Characterizations of Aluminum Alloy Sheet Materials Numisheet 25 John C. Brem, Frederic Barlat, Robert E. Dick, and Jeong-Whan Yoon Alcoa Technical Center, PA, 1569-1, USA Abstract. This report reproduces the contents of a document provided in the Numisheet 25 Benchmark Study for the characterization of aluminum alloys. DOCUMENTATION Samples of aluminum alloys 622-T43 (t = 1.-mm) and 5182-O (t = 1.625-mm) were provided by ALCOA. Evaluations to determine the material characteristics of these sheet samples were performed at Alcoa Laboratories. The equal biaxial mechanical behaviors were assessed using the instrumented hydraulic bulge test system. All tests were conducted at a constant true strain rate =.5/second with the extensometer oriented at 45 degrees to the rolling direction of the sheet. The maximum values of stress and strain measured in these tests are summarized in Table 1, along with the Hollomon and Voce descriptions of the true membrane stress true thickness strain behaviors determined for each material. Uniaxial tension tests were conducted using specimens taken at, 15, 3, 45, 6, 75, and 9 degrees to the rolling direction of each sheet material. The.2% offset yield strength (YS), ultimate tensile strength (UTS), uniform and total elongations, and plastic strain ratios (r values) that were measured in the directional tests that were conducted for each material are listed in Table 2. The optimized Hollomon and Voce constants that describe the directional true stress true plastic strain behaviors of the materials are listed in Table 3. Microsoft Excel worksheets listing the directional engineering and true stress-strain relationships for each material are being sent in separate attachments. Analyses of the elastic region of the engineering stress-strain curves measured in a separate series of tests in uniaxial tension for the longitudinal ( degrees) and transverse (9 degrees) test directions were conducted to determine the modulus of elasticity and Poisson s ratio. These results are shown in Table 4. Disk compression tests were performed for each sample using nominal 12.7-mm diameter x t specimens. Table 5 lists the biaxial r value ( r TD / r RD ) that was determined for each of the materials. The mechanical behavior characteristics that were experimentally determined for each material in the various modes of deformation were used as input to yield criteria developed by F.Barlat, Alcoa in 2 (Yld2-2d) [1] and, most recently, in 24 (Yld24-18p) [2]. The material constants that describe the respective anisotropic yield surface shapes that were calculated for each sample based on the Yld2-2d [1] yield function are shown in Table 6, while those for the Yld24-18p [2] yield criterion are listed in Table 7. Additionally, Tables 8 and 9 list the material constants that define the yield surface shapes per Yld89 [3] using both stress-based and r value-based calculations. The stress-based and r value-based constants determined using Yld91 [4] are shown in Tables 1 and 11, respectively. The material constants determined with the Yld96 [5] criterion are shown in Table 12. See the attached Appendix for the material constants that are associated with each of the respective yield functions. In all 1179
