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New biaxial yield function for aluminum alloys based on plastic work and work-hardening analyses
Acta Materialia 118 (2016) 109e119
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New biaxial yield function for aluminum alloys based on plastic work and work-hardening analyses S. Saimoto a, *, P. Van Houtte b, K. Inal c, M.R. Langille a a
Mechanical and Materials Engineering, Queen's University, Kingston, ON K7L 3N6, Canada MTM, Katholieke Universiteit, Lueven, Belgium c Department of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 4 May 2016 Received in revised form 11 May 2016 Accepted 19 July 2016
Yield locus measurements and their analytical descriptions has been the bases for modeling metal processing. These analytical descriptions play a role in models to predict limit strains observed during determination of forming limit diagrams of flat metal stock as means to evaluate their fabrication performance. Some of these analytical descriptions use the isotropic or anisotropic plastic potentials that take into account the crystallographic texture of the material. The applicability of such potentials is validated by comparing their predictions to that of unidirectional tensile data. In the current work, this approach is reversed to examine whether the constitutive relations, which replicate the measured stressstrain diagrams, can generate a two-dimensional section of the yield locus. The strategy is to sum the computed plastic work (PW) from unidirectional mechanical tests in two principal directions which also accommodates the biaxial interaction strains. The resulting yield function includes fq and f4, the prescribed stress ratios and the texture parameters R4 and Rq in their respective principal directions. The prescribed PW at an arbitrary strain during unidirectional tensile test is equated to the work sum from the biaxial stresses on the premise that the plastic flow-stress registers only the increasing density of obstacles it generates. The computed biaxial yield stresses showed good fits for AA5154 and AA5754 with only small modification due to latent work hardening. © 2016 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Constitutive modeling Yield locus prediction Plastic work Biaxial stress-strain diagram Work-hardening modeling
1. Introduction Formability of metallic sheet for stamping and deep-drawing operations has been of high industrial interest in order to optimize die-designs for fabrication processes and the progress of such studies to 1977 has been compiled by Hecker et al. [1]. The practical test of examining etched patterns on sheet surface to determine the principal strains at which limit strains of localized necking occur has been developed by Keeler [2] and Goodwin [3]. This mapping of limit strains for sheet fabrication has become the industrial standard, referred to as the forming limit diagram (FLD). The theoretical basis for the analysis is based on the observation that plastic shapechange is independent of hydrostatic stresses whereby the principal stresses are all equal. A very simple yield locus is the wellknown von Mises criterion for the occurrence of plastic flow. Today's experts in plasticity theory know that it merely represents a
* Corresponding author. E-mail address: [email protected] (S. Saimoto). http://dx.doi.org/10.1016/j.actamat.2016.07.036 1359-6454/© 2016 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
hypersphere in stress deviator space, i.e. it is the simplest possible closed surface in that sphere. Reid [4] points out that it can be based on the notion that plastic flow begins when the elastic distortion energy reaches a critical limit, usually designated at 0.2% off-set strain. However, from a physical point of view, this criterion is only meaningful for a material which has ideal isotropic elastic properties such as tungsten but not for the other crystalline materials. Nevertheless, in materials engineering the von Mises approximation is often used for cubic materials with a weak crystallographic texture as a reasonable first estimate. This analytical basis for a yield locus was accompanied by experimental observations that the incremental component of in-elastic strain may be computed by differentiating the yield function, in terms of stresses, with that specific stress component. This initial Levy-Mises criterion [4] has become known as the associated flow rule whereby the total plastic strain increment is normal to the yield surface. This extension of the von Mises criterion based on the existence of a critical magnitude of distortion energy for initiation of plastic flow to pre-strained work-hardened structure, as conventionally perceived, inherently assumes that the measured plastic work
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S. Saimoto et al. / Acta Materialia 118 (2016) 109e119
directly corresponds to the elastic distortion energy and that elastic anisotropy can be ignored (reasonable for tungsten, not unreasonable for aluminum, as the elastic anisotropy of aluminum single crystals is not very large). The current work suggests that plastic work initially follows a quadratic stress relation which becomes cubic and returns to quadratic at very large strains and inversely correlates with the work-hardening coefficient as will be examined in the discussion section. The seminal advance in the application of mathematical theory of plasticity to sheet formability was initiated by Marciniak and Kuczynski [5] in order to predict localized necking in biaxial stretch cases. However in order to initiate this strain localization, an initial existing groove in the plane of the sheet was envisioned. At present, this analysis is widely invoked to introduce the role of texture in FLD [6] and recent hypotheses invoke yield functions which do not retain the quadratic stress relation and hence the concept of distortion energy is relaxed. Overviews of such works are found in the following reports [7,8]. The validation of yield function is experimentally difficult since for cases other than unidirectional tests, the determination of the flow stress components and its accompanying plastic strains are imprecise. One well-defined test is that of stretching thin-walled tubular specimens with correlated internal pressure. The difficulty of precise strain measurement has been overcome by Kuwabara's group [9] using a spherometer, a specially developed displacement device for large in-elastic strains. In their study the stresses along loading paths of set biaxial ratios parallel to the cylinder axis (4) and its circumference (q) were determined and the total energy (TW) required for specific deformation modes were equated to the area defined by the stress-strain diagram for unidirectional tension to attain a given tensile strain to define the evolution of yield loci. Using extruded tubing of AA5154, the yield loci data at increasing degree of cold work were compared to various theoretical yield functions mentioned above [7,8]. Another notable study is due to Iadicola et al. [10] which measured the inplane stresses of biaxial stressed sheets of AA5754 using in-situ X-ray diffraction. From the quasi-static recording during unidirectional tests (UD), balanced biaxial (BB) and plane-strain (PS) cases, the principal stresses and strains were reported. These short-hand notations have been retained in this work for the ease of crossreferencing. The strategy to derive constitutive relations for in-plane biaxialstressed sheet in this work is to reverse that of conventional practice of yield function determination in which the fit parameters are matched with measured in-plane stresses at points of constant total work. The constitutive relation derived from this yield function is compared to the unidirectional tensile one to attest to its validity. In the current work, constitutive relation analyses (CRA) which can replicate the measured stress-strain diagram using at least two-fit loci become the starting point from which the plastic work due to biaxial stresses are calculated. The CRA procedure defines a new yield stress (sfinal ) by back-extrapolating the plastic 0 response from beyond the yield phenomenon and the associated yield point elongation (YPE). This determination corresponds to an off-set strain of 0.02%, slightly above the proportional limit as described elsewhere [11]. Hence the plastic work (PW) is defined as !(ssfinal ) dεp wherein s is the applied stress, (s-sfinal ), the flow 0 0 stress and εp, the logarithmic (true) plastic strain. Note that the total work (TW) includes !sfinal dεp which give rise to heat as in the case 0 of friction stress and contributions from the yield phenomenon are removed. Since the new constitutive relations by Saimoto and Van Houtte (S-VH) [11] based on the Taylor slip model directly correlate the flow shear stress to the shear strain [11,12], conversion to normal stress and strain was performed in order to use the width to thickness strain ratio R (Lankford ratio), as an indirect measure of textural effects on plastic flow. The hypothesis invokes that under
biaxial stress-states the incurred principal plastic strains are additive and also plastic work (PW) under proportional loading are additive. The net result for sheet stock is that the derivative of PW at a given thickness strain gives rise to the effective compression stress at that point. This slope-method offers the possibility of comparing the yield locus at constant PW to that of constant thickness strain. However the usual procedure is to use constant TW for the cases reported [9,10] which includes the yield stress and explicitly described by Barlat et al. [13,14]. Comparison of this criterion of constant TW to that of constant PW will be presented. Hence the new biaxial yield function is based on this PW relation that can be analytically derived using the constitutive relation which replicates the stress-strain diagram and integrated with respect to the in-plane stresses which are proportionally related. By equating PW from tensile tests at a given strain and using this relation, the biaxial stresses and strain can be calculated. The attribute of analytical prediction of yield locus from constitutive relations is two-fold. 1) The yield locus can be continuously mapped and compared with the data to assess the degree of fit. Moreover the effect of the texture-parameter R on the point-by-point curvature or tangent at any given stress ratio can be assessed. 2) From the calculated in-plane stresses for any given stress ratio, the strains corresponding to those stresses can also be calculated from which the principal major (minor) stress-principal major (minor) strain loci can be generated. This procedure also implies that CRA of such measured loci for the in-plane stresses can be analyzed to predict the strains in three principal directions. For example, such analysis for PS case can be used to ascertain if the actual test met the condition of continuous zero strain in the minor axis. The evolution of stresses under PS or BB conditions can be CRA analyzed to assess ductile failure and FLD prediction according to prior-described methodology [15].
2. Analytical procedures 2.1. Derivation of yield function based on plastic work Fig. 1 shows CRA of UD test in 4 direction whereby the two-fit
Fig. 1. Stress-strain diagram for AA5154 for UD 4 axis from Kuwabara et al. [9] showing the two-fitted loci using CRA. Note the definition of sfinal and the location of corre0 sponding coordinates to t3 and g3 which is slightly off-line giving rise to transition region in derived expended-energy plots.
S. Saimoto et al. / Acta Materialia 118 (2016) 109e119
loci results in sfinal by back extrapolation from beyond the YPE and 0 the two loci with b1 and b2 power-law exponents intersect at t3 and g3. This transition in the plastic flow curve means that upon applying biaxial stresses to the start material, the major stress will attain t3 prior to that for the minor. In this situation it is assumed the minor one will start to follow its prescribed b2 locus rather than continue on b1. The rationale being that the minor strain is confronted by the obstacles generated by the major one. The kinks in the following graphs indicate this transition point but the region of ambiguity is small. For PW ¼ !t dg ¼ !(ssfinal ) dε and 0 !sfinal dε, as the lattice friction work; the elastic strain components 0 have been subtracted from the total strain, a typical procedure in such analysis. Furthermore it is understood that work means the rate of doing work. The sum of these two terms is TW. Note that (ssfinal )/M is the shear flow stress t and Mε ¼ g wherein M is the 0 Taylor factor. The CRA parameters for all the mechanical tests are listed in Table 1. The Saimoto-Van Houtte (S-VH) constitutive relation [11] is given below with mean slip distance defined as l ¼ C1 tb,
t¼
1 P 1 2þb 2þ1 b ð2 þ bÞðambÞ2 g 4A C1 b
(1)
wherein P is a geometrical ratio of the additional dislocation length encompassing the incremental slip area, typically near unity, and A is the dynamic annihilation factor. Since the optimum fit requires two loci, the first power-law exponent is designated b1 and the second, b2. For the current analytical purpose, tj is converted from (sjsfinal )/M to Dsj/Mj and gj to Mj εj such that the texture effect 0 can be taken into account using Rj values wherein the script notation j indicates the direction of applied stress. In the subsequent analysis it will become evident that conversion of multi-slip tensile stress to effective flow shear stress is analogous to using the Tresca criterion for plastic flow. The notation Dskh refers to the reduced effective stress required 4 final to initiate plastic flow whereby Ds4 ) and 4 ¼ (s4s0 q 4 Dsqq ¼ (sqqfqsfinal ) and fq ¼ D sqq/D s4 0 4 ¼ sq/s4 becomes the condition for proportional loading, in which the concept of yield stress k is elucidated. At the origin of Dsqq versus Ds4 4, any increase of Dsh initiates plastic flow. To maintain clarity of which components are derived from which stress or strain states, the convention is adopted whereby the subscript h indicates direction of measurement and superscript k, that due to applied stress. Fig. 2 shows the 4 t conversion of Ds4 4 versus ε4 to effective compressive stress Dst versus thickness strain εtt using the indicated parameters in the following analyses. Thus equation (1) becomes
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4 4 Fig. 2. Conversion of UD locus to iso-flow stress model Ds4 4 ¼ Dsx4 using εx4 and then to Dstt and εtt as illustrated. The elevation of stress is due to additional compressive stress to account for lateral contraction during UD test. The vertical lines indicate lo4 t t cations corresponding to t3 and g3. Note !Ds4 4 dε4 ¼ !Dst dεt.
