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An experimental and numerical study of prestrained AA5754 sheet in bending
Journal of Materials Processing Technology 213 (2013) 1–10
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Journal of Materials Processing Technology journal homepage: www.elsevier.com/locate/jmatprotec
An experimental and numerical study of prestrained AA5754 sheet in bending Diane Wowk ∗,1 , Keith Pilkey. Department of Mechanical and Materials Engineering, McLaughlin Hall, Queen’s University, Kingston, Ontario, Canada K7L 3N6
a r t i c l e
i n f o
Article history: Received 18 July 2011 Received in revised form 20 June 2012 Accepted 10 August 2012 Available online 19 August 2012 Keywords: AA5754 Strain-rate sensitivity Prestrain Bending Buckling Sheet forming
a b s t r a c t This study focuses on the effects of prestrain magnitude on 3 mm AA5754 sheet in bending at nominal applied strain rates of 0.001/s and 0.1/s. The necessity of incorporating prestrain and strain rate effects into numerical simulations of bending is also evaluated. A series of experimental bend tests using axial compression in the longitudinal material direction were performed following plane strain prestrain in the transverse material direction. An inelastic buckling mode of deformation was produced with the peak buckling load increasing and the minimum load decreasing with larger magnitudes of prestrain. A semi-empirical material model, referred to as the Voce-MA model, was developed which incorporates strain rate-sensitivity of the flow stress, the prestrain magnitude and their interaction. Simulations of the bend tests using this material model were then performed in LS-DYNA at nominal applied strain rates of 0.001/s and 0.1/s for samples with 0, 3, 6 and 12% plane strain prestrain. It was shown that for AA5754 sheet in bending, prestrain effects must be considered in terms of current sheet thickness and material hardening. While the peak and minimum loads are not strain rate sensitive at the low rates used in this study, a rate-dependant material model is still necessary in order to account for the deviations in local strain rate from the applied strain rate. The Voce-MA material model was capable of representing prestrain and strain rate effects for all cases of AA5754 sheet in bending considered in this study. © 2012 Elsevier B.V. All rights reserved.
1. Introduction With the automotive industry striving to decrease vehicle weight and increase fuel efficiency, aluminum alloys such as AA5754 have become an attractive option for structural members in automobiles due to their high strength-to-weight ratio. During the manufacture of automotive components, forming may proceed in multiple steps where residual strains could influence material behaviour in the subsequent loading stages. In addition, forming processes can occur at different strain rates, and in some techniques such as bending, the local strain rates may differ significantly from the nominal applied rate. It is therefore important to understand the effects that prestrain and strain rate have on the behaviour of AA5754 sheet during forming and be able to incorporate them into numerical simulations. It is well known that the fabrication history has a significant effect on formability and must be included in simulations in order to achieve accurate predictions of forming limits. Toros et al. (2011) examined the effects of prestrain on springback in AA5754 sheet and noted that the springback ratio increases with increasing prestrain. Williams et al. (2010) have shown that for AA5754
∗ Corresponding author. Tel.: +1 613 541 6000x6457. E-mail address: [email protected] (D. Wowk). 1 Now at the Royal Military College of Canada, Kingston, Ontario, Canada K7K 7B4. 0924-0136/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmatprotec.2012.08.002
hydroformed tubes, the increase in the load required for deformation was on the order of 50% when residual prestrain was present. In both studies, the yield stress following 11% prestrain more than doubled. Prestrain can be incorporated into numerical simulations in two stages: the forming process is first simulated and then the residual stresses and strains are exported for the subsequent analysis. The present work makes use of a single stage simulation where the prestrain is incorporated into the material model via the hardening curve, and the sample geometry accounts for the reduced thickness resulting from a prior forming process. In addition to prestrain, the effect of strain rate on the formability and the forces required during the forming process may be significant as discussed by Verleysen et al. (2011). Most often, strain rate effects are considered during high speed forming processes such as electromagnetic forming, although strain rate effects may be present during lower rate operations such as bending and deep drawing due to locally high strain rates. Gedikli et al. (2011a) and Mahabunphachai et al. (2011) showed that for the warm hydroforming process, formability of AA5754 decreased with increasing strain rates between 0.0013/s and 0.13/s. Steel generally shows positive rate sensitivity characterised by higher flow stress and yield stress with increasing strain rate, whereas AA5754 transitions from negative to positive rate sensitivity with increasing strain rate. Smerd et al. (2005) have shown that this changeover occurs between strain rates of 0.0033/s and 600/s and coincides with the disappearance of the Portevin-Le Chatelier
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effect in the deformation process. In AA5754 sheet, a 5.2% reduction in the flow stress has been reported by Wowk and Pilkey (2009) at 13% equivalent strain when the strain rate was increased from 0.001/s to 0.1/s, while Halim et al. (2007) reported a decrease of approximately 10% in the flow stress when the strain rate was increased from 0.0006/s to 0.06/s. Increasing the strain rate from 0.0033/s to 600/s resulted in an increase of 9.7% in the flow stress at 5% strain as presented by Smerd et al. (2005). Strain rates seen during forming operations may not be uniform throughout the component, and in the case of bending can range from zero at the neutral axis of the sheet to a peak at the outer fibre of the bend. In addition, the local strain rates also change as the bend progresses, even when the remote applied strain rate remains constant. In finite element simulations, it is therefore necessary to have a rate sensitive material model that can capture the local material response throughout the forming process. The Voce material model (Voce, 1955) is commonly used for aluminium alloys such as AA5754 but does not incorporate strain rate. The classic powerlaw hardening equation, = Cεn can be adapted to account for strain rate sensitivity through multiplication by a rate term, ε˙ m , requiring a different set of coefficients for each strain rate. This rate powerlaw material model has been used by Mahabunphachai et al. (2011) and Gedikli et al. (2011b) to simulate the response of AA5754 under warm hydroforming conditions, by assuming a constant strain rate throughout the forming process. The rate-sensitive material models available in commercial finite element method (FEM) codes, such as Johnson and Cook (1985) and Zerilli and Armstrong (1987) can be used for a wide range of materials and strain rates, but are not able to capture the transition between negative and positive strain rate sensitivity exhibited by AA5754. Benallal et al. (2006) developed a model for materials exhibiting negative rate sensitivity which were implemented in FEM codes, but the large number of elements needed to represent the propagating PLC bands is not practical for large scale simulations such as automotive components. Therefore, it is often necessary to modify existing material models to suit the specific material, application of interest and to provide the level of detail required. The current study adapts the classic Voce material model to capture the transition from negative to positive rate sensitivity, the effects of prestrain magnitude (i.e. plane strain in the transverse material direction) and the interaction between prestrain and rate sensitivity. Studies such as that by Huh et al. (2003) have focused on the effects of prestrain during vehicle impact at high strain rates, but for AA5754, there is little data considering prestrain and strain rate effects in the region of negative rate sensitivity between 0.001/s and 0.1/s. These are the strain rates that are of interest in many forming operations, but their effects are often considered to be negligible. One forming parameter that is of interest in sheet metals is bendability, which is of primary importance in hemming or flanging operations. Localized bending may also occur as wrinkling, which is a failure mode often seen in sheet forming and in the compressive zone during tube bending where local buckling or bending instability occurs. In many cases, bending occurs in the later forming stages, where the existing residual strains can cause accelerated strain localization and reduce bendability as shown by Sarkar et al. (2001) and Lloyd et al. (2002). Bending of AA5754 sheet has been performed and simulated using techniques such wrap bending (Hu et al., 2010) and V-bending (Toros et al., 2011), both of which produce inhomogeneous deformation due to the interaction with the loading tool. This present study focuses on small prestrained rectangular sheet samples loaded in axial compression to produce unrestricted bending or inelastic buckling. Doege et al. (1995) loaded rectangular samples with a compressive axial load to induce bending, which provided insight into the development of wrinkles during the deep-drawing process. The primary objectives of the current study were to observe the
Table 1 Composition of AA5754 (weight percent). Material
Mg
Mn
Si
Fe
Ti
Zn
Cu
Cr
AA5754
2.76
0.27
0.20
0.34
0.01
0.01
0.02
0.02
response of 3 mm AA5754 sheet in bending at different strain rates and prestrain magnitudes, and to predict various characteristics of these bend tests such as the peak load and minimum load. Plane strain prestrains of magnitude 0, 3, 6 and 12% were applied to blanks in the transverse material direction, which represents the strain state typically seen in hydroformed tubes. Experimental bend tests were then performed by applying axial compressive load to rectangular sheet samples in the longitudinal material direction at nominal applied strain rates of 0.001/s and 0.1/s. A semi-empirical material model was developed that captured the rate sensitivity of AA5754 for different prestrain magnitudes and was implemented in the commercial finite element code, LS-DYNA. Simulations of these tests at nominal applied strain rates of 0.001/s and 0.1/s showed that the implemented Voce-MA material model was able to successfully predict the response of AA5754 sheet in bending for the cases considered. 2. Material The material under study is 3 mm thick AA5754 sheet in the O-temper condition with the composition presented in Table 1. 2.1. Tensile testing A series of tensile tests using prestrained samples were performed at strain rates of 0.001/s, 0.1/s, 500/s and 1500/s. Strain rates higher than the nominal applied strain rates of 0.001/s and 0.1/s used for the bending study were considered because during bending, the local strain rate at the outer fiber of the specimen is higher than the applied axial strain rate. All tensile tests were performed in the longitudinal material direction at room temperature, and employed a subsize version of the ASTM E8 (1997) tensile sample geometry machined from either the prestrained blanks or the as-received sheet. The subsize tensile samples had a gauge length of 12.5 mm and a width of 1.75 mm. Plane strain prestrain was applied to notched rectangular blanks using a servo-hydraulic Instron test frame and a wide-grip configuration. The sheet blanks with width of 95.3 mm and a height of 33.6 mm were loaded in tension in the transverse material direction at a strain rate of 0.001/s. A grid pattern of dots was marked on the surface of the blanks to enable measurement of the prestrain using the image analysis software ImagePro. Six levels of nominal prestrain were applied: 2, 3, 6, 8, 10 and 12%, with standard deviations between 0.15 and 0.3%. Uniaxial tensile testing at strain rates of 0.001/s and 0.1/s was performed by applying a constant cross-head velocity using a servo-hydraulic Instron machine. A tensile split-Hopkinson bar apparatus was used for strain rates of 500/s and 1500/s following the method employed by Smerd et al. (2005) in their dynamic testing of 1.6 mm AA5754 sheet. Samples with 0, 2, 3, 6, 8, 10 and 12% nominal prestrain were tested at strain rates of 0.001/s and 0.1/s with at least five repeats of each combination being performed. Testing at 500/s and 1500/s was limited to three repeats of samples with 0 and 12% prestrain. 2.2. Experimental test results The Voce-type hardening rule of the following form, f = B − (B − A)e−nεp
(1)
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Fig. 3. Assembled bend test fixture. Fig. 1. Flow stress curves of true stress versus true plastic strain in uniaxial tension, with no prestrain, for a range of strain rates between 0.001/s and 1500/s.
