Validation of a new anisotropic yield criterion through bulge test

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1 1 Validation of a new anisotroic yield criterion through bulge test Prof. Dr.-Ing. D. Banabic, Dil.-Ing. G. A. Cosovici, Dil.-Ing. D.S. Comsa Technical University of Cluj-Naoca, Romania Dr.-Ing. S. Wagner, Prof. Dr. Ing. K. Siegert Institute for Metal Forming Technology, University of Stuttgart, Germany Abstract The aer resents a new anisotroic yield criterion and its imlementation in the ABAQUS/Standard finite-element code. The yield criterion is an extension of the formulation roosed by Barlat and Lian in In order to obtain a better reresentation of the lastic behaviour of the orthotroic sheet metals, some additional coefficients have been included in the ression of the equivalent stress. The constitutive model has been used in the simulation of a hydraulic bulging rocess. The numerical results have been comared with erimental data. 1 Introduction The mechanics of the sheet metal forming rocesses is greatly influenced by the lastic anisotroy of the material. In order to obtain a roer descrition of the anisotroy, the classical von Mises yield criterion should be modified. Hill roosed in 1948 /1/ a simle anisotroic yield criterion in the form of a quadratic function. Since then, several scientists have develoed more and more sohisticated yield functions for anisotroic materials. Hill himself successively imroved his criterion in 1979, 199 and 1993 /, 3, 4/. Hosford /5/ initiated another interesting research direction by introducing an isotroic yield function based on crystallograhic calculations. He also succeeded in extending this criterion to anisotroy /6/. During the last two decades, many other yield functions have been roosed aiming to imrove the fitting with erimental data. Among the recent achievements in this field, one may notice the formulations roosed by Barlat and Lian /7/, Barlat et al. /8, 9, 1/, Karafillis and Boyce /11/, as well as Cazacu and Barlat /1/. A comrehensive descrition of the most imortant yield criteria can be found in /13/.

2 Validation of a new anisotroic yield criterion through bulge test Banabic The yield criterion used in this aer has been derived from the one roosed by Barlat and Lian /7/. The new coefficients included in the ression of the equivalent stress imrove its caability to reresent the anisotroy of sheet metals. The identification rocedure needs seven material arameters: the uniaxial yield stresses and the coefficients of lastic anisotroy associated to three lanar directions (defined by an angle of, 45 and 9 measured from the rolling direction), as well as the equibiaxial yield stress associated to the rolling and transverse directions. The authors have included the new yield criterion into an elastolastic constitutive model for membranes under lane-stress conditions. The constitutive model has been imlemented as a UMAT subroutine in ABAQUS/Standard. The numerical simulation of a hydraulic bulging rocess has been erformed in order to evaluate the erformances of the model. Descrition of the new yield criterion.1 Equation of the yield surface A yield surface is generally described by an imlicit equation having the form Φ( σ, Y) : = where σ Y = (1) σ is the equivalent stress and Y is a yield arameter. The sheet metal is assumed to behave as an orthotroic membrane under lane-stress conditions. Using this hyothesis, the equivalent stress is defined as follows: 1 k k k [ a( Γ + Ψ) + a( Γ Ψ) + ( 1 a)( Ψ) ] k σ = () where a 1 and k Ν are material arameters, while Γ and Ψ are functions deending on the non-zero comonents of the stress tensor: ( Pσ 11 Qσ ) + R σ 1σ1 Γ = Mσ (3) 11 + N σ, Ψ = The coefficients M, N, P, Q and R in equation (3) are also material arameters. The stress comonents ( α, β = 1,) are ressed in an orthonormal basis coincident with the σ αβ axes of lastic orthotroy (1 is the rolling direction RD, is the transverse direction TD, while 3 is the normal direction ND). One may rove that the yield surface defined by equations (1) (3) is always convex in the stress sace if a 1 and k is a strictly ositive integer number.

