Modeling and Simulation of Induced Anisotropic ... - Extras Springer

Modeling and simulation of induced anisotropic hardening and springback in sheet metals during non-proportional loading J. Wang, V. Levkovitch, F. Reusch, B. Svendsen Chair of Mechanics, University of Dortmund, D-44221 Dortmund, Germany Abstract. Sheet metal forming processes invariably involve non-proportional loading and changes in deformation/loading direction, resulting in a very complex material and structural behaviour. Much of this complexity can be traced back to the reaction of the material microstructure to such loading. In the case of hardening behaviour, for example, the interplay between the direction of inelastic deformation, the orientation of dislocation structures, and the current deformation/loading direction, plays an important role. The purpose of the current work is the formulation, numerical implementation and application of a thermodynamically-consistent material model for anisotropic hardening in such materials taking this interplay into account. Direction- and orientation-dependent quantities are represented in the model with the help of evolving structure tensors. These include the strength of dislocation structures, dislocation polarization, and kinematic hardening. Together with the initial texture-based inelastic flow anisotropy, these govern the evolving anisotropic response of the model to changes in loading direction. To exemplify this, the behaviour of the model subject to typical two-stage loading histories involving monotonic, reversing and orthogonal changes in loading/deformation direction are investigated for various parameter values. These simulations demonstrate clearly the effect of severe and abrupt changes in deformation/loading direction on the hardening behaviour and so on the resulting springback process.

INTRODUCTION As is well-known, the classical effect of springback in sheet metals after forming is sensitive to many parameters having to do both with the structure as well as with the material involved. In particular, on the material side, the effect of changes in the loading/deformation direction during metal forming processes on the direction of inelastic deformation and on the orientation of dislocation structures influencing the hardening behaviour can play a signficant role. Indeed, experimental studies have shown that severe non-proportional loading path changes, common in complicated multiple-stage forming processes, can induce strongly anisotropic material behaviour. In particular, [1] have shown that this results from the evolution of dislocation structures driven by changes in loading direction. Among the material models accouting for this, the incremental model of Teodosiu and Hu [2] has been used by a number of authors [e.g., 3, 4] to model induced anisotropic hardening behaviour and its effect on the springback process. In this approach, evolving structure tensors are used to model directional hardening effects resulting from a reorientation of persistent planar

dislocation structures upon changes in loading direction. One problem with the incremental form is the choice of back-rotated local configuration with respect to which the material symmetry directions are defined. In the current work, we show that this problem can be alleviated by working with a thermodynamically-consistent finite formulation of the model in which the material symmetry directions are defined with respect to the intermediate configuration. After comparing these two formulations with each other and with experimental observations, the model is briefly applied to model the effects of non-proportional loading on the anisotropic behaviour and on the springback behaviour of sheet metal.

INELASTIC MODEL The purpose of this section is to briefly summarize the inelastic part of the model under consideration in this work. As mentioned above, this is based on those of Teodosiu and Hu [2], and more recently Li et al. [3], which take into account the effect of changes in deformation/loading direction on the hardening behaviour. In particular, this

CP778 Volume A, Numisheet 2005, edited by L. M. Smith, F. Pourboghrat, J.-W. Yoon, and T. B. Stoughton © 2005 American Institute of Physics 0-7354-0265-5/05/$22.50

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is achieved with the help of a stress-like fourth-order symmetric tensor-valued internal variable S representing the effect of a persistent planar or sheet-like dislocation structures on the hardening behaviour. In addition, this behaviour is influenced by the effective polarity of excess dislocations building up at these structures, represented mathematically in the model via a second-order symmetric traceless tensor-valued internal variable P . To these are added “standard” kinematic and isotropic hardening mechanisms. The former is represented as usual by a second-order symmetric traceless tensor-valued internal variable X , i.e., the back stress, and the latter by a scalarvalue internal variable r. Among other reasons, the evolution of all of the tensor-valued internal variables here is defined with respect to a local configuration rotating relative to the current configuration. In effect, this local configuration is tacitly assumed in [2] as that in which the material symmetry directions for inelastic flow, as embodied in the fourth-order symmetric traceless tensor A , are constant. In other words, Q represents the rotation of these directions with respect to the current configuration. On this basis, consider first the yield function φ = σHill (M − X , A ) − σY0 − r − m |S | in the current context. Here, q σHill (M − X , A ) := (M − X ) · A [M − X ]

(1)

(2)

represents the Hill equivalent stress, M = Q K Q the symmetric, traceless back-rotated Kirchhoff stress K , and X the symmetric, traceless back-stress. Further, m (1 − m) represents a material parameter determining the fraction of |S | contributing to isotropic (kinematic) hardening processes via the effective yield stress. Now, in the current associated context, φ determines in particular the inelastic deformation rate

A [M − X ] σHill

.