was used to fit the flow stress ( f ) versus equivalent plastic strain (εp ) data for each sample, as this relationship is able to capture the saturation in hardening rate observed at higher strains in AA5754. One characteristic curve was identified for each combination of prestrain and strain rate as having parameters A, B, n and an equivalent prestrain lying within the 95% confidence interval of the entire data set. Equivalent prestrain was determined using a von Mises yield function. For samples without prestrain, Fig. 1 shows a region of negative strain rate sensitivity below 0.1/s due to the Portevin-Le Chatelier effect, where mobile dislocations encounter obstacles and are pinned by diffusing Mg atoms. As the strain rate increases, the Mg atoms cannot diffuse fast enough to pin dislocations before they overcome obstacles resulting in a decrease in the flow stress and hardening rate. For AA5754 sheet, a transition from negative to positive rate sensitivity occurs with increasing strain rate, with the changeover occurring between 0.1/s and 500/s. Accurate stress–strain data is difficult to obtain within this strain rate range due to the loading limitations of the Instron and the Hopkinson bar testing apparatus. The change from negative to positive rate sensitivity is thought to mark the end of the Portevin-Le Chatelier effect, and the start of another deformation mechanism, where dislocations cannot overcome obstacles in the shorter time period present at higher strain rates. Positive rate sensitivity is observed between the strain rates of 500/s and 1500/s with an increase in both the yield stress and the hardening rate. Note that the results for 500/s were truncated at the end of the first loading pulse prior to reaching maximum load, whereas the results for 1500/s reached maximum load during the first loading pulse. Detailed results for prestrains of 2, 3, 6, 8, 10 and 12% at strain rates of 0.001/s and 0.1/s have been previously published by Wowk and Pilkey (2009) and showed that the yield stress is rate independent and increases linearly with increasing prestrain. Fig. 2
illustrates that, following 12% plane strain prestrain in the transverse material direction, there appears to be a modest decrease in the yield stress between 0.001/s and 0.1/, but this difference is attributed to the method used to identify the characteristic curves in the current study and the scatter in the experimental data. A comparison between strain rates of 0.1/s and 1500/s indicates evidence of positive rate sensitivity with an increase in the hardening rate. The characteristic curve for prestrained samples tested at 500/s does not have the typical shape of the other experimental curves presented in Fig. 2. This irregularity is caused by the fact that the Voce curve was fit to data representing only the initial part of the flow stress curve following prestrain as results were only available for the first load pulse. It has been shown by Fernandes et al. (1998) that during subsequent loading, the initial hardening rate of the flow stress curve is very high, and a characteristic curve based on this small range of data does not incorporate the reduced hardening rate that develops as loading progresses. 3. Bend tests for prestrained AA5754 3.1. Bend test procedure Plane strain prestrain magnitudes of 0, 3, 6, and 12% with standard deviations between 0.17 and 0.31% were applied to sheet blanks using the same method discussed in Section 2.1 for tensile testing. Rectangular samples having a width of 19.1 mm (transverse material direction) and a height of 25.4 mm (longitudinal material direction) were machined from the central region of the prestrained blanks. A compressive axial load was applied in the longitudinal material direction at nominal applied strain rates of 0.001/s and 0.1/s to induce bending about the transverse material axis. Three repeats of each combination of strain rate and prestrain level were completed. The bend testing of specimens at applied strain rates of 0.001/s and 0.1/s, was performed using a servo-hydraulic Instron testing apparatus and the bend test fixture depicted in Fig. 3. This fixture includes inserts which allow for an unobstructed bend to occur, while the controlled alignment and slight initial compressive load provided by the bolts ensure repeatability of the bend location. The geometry of the sheet samples and the configuration of the test set up allowed for tight radius bends to develop freely, similar to that seen in sheet wrinkling, although the stress state may not be representative of the one developed in a full-size component. The applied strain rate was controlled through the displacement of the cross-head, and loading continued until a maximum compressive load of 20 kN was reached. 3.2. Bend test results
Fig. 2. Flow stress curves of true stress versus true equivalent plastic strain in uniaxial tension, with 12% prestrain, for strain rates between 0.001/s and 1500/s.
The plot of load versus sample end displacement shown in Fig. 4 is typical for all samples loaded at either 0.001/s or 0.1/s. This shape of curve is characteristic of Type II structures in bending as
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Fig. 4. Typical response of a sample in bending at nominal applied strain rates of either 0.001/s or 0.1/s.
Fig. 7. Correlation between the reduced modulus and the corresponding peak load at buckling at stain rates of 0.001/s and 0.1/s. Nominal prestrain levels are indicated below each data pair.
3.3. Discussion of experimental results The peak loads plotted in Fig. 5 correspond to critical buckling loads, Pcr , which are typically determined for inelastic buckling using the classical Euler formula as a function of the reduced modulus, Er : Pcr =
2 Er I (Kl)
2
4EEt Er = √ 2 ( E + Et )
(2)
(3)
described by Calladine and English (1984), where the load drops steeply following buckling. Initially, the load increases as the sample is loaded primarily in axial compression. Following yielding, inelastic buckling occurs, with the peak value corresponding to the critical buckling load. The mode of loading transitions from direct compression to bending, and as a result, the load decreases as the plastic bend becomes more severe. The minimum load is reached just prior to the two ends of the sample coming into contact. Fig. 5 shows that the peak load increases with increasing prestrain magnitude and Fig. 6 illustrates that the minimum load decreases with increasing prestrain for all strain rates. The minimum load was calculated using the average load corresponding to the final 3 mm of displacement.
where I is the moment of inertia, K the column effective length, l is the unsupported length, E is the Young’s modulus and Et represents the tangent modulus at the peak buckling stress. Fig. 7 shows the test results for the buckling load and the corresponding reduced modulus resulting from tensile tests on prestrained uniaxial samples. Individual points on this graph result from strain rates of 0.001/s and 0.1/s and different prestrain levels (0, 3, 6 and 12%). The buckling load was taken as an average of the peak values within the result set, and the reduced modulus was determined from the characteristic curve at the peak load, for the specified strain rate and prestrain magnitude as defined by Wowk and Pilkey (2009). It is clear from Fig. 7 that the linear trend expected using the reduced modulus theory is not supported by the experimental results in this study. One possible explanation is that the reduced modulus theory may not be applicable to material that has been prestrained into the Luders regime as shown by Abel (1988). In addition, the calculation of the reduced modulus is extremely sensitive to the local hardening rate of the characteristic curve fit, and may be affected by the scatter in the experimental data.
Fig. 6. Linear trend for minimum load versus true equivalent prestrain for test repeats at nominal applied strain rates of 0.001/s and 0.1/s.
Fig. 8. Correlation between the yield stress and the peak load at buckling at strain rates of 0.001/s and 0.1/s. Nominal prestrain levels are indicated below each data pair.
Fig. 5. Linear trend for peak load versus true equivalent prestrain for test repeats at nominal applied strain rates of 0.001/s and 0.1/s.
D. Wowk, K. Pilkey. / Journal of Materials Processing Technology 213 (2013) 1–10
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Table 2 Summary of LS-DYNA simulations. Model number
Applied strain rate
Prestrain magnitude (%)
Prestrained sheet thickness (mm)
1 2 3 4 5 6 7 8
0.001/s 0.1/s 0.001/s 0.1/s 0.001/s 0.1/s 0.001/s 0.1/s
0 0 3.3 3.4 6.3 6.3 12.8 13.0
2.91 2.91 2.87 2.86 2.78 2.78 2.64 2.63
In Fig. 8, however, the peak load has been plotted against the yield stress from the characteristic curves, indicating a linear trend between these two parameters. This relationship is not surprising since these particular samples undergo yielding prior to inelastic buckling. The linear trend between peak load and yield stress also explains why the peak load increases with increasing prestrain (Fig. 5), since the work hardening that occurs during prestrain results in an increased yield stress upon reloading. Further discussions of the relationship between prestrain and minimum load, as well as strain rate sensitivity can be found in Section 4.5 as the numerical simulations provide further insight and clarity to these results. 4. FEM simulation of bend tests 4.1. Simulation procedure A series of simulations of the experimental bend tests were performed using the explicit dynamic FEM software LS-DYNA. Nominal applied strain rates of 0.001/s and 0.1/s were applied to samples with prestrain magnitudes corresponding to the average of the three test repeats discussed in Section 3.1. Since the magnitude of prestrain does not change throughout the simulation, the prestrain process itself was not simulated, rather the resulting hardening curve was defined in the material model and the reduced, uniform sheet thickness was represented in the model geometry. A semiempirical rate sensitive material model, was developed specifically for prestrained AA5754, which incorporates the transition between negative and positive strain rate sensitivity seen by this material, the prestrain magnitude and the interaction between prestrain and rate sensitivity. The authors refer to this material model as the Voce-MA model to differentiate it from the classic Voce relationship. The simulation matrix for the eight models is presented in Table 2. 4.2. Material model implementation 4.2.1. General formulation of the Voce-MA material model The material model used in this study was first presented by Wowk and Pilkey (2009) as a Voce-type hardening model, where a rate-dependant expression for the flow stress (hardening rule) is given in terms of the strain rate and the true equivalent plastic strain (von Mises strain). f = (B − (B − A)e−nεp )
Voce
ε˙ m ε˙
0 multiplicative
ε˙
+ m ln
ε˙ 0
(4)
additive
The effect of the multiplicative term (m) is primarily to change the hardening rate of the curve, but it has the secondary effect of offsetting the flow stress. The additive term (m ) shifts the curve vertically to reflect changes in the yield stress. In the present work, both terms are necessary as changes in strain rate result in significant changes to the hardening rate and the yield stress. In order to
Fig. 9. The variation of the Voce-MA rate parameters (a) m and (b) m with strain rate.