εff ¼ P 4A
Dsff 2 þ bf
2 þ bf ðambÞ2
1 C1 b
ð3þbf Þ
f
¼ 1 þ Rf εft
(2)
Mf
Another notation problem occurs for the following in which the stresses and strains in the direction of measurement needs to be distinctly separated from those expected during measurement to those used for the construction of the model. For this purpose, subscript x4 and xq will be used wherein x stands for experimental. Upon simultaneous application of biaxial stresses, the measured 4 total strain ε4 x4 in 4 direction is comprised of contributions from ε4 q q q q and that due to lateral contraction from sq, ε4 ¼ R εt , resulting in 4 4 4 q q q q q q ε4 x4 ¼ ε4 þ ε4 ¼ ε4 þ R εt ¼ ε4 R εq/(1 þ R ). 4 Thus PW in 4 direction is given by !Ds4 x4 dεx4 and in q as q q q q 4 4 4 !Dsxq dεxq with dεxq ¼ dεq R dε4/(1 þ R ). For the sake of clarity, the initial model invokes equality be4 tween Ds4 4 and Dsx4 and henceforth referred to as the iso-flow stress model shortened to iso-model:
Table 1 Fitting parameters from s-ε diagrams generated from the CRA for the unidirectional XRD and continuous tests of AA5754 and those for AA5154 for both RD (4) and TD (q). Sample AA5754-continuous RD AA5754-XRD UD-RD AA5754-continuous TD AA5754-XRD UD-TD AA5154 UD-4 AA5154 UD-q Sample AA5754-continuous RD AA5754-XRD UD-RD AA5754-continuous TD AA5754-XRD UD-TD AA5154 UD-4 AA5154 UD-q
a used
m (MPa)
sfinal (MPa) o
1/4
C1 (b1) (mm)
b1
0.4 0.4 0.4 0.4 0.4 0.4
32012 32012 32012 32012 32012 32012
68.69 78.83 66.15 83.62 84.62 85.86
131 150 149 120 192 138
0.333 0.201 0.574 0.303 2.607 1.824
0.117 0.720 0.285 0.595 0.829 0.635
P/A
t3 (MPa)
g3
l3 (mm)
b2
C1 (b2) (nm)
0.0952 0.0779 0.0839 0.1004 0.0650 0.0907
54.52 37.26 50.30 47.21 43.19 43.89
0.2941 0.1833 0.2817 0.3487 0.2401 0.2192
0.261 0.211 0.242 0.328 0.167 0.225
0.979 0.110 0.746 0.197 0.334 0.238
6.60 340.00 16.20 190.00 65.20 110.40
112
Z
S. Saimoto et al. / Acta Materialia 118 (2016) 109e119
Dsfxf dεfxf
"
# ) Z Rq q and ¼ Dsqxq dεqxq dε q 1 þ Rq Z Rf f dε ¼ Dsqq dεqq f 1 þ Rf Z
(
Dsff
dεff
(3) Hence collecting terms
Z PW ¼
Dsfxf dεfxf þ
Dsqxq dεqxq
Rf f dε f 1 þ Rf ( " # ) Z Rq f q q q Dsq dεq Dsf þ dεq 1 þ Rq Z
¼
Z
Dsff dεff Dsqq
(4)
q Although CRA can be performed to determine Ds4 x4 and Dsxq, for computational simplicity and for retention of model contributing components, the longer form will be used. It will be kept in mind that initiation of biaxial straining retards slip systems which contributes to q contraction resulting in a dynamic latent hardening effect but its role will manifest itself in the final calculated biaxial stresses in comparison with experiment. Necessary modifications will be incorporated in case by case basis depending on the workhardening trends for different alloys. For proportional loading exq periments, Dsqq/Ds4 4 is kept at a constant value, f and its inverse as 4 f . Thus PW under biaxial stress state whereby Dstt is the effective compressive stress for any specific imposed stress ratio is given as:
Z PW ¼
Dstt dεtt ¼ "
Z
f q Rf ¼ 1 1þRf
Dsff dεff #Z
Z q f f f f R Dsf dεf f
1þR "
Dsff dεff þ
1
f f Rq 1þRq
Z þ
#Z
Dsqq dεqq
Z
(6) dεqq 1þRf 1þRq dεff
whereby dεft ¼ fε q ¼ ¼ M 2 f 1þRf and M ¼ M4/Mq, the t Taylor factor ratio. The role of Mj does not manifest itself in equation (5) since the original stress-strain data is in normal coordinates which is converted to shear terms in order to perform CRA. The constitutive q 1þRq
(5)
Dsqq dεqq
h
i Dsf i fε q Dsq h f q þ 1 þ Rq f f Rq
1 þ Rf f q Rf
1 þ fε q 1 þ fε q
dεq
relation in shear terms are reconstructed to normal coordinates using constant Mj of 3.0. (Nevertheless if quantitative texture analysis for Mj and Rj were available from orientation distribution function (ODF) computation, the predictions would be slightly more precise than using constant Mj.) The near identical plots using equations (5) and (6) affirm the validity of the Levy-Mises relation. Computationally it is more transparent to use equation (5) to
f f Rq Dsqq dεqq 1þRq
By substitution of constitutive relations, equation (2), and integrating using the stress terms, PW can be calculated as function of εtt. The slope of this plot results in Dstt the effective compressive stress under biaxial stresses. Fig. 3 demonstrates this analysis. An analytical way to determine Dstt is by differentiating the t above relation with respect to dεtt for which the functions dε4 4/dεt q t and dεq/dεt are required. The application of Levy-Mises relation results in
Dstt ¼
Fig. 3. Plastic work versus thickness strain derived from Fig. 2 comparing UD in 4 to BB, equal biaxial stresses. The horizontal arrow indicate constant PW equivalent to 0.15 UD strain. Break in curves due to b1 to b2 transition.
determine the in-plane stresses and hence equation (6) is not used in the subsequent analyses. 2.2. Calculation of the in-plane stress component In order to delineate the model encompassed in equation (5), a graphical illustration of the analysis is presented using the BB stress state. Briefly the desired PW is defined by the unidirectional tensile test in 4 or q direction corresponding to rolling (RD) or transverse (TD) direction at a selected strain level. The problem is to find that combination of biaxial stresses in the imposed ratio which generates plastic work components the sum of which is the prescribed PW. The method selected to represent the evolution is to use the thickness strain εtt as the common abscissa. By incremental inq creases in PW, loci for Ds4 4 and Dsq are constructed under the q q desired f conditions and can be compared to that for ε4 x4 or εxq; as shown in Fig. 4 for one to one BB stress ratio. Using Excel spreadsheet, this procedure is readily possible and very illustrative but a simplified way is to directly apply equation (5) by using the prescribed values for PW, f q, f4 and the tabulated parameters in Table 1.
S. Saimoto et al. / Acta Materialia 118 (2016) 109e119
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4 Fig. 4. (a) Conversion of constant PW ¼ 19.5 from Fig. 3 to their respective reduced stress and strain axes. The sum of strains denoted by εqt ¼ 0.073 for Dsqq and by ε4 t ¼ 0.10 for Ds4 q q 4 is equal to that of εtt ¼ 0.173 which defines the BB stresses at 162 MPa, the difference in strains are due to texture. (b) For clarity, the comparison of Ds4 4 vs εx4 and Dsq vs εxq for equal q UD stress BB to initial UD test is shown. The sum of the areas at ε4 x4 ¼ 0.092 and εxq ¼ 0.08 in the BB case are equal to that prescribed by the UD-4 case at εUD ¼ 0.15. These output values of stress are those recorded for the yield locus.
Fig. 5. Reduced stress plot Dsqq versus Ds4 4 for AA5154 tests from Kuwabara et al. [9]. Determined at indicated UD strains. The computed iso-model yield predictions from equation (5) are shown as continuous lines. The subsequent analysis focus on εUD UD ¼ 0.15.
The accompanying principal strains are determined from the calculated stresses using equation (2).
3. Results 3.1. Predictions using biaxial test results of Kuwabara et al. [9]. The tubular tests of AA5154-H112 in this work [9] were notable in that proportional loading ranging from 1:0 (UD), 4:1, 2:1 4:3, 20:23 and 1:1 (BB) were carried out. The designation of 2:1 as PS is not indicated since this identity only occurs for an ideal plastic material whereas the means to derive the precise values will be described in the next section. In their publication, the UD stressstrain diagrams for 4 and q directions were presented including R4 ¼ 0.36 and Rq ¼ 0.59. After digital scanning the measured stresses (from their Fig. 7 [9]) at equivalent TW using prescribed values at εUD UD, the reduced stress terms are plotted in Fig. 5; that is, the origin is the point of eminent plastic flow with any incremental biaxial stress and the locus expands with work hardening. Hence the constitutive model can generate analytically the continuous
Fig. 6. Optimum predicted yield loci of AA5154 for PW and TW at εUD UD ¼ 0.15 conditions using UD from 4 and q axes together with R4 ¼ 0.36 and Rq ¼ 0.59. The iso-models were modified as described in text to account for inferred latent hardening using LHF factor.
loci. Since the current focus is the result at high strains, the apparent large deviations at low strains are not examined at present due to the complication of the b1 to b2 transition. At equivalent PW and TW to εUD UD at 0.15, the iso-model locus using equation (5) (as shown in Fig. 5) is adjusted using PW/LHF or TW/LHF and compared to experiment in Fig. 6. Note that near BB, TW predictions are lower than the PW ones but both are within 5% of the measured ones based on TW. Thus the derived functional form appears to encompass the yield locus evolution but the stresses near BB need to be increased. Referring to conventional predictions, for R smaller than unity should result in BB stresses below the UD yield stress; whereby the yield locus becomes circular for R ¼ 0.0. This effect is sometimes referred to as the “anomalous” behaviour of aluminum alloys [13] and will be treated in the discussion section. The procedure to attain these optimum fits will be described below. In order to consider the PW predictions and means to improve the fit near BB as illustrated in Fig. 6, the role of texture was assessed by determining the effective plastic Poisson ratio (defined q 4 as εxq q/ε4 x4) for BB case by plotting εxq versus εx4 plot at BB in Fig. 7a as suggested by Barlat et al. [14]. Due to the two-loci fit, a break
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S. Saimoto et al. / Acta Materialia 118 (2016) 109e119
Fig. 7. The predicted εxq q versus ε4 x4 for stress BB. Note for equal stresses the accompanying strains are not equal, that is slope is not unity. (a) The break indicates regions of b1 and b2. (b) The variation of strain ratios with that of stress are illustrated in the b2 regions.
(static) latent hardening ratio is √2 and those measured after rolling at 450 K was 1.40 [16]. After several trials, the factor q LHFq ¼ (1 þ K √fq) when s4 4 > sq (the corresponding q 4 4 4 LHF ¼ (1 þ K √f ) when sq > s4) were introduced such that Ds4x4 ¼ LHF Ds44.The necessary inclusion of the stress ratio is evident and the square root function was an improvement over the linear one, fq (not shown). This modification needs inclusion in equation (5) wherein PW is modified as PW/LHF and the corresponding biaxial calculated-stresses are designated as unit-model and multiplied by LHF to result in the optimum fits as shown in Fig. 6. The fitting magnitude for K was 0.2 for LHF ¼ 1.2 for TW locus and 0.075 for PW; LHF ¼ 1.075. The plots of Dskh versus εkxk for the pertinent BB and PS cases in relation to UD will be presented in a separate section in comparison with the AA5754 analysis to follow. These analyses are required in order to compare the calculated yield stresses to those of the measured ones at all biaxial ratios. 3.2. Comparison of model predictions to Iadicola et al. [10]. Fig. 8. Yield loci prediction for PW equivalent to UD strain of 0.15 using the iso-model and modifications by adopting constant R's which increases the BB stresses.
between b1 and b2 occurs but for the measure of ductility comparison the higher strain region is used. In this iso-model, the balanced stresses and balanced strains are not coincident and three cases at higher strains are illustrated in Fig. 7b. The stress ratio q corresponding to equal biaxial strains occurred at s4 4/sq ¼ 0.933 (model prediction) compared to that observed at 0.87 (experimental) [9] and εqq/ε4 4 ¼ 0.744 for model at equal-stresses. These differences are a manifestation of a large texture effect. This role of texture was noted by Barlat et al. [14] in performing compression tests along the sheet normal direction. To illustrate this effect, constant R ¼ 0.59 and 1.0 were determined using constant PW to compare with the iso-model and plotted in Fig. 8 indicating the increase in BB stress values. However this adjustment is curvefitting and not physically based. Thus the improvement shown in Fig. 6 invoked a dynamic latent hardening factor (LHF). Although the predictive fit to experiment of the iso-model, wherein Dskxk ¼ Dskk is assumed, is remarkable as demonstrated in Fig. 8, measurable differences which occur may be due to a dynamic latent hardening effect. As previously pointed out, the iso-model does not directly account for the modified obstacle structure as the second in-plane strain is proportionally applied. This application may involve accommodation of a LHF. For multiple slip situation of a rolled (110) [001] copper single crystal (25 mm wide by 2.5 mm thick) with reduction of 70%, the theoretical maximum
This work on AA5754 alloy was performed using an ingenious apparatus whereby the central dome section on either side were free surfaces and the stresses at various ratios were in-situ determined by standard X-ray line-shift method. For this reason by
Fig. 9. Predicted yield locus using CRA of AA5754 data [10] and the iso-model for PW equivalent to tensile strain of 0.15. The arrows indicate shift of biaxial stresses based on the current TW calculation method. The recalculated data points of TW and PW as described in text showed much better fits.
