Assessment of the Modified Maximum Force Criterion

Key Engineering Materials Vols. 410-411 (2009) pp 511-520 online at http://www.scientific.net © (2009) Trans Tech Publications, Switzerland Online available since 2009/Mar/02

Assessment of the Modified Maximum Force Criterion for Aluminum Metallic Sheets Dorel Banabic1, a, Stefan Soare1,b 1

Technical University of Cluj-Napoca, C. Daicoviciu 15, 400020 Cluj-Napoca, Romania a

b

[email protected], [email protected]

Keywords: sheet metal, normal anisotropy, yield function, limit strains

Abstract. The paper presents an analysis of a recently proposed failure criterion for thin sheets. According to Aretz [1], this criterion becomes numerically unstable for yield surfaces with locally constant exterior normal fields. Here we make more precise statements about the nature of this instability, asses the predictive capabilities of the criterion, and introduce a fitting parameter for its plane strain calibration. Introduction To date, one of the most accurate models for predicting the failure strains during biaxial stretching of thin sheets is the M-K model, [2]. However, it is well known that this accuracy requires a considerable number of arithmetic operations which number could multiply thousand of times if the M-K model is used in finite element codes to guard against the failure strains, leading to a significant increase of the simulation time. Simpler failure models, with minimum arithmetic, would then gain priority for usage in finite element codes with industrial applications. Unfortunately, such models are scarce and their predictions are in general at best orientational (or, qualitative). In this direction we mention the now classical models of Swift, [3], and Hill, [4], which are the departure point for most of the past and current research on analytical models of sheet failure. Swift and Hill based their analyses on the maximum force principle: the sheet fails when the applied force begins to decrease, that is, at failure the force reaches its maximum. When coupled with other assumptions upon the deformation process, this principle can lead to an analytic expression for the particular case of proportional loading. However, these "other assumptions" result in models that either underestimate the failure strains, like Swift's model, or even hamper the abilities of the model to correctly predict the failure strains for positive strain ratios, Hill's model. Recently Hora and his coworkers, [5], have proposed a phenomenological modification of the Swift-Hill approach. Their failure criterion is capable of prediction for the entire range of strain ratios. In this section we try to asses the accuracy of the model, investigate the point raised by Aretz, [1], regarding the applicability of the criterion, and propose possible improvements. The MMFC Criterion Consider a plane thin metallic sheet subjected to in-plane biaxial stretching. The sheet is assumed to have orthotropic symmetry (including the cases of in-plane isotropy). The Cartesian coordinate system will then be aligned with the symmetry axes. That is, the x-axis is along the rolling direction, the y-axis is along the transverse direction, and the z-axis is normal to the plane of the sheet. The loading path is such that the strain increments (and thus the strains) are parallel with the symmetry axes and proportional:

β := d ε 22 / d ε11

(1)

The modified maximum force criterion (for short, MMFC) proposed in [5] assumes that at failure the following relationship holds: All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of the publisher: Trans Tech Publications Ltd, Switzerland, www.ttp.net. (ID: 92.82.62.166-07/07/09,14:02:48)

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∂σ 11 ∂σ 11 ∂β + ⋅ = σ 11 ∂ε 11 ∂β ∂ε11

(2)

We assume the plastic deformation of the sheet modeled with the classical (associative) plasticity theory. The yield function will be denoted with f. Due to the material symmetry and the particular loading path, the shear stresses are zero. Then, from the normality rule, and first order homogeneity of f (thus zeroth order homogeneity of its first order partial derivatives) it follows that

 ∂f



β = (σ 11 , σ 22 )   ∂σ 22 

 ∂f   ∂f   ∂σ (σ 11 , σ 22 )  =  ∂σ (1, Ω)   11   22 

 ∂f   ∂σ (1, Ω)   11 

(3)

where we have defined Ω := σ 22 σ 11

(4)

If the yield surface (note that only the biaxial yield curve is actually needed) is continuous then for each β ranging from uniaxial to equi-biaxial tension there exists an Ω ∈ [0,1] satisfying the above equation. We further assume that there is a one-to-one relationship β = β (Ω) and that this correspondence is continuously differentiable. We note here that this one-to-one relationship is not satisfied by yield surfaces for which their restriction to the (σ 11 , σ 22 ) plane has straight portions, parallel with the σ 22 stress axis. One such example is Tresca’s yield criterion. However, if the biaxial yield curve is strictly convex, the one-to-one chracter of the correspondence β = β (Ω) is assured. Neglecting the elastic strains, the material deforms plastically during its entire loading history and therefore we have

()

