On the rotation of orthotropic axes under uniaxial off-axis tension in sheet metals

Transcription

1 64 ISSN MECHANIKA. 03 Volume 9(3): On the rotation of orthotroic axes under uniaxial off-axis tension in sheet metals Mohammad Zehsaz*, Hadi Mehdiour**, Alireza Alimohammadi*** *Deartment of Mechanical Engineering, University of Tabriz, Tabriz 5666, Iran, **Deartment of Mechanical Engineering, University of Tabriz, Tabriz 5666, Iran, ***Deartment of Mechanical Engineering, University of Tabriz, Tabriz 5666, Iran, htt://dx.doi.org/0.5755/j0.mech Introduction The re-orientation of orthotroic axes due to the off-axis uniaxial tensile load has a major imortance when the rate of this rotation is significant while the system of axes reserved. Kim observed that during the twisting of the cold drawn tubes [], the orthotroic symmetry is maintained but the orthotroic axes rotate in the twisting direction. Boehler et al., Kim and in and Losilla et al. have shown that the cold rolled sheet metals under uniaxial off-axes tension loading remain aroximately orthotroic but there is a large in-lane re-orientation of the orthotroic symmetric axes [-4]. Bunge and Nielsen have used both exerimental and orientation distribution function (ODF) measurement techniques to obtain the magnitude of the texture rotation of the orthotroic Aluminium cold rolled secimens subjected to off-axis stretching [5]. Tugcu and Neale have studied the orthotroic axes rotation using the orthogonal tensor R that secifies the olar decomosition of the deformation [6]. Attemts have also been made to solve the roblem of orthotroic reorientation by Dafalias, Dafalias and Rashid and Cho and Dafalias [7-9]. The main theoretical tool to exlain orientation of the orthotroic axes is the concet of lastic sin. This henomenon is described and simulated by Dafalias using a simle theory of lasticity [0], which combines Hill s quadratic yield criterion for orthotroic sheet metals with the concet of lastic sin as an essential constitutive comonent for the orientational evolution of the anisotroic tensorial internal variables []. Itskov and Aksel have resented a constitutive model for orthotroic materials using lastic sin concet assuming that the anisotroies are reserved but the anisotroy axes can undergo a rigid rotation due to the large lastic deformations []. Tong et al. resented a simlified anisotroic lasticity theory which has been used to exlain the anisotroic flow of the orthotroic olycrystalline sheet metals under off-axis uniaxial tension. Their theory was formulated in terms of the intrinsic variables of the rincial stresses and a loading orientation angle for the uniaxial tension. They acquired a suitable analytical formula for the macroscoic lastic sin for orthotroic sheet metals with reserved but rotated orthotroic symmetry axes under offaxis uniaxial tension [3]. Also Tong has resented a henomenological theory based on the lastic sin concet, Fourier series and concets of microscoic olycrystalline lasticity for describing the anisotroic lastic flow of the orthotroic olycrystalline aluminium sheet metals under lane stress [4-5]. Hahm and Kim develoed the Kim and in s work regarding to the work hardening of cold rolled sheet metals. They showed that Lankford s Ratio (R) under uniaxial off-axis tension remain aroximately constant [6]. Kim D.-N. et al. roosed a constitutive model based on continuum energy considerations, the Lee decomosition and an anisotroic stored energy function of the logarithmic strains in which the rotation of the orthotroic axes is also considered [7]. In this aer, a re-orientation of the axes when orthotroic sheet metals are subjected to off-axis uniaxial tension is studied and formulated by using the following assumtions: The rincial direction of lastic strain increment is the same as the sheet metal orthotroic axes, and Lankford s number remain unchanged. In many works, it was tacitly assumed that Hill s [] suggestion of having the orthotroic directions aligned with the rincial lastic stretch directions is valid, and this is indeed the case when loading maintains fixed rincial stress directions from the onset of lastic deformation. The validity of the first assumtion has been extensively discussed by Dafalias [0]. The second assumtion has been considered using Hahm and Kim s