capture the transition from negative to positive strain rate sensitivity for AA5754, both m and m’ are expressed as functions of the strain rate, to obtain:
ε˙ m(ε)˙
f = (B − (B − A)e−nεp )
ε˙ 0
˙ ln + m (ε)
ε˙ ε˙ 0
(5)
˙ are the rate-sensitive versions of the mul˙ and m (ε) where m(ε) tiplicative and additive rate parameters. Eq. (5) is referred to hereafter as the Voce-MA material model and uses a base strain rate ε˙ 0 of 0.001/s in this study. Adiabatic temperature changes were not considered in the Voce-MA material model, as it has been shown by Smerd et al. (2005) that their effects can be considered negligible for AA5754 at strain rates up to 1500/s. The overall relationships for the rate sensitivity parameters m and m were determined by fitting the characteristic flow stress results at 0.001/s, 0.1/s, 500/s and 1500/s, using the Voce parameters A, B and n determined for the base strain rate (ε˙ 0 ) of 0.001/s. Fig. 9 shows the parameters m and m plotted against the natural logarithm of strain rate for samples with no prestrain and for 12% transverse prestrain. The negative rate sensitivity between 0.001/s and 0.1/s from Fig. 1 produces a corresponding decrease in the multiplicative rate parameter and the positive rate sensitivity above 0.1/s is shown by an increase in the multiplicative rate parameter. The additive rate parameter compensates for the shift in the flow stress caused by the multiplicative rate parameter. While results at 500/s were used in producing these quadratic trends, the 500/s data was not used in determining the material constants due to the fact that only a portion of the stress–strain data was available as previously discussed in Section 2.2.
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Fig. 10. The variation of the Voce parameter, A, with prestrain, for a strain rate of 0.001/s.
Fig. 12. The variation of the Voce parameter, n, with prestrain, for a strain rate of 0.001/s.
Assuming that the multiplicative and additive rate parameters both follow a quadratic relationship with the natural logarithm of strain rate, they can be written as: ˙ 2 + a2 (ln ε) ˙ + a3 ˙ = a1 (ln ε) m(ε)
2
˙ = a4 (ln ε) ˙ + a5 (ln ε) ˙ + a6 m (ε)
(6) (7)
where a1 ,. . ., a6 are fitting constants. While the prestrain magnitude is not a parameter within the Voce-MA material model itself, the Voce constants A, B and n, as well as the fitting constants, a1 ,. . ., a6 are determined specifically for the prestrain magnitude of interest as described in Section 4.2.2. This approach is considered reasonable because the prestrain magnitude is an initial condition and does not change during loading; therefore, its effect can simply be incorporated into a set of material constants. 4.2.2. Relationship between the material constants and prestrain magnitude The Voce parameters A, B and n define the basic shape of the flow stress curve, and their relationship with prestrain was determined using curve fitting techniques. The parameter A is the yield stress, B is the saturation stress, and n is a general representation of the overall hardening of the Voce curve. Figs. 10–12 show these parameters for the base strain rate (ε˙ 0 ) of 0.001/s plotted against equivalent prestrain. The best fit curves correspond to second-order polynomials for parameters A and B, and a third-order polynomial for the parameter n. Parameters A and n increase with increasing prestrain, due to work hardening, as well as an increased hardening rate following a path change (Wowk and Pilkey, 2009). Since the flow stress following prestrain eventually converges to the monotonic curve, the parameter B must follow a decreasing relationship to compensate for the increase in n. Similar trends were observed at a strain rate of 0.1/s.
Fig. 11. The variation of the Voce parameter, B, with prestrain, for a strain rate of 0.001/s.
Fig. 13. The variation of the Voce-MA multiplicative rate parameter (m) with prestrain, at a strain rate of 0.1/s, using a base strain rate of 0.001/s. Predicted values of A, B and n were used.
At a strain rate of 0.1/s, the magnitude of the multiplicative rate parameter, m, is found to decrease with increasing prestrain, as illustrated in Fig. 13. This decrease is expected since the multiplicative rate parameter is a measure of the rate sensitivity of the hardening rate, which has been shown to decrease with increasing amounts of prestrain following a path change for AA5754 (Wowk and Pilkey, 2009). Fig. 14 demonstrates that the magnitude of the additive rate parameter m’ also decreases with increasing prestrain, as it compensates for the offset in the flow stress caused by the parameter m. Testing at 1500/s was only performed at prestrain magnitudes of 0 and 12%; hence the linear relationship that exists between the rate parameters and equivalent prestrain at 0.1/s was assumed to be present at higher strain rates as well.
Fig. 14. The variation of the Voce-MA additive rate parameter (m ) with prestrain, at a strain rate of 0.1/s, using a base strain rate of 0.001/s. Predicted values of A, B and n were used.
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Table 3 Voce-MA material constants used in the LS-DYNA simulations. Prestrain
0%
3.3%
3.4%
6.3%
12.8%
13.0%
Strain rate A B n a1 a2 a3 a4 a5 a6
0.001/s, 0.1/s 105.6 280.4 14.54 0.00044 0.00012 −0.020 −0.042 0.098 2.68
0.001/s 142.2 260.2 17.55 0.00039 0.00065 −0.014 −0.044 −0.061 1.66
0.1/s 143.1 259.8 17.79 0.00039 0.00066 −0.014 −0.044 −0.066 1.63
0.001/s, 0.1/s 168.2 251.2 27.87 0.00035 0.0011 −0.0086 −0.045 −0.21 0.70
0.001/s 196.5 264.2 43.79 0.00025 0.0022 0.0031 −0.048 −0.523 1.32
0.1/s 196.7 265.1 43.68 0.00025 0.0022 0.0034 −0.048 −0.53 −1.37
4.2.3. Determining the material constants A total of nine material constants are required for the Voce-MA material model: the Voce parameters A, B and n, and the six con˙ and m (ε). ˙ The parameters stants a1 through a6 which define m(ε) A, B and n were determined for the specified value of prestrain at the base strain rate of 0.001/s (Figs. 10–12). The rate parameters m and m’ were then calculated for the specified prestrain value at strain rates of 0.1/s, and 1500/s (Figs. 13 and 14). Finally, a1 through ˙ and a6 were determined using the quadratic relationship for m(ε) ˙ in Eqs. (6) and (7), knowing that both parameters are zero at m (ε) the base strain rate of 0.001/s. The resulting constants for the specific prestrain values used in the bending simulations are presented in Table 3. 4.2.4. Flow stress predictions Fig. 15 presents the predicted stress–strain curve for a range of strain rates and prestrain magnitudes compared with the
experimental results, plotted up to the Considère condition. When no prestrain is applied, the material model predicts the flow stress at all three strain rates to within 3%, except near the yield point where the difference can be up to 7.3%. Differences between the predicted stress–strain curves from the material model and the test data can be attributed to the curve fitting techniques used to determine the material constants as well as inherent scatter in the data. When 12% prestrain is applied, the material model predicts the flow stress at 0.001/s and 0.1/s to within 2.5%, but at 1500/s, the overall shape of the curve is not captured. The material model uses a base strain rate of 0.001/s, which reflects the increase in the initial hardening rate following prestrain that occurs at lower strain rates, but it is evident in Fig. 15b, that the flow stress curve at 1500/s is much flatter and therefore cannot be accurately represented by this base curve. While the multiplicative rate parameter is able to modify the slope of flow stress curve, it is unable change the overall shape and characteristics of the underlying Voce relationship. The inability of the model to represent the flow stress at 1500/s will not be a factor for the bending simulations in this study as the maximum strain rate seen in the models was 0.24/s for an applied strain rate of 0.1/s. When 2, 3, 6, 8, and 10% prestrain is applied, the maximum difference between the material model prediction of the flow stress and the experimental results at strain rates of 0.001/s and 0.1/s is 4.8% (Wowk and Pilkey, 2009).
4.3. Bend test simulation
Fig. 15. Predicted and experimental flow stress curves (a) without prestrain, and (b) with 12% prestrain for strain rates between 0.001/s and 1500/s.
