well as earing during cup drawing (Fig. 10).
Fig. 12. Correlation of texture and earing in AA 5005, (a) simulation based on experimental textures, (b) simulation based on modelled textures (see text for details).
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Evolution of texture and texture-induced anisotropy can be modelled with the help of polycrystal-plasticity codes like the familiar Taylor model. In the present study, softening during backannealing was modelled with an analytical softening model named AlSoft treating both recovery and recrystallisation including the resulting texture changes. Texture evolution during subsequent cold rolling was tackled with the GIA polycrystal-plasticity code considering grain-interaction [1]. Then, the visco-plastic selfconsistent (VPSC) model was employed to simulate the resulting texture-induced anisotropy [9–11]. For the simulation of tensile tests in different in-plane directions, the load case (i.e. strains and stresses) of uniaxial tensile deformation under an angle α to the original sheet rolling direction is imposed on the sheet texture (Eq. (4)). The load case upon cup drawing is approximated by a plane-stress state with a constant ratio of radial and tangential stresses in the flange (Eq. (7)). VPSC then computes the crystal response in terms of reorientation and shape change of the individual grains as well as the resulting macroscopic stresses and strains. Best results were obtained with an affine formulation to solve the interaction between inclusion and matrix in the VPSC code [18]. For the work-hardened material in state H16 a slight improvement of the simulations could be achieved by taking the anisotropic substructure topology into consideration [21]. By contrast, the well-recovered state H24 comprised an equiaxed subgrain structure [1]. Hence, despite its strongly layered microstructure (Fig. 2(b)), best simulation results were achieved by assuming a spherical substructure. For the soft-annealed material in state O likewise a spherical grain structure (Fig. 2(a)) was utilised. Comparison of the simulated anisotropy in materials properties with the experimental results showed that the VPSC model is able to predict the in-plane anisotropy obtained in tensile tests and the earing behaviour observed in cup drawing tests with reasonable accuracy (Figs. 8, 12(a)). In some cases, most notably for r-values at α¼ 901, quite large discrepancies were obtained, however. Despite these discrepancies we presume that the accuracy obtained will be sufficient to utilise texture-based anisotropy data as an input for subsequent forming simulations (e.g. [33–35]). With a view to a comprehensive through-process simulation of texture and texture-related properties we have used simulated textures in place of the experimental ones as an input for the VPSC model. For both the in-plane anisotropy obtained in tensile tests and for the earing behaviour in cup drawing tests the predictions based on modelled textures largely resemble the simulations based on experimental textures (see Figs. 8 and 9 as well as Fig. 12(a) and (b)); however, the deviations between simulation and experiments are generally larger. Presumably, the errors of texture modelling and subsequent simulation of anisotropic properties tend to add up, leading to increasing discrepancies from the experimental results. Nonetheless, the VPSC simulations based on modelled textures are capable of reproducing the overall tendencies of anisotropic properties in AA 5005 alloy sheet in a variety of tempers and textures. Accordingly, a combined application of the present models yields valuable information on the evolution of
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texture and earing as a function of process parameters that can be used to improve the quality of aluminium sheet alloys.
Acknowledgements The VPSC software was provided by Dr. C.N. Tomé (Los Alamos Nat. Lab., USA). The authors would like to thank their colleagues Dr. H. Aretz and Prof. J. Hirsch for helpful discussions.
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