(3)

Since σHill is positive homogeneous of order one in M − X , this last relation implies the form (M − X ) · DP = σHill ²˙

(4)

for the inelastic dissipation rate. Consider next the the saturation form P˙ = cp {sgn(DP ) − P } ²˙

(5)

for the evolution of effective dislocation polarity as determined by the saturation rate cp and the direction sgn(DP ) :=

DP |DP |

for the back stress, where cx represents its saturation rate, −1 and xsat A [DP ] its saturation value, with q (8) xsat = x0 + (1 − m) |Sd |2 + f |Sl |2 as based on the split

S = Sd + Sl = sd sgn(DP ) ⊗ sgn(DP ) + Sl

(6)

235

(9)

of S into dynamic Sd and latent Sl parts. In particular, the former contribution is due to directional hardening on currently active slip systems, while the latter represents the cumulative effect of changes in loading direction on the current directional hardening behaviour, i.e., due to dislocation structures oriented in directions other than sgn(DP ). Indeed, for simpicity, Teodosiu and Hu [2] posit the split (9) to be orthogonal, such that sd = sgn(DP ) · S [sgn(DP )] holds. In this context, the evolution relations ¾n ½ |Sl | l Sl ²˙ , S˙ l = −cl ssat (10) s˙d

T

DP = ²˙ (∂M −X φ) = ²˙

of inelastic flow. Also of this form is the evolution relation ˙ = c {x A−1 [D ] − ²˙ X } X (7) x sat P

=

cd [hp (ssat − sd ) − hx sd ] ²˙ ,

for Sl and sd , respectively, apply. Here, cl and cd represent the corresponding saturation rates, and ssat the saturation value of sd . The exponent nl determines the influence of prestraining on the evolution of Sl . Further, the material functions hp and hx govern the contribution to active or dynamic directional hardening from dislocation polarity and kinematic hardening, respectively. In particular, the magnitude of these contributions depends on the projections dp := sgn(DP ) · P and dx := sgn(DP ) · X of P and X , respectively, onto the current inelastic flow direction. We have ¯ ¯  ¯ cp ¯ sd  ¯ ¯  if dp ≥ 0 1 − − d  p¯ ¯  cd + cp ssat hp = µ ¶  cp sd  np   (1 + dp ) 1− if dp < 0 cd + cp ssat (11) and µ ¶ dx 1 1− , (12) hx = 2 xsat N · X with

A [DP ] M −X = −1 (M − X ) · DP DP · A [DP ] −1

N :=

(13)

following from the fact that A is symmetric and positivedefinite, such that (3) can be inverted.

Lastly, the evolution of r is described by the Voce-type form r˙ = cr (rsat − r)²˙ (14) in terms of the accumulated equivalent inelastic deformation ². Here, rsat represents the saturation value of r (i.e., at ² = ∞), and cr the saturation rate. In addition, the evolution of the accumulated equivalent inelastic deformation ² is determined in the rate-independent context by the plastic multiplier γ , i.e., ²˙ = γ , enforcing the yield condition φ ≤ 0, and in the rate-dependent context via an evolution relation. For simplicity, we restrict attention in this work to the former case. In summary, then, the above relations contain a total of 19 material parameters. These include the six Hill anisotropy parameters F , G, H, L, M , and N determinining A , the saturation parameters cp , cx , cd , cl , and cr , as well as the remaining hardening parameters σY0 , m, x0 , f , rsat , ssat , nl and np .

