A modified Burzynski criterion for anisotropic pressure-dependent materials

Transcription

1 Sādhanā Vol. 4, No., January 7, pp DOI.7/s Indian Academy of Sciences A modified Burzynski criterion for anisotropic pressure-dependent materials FARZAD MOAYYEDIAN and MEHRAN KADKHODAYAN, * Department of Mechanical Engineering, Eqbal Lahoori Institute of Higher Education (ELIHE), Mashhad, Iran Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran farzad.moayyedian@eqbal.ac.ir; kadkhoda@um.ac.ir MS received 9 October 4; revised April 6; accepted August 6 Abstract. In this paper the Burzynski criterion, which was introduced for isotropic pressure-dependent materials, is modified for anisotropic pressure-dependent materials in plane-stress condition. The modified criterion can be calibrated with experimental data points such as tensile stress at,45and 9, compressive stress at and 9 and R-values in tensile stress at,45 and 9 from rolling direction and also biaxial tensile stress and tensile R-value. To identify the anisotropic parameters an error function is set up through comparison of the predicted yield stresses and R-values with those from experiments. Then the Downhill simplex method is applied to solve high-nonlinearity equations. Finally, considering Al -T4 (BCC), Al 9-T (FCC), AZ (HCP) and also Mg. Th alloy, Mg 4% Li alloy, pure textured magnesium, textured magnesium and Ti 4Al /4O, which are HCP materials with e p ¼ %; ; % as case studies and comparing the results for the modified Burzynski criterion with experiments, it is shown that the Burzynski criterion is appropriate for pressure-dependent anisotropic materials with proper accuracy. Keywords. criterion; Burzynski criterion; pressure-dependent anisotropic materials; Downhill simplex method.. Introduction In new research works to accurately model the behaviour of materials, anisotropic behaviour along with pressure dependency is adopted. In the current study a criterion used for pressure-dependent isotropic materials is newly developed for anisotropic pressure-dependent materials and compared with other related criteria and experimental results. In the following the effects of anisotropy and pressure dependency on yielding of materials are reviewed briefly. Spitzig and Richmond [] presented experimental results on iron-based materials and aluminium and showed that there was no need for the pressure dependency of yielding to be associated with irreversible plastic dilatancy. Liu et al [] presented a new pressure dependency criterion based on the criterion given by Hill for anisotropic solids and the criterion proposed by Drucker and Prager for soil that tensile and compressive strengths are widely different. Barlat et al [] proposed a generalized yield description to account for binary aluminium magnesium sheet samples that were fabricated by different processing paths to obtain different microstructures. It was subsequently shown that this yield function was suitable for description of the plastic *For correspondence behaviour of any aluminium alloy sheet. Yoon et al [4] implemented a non-quadratic yield function that simultaneously accounted for the anisotropy of uniaxial yield stresses and R-values in a finite-element code. Barlat et al [5] proposed a new plane stress yield function that described the anisotropic behaviour of sheet metals, in particular, aluminium alloy sheets. The anisotropy of the function was introduced in the formulation using two linear transformations on the Cauchy stress tensor. Stoughton and Yoon [6] proposed a non-associated flow rule based on a pressure-sensitive yield criterion with isotropic hardening that was consistent with the Spitzig and Richmond [] results. The significance of their work was that the model distorted the shape of the yield function in tension and compression, fully accounting for the strength-differential effect (SDE). Hu and Wang [7] proposed a yield criterion with three independent invariants and deviatoric stress tensors. They considered strength-differential in tension and compression and predicted that yielding behaviours of many isotropic materials exhibit both the pressure and stress-state dependence. Hu [] suggested a yield criterion derived with the use of invariants of the stress tensor to consider anisotropy. Anisotropic properties of the predicted yield surface were characterized by seven experimental data points obtained from three standard uniaxial-tension tests and one equibiaxial-tension test. Aretz [9] proposed a yield function that 95