()

()

σ = f (σ 11 , σ 22 ) = H ε ⇔ σ 11 f (1, Ω) = H ε ⇒ σ 11 = F (Ω) H ε

(5)

where we have denoted F (Ω) := 1/ f (1, Ω) , and with H the hardening law of the material. The hardening parameter ε is defined by the work equivalence principle:

σ : d ε = σ ⋅ d ε ⇔ (σ 11 + βσ 22 )d ε11 = σ ⋅ d ε ⇔ σ 11 (1 + βΩ) = σ 11 f (1, Ω)d ε

(6)

from where we obtain

d ε11 d ε = f (1, Ω) (1 + β Ω)

(7)

For a given strain ratio β the right-hand side of the above equation is constant. Then taking into account that both ε 11 and ε start with zero initial values we obtain

ε11 1 = =: g (Ω) ⇔ ε11 = g (Ω)ε ε (1 + βΩ) F (Ω)

(8)

The partial derivatives involved in the MMFC criterion (2) are computed as follows. Using eqs (5) and (7),

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()

F ( Ω) H ' ε ∂σ 11 dε ∂σ = F ( Ω) H ' ε ⇒ 11 = ∂ε 11 d ε11 ∂ε11 g ( Ω)

()

(9)

()

F '(Ω) H ε ∂σ 11 dΩ = F '(Ω) (β )H ε = dβ ∂β dβ (Ω ) dΩ

()

513

(10)

and using eqs. (1) and (8)

β=

ε 22 ∂β −ε −β −β ⇒ = 222 = = ∂ε11 ε11 ε11 ε11 g (Ω)ε

(11)

Substituting eqs. (9), (10) and (11) back into eq. (2) we obtain

 F '(Ω) β (Ω)  FH ' F ' H (− β ) F (Ω ) + = FH ⇔ H ' ε − + F (Ω )  H ε = 0 β ' gε g g ( Ω)  β '(Ω) g (Ω)ε 

()

()

(12)

Finally,

 F '(Ω) β (Ω)  H ' ε −  g (Ω ) + H ε =0 F (Ω) β '(Ω)ε  

()

()

(13)

Since the correspondence β = β (Ω) is one-to-one, instead of the strain ratio β we can take the stress ratio Ω as input. Then, for a given Ω , equation (13) is to be solved for the accumulated equivalent plastic strain ε at the moment of failure. Then the strains at failure follow from

ε11 = g (Ω)ε , ε 22 = βε11

(14)

In-Plane Isotropic Yield Function We shall use the following in-plane isotropic sixth order polynomial yield function 6

5 σ = a1 (σ 116 + σ 226 ) + a2 (σ 115 σ 22 + σ 11σ 22 ) + a3 (σ 114 σ 222 + σ 112 σ 225 ) + a4σ 113 σ 223 =: Q

(15)

It is shown in [6] that the above function can model a wide variety of shapes for yield surfaces and therefore it will be a good testing function for the MMFC criterion. The coefficients of the criterion are identified through the formulas, [6]: 6

(

)

a1 = 1 σ 0 , a2 = −6a1 R 1 + R ,

{

(16)

a3 = P 6 − a1 (1 + t 6 ) − a2t (1 + t 4 ) − t 3 σ b 

( −6)

}

− 2(a1 + a2 )  

t 2 (1 − t ) 2  ,  

(17)

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{

}

( −6) a4 = t 2 (1 + t 2 ) σ b − 2(a1 + a2 )  − 2  P 6 − a1 (1 + t 6 ) − a2t (1 + t 4 )   

t 2 (1 − t ) 2  ,  

(18)

where we have denoted

(

)

R := ( r0 + r45 + r90 ) 4, t := σ b σ b + σ 0 ,

(19)

and σ 0 , σ b are the normalized yield stresses for the uniaxial tensile test along the rolling direction (RD) and the balanced-biaxial stretch along the RD and TD; rθ is the r-value for a sample cut from the sheet at an angle θ from the RD and tested in uniaxial tension. The parameter P is a shape parameter for the yield surface (here, for the biaxial yield curve). It can be shown that, for any given input data set σ 0 , σ b , R  there exist lower and upper bounds M 1 and M 2 such that for

M 1 ≤ P ≤ M 2 the yield surface is convex. The algorithm for the numerical

computation of the convexity interval

[ M1 , M 2 ]

is given in [6]. The important thing to be

mentioned here is that P has a similar role with the variable exponent of Hosford's criterion, [7]. That is, with values of P from M 1 to M 2 the shape of the biaxial yield curve varies from the oval shape of an ellipse (resembling Hill's quadratic) to a hexagon with rounded corners (a smoothing of Tresca's hexagon). This will be illustrated in the next section. For an Ω fixed within the interval [0,1] we have