S. Saimoto et al. / Acta Materialia 118 (2016) 109e119
necessity the tests were not continuous but quasi-static. From the digital data set kindly made available by Dr. Iadicola, those performed by X-ray determination for UD, PS and BB were used in the initial trials. As will become evident, the scatter, noted by the authors [10], becomes large using the quasi-static tests but for selfeconsistent comparison, the analyses using these results follow. Applying equation (5), the iso-model locus in Fig. 9 was generated. In comparison with the published data, a large discrepancy is evident. Examination of the method used by Iadicola et al. [10] indicated that the calculated expended work was based on deviator tensor instead of the direct comparison of TW under UD with the sum of work due to biaxial stresses as used by Kuwabara et al. [9] and Barlat et al. [13]. Hence BB stress-strain data were transformed to constitutive relations using CRA and the biaxial stresses at BB for equivalent TW and PW as denoted at UD strain of 0.15 were calculated and plotted in Fig. 9. This procedure in effect converted the AA5754 data to the same basis as for the AA5154 one, as indicated in Fig. 9. The fit to the iso-model was remarkable and LHF was almost 1.0. This illustration reveals that the selected criterion for yield-locus evolution with strain requires careful stipulation and calculation. One way to examine the issues is to use the CRA to predict the evolution of strains with the respective biaxial stresses as follows.
3.3. Prediction of principal strains to corresponding principal stresses The model predicts the pertinent biaxial stresses which by means of equation (2) can generate the corresponding strains. Thus a continuous plot of Dskh versus εkxh for various denoted stress ratios can be generated as shown for the iso-model predictions using the UD data of Kuwabara et al. [9] in Fig. 10. The major stress-strain 4 locus is stronger than that for UD and the minor strain in q is almost perfectly zero, indicating ideal plane strain, if the stress ratio is fq ¼ 0.605 (Fig. 10a) and similarly for f4 ¼ 0.6 using q as the major strain. In Fig. 10b the stress ratios of 1:1 are shown wherein separation between UD and BB (4) and BB (q) are evident. These effects are attributed to the role of texture and dynamic latent hardening. In the case of Iadicola et al. [10] data with the previously noted wide scatter due to quasi-static nature of stress-strain determination, Fig. 11a for PS shows very small difference between model PS using the denoted fq ¼ 0.3 and UD but a noticeable difference from the measured data. To examine the possible causes, the model
115
using fq ¼ 0.605 better fits the data and from this model the corresponding minor stress-minor strain were generated as shown in Fig. 11b in which neither of these cases result in ideal plane strain as demonstrated in Fig. 10a. This situation arose since the apparatus used for these tests controlled the strain path and the measured stresses were in response to this strain control. The implication is that the current data of Iadicola et al. [10] in their Fig. 7 cannot be used to validate the current model. To overcome the quasi-static tests results from X-ray analysis, predictions from the continuous tests supplied by the authors, generated analogous plots to those of Fig. 10 in Fig. 12 which indicate that near plane strain condition is achieved near fq ¼ 0.688. Thus although the study of Iadicola et al. [10] is of great interest, one can only qualitatively conclude that strengthening during biaxial straining is positive but its precise measure requires tests with controlled plane strain. The analyses of the above results suggest that current constitutive relations prediction of yield locus is viable but it requires one of two ways to validate the fitting factor of LHF. This can be done by careful burst tests as proposed by Barlat et al. [14] to precisely determine the strains at balanced biaxial stresses or to perform precise plane strain measurements. The constitutive relations analyses of UD in RD and TD tests should be supplemented by through-thickness texture analysis using ODF to determine the mean Rj and Mj. If through such means a reproducible yield locus and the stress-strain loci for any desired stress ratio are validated, this methodology can be used to convert the strain maps of FLD into point-by-point stress maps.
4. Discussion 4.1. Summary of plastic work derivation and predictions The fundamental assumption of constitutive yield-locus model is that plastic work due to each principal strain can be algebraically summed and equated with that reference energy identified by the unidirectional tests in the principal strain direction; typically taken to be RD. The principal strains include the summation of negative contribution from the simultaneously applied second axial stress. This negative term is explicitly evident in equation (5). The analytical unidirectional stress-strain relation for each stress axis is determined using CRA. Since the usual tests are performed by proportional loading, the plastic work integral can be calculated by converting differential strain into that of stresses by means of CRA
Fig. 10. (a) Reduced stresses with their respective strain axes for AA5154 [9] comparing the loci for UD with those for biaxial stress with ratio fq ¼ 0.6 which results in plane strain condition. (b) Comparison of equal stress evolution with strain showing deviation between 4 and q axes.
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Fig. 11. (a) Analogous to Fig. 10, predicted Dskh versus εkxh for designated plane strain at fq ¼ 0.3 and that for UD compared to the measured values [10]. Note the lack of agreement and the expected separation between predicted PS and UD increases as fq is increased. (b) Analogous plot to Fig. 10a, showing that the minor strain does not approach plane strain until fq is increased as indicated.
Fig. 12. Reduced stresses versus respective strain plots generated from continuous tensile tests for UD. Note the predicted strengthening as the condition of plane strain is approached.
relations. This plastic work can be directly related to the integral of the equivalent compressive stress with incremental thickness change. Hence differentiation of the plastic work with respect to thickness strain (slope method) results in Dstt. By applying the Levy-Mises relation, an analytical relation can also be derived and the comparison of the resultant plot show good agreement; hence validating this conventionally-accepted relation which is the basis of plasticity theory. The iso-model assumed that the flow stress under biaxial stress states were the same as for unidirectional at the net strain due to the biaxial strains wherein one component is negative. This assumption resulted in a general-form yield locus which approached the measured data in shape and magnitude. The degree of fit, however, appeared to be a function of the measured Rvalues and implied the effect of latent hardening. Consideration of this issue indicated that the magnitude of the predicted biaxial stresses required to be larger by a factor LHF; this adjustable factor which is alloy dependent can be directly determined from the measured BB test, usually performed by bulge testing. The functional stress forms which comprise the proposed yield function presented in equation (5) is examined and for the sake of
clarity only the work term expressed in shear terms are used. Furthermore the material constants will not be explicitly denoted but the developing relations will be designated as proportional to (») PW. From equation (1) referring to only one term of PW as » !t dg » !t (t1þb) dt » t3þb. Hence for linear hardening wherein b ¼ 1.0, PW » t2 which is analogous to the von Mises function. Typically for aluminum alloys b1 ¼ 0.0, indicating parabolic hardening [12] and hence PW » t3. In the case of copper specimens, the b values at 20 K are near 1.0 increasing to b1 ¼ 0 and b2 ¼ 0.9 at 473 K [17]. Moreover since b1 < b2, a break occurs in Fig. 3. However to elucidate the evolution of the work function, t3þb can be rewritten as (t3 tb) whereby tb » (t dt/dg)1 and hence PW » t2/ (dt/dg). Thus as strain evolves the incremental increase in expended work approaches the quadratic form when (dt/dg) ¼ constant upon attaining Stage IV deformation regime. The inference of this observation is that initially the t2 term corresponds to the dislocation density but as strain progresses deformation debris is created, some of which dynamically recovers as volume fraction of point defects (CV) increase with strain. This progression was indicated during cold rolling of AA3003 with intermediate tensile tests till Stage IV was reached when CV attained 0.02 [18]. These inherent microstructure issues may be circumvented by relaxing the quadratic deviator stress function in formulating the plastic potential function. However identifying the cause for such deviation will be more useful in analyzing actual material performance during fabrication, especially for failure analyses. The attribute of the current model is that constitutive relations for any stress ratio and its corresponding strains can be calculated such that precise prediction for balanced biaxial stresses at unequal strains and precise plane-strain stress ratios can be predicted. The usefulness of such an approach to sheet formability modeling using local strain measurements to deduce stress states are apparent. In such analysis the role of evolving texture can be better assessed. Furthermore, the calculation of yield loci at constant PW rather than TW appears to better encompass the strain-hardening evolution; the method used in performing crystal plasticity finite element modeling. It can be readily shown by simplistic modeling using linear hardening, the use of TW results in slightly modifying the quadratic stress relation and this effect will become much larger for age-hardened alloys. There are some cautionary footnotes to the current derivation which needs to be kept in mind. First, the Taylor factor M was assumed to be constant at 3.0 but as previously
S. Saimoto et al. / Acta Materialia 118 (2016) 109e119
pointed out it does not explicitly affect the work relation of equation (5). However the analytical derivation of the effective normal stress via equation (6) varies with the M ratio. In future studies, Mj and Rj derived from orientation distribution function can be used. Secondly, the transition region at cross-over of b1 to b2 loci exists at low strains but for considerations of FLD at higher strains it does not hinder the analysis. Thirdly, the biaxial stresses were derived from the work relation PW ¼ stress-work integral which is also equal to strain-work integral and hence by means of CRA, strain maps should be convertible to stress maps on a point-by-point determination of the grid-based experimental data. Aside from generating the uni-directional stress-strain locus by plasticity theory, there have been many studies to derive the work-hardening locus either from dislocation mechanics models [19] or by using mesoscopic functional relations with adjustable parameters to result in the best fit [20]. However none of these works can replicate the stress-strain loci as well as the current S-VH relation which correlates the fit parameters to the evolution of work-hardening. This procedure is unique in that the fit-parameters are the precisely fitted modeled ones from which the microstructure evolution is assessed. 4.2. How does this simple derivation result in credible yield-loci predictions? The simple yield function from this analysis is reproduced below whereby fq and f4 are the prescribed stress ratios and prescribed PW incorporating texture and work-hardening.
PW ¼ LHF
(
f q Rf 1 1 þ Rf
)Z
(
Dsff dεff
þ
f f Rq 1 1 þ Rq
)Z
Dsqq dεqq
(7)
4 final Thus biaxial stresses become s4 and x4 ¼ LHF Ds4 þ s0
sqxq ¼ LHF Dsqq þ fq sfinal and similarly in the stress space whereby f4 0
applies. The final derivation means that from the right hand side which is the iso-model its stress predictions are multiplied by the appropriate LHF ¼ (1 þ K √fk). Although the answer to the above question may take much discussion, the bases of the rationale must be due to the nature of work hardening. The implications of the Cottrell-Stokes relation with respect to biaxial deformation have recently become apparent to the current authors. Basically it reveals that work hardening proceeds by slip processes the by-
Fig. 13. Normalized t/t3 versus g/g3 plot for AA5154 and AA5754 after CRA of continuous UD results showing that the 4 (RD) and q (TD) loci become coincidental although the R values are not identical. The differences between the alloys are attributed to disproportioning of cells to Taylor lattice structures.