2 96 Farzad Moayyedian and Mehran Kadkhodayan included eight anisotropy parameters, which could be fitted to experimental data. Furthermore, it was shown that the proposed yield function had nearly the same flexibility like that of Yld-D. Burzynski [] presented an energybased hypothesis, called by the author as the hypothesis of variable volumetric-distortional limit energy. Lee et al [] extended continuum plasticity models considering the unusual plastic behaviour of magnesium alloy sheet. Hardening law based on two-surface model was further extended to consider the general stress strain response of metal sheets, including the Bauschinger effect, transient behaviour and the unusual asymmetry. Aretz [] presented the issue of yield function convexity in the presence of a hydrostatic-pressure-sensitive yield stress. Hu and Wang [] proposed a new theory for pressure-dependent materials, in which a corresponding constitutive model could be constructed and characterized experimentally via two steps, i.e., one related to the characterization of yielding behaviour of material, and the other to the plastic flow of material deformation. Huh et al [4] evaluated the accuracy of popular anisotropic yield functions based on the rootmean-square error (RMSE) of the yield stresses and R- values. The yield functions included were Hill4, Yld9, Yld9, Yld96, Yld-d, BBC and Yld-p yield criteria. They concluded that the Yld- yield function was the best criterion to accurately describe the yield stress and R-value directionalities of sheet metals. Vadillo et al [5] formulated an implicit integration of elastic plastic constitutive equations for the paraboloid case of Burzynski yield condition and the tangent operator consistent with the integration algorithm was developed. Taherizadeh et al [6] developed a generalized finite-element formulation of stress integration method for nonquadratic yield functions and potentials with mixed nonlinear hardening under non-associated flow rule to analyse the anisotropic behaviour of sheet materials. Gao et al [7] described a plasticity model for isotropic materials, which was a function of the hydrostatic stress as well as the second and third invariants of the stress deviator. They presented its finite-element implementation, including integration of the constitutive equations using the backward Euler method and formulation of the consistent tangent moduli. Lou et al [] proposed a simple approach to extend symmetric yield functions for the consideration of SDE in sheet metals. The SDE was coupled with symmetric yield functions by simply adding a weighted pressure term for anisotropic materials. This approach was applied to the symmetric Yld-d and the yield function was modified to describe the anisotropic and asymmetric yielding of two aluminium alloys with small and strong SD effects. Yoon et al [9] proposed a general asymmetric yield function with dependence on the stress invariants for pressure-sensitive metals. The proposed function was transformed in the space of the stress, the von Mises stress and the normalized invariant to theoretically investigate the possible reason of SD effect. The yield function reasonably modeled the evolution of yield surfaces for a zirconium clock-rolled plate during in-plane and through-thickness compression. The yield function was also applied to describe the orthotropic behaviour of a face-centred cubic metal of AA- T4 and two hexagonal close-packed metals of high-purity a-titanium and AZ magnesium alloy. In the current research, a pressure-dependent isotropic criterion, the Burzynski criterion, for isotropic materials presented by Burzynski [], is developed to consider the anisotropy effects along with the pressure dependency in plane-stress condition called the modified Burzynski criterion. The obtained results are compared with Hill 4, Yld-d, Modified Yld-d and experimental data points. It is shown that the presented criterion can be adopted with proper errors by comparing to experimental results for anisotropic pressure-dependent materials such as Al -T4, Al 9-T and AZ and also Mg. Th alloy, Mg 4% Li alloy, pure textured magnesium, textured magnesium and Ti 4Al /4O with e p ¼ %; ; %.. Burzynski criterion The Burzynski criterion for three-dimensional states of stress- and pressure-dependent isotropic materials has the following form Vadillo et al [5]: U ¼ Ar e þ Br m þ Cr m ¼ where r e is the effective stress: ðþ rffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r e ¼ s xx þ s yy þ s zz þ s xy þ s xz þ s yz ðþ in which s ij is the deviatoric stress tensor and its components are s xx ¼ r xx r yy r zz >< s yy ¼ r yy r xx r zz ðþ s zz ¼ r zz r xx r yy >: s xy ¼ s xy ; s xz ¼ s xz ; s yz ¼ s yz and r m is the hydrostatic stress: r m ¼ r xx þ r yy þ r zz : ð4þ A C can be determined by three experimental data points such as r T Y (uniaxial tensile test), rc Y (uniaxial compressive test) and s S (simple-shear test). For plane-stress problems, (i.e., r zz ¼ s xz ¼ s yz ¼ ), Eq. () still takes its current form and the effective stress, considering s z ¼s x s y, becomes p r e ¼ ffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s x þ s y þ s xs y þ s xy ð5þ