CONSTITUTIVE ROTATION AND STRESS MODEL

(15)

via the corresponding spin Ω . As it turns out, the choice of rotation here is related to the choice of model for the stress. In the case of a hyperelastic-based formulation, for example, the material model, and in particular the material symmetry, is defined with respect to the intermediate configuration in the context of the elastoplastic decompositions F

=

FE FP ,

L

=

LE + FE LP FE−1 ,

(16)

(17)

gradients, respectively. To delve into this a bit, consider the polar decompositions

LE

RE ,

DE

≈

˙ RT , RE lnU E E

≈

R˙ E R .

WE

=

RE UE ,

=

R˙ E R + RE U˙ E UE R , −1

T E

(18)

of the local elastic deformation FE and corresponding rate LE . In particular, in the case of small elastic strain,

˙ lnU E

=

RT E DRE − DP ,

R˙ E

=

W RE − RE WP ,

(20)

for lnUE and RE , respectively, the latter via the skewsymmetric part of (16)2 together with (19)1 . This is of course contingent on having a constitutive relation for the plastic spin WP . Consider next the case of isotropic (hyper)elasticity for small elastic strain with respect to the intermediate configuration. We have (21)

for the (in this context) Mandel stress M as a function of the elastic right logarithmic stretch lnUE , with λ and µ the usual elasticity constants. Taking the time derivative of this, one obtains the corresponding incremental form ˙ = λ (I · D) I + 2 µ [RT DR − D ] M E E P

(22)

via the connection (20)1 and the fact that DP is traceless. In this case, we clearly have the identification Q ≡ RE , i.e., the material symmetry is formulated with respect to the intermediate configuration. Then Ω ≡ W − RE WP RT E

(23)

follows for Ω from (15). This case of small elastic strain can be compared with the rigid-plastic case, i.e., UE = I and U˙ E = 0. Here, the results FE = RE , (24)

=

0,

=

R˙ E R , T E

follow from (19). In turn, these imply FP

=

RT EF ,

DP

=

RT E DRE ,

(25)

and again (20)2 , via (16). Since F , D and W are known, FP and DP are both determined in this case once (20)2 is solved for RE . Again, this is contingent on having a constitutive model for the plastic spin WP . In addition, one then has the constitutive form M = X + σY ²˙ N

236

(19)

T E

In particular, the last two of these results imply the evolution relations

WE

L := F˙ F −1 = D + W = sym(L) + skw(L)

T E

≈

DE

of the deformation F and velocity

FE

FE

M = λ (I · lnUE ) I + 2 µ lnUE

As mentioned at the beginning of the last section, the evolution of the tensor-valued internal variables is formulated with respect to the rotation Q, itself a constitutive quantity determined by the evolution relation ˙ = ΩQ Q

i.e., UE ≈ I + lnUE , and these reduce to

(26)

for M = RT E K RE at yield φ = 0 via (13). For example, Teodosiu and Hu [2] assumed for simplicity that the elasticity and rate-dependence of the material behaviour can be neglected and worked with a rigid-plastic formulation. In this context, they worked with the Jaumann spin Ω ≡ W , where W = skw(L) represents the continuum spin, i.e., the skew-symmetric part of the velocity gradient L := F˙ F −1 . This was also done in the more general elastoplastic context by Li et al. [3]. From (23), we see that this is consistent with Q ≡ RE only when the plastic spin WP vanishes identically. On the other hand, as pointed out by Dafalias [5], the rotation of the material symmetry directions of the microstructure with respect to the current configuration as embodied in a non-zero value of WP is in general significant and should be included in any complete model. This represents work in progress and will be reported on at the meeting.

MODEL VERIFICATION

FIGURE 1. Experimental results for TRIP600 steel subjected to a Bauschinger strain path change (Bouvier et al. [4])

of the second stage. This is attributed to the deformation localization in the so-called microbands. In addition, the experiments also show that the softening effect becomes stronger with the increasing amount of pre-strain. After the stagnation or softening, further hardening takes place due to formation of new dislocation walls.