Simulations of the bend tests consisted of two steel loading plates in frictionless contact with the 19.1 mm × 25.4 mm AA5754 sample. Eight node solid elements with reduced integration and hourglass control were used, and a convergence study showed that four elements through the thickness were sufficient. Buckling was initiated by applying a lateral offset of 0.26 mm to the middle of the sample geometry. This offset was determined through calibration with the buckling load for a sample without prestrain deformed at a strain rate of 0.001/s. This offset value was then used for all models as it represented the buckling load for this particular combination of material, length of sample and end conditions. Loading was prescribed by a velocity–time curve to produce constant applied nominal strain rates of 0.001/s and 0.1/s. The rate-sensitive Voce-MA material model from Eq. (5) was implemented in the explicit FEM software LS-DYNA through a userdefined-material model (UMAT). This implementation allowed for a dynamic model to be run, with the UMAT updating the stress–strain curve for each element in response to changes in the local strain rate during the simulation. An isotropic hardening scheme was adopted, along with the plastic return method where the flow stress returns to the yield surface along the radial direction. The plastic strain rate was determined using the Von Mises equivalent plastic strain increment. Blocks of 1000 time steps were used for the time increment in the calculation of the plastic strain rate in order to reduce numerical oscillations, as the strain rate could
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Fig. 16. Comparison of the overall deformed shape between test and simulation.
change by orders of magnitude when calculated using individual time steps. 4.4. Numerical predictions The experimentally deformed sample and the predicted deformed shape are shown in Fig. 16. It is evident that the flaring out of the sample edges is captured by the simulation, and the overall width and height of the deformed sample was predicted to within 2.9% and 8.0%, respectively, for all load cases. Output from the FEM simulations, in the form of load versus displacement curves, is compared with the average of the three test repeats in Fig. 17. The displacements for the test results have been shifted in order to match the peak loads for both the test and simulation. While only the four bounding cases are shown, the FEM simulations predict similar overall response, peak loads and minimum loads for all prestrain magnitudes and strain rates. The FEM simulations were able to predict the increase in peak load and the decrease in minimum load with increasing prestrain, as depicted in Figs. 18 and 19 for tests at 0.001/s and 0.1/s. The peak load was predicted to within 3.0% of the experimental results for all combinations of strain rate and prestrain magnitude. The maximum difference in the minimum load, however, was much larger at 9.1%. 4.5. Discussion of FEM predictions Overall, the FEM simulations over-predict the peak load and minimum load as seen in 13 out of the 16 cases presented in Figs. 18 and 19. This is to be expected in finite element simulations because the continuum of the sample is represented by a finite number of degrees of freedom. In addition, the material model itself overpredicts the flow stress by up to 5.3% for the four levels of prestrain and the two strain rates considered. The over-prediction of the minimum load may also be due to the fact that the material model did not take into account the reduction in load carrying capacity associated with the development of shear bands at the outer surface of the bend. This mode of strain localization develops from surface undulations, which were visible to the naked eye as “roughness” as discussed by Lievers et al. (2003). The decrease in minimum load with increasing prestrain, observed in Fig. 19, is a result of sheet thinning following prestrain. The sheet material had a starting thickness of 2.91 mm, but following 12% prestrain, for example, the thickness was reduced to 2.63 mm. The reduced thickness with increasing prestrain allows for a tight-radius bend to develop more easily, thereby decreasing the load through the sample. This reasoning is illustrated in Fig. 20 where the predicted minimum loads are similar for samples with the same sheet thickness, even though they have different material flow stress curves. Although the trend of a decreasing minimum load for higher prestrains could be caused by the development of shear bands, this is not the case here, as Fig. 19 shows that the simulations predict the same decreasing trend even though localization was not included in the material model. The requirement for including prestrain effects into the simulation through both the hardening curve and the reduced sheet thickness is highlighted in Fig. 20. For 12% prestrain at 0.001/s, a 14.4% under prediction of the peak load results when hardening is not accounted for and the
Fig. 17. Comparison between FEM simulation and experimental test results for samples with (a) no prestrain at 0.001/s, (b) no prestrain at 0.1/s, (c) 12% prestrain at 0.001/s, and (d) 12% prestrain at 0.1/s.
D. Wowk, K. Pilkey. / Journal of Materials Processing Technology 213 (2013) 1–10
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Fig. 18. Comparison of peak loads between the average of experimental bend tests and FEM simulation at applied strain rates of 0.001/s and 0.1/s. Fig. 21. Plastic strain rate versus crosshead displacement comparison between elements at two different locations within a sample with no prestrain at an applied strain rate of 0.1/s.
Fig. 19. Comparison of minimum loads between the average of experimental bend tests and FEM simulation at applied strain rates of 0.001/s and 0.1/s.
original sheet thickness is used. If the reduction in sheet thickness is accounted for, but the hardening effect is ignored, the peak load is under predicted by 31.9%. For a tensile test conducted without prestrain, an increase in the strain rate from 0.001/s to 0.1/s has been shown to produce up to a 5% reduction in the flow stress for this particular AA5754 sheet (Wowk and Pilkey, 2009). In the bending cases considered here, this softening did not lead to significant differences in the peak load or minimum load. The rate insensitivity of the peak load is attributed to the yield stress being rate independent between 0.001/s and 0.1/s as discussed in Section 2.2. The rate insensitivity of the minimum load is attributed to the fact that the local strain rates in the sample during bending differ from the nominal applied strain rate. The strain rates at the outer surface of the bend, and away from the bend are plotted in Fig. 21, for the duration of the simulation for a nominal applied strain rate of 0.1/s. At locations away from the bend, the strain rate is very low as rigid body rotation is occurring, whereas at the middle of the sample at the outer fibre, the strain
Fig. 22. Load versus crosshead displacement comparison between predictions using the Voce-MA material model and a rate-insensitive material model for a sample with no prestrain at an applied strain rate of 0.1/s.
rate can be up to a factor of 1.5 higher than the nominal applied rate. Hence, even when a nominal strain rate of 0.1/s is applied, the average local strain rate within the sample can be much lower, thus reducing the overall effect of material rate sensitivity. The need for a material model that can adapt to changes in strain rate throughout the FEM simulation was evaluated by performing a simulation using a rate-insensitive, piecewise linear material model to represent the stress–strain curve for a strain rate of 0.1/s. This setup evaluates the effect of excluding differences in strain rate within the sample and how the local rates change as deformation progresses. Fig. 22 compares the predictions from this simulation with those obtained using the rate-sensitive Voce-MA material model when no prestrain was applied. The peak loads differ by only 0.8% due to the rate insensitivity of the yield stress. The rateinsensitive model underpredicts the minimum load by 5.4% when compared with the rate sensitive model. This underprediction is a result of the rate-insensitive model defining all elements to have the lower hardening rate corresponding to a strain rate of 0.1/s. In reality, the elements away from the bend experience very low strain rates (Fig. 21) and, as a result, have a higher flow stress than if all of the elements are simply assumed to have the properties of the applied strain rate.
5. Conclusions
Fig. 20. Comparison of load versus crosshead displacement during bending between simulations at an applied strain rate of 0.001/s for samples with varying prestrain and sheet thickness.
In this study, experimental and numerical bend tests of 3 mm AA5754 sheet were performed at strain rates of 0.001/s and 0.1/s with plane strain prestrain magnitudes of 0, 3, 6 and 12% applied in the transverse material direction. The following conclusions were drawn from this work:
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1. The peak load increases with increasing magnitudes of plane strain prestrain in the transverse direction due to the direct relationship that exists between the yield stress upon reloading and the peak load for plastic buckling of prestrained AA5754 sheet. 2. The minimum load decreases with increasing magnitudes of plane strain prestrain in the transverse direction due to the reduction in sheet thickness during the prestrain process. 3. The change in the flow stress and the change in thickness of the sheet due to prestrain must both be considered in the numerical simulations as differences of up to 32% in the peak load can occur for the bending cases considered here. 4. There are no differences in the peak load or the minimum load between applied strain rates of 0.001/s and 0.1/s due to the rate insensitivity of the yield stress and the distribution of local strain rates during bending. 5. At nominal applied strain rates of 0.001/s and 0.1/s, a rate-dependent material model is not necessary for bending simulations when predicting the peak-buckling load, but should be included when predicting the minimum load. An underprediction of up to 5.4% in the minimum load can occur at 0.1/s with no prestrain, due to the differences between the local strain rates and the nominal applied strain rate. 6. The Voce-MA material model was able to predict the peak load and minimum load from the bend tests to within 3.0% and 9.1% respectively for all cases considered. Acknowledgements The authors wish to thank Mike Worswick at the University of Waterloo for the use of his tensile split-Hopkinson bar facility. This research was funded by the Natural Sciences and Engineering Research Council (NSERC) and General Motors of Canada Limited. References Abel, A., 1988. Plastic prestrain effects on the buckling characteristics of steel. In: Proceeding of the 8th International Conference of Strength of Metals and Alloys, vol. 1, pp. 259–264. ASTM E8, 1997. Standard Test Methods for Tensile Testing of Metallic Materials. Annual Book of ASTM Standards, 03.01.1997. Benallal, A., Berstad, T., Borvik, T., Clausen, A.H., Hopperstad, O.S., 2006. Dynamic strain aging and related instabilities: experimental, theoretical and numerical aspects. European Journal of Mechanics A – Solids 25, 397–424.