3 A modified Burzynski criterion for anisotropic 97 in which s x ¼ r x r y >< s y ¼ r y r x >: s xy ¼ s xy and the hydrostatic stress becomes ð6þ r m ¼ r xx þ r yy : ð7þ. criterion for plane-stress problems To develop the Burzynski criterion to consider anisotropic materials, a linear transformation L, for modified deviatoric stress in terms of stress components with five independent parameters can be defined as where L 66 < : s xx s yy s xy s ¼ Lr ðþ 9 = L L ; ¼ 4 L L 5 r 9 < xx = r yy : ; : ð9þ L 66 s xy Then L ij can be defined in terms of a i []: 9 9 L a >< L >= L ¼ >< a >= a : ðþ L >: 4 5 a >; >: 4 >; 9 It should be noted that a i ði ¼ ; ; ; 4; 5Þ parameters can give the ability to consider anisotropy effects for deviatoric stress tensor. Therefore, the modified deviatoric stress can be expressed in terms of five independents parameters of a i as 9 >< s xx >= s yy >: >; ¼ 9 s xy >< ða þ a þ 4a a 4 Þr xx þ ða 4a 4a þ 4a 4 ð4a 4a 4a þ a 4 Þr xx þ ða þ 4a þ a a 4 >: 9a 5 s xy a 5 Þr yy Þr yy 9 >= >; : ðþ This idea has arisen from one of the linear transformation and the Yld-d criterion as in Barlat et al [5]. In this case the modified effective stress can take the following form: p r e ¼ ffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s xx þ s yy þ s xxs yy þ s xy ðþ and to modify the hydrostatic stress the following form can be employed []: r m ¼ a 6r xx þ a 7 r yy : ðþ a 6 and a 7 give the ability to consider the anisotropy effects for hydrostatic stress. Finally the Burzynski criterion in Eq. () can be modified to consider the anisotropic material effects as U ¼ a r e þ a 9 r m þ a r m ¼ : ð4þ In Eq. (5), the parameter a weights the modified effective deviatoric stress ðr e Þ, and the parameters a 9 and a weight the modified hydrostatic pressure ðr m Þ in the modified Burzynski criterion. Inserting Eq. () into Eq. () and the result in Eq. (5) and also using Eq. (4), the following form for the modified Burzynski criterion is obtained in terms of stress components: U r xx ; r yy ; s xy ða þ a þ 4a a 4 Þr xx þ ða 4a 4a þ 4a 4 Þr yy þ ð4a 4a 4a þ a 4 Þr xx þ ða þ 4a þ a a 4 Þr yy þ ða 7 þ a þ 4a a 4 Þr xx þ ða 4a 4a þ 4a 4 Þr yy B 6 ð4a 4a 4a þ a 4 Þr xx þ ða þ 4a þ a a 4 Þr 4 yy 5A þ 9a 5 s xy a 6 r xx þ a 7 r þa yy a 6 r xx þ a 7 r yy þ a 9 ¼ : ð5þ These material parameters can be determined by experimental data, which is explained in section. Furthermore, the first differentiation of the proposed criterion is useful for calibration. Therefore, from Eq. (6) this differentiation can be obtained as " # o U ða þ a þ 4a a 4 Þ ð4a a 4 Þr xx þ ða þ a 4 Þr yy ¼ a or xx þð4a 4a 4a þ a 4 Þ ða a Þr xx þða þ 4a Þr yy þ h a a 6 r xx þ a 7 r i yy 6 a 9 þ a >< " # o U ða 4a 4a þ 4a 4 Þ ð4a a 4 Þr xx þ ða þ a 4 Þr yy ¼ a : or yy þða þ 4a þ a a 4 Þ ða a Þr xx þða þ 4a Þr yy h a a 6 r xx þ a 7 r i yy 7 a 9 þ a o U >: ¼ 54a a 5 os s xy xy ð6þ 4. Calibration of the modified Burzynski criterion To calibrate the proposed criterion, experimental data points are required [], which consist of six data in stresses such as uniaxial tensile tests at,45 and 9 from rolling direction, the tensile biaxial stress test and uniaxial compressive test at and 9 from rolling direction, four