cases, the reference stress state used to compute the yield function coefficients is the balanced biaxial state (i.e. The yield surface shape, as described by the Yld2-2d [1] yield function, and the instantaneous strain hardening behavior measured experimentally in balanced biaxial tension for each respective sample were used as input to calculate a forming limit diagram (FLD). The limit strains that comprise each FLD were calculated according to the Marciniak-Kuczynski (MK) imperfection theory for strain localization [6]. The forming limit diagrams are illustrated in terms of true strains in Figures 1 and 2. The raw data are being sent separately as a Microsoft Excel attachment. Standard x-ray techniques were used to assess the crystallographic texture attributes of each material. The Al(lll), Al(2), and Al(22) pole figures and phi-sections that were measured in these analyses are shown in Figures 3 and 4, respectively, for each of the sheet materials. Collections of orientations (1 grains) that are representative of the microstructures for these materials are being sent separately. The Euler angles, expressed in Bunge notation (phi-1, phi, and phi-2), that are contained in these files can be used for polycrystal-based modeling. TABLE 1. 45 Equal Biaxial Tension Test Data (True Strain Rate =.5/s) Maximum Values Hollomon: a = K e n Voce: a = A - B exp (-Ce) a,mpa e n K,MPa A,MPa B,MPa C 622-T43 t = 1.-mm 362.5.516.255 448.58 363.44 234.67 7.278 5182-O t = 1.625-mm 431.8.58.315 555.88 437.28 312.26 6.179 TABLE 2. Uniaxial Tension Test Data Test YS, UTS, % Elongation Direction MPa MPa Uniform Total 622-T43 t = 1.-mm 136. 256.9 22.2 27.2 15 136. 253.4 22.8 26.9 3 134.7 251.7 24. 28.4 45 131.2 247.6 24.8 29.4 6 129.8 241.9 25. 28.2 75 128.9 239.8 24.9 28.5 9 127.6 238.3 24. 27.6 15 3 45 6 75 9 13. 128. 126.1 124.9 124.8 126.4 128.2 NOTE: r-bar = (r + 2r 45 o + r 9 ) / 4 5182-O t = 1.625-mm 19.9 23.1 21.9 23.7 23.1 22.7 24.6 281.1 279.8 272.9 268.9 267.5 269.5 273.8 23.4 24.1 26.7 27.6 27. 28.1 26. r Value 1.29 1.1.73.532.553.689.728.957.93.916.934.947.981 1.58 r-bar.75.971 118
TABLE 3. True Stress True Plastic Strain Descriptions Uniaxial Tension Test Max Hollomon: σ = K ε n Voce: σ = A B exp (-Cε) Direction ε P n K, MPa A, MPa B, MPa C 622-T43 --- t = 1.-mm.196.258 479.92 339.5 22.5 1.357 15.21.253 468.71 336.5 198.74 1.53 3.21.252 463.44 336.95 199.26 9.557 45.217.254 455. 335.4 199.29 8.975 6.218.252 443.31 328.64 194.78 8.864 75.218.256 442.72 325.9 194.23 9.42 9.21.258 442.45 322.13 193.64 9.196 5182-O --- t = 1.625-mm.177.319 586.72 366.84 251.7 11.166 15.22.32 574.24 366.87 252.76 1.462 3.193.322 561.38 361.29 248.42 1.62 45.28.322 55.75 358.74 247.11 9.719 6.23.323 545.76 355.56 244.73 9.638 75.2.323 553.45 36.91 248.76 9.569 9.216.318 557.1 362.39 248.11 9.981 TABLE 4. Engineering Stress-Strain Curve Analyses (Elastic Region) Modulus, GPa Poisson s Ratio Material L T Avg. L T Avg. 622-T43 (1.-mm) 7.9 69.4 7.2.368.358.363 5182-O (1.625-mm) 7.2 7.9 7.6.338.344.341 TABLE 5. Disk Compression Test Data r Value (Biaxial) 622-T43 (1.-mm) 5182-O (1.625-mm) 1.149.948 TABLE 6. Material Constants for Yield Function Yld2-2d (Exponent a=8) Material α 1 α 2 α 3 α 4 α 5 α 6 α 7 α 8 622-T43.93849 1.45181.929135 1.29875.987446 1.35941.952861 1.1199 5182-O.93633 1.7871.966889 1.4853 1.269 1.16975 1.32625 1.114336 1181