F (Ω) =  a1 (1 + Ω6 ) + a2 ( Ω + Ω5 ) + a3 ( Ω 2 + Ω 4 ) + a4 Ω3  F '(Ω) =

( −1/6)

= Q ( −1/6) ,

− F dQ dQ , = 6a1Ω5 + a2 (1 + 5Ω 4 ) + 2a3 ( Ω + 2Ω3 ) + 3a4 Ω 2 , 6Q d Ω d Ω

5 4 3 2 ∂f ∂σ 22 6a1Ω + a2 (1 + 5Ω ) + 2a3 ( Ω + 2Ω ) + 3a4 Ω Q β= = =: 1 5 2 4 3 ∂f ∂σ 11 6a1 + a2 ( 5Ω + Ω ) + 2a3 ( 2Ω + Ω ) + 3a4 Ω Q2

β '(Ω) = ( Q1 '− β Q2 ') Q2 ,

(20)

(21)

(22)

(23)

Q1 ' = 30a1Ω 4 + 20a2 Ω3 + 2a3 (1 + 6Ω 2 ) + 6a4 Ω,

(24)

Q2 ' = 5a2 (1 + Ω 4 ) + 8a3Ω (1 + Ω 2 ) + 9a4 Ω 2 .

(25)

The left hand side member of equation (13) is a strictly decreasing convex function. A Newton algorithm for finding its unique zero is then guaranteed to work. As initial guess we used ε = 0.01 (since here we are dealing with metals, the equivalent plastic strain at failure is always greater than 0.01). With this initial guess the number of iterations until the values of the function become less than 10( −6) , the stopping criterion (combined with a check of near zero variation of the derivative), is about six or seven.

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Applications to Two Aluminum Alloys The first application is to the prediction of the FLD of AA3104-H19. This alloy is described in [8]. There the hardening law was provided in the form

(

σ = 390.4 0.0036 + ε

)

0.07

MPa

(26)

Fig. 1. Influence of the shape parameter P upon the biaxial yield curves in the case of AA3104-H19. The experimental data needed for the identification of the Poly6 criterion is listed in Table 1. The convexity interval for this alloy is found to be 0.8386 ≤ P ≤ 0.9486 . The influence of P upon the shape of the biaxial yield curve is shown in Figure 1. The projection of the entire yield surface on the biaxial plane for two values of P is shown in Figure 2. Predictions of the FLD for several values of the P parameter are shown in Figures 3 and 4. The shape of the yield surface clearly has a strong influence upon the predicted FLD using the MMFC. This is also true for the M-K model. However, the point raised by Aretz in [1] is also confirmed to be true. In Figure 4 it is to be noticed that in the extreme case of a hexagonal shape of the yield surface the predicted FLD has a sharp oscillation localized near the plane strain point. The nature of this phenomenon is numerical, as pointed out in the cited paper, and is caused by the zero value of β ' near Ω = 0.5 , see Fig. 5.

Fig. 2. In-plane isotropic Poly6 yield surfaces of AA3104-H19 for two values of the shape parameter P.

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Table 1. Experimental input data for Poly6 description of AA3104-H19 and AA3103-O

σ0

σb

r0

r45

r90

AA3104-H19

1.0000

0.9609

0.4300

0.8400

1.2600

AA3103-O

1.0000

1.0909

0.6390

0.5130

0.6050

Fig. 3. MMFC criterion: influence of the shape of the yield surface upon the predicted FLD for AA3104-H19.

Fig. 4. MMFC criterion: FLD prediction for AA3104-H19 when the biaxial yield curve takes the extreme hexagonal shape (the shape parameter P takes the upper bound value).

Fig. 5. MMFC criterion: graphs of the β = β (Ω) function for several values of the shape parameter P.

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The second application concerns AA3103-O for which experimental input data for material parameters is listed in Table 1. The hardening curve of this alloy is characterized in the form

(

σ = 188.0 0.044 + ε

)

0.226

MPa

(27)

The convexity interval for the in-plane isotropic Poly6 modeling of this alloy is 0.8365 ≤ P ≤ 0.8987 . Several biaxial yield curves for this alloy are shown in Figure 6, whereas Figure 7 shows two instances of the yield surface. The predicted FLD's are shown in Figures 8 and 9. The singularity is also present for the shape parameter associated with the hexagonal shape of the yield surface, only this time on the left hand side of the FLD, see also Figure 10.