4 9 Farzad Moayyedian and Mehran Kadkhodayan experimental tests on tensile plastic strain increment ratio R-values R ¼ dep y at, 45 and 9 and also tensile de p z biaxial R-value R ¼ dep y. The effect of pressure dependency can be automatically satisfied because of existence of de p x modified hydrostatic stress inherently. Hence, the proposed criterion can have the effect of anisotropy effect and pressure dependency simultaneously in a different way from that of Modified Yld-d in Lou et al []. 4. Yield stress tests Because of pressure dependency of the proposed criterion, the behaviour of material is different in tension and compression and therefore the uniaxial experimental result in both tension and compression cases is required. For tensile yield stress tests in h from the rolling direction it is considered that < r xx ¼ r T h cos h r yy ¼ r T h sin h ð7þ : s xy ¼ r T h sin h cos h where h is the angle from the rolling direction and r T h is the tensile yield stress in h direction. Substituting these values in Eq. (6), a second-order equation in terms of r T h can be obtained as A h r T þbh h r T h ¼ : ðþ Considering the positive root of this equation, r T h can be found as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r T h ¼ B h þ B h þ 4A h ð9þ A h in which For the compressive yield stress tests it is considered that < r xx ¼r C h cos h r yy ¼r C h sin h : ðþ : s xy ¼r C h sin h cos h With the same process, the following second-order equation for r C h is obtained: A h r C Bh h r C h ¼ ðþ in which qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r C h ¼ B h þ B h þ 4A h : ðþ A h For balance biaxial yield stress test it is considered that < r xx ¼ r T b r yy ¼ r T b : ð4þ : s xy ¼ By substituting these values in Eq. (6), a second-order equation in terms of r T b can be obtained as A b r T þbb b r T b ¼ ð5þ and its positive root is where r T b ¼ B p b þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B b þ 4A b A b ð6þ "!# ½a a þ4a þa 4 þ½a þ4a a a 4 a 6 þa >< 7 A b ¼a þa 9 7 þa ½ a þ4a þa 4 ½a þ4a a a 4 a 6 þa : >: 7 B b ¼a ð7þ ða þ a þ 4a a 4 Þcos h þ ða 4a 4a þ 4a 4 Þsin h þ ð4a 4a 4a þ a 4 Þcos h þ ða þ 4a þ a a 4 Þsin h A h ¼ a þ ða 7 þ a þ 4a a 4 Þcos h þ ða 4a 4a þ 4a 4 Þsin h a 6 cos h þ a 7 sin h >< þ a 9 ð4a 4a 4a þ a 4 Þcos h þ ða þ 4a þ a a 4 Þsin 6 h C7 : 4 A5 þ½9a 5 sin h cos h a 6 cos h þ a 7 sin h >: B h ¼ a ðþ

5 A modified Burzynski criterion for anisotropic R-value tests along with associated flow rule Although the presented criterion is pressure dependent, the associated flow rule for plane stress problems can be used as >< >: de p xx ¼ dk o U or xx de p yy ¼ dk o U or yy de p xy ¼ dk o U os xy ðþ The thickness strain is calculated by the assumption of incompressibility as de p zz ¼dep xx dep yy ð9þ The R-value in h-direction from rolling direction under tension is denoted as R T h, which is calculated as follows: de p yy R T h ¼ dep yy de p ¼ zz de p xx þ de p yy o U or ¼ xx sin h þ o U or yy cos h o U or xy sin h cos h : ðþ o U or xx þ o U or yy The R-value in the balanced biaxial tension is defined as the ratio of the strain increments in transverse direction to that in rolling direction in the balanced biaxial tension using Eq. (4), which is obtained as R T b ¼ dep yy de p ¼ xx o U or yy o U or xx ðþ 5. Parameter evaluation and RMSE of the yield stresses and R-values Ten material constants denoted as a i ði ¼ Þ in the modified Burzynski yield function of Eq. (6) are calibrated using experimental data points for numerical analysis. These material constants are calculated by experimental results including r T ; rt 45 ; rt 9 ; rt b ; rc ; rc 9 ; R T ; RT 45 ; RT 9 ; RT b andthentheyareutilizedtosetupanerror function as " # " # " # E ¼ rt exp : r T þ rt 45 exp : r T þ rt 9 exp : 45 r T 9 " # " # " # þ rt b exp : þ rc exp : þ rc 9 exp : r T b r C " # " # þ RT 45 þ RT R T R T 45 " # " # þ RT b þ RT 9 R T 9 R T b r C 9 ðþ The error function is minimized by the Downhill simplex method to identify the material parameters. The RMSEs of tensile and compressive yield stresses and also tensile R- values are computed as The yield stresses and R-values are computed from experiments for Al -T4 and Al 9-T []. Er T ¼ 7 Er C ¼ 7 " # " rt 5 r T þ rt # rt 5 r T " # " rt 45 r T þ rt 6 # " rt 6 45 r T þ rt 75 # rt 75 6 r T 75 " # rt C 9 A " r T # rt r T þ rt 5 þ rt 45 þ rt 9 r T 9 " # " rc 5 r C þ rc # rc 5 r C " # " rc 45 r C þ rc 6 # " rc 6 45 r C þ rc 75 # rc 75 6 r C 75 " # rc C 9 A " r C # rc r C þ rc 5 þ rc 45 þ rc 9 r C 9 ðþ ð4þ