Experimental observations In recent years, a lot of microstructural studies were carried out in order to understand the macroscopic material behaviour of polycrystalline metals under strain-path changes at large deformations. The results of these observations can be summarized as follows (e.g., [4]). 1. Monotonic strain path. The hardening mechanisms are the same during the pre-loading and the subsequent loading. From the microstructural point of view, this type of strain path change consists in a continuous reinforcement of the dislocation wall structure. The stress-strain curves after re-loading are identical with these of a monotonic loading. 2. Bauschinger strain path change (load reversal). Figure 1 (Bouvier et al. [4])1 shows the material response in the case of a load reversal. Besides the lower yield stress usually observed after reloading (Bauschinger effect), there exists a stage of hardening stagnation, which is assumed to be associated with the annihilation of the dislocation walls created during pre-loading. 3. Orthogonal strain path change. Figure 2 (Bouvier et al. [4]) shows an increase in the yield stress after the strain path change. This behavior can be explained by the assumption that the preformed dislocation structures act as an obstacles to slip on the newly activated slip systems. It is important to note that the work-softening is observed at the beginning 1

Figure 1 and Figure 2 reprinted with permission from Elsevier

237

FIGURE 2. Experimental results for FeP06 steel subjected to an orthogonal strain path change (Bouvier et al. [4])

One element verification Models based on both the Jaumann (WP = 0) and Green-Naghdi (RP = I) rates have been implemented into the finite-element program ABAQUS via the user material interface UMAT. Figure 3 compares the stressstrain responses obtained using Jaumann and GreenNaghdi rates respectively in the case of simple shear up to 400% using Hill flow anisotropy model (e.g., material symmetry is cubic). In order to concentrate on the rate effect on the stress-strain response, the ideal plasticity was used. As shown for deformations under 100%, there is no significant differences between these two rates, but as the deformation increases, Jaumann case begins to oscillate while Green-Naghdi case decreases continuously. It should be pointed out that if an isotropic yield function (e.g., von Mises) is used, both Jaumann and GreenNaghdi formulations yield almost the same stress responses in the case of saturating hardening irrespective of the amount of shear.

Mises stress SD |SL| |S|

Mises stress |S| SD SL

Jaumann Green-Naghdi

FIGURE 4. Bauschinger strain path change (left:AKQD mild steel, right: IF steel) FIGURE 3. Comparison of Jaumann and Green-Naghdi in the case of simple shear

To demonstrate the capability of the model we simulate the two-stage tests described in the previous section using the Jaumann rate formulation. We work with the material parameter values from [2] for AKQD mild steel at room temperature. The initial Hill flow anisotropy parameters are chosen in such a way that the yield function becomes the von Mises one (F = G = H = 0.5, L = M = N = 1.5). The elastic constants are: λ = 121153.8 MPa and µ = 80769.2 MPa. In addition, the hardening behaviour is determined by the values cp = 2.7, cx = 50.0, cd = 2.7, cl = 2.7, and cr = 0.0 for the saturation parameters. In addition, we have m = 0.59, f = 2.4, nl = 3.0 and np = 0.2 for the remaining dimensionless hardening parameters. Lastly, σY0 = 51.0 MPa, x0 = 64.0 MPa, rsat = 0.0 MPa, ssat = 145.0 MPa were assumed for the dimensional hardening parameters. For the monotonic strain path changes, the stress response before unloading and after reloading remains the same. Since no strain path changes take place (direction of deformation rate remains the same between the previous deformation and the subsequent deformation), S l retains its initial value 0. In the case of reversed loading Sl does not evolve either. According to Teodosiu and Hu (1998), the evolution of Sl is attributed to the development of microbands due to the interaction between newly activated slip systems and the previously developed microstructrues. In the reversed loading no microbands evolve. In the Bauschinger case, the back stress X plays the key role so that the yield flow stress is lower than that before unloading (Figure 4 left). In order to demonstrate the influence of the corresponding parameters, we used another set of hardening parameters mainly based on the values for IF steel from [3]: cp = 1.17, cd = 1.9, cl = 4.77, cr = 9.55, m = 0.6, f = 3.93, nl = 1.88, np = 27.1, σY0 = 40.73 MPa, rsat = 27.79 MPa and ssat = 83.02 MPa. Furthermore, we assumed cx = 38.86, x0 = 24.86 MPa. Figure 4 right shows that the model is capable to reproduce the strong hardening stagnation which is observed in experiments. Finally, in the case of orthogonal strain path change, it is of interest to inspect the evolution variable S describ-