Calladine, C.R., English, R.W., 1984. Strain-rate and inertia effects in the collapse of two types of energy-absorbing structure. International Journal of Mechanical Sciences 26 (11/12), 689–701. Doege, E., El-Dsoki, T., Seibert, D., 1995. Prediction of necking and wrinkling in sheetmetal forming. The Journal of Materials Processing Technology 50, 197–206. Fernandes, J.V., Rodrigues, D.M., Menezes, L.F., Vieira, M.F., 1998. A modified swift law for prestrained materials. International Journal of Plasticity 14 (6), 537–550. Gedikli, H., Necati Cora, O., Koc¸, M., 2011a. Parametric investigation of circular and elliptical bulge tests in warm hydroforming process for AA5754-O sheet. Key Engineering Materials 473, 594–601. Gedikli, H., Necati Cora, O., Koc¸, M., 2011b. Comparative investigations on numerical modeling for warm hydroforming of AA5754-O aluminum sheet alloy. Materials & Design 32, 2650–2662. Halim, H., Wilkinson, D., Niewczas, M., 2007. The Portevin-Le Chatelier (PLC) effect and shear band formation in an AA5754 alloy. Acta Materialia 55, 4151–4160. Hu, X., Jain, M., Wu, P., Wilkinson, D., Mishra, R., 2010. A macro–micro-multi-level modeling scheme to study the effect of particle distribution on wrap-bendability of AA5754 sheet alloys. Journal of Materials Processing Technology 210, 1232–1242. Huh, H., Kim, K.P., Kim, S.H., Song, J.H., Kim, H.S., Hong, S.K., 2003. Crashworthiness assessment of front side members in an auto-body considering the fabrication histories. International Journal of Mechanical Sciences 45, 1645–1660. Johnson, G.R., Cook, W.H., 1985. Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Engineering Fracture Mechanics 21 (1), 31–48. Lievers, W.B., Pilkey, A.K., Lloyd, D.J., 2003. The influence of iron content on the bendability of AA6111 sheet. Materials Science and Engineering A 361, 312–320. Lloyd, D.J., Evans, D., Pelow, C., Nolan, P., Jain, M., 2002. Bending in aluminium alloys AA6111 and AA5754 using the cantilever bend test. Materials Science and Technology 18, 621–628. Mahabunphachai, S., Koc, M., Carsley, J., 2011. Investigations on deformation behavior of AA5754 sheet alloy under warm hydroforming conditions. Journal of Manufacturing Science and Engineering ASME 133, 051007-1-9. Sarkar, J., Kutty, T., Conlon, K., Wilkinson, D., Embury, J., Lloyd, D., 2001. Tensile and Bending properties of AA5754 aluminum alloys. Materials Science and Engineering: A 316, 52–59. Smerd, R., Winkler, S., Salisbury, C., Worswick, M., Lloyd, D., Finn, M., 2005. High strain rate tensile testing of automotive aluminum sheet. International Journal of Impact Engineering 32, 541–560. Toros, S., Alkan, M., Ecmel Ece, R., Ozturk, F., 2011. Effect of pre-straining on the springback behavior of the AA5754-O alloy. Material Technology 45 (6), 613–618. Verleysen, P., Peris, J., Van Slycken, J., Faes, K., Duchene, L., 2011. Effect of strain rate on the forming behaviour of sheet metals. Journal of Materials Processing Technology 211, 1457–1464. Voce, E.A., 1955. Practical strain hardening function. Metallurgia 51, 219–226. Williams, B., Simha, C., Abedrabbo, N., Mayer, R., Worswick, M., 2010. Effect of anisotropy, kinematic hardening, and strain-rate sensitivity on the predicted axial crush response of hydroformed aluminium alloy tubes. International Journal of Impact Engineering 37, 652–661. Wowk, D., Pilkey, K., 2009. Effect of prestrain with a path change on the strain rate sensitivity of AA5754 sheet. Materials Science and Engineering: A 520, 174–178. Zerilli, F.J., Armstrong, R.W., 1987. Dislocation-mechanics-based constitutive relations for material dynamics calculations. Journal of Applied Physics 61 (5), 1816–1825.
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Journal of Materials Processing Technology journal homepage: www.elsevier.com/locate/jmatprotec
An experimental and numerical study of prestrained AA5754 sheet in bending Diane Wowk ∗,1 , Keith Pilkey. Department of Mechanical and Materials Engineering, McLaughlin Hall, Queen’s University, Kingston, Ontario, Canada K7L 3N6
a r t i c l e
i n f o
Article history: Received 18 July 2011 Received in revised form 20 June 2012 Accepted 10 August 2012 Available online 19 August 2012 Keywords: AA5754 Strain-rate sensitivity Prestrain Bending Buckling Sheet forming
a b s t r a c t This study focuses on the effects of prestrain magnitude on 3 mm AA5754 sheet in bending at nominal applied strain rates of 0.001/s and 0.1/s. The necessity of incorporating prestrain and strain rate effects into numerical simulations of bending is also evaluated. A series of experimental bend tests using axial compression in the longitudinal material direction were performed following plane strain prestrain in the transverse material direction. An inelastic buckling mode of deformation was produced with the peak buckling load increasing and the minimum load decreasing with larger magnitudes of prestrain. A semi-empirical material model, referred to as the Voce-MA model, was developed which incorporates strain rate-sensitivity of the flow stress, the prestrain magnitude and their interaction. Simulations of the bend tests using this material model were then performed in LS-DYNA at nominal applied strain rates of 0.001/s and 0.1/s for samples with 0, 3, 6 and 12% plane strain prestrain. It was shown that for AA5754 sheet in bending, prestrain effects must be considered in terms of current sheet thickness and material hardening. While the peak and minimum loads are not strain rate sensitive at the low rates used in this study, a rate-dependant material model is still necessary in order to account for the deviations in local strain rate from the applied strain rate. The Voce-MA material model was capable of representing prestrain and strain rate effects for all cases of AA5754 sheet in bending considered in this study. © 2012 Elsevier B.V. All rights reserved.
1. Introduction With the automotive industry striving to decrease vehicle weight and increase fuel efficiency, aluminum alloys such as AA5754 have become an attractive option for structural members in automobiles due to their high strength-to-weight ratio. During the manufacture of automotive components, forming may proceed in multiple steps where residual strains could influence material behaviour in the subsequent loading stages. In addition, forming processes can occur at different strain rates, and in some techniques such as bending, the local strain rates may differ significantly from the nominal applied rate. It is therefore important to understand the effects that prestrain and strain rate have on the behaviour of AA5754 sheet during forming and be able to incorporate them into numerical simulations. It is well known that the fabrication history has a significant effect on formability and must be included in simulations in order to achieve accurate predictions of forming limits. Toros et al. (2011) examined the effects of prestrain on springback in AA5754 sheet and noted that the springback ratio increases with increasing prestrain. Williams et al. (2010) have shown that for AA5754
∗ Corresponding author. Tel.: +1 613 541 6000x6457. E-mail address: [email protected] (D. Wowk). 1 Now at the Royal Military College of Canada, Kingston, Ontario, Canada K7K 7B4. 0924-0136/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmatprotec.2012.08.002
hydroformed tubes, the increase in the load required for deformation was on the order of 50% when residual prestrain was present. In both studies, the yield stress following 11% prestrain more than doubled. Prestrain can be incorporated into numerical simulations in two stages: the forming process is first simulated and then the residual stresses and strains are exported for the subsequent analysis. The present work makes use of a single stage simulation where the prestrain is incorporated into the material model via the hardening curve, and the sample geometry accounts for the reduced thickness resulting from a prior forming process. In addition to prestrain, the effect of strain rate on the formability and the forces required during the forming process may be significant as discussed by Verleysen et al. (2011). Most often, strain rate effects are considered during high speed forming processes such as electromagnetic forming, although strain rate effects may be present during lower rate operations such as bending and deep drawing due to locally high strain rates. Gedikli et al. (2011a) and Mahabunphachai et al. (2011) showed that for the warm hydroforming process, formability of AA5754 decreased with increasing strain rates between 0.0013/s and 0.13/s. Steel generally shows positive rate sensitivity characterised by higher flow stress and yield stress with increasing strain rate, whereas AA5754 transitions from negative to positive rate sensitivity with increasing strain rate. Smerd et al. (2005) have shown that this changeover occurs between strain rates of 0.0033/s and 600/s and coincides with the disappearance of the Portevin-Le Chatelier
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effect in the deformation process. In AA5754 sheet, a 5.2% reduction in the flow stress has been reported by Wowk and Pilkey (2009) at 13% equivalent strain when the strain rate was increased from 0.001/s to 0.1/s, while Halim et al. (2007) reported a decrease of approximately 10% in the flow stress when the strain rate was increased from 0.0006/s to 0.06/s. Increasing the strain rate from 0.0033/s to 600/s resulted in an increase of 9.7% in the flow stress at 5% strain as presented by Smerd et al. (2005). Strain rates seen during forming operations may not be uniform throughout the component, and in the case of bending can range from zero at the neutral axis of the sheet to a peak at the outer fibre of the bend. In addition, the local strain rates also change as the bend progresses, even when the remote applied strain rate remains constant. In finite element simulations, it is therefore necessary to have a rate sensitive material model that can capture the local material response throughout the forming process. The Voce material model (Voce, 1955) is commonly used for aluminium alloys such as AA5754 but does not incorporate strain rate. The classic powerlaw hardening equation, = Cεn can be adapted to account for strain rate sensitivity through multiplication by a rate term, ε˙ m , requiring a different set of coefficients for each strain rate. This rate powerlaw material model has been used by Mahabunphachai et al. (2011) and Gedikli et al. (2011b) to simulate the response of AA5754 under warm hydroforming conditions, by assuming a constant strain rate throughout the forming process. The rate-sensitive material models available in commercial finite element method (FEM) codes, such as Johnson and Cook (1985) and Zerilli and Armstrong (1987) can be used for a wide range of materials and strain rates, but are not able to capture the transition between negative and positive strain rate sensitivity exhibited by AA5754. Benallal et al. (2006) developed a model for materials exhibiting negative rate sensitivity which were implemented in FEM codes, but the large number of elements needed to represent the propagating PLC bands is not practical for large scale simulations such as automotive components. Therefore, it is often necessary to modify existing material models to suit the specific material, application of interest and to provide the level of detail required. The current study adapts the classic Voce material model to capture the transition from negative to positive rate sensitivity, the effects of prestrain magnitude (i.e. plane strain in the transverse material direction) and the interaction between prestrain and rate sensitivity. Studies such as that by Huh et al. (2003) have focused on the effects of prestrain during vehicle impact at high strain rates, but for AA5754, there is little data considering prestrain and strain rate effects in the region of negative rate sensitivity between 0.001/s and 0.1/s. These are the strain rates that are of interest in many forming operations, but their effects are often considered to be negligible. One forming parameter that is of interest in sheet metals is bendability, which is of primary importance in hemming or flanging operations. Localized bending may also occur as wrinkling, which is a failure mode often seen in sheet forming and in the compressive zone during tube bending where local buckling or bending instability occurs. In many cases, bending occurs in the later forming stages, where the existing residual strains can cause accelerated strain localization and reduce bendability as shown by Sarkar et al. (2001) and Lloyd et al. (2002). Bending of AA5754 sheet has been performed and simulated using techniques such wrap bending (Hu et al., 2010) and V-bending (Toros et al., 2011), both of which produce inhomogeneous deformation due to the interaction with the loading tool. This present study focuses on small prestrained rectangular sheet samples loaded in axial compression to produce unrestricted bending or inelastic buckling. Doege et al. (1995) loaded rectangular samples with a compressive axial load to induce bending, which provided insight into the development of wrinkles during the deep-drawing process. The primary objectives of the current study were to observe the