exerimental results on the low carbon steel sheet metals [6]. Based on these assumtions, a simle and alicable method is develoed to obtain the rate of orthotroic axes rotation under uniaxial off-axis tension with lastic strain increment. Also a function of second re-strain magnitude has been obtained which is similar to the one roosed by Kim and in [3]. The roer and exlicit analytical form for rotation angle of orthotroic symmetry axes sheet metals under off-axis uniaxial tension is obtained and the associated material anisotroic constant are determined and comared with the exerimental data of Kim and in [3]. Also, to determine the direction of the rotation of the orthotroic axes under mentioned loading and stress state a new simle method is resented.. Descrition of the roblem Are the orthotroic symmetries reserved when orthotroic sheet metals are subjected to in-lane off-axis uniaxial tension loading where the direction of loading is

2 65 fixed with resect to the initial orthotroic axes? Moreover, if orthotroic symmetries are reserved, do the orthotroic axes remain the same, or they rotate? If yes, towards which direction? Finally, how fast do they rotate in relation to the lastic strain induced by the non-coaxial loading? The above questions should be answered by exerimental observations before develoing a theory of anisotroic re-orientation. These answers can show whether the theoretical objective is worth ursuing or not. Kim demonstrated that the answer for the first question is ositive []. Kim and in s [3] exeriments corroborate the results of Kim s exeriment []. In the next ste, they have rovided answers to the second and third questions. If the answer to the third question is that the orthotroic axes do rotate but very slowly, again one may reason that for ractical uroses, the orthotroic orientation may be assumed to remain fixed. yield stress distribution has been carried out for the small secimens at each ε and ψ. It was shown that the answers for two questions mentioned at the beginning of this section are ositive. Also, it is ossible to investigate the evolution of the orientation of the orthotroic axes X and with resect to the second re-strain ε. This can be done by calculating the shift of the symmetrical yield stress distribution using the angle β from the direction of ε. (see Fig. ). 3. Calculating the magnitude of rotation of orthotroic axes In simulating the orientation of orthotroic axes of sheet metals due to the uniaxial off-axis tension (second re-strain), Hahm and Kim s investigation [6] shows that the incremental lastic strain ratio (R) which is defined as: d R d d, () remains unchanged. So that d and d are the lastic strain increments in the second re-strain and transverse directions, resectively. Using the rincial axes of the alied stress tensor in the selected Cartesian coordinate system which has been used in Kim and in's [3] exerimental work (see Fig. ), the exressions for the stress and strain increment tensors under uniaxial tension are: 0 0 σ () Fig. Schematic resentation of the tensile secimen and the different directions and related angles, ψ, θ Kim and in erformed an exerimental method to study the cold rolled sheet metals anisotroy with tensile tests at different angles to the rolling direction [3]. They have utilized the variation of the uniaxial yield stresses with the orientation of the tensile loading axis which can be used to set u orthotroic symmetry. For their tests, they have selected cold rolled sheets of low carbon steel which is widely used in the automotive industry. This alloy has moderate initial orthotroy. To increase the degree of orthotroy, full size sheets were stretched along the rolling direction u to 3% and 6% of tensile strains. Then, the stretched secimens were cut at an angle of ψ from the Rolling Direction (). Fig. resents a schematic diagram of the secimens. The and T.D. (Transverse Direction) are the initial orthotroic axes. Three values of ψ were chosen: 30, 45 and 60 degrees and for each value of ψ, the secimens were subjected to a tensile second restrain ε along their axis with the magnitude of,, 5, and 0 ercent. To investigate the ossible re-orientation of the initial orthotroic directions and T.D. due to the mentioned re-straining, small size secimens were cut from the re-strained