Fig. 6. Influence of the shape parameter P upon the biaxial yield curves in the case of AA3103-O.

Fig. 7. In-plane isotropic Poly6 yield surfaces of AA3104-H19 for two values of the shape parameter P.

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Fig. 8. MMFC criterion: influence of the shape of the yield surface upon the predicted FLD for AA3103-O.

Fig. 9. MMFC criterion: FLD prediction for AA3103-O when the biaxial yield curve takes the extreme hexagonal shape (the shape parameter P takes the upper bound value).

Fig. 10. MMFC criterion: graphs of the β = β (Ω) function for several values of the shape parameter P. Discussion In this subsection the predictive capabilities of the MMFC proposed by Hora and his coworkers, [5], have been tested on a new class of in-plane isotropic criteria, [6]. The following conclusions can be drawn. The shape of the biaxial yield curve has a great influence upon the MMFC predicted FLC. This is also true for the M-K model, and it is a feature that should be possessed by any good failure criterion, if the criterion is to be applied to different crystal classes.

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The MMFC can "predict" a localized pulse on the FLC when the shape of the yield surface resembles a hexagon. This is a numerical issue of the MMFC and it could be solved by washing out the pulse with values on the FLC situated in the immediate neighborhood of the pulse. However, we should stress here that this numerical issue is intimately connected with the one-to-one character of the β = β (Ω) correspondence. The pulse on the predicted FLC appears on those loading paths that pass through locations in the biaxial plane where the biaxial yield curve has almost straight edges. Along these edges the strain ratio is nearly constant and so the function β = β (Ω) is difficult to invert, at least locally, and, even if possible numerically, the derivative Ω '( β ) would be infinite (a very large number, in practice). In the extreme case of Tresca’s criterion, the function β = β (Ω) is not invertible. Thus, a better alternative to just washing out the pulse on the FLC is to adjust properly the shape of the biaxial yield surface so that it is strictly convex. Finally, for both aluminum alloys studied here the MMFC predictions did not provide an overall agreement with the experimental data, although the topology of the predicted FLC follows the experimental trend. In particular, it failed to predict properly the plane strain point of the FLC. From Figures 3, 4, 8 and 9 it is clear that an overall good agreement with the experimental data can be achieved by simply translating the MMFC prediction with a proper amount of strain along the (vertical) major strain axis. This points to the necessity of introducing a fitting parameter into the MMFC, similarly with the size of the in initial imperfection of the M-K model. A first step in this direction would be to consider the following slight generalization of the MMFC:

 ∂σ 11 ∂σ 11 ∂β F '(Ω) β (Ω)  + ⋅ = Λσ 11 ⇒ H ' ε −  Λg (Ω) + H ε =0 ∂ε 11 ∂β ∂ε11 F (Ω) β '(Ω)ε  

()

()

(28)

The parameter Λ can be adjusted so that the MMFC prediction of the plane strain limit strain best fits the experimental input. Figures 11 and 12 show improvements in the overall agreement between predictions and data, when the parameterized MMFC is used.

Fig. 11. MMFC criterion with an extra parameter Λ for fitting the plane strain point. FLD predictions for AA3104-H19.

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Fig. 12. MMFC criterion with an extra parameter Λ for fitting the plane strain point. FLD predictions for AA3103-O. References [1] H. Aretz: Modelling Simul. Mater. Sci. Eng. Vol. 12 (2004), p. 667. [2] Z. Marciniak, K.Kuczynski: Int. J. Mech. Sci. Vol. 9 (1967), p. 609. [3] H.W. Swift: J. Mech. Phys. Solids Vol. 1 (1952), p. 1. [4] R. Hill: J. Mech. Phys. Solids Vol.1 (1952), p. 19. [5] P. Hora, L. Tong, J. Reissner: Proceedings of the Numisheet 1996 Conference, Detroit, p. 252. [6] S. Soare, D. Banabic: In-plane isotropic polynomials and applications, (submitted to ESAFORM 2009 Conf.) Twente, 2009. [7] W.F. Hosford: Journal of Applied Mechanics. Trans. ASME Vol. 39 (1972), p. 607. [8] P.D. Wu, M. Jain, J. Savoie, S.R. MacEwen, P. Tugcu, K.W.Neale: Int. J. Plasticity 19 (2003), p. 121.

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Sheet Metal 2009 doi:10.4028/www.scientific.net/KEM.410-411 Assessment of the Modified Maximum Force Criterion for Aluminum Metallic Sheets doi:10.4028/www.scientific.net/KEM.410-411.511