Table 1 Composition of AA5754 (weight percent). Material
Mg
Mn
Si
Fe
Ti
Zn
Cu
Cr
AA5754
2.76
0.27
0.20
0.34
0.01
0.01
0.02
0.02
response of 3 mm AA5754 sheet in bending at different strain rates and prestrain magnitudes, and to predict various characteristics of these bend tests such as the peak load and minimum load. Plane strain prestrains of magnitude 0, 3, 6 and 12% were applied to blanks in the transverse material direction, which represents the strain state typically seen in hydroformed tubes. Experimental bend tests were then performed by applying axial compressive load to rectangular sheet samples in the longitudinal material direction at nominal applied strain rates of 0.001/s and 0.1/s. A semi-empirical material model was developed that captured the rate sensitivity of AA5754 for different prestrain magnitudes and was implemented in the commercial finite element code, LS-DYNA. Simulations of these tests at nominal applied strain rates of 0.001/s and 0.1/s showed that the implemented Voce-MA material model was able to successfully predict the response of AA5754 sheet in bending for the cases considered. 2. Material The material under study is 3 mm thick AA5754 sheet in the O-temper condition with the composition presented in Table 1. 2.1. Tensile testing A series of tensile tests using prestrained samples were performed at strain rates of 0.001/s, 0.1/s, 500/s and 1500/s. Strain rates higher than the nominal applied strain rates of 0.001/s and 0.1/s used for the bending study were considered because during bending, the local strain rate at the outer fiber of the specimen is higher than the applied axial strain rate. All tensile tests were performed in the longitudinal material direction at room temperature, and employed a subsize version of the ASTM E8 (1997) tensile sample geometry machined from either the prestrained blanks or the as-received sheet. The subsize tensile samples had a gauge length of 12.5 mm and a width of 1.75 mm. Plane strain prestrain was applied to notched rectangular blanks using a servo-hydraulic Instron test frame and a wide-grip configuration. The sheet blanks with width of 95.3 mm and a height of 33.6 mm were loaded in tension in the transverse material direction at a strain rate of 0.001/s. A grid pattern of dots was marked on the surface of the blanks to enable measurement of the prestrain using the image analysis software ImagePro. Six levels of nominal prestrain were applied: 2, 3, 6, 8, 10 and 12%, with standard deviations between 0.15 and 0.3%. Uniaxial tensile testing at strain rates of 0.001/s and 0.1/s was performed by applying a constant cross-head velocity using a servo-hydraulic Instron machine. A tensile split-Hopkinson bar apparatus was used for strain rates of 500/s and 1500/s following the method employed by Smerd et al. (2005) in their dynamic testing of 1.6 mm AA5754 sheet. Samples with 0, 2, 3, 6, 8, 10 and 12% nominal prestrain were tested at strain rates of 0.001/s and 0.1/s with at least five repeats of each combination being performed. Testing at 500/s and 1500/s was limited to three repeats of samples with 0 and 12% prestrain. 2.2. Experimental test results The Voce-type hardening rule of the following form, f = B − (B − A)e−nεp
(1)
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3
Fig. 3. Assembled bend test fixture. Fig. 1. Flow stress curves of true stress versus true plastic strain in uniaxial tension, with no prestrain, for a range of strain rates between 0.001/s and 1500/s.
was used to fit the flow stress ( f ) versus equivalent plastic strain (εp ) data for each sample, as this relationship is able to capture the saturation in hardening rate observed at higher strains in AA5754. One characteristic curve was identified for each combination of prestrain and strain rate as having parameters A, B, n and an equivalent prestrain lying within the 95% confidence interval of the entire data set. Equivalent prestrain was determined using a von Mises yield function. For samples without prestrain, Fig. 1 shows a region of negative strain rate sensitivity below 0.1/s due to the Portevin-Le Chatelier effect, where mobile dislocations encounter obstacles and are pinned by diffusing Mg atoms. As the strain rate increases, the Mg atoms cannot diffuse fast enough to pin dislocations before they overcome obstacles resulting in a decrease in the flow stress and hardening rate. For AA5754 sheet, a transition from negative to positive rate sensitivity occurs with increasing strain rate, with the changeover occurring between 0.1/s and 500/s. Accurate stress–strain data is difficult to obtain within this strain rate range due to the loading limitations of the Instron and the Hopkinson bar testing apparatus. The change from negative to positive rate sensitivity is thought to mark the end of the Portevin-Le Chatelier effect, and the start of another deformation mechanism, where dislocations cannot overcome obstacles in the shorter time period present at higher strain rates. Positive rate sensitivity is observed between the strain rates of 500/s and 1500/s with an increase in both the yield stress and the hardening rate. Note that the results for 500/s were truncated at the end of the first loading pulse prior to reaching maximum load, whereas the results for 1500/s reached maximum load during the first loading pulse. Detailed results for prestrains of 2, 3, 6, 8, 10 and 12% at strain rates of 0.001/s and 0.1/s have been previously published by Wowk and Pilkey (2009) and showed that the yield stress is rate independent and increases linearly with increasing prestrain. Fig. 2
illustrates that, following 12% plane strain prestrain in the transverse material direction, there appears to be a modest decrease in the yield stress between 0.001/s and 0.1/, but this difference is attributed to the method used to identify the characteristic curves in the current study and the scatter in the experimental data. A comparison between strain rates of 0.1/s and 1500/s indicates evidence of positive rate sensitivity with an increase in the hardening rate. The characteristic curve for prestrained samples tested at 500/s does not have the typical shape of the other experimental curves presented in Fig. 2. This irregularity is caused by the fact that the Voce curve was fit to data representing only the initial part of the flow stress curve following prestrain as results were only available for the first load pulse. It has been shown by Fernandes et al. (1998) that during subsequent loading, the initial hardening rate of the flow stress curve is very high, and a characteristic curve based on this small range of data does not incorporate the reduced hardening rate that develops as loading progresses. 3. Bend tests for prestrained AA5754 3.1. Bend test procedure Plane strain prestrain magnitudes of 0, 3, 6, and 12% with standard deviations between 0.17 and 0.31% were applied to sheet blanks using the same method discussed in Section 2.1 for tensile testing. Rectangular samples having a width of 19.1 mm (transverse material direction) and a height of 25.4 mm (longitudinal material direction) were machined from the central region of the prestrained blanks. A compressive axial load was applied in the longitudinal material direction at nominal applied strain rates of 0.001/s and 0.1/s to induce bending about the transverse material axis. Three repeats of each combination of strain rate and prestrain level were completed. The bend testing of specimens at applied strain rates of 0.001/s and 0.1/s, was performed using a servo-hydraulic Instron testing apparatus and the bend test fixture depicted in Fig. 3. This fixture includes inserts which allow for an unobstructed bend to occur, while the controlled alignment and slight initial compressive load provided by the bolts ensure repeatability of the bend location. The geometry of the sheet samples and the configuration of the test set up allowed for tight radius bends to develop freely, similar to that seen in sheet wrinkling, although the stress state may not be representative of the one developed in a full-size component. The applied strain rate was controlled through the displacement of the cross-head, and loading continued until a maximum compressive load of 20 kN was reached. 3.2. Bend test results
Fig. 2. Flow stress curves of true stress versus true equivalent plastic strain in uniaxial tension, with 12% prestrain, for strain rates between 0.001/s and 1500/s.
The plot of load versus sample end displacement shown in Fig. 4 is typical for all samples loaded at either 0.001/s or 0.1/s. This shape of curve is characteristic of Type II structures in bending as
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Fig. 4. Typical response of a sample in bending at nominal applied strain rates of either 0.001/s or 0.1/s.
Fig. 7. Correlation between the reduced modulus and the corresponding peak load at buckling at stain rates of 0.001/s and 0.1/s. Nominal prestrain levels are indicated below each data pair.
3.3. Discussion of experimental results The peak loads plotted in Fig. 5 correspond to critical buckling loads, Pcr , which are typically determined for inelastic buckling using the classical Euler formula as a function of the reduced modulus, Er : Pcr =
2 Er I (Kl)
2
4EEt Er = √ 2 ( E + Et )
(2)
(3)
described by Calladine and English (1984), where the load drops steeply following buckling. Initially, the load increases as the sample is loaded primarily in axial compression. Following yielding, inelastic buckling occurs, with the peak value corresponding to the critical buckling load. The mode of loading transitions from direct compression to bending, and as a result, the load decreases as the plastic bend becomes more severe. The minimum load is reached just prior to the two ends of the sample coming into contact. Fig. 5 shows that the peak load increases with increasing prestrain magnitude and Fig. 6 illustrates that the minimum load decreases with increasing prestrain for all strain rates. The minimum load was calculated using the average load corresponding to the final 3 mm of displacement.
where I is the moment of inertia, K the column effective length, l is the unsupported length, E is the Young’s modulus and Et represents the tangent modulus at the peak buckling stress. Fig. 7 shows the test results for the buckling load and the corresponding reduced modulus resulting from tensile tests on prestrained uniaxial samples. Individual points on this graph result from strain rates of 0.001/s and 0.1/s and different prestrain levels (0, 3, 6 and 12%). The buckling load was taken as an average of the peak values within the result set, and the reduced modulus was determined from the characteristic curve at the peak load, for the specified strain rate and prestrain magnitude as defined by Wowk and Pilkey (2009). It is clear from Fig. 7 that the linear trend expected using the reduced modulus theory is not supported by the experimental results in this study. One possible explanation is that the reduced modulus theory may not be applicable to material that has been prestrained into the Luders regime as shown by Abel (1988). In addition, the calculation of the reduced modulus is extremely sensitive to the local hardening rate of the characteristic curve fit, and may be affected by the scatter in the experimental data.
Fig. 6. Linear trend for minimum load versus true equivalent prestrain for test repeats at nominal applied strain rates of 0.001/s and 0.1/s.
Fig. 8. Correlation between the yield stress and the peak load at buckling at strain rates of 0.001/s and 0.1/s. Nominal prestrain levels are indicated below each data pair.
Fig. 5. Linear trend for peak load versus true equivalent prestrain for test repeats at nominal applied strain rates of 0.001/s and 0.1/s.