TABLE 7. Material Constants for Yield Function Yld24-18p (Exponent a=8) 622-T43 c 12 c 13 c 21 c 23 c 31 c 32 (1.-mm).949886 1.19946 1.64143 1.32861 1.14313 1.253713 c 44 c 55 c 66 c 12 c 13 c 21 1.331 1.2471 1.275242.91719.846874.98558 c 23 c 31 c 32 c 44 c 55 c 66.62173.735854.797399 1.13388.988851.52191 5182-O (1.625-mm) c 12 c 13 c 21 c 23 c 31 c 32.983762.7176.92832.57562.97932.98527 c 44 c 55 c 66 c 12 c 13 c 21 1.5394.99462.897646.894861 1.123328 1.6215 c 23 c 31 c 32 c 44 c 55 c 66 1.15556.751489.763667.998866 1.15 1.94181 TABLE 8. Material Constants for Yield Function Yld89 (Exponent m=8) - Stress-Based Material a c h p 622-T43.741864.858579 1.68256 1.39155 5182-O.867393.991319 1.3391 1.65923 TABLE 9. Material Constants for Yield Function Yld89 (Exponent m=8) - r Value-Based Material a c h p 622-T43 1.34446 1.155261 1.97167.961368 5182-O 1.731 1.7918.97535.973394 TABLE 1. Material Constants for Yield Function Yld91 (Exponent m=8 - Stress-Based Material a b c f g h 622-T43 1.6543.9315 1.1225 Isotropic Values = 1 1.13459 5182-O 1.3337.965741 1.1554 1.59716 TABLE 11. Material Constants for Yield Function Yld91 (Exponent m=8) - r Value-Based Material a b c f g h 622-T43 1.52492.946928.95471 Isotropic Values = 1.8953 5182-O.984944 1.1511 1.1573.986544 1182
TABLE 12. Material Constants for Yield Function Yld96 (Exponent a=8 and c 4 =c 5 =1=Isotropic Values) 622-T43 (1.-mm) 5182-O (1.625-mm) c 1 c 2 c 3 c 6 1.26169.887357 1.1265 1.55135 α x α y α z α z1 1.34165 1.4889 1..4415 c 1 c 2 c 3 c 6 1.57924.92731 1.16333 1.92887 α x α y α z α z1.824 1.44 1..63756 REFERENCES [1] Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dick, R.E., Lege, D.J., Pourboghrat, F., Choi, S.-H., Chu, E., 23. Plane Stress Yield Function for Aluminum Alloy Sheets Part I: Theory. Int. J. Plasticity 19, 1297-1319. [2] Barlat, F., Aretz, H., Yoon, J.W., Karabin, M.E., Brem, J.C., Dick, R.E., 24. Linear Transfomation-Based Anisotropic Yield Functions. Int. J. Plasticity, in press. [3] Barlat, F., Lian, J., 1989. Plastic Behavior and Stretchability of Sheet Metals. Part I: A Yield Function for Orthotropic Sheets Under Plane Stress Conditions, Int. J. Plasticity 5, 51-66. [4] Barlat, F., Lege, D.J., Brem, J.C., 1991. A Six-Component Yield Function for Anisotropic Materials, Int. J. Plasticity 7, 693-712. [5] Barlat, F., Maeda, Y., Chung, K., Yanagawa, M., Brem, J.C., Hayashida, Y., Lege, D.J., Matsui, K., Murtha, S.J., Hattori, S., Becker, R.C., Makosey, S., 1997. Yield Function Development for Aluminum Alloy Sheets, J. Mech. Phys. Solids 45, 1727-1763. [6] Marciniak, Z., Kuczynski, K., 1967. Limit Strain in the Process of Stretch Forming Sheet Metal. Int. J. Mech. Sci. 9, 69-62. APPENDIX YLD89 m m 1 2 1 2 2 m 2 2σ m φ = a K + K + a K K + c K + = where K 1 σ xx + hσ yy = 2 1183
K 2 2 σ xx hσ yy = + p σ 2 2 2 xy The anisotropy coefficients are a, c, h and p. The exponent m is 6 for bcc materials and 8 for fcc materials. YLD91 s ( ) s1 s2 s2 s3 s3 s1 2 m m m m φ = φ = + + = σ where s k are the principal values of the tensor s defined by a linear transformation on the stress tensor b + c c b 1 c c + a a b a a + b s = L = 3 3 f 3g 3h The anisotropy coefficients are a, b, c, f, g and h. The exponent m is 6 for bcc materials and 8 for fcc materials. YLD96 (PLANE STRESS) s ( ) 1 s2 s3 2 s3 s1 3 s1 s2 2 a a a a φ = φ = α + α + α = σ where s k are the principal values of the tensor s defined by a linear transformation on the stress tensor 1184