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product of which is the increased density of obstacles. Furthermore if the nano-scale thermally activatable atomic sites are identical although the line lengths between them shorten, the strain-rate sensitivity S is constant. Thus the relation between activation work t n ¼ k/S (k, the Boltzmann constant) and S indicate that t n is constant. The deduced conclusion is that since t is inversely proportional to [, wherein n ¼ [ b d with [, being the interobstacle length and b, the Burgers vector, the activation distance, d, must be constant. Under constant temperature and strain rate, it is evident that if the obstacle type does not change with strain, d remains constant and only the obstacle-density increases. Moreover it can be readily shown that positive local internal stresses assist in overcoming dense patches of obstacles whereas negative local internal stresses reduce the applied stress such that varying densities of obstacles can co-exist. For the lowest free energy state, the compressive volume fractions must be higher than that of the tension regions. This structure essentially gives rise to cells which are self-organized during the entropy production process. The upshot is that local tloc nloc is equal to the product of mean t and mean n which is directly experimentally measurable. To demonstrate such a microstructural evolution takes place, the constitutive relations involving unidirectional tests for AA5154 and AA5754 are plotted after normalization with t3 and g3 in Fig. 13. Note these parameters satisfy both b1 and b2 regions of CRA. The resulting near-master curve is found seemingly unaffected by different test temperatures [12]. Thus the activation distance controls the thermodynamic response of plastic deformation and it will be constant even under biaxial stresses if the obstacle type for these strain states does not change. However compared to unidirectional tests, upon simultaneous application of the second principal strain, the slip systems which contribute not only to thickness strain but also lateral contraction strains are retarded. Thus the evolving work hardening structure will differ somewhat from that of unidirectional. Nevertheless the structure evolution will only change in scale not distribution within the constant b region. Furthermore depending on the alloy the degree of cell versus Taylor lattice structure may change. The outstanding fact of plastic flow is that t ¼ a m b/[ ¼ a m b r1/2 whereby r is the mean dislocation density and a, the obstacle strength factor. Thus the constant expended plastic work under biaxial strain is related to the amount of dislocation generated but since this energy is scalar, the plastic work is just the sum of its component parts. Thus if the self-organizing internal stresses arrange the obstacles in a minimum energy configuration, the stored work may vary resulting in mean dislocation density difference. To illustrate this deduction, the predicted PW versus εtt for BB and PS for AA5154 is compared to that of UD in Fig. 14a. Using the constant PW at UD strain of 0.15, the corresponding εtt and Dstt for BB and PS are converted to skxk versus εkxk as presented in Fig. 14b. Future experiments generating such plots will not only validate the analysis but reveal the work hardening differences depending on the biaxial ratios. Using CRA on PS 4, BB 4 and UD 4 with fit parameters in Table 2, the normalized plot in Fig. 14c show that PS and BB are very similar but measurable difference occur with respect to UD due to application of LHF ¼ 1.075. The rate of doing work is related to the activation work as fol _ lows: tg_ ¼ tn gn ; that is, it is a product of the activation work and the density of dynamic activation sites [21]. Hence flow stress is controlled by thermal activation of the mean obstacle density. However if the activation distance is constant the evolution of strain is basically crystallographic and follows the Taylor slip analysis which is the basis for crystal plasticity finite element modeling. The activation distance factor is taken into account in equation (1) in the proportionality constant which is strain-rate
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S. Saimoto et al. / Acta Materialia 118 (2016) 109e119
Fig. 14. (a) Derived plastic work for AA5154 [9] at which magnitudes of constant PW ¼ 19.5 with minimum of 18.14 were used to determine the biaxial stresses at varying fq. (b) Inplane stresses for selected PW for PS and BB conditions. Note the PS is slightly larger than UD whereas BB is slightly lower as shown in Fig. 6. (c) Normalized shear stress e shear strain plots showing that the nature of work hardening under UD generated obstacles and BB (PS) ones are not identical as indicated by LHF factor.
Table 2 CRA fit parameters from Fig. 14b.
s ratio and direction
Sample 4 x4
AA5154 s
vs
ε4 x4
vs
ε4 x4
q
f - UD, f ¼ 0 f - PS, fq ¼ 0.608 f - BB, f4 ¼ 0.933
s ratio and direction
Sample 4 x4
AA5154 s
q
f - UD, f ¼ 0 f - PS, fq ¼ 0.608 f - BB, f4 ¼ 0.933
a used
m (MPa)
sfinal (MPa) o
1/4
C1 (b1) (mm)
b1
0.4 0.4 0.4
32012 32012 32012
84.62 70.15 60.48
192 113 86
2.607 0.961 0.903
0.829 0.475 0.481
LHF
P/A
t3 (MPa)
g3
l3 (mm)
b2
C1 (b2) (nm)
1.000 1.075 1.075
0.0650 0.1107 0.1446
43.19 55.78 63.55
0.2401 0.2234 0.1921
0.167 0.186 0.162
0.334 0.520 0.576
65.20 29.00 19.05
and temperature dependent and reduces to the modified Hollomon relation. Thus, if the thermodynamic term is inherently included as well as the dynamic annihilation (A term in equation (1)), incremental shear due to local shear stress is the same for such combination to comply with the imposed strain. Fig. 13 appears to affirm this deduction. The result is the Levy-Mises relation becomes just the utilization of slip systems dealing with a shape-changing incompressible solid. Noteworthy demonstrations are the tensile testing of [001] single crystals wherein R is unity or the rolling of (110) [001] crystal slabs [16]. The conclusion is that the rate of doing
work is determined by thermally activated flow but the density of obstacles are derived from the shape-change geometry and hence a simple addition of plastic work of the component stress-strain states can describe the evolving structure. The inference is that although entropy production is large, to attain a given total work magnitude its variation with different unidirectional straining conditions is small according to the variation in P/A (Table 1). The deduction is that the expended energy for shape change is directly related to the stored work. This premise is basic to the S-VH constitutive relation whereby the energy expended is related to
S. Saimoto et al. / Acta Materialia 118 (2016) 109e119
dislocation creation and equated to the retained dislocation density by invoking an annihilation factor A. Thus the credible predictions using this Taylor slip-model relation are self-consistent and satisfy the shape-change and thermodynamic conditions. Table 2 shows that if the change in LHF from 1.0 to 1.075 the P/A increases under biaxial stresses. This observation indicates that the increase in strength is due to some change in the nature of obstacles as suggested by Fig. 14c. On considering the representative elemental volume (REV), the shape change response is assumed to be homogeneous of all such elements and its response to the bounding traction forces are the same, neglecting micro-band formation. Although electron microscopy show vast differences in array of dislocations and barriers within each REV such as dispersoids and grain boundaries, each REV is homogenized by the selforganized internal stresses and the shear flow stresses evolve in almost isotropic manner and crystallographic slip conform to the imposed strain. The crystal plastic finite element analysis using this relation for UD to predict that of PS and BB have been recently demonstrated [22]. The current analysis is analogous to that of non-linear elasticity whereby the stress-strain relation is precisely known throughout the given strain range. Complication may occur if the plastic true strain rate becomes a function of strain or if the friction heat starts to affect the isothermal testing conditions. The current status is that t3 is dependent on temperature and volume fraction of generated defects which give rise to osmotic stress which in turn assist in overcoming shear resistant obstacles. Thus the long range intent of this study is to generate stress-strain relations under plane strain conditions to examine various criteria of limit strain predictions for FLD as previously described [15]. 5. Conclusions The concept of effective compressive stress versus the thickness strain suggested that components of the plastic work due to the biaxial stresses can be represented on a common basis of thickness strain since each contributes to it in an assumed additive manner. From this premise and using the constitutive relations which can precisely replicate the measured stress-strain diagram, a simple formulation for the in-plane stresses applied under proportional loading was derived. The comparison of predictions to the published yield determinations for AA5154 [9] and AA5754 [10] resulted in good agreement especially since the same formulation were applied to both alloys. The “anomalous” BB yield behaviour in aluminum alloy is inherent to strain-hardening mechanisms attributed to the thermally activated flow mechanisms found in face-centred cubic metals; that is since the flow stress is characterized by the mean dislocation density unaffected by internal stresses, yielding tend to obey Tresca criterion if the texture is not highly developed. This analysis permits generation of the component stresses versus its accompanying strains such that evolution of work hardening under different stress states can be examined. Such predictions will become useful in modeling change of strain direction during sheet fabrication as well as for ductile failure analysis.
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Acknowledgements The authors thank the Natural Sciences and Engineering Research Council of Canada through support of the Automobile Partnership Cooperation program, a consortium effort commissioned at the University of Waterloo. For one of us (SS), it has been a long journey through crystal plasticity studies before trying to integrate work-hardening with yield locus evolution, a topic of great interest to his thesis supervisors: Professors W. A. Backofen and W. F. Hosford; originators of texture strengthening analyses.
References [1] S.S. Hecker, A.K. Ghosh, H.L. Gegel (Eds.), Formability Analysis, Modeling and Experimentation, Met. Soc. AIME, 1978. [2] S.P. Keeler, Determination of forming limit in automotive stamping, Met. Ind. 42 (1965) 683e691. [3] G.M. Goodwin, Application of strain analysis to sheet metal forming problems, La, Metall, Italiana 8 (1968) 7657e7774. [4] C.N. Reid, Deformation Geometry for Materials Scientists, Pergamon, Oxford, UK, 1973. [5] Z. Marciniak, K. Kuczynski, Limit strains in the processes of stretch-forming sheet metal, Int. J. Mech. Sci. 9 (1967) 609e620. [6] P. Van Houtte, A. Van Bael, S. He, Anisotropy and formability in sheet metal forming, in: J.M.A. Cesar de Sa, A.D. Santos (Eds.), Proc. NUMIFORM. Amer. Inst. Phys. NY, USA, 2007, pp. 159e164. [7] T. Balan, B. Bacroix, C. Teodosiu, Cold metal forming, in: D. Banabic (Ed.), Proc. TPR 2000, Techn. Univ., Cluj-Napoca, Romania, 2000, pp. 73e84. [8] P. Dasappa, K. Inal, R.K. Mishra, The effects of anisotropic yield functions and their parameters on prediction of forming limit diagrams, Int. J. Solid Struct. 49 (2012) 3528e3550. [9] T. Kuwabara, K. Yoshida, K. Narihara, S. Takahashi, Anisotropic plastic deformation of extruded aluminum alloy tube under axial forces and internal pressure, Int. J. Plast. 21 (2005) 101e117. [10] M. Iadicola, T. Foecke, S.W. Banovic, Experimental observations of evolving yield loci in biaxially strained {AA5754-O}, Int. J. Plast. 24 (2008) 2084e2101. [11] S. Saimoto, P. VanHoutte, Constitutive relation based on Taylor slip analysis to replicate work-hardening evolution, Acta Mater. 59 (2011) 602e612. [12] S. Saimoto, D.J. Lloyd, A new analysis of yielding and work hardening in AA1100 and AA5754 at low temperatures, Acta Mater. 60 (2012) 6352e6361. [13] F. Barlat, Y. Maeda, K. Chung, M. Yanagawa, J.C. Brem, Y. Hayashida, D.J. Lege, K. Matsui, S.J. Murtha, S. Hattori, R.C. Becker, S. Makosey, Yield function development for aluminum alloy sheets, J. Mech. Phys. Solids 45 (1997) 1727e1763. [14] F. Barlat, J.C. Brem, J.W. Yoon, K. Chung, R.E. Dick, D.J. Lege, F. Pourboghrat, S.H. Choi, E. Chu, Plane stress yield function for aluminum alloy sheetsdpart 1: theory, Int. J. Plast. 19 (2003) 1297e1319. [15] S. Saimoto, B.J. Diak, D.J. Lloyd, Point defect generation, nano-void formation and growth. II. Criterion for ductile failure, Philos. Mag. 92 (2012) 1915e1936. [16] L.E. Collins, Plane Strain Deformation and Subsequent Recovery of Ideally Oriented Copper Single Crystals, M.Sc. Thesis, Queen's University, Kingston, Canada, 1977. [17] S. Saimoto, B.J. Diak, Point defect generation, nano-void formation and growth. I. Validation, Philos. Mag. 92 (2012) 1890e1914. [18] S. Saimoto, K. Inal, H. Jin, Constitutive analyses of tensile tests on pre-rolled AA3003 sheets, Mater. Sci. Forum 794e796 (2014) 710e715. [19] Y. Aoyagi, K. Shizawa, Multiscale crystal plasticity modeling based on geometrically necessary crystal defects and simulation on fine-graining for polycrystal, Int. J. Plast. 23 (2007) 1022e1040. di, N.Q. Chinh, J. Gubicza, T.G. Langdon, Plastic behavior of fcc metals [20] T. Csana over a wide range of strain: macroscopic and microscopic descriptions and their relationship, Acta Mater. 59 (2011) 2385e2391. [21] S. Saimoto, Dynamic dislocationedefect analysis, Philos. Mag. 86 (2006) 4213e4233. [22] A.P. Brahme, K. Inal, R.K. Mishra, S. Saimoto, A new strain hardening model for rate-dependent crystal plasticity, Comput. Mater. Sci. 50 (2011) 2898e2908.