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Table 2 Summary of LS-DYNA simulations. Model number
Applied strain rate
Prestrain magnitude (%)
Prestrained sheet thickness (mm)
1 2 3 4 5 6 7 8
0.001/s 0.1/s 0.001/s 0.1/s 0.001/s 0.1/s 0.001/s 0.1/s
0 0 3.3 3.4 6.3 6.3 12.8 13.0
2.91 2.91 2.87 2.86 2.78 2.78 2.64 2.63
In Fig. 8, however, the peak load has been plotted against the yield stress from the characteristic curves, indicating a linear trend between these two parameters. This relationship is not surprising since these particular samples undergo yielding prior to inelastic buckling. The linear trend between peak load and yield stress also explains why the peak load increases with increasing prestrain (Fig. 5), since the work hardening that occurs during prestrain results in an increased yield stress upon reloading. Further discussions of the relationship between prestrain and minimum load, as well as strain rate sensitivity can be found in Section 4.5 as the numerical simulations provide further insight and clarity to these results. 4. FEM simulation of bend tests 4.1. Simulation procedure A series of simulations of the experimental bend tests were performed using the explicit dynamic FEM software LS-DYNA. Nominal applied strain rates of 0.001/s and 0.1/s were applied to samples with prestrain magnitudes corresponding to the average of the three test repeats discussed in Section 3.1. Since the magnitude of prestrain does not change throughout the simulation, the prestrain process itself was not simulated, rather the resulting hardening curve was defined in the material model and the reduced, uniform sheet thickness was represented in the model geometry. A semiempirical rate sensitive material model, was developed specifically for prestrained AA5754, which incorporates the transition between negative and positive strain rate sensitivity seen by this material, the prestrain magnitude and the interaction between prestrain and rate sensitivity. The authors refer to this material model as the Voce-MA model to differentiate it from the classic Voce relationship. The simulation matrix for the eight models is presented in Table 2. 4.2. Material model implementation 4.2.1. General formulation of the Voce-MA material model The material model used in this study was first presented by Wowk and Pilkey (2009) as a Voce-type hardening model, where a rate-dependant expression for the flow stress (hardening rule) is given in terms of the strain rate and the true equivalent plastic strain (von Mises strain). f = (B − (B − A)e−nεp )
Voce
ε˙ m ε˙
0 multiplicative
ε˙
+ m ln
ε˙ 0
(4)
additive
The effect of the multiplicative term (m) is primarily to change the hardening rate of the curve, but it has the secondary effect of offsetting the flow stress. The additive term (m ) shifts the curve vertically to reflect changes in the yield stress. In the present work, both terms are necessary as changes in strain rate result in significant changes to the hardening rate and the yield stress. In order to
Fig. 9. The variation of the Voce-MA rate parameters (a) m and (b) m with strain rate.
capture the transition from negative to positive strain rate sensitivity for AA5754, both m and m’ are expressed as functions of the strain rate, to obtain:
ε˙ m(ε)˙
f = (B − (B − A)e−nεp )
ε˙ 0
˙ ln + m (ε)
ε˙ ε˙ 0
(5)
˙ are the rate-sensitive versions of the mul˙ and m (ε) where m(ε) tiplicative and additive rate parameters. Eq. (5) is referred to hereafter as the Voce-MA material model and uses a base strain rate ε˙ 0 of 0.001/s in this study. Adiabatic temperature changes were not considered in the Voce-MA material model, as it has been shown by Smerd et al. (2005) that their effects can be considered negligible for AA5754 at strain rates up to 1500/s. The overall relationships for the rate sensitivity parameters m and m were determined by fitting the characteristic flow stress results at 0.001/s, 0.1/s, 500/s and 1500/s, using the Voce parameters A, B and n determined for the base strain rate (ε˙ 0 ) of 0.001/s. Fig. 9 shows the parameters m and m plotted against the natural logarithm of strain rate for samples with no prestrain and for 12% transverse prestrain. The negative rate sensitivity between 0.001/s and 0.1/s from Fig. 1 produces a corresponding decrease in the multiplicative rate parameter and the positive rate sensitivity above 0.1/s is shown by an increase in the multiplicative rate parameter. The additive rate parameter compensates for the shift in the flow stress caused by the multiplicative rate parameter. While results at 500/s were used in producing these quadratic trends, the 500/s data was not used in determining the material constants due to the fact that only a portion of the stress–strain data was available as previously discussed in Section 2.2.
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Fig. 10. The variation of the Voce parameter, A, with prestrain, for a strain rate of 0.001/s.
Fig. 12. The variation of the Voce parameter, n, with prestrain, for a strain rate of 0.001/s.
Assuming that the multiplicative and additive rate parameters both follow a quadratic relationship with the natural logarithm of strain rate, they can be written as: ˙ 2 + a2 (ln ε) ˙ + a3 ˙ = a1 (ln ε) m(ε)
2
˙ = a4 (ln ε) ˙ + a5 (ln ε) ˙ + a6 m (ε)
(6) (7)
where a1 ,. . ., a6 are fitting constants. While the prestrain magnitude is not a parameter within the Voce-MA material model itself, the Voce constants A, B and n, as well as the fitting constants, a1 ,. . ., a6 are determined specifically for the prestrain magnitude of interest as described in Section 4.2.2. This approach is considered reasonable because the prestrain magnitude is an initial condition and does not change during loading; therefore, its effect can simply be incorporated into a set of material constants. 4.2.2. Relationship between the material constants and prestrain magnitude The Voce parameters A, B and n define the basic shape of the flow stress curve, and their relationship with prestrain was determined using curve fitting techniques. The parameter A is the yield stress, B is the saturation stress, and n is a general representation of the overall hardening of the Voce curve. Figs. 10–12 show these parameters for the base strain rate (ε˙ 0 ) of 0.001/s plotted against equivalent prestrain. The best fit curves correspond to second-order polynomials for parameters A and B, and a third-order polynomial for the parameter n. Parameters A and n increase with increasing prestrain, due to work hardening, as well as an increased hardening rate following a path change (Wowk and Pilkey, 2009). Since the flow stress following prestrain eventually converges to the monotonic curve, the parameter B must follow a decreasing relationship to compensate for the increase in n. Similar trends were observed at a strain rate of 0.1/s.
Fig. 11. The variation of the Voce parameter, B, with prestrain, for a strain rate of 0.001/s.
Fig. 13. The variation of the Voce-MA multiplicative rate parameter (m) with prestrain, at a strain rate of 0.1/s, using a base strain rate of 0.001/s. Predicted values of A, B and n were used.
At a strain rate of 0.1/s, the magnitude of the multiplicative rate parameter, m, is found to decrease with increasing prestrain, as illustrated in Fig. 13. This decrease is expected since the multiplicative rate parameter is a measure of the rate sensitivity of the hardening rate, which has been shown to decrease with increasing amounts of prestrain following a path change for AA5754 (Wowk and Pilkey, 2009). Fig. 14 demonstrates that the magnitude of the additive rate parameter m’ also decreases with increasing prestrain, as it compensates for the offset in the flow stress caused by the parameter m. Testing at 1500/s was only performed at prestrain magnitudes of 0 and 12%; hence the linear relationship that exists between the rate parameters and equivalent prestrain at 0.1/s was assumed to be present at higher strain rates as well.
Fig. 14. The variation of the Voce-MA additive rate parameter (m ) with prestrain, at a strain rate of 0.1/s, using a base strain rate of 0.001/s. Predicted values of A, B and n were used.
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Table 3 Voce-MA material constants used in the LS-DYNA simulations. Prestrain
0%
3.3%
3.4%
6.3%
12.8%
13.0%
Strain rate A B n a1 a2 a3 a4 a5 a6
0.001/s, 0.1/s 105.6 280.4 14.54 0.00044 0.00012 −0.020 −0.042 0.098 2.68
0.001/s 142.2 260.2 17.55 0.00039 0.00065 −0.014 −0.044 −0.061 1.66
0.1/s 143.1 259.8 17.79 0.00039 0.00066 −0.014 −0.044 −0.066 1.63
0.001/s, 0.1/s 168.2 251.2 27.87 0.00035 0.0011 −0.0086 −0.045 −0.21 0.70
0.001/s 196.5 264.2 43.79 0.00025 0.0022 0.0031 −0.048 −0.523 1.32
0.1/s 196.7 265.1 43.68 0.00025 0.0022 0.0034 −0.048 −0.53 −1.37
4.2.3. Determining the material constants A total of nine material constants are required for the Voce-MA material model: the Voce parameters A, B and n, and the six con˙ and m (ε). ˙ The parameters stants a1 through a6 which define m(ε) A, B and n were determined for the specified value of prestrain at the base strain rate of 0.001/s (Figs. 10–12). The rate parameters m and m’ were then calculated for the specified prestrain value at strain rates of 0.1/s, and 1500/s (Figs. 13 and 14). Finally, a1 through ˙ and a6 were determined using the quadratic relationship for m(ε) ˙ in Eqs. (6) and (7), knowing that both parameters are zero at m (ε) the base strain rate of 0.001/s. The resulting constants for the specific prestrain values used in the bending simulations are presented in Table 3. 4.2.4. Flow stress predictions Fig. 15 presents the predicted stress–strain curve for a range of strain rates and prestrain magnitudes compared with the
experimental results, plotted up to the Considère condition. When no prestrain is applied, the material model predicts the flow stress at all three strain rates to within 3%, except near the yield point where the difference can be up to 7.3%. Differences between the predicted stress–strain curves from the material model and the test data can be attributed to the curve fitting techniques used to determine the material constants as well as inherent scatter in the data. When 12% prestrain is applied, the material model predicts the flow stress at 0.001/s and 0.1/s to within 2.5%, but at 1500/s, the overall shape of the curve is not captured. The material model uses a base strain rate of 0.001/s, which reflects the increase in the initial hardening rate following prestrain that occurs at lower strain rates, but it is evident in Fig. 15b, that the flow stress curve at 1500/s is much flatter and therefore cannot be accurately represented by this base curve. While the multiplicative rate parameter is able to modify the slope of flow stress curve, it is unable change the overall shape and characteristics of the underlying Voce relationship. The inability of the model to represent the flow stress at 1500/s will not be a factor for the bending simulations in this study as the maximum strain rate seen in the models was 0.24/s for an applied strain rate of 0.1/s. When 2, 3, 6, 8, and 10% prestrain is applied, the maximum difference between the material model prediction of the flow stress and the experimental results at strain rates of 0.001/s and 0.1/s is 4.8% (Wowk and Pilkey, 2009).