3 Banabic Validation of a new anisotroic yield criterion through bulge test 3 The shae of the yield surface is controlled by seven material arameters: k, a, M, N, P, Q and R. The integer onent k has a secial status. Its value is established in accordance with the crystallograhic structure of the material /5/: k = 3 for BCC alloys k = 4 for FCC alloys. The other six arameters (a, M, N, P, Q and R) are established in such a way that the constitutive equations associated to the yield surface reroduce as well as ossible the lastic behaviour of the sheet metal. The identification rocedure is described in.4. One may notice that the equivalent stress defined by equations () (3) is very similar to the ression roosed by Barlat and Lian /7/ for orthotroic sheet metals under lanestress conditions. In fact, the Barlat-Lian formulation may be obtained assigning articular values to the coefficients M, N, P, Q and R.. Flow rule The flow rule associated to the yield surface described by equations (1) (3) is /14/ Φ ε = ε αβ, α, β = 1, (4) σ where ε αβ αβ ( α, β = 1, ) are the lanar comonents of the lastic strain-rate tensor (also ressed in the system of orthotroy axes), and ε is the equivalent lastic strain-rate. The last quantity is defined by the ower law σ σ ε = αβ ε αβ (5) Here and in the subsequent equations, the tensor summation rule is used (the Greek indices take the values 1 and, while the Latin ones take the values 1, and 3). The non-lanar comonents of the lastic strain-rate tensor are restricted by the lanestress condition for membranes and the isochoric character of the lastic deformation: ε α3 = ε 3α =, α = 1, (6) ε = ε 33 γγ.3 Hardening law Assuming a urely isotroic hardening of the sheet metal, only one scalar arameter is needed for describing the evolution of the yield surface. This is the so-called lastic equivalent strain comuted as the time-integral of the equivalent lastic strain-rate:

4 4 Validation of a new anisotroic yield criterion through bulge test Banabic ε T = ε dt (7) The evolution of the yield surface has been taken into account by means of a Swift hardening law /14/: ( ) ( ) n Y ε = K ε + ε (8) where Y is the yield arameter (see equation (1)), while K, and n are material arameters. In this formulation, Y is chosen to be the uniaxial yield stress associated to the rolling direction. ε.4 Identification rocedure The arameters a, M, N, P, Q and R in the ression of the equivalent stress are comuted in such a way that the constitutive equations associated to the yield surface reroduce as well as ossible the following characteristics of the sheet metal: yield stress obtained by a uniaxial tensile test along RD σ σ 9 σ 45 σ b r r 9 r 45 yield stress obtained by a uniaxial tensile test along TD yield stress obtained by a uniaxial tensile test along a direction equally inclined to RD and TD yield stress obtained by an equibiaxial tensile test along RD and TD coefficient of lastic anisotroy associated to RD coefficient of lastic anisotroy associated to TD coefficient of lastic anisotroy associated to a direction equally inclined to RD and TD. One may notice that seven conditions act on six material coefficients. Due to this overconstraint, the authors have adoted an identification rocedure based on the minimisation of the following error-function: f ( a, M, N, P,Q, R) σ = σ r r 1 1 σ + σ r + r σ + σ r + r σ + σ b b 1 + (9)

5 Banabic Validation of a new anisotroic yield criterion through bulge test 5 where σ, σ, σ, σ, and r are the uniaxial yield stresses, the equibiaxial yield 9 45 b r, r 9 45 stress and the coefficients of lastic anisotroy redicted by the constitutive equations. The identification rocedure needs formulas for evaluating these quantities..4.1 Prediction of the uniaxial yield stress Let o σ 9 σ σ 11 1 = σ = σ 1 o ( ) be the uniaxial yield stress associated to a direction inclined at an angle cos, = σ to RD. The corresonding non-zero comonents of the stress tensor are σ = σ sin cos sin, Equations (1) (3) and (1) lead to the following formula for the uniaxial yield stress: σ = Y F (11) where k [ ] k k k ( ) = a( Γ + Ψ ) + a( Γ Ψ ) + ( 1 a)( Ψ ) F (1) and Γ = M cos + Nsin, Ψ = ( Pcos Qsin ) + R sin cos 1 (1) (13).4. Prediction of the equibiaxial yield stress In case of an equibiaxial tension along RD and TD, the lanar comonents of the stress tensor are σ = σ = σ, σ = = (14) 11 b 1 σ 1 Equations (1) (3) and (14) lead to a formula for evaluating the equibiaxial yield stress: σb = Y (15) F b where F 1 k k k [ a( Γ ) ( ) ( )( ) ] k b + Ψb + a Γb Ψb + 1 a Ψb = (16) b and Γ = M + N, Ψ P Q (17) b b =.4.3 Prediction of the r-coefficient The r-coefficient associated to a direction inclined at an angle 9 to RD is defined as follows: o o