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Full length article
New biaxial yield function for aluminum alloys based on plastic work and work-hardening analyses S. Saimoto a, *, P. Van Houtte b, K. Inal c, M.R. Langille a a
Mechanical and Materials Engineering, Queen's University, Kingston, ON K7L 3N6, Canada MTM, Katholieke Universiteit, Lueven, Belgium c Department of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 4 May 2016 Received in revised form 11 May 2016 Accepted 19 July 2016
Yield locus measurements and their analytical descriptions has been the bases for modeling metal processing. These analytical descriptions play a role in models to predict limit strains observed during determination of forming limit diagrams of flat metal stock as means to evaluate their fabrication performance. Some of these analytical descriptions use the isotropic or anisotropic plastic potentials that take into account the crystallographic texture of the material. The applicability of such potentials is validated by comparing their predictions to that of unidirectional tensile data. In the current work, this approach is reversed to examine whether the constitutive relations, which replicate the measured stressstrain diagrams, can generate a two-dimensional section of the yield locus. The strategy is to sum the computed plastic work (PW) from unidirectional mechanical tests in two principal directions which also accommodates the biaxial interaction strains. The resulting yield function includes fq and f4, the prescribed stress ratios and the texture parameters R4 and Rq in their respective principal directions. The prescribed PW at an arbitrary strain during unidirectional tensile test is equated to the work sum from the biaxial stresses on the premise that the plastic flow-stress registers only the increasing density of obstacles it generates. The computed biaxial yield stresses showed good fits for AA5154 and AA5754 with only small modification due to latent work hardening. © 2016 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Constitutive modeling Yield locus prediction Plastic work Biaxial stress-strain diagram Work-hardening modeling
1. Introduction Formability of metallic sheet for stamping and deep-drawing operations has been of high industrial interest in order to optimize die-designs for fabrication processes and the progress of such studies to 1977 has been compiled by Hecker et al. [1]. The practical test of examining etched patterns on sheet surface to determine the principal strains at which limit strains of localized necking occur has been developed by Keeler [2] and Goodwin [3]. This mapping of limit strains for sheet fabrication has become the industrial standard, referred to as the forming limit diagram (FLD). The theoretical basis for the analysis is based on the observation that plastic shapechange is independent of hydrostatic stresses whereby the principal stresses are all equal. A very simple yield locus is the wellknown von Mises criterion for the occurrence of plastic flow. Today's experts in plasticity theory know that it merely represents a
* Corresponding author. E-mail address: [email protected] (S. Saimoto). http://dx.doi.org/10.1016/j.actamat.2016.07.036 1359-6454/© 2016 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
hypersphere in stress deviator space, i.e. it is the simplest possible closed surface in that sphere. Reid [4] points out that it can be based on the notion that plastic flow begins when the elastic distortion energy reaches a critical limit, usually designated at 0.2% off-set strain. However, from a physical point of view, this criterion is only meaningful for a material which has ideal isotropic elastic properties such as tungsten but not for the other crystalline materials. Nevertheless, in materials engineering the von Mises approximation is often used for cubic materials with a weak crystallographic texture as a reasonable first estimate. This analytical basis for a yield locus was accompanied by experimental observations that the incremental component of in-elastic strain may be computed by differentiating the yield function, in terms of stresses, with that specific stress component. This initial Levy-Mises criterion [4] has become known as the associated flow rule whereby the total plastic strain increment is normal to the yield surface. This extension of the von Mises criterion based on the existence of a critical magnitude of distortion energy for initiation of plastic flow to pre-strained work-hardened structure, as conventionally perceived, inherently assumes that the measured plastic work
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directly corresponds to the elastic distortion energy and that elastic anisotropy can be ignored (reasonable for tungsten, not unreasonable for aluminum, as the elastic anisotropy of aluminum single crystals is not very large). The current work suggests that plastic work initially follows a quadratic stress relation which becomes cubic and returns to quadratic at very large strains and inversely correlates with the work-hardening coefficient as will be examined in the discussion section. The seminal advance in the application of mathematical theory of plasticity to sheet formability was initiated by Marciniak and Kuczynski [5] in order to predict localized necking in biaxial stretch cases. However in order to initiate this strain localization, an initial existing groove in the plane of the sheet was envisioned. At present, this analysis is widely invoked to introduce the role of texture in FLD [6] and recent hypotheses invoke yield functions which do not retain the quadratic stress relation and hence the concept of distortion energy is relaxed. Overviews of such works are found in the following reports [7,8]. The validation of yield function is experimentally difficult since for cases other than unidirectional tests, the determination of the flow stress components and its accompanying plastic strains are imprecise. One well-defined test is that of stretching thin-walled tubular specimens with correlated internal pressure. The difficulty of precise strain measurement has been overcome by Kuwabara's group [9] using a spherometer, a specially developed displacement device for large in-elastic strains. In their study the stresses along loading paths of set biaxial ratios parallel to the cylinder axis (4) and its circumference (q) were determined and the total energy (TW) required for specific deformation modes were equated to the area defined by the stress-strain diagram for unidirectional tension to attain a given tensile strain to define the evolution of yield loci. Using extruded tubing of AA5154, the yield loci data at increasing degree of cold work were compared to various theoretical yield functions mentioned above [7,8]. Another notable study is due to Iadicola et al. [10] which measured the inplane stresses of biaxial stressed sheets of AA5754 using in-situ X-ray diffraction. From the quasi-static recording during unidirectional tests (UD), balanced biaxial (BB) and plane-strain (PS) cases, the principal stresses and strains were reported. These short-hand notations have been retained in this work for the ease of crossreferencing. The strategy to derive constitutive relations for in-plane biaxialstressed sheet in this work is to reverse that of conventional practice of yield function determination in which the fit parameters are matched with measured in-plane stresses at points of constant total work. The constitutive relation derived from this yield function is compared to the unidirectional tensile one to attest to its validity. In the current work, constitutive relation analyses (CRA) which can replicate the measured stress-strain diagram using at least two-fit loci become the starting point from which the plastic work due to biaxial stresses are calculated. The CRA procedure defines a new yield stress (sfinal ) by back-extrapolating the plastic 0 response from beyond the yield phenomenon and the associated yield point elongation (YPE). This determination corresponds to an off-set strain of 0.02%, slightly above the proportional limit as described elsewhere [11]. Hence the plastic work (PW) is defined as !(ssfinal ) dεp wherein s is the applied stress, (s-sfinal ), the flow 0 0 stress and εp, the logarithmic (true) plastic strain. Note that the total work (TW) includes !sfinal dεp which give rise to heat as in the case 0 of friction stress and contributions from the yield phenomenon are removed. Since the new constitutive relations by Saimoto and Van Houtte (S-VH) [11] based on the Taylor slip model directly correlate the flow shear stress to the shear strain [11,12], conversion to normal stress and strain was performed in order to use the width to thickness strain ratio R (Lankford ratio), as an indirect measure of textural effects on plastic flow. The hypothesis invokes that under
biaxial stress-states the incurred principal plastic strains are additive and also plastic work (PW) under proportional loading are additive. The net result for sheet stock is that the derivative of PW at a given thickness strain gives rise to the effective compression stress at that point. This slope-method offers the possibility of comparing the yield locus at constant PW to that of constant thickness strain. However the usual procedure is to use constant TW for the cases reported [9,10] which includes the yield stress and explicitly described by Barlat et al. [13,14]. Comparison of this criterion of constant TW to that of constant PW will be presented. Hence the new biaxial yield function is based on this PW relation that can be analytically derived using the constitutive relation which replicates the stress-strain diagram and integrated with respect to the in-plane stresses which are proportionally related. By equating PW from tensile tests at a given strain and using this relation, the biaxial stresses and strain can be calculated. The attribute of analytical prediction of yield locus from constitutive relations is two-fold. 1) The yield locus can be continuously mapped and compared with the data to assess the degree of fit. Moreover the effect of the texture-parameter R on the point-by-point curvature or tangent at any given stress ratio can be assessed. 2) From the calculated in-plane stresses for any given stress ratio, the strains corresponding to those stresses can also be calculated from which the principal major (minor) stress-principal major (minor) strain loci can be generated. This procedure also implies that CRA of such measured loci for the in-plane stresses can be analyzed to predict the strains in three principal directions. For example, such analysis for PS case can be used to ascertain if the actual test met the condition of continuous zero strain in the minor axis. The evolution of stresses under PS or BB conditions can be CRA analyzed to assess ductile failure and FLD prediction according to prior-described methodology [15].
2. Analytical procedures 2.1. Derivation of yield function based on plastic work Fig. 1 shows CRA of UD test in 4 direction whereby the two-fit
Fig. 1. Stress-strain diagram for AA5154 for UD 4 axis from Kuwabara et al. [9] showing the two-fitted loci using CRA. Note the definition of sfinal and the location of corre0 sponding coordinates to t3 and g3 which is slightly off-line giving rise to transition region in derived expended-energy plots.
S. Saimoto et al. / Acta Materialia 118 (2016) 109e119
loci results in sfinal by back extrapolation from beyond the YPE and 0 the two loci with b1 and b2 power-law exponents intersect at t3 and g3. This transition in the plastic flow curve means that upon applying biaxial stresses to the start material, the major stress will attain t3 prior to that for the minor. In this situation it is assumed the minor one will start to follow its prescribed b2 locus rather than continue on b1. The rationale being that the minor strain is confronted by the obstacles generated by the major one. The kinks in the following graphs indicate this transition point but the region of ambiguity is small. For PW ¼ !t dg ¼ !(ssfinal ) dε and 0 !sfinal dε, as the lattice friction work; the elastic strain components 0 have been subtracted from the total strain, a typical procedure in such analysis. Furthermore it is understood that work means the rate of doing work. The sum of these two terms is TW. Note that (ssfinal )/M is the shear flow stress t and Mε ¼ g wherein M is the 0 Taylor factor. The CRA parameters for all the mechanical tests are listed in Table 1. The Saimoto-Van Houtte (S-VH) constitutive relation [11] is given below with mean slip distance defined as l ¼ C1 tb,
t¼
1 P 1 2þb 2þ1 b ð2 þ bÞðambÞ2 g 4A C1 b
(1)
wherein P is a geometrical ratio of the additional dislocation length encompassing the incremental slip area, typically near unity, and A is the dynamic annihilation factor. Since the optimum fit requires two loci, the first power-law exponent is designated b1 and the second, b2. For the current analytical purpose, tj is converted from (sjsfinal )/M to Dsj/Mj and gj to Mj εj such that the texture effect 0 can be taken into account using Rj values wherein the script notation j indicates the direction of applied stress. In the subsequent analysis it will become evident that conversion of multi-slip tensile stress to effective flow shear stress is analogous to using the Tresca criterion for plastic flow. The notation Dskh refers to the reduced effective stress required 4 final to initiate plastic flow whereby Ds4 ) and 4 ¼ (s4s0 q 4 Dsqq ¼ (sqqfqsfinal ) and fq ¼ D sqq/D s4 0 4 ¼ sq/s4 becomes the condition for proportional loading, in which the concept of yield stress k is elucidated. At the origin of Dsqq versus Ds4 4, any increase of Dsh initiates plastic flow. To maintain clarity of which components are derived from which stress or strain states, the convention is adopted whereby the subscript h indicates direction of measurement and superscript k, that due to applied stress. Fig. 2 shows the 4 t conversion of Ds4 4 versus ε4 to effective compressive stress Dst versus thickness strain εtt using the indicated parameters in the following analyses. Thus equation (1) becomes
111
4 4 Fig. 2. Conversion of UD locus to iso-flow stress model Ds4 4 ¼ Dsx4 using εx4 and then to Dstt and εtt as illustrated. The elevation of stress is due to additional compressive stress to account for lateral contraction during UD test. The vertical lines indicate lo4 t t cations corresponding to t3 and g3. Note !Ds4 4 dε4 ¼ !Dst dεt.