secimens at different angles and tested under tensile loading. For tracing the evolution and rotation of the orthotroic symmetries by following the shift, with resect to the second re-strain ε, the record of tensile And the lastic strain increment tensor is: d d 0 dε d d 0. (3) 0 0 d d Like many other works, also it is assumed that Hill s suggestion of having the orthotroic directions aligned with the rincial lastic strain directions is valid []. Under this assumtion it can be written that: d d cot. (4) d In which the subscrit denotes the second loading direction and subscrit shows the normal direction to the second loading direction (see Fig. ). Kim and in suggest that in the absence of an aroriate theory [3], the rate of orientation change of the orthotroic axes is roortional to the shear strain rate with resect to the rincial directions of stress as below: Cd d. (5) where β is the angle between the X-axis direction and the tensile loading direction, as shown in Fig., and C is a constant, the value of which is deendent uon the state of

3 66 hardening as Kim and in ointed out [3]. It seems that this assumtion is more secific and a more general form of the above uation can be used as: d f d d, (6) where f is an arbitrary function of second re-strain magnitude. It should be highlighted that the orthotroic axes rotation under uniaxial off-axis tension in addition to the second re-strain magnitude also deends on the loading angle (ψ). Last-mentioned deendence can be introduced in orthotroic axes re-orientation roblem by imlementing the boundary conditions for the governing differential uations. As mentioned before, Hahm and Kim showed that for steel sheets which are widely used in automotive industries [6], the incremental lastic strain ratio R remains unchanged during the orthotroic axes re-orientation, aroximately. Dividing the numerator and denominator of Eq. (4) by d and using Eq. (6) leads to: R R cot d f 0 d. (7) In which the uer bound of integration, ε, is the second re-strain magnitude in Kim and in's exeriments [3]. Eq. (7) after analytical integration leads to: R arc cot cot ex f d 0 R, (8) where ψ = θ + β. The simlest suggestion for function of f is the one given by Kim and in [3], i.e. f C and by the substitution of this form of function into Eq. (8) and knowing that R.65, it can be written as: (+C) arccot cot ex. (9) In the above exression C is a material constant and Eq. (9) must satisfy two following boundary conditions: If ε = 0 then θ = 0 where this condition was satisfied in Eq. (9). For every ε when ψ = 0 or ψ = π / then θ = 0 and this condition is satisfied in Eq. (9). 4. The direction of the re-orientation of orthotroic axes under uniaxial off-axis tensile loading In addition to the rolling and transverse directions, there is another direction in the orthotroic sheet metals that if loading is alied on that direction, the orthotroic axes don t rotate. In this aer this direction that is shown in Fig., called uivalent angle (ψ ). To define the re-orientation of the orthotroic axes under uniaxial off-axis tension, both the magnitude and direction of the orientation should be determined. Dafalias [0] has discussed the direction of the re-orientation of the orthotroic axes under uniaxial off-axis tension using the relation which Hill reresented in the form of tan ψ = (g + h - ) / (f + h -) where f, g and h are normalized coefficients of Hill s quadratic criteria []. Dafalias [0] showed that for the reorted values of f, g and h by Kim and in [3], ψ = 44.68º. In this research work, we roose a simle relationshi which can be used to determine the direction of the re-orientation of the orthotroic axes under off-axis uniaxial tension. To determine the relation for ψ, we suose that the yield stress for the initial sheet in ψ is σ, so the following uilibrium uations are satisfied: sin T.D. cos, (0) where and are the yielding stress for initial T.D. sheet in the rolling and transverse directions resectively. Dividing both sides of the uilibrium uations gives: 5. Results arctan. () T.D. The re-orientation of the orthotroic axis of the sheet metals under uniaxial off-axis tension is obtained using Eq. (9) with different material constants. The results are resented in Figs. -4. These results are comared with the exerimental data given by Kim and