6 6 Validation of a new anisotroic yield criterion through bulge test Banabic r = where ε + 9 ε ND ε +9 o direction inclined at + 9 to RD, and ε ND is the comonent of the same tensor associated to ND. After using the condition of lastic incomressibility ε ε + ε ND = (19) (18) is the lanar comonent of the lastic strain-rate tensor associated to a Equation (18) becomes ε r = 1 () ε where ND ε is the comonent of the lastic strain-rate tensor associated to the direction inclined at the angle to RD. ε and ε ND may be rewritten using the tensor comonents ressed in the system of orthotroy axes: = ε cos + ε sin + ε sin cos + ε ε ε ND 11 = ε 33 = ε 11 ε 1 1 sin cos, Because the r-coefficient is also defined for a uniaxial stress state, equations (1) are valid. It is then ossible to write σ 11 σ σ 1 σ 1 cos =, sin =, sin cos = = () σ σ σ σ Equations () (), (1) (5), and (11) (13) lead to the desired formula for evaluating the coefficient of lastic anisotroy: k r = 1 (3) where G [ F( ) ] G( ) k 1 [ ] ( ) [ a( Γ + Ψ ) a( Γ Ψ ) k 1 k 1 ( ) = a( Γ + Ψ ) + a( Γ Ψ ) ( )( ) ] ( )( M + N + ) k 1 P Q Pcos Qsin 1 a Ψ Ψ k 1 + (1) (4) The identification rocedure uses equations (11) (13), (15) (17) and (3) (4) to comute and. Due to the comlexity of these formulas, the σ, σ9, σ45, σb, r, r 9 r45 authors have develoed a numerical minimisation strategy based on the downhill simlex method /15/.

7 Banabic Validation of a new anisotroic yield criterion through bulge test 7.5 Numerical results of the identification rocedure The identification rocedure has been tested for an alluminium alloy (AA33-, thickness 1 mm). The mechanical arameters used as inut data are listed in Table 1: σ σ 9 σ 45 σ b r r 9 r 45 Mechanical arameters AA33- (thickness 1 mm) 57 MPa 55.6 MPa 57.5 MPa 58.5 MPa Table. 1 Mechanical arameters of an aluminium alloy (AA33-) Table shows the coefficients rovided by the identification rocedure. The corresonding values of the arameters k and Y are also resented. Coefficients of the yield criterion AA33- (thickness 1 mm) A.6576 M.4676 N.4735 P.577 Q.53 R 1.46 k 4 Y 57 MPa Table. Coefficients of the yield criterion aluminium alloy (AA33-) These coefficients will be inut data for the numerical simulation of a hydraulic bulging test.

8 8 Validation of a new anisotroic yield criterion through bulge test Banabic.6 Mechanical model of a hydraulic bulging test The sheet metal subjected to hydraulic bulging is assumed to behave as an elastolastic membrane under lane-stress conditions. The elastic comonent of the strain is generally very small as comared to the lastic art, while the rotations of the material volumes are large. Due to the small amount of elastic deformation, the incomressibility condition is used when udating the thickness of the membrane. A linearly isotroic constitutive model describes the elastic behaviour of the sheet metal. On the other hand, its lastic behaviour is assumed to be orthotroic. The local axes of lastic orthotroy change continuously during the forming rocess (their current orientation is given by the rotational comonent of the deformation gradient tensor). The new yield criterion, its associated flow rule and the Swift hardening law resented in describe the lastic behaviour of the sheet metal. A global coordinate system is used to ress the current ositions of the articles belonging to the membrane, as well as the external loads (generated by ressure in case of a hydraulic bulging rocess). Its unit vectors (denoted by e 1, e, e 3 in Figure 1) are arallel to the initial axis of lastic orthotroy. Fig. 1 Vector bases used in the finite-element model