εff ¼ P 4A
Dsff 2 þ bf
2 þ bf ðambÞ2
1 C1 b
ð3þbf Þ
f
¼ 1 þ Rf εft
(2)
Mf
Another notation problem occurs for the following in which the stresses and strains in the direction of measurement needs to be distinctly separated from those expected during measurement to those used for the construction of the model. For this purpose, subscript x4 and xq will be used wherein x stands for experimental. Upon simultaneous application of biaxial stresses, the measured 4 total strain ε4 x4 in 4 direction is comprised of contributions from ε4 q q q q and that due to lateral contraction from sq, ε4 ¼ R εt , resulting in 4 4 4 q q q q q q ε4 x4 ¼ ε4 þ ε4 ¼ ε4 þ R εt ¼ ε4 R εq/(1 þ R ). 4 Thus PW in 4 direction is given by !Ds4 x4 dεx4 and in q as q q q q 4 4 4 !Dsxq dεxq with dεxq ¼ dεq R dε4/(1 þ R ). For the sake of clarity, the initial model invokes equality be4 tween Ds4 4 and Dsx4 and henceforth referred to as the iso-flow stress model shortened to iso-model:
Table 1 Fitting parameters from s-ε diagrams generated from the CRA for the unidirectional XRD and continuous tests of AA5754 and those for AA5154 for both RD (4) and TD (q). Sample AA5754-continuous RD AA5754-XRD UD-RD AA5754-continuous TD AA5754-XRD UD-TD AA5154 UD-4 AA5154 UD-q Sample AA5754-continuous RD AA5754-XRD UD-RD AA5754-continuous TD AA5754-XRD UD-TD AA5154 UD-4 AA5154 UD-q
a used
m (MPa)
sfinal (MPa) o
1/4
C1 (b1) (mm)
b1
0.4 0.4 0.4 0.4 0.4 0.4
32012 32012 32012 32012 32012 32012
68.69 78.83 66.15 83.62 84.62 85.86
131 150 149 120 192 138
0.333 0.201 0.574 0.303 2.607 1.824
0.117 0.720 0.285 0.595 0.829 0.635
P/A
t3 (MPa)
g3
l3 (mm)
b2
C1 (b2) (nm)
0.0952 0.0779 0.0839 0.1004 0.0650 0.0907
54.52 37.26 50.30 47.21 43.19 43.89
0.2941 0.1833 0.2817 0.3487 0.2401 0.2192
0.261 0.211 0.242 0.328 0.167 0.225
0.979 0.110 0.746 0.197 0.334 0.238
6.60 340.00 16.20 190.00 65.20 110.40
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Z
S. Saimoto et al. / Acta Materialia 118 (2016) 109e119
Dsfxf dεfxf
"
# ) Z Rq q and ¼ Dsqxq dεqxq dε q 1 þ Rq Z Rf f dε ¼ Dsqq dεqq f 1 þ Rf Z
(
Dsff
dεff
(3) Hence collecting terms
Z PW ¼
Dsfxf dεfxf þ
Dsqxq dεqxq
Rf f dε f 1 þ Rf ( " # ) Z Rq f q q q Dsq dεq Dsf þ dεq 1 þ Rq Z
¼
Z
Dsff dεff Dsqq
(4)
q Although CRA can be performed to determine Ds4 x4 and Dsxq, for computational simplicity and for retention of model contributing components, the longer form will be used. It will be kept in mind that initiation of biaxial straining retards slip systems which contributes to q contraction resulting in a dynamic latent hardening effect but its role will manifest itself in the final calculated biaxial stresses in comparison with experiment. Necessary modifications will be incorporated in case by case basis depending on the workhardening trends for different alloys. For proportional loading exq periments, Dsqq/Ds4 4 is kept at a constant value, f and its inverse as 4 f . Thus PW under biaxial stress state whereby Dstt is the effective compressive stress for any specific imposed stress ratio is given as:
Z PW ¼
Dstt dεtt ¼ "
Z
f q Rf ¼ 1 1þRf
Dsff dεff #Z
Z q f f f f R Dsf dεf f
1þR "
Dsff dεff þ
1
f f Rq 1þRq
Z þ
#Z
Dsqq dεqq
Z
(6) dεqq 1þRf 1þRq dεff
whereby dεft ¼ fε q ¼ ¼ M 2 f 1þRf and M ¼ M4/Mq, the t Taylor factor ratio. The role of Mj does not manifest itself in equation (5) since the original stress-strain data is in normal coordinates which is converted to shear terms in order to perform CRA. The constitutive q 1þRq
(5)
Dsqq dεqq
h
i Dsf i fε q Dsq h f q þ 1 þ Rq f f Rq
1 þ Rf f q Rf
1 þ fε q 1 þ fε q
dεq
relation in shear terms are reconstructed to normal coordinates using constant Mj of 3.0. (Nevertheless if quantitative texture analysis for Mj and Rj were available from orientation distribution function (ODF) computation, the predictions would be slightly more precise than using constant Mj.) The near identical plots using equations (5) and (6) affirm the validity of the Levy-Mises relation. Computationally it is more transparent to use equation (5) to
f f Rq Dsqq dεqq 1þRq
By substitution of constitutive relations, equation (2), and integrating using the stress terms, PW can be calculated as function of εtt. The slope of this plot results in Dstt the effective compressive stress under biaxial stresses. Fig. 3 demonstrates this analysis. An analytical way to determine Dstt is by differentiating the t above relation with respect to dεtt for which the functions dε4 4/dεt q t and dεq/dεt are required. The application of Levy-Mises relation results in
Dstt ¼
Fig. 3. Plastic work versus thickness strain derived from Fig. 2 comparing UD in 4 to BB, equal biaxial stresses. The horizontal arrow indicate constant PW equivalent to 0.15 UD strain. Break in curves due to b1 to b2 transition.
determine the in-plane stresses and hence equation (6) is not used in the subsequent analyses. 2.2. Calculation of the in-plane stress component In order to delineate the model encompassed in equation (5), a graphical illustration of the analysis is presented using the BB stress state. Briefly the desired PW is defined by the unidirectional tensile test in 4 or q direction corresponding to rolling (RD) or transverse (TD) direction at a selected strain level. The problem is to find that combination of biaxial stresses in the imposed ratio which generates plastic work components the sum of which is the prescribed PW. The method selected to represent the evolution is to use the thickness strain εtt as the common abscissa. By incremental inq creases in PW, loci for Ds4 4 and Dsq are constructed under the q q desired f conditions and can be compared to that for ε4 x4 or εxq; as shown in Fig. 4 for one to one BB stress ratio. Using Excel spreadsheet, this procedure is readily possible and very illustrative but a simplified way is to directly apply equation (5) by using the prescribed values for PW, f q, f4 and the tabulated parameters in Table 1.
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4 Fig. 4. (a) Conversion of constant PW ¼ 19.5 from Fig. 3 to their respective reduced stress and strain axes. The sum of strains denoted by εqt ¼ 0.073 for Dsqq and by ε4 t ¼ 0.10 for Ds4 q q 4 is equal to that of εtt ¼ 0.173 which defines the BB stresses at 162 MPa, the difference in strains are due to texture. (b) For clarity, the comparison of Ds4 4 vs εx4 and Dsq vs εxq for equal q UD stress BB to initial UD test is shown. The sum of the areas at ε4 x4 ¼ 0.092 and εxq ¼ 0.08 in the BB case are equal to that prescribed by the UD-4 case at εUD ¼ 0.15. These output values of stress are those recorded for the yield locus.
Fig. 5. Reduced stress plot Dsqq versus Ds4 4 for AA5154 tests from Kuwabara et al. [9]. Determined at indicated UD strains. The computed iso-model yield predictions from equation (5) are shown as continuous lines. The subsequent analysis focus on εUD UD ¼ 0.15.
The accompanying principal strains are determined from the calculated stresses using equation (2).
3. Results 3.1. Predictions using biaxial test results of Kuwabara et al. [9]. The tubular tests of AA5154-H112 in this work [9] were notable in that proportional loading ranging from 1:0 (UD), 4:1, 2:1 4:3, 20:23 and 1:1 (BB) were carried out. The designation of 2:1 as PS is not indicated since this identity only occurs for an ideal plastic material whereas the means to derive the precise values will be described in the next section. In their publication, the UD stressstrain diagrams for 4 and q directions were presented including R4 ¼ 0.36 and Rq ¼ 0.59. After digital scanning the measured stresses (from their Fig. 7 [9]) at equivalent TW using prescribed values at εUD UD, the reduced stress terms are plotted in Fig. 5; that is, the origin is the point of eminent plastic flow with any incremental biaxial stress and the locus expands with work hardening. Hence the constitutive model can generate analytically the continuous
Fig. 6. Optimum predicted yield loci of AA5154 for PW and TW at εUD UD ¼ 0.15 conditions using UD from 4 and q axes together with R4 ¼ 0.36 and Rq ¼ 0.59. The iso-models were modified as described in text to account for inferred latent hardening using LHF factor.
loci. Since the current focus is the result at high strains, the apparent large deviations at low strains are not examined at present due to the complication of the b1 to b2 transition. At equivalent PW and TW to εUD UD at 0.15, the iso-model locus using equation (5) (as shown in Fig. 5) is adjusted using PW/LHF or TW/LHF and compared to experiment in Fig. 6. Note that near BB, TW predictions are lower than the PW ones but both are within 5% of the measured ones based on TW. Thus the derived functional form appears to encompass the yield locus evolution but the stresses near BB need to be increased. Referring to conventional predictions, for R smaller than unity should result in BB stresses below the UD yield stress; whereby the yield locus becomes circular for R ¼ 0.0. This effect is sometimes referred to as the “anomalous” behaviour of aluminum alloys [13] and will be treated in the discussion section. The procedure to attain these optimum fits will be described below. In order to consider the PW predictions and means to improve the fit near BB as illustrated in Fig. 6, the role of texture was assessed by determining the effective plastic Poisson ratio (defined q 4 as εxq q/ε4 x4) for BB case by plotting εxq versus εx4 plot at BB in Fig. 7a as suggested by Barlat et al. [14]. Due to the two-loci fit, a break
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Fig. 7. The predicted εxq q versus ε4 x4 for stress BB. Note for equal stresses the accompanying strains are not equal, that is slope is not unity. (a) The break indicates regions of b1 and b2. (b) The variation of strain ratios with that of stress are illustrated in the b2 regions.
(static) latent hardening ratio is √2 and those measured after rolling at 450 K was 1.40 [16]. After several trials, the factor q LHFq ¼ (1 þ K √fq) when s4 4 > sq (the corresponding q 4 4 4 LHF ¼ (1 þ K √f ) when sq > s4) were introduced such that Ds4x4 ¼ LHF Ds44.The necessary inclusion of the stress ratio is evident and the square root function was an improvement over the linear one, fq (not shown). This modification needs inclusion in equation (5) wherein PW is modified as PW/LHF and the corresponding biaxial calculated-stresses are designated as unit-model and multiplied by LHF to result in the optimum fits as shown in Fig. 6. The fitting magnitude for K was 0.2 for LHF ¼ 1.2 for TW locus and 0.075 for PW; LHF ¼ 1.075. The plots of Dskh versus εkxk for the pertinent BB and PS cases in relation to UD will be presented in a separate section in comparison with the AA5754 analysis to follow. These analyses are required in order to compare the calculated yield stresses to those of the measured ones at all biaxial ratios. 3.2. Comparison of model predictions to Iadicola et al. [10]. Fig. 8. Yield loci prediction for PW equivalent to UD strain of 0.15 using the iso-model and modifications by adopting constant R's which increases the BB stresses.
between b1 and b2 occurs but for the measure of ductility comparison the higher strain region is used. In this iso-model, the balanced stresses and balanced strains are not coincident and three cases at higher strains are illustrated in Fig. 7b. The stress ratio q corresponding to equal biaxial strains occurred at s4 4/sq ¼ 0.933 (model prediction) compared to that observed at 0.87 (experimental) [9] and εqq/ε4 4 ¼ 0.744 for model at equal-stresses. These differences are a manifestation of a large texture effect. This role of texture was noted by Barlat et al. [14] in performing compression tests along the sheet normal direction. To illustrate this effect, constant R ¼ 0.59 and 1.0 were determined using constant PW to compare with the iso-model and plotted in Fig. 8 indicating the increase in BB stress values. However this adjustment is curvefitting and not physically based. Thus the improvement shown in Fig. 6 invoked a dynamic latent hardening factor (LHF). Although the predictive fit to experiment of the iso-model, wherein Dskxk ¼ Dskk is assumed, is remarkable as demonstrated in Fig. 8, measurable differences which occur may be due to a dynamic latent hardening effect. As previously pointed out, the iso-model does not directly account for the modified obstacle structure as the second in-plane strain is proportionally applied. This application may involve accommodation of a LHF. For multiple slip situation of a rolled (110) [001] copper single crystal (25 mm wide by 2.5 mm thick) with reduction of 70%, the theoretical maximum
This work on AA5754 alloy was performed using an ingenious apparatus whereby the central dome section on either side were free surfaces and the stresses at various ratios were in-situ determined by standard X-ray line-shift method. For this reason by
Fig. 9. Predicted yield locus using CRA of AA5754 data [10] and the iso-model for PW equivalent to tensile strain of 0.15. The arrows indicate shift of biaxial stresses based on the current TW calculation method. The recalculated data points of TW and PW as described in text showed much better fits.