6 Farzad Moayyedian and Mehran Kadkhodayan ER T ¼ 7 " # " RT 5 R T þ RT # RT 5 R T " # " RT 45 R T þ RT 6 # " RT 6 45 R T þ RT 75 # RT 75 6 R T 75 " # RT C 9 A " R T # RT R T þ RT 5 þ RT 45 þ RT 9 R T 9 ð5þ 6. Results and discussion The yield surfaces are constructed by Hill4, Yld-d, Modified Yld-d and presented modified Burzynski yield functions and compared with experimental results for Al-T4 (a BCC material) and Al 9-T (a FCC material) in figures,,, 4, 5 and 6. The mechanical Figure. 4 5 Hill 4 Yld -d Modified Yld -d Comparison of the yield surfaces for Al9-T. Figure. 5 5 Hill 4 Yld -d Modified Yld -d Exp. datapoints Comparison of the yield surfaces for Al-T Figure 4. The Burzynski yield condition for Al 9-T in r e r m plane and the modified Burzynski ellipsoid Figure. yield condition for Al -T4 in r e r m plane, modified Burzynski Torre paraboloid. properties of these materials in different directions are available in table. Table shows the a i (i = ) parameters computed by minimizing the error function of Eq. () using the Downhill simplex method. In Hill4 and Yld-d, yield surfaces are symmetric with respect to the stress-free condition (pressure-independent yield

7 A modified Burzynski criterion for anisotropic (a) Tensile yield stress [MPa] Hill 4 Yld -d Modified Yld -d (a) Tensile yield stress [MPa] 9 Hill 4 Yld -d Modified Yld -d (b) Compressive yield stress [MPa] Hill 4 Yld -d Modified Yld -d (b) Compressive yield stress [MPa] Hill 4 Yld -d Modified Yld -d Figure 5. Comparison of the yield stress directionality for Al -T4 for (a) uniaxial tensile yield stress and (b) uniaxial compressive yield stress. criteria) but Modified Yld-d and the presented modified Burzynski are asymmetric ones. The Modified Yld-d has been presented for anisotropic pressure-dependent materials newly by Lou et al [] and introduced as a powerful criterion to predict experimental results rather than pressure-dependent criteria such as Hill 4 and Yld--d. Figure shows that the presented criterion predicts the experimental results satisfactorily and it is also close to Modified Yld-d especially in the third quadrant. It may be deduced that the presented criterion is quite successful in predicting the experimental results for Al -T4 as a pressure-dependent anisotropic criterion. Figure shows the modified deviatoric stress versus modified hydrostatic pressure using parameters a, a 9 and a of table. For the pressure-independent materials, the modified deviatoric and modified hydrostatic stresses are independent of each other; however, one can observe their dependency for pressure-dependent ones. It is seen that for Al -T4 a modified Burzynski Torre paraboloid by Burzynski [] and Vadillo et al [5] is obtained and it may be deduced that the presented modified Burzynski is a Figure 6. Comparison of the yield stress directionality for Al 9-T for (a) uniaxial tensile yield stress and (b) uniaxial compressive yield stress. proper criterion in r xx r yy plane compared to the experimental results for Al -T4 (a BCC material). Figure displays the situation of modified Burzynski criterion compared to other criteria for Al 9-T. The modified Burzynski can be a reasonable criterion in r xx r yy plane compared to the experimental results for Al 9- T (a FCC material); however, it may be concluded that the presented modified Burzynski criterion can estimate the yield surface in r xx r yy plane for BCC materials more accurately rather than FCC materials when the modified deviatoric stress versus modified hydrostatic pressure has an ellipsoid shape (figure 4). Figure 5 shows the variations of tensile and compressive yield stresses of four criteria versus the angle from the rolling direction compared to experimental results. It is seen that in predicting the tensile yield stress the Hill 4, Yld-d and Modified Yld-d are nearly the same and more accurate than the presented modified Burzynski criterion. However, the presented criterion is quite successful in predicting the compressive yield stress, which may be attributed to its dependency to hydrostatic pressure in its