9 Banabic Validation of a new anisotroic yield criterion through bulge test 9 In order to simlify the maniulation of the lane-stress condition as well as the evolution of the lastic orthotroy, two orthonormal vector bases are attached to each article. They are denoted by ı 1, ı, ı 3 and î 1, î, î 3 in Figure 1 and change continuously during the simulation of the forming rocess. Both bases have two vectors tangent to the membrane surface (ı 1, ı and î 1, î, resectively). ı 1 is obtained by rojecting e 1 onto the tangent lane. In case that e 1 is almost erendicular to the surface of the membrane, ı 1 is calculated as the rojection of e 3. The unit vector ı 3 is always erendicular to the membrane, so its construction is straightforward. At last, ı results as the cross roduct of ı 3 and ı 1. The vectors î 1, î and î 3 are resectively coincident with the local axes of lastic orthotroy in the configuration under analysis. As mentioned above, the orientation of these axes is given by the rotational comonent of the deformation gradient tensor. The vectors î 1, î and î 3 can thus be obtained using the olar decomosition theorem /17/. The local bases are used to ress the tensor comonents restricted by the lane-stress assumtion, as well as to simlify the maniulation of the constitutive equations (see below). The finite-element model included in ABAQUS/Standard is based on the theorem of virtual work /17/: ext δε d Ω = W ( ui,δu σ αβ αβ i ) (5) Ω where Ω is the satial domain occuied by the sheet metal in the current configuration, u i i = 1,,3 are comonents of the dislacement vectors connecting the reference and ( ) current ositions of the articles, δu i ( i = 1,,3) field, ( α, β = 1, ) δε αβ σ αβ 1 δu = x α β δu + x are comonents of a virtual dislacement are non-zero comonents of the Cauchy stress tensor, ext is a virtual strain field, and ( u, u ) β α i i (6) W δ is the virtual work associated to the external loads. In case of a hydraulic bulging rocess, this work is done by the ressure acting on one face of the sheet metal: W ext ( u, δu ) = n δ i i A i u i da where A is the current area of the membrane surface loaded by the ressure, and n i i = 1,,3 are comonents of the unit vector erendicular to this surface (this vector ( ) has the same orientation as the external load). (7)

10 1 Validation of a new anisotroic yield criterion through bulge test Banabic The vector comonents in equation (7) are ressed in the global basis e 1, e, e 3. On the other hand, the tensor comonents in equation (5) are ressed in the local basis ı 1, ı, ı 3. The mechanical model defined by equations (5)-(7) contains both kinematical and material non-linearities. Its solution is obtained using a Newton-Rahson rocedure /17/. The linearization of equation (5) is constructed by retaining the first and second terms of its Taylor ansion in the vicinity of an aroximate current configuration /17/: e δu i u i ( δεαβ ) Cαβηλ ( εηλ ) dω ( δεαβ ) σαη( εηβ ) dω + σαβ dω = Ω Ω Ω x α xβ (8) ext ext W W ( ui, δui ) + ( ui ) ( δεαβ ) σαβ dω u where u i ε αβ ( i = 1,,3) 1 u = xβ α i + x Ω are corrections of the dislacement field comonents, u β α are the corresonding corrections of the strain comonents, and C e ( α, β, η, λ = 1,) comonents of the so-called elastolastic modulus consistent with the Newton-Rahson scheme. This last quantity defines a relationshi between the corrections of the stress and strain tensors /17/: σ αβ + σ αη e ( ) R ( R ) R σ = C ( ε ) ηλ βλ αη λη λβ αβηλ R αβ ( α, β = 1, ) are comonents of the rotation tensor and R ( α, β = 1, ) corrections. αβηλ ηλ (9) R (3) αβ are are their Due to the fact that all the constitutive equations discussed in contain tensors ressed in the system of lastic orthotroy axes, it is more convenient to start the evaluation of the elastolastic modulus in the co-rotational basis î 1, î, î 3. The transfer of e C α, β, η, λ = 1, to the general basis (ı 1, ı, ı 3 ) consists the co-rotational comonents ( ) αβηλ in erforming the following transformation /17/: C e αβηλ = R αω R βψ R η R λξ Ĉ e ωψξ The co-rotational comonents Ĉ e ( α, β, η, λ = 1,) T e {}[ ĝ Ĉ ]{} ĝ (31) αβηλ can be calculated using the matrix formula /17/ e e 1 e T e [ Ĉ ] = [ Ĉ ] ( [ Ĉ ]{} ĝ )({} ĝ [ Ĉ ]) + H (3)