S. Saimoto et al. / Acta Materialia 118 (2016) 109e119
necessity the tests were not continuous but quasi-static. From the digital data set kindly made available by Dr. Iadicola, those performed by X-ray determination for UD, PS and BB were used in the initial trials. As will become evident, the scatter, noted by the authors [10], becomes large using the quasi-static tests but for selfeconsistent comparison, the analyses using these results follow. Applying equation (5), the iso-model locus in Fig. 9 was generated. In comparison with the published data, a large discrepancy is evident. Examination of the method used by Iadicola et al. [10] indicated that the calculated expended work was based on deviator tensor instead of the direct comparison of TW under UD with the sum of work due to biaxial stresses as used by Kuwabara et al. [9] and Barlat et al. [13]. Hence BB stress-strain data were transformed to constitutive relations using CRA and the biaxial stresses at BB for equivalent TW and PW as denoted at UD strain of 0.15 were calculated and plotted in Fig. 9. This procedure in effect converted the AA5754 data to the same basis as for the AA5154 one, as indicated in Fig. 9. The fit to the iso-model was remarkable and LHF was almost 1.0. This illustration reveals that the selected criterion for yield-locus evolution with strain requires careful stipulation and calculation. One way to examine the issues is to use the CRA to predict the evolution of strains with the respective biaxial stresses as follows.
3.3. Prediction of principal strains to corresponding principal stresses The model predicts the pertinent biaxial stresses which by means of equation (2) can generate the corresponding strains. Thus a continuous plot of Dskh versus εkxh for various denoted stress ratios can be generated as shown for the iso-model predictions using the UD data of Kuwabara et al. [9] in Fig. 10. The major stress-strain 4 locus is stronger than that for UD and the minor strain in q is almost perfectly zero, indicating ideal plane strain, if the stress ratio is fq ¼ 0.605 (Fig. 10a) and similarly for f4 ¼ 0.6 using q as the major strain. In Fig. 10b the stress ratios of 1:1 are shown wherein separation between UD and BB (4) and BB (q) are evident. These effects are attributed to the role of texture and dynamic latent hardening. In the case of Iadicola et al. [10] data with the previously noted wide scatter due to quasi-static nature of stress-strain determination, Fig. 11a for PS shows very small difference between model PS using the denoted fq ¼ 0.3 and UD but a noticeable difference from the measured data. To examine the possible causes, the model
115
using fq ¼ 0.605 better fits the data and from this model the corresponding minor stress-minor strain were generated as shown in Fig. 11b in which neither of these cases result in ideal plane strain as demonstrated in Fig. 10a. This situation arose since the apparatus used for these tests controlled the strain path and the measured stresses were in response to this strain control. The implication is that the current data of Iadicola et al. [10] in their Fig. 7 cannot be used to validate the current model. To overcome the quasi-static tests results from X-ray analysis, predictions from the continuous tests supplied by the authors, generated analogous plots to those of Fig. 10 in Fig. 12 which indicate that near plane strain condition is achieved near fq ¼ 0.688. Thus although the study of Iadicola et al. [10] is of great interest, one can only qualitatively conclude that strengthening during biaxial straining is positive but its precise measure requires tests with controlled plane strain. The analyses of the above results suggest that current constitutive relations prediction of yield locus is viable but it requires one of two ways to validate the fitting factor of LHF. This can be done by careful burst tests as proposed by Barlat et al. [14] to precisely determine the strains at balanced biaxial stresses or to perform precise plane strain measurements. The constitutive relations analyses of UD in RD and TD tests should be supplemented by through-thickness texture analysis using ODF to determine the mean Rj and Mj. If through such means a reproducible yield locus and the stress-strain loci for any desired stress ratio are validated, this methodology can be used to convert the strain maps of FLD into point-by-point stress maps.
4. Discussion 4.1. Summary of plastic work derivation and predictions The fundamental assumption of constitutive yield-locus model is that plastic work due to each principal strain can be algebraically summed and equated with that reference energy identified by the unidirectional tests in the principal strain direction; typically taken to be RD. The principal strains include the summation of negative contribution from the simultaneously applied second axial stress. This negative term is explicitly evident in equation (5). The analytical unidirectional stress-strain relation for each stress axis is determined using CRA. Since the usual tests are performed by proportional loading, the plastic work integral can be calculated by converting differential strain into that of stresses by means of CRA
Fig. 10. (a) Reduced stresses with their respective strain axes for AA5154 [9] comparing the loci for UD with those for biaxial stress with ratio fq ¼ 0.6 which results in plane strain condition. (b) Comparison of equal stress evolution with strain showing deviation between 4 and q axes.
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S. Saimoto et al. / Acta Materialia 118 (2016) 109e119
Fig. 11. (a) Analogous to Fig. 10, predicted Dskh versus εkxh for designated plane strain at fq ¼ 0.3 and that for UD compared to the measured values [10]. Note the lack of agreement and the expected separation between predicted PS and UD increases as fq is increased. (b) Analogous plot to Fig. 10a, showing that the minor strain does not approach plane strain until fq is increased as indicated.
Fig. 12. Reduced stresses versus respective strain plots generated from continuous tensile tests for UD. Note the predicted strengthening as the condition of plane strain is approached.
relations. This plastic work can be directly related to the integral of the equivalent compressive stress with incremental thickness change. Hence differentiation of the plastic work with respect to thickness strain (slope method) results in Dstt. By applying the Levy-Mises relation, an analytical relation can also be derived and the comparison of the resultant plot show good agreement; hence validating this conventionally-accepted relation which is the basis of plasticity theory. The iso-model assumed that the flow stress under biaxial stress states were the same as for unidirectional at the net strain due to the biaxial strains wherein one component is negative. This assumption resulted in a general-form yield locus which approached the measured data in shape and magnitude. The degree of fit, however, appeared to be a function of the measured Rvalues and implied the effect of latent hardening. Consideration of this issue indicated that the magnitude of the predicted biaxial stresses required to be larger by a factor LHF; this adjustable factor which is alloy dependent can be directly determined from the measured BB test, usually performed by bulge testing. The functional stress forms which comprise the proposed yield function presented in equation (5) is examined and for the sake of
clarity only the work term expressed in shear terms are used. Furthermore the material constants will not be explicitly denoted but the developing relations will be designated as proportional to (») PW. From equation (1) referring to only one term of PW as » !t dg » !t (t1þb) dt » t3þb. Hence for linear hardening wherein b ¼ 1.0, PW » t2 which is analogous to the von Mises function. Typically for aluminum alloys b1 ¼ 0.0, indicating parabolic hardening [12] and hence PW » t3. In the case of copper specimens, the b values at 20 K are near 1.0 increasing to b1 ¼ 0 and b2 ¼ 0.9 at 473 K [17]. Moreover since b1 < b2, a break occurs in Fig. 3. However to elucidate the evolution of the work function, t3þb can be rewritten as (t3 tb) whereby tb » (t dt/dg)1 and hence PW » t2/ (dt/dg). Thus as strain evolves the incremental increase in expended work approaches the quadratic form when (dt/dg) ¼ constant upon attaining Stage IV deformation regime. The inference of this observation is that initially the t2 term corresponds to the dislocation density but as strain progresses deformation debris is created, some of which dynamically recovers as volume fraction of point defects (CV) increase with strain. This progression was indicated during cold rolling of AA3003 with intermediate tensile tests till Stage IV was reached when CV attained 0.02 [18]. These inherent microstructure issues may be circumvented by relaxing the quadratic deviator stress function in formulating the plastic potential function. However identifying the cause for such deviation will be more useful in analyzing actual material performance during fabrication, especially for failure analyses. The attribute of the current model is that constitutive relations for any stress ratio and its corresponding strains can be calculated such that precise prediction for balanced biaxial stresses at unequal strains and precise plane-strain stress ratios can be predicted. The usefulness of such an approach to sheet formability modeling using local strain measurements to deduce stress states are apparent. In such analysis the role of evolving texture can be better assessed. Furthermore, the calculation of yield loci at constant PW rather than TW appears to better encompass the strain-hardening evolution; the method used in performing crystal plasticity finite element modeling. It can be readily shown by simplistic modeling using linear hardening, the use of TW results in slightly modifying the quadratic stress relation and this effect will become much larger for age-hardened alloys. There are some cautionary footnotes to the current derivation which needs to be kept in mind. First, the Taylor factor M was assumed to be constant at 3.0 but as previously
S. Saimoto et al. / Acta Materialia 118 (2016) 109e119
pointed out it does not explicitly affect the work relation of equation (5). However the analytical derivation of the effective normal stress via equation (6) varies with the M ratio. In future studies, Mj and Rj derived from orientation distribution function can be used. Secondly, the transition region at cross-over of b1 to b2 loci exists at low strains but for considerations of FLD at higher strains it does not hinder the analysis. Thirdly, the biaxial stresses were derived from the work relation PW ¼ stress-work integral which is also equal to strain-work integral and hence by means of CRA, strain maps should be convertible to stress maps on a point-by-point determination of the grid-based experimental data. Aside from generating the uni-directional stress-strain locus by plasticity theory, there have been many studies to derive the work-hardening locus either from dislocation mechanics models [19] or by using mesoscopic functional relations with adjustable parameters to result in the best fit [20]. However none of these works can replicate the stress-strain loci as well as the current S-VH relation which correlates the fit parameters to the evolution of work-hardening. This procedure is unique in that the fit-parameters are the precisely fitted modeled ones from which the microstructure evolution is assessed. 4.2. How does this simple derivation result in credible yield-loci predictions? The simple yield function from this analysis is reproduced below whereby fq and f4 are the prescribed stress ratios and prescribed PW incorporating texture and work-hardening.
PW ¼ LHF
(
f q Rf 1 1 þ Rf
)Z
(
Dsff dεff
þ
f f Rq 1 1 þ Rq
)Z
Dsqq dεqq
(7)
4 final Thus biaxial stresses become s4 and x4 ¼ LHF Ds4 þ s0
sqxq ¼ LHF Dsqq þ fq sfinal and similarly in the stress space whereby f4 0
applies. The final derivation means that from the right hand side which is the iso-model its stress predictions are multiplied by the appropriate LHF ¼ (1 þ K √fk). Although the answer to the above question may take much discussion, the bases of the rationale must be due to the nature of work hardening. The implications of the Cottrell-Stokes relation with respect to biaxial deformation have recently become apparent to the current authors. Basically it reveals that work hardening proceeds by slip processes the by-
Fig. 13. Normalized t/t3 versus g/g3 plot for AA5154 and AA5754 after CRA of continuous UD results showing that the 4 (RD) and q (TD) loci become coincidental although the R values are not identical. The differences between the alloys are attributed to disproportioning of cells to Taylor lattice structures.