8 Farzad Moayyedian and Mehran Kadkhodayan Table. Experimental data points of Al -T4 and Al 9-T []. Material r T r T 45 r T 9 r T b r C r C 9 R T R T 45 R T 9 R T b Al -T Al 9-T Table. Material parameters in modified Burzynski criterion of Al -T4 and Al 9-T. Material a a a a 4 a 5 a 6 a 7 a a 9 a Al -T Al 9-T Hill 4 Yld -d Modified Yld -d Hill 4 Yld -d Modified Yld -d R-value R-value Figure 7. Comparison of the R-value directionality for Al -T4. mathematical presentation as in Eq. (5). But, the Hill 4 and Yld-d are close to each other and compute the compressive yield stress far from the experimental results. Moreover, it is noted that the Modified Yld-d predicts the compressive yield stress nearly independent of angle of rolling direction which cannot be true for a direction-dependent material criterion (figure 5). Figure 6 shows that the modified Burzynski has not enough accuracy to predict the yield and compressive yield stress for Al 9-T; however, as mentioned earlier the modified Burzynski can be used for BCC materials more accurately rather than FCC materials. In figures 7 and, R-values versus angle from the rolling direction for Al -T4 and Al 9-T are shown. Compared to other criteria, the presented modified Burzynski is nearly well fitted with experimental results, especially for Al 9-T, in predicting R-values. In table a, b the differences between the current results and experimental data are computed for two materials from Eqs. () (5). As seen, the discrepancy of predicted compressive yield stress for Al -T4 is about.6%, which is the lowest compared with other criteria. For the tensile yield stress; however, the discrepancy is about.% which is higher than from other criteria but it is still Figure. Comparison of the R-value directionality for Al 9-T4. Table. Discrepancy of different criteria compared with experimental results for (a) Al -T4, (b) Al 9-T and (c) AZ (in percentage). Criterion E T r E C r E T R (a) Hill Yld-d Modified Yld-d (b) Hill Yld-d Modified Yld-d (c) Modified Yld -d small enough compared to experimental results. In addition, the discrepancy of obtained tensile R-value is about 5.9%, which is less than that of Hill criterion.

9 A modified Burzynski criterion for anisotropic 4 (a) 95 4 Tensile yield stress [MPa] Modified Yld-d Modified Birzynski Modified Yld-d Exp. datapoints Figure 9. Comparison of the yield surfaces for AZ. (b) Compressive yield stress [MPa] Modified Yld-d Figure. The Burzynski yield condition for AZ in r e r m plane and the modified Burzynski ellipsoid. It is also seen that the differences of obtained results for Al 9-T with experimental data are about.% and.47% for compressive and tensile yield stresses, respectively. However, it is about % for R-values, which is still less than that of Hill 4 but more than those of Yld-d and Modified Yld-d (table a, b). To check the proposed criterion for strong SDE, AZ, which is a HCP material, at % plastic strain is selected for case study. Its experimental data points are utilized from Yoon et al [9]. Figure 9 shows the yield function for AZ in r xx r yy and as seen the modified Burzynski predicts experimental results with good accuracy and more realistically than Modified Yld-d. In figure, r e r m is plotted for AZ and in figure tensile and compressive yield stresses are compared with experimental results and as seen the is successful in predicting the mechanical behaviour of a HCP material like FCC and BCC materials Figure. Comparison of the yield stress directionality for AZ: (a) uniaxial tensile yield stress and (b) uniaxial compressive yield stress. In table c, the obtained errors of modified Burzynski and Modified Yld-d in predicting tensile and compressive yield stresses, compared with experimental results, are shown and it is seen that the proposed criterion is also successful for a HCP material. Furthermore, the initial yield surface is not sufficient for the correctness of a selected function and it is important to find the subsequent yield surface in an appropriate yield function [, 5]. Therefore, five more HCP anisotropic materials such as Mg. Th alloy, Mg 4% Li alloy and pure textured magnesium, textured magnesium and Ti 4Al /4O are taken into account as case studies with e p ¼ %; ; %. The experimental data points are presented in table 4. Because only five experimental data points have been reported, the other five experimental data points are assumed to be isotropic. That is, four R-values of R T ; RT 45 ; RT 9 ; RT b are set to unity and the uniaxial tensile yield stress in the diagonal direction denoted as r T 45 is assumed to be identical to that in RD denoted as r T []. Then these data are used to calibrate material parameters in the modified Burzynski criterion yield function as presented in tables 4, 5 and 6 for Mg. Th alloy, Mg 4% Li alloy, pure textured magnesium, textured