11 Banabic Validation of a new anisotroic yield criterion through bulge test 11 where e [ Ĉ ] {} ĝ Ĉ = Ĉ Ĉ e 1111 e 11 e 111 Ĉ Ĉ Ĉ e 11 e e 1 Φ Φ =,, σ Φ σ 11 σ 1 e e ( Ĉ111 + Ĉ111 ) e e ( Ĉ 1 + Ĉ 1 ) ( ) e e Ĉ + Ĉ T (33) (34) dy H = dε e 1 e 1 e [ Ĉ ] = ([ U] + ( ε ε )[ C ][ M ]) [ C ] The equations (34) (36) need some lanations. The column-vector { ĝ} Φ (35) (36) defines the gradient of the yield surface described by equations (1) (3). It contains the first derivatives of the function with resect to the co-rotational comonents of the stress tensor in the current aroximate configuration. H is the strain-hardening modulus. It is the first derivative of the Swift hardening law with resect to the equivalent lastic strain in the current aroximate configuration (see equation (8)). 1 ε and ε are the equivalent lastic strains associated to the reference configuration and current aroximate configuration, resectively. [ U ] is the third-order unit matrix, while the other matrices used in equation (36) have the following structure: [ M ] Φ σ 11 = symm. Φ σ 11 σ Φ σ Φ σ 11 σ 1 Φ σ σ 1 Φ Φ + σ 1 σ 1 σ 1 (37) e [ ] E = 1 ν C 1 ν ν 1 1 ν [ M ] (38) is related to the curvature of the yield surface. Its

12 1 Validation of a new anisotroic yield criterion through bulge test Banabic elements are the second order derivatives of the function Φ with resect to the corotational comonents of the stress tensor in the current aroximate configuration. Finally, e [ C ] is the classical elasticity matrix for lane-stress roblems. Its elements deend on the Young s modulus E and the Poisson s ratio ν. The simulation of the hydroforming rocess is divided into small time increments. For each increment, a finite-element aroximation of equation (8) is solved numerically using a Newton-Rahson rocedure. After achieving the convergence, the nodal coordinates and the state arameters of the elements are udated. These arameters are the lastic equivalent strain, the thickness of the membrane and the comonents of the stress tensor associated to the integration oints. After udating, the current configuration of the finite-element model is taken as a reference status for the next increment. The simulation is stoed when the ressure acting on the sheet metal reaches its maximum value..7 Results and discutions The authors have imlemented the elastolastic material model in ABAQUS/Standard by means of a UMAT subroutine. This subroutine has been used for the simulation of a hydraulic bulging test. The forming rocess is schematically shown in Figure. The mechanical arameters of the sheet metal (AA33-; thickness 1 mm) are listed in Table 1. The circular secimen comes into contact with the claming ring in the region of the fillet. The frictional comonent of this interaction is described by the classical Coulomb model imlemented in ABAQUS/Standard (a friction coefficient µ =.15 is assigned to the contact surface). The ressure of the liquid has been used as a control arameter. Its value is considered to have a linear variation from to 4 MPa. Fig. Princile of the hydraulic bulging test

13 Banabic Validation of a new anisotroic yield criterion through bulge test 13 The elastic behaviour of the sheet metal is defined by the Young s modulus E = MPa and the Poisson s ratio ν =. 33. The lastic behaviour is defined by the coefficients of the yield criterion (see Table ) and the coefficients of the Swift hardening law: K = MPa, ε =. 56 and n =. 5 (see equation (8)). Due to the lastic orthotroy of the sheet metal, as well as the axial symmetry of the tools, only one quarter of the secimen needs to be meshed. The external boundary of the mesh corresonds to the claming circle (see Figure ). The radius of this circle is 58 mm. All the nodes belonging to the external contour are comletely restrained. The fillet radius of the claming ring is 5 mm (see Figure ). The mesh used in the simulation consists of 18 M3D3 elements and 54 M3D4 elements /17/, with a total number of 553 nodes. The erimental work has been erformed in the Institute of Metal Forming (IFU) of the University of Stuttgart. Three secimens have been subjected to the hydraulic bulging test. The olar deflection corresonding to different values of the liquid ressure has been recorded for each secimen. The results of these measurements are lotted in Figure 3 as discrete oints. The deendence ressure vs. olar deflection redicted by the finiteelement rogramme is suerimosed on the same diagram. One may notice a very good agreement between the numerical results and the erimental data. ABAQUS/Standard also comutes the satial distributions of the strain and stress comonents. A rectangular grid has been rinted on each secimen. The distribution of the rincial logarithmic strains at the end of the bulging test can thus be measured. Figures 4 and 5 show a comarison of the erimental and comuted values of the rincial logarithmic strains. As one may notice when examining the diagrams resented in Figures 4 and 5, the redictions of the elastolastic model are in good agreement with the data obtained from eriments. Little differences between the comuted and measured values of the rincial logarithmic strains aear in the vicinity of the claming edge. The frictional effects associated to the contact interactions between the secimen and the metallic ring are robably resonsible for this inaccuracy. Another cause of the discreancies may be the errors of the strain measurements in the claming region (due to the high curvature of the sheet metal secimen at the end of the forming rocess).