238

ing the directional strength of dislocation structure. As shown in the Figures 5 and 6, the value of the norm of Sl remains zero during the first deformation stage and gains a value only after the strain-path change. In contrast, the value of sd increases continuously during the first stage and drops to zero at the beginning of the subsequent strain path. At the beginning of the second stage, Sl decreases more rapidly than sd increases, thus, leading to a short softening area and a subsequent hardening. The softening effect during the second stage is stronger when the pre-straining increases, which is confirmed by experiments. We also can see how different values of corresponding parameters influences the softening effects (compare Figures 5 and 6). Mises: prestrain 59% Mises: prestrain 34% |S| SD |SL|

Effective plastic strain

FIGURE 5. steel)

Orthogonal strain path change (AKQD mild

Mises: 33% prestrain Mises: 69% prestrain |S| SD |SL|

Effective plastic strain

FIGURE 6.

Orthogonal strain path change (IF steel)

SIMULATION EXAMPLES The ultimate goal of the present work is the application of models such as that described above to the mod-

eling and simulation of metal forming processes. As a first example of this, consider the case of deep drawing. The blank has an initially square shape (200 mm by 200 mm) and is 0.82 mm thick. The die is modelled as a rigid surface with a square hole of 102.5 mm by 102.5 mm rounded at the edges by a radius of 10 mm. The rigid square punch measures 100 mm by 100 mm and is rounded at the edges by the same 10 mm radius. A mass of 0.6396 kg is attached to the blank holder, and a concentrated load of 2.287×104 N is applied to the reference node of the blank holder. The friction coefficient between the sheet and the punch is taken to be 0.25, and that between the sheet and the die 0.125. It is assumed that there is no friction between the blank and the blank holder. For the finite-element simulation, one-quarter of the structure was discretized using threedimensional brick elements (C3D8) for the blank, and three-dimensional rigid surface elements (R3D4) for the die, the punch, and the blank holder. Since the main purpose of this study was a qualitative comparison of different material models, only one layer of elements over the thickness was used, although this may have affected the accuracy of the results. Figure 7 shows the springback effect in terms of the displacement U 3 in z direction obtained using three different models: (i), isotropic von Mises yielding and non-linear isotropic-kinematic hardening, (ii), isotropic von Mises yielding and Teodosiu-Hu anisotropic hardening, and (iii), orthotropic Hill yielding and non-linear isotropic-kinematic hardening. To this end, the initial Hill flow anisotropy parameters are chosen as F = 0.414, G = 0.558, H = 0.442, L = M = 1.5, and N = 1.52. In all three simulations, the initial yield stress σY0 is set to 91.3 MPa, rsat to 220 MPa, and x0 to 230 MPa. In addition, the Teodosiu-Hu anisotropic hardening parameters are those for IF steel as specified in previous sections. The Jaumann corotational rate was chosen for all model combinations. The springback results are obtained in two steps: 1) removing the punch while fixing the holder and die; 2) fixing the center of the sheet and removing the holder and die.

Consider now the results shown in Figure 7. Particularly, regarding the springback in the flange area, Teodosiu-Hu’s model (ii) shows less springback than the model(i). An explanation for this could be that the model (ii) takes into account the strain path change and hence triggers extra hardening effects. In addition, the model (iii) shows that the initial texture plays an important role too as it yields a non-symmetric response of the structure and the limits of the springback being larger than in the case of the first two models. Although these differences basically agree with the tendency observed in other forming processes (Bouvier et al.[4]), more detail work will be carried out to investigate the influences of each specific hardening parameter on springback behaviour in future.

CONCLUSIONS The material model with evolving structure tensors associated with microstructural intragranular mechanisms describes macroscopic behaviour better than classical phenomenological models. The verifications show that the implementation of the model is qualitatively reasonable. Initial texture anisotropy plays a key role during sheet forming process, while induced work-hardening represented by evolving structure tensors is also important once abrupt strain path changes take place. It should be pointed out that future work will be carried out to take into account the effect of plastic spin describing rotational hardening effects in the material and the texture development.

REFERENCES 1. 2.

3. Mises+combined hardening

Mises+anisotropic hardening

4.

5. Hill+combined hardening

FIGURE 7.

6.

Springback in deep drawing

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