4.3. Bend test simulation
Fig. 15. Predicted and experimental flow stress curves (a) without prestrain, and (b) with 12% prestrain for strain rates between 0.001/s and 1500/s.
Simulations of the bend tests consisted of two steel loading plates in frictionless contact with the 19.1 mm × 25.4 mm AA5754 sample. Eight node solid elements with reduced integration and hourglass control were used, and a convergence study showed that four elements through the thickness were sufficient. Buckling was initiated by applying a lateral offset of 0.26 mm to the middle of the sample geometry. This offset was determined through calibration with the buckling load for a sample without prestrain deformed at a strain rate of 0.001/s. This offset value was then used for all models as it represented the buckling load for this particular combination of material, length of sample and end conditions. Loading was prescribed by a velocity–time curve to produce constant applied nominal strain rates of 0.001/s and 0.1/s. The rate-sensitive Voce-MA material model from Eq. (5) was implemented in the explicit FEM software LS-DYNA through a userdefined-material model (UMAT). This implementation allowed for a dynamic model to be run, with the UMAT updating the stress–strain curve for each element in response to changes in the local strain rate during the simulation. An isotropic hardening scheme was adopted, along with the plastic return method where the flow stress returns to the yield surface along the radial direction. The plastic strain rate was determined using the Von Mises equivalent plastic strain increment. Blocks of 1000 time steps were used for the time increment in the calculation of the plastic strain rate in order to reduce numerical oscillations, as the strain rate could
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Fig. 16. Comparison of the overall deformed shape between test and simulation.
change by orders of magnitude when calculated using individual time steps. 4.4. Numerical predictions The experimentally deformed sample and the predicted deformed shape are shown in Fig. 16. It is evident that the flaring out of the sample edges is captured by the simulation, and the overall width and height of the deformed sample was predicted to within 2.9% and 8.0%, respectively, for all load cases. Output from the FEM simulations, in the form of load versus displacement curves, is compared with the average of the three test repeats in Fig. 17. The displacements for the test results have been shifted in order to match the peak loads for both the test and simulation. While only the four bounding cases are shown, the FEM simulations predict similar overall response, peak loads and minimum loads for all prestrain magnitudes and strain rates. The FEM simulations were able to predict the increase in peak load and the decrease in minimum load with increasing prestrain, as depicted in Figs. 18 and 19 for tests at 0.001/s and 0.1/s. The peak load was predicted to within 3.0% of the experimental results for all combinations of strain rate and prestrain magnitude. The maximum difference in the minimum load, however, was much larger at 9.1%. 4.5. Discussion of FEM predictions Overall, the FEM simulations over-predict the peak load and minimum load as seen in 13 out of the 16 cases presented in Figs. 18 and 19. This is to be expected in finite element simulations because the continuum of the sample is represented by a finite number of degrees of freedom. In addition, the material model itself overpredicts the flow stress by up to 5.3% for the four levels of prestrain and the two strain rates considered. The over-prediction of the minimum load may also be due to the fact that the material model did not take into account the reduction in load carrying capacity associated with the development of shear bands at the outer surface of the bend. This mode of strain localization develops from surface undulations, which were visible to the naked eye as “roughness” as discussed by Lievers et al. (2003). The decrease in minimum load with increasing prestrain, observed in Fig. 19, is a result of sheet thinning following prestrain. The sheet material had a starting thickness of 2.91 mm, but following 12% prestrain, for example, the thickness was reduced to 2.63 mm. The reduced thickness with increasing prestrain allows for a tight-radius bend to develop more easily, thereby decreasing the load through the sample. This reasoning is illustrated in Fig. 20 where the predicted minimum loads are similar for samples with the same sheet thickness, even though they have different material flow stress curves. Although the trend of a decreasing minimum load for higher prestrains could be caused by the development of shear bands, this is not the case here, as Fig. 19 shows that the simulations predict the same decreasing trend even though localization was not included in the material model. The requirement for including prestrain effects into the simulation through both the hardening curve and the reduced sheet thickness is highlighted in Fig. 20. For 12% prestrain at 0.001/s, a 14.4% under prediction of the peak load results when hardening is not accounted for and the
Fig. 17. Comparison between FEM simulation and experimental test results for samples with (a) no prestrain at 0.001/s, (b) no prestrain at 0.1/s, (c) 12% prestrain at 0.001/s, and (d) 12% prestrain at 0.1/s.
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Fig. 18. Comparison of peak loads between the average of experimental bend tests and FEM simulation at applied strain rates of 0.001/s and 0.1/s. Fig. 21. Plastic strain rate versus crosshead displacement comparison between elements at two different locations within a sample with no prestrain at an applied strain rate of 0.1/s.
Fig. 19. Comparison of minimum loads between the average of experimental bend tests and FEM simulation at applied strain rates of 0.001/s and 0.1/s.
original sheet thickness is used. If the reduction in sheet thickness is accounted for, but the hardening effect is ignored, the peak load is under predicted by 31.9%. For a tensile test conducted without prestrain, an increase in the strain rate from 0.001/s to 0.1/s has been shown to produce up to a 5% reduction in the flow stress for this particular AA5754 sheet (Wowk and Pilkey, 2009). In the bending cases considered here, this softening did not lead to significant differences in the peak load or minimum load. The rate insensitivity of the peak load is attributed to the yield stress being rate independent between 0.001/s and 0.1/s as discussed in Section 2.2. The rate insensitivity of the minimum load is attributed to the fact that the local strain rates in the sample during bending differ from the nominal applied strain rate. The strain rates at the outer surface of the bend, and away from the bend are plotted in Fig. 21, for the duration of the simulation for a nominal applied strain rate of 0.1/s. At locations away from the bend, the strain rate is very low as rigid body rotation is occurring, whereas at the middle of the sample at the outer fibre, the strain
Fig. 22. Load versus crosshead displacement comparison between predictions using the Voce-MA material model and a rate-insensitive material model for a sample with no prestrain at an applied strain rate of 0.1/s.
rate can be up to a factor of 1.5 higher than the nominal applied rate. Hence, even when a nominal strain rate of 0.1/s is applied, the average local strain rate within the sample can be much lower, thus reducing the overall effect of material rate sensitivity. The need for a material model that can adapt to changes in strain rate throughout the FEM simulation was evaluated by performing a simulation using a rate-insensitive, piecewise linear material model to represent the stress–strain curve for a strain rate of 0.1/s. This setup evaluates the effect of excluding differences in strain rate within the sample and how the local rates change as deformation progresses. Fig. 22 compares the predictions from this simulation with those obtained using the rate-sensitive Voce-MA material model when no prestrain was applied. The peak loads differ by only 0.8% due to the rate insensitivity of the yield stress. The rateinsensitive model underpredicts the minimum load by 5.4% when compared with the rate sensitive model. This underprediction is a result of the rate-insensitive model defining all elements to have the lower hardening rate corresponding to a strain rate of 0.1/s. In reality, the elements away from the bend experience very low strain rates (Fig. 21) and, as a result, have a higher flow stress than if all of the elements are simply assumed to have the properties of the applied strain rate.
5. Conclusions
Fig. 20. Comparison of load versus crosshead displacement during bending between simulations at an applied strain rate of 0.001/s for samples with varying prestrain and sheet thickness.
In this study, experimental and numerical bend tests of 3 mm AA5754 sheet were performed at strain rates of 0.001/s and 0.1/s with plane strain prestrain magnitudes of 0, 3, 6 and 12% applied in the transverse material direction. The following conclusions were drawn from this work:
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1. The peak load increases with increasing magnitudes of plane strain prestrain in the transverse direction due to the direct relationship that exists between the yield stress upon reloading and the peak load for plastic buckling of prestrained AA5754 sheet. 2. The minimum load decreases with increasing magnitudes of plane strain prestrain in the transverse direction due to the reduction in sheet thickness during the prestrain process. 3. The change in the flow stress and the change in thickness of the sheet due to prestrain must both be considered in the numerical simulations as differences of up to 32% in the peak load can occur for the bending cases considered here. 4. There are no differences in the peak load or the minimum load between applied strain rates of 0.001/s and 0.1/s due to the rate insensitivity of the yield stress and the distribution of local strain rates during bending. 5. At nominal applied strain rates of 0.001/s and 0.1/s, a rate-dependent material model is not necessary for bending simulations when predicting the peak-buckling load, but should be included when predicting the minimum load. An underprediction of up to 5.4% in the minimum load can occur at 0.1/s with no prestrain, due to the differences between the local strain rates and the nominal applied strain rate. 6. The Voce-MA material model was able to predict the peak load and minimum load from the bend tests to within 3.0% and 9.1% respectively for all cases considered. Acknowledgements The authors wish to thank Mike Worswick at the University of Waterloo for the use of his tensile split-Hopkinson bar facility. This research was funded by the Natural Sciences and Engineering Research Council (NSERC) and General Motors of Canada Limited. References Abel, A., 1988. Plastic prestrain effects on the buckling characteristics of steel. In: Proceeding of the 8th International Conference of Strength of Metals and Alloys, vol. 1, pp. 259–264. ASTM E8, 1997. Standard Test Methods for Tensile Testing of Metallic Materials. Annual Book of ASTM Standards, 03.01.1997. Benallal, A., Berstad, T., Borvik, T., Clausen, A.H., Hopperstad, O.S., 2006. Dynamic strain aging and related instabilities: experimental, theoretical and numerical aspects. European Journal of Mechanics A – Solids 25, 397–424.
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