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product of which is the increased density of obstacles. Furthermore if the nano-scale thermally activatable atomic sites are identical although the line lengths between them shorten, the strain-rate sensitivity S is constant. Thus the relation between activation work t n ¼ k/S (k, the Boltzmann constant) and S indicate that t n is constant. The deduced conclusion is that since t is inversely proportional to [, wherein n ¼ [ b d with [, being the interobstacle length and b, the Burgers vector, the activation distance, d, must be constant. Under constant temperature and strain rate, it is evident that if the obstacle type does not change with strain, d remains constant and only the obstacle-density increases. Moreover it can be readily shown that positive local internal stresses assist in overcoming dense patches of obstacles whereas negative local internal stresses reduce the applied stress such that varying densities of obstacles can co-exist. For the lowest free energy state, the compressive volume fractions must be higher than that of the tension regions. This structure essentially gives rise to cells which are self-organized during the entropy production process. The upshot is that local tloc nloc is equal to the product of mean t and mean n which is directly experimentally measurable. To demonstrate such a microstructural evolution takes place, the constitutive relations involving unidirectional tests for AA5154 and AA5754 are plotted after normalization with t3 and g3 in Fig. 13. Note these parameters satisfy both b1 and b2 regions of CRA. The resulting near-master curve is found seemingly unaffected by different test temperatures [12]. Thus the activation distance controls the thermodynamic response of plastic deformation and it will be constant even under biaxial stresses if the obstacle type for these strain states does not change. However compared to unidirectional tests, upon simultaneous application of the second principal strain, the slip systems which contribute not only to thickness strain but also lateral contraction strains are retarded. Thus the evolving work hardening structure will differ somewhat from that of unidirectional. Nevertheless the structure evolution will only change in scale not distribution within the constant b region. Furthermore depending on the alloy the degree of cell versus Taylor lattice structure may change. The outstanding fact of plastic flow is that t ¼ a m b/[ ¼ a m b r1/2 whereby r is the mean dislocation density and a, the obstacle strength factor. Thus the constant expended plastic work under biaxial strain is related to the amount of dislocation generated but since this energy is scalar, the plastic work is just the sum of its component parts. Thus if the self-organizing internal stresses arrange the obstacles in a minimum energy configuration, the stored work may vary resulting in mean dislocation density difference. To illustrate this deduction, the predicted PW versus εtt for BB and PS for AA5154 is compared to that of UD in Fig. 14a. Using the constant PW at UD strain of 0.15, the corresponding εtt and Dstt for BB and PS are converted to skxk versus εkxk as presented in Fig. 14b. Future experiments generating such plots will not only validate the analysis but reveal the work hardening differences depending on the biaxial ratios. Using CRA on PS 4, BB 4 and UD 4 with fit parameters in Table 2, the normalized plot in Fig. 14c show that PS and BB are very similar but measurable difference occur with respect to UD due to application of LHF ¼ 1.075. The rate of doing work is related to the activation work as fol _ lows: tg_ ¼ tn gn ; that is, it is a product of the activation work and the density of dynamic activation sites [21]. Hence flow stress is controlled by thermal activation of the mean obstacle density. However if the activation distance is constant the evolution of strain is basically crystallographic and follows the Taylor slip analysis which is the basis for crystal plasticity finite element modeling. The activation distance factor is taken into account in equation (1) in the proportionality constant which is strain-rate
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Fig. 14. (a) Derived plastic work for AA5154 [9] at which magnitudes of constant PW ¼ 19.5 with minimum of 18.14 were used to determine the biaxial stresses at varying fq. (b) Inplane stresses for selected PW for PS and BB conditions. Note the PS is slightly larger than UD whereas BB is slightly lower as shown in Fig. 6. (c) Normalized shear stress e shear strain plots showing that the nature of work hardening under UD generated obstacles and BB (PS) ones are not identical as indicated by LHF factor.
Table 2 CRA fit parameters from Fig. 14b.
s ratio and direction
Sample 4 x4
AA5154 s
vs
ε4 x4
vs
ε4 x4
q
f - UD, f ¼ 0 f - PS, fq ¼ 0.608 f - BB, f4 ¼ 0.933
s ratio and direction
Sample 4 x4
AA5154 s
q
f - UD, f ¼ 0 f - PS, fq ¼ 0.608 f - BB, f4 ¼ 0.933
a used
m (MPa)
sfinal (MPa) o
1/4
C1 (b1) (mm)
b1
0.4 0.4 0.4
32012 32012 32012
84.62 70.15 60.48
192 113 86
2.607 0.961 0.903
0.829 0.475 0.481
LHF
P/A
t3 (MPa)
g3
l3 (mm)
b2
C1 (b2) (nm)
1.000 1.075 1.075
0.0650 0.1107 0.1446
43.19 55.78 63.55
0.2401 0.2234 0.1921
0.167 0.186 0.162
0.334 0.520 0.576
65.20 29.00 19.05
and temperature dependent and reduces to the modified Hollomon relation. Thus, if the thermodynamic term is inherently included as well as the dynamic annihilation (A term in equation (1)), incremental shear due to local shear stress is the same for such combination to comply with the imposed strain. Fig. 13 appears to affirm this deduction. The result is the Levy-Mises relation becomes just the utilization of slip systems dealing with a shape-changing incompressible solid. Noteworthy demonstrations are the tensile testing of [001] single crystals wherein R is unity or the rolling of (110) [001] crystal slabs [16]. The conclusion is that the rate of doing
work is determined by thermally activated flow but the density of obstacles are derived from the shape-change geometry and hence a simple addition of plastic work of the component stress-strain states can describe the evolving structure. The inference is that although entropy production is large, to attain a given total work magnitude its variation with different unidirectional straining conditions is small according to the variation in P/A (Table 1). The deduction is that the expended energy for shape change is directly related to the stored work. This premise is basic to the S-VH constitutive relation whereby the energy expended is related to
S. Saimoto et al. / Acta Materialia 118 (2016) 109e119
dislocation creation and equated to the retained dislocation density by invoking an annihilation factor A. Thus the credible predictions using this Taylor slip-model relation are self-consistent and satisfy the shape-change and thermodynamic conditions. Table 2 shows that if the change in LHF from 1.0 to 1.075 the P/A increases under biaxial stresses. This observation indicates that the increase in strength is due to some change in the nature of obstacles as suggested by Fig. 14c. On considering the representative elemental volume (REV), the shape change response is assumed to be homogeneous of all such elements and its response to the bounding traction forces are the same, neglecting micro-band formation. Although electron microscopy show vast differences in array of dislocations and barriers within each REV such as dispersoids and grain boundaries, each REV is homogenized by the selforganized internal stresses and the shear flow stresses evolve in almost isotropic manner and crystallographic slip conform to the imposed strain. The crystal plastic finite element analysis using this relation for UD to predict that of PS and BB have been recently demonstrated [22]. The current analysis is analogous to that of non-linear elasticity whereby the stress-strain relation is precisely known throughout the given strain range. Complication may occur if the plastic true strain rate becomes a function of strain or if the friction heat starts to affect the isothermal testing conditions. The current status is that t3 is dependent on temperature and volume fraction of generated defects which give rise to osmotic stress which in turn assist in overcoming shear resistant obstacles. Thus the long range intent of this study is to generate stress-strain relations under plane strain conditions to examine various criteria of limit strain predictions for FLD as previously described [15]. 5. Conclusions The concept of effective compressive stress versus the thickness strain suggested that components of the plastic work due to the biaxial stresses can be represented on a common basis of thickness strain since each contributes to it in an assumed additive manner. From this premise and using the constitutive relations which can precisely replicate the measured stress-strain diagram, a simple formulation for the in-plane stresses applied under proportional loading was derived. The comparison of predictions to the published yield determinations for AA5154 [9] and AA5754 [10] resulted in good agreement especially since the same formulation were applied to both alloys. The “anomalous” BB yield behaviour in aluminum alloy is inherent to strain-hardening mechanisms attributed to the thermally activated flow mechanisms found in face-centred cubic metals; that is since the flow stress is characterized by the mean dislocation density unaffected by internal stresses, yielding tend to obey Tresca criterion if the texture is not highly developed. This analysis permits generation of the component stresses versus its accompanying strains such that evolution of work hardening under different stress states can be examined. Such predictions will become useful in modeling change of strain direction during sheet fabrication as well as for ductile failure analysis.
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Acknowledgements The authors thank the Natural Sciences and Engineering Research Council of Canada through support of the Automobile Partnership Cooperation program, a consortium effort commissioned at the University of Waterloo. For one of us (SS), it has been a long journey through crystal plasticity studies before trying to integrate work-hardening with yield locus evolution, a topic of great interest to his thesis supervisors: Professors W. A. Backofen and W. F. Hosford; originators of texture strengthening analyses.
References [1] S.S. Hecker, A.K. Ghosh, H.L. Gegel (Eds.), Formability Analysis, Modeling and Experimentation, Met. Soc. AIME, 1978. [2] S.P. Keeler, Determination of forming limit in automotive stamping, Met. Ind. 42 (1965) 683e691. [3] G.M. Goodwin, Application of strain analysis to sheet metal forming problems, La, Metall, Italiana 8 (1968) 7657e7774. [4] C.N. Reid, Deformation Geometry for Materials Scientists, Pergamon, Oxford, UK, 1973. [5] Z. Marciniak, K. Kuczynski, Limit strains in the processes of stretch-forming sheet metal, Int. J. Mech. Sci. 9 (1967) 609e620. [6] P. Van Houtte, A. Van Bael, S. He, Anisotropy and formability in sheet metal forming, in: J.M.A. Cesar de Sa, A.D. Santos (Eds.), Proc. NUMIFORM. Amer. Inst. Phys. NY, USA, 2007, pp. 159e164. [7] T. Balan, B. Bacroix, C. Teodosiu, Cold metal forming, in: D. Banabic (Ed.), Proc. TPR 2000, Techn. Univ., Cluj-Napoca, Romania, 2000, pp. 73e84. [8] P. Dasappa, K. Inal, R.K. Mishra, The effects of anisotropic yield functions and their parameters on prediction of forming limit diagrams, Int. J. Solid Struct. 49 (2012) 3528e3550. [9] T. Kuwabara, K. Yoshida, K. Narihara, S. Takahashi, Anisotropic plastic deformation of extruded aluminum alloy tube under axial forces and internal pressure, Int. J. Plast. 21 (2005) 101e117. [10] M. Iadicola, T. Foecke, S.W. Banovic, Experimental observations of evolving yield loci in biaxially strained {AA5754-O}, Int. J. Plast. 24 (2008) 2084e2101. [11] S. Saimoto, P. VanHoutte, Constitutive relation based on Taylor slip analysis to replicate work-hardening evolution, Acta Mater. 59 (2011) 602e612. [12] S. Saimoto, D.J. Lloyd, A new analysis of yielding and work hardening in AA1100 and AA5754 at low temperatures, Acta Mater. 60 (2012) 6352e6361. [13] F. Barlat, Y. Maeda, K. Chung, M. Yanagawa, J.C. Brem, Y. Hayashida, D.J. Lege, K. Matsui, S.J. Murtha, S. Hattori, R.C. Becker, S. Makosey, Yield function development for aluminum alloy sheets, J. Mech. Phys. Solids 45 (1997) 1727e1763. [14] F. Barlat, J.C. Brem, J.W. Yoon, K. Chung, R.E. Dick, D.J. Lege, F. Pourboghrat, S.H. Choi, E. Chu, Plane stress yield function for aluminum alloy sheetsdpart 1: theory, Int. J. Plast. 19 (2003) 1297e1319. [15] S. Saimoto, B.J. Diak, D.J. Lloyd, Point defect generation, nano-void formation and growth. II. Criterion for ductile failure, Philos. Mag. 92 (2012) 1915e1936. [16] L.E. Collins, Plane Strain Deformation and Subsequent Recovery of Ideally Oriented Copper Single Crystals, M.Sc. Thesis, Queen's University, Kingston, Canada, 1977. [17] S. Saimoto, B.J. Diak, Point defect generation, nano-void formation and growth. I. Validation, Philos. Mag. 92 (2012) 1890e1914. [18] S. Saimoto, K. Inal, H. Jin, Constitutive analyses of tensile tests on pre-rolled AA3003 sheets, Mater. Sci. Forum 794e796 (2014) 710e715. [19] Y. Aoyagi, K. Shizawa, Multiscale crystal plasticity modeling based on geometrically necessary crystal defects and simulation on fine-graining for polycrystal, Int. J. Plast. 23 (2007) 1022e1040. di, N.Q. Chinh, J. Gubicza, T.G. Langdon, Plastic behavior of fcc metals [20] T. Csana over a wide range of strain: macroscopic and microscopic descriptions and their relationship, Acta Mater. 59 (2011) 2385e2391. [21] S. Saimoto, Dynamic dislocationedefect analysis, Philos. Mag. 86 (2006) 4213e4233. [22] A.P. Brahme, K. Inal, R.K. Mishra, S. Saimoto, A new strain hardening model for rate-dependent crystal plasticity, Comput. Mater. Sci. 50 (2011) 2898e2908.