in for a metal sheet by 3% initial re-strain [3]. Note when the loading angle is greater than the uivalent angle (ψ ) and consuently the orthotroic axes rotate clockwise then the ositive direction of θ should be calculated from the initial T.D. towards the -axis which rotates clockwise. Therefore, the value of θ is ositive in every condition. The Eq. (9) is exlicit formulation of the state and orientation of the anisotroic orthotroy under off-axis uniaxial tensile loads. The rotation of the orthotroic axes can be obtained using this uation based on two variables of loading angle ψ and secondary re-strain ε. The secondary re-strain consists of elastic and lastic strains but the former is neglected due to its low value. In addition to these variables, one other constant C is ruired which can be obtained using exerimental data with given initial anisotroy for any secific material. This material constant is exactly the same as the one used by Kim and in [3]. To comare the calculated values with exerimental data, the constants of Eq. (9) are extracted for the orthotroic sheet metal using Kim and in s exerimental data [3]. Figs. -4 show the rotation of the orthotroic axis

4 67 close to the uivalent angle, which we can consider that these two angles are coinciding. 6. Discussion Fig. The rotation of the orthotroic axis for loading angles of 30 In the course of a work on the subject of this aer, Dafalias elaborately discussed the re-orientation of orthotroic axes under uniaxial off-axis tensile loading [0]. He entirely connected the roblem of the reorientation of orthotroic axes to lastic sin and also the concet of rotation of substructure texture. He esecially discussed the concet of uivalent angle and direction of re-orientation of orthotroic axes. However, the model roosed by Dafalias is difficult to use due to its comlexity [0]. On the contrary, the formulation resented in this aer is very simle, succinct and also alicable. The model resented in this work is based on Kim and in s exerimental data and so Dafalias s work [3, 0]. To make the roosed model more flexible, other f can be used which may ruire forms of function more material constants rather than only one C and this can be regarded as a future work for this research rogram. 7. Conclusion Fig. 3 The rotation of the orthotroic axis for loading angles of 45 The value and direction of the re-orientation of orthotroic axis under off-axis uni-axial loading is obtained using a simle formulation. This method is based on the following assumtions: The rincial direction of lastic strain increment is the same as the sheet metal orthotroic axes, and Lankford s number remain unchanged. Also, associated material anisotroic constant has been obtained and comared with those given by Kim and in which has been obtained based on exerimental tests [3]. The results show that by using this formulation, the orientation and intensity of the rotation of the orthotroic axis can be calculated for uniaxial off-axis loading with good agreement with exerimental data. To use this model, only one material constant should be determined based on affordable number of tests and this adds to the advantages of the model. Acknowledgement Fig. 4 The rotation of the orthotroic axis for loading angles of 60 for three loading angles of 30, 45 and 60 resectively. These results have been obtained using Eq. (9) for a low carbon steel sheet with 3% initial re-strain. The material constant is given in each figure. The agreement with exerimental data is good for the loading angle of 30, because of loading angle is smaller than the uivalent angle (ψ < ψ ); and, the difference between the loading angle and uivalent angle is significant. For the same reasons, for the loading angle of 60, the results comare well with the exerimental data in site of the fact that the uivalent angle is larger than the loading angle (ψ > ψ ). But there is a relatively large difference between the exerimental data and the calculated values for the loading angle of 45. The main reason for this difference is that the loading angle is The authors would like to acknowledge and areciate the cooeration, hel and suort of MAPNA,Grou Mr. Mehdi eganeh and also the authors would like to exress their gratitude for the financial suort rovided by the University of Tabriz. References. Kim, K.H. 99. Evolution of anisotroy during twisting of cold drawn tube, J. Mech. Phys. Solids. 40: Boehler, J.P.; Koss, S. 99. Evolution of anisotroy in sheet-steels subjected to off-axes large deformation, in: Brueller, O., Mannl, V., and Najar, J. (Eds.), Advances in Continuum Mechanic. Sringer, Kim, K.H.; in, J.J Evolution of anisotroy under lane stress, J. Mech. Phys. Solids. 45: Losilla, G.; Boehler, J.P.; Zheng, Q.S A generally anisotroichardening model for big offset-strain