s = L 2 3 3 2 3 3 1 1 1 c2 c1 c1 + c2 = 3 c + c c c c c + c c 3c 3c 3c 4 5 6 The coefficient α k are defined by α = α p + α p + α p 2 2 2 k x 1k y 2k z 3k α = α cos 2β + α sin 2β 2 2 z z z1 where α z = 1, the p ij are the components of the transformation matrix from the principal axes of anisotropy (x, y, z) to the principal axes of s (1, 2, 3), and β represents the angle between the rolling direction x and the direction associated with the principal value of s, s1 and s3 ( s 1 s 3 s 3 ) cos β = x 1 if s 1 3 x 3 if s < s 1 3 s or Here, the dot denotes the scalar product. The anisotropy coefficients are the c i, α x, α y and α z1. The exponent a is 6 for bcc materials and 8 for fcc materials. where s k and stress deviator s s k YLD2-2D ( s,s ) s 1 s 2 s 2 s 1 s 1 s 2 a a a a φ = φ = + 2 + + 2 + = 2σ are the principal values of the tensor s and s defined by two linear transformations on the " " = = = =!! α1 α3 2α 4 α2 and 2α 5 α6 s A s s s A s s The anisotropy coefficients are the eight α 7 8 α k. The exponent a is 6 for bcc materials and 8 for fcc materials. α 1185
# $ $ $!!! s where i and stress deviator s s j YLD24-18P ## # # # # # # # # # # # # # # # # # # ( s,s ) s 1 s 1 s 1 s 2 s 1 s 3 s 2 s 1 s 2 s 2 s 2 s 3 a a a a a a φ = φ = + + + + + a a a a 3 1 3 2 3 3 4σ + s s + s s + s s = are the principal values of the tensors s $ and s $ defined by two linear transformations on the s = C s = C T = L s = C s = C T = L is the Cauchy stress tensor. C and C are the tensors containing the anisotropy coefficients C c c 12 13 c c 21 23 c 31 c 32 = c 44 c 55 c 66 C c c 12 13 c c 21 23 c 31 c 32 = c 44 c 55 c 66 T is the transformation used to get the stress deviator from the stress tensor, i.e., 2 1 1 1 2 1 1 1 1 2 T = 3 3 3 3 The 18 anisotropy coefficients are the nine c pq and the nine c pq. The exponent a is 6 for bcc materials and 8 for fcc materials. 1186
.4.3.2.1 ε RD -.1 -.2 -.3 -.3 -.2 -.1.1.2.3.4 ε TD FIGURE 1. Predicted Forming Limit Diagram Al Alloy 622-T43 Sheet (t = 1.-mm) 1187
.4.3.2.1 ε RD -.1 -.2 -.3 -.3 -.2 -.1.1.2.3.4 ε TD FIGURE 2. Predicted Forming Limit Diagram Al Alloy 5182-O Sheet (t = 1.625-mm) 1188
Recalculated Pole Figures 111 2 22 Orientation Distribution Function Phi2 == deg Phi2 == 5 deg Phi2 == 1 deg Phi2 == 15 deg Phi2 == 2 deg Phi2 == 25 deg Phi2 == 3 deg Phi2 == 35 deg Phi2 == 4 deg Phi2 == 45 deg Phi2 == 5 deg Phi2 == 55 deg Phi2 == 6 deg Phi2 == 65 deg Phi2 == 7 deg Phi2 == 75 deg Phi2 == 8 deg Phi2 == 85 deg Phi2 == 9 deg FIGURE 3. Crystallographic Texture 622-T43 1189
Recalculated Pole Figures 111 2 22 Orientation Distribution Function Phi2 == deg Phi2 == 5 deg Phi2 == 1 deg Phi2 == 15 deg Phi2 == 2 deg Phi2 == 25 deg Phi2 == 3 deg Phi2 == 35 deg Phi2 == 4 deg Phi2 == 45 deg Phi2 == 5 deg Phi2 == 55 deg Phi2 == 6 deg Phi2 == 65 deg Phi2 == 7 deg Phi2 == 75 deg Phi2 == 8 deg Phi2 == 85 deg Phi2 == 9 deg FIGURE 4. Crystallographic Texture 5182-O 119