10 4 Farzad Moayyedian and Mehran Kadkhodayan Table 4. Experimental data points of (a) Mg. Th alloy, (b) Mg 4% Li alloy, (c) pure textured magnesium, (d) textured magnesium and (e) Ti 4Al /4O []. Material (%) r T r T b r T 9 r C r C 9 (a) (b) (c) (d) (e) Table 5. Material parameters for modified Burzynski of (a) Mg. Th alloy, (b) Mg 4% Li alloy, (c) pure textured magnesium, (d) textured magnesium and (e) Ti 4Al /4O titanium alloy. e p ð%þ a a a a 4 a 5 a 6 a 7 a a 9 a (a) (b) (c) (d) (e) magnesium and Ti 4Al /4O. The coefficients a i ði ¼ Þ of these anisotropic HCP materials in e p ¼ %; ; % are collected in table 5 for the modified Burzynski criterion. The yield functions in r xx r yy plane, the diagrams of modified Burzynski criterion in r e r m plane and also the tensile/compressive yield stress directionality of these materials with e p ¼ %; ; % for these materials can be shown according the procedure explained before. In figures, 5,, and 4 the yield functions, in figures, 6, 9, and 5 the modified Burzynski criterion in r e r m plane and finally in figures 4, 7,,

11 A modified Burzynski criterion for anisotropic 5 Table 6. Discrepancy of different e p compared to experimental results for (a) Mg. Th alloy, (b) Mg 4% Li alloy, (c) pure textured magnesium, (d) textured magnesium and (e) Ti 4Al / 4O titanium alloy (in percentage) and the modified Burzynski. e p ð%þ E T r E C r E b r 4 (a) (b) (c) (d) (e) % % Experimental % Experimental Experimental % % % Figure. The Burzynski yield condition for Mg. Th alloy in r e r m plane for e p ¼ %; ; %. Directional yield stress [MPa] Tensile % Tensile Tensile % Compressive % Compressive Compressive % Experimental tensile % Experimental tensile Experimental tensile % Experimental compressive % Experimenta compressive Experimental compressive % Figure 4. Comparison of the tensile/compressive yield stress directionality for Mg. Th alloy with e p ¼ %; ; %. 5 Figure. Comparison of the yield surfaces for Mg. Th alloy with e p ¼ %; ; %. % % Experimental % Experimental Experimental % 5 5 and 6 the tensile/compressive yield stress directionality of these materials for e p ¼ %; ; % of Mg. Th alloy, Mg 4% Li alloy, pure textured magnesium, textured magnesium and Ti 4Al /4O are shown. It is observed that all anisotropic materials follow an identical pattern in subsequent yielding with e p ¼ %; ; % except for pure textured magnesium and textured magnesium. These two materials have the same pattern for e p ¼ ; % and vary from e p ¼ % for pure textured magnesium, figures and 9, and for textured Figure 5. Comparison of the yield surfaces for Mg 4% Li alloy with e p ¼ %; ; %.

12 6 Farzad Moayyedian and Mehran Kadkhodayan 4 4 % % % % Figure 6. The Burzynski yield condition for Mg 4% Li alloy in r e r m plane with e p ¼ %; ; % and the modified Burzynski hyperboloid Figure 9. The Burzynski yield condition for pure textured magnesium in r e r m plane for e p ¼ %; ; %. For e p ¼ % the modified Burzynski Torre paraboloid and for e p ¼ ; % the modified Burzynski hyperboloid. Directional yield stress [MPa] 6 4 Tensile % Tensile Tensile % Compressive % Compressive Compressive % Experimental tensile % Experimental tensile Experimental tensile % Experimental compressive % Experimental compressive Experimental compressive % Directional yield stress [MPa] Tensile % Tensile Tensile % Compressive % Compressive Compressive % Experimental tensile % Experimental tensile Experimental tensile % Experimental compressive % Experimental compressive Experimental compressive % Figure 7. Comparison of the tensile/compressive yield stress directionality for Mg 4% Li alloy with e p ¼ %; ; %. % % Experimental % Experimental Experimental % Figure. Comparison of the yield surfaces for pure textured magnesium with e p ¼ %; ; % Figure. Comparison of the tensile/compressive yield stress directionality for pure textured magnesium with e p ¼ %; ; %. magnesium, figures and. This issue may be attributed to the enormous change in the mechanical structure of these anisotropic materials from e p ¼ % to e p ¼. Finally, to validate the modified Burzunski to predict the subsequent yield stresses, the three relative errors are proposed as Eq. (6). In this relation, Er T, EC r and Eb r are the relative errors of directional tensile and compressive yield stresses and biaxial tensile stress, respectively. It should be mentioned that due to lack of experimental data points for these materials it is assumed that r T 45 is identical to r T []. The relative errors are presented in table 6 for Mg. Th alloy, Mg 4% Li alloy, pure textured magnesium, textured magnesium and Ti 4Al /4O. It is seen