14 14 Validation of a new anisotroic yield criterion through bulge test Banabic Pressure [bar] Fig Polar deflection [mm] Pressure vs. olar deflection (comarison between numerical results and erimental data Exeriments Theory.3.5 Theory Exeriments. εmin Fig Initial radius [mm] Comuted and erimental distributions of the minor rincial logarithmic strain at the end of the bulging test

15 Banabic Validation of a new anisotroic yield criterion through bulge test Theory Exeriments.3.5 εmax Fig Initial radius [mm] Comuted and erimental distributions of the major rincial logarithmic strain at the end of the bulging test 3 Conclusions The hydraulic bulging test is articularly suitable for studying the mechanical resonse of sheet metals subjected to biaxial loads. The authors have used this test to evaluate the erformances of a new yield criterion. The equibiaxial yield stress of the sheet metal is one of the arameters used in the identification rocedure. Due to the close relation between this arameter and the mechanics of the hydraulic bulging rocess, the new yield criterion gives accurate redictions. The distributions of the rincial logarithmic strains resulted from comutations are in very good agreement with erimental data. The authors have noticed that the accuracy of the ressure vs. olar deflection curve is highly influenced by the hardening rule used in the model.

16 16 Validation of a new anisotroic yield criterion through bulge test Banabic References /1/ Hill, R. The Mathematical Theory of Plasticity Oxford: Clarendon Press, 195 // Hill, R. Theoretical lasticity of textured aggregates In: Math. Proc. Cambridge Phil. Soc. 85 (1979) 179 /3/ Hill, R. Constitutive modelling of orthotroic lasticity in sheet metals In: J. Mech. Phys. Solids 38 (199) 45 /4/ Hill, R. A user-friendly theory of orthotroic lasticity in sheet metals In: Int. J. Mech. Sci. 35 (1993) 19 /5/ Hosford, W. F. A generalized isotroic yield criterion In: J. Al. Mech. 39 (197) 67 /6/ Hosford, W. F. On yield loci of anisotroic cubic metals In: Proc. 7th North American Metalworking Conf.- Dearborn (1979) 191 /7/ Barlat, F. Lian, J. Plastic behavior and stretchability of sheet metals. Part I: Yield function for orthotroic sheets under lane stress conditions In: Int. J. Plasticity 5 (1989) 51 /8/ Barlat, F. Lege, D. J. A six-comonent yield function for anisotroic materials In: Int. J. Plasticity 7 (1991) 693 Brem, J. C. /9/ Barlat, F. Lege, D. J. Yield function develoment for aluminium alloy sheets In: J. Mech. Phys. Solids 45 (1997) 177 Matsui, K. Murtha, S. J. Hattori, S. Becker, R.C. Makosey, S. /1/ Barlat, F. Brem, J. C. Plane stress yield function for aluminium alloy sheets. Part I: Formulation

17 Banabic Validation of a new anisotroic yield criterion through bulge test 17 Yoon, J. W. In: Int. J. Plasticity (to be ublished) Chung, K. Dick, R. E. Lege, D. J. Pourboghrat, F. Choi, S. H. Chu, E. /11/ Karafillis, A. P. Boyce, M. C. A general anisotroic yield criterion using bounds and a transformation weighting tensor In: J. Mech. Phys. Solids 41 (1993) 1859 /1/ Cazacu, O. Barlat, F. Alication of the theory of reresentation to describe yielding of anisotroic aluminum alloys In: Int. J. Engng Sci. (to be ublished) /13/ Banabic, D. Bunge, H. J. Poehlandt, K. Formability of Metallic Materials. Plastic Anisotroy, Formability Testing, Forming Limits Berlin: Sringer Verlag, Tekkaya, A. E. /14/ Lemaître, J. Chaboche, J. L. Mécanique des matériaux solides Paris: Dunod, 1988 /15/ Press, W. H. Teukolsky, S. H. Numerical Recies in C. The Art of Scientific Comuting Cambridge: Cambridge University Press, 199 Vetterling, W. T. Flannery, B. P. /16/ Vial, C. Hosford, W. F. Yield loci of anisotroic sheet metals In: Int. J. Mech. Sci. 5 (1983) 899 Caddell, R. M. /17/ ABAQUS, Inc. ABAQUS Theory Manual. Version 6..3 Electronic documentation