5 68 yield stresses, Acta Mech. 44(3-4): Bunge, H.J.; Nielsen, I Exerimental determination of lastic sin in olycrystalline materials, Int. J. Plasticity. 3: Tugcu, P.; Neale, K.W On the imlementation of anisotroic yield functions into finite strain roblems of sheet metal forming, Int. J. Plasticity. 5: Dafalias,.F The lastic sin, J. Al. Mech. 5: Dafalias,.F.; Rashid, M.M The effect of lastic sin on anisotroic material behavior, Int. J. Plasticity. 5: Cho, H.S.; Dafalias,.F Distortional and orientational hardening at large viscolastic deformations, Int. J. Plasticity. : Dafalias,.F Orientational evolution of lastic orthotroy in sheet metal, Int. J. Plasticity. 48: Hill, R The Mathematical Theory of Plasticity. Clarendon Press, Oxford, Itskov, M.; Aksel, N A constitutive model for orthotroic elasto-lasticity at large strains, Arch. Al. Mech. 74: htt://dx.doi.org/0.007/s Tong, W.; Tao, H.; Jiang, X Modeling of rotation of orthotroic symmetry axes of sheet metals subjected to off-axis tension, ASME Trans. J. Al. Mech. 7(4): htt://dx.doi.org/0.5/ Tong, W A lanar lastic flow theory of orthotroic sheets and the exerimental rocedure for its evaluations, Proc. R. Soc. Lond. A 46: htt://dx.doi.org/0.098/rsa Tong, W A lane stress anisotroic lastic flow theory for orthotroic sheet metals, Int. J. Plasticity. : htt://dx.doi.org/0.06/j.ijlas Hahm, J.H.; Kim, K.H Anisotroic work hardening of steel sheets under lane stress, Int. J. Plasticity. 4: htt://dx.doi.org/0.06/j.ijlas Kim, D.-N.; Montáns, F.J.; Bathe, K.-J Insight into a model for large strain anisotroic elastolasticity, Comut. Mech. 44: htt://dx.doi.org/0.007/s z. Mohammad Zehsaz, Hadi Mehdiour, Alireza Alimohammadi ORTOTROPINIŲ AŠIŲ SUKIMASIS SU ĮTEMPIŲ KRPTIMI NESUTAMPANČIUOSE METALO LAKŠTUOSE R e z i u m ė Ekserimentiniai tyrimai rodo, kad kai ortotroiniai metalo lakštai yra veikiami įtemių, nesutamančių su jų krytimi, ašys ersiorientuoja, o ortotroinė simetrija išsaugoma. Šiame darbe šis reiškinys tiriamas ir formuluojamas remiantis tokia rielaida: agrindinė lastinių deformacijų didėjimo krytis yra ta ati kai ir metalo lakšto ortotroinių ašių krytis ir Lankfordo skaičius lieka neakitęs. Pateikiamas arastas ir ritaikomas metodas, kuris susieja ortotroinės ašies sukimosi greitį veikiant neašiniam įtemiui, sukeliančiam lastinės deformacijos rieaugį. Tai at asiūlytas naujas suarastintas metodas ortotroinės ašies sukimosi kryčiai nustatyti, kai įtemiai nesutama su ašies krytimi. Parodyta, kad asiūlytieji metodai gali būti sėkmingai taikomi ortotroinių ašių sukimosi greičiui ir kryčiai nustatyti, o gaunami rezultatai gerai sutama su ekserimentiniais. Mohammad Zehsaz, Hadi Mehdiour, Alireza Alimohammadi ON THE ROTATION OF ORTHOTROPIC AXES UNDER UNIAXIAL OFF-AXIS TENSION IN SHEET METALS S u m m a r y Exerimental investigations show that when orthotroic sheet metals are subjected to off-axis uniaxial tension, a re-orientation of the axes occurs, while orthotroic symmetries are reserved. In this aer this henomenon is investigated and formulated based on the following assumtions: the rincial direction of the lastic strain increment is the same as the direction of the orthotroic axes of the sheet metal and Lankford s number remains unchanged. A simle and alicable method is resented which relate the rate of change of orthotroic axes rotation under uniaxial off-axis tension with lastic strain increment. Also, a new and simle method is develoed to obtain the direction of the orthotroic axes rotation under uniaxial off-axis tension. It is shown that the roosed methods can successfully be used to obtain the magnitude and direction of the rotation of the orthotroic axes and the results show good agreement with the exerimental. Keywords: Orthotroic axes, Plastic Strain, Re-Orientation, Sheet metal. Received October 8, 0 Acceted June 7, 03