13 A modified Burzynski criterion for anisotropic 7 % % Experimental % Experimental Experimental % Directional yield stress [MPa] Tensile % Tensile Tensile % Compressive % Compressive Compressive % Experimental tensile % Experimental tensile Experimental tensile % Experimental compressive % Experimental compressive Experimental compressive % 4 Figure. Comparison of the yield surfaces for textured magnesium with e p ¼ %; ; % Figure. Comparison of the tensile/compressive yield stress directionality for textured magnesium with e p ¼ %; ; %. % % Experimental % Experimental Experimental % % % Figure. The Burzynski yield condition for textured magnesium in r e r m plane with e p ¼ %; ; %. For e p ¼ % the modified Burzynski Torre paraboloid and for e p ¼ ; % the hyperbolid. that the values of relative errors are very small and the modified Burzynski is successful for predicting subsequent yield stresses for these materials. Figure 4. Comparison of the yield surfaces for Ti 4Al /4O titanium alloy with e p ¼ %; ; %. 7. Conclusions An asymmetric yield function, the Burzynski criterion for isotropic materials, is extended for an asymmetric anisotropic yield function, the modified Burzynski criterion. In " Er T ¼ r T # " rt þ rt # " rt 45 þ rt 9 # rt A r T " Er C ¼ r C # " rc r C þ rc 9 # >< rc A r C 9 Er b ¼ rt b rt b >: r T b r T r T 9 ð6þ

14 Farzad Moayyedian and Mehran Kadkhodayan % % other HCP anisotropic materials such as Mg. Th alloy, Mg 4% Li alloy and pure textured magnesium, textured magnesium and Ti 4Al /4O are examined for e p ¼ %; ; % and it is shown that the modified Burzynski is a powerful criterion to predict the subsequent yield surfaces as well Figure 5. The Burzynski yield condition for Ti 4Al /4O titanium alloy in r e r m plane with e p ¼ %; ; % and the modified Burzynski ellipsoid. Nomenclature L ij s ij s ij R R T b R T h linear transformation matrix deviatoric stress tensor modified deviatoric stress tensor plastic strain increment ratio tensile equibiaxial R-value tensile R-value in h direction Greek symbols Directional yield stress [MPa] Tensile % Tensile Tensile % Compressive % Compressive Compressive % Experimental tensile % Experimental tensile Experimental tensile % Experimental compressive % Experimental compressive Experimental compressive % a i e p de p ij h U U r e r e r ij r m r m r T b r C h r T h anisotropy parameters effective plastic strain increment of plastic strain tensor angle from the rolling direction Burzynski criterion modified Burzynski criterion effective stress modified effective stress stress tensor hydrostatic stress modified hydrostatic stress tensile equibiaxial yield stress compressive yield stress in h direction tensile yield stress in h direction Figure 6. Comparison of the tensile/compressive yield stress directionality for Ti 4Al /4O titanium alloy with e p ¼ %; ; %. this extension a linear transformation for transforming isotropic deviatoric stress with five coefficients to anisotropic modified deviatoric stress and two coefficients for transforming isotropic hydrostatic stress to anisotropic hydrostatic stress are considered. These modified anisotropic deviatoric and hydrostatic stresses are weighted in the modified Burzynski criterion with three coefficients. Finally with experimental data points these parameters are evaluated for Al -T4 (BCC), Al 9-T (FCC) and AZ (HCP) and it is shown that this criterion is well fitted especially for materials with the modified Burzynski Torre paraboloid. In addition, to show the accuracy of the modified Burzynski in predicting the subsequent yield stresses, five References [] Spitzig W A and Richmond O 94 The effect of pressure on the flow stress of metals. Acta Metall. : [] Liu C, Huang Y and Stout M G 997 On the asymmetric yield surface of plastically orthotropic materials: a phenomenological study. Acta Metall. 45: [] 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 and Makosey S 997 Yield function development for aluminium alloy sheets. J. Mech. Phys. Solids 45: [4] Yoon J W, Barlat F, Chung K, Pourboghrat F and Yang D Y Earing predictions based on asymmetric nonquadratic yield function. Int. J. Plasticity 6: 75 4 [5] Barlat F, Brem J C, Yoon J W, Chung K, Dick R E, Lege D J, Pourboghrat F, Choi S H and Chu E Plane stress yield

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