Modelling the Stress-Strain Behaviour for Aluminum Alloy AA6111

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1 Proeedings of the 9 th International Conferene on Aluminium Alloys (4) Edited by J.F. Nie, A.J. Morton and B.C. Muddle Institute of Materials Engineering Australasia Ltd 939 Modelling the Stre-Strain Behaviour for Aluminum Alloy AA6 W.J. Poole,.J. Lloyd epartment of Metals and Materials Engineering, The University of British Columbia, Stores Rd, Vanouver, BC, V6T Z4, Canada Alan International, Kingston Researh and evelopment Centre, Kingston, ON, K7L 5L9, Canada Keywords: preipitation, AA6, work hardening model, uniform elongation, yield stre model Abstrat It is beoming inreasingly important to have physially based preditive models for mehanial properties suh as yield stre and work hardening behaviour. In this work, a yield stre model for heat treatments after the solution treatment has been developed in the internal state variable framework. The yield strength response is determined from the key mirostrutural parameters, i.e. the volume fration and size of the strengthening phase. The work hardening behaviour as a funtion of the preipitate state has been onsidered in terms of the disloation/preipitate interations (i.e. whether the preipitates are shearable or not) and the appropriate flow stre superposition law for disloation and preipitation hardening.. Introdution There has reently been a resurgene of interest in examining 6 series aluminum alloys for automotive and aerospae appliations. Over the past several years, signifiant progre has been made in understanding the preipitation sequene and the effet of multi-step heat treatment on age hardening response. For example, detailed preipitation strengthening models have been developed [-3] for the automotive alloy AA6, and a areful onsideration of the hange in work hardening behaviour for different preipitate states [4] has been determined. The urrent work attempts to ombine the models for preipitation strengthening and work hardening to produe a omprehensive model whih desribes the stre-strain behaviour up to the neking point for a variety of underaged, peak aged and overaged onditions in AA6. This is of value from a number of perspetives inluding i) the potential use of the model as an input to finite elements models whih are used to simulate metal forming or rash worthine or ii) as a tool to aid alloy development. The approah taken in the work follows the internal state variable framework where the key mirostrutural parameters, i.e. the preipitate size, volume fration and size distribution, are expliitly used as input parameters. Work hardening is understood in terms of the ompetition between disloation storage and dynami reovery and how this balane is affeted by the preipitate population. Finally, areful attention is paid to the question of how the various flow stre ontributions should be summed to give the overall mehanial response.

2 94. Model evelopment. Yield Stre Model The yield stre for a preipitation hardening alloy an be desribed by a summation of the intrinsi strengthening (i.e. lattie resistane and grain size strengthening), the solid solution ontribution and the preipitation hardening ontribution, i.e. = + + () o The solid solution and preipitation hardening omponents are fundamentally linked by a ma balane of the alloying additions and are, thus, interdependent. The preipitation hardening ontribution is a funtion of the average strength of preipitates as obstales to disloation motion, F, and the spaing between these obstales, L, and is given by: MF ppt = () bl where M is the Taylor fator (3.6) and b is the magnitude of the Burgers vetor (i.e..86 nm for Al). In the ase of AA6, areful TEM examination has revealed that β and Q preipitates oexist in various ratios during ageing at temperatures between 8 and 5 o C(see ref. [,]). In this work, the simplifying aumption will be made that the two preipitates an be treated as a single preipitation population. Further, sine these preipitates form needles or laths parallel to the <> diretion, the average spaing on the () glide plane, L, an be determined as: π L = f where R is the average equivalent radius and f is the total volume fration of preipitates [, 3]. The average obstale strength is determined using the result of eshamps and Brehet [5] whih aumes: i) a linear dependeny of preipitate strength on size for shearable preipitates; ii) a size independent strength for non-shearable preipitates and; iii) a Gauian distribution of preipitate sizes. The result for average obstale strength is then: K ( ) R R R K R π R R + + R exp exp erf erf F = kgb + kgbr R ppt K π R erf R where R is the ritial radius for the transition from disloation shearing to disloation bypaing, is the standard deviation of the preipitate radius distribution, G is the shear modulus of aluminum (7 GPa) and where k and K are given by: b K = (4b) π + erf R / k = R and ( ) ( ) Substituting equations (3) and (4) into equation () allows for a predition of the preipitate strengthening ontribution. The values for f, R and have been determined for a wide (3) (4a)

3 range of ageing onditions (see Table ) in a previous study using quantitative transmiion eletron mirosopy. This leaves only a single unknown variable in the preipitation hardening model, the shearable/non-shearable transition radius. In this work, a good fit to the data was found with R =. 6 nm. Finally, the solid solution ontribution to strengthening,, is estimated as: f = o α (5) f eq where o is the solid solution ontribution for the solution treated material, f eq is the equilibrium volume fration of the preipitating phase (i.e..9) and α is onstant. The overall model results are rather insensitive to the value of α and this was hosen to be, i.e. when f = f, = eq Table : Summary of data from quantitative TEM haraterizing preipitate equivalent radius, volume fration and size distribution [,]. Notes: i) / R is the ratio of the standard deviation of the size distribution to the mean radius and ii) the values marked with asteri are estimates. volume fration mean equivalent Ageing Condition ( β and Q ) radius / nm / R.5 8 o C o C o C o C o C.7 * o C o C.8 * 9.5 * -. Work Hardening Model The work hardening behaviour of a preipitation hardening alloy is a funtion of two main fators, i.e. i) the modifiation of disloation storage by preipitates and ii) the appropriate superposition law for adding preipitate and disloation ontributions to the flow stre. The evolution of disloation density with strain an be written in the Koks/Meking/Estrin framework [6,7] as: ρ = ( k ρ k ρ + k ) P (6) ε where ρ is the disloation density, 3 94 p ε is the plasti strain, k is related to the storage of disloation vis disloation/disloation interations, k is the rate of dynami reovery and k is an additional storage term due to disloation/preipitate interations. The flow stre ontribution from disloation hardening is given by: where α is a onstant of magnitude.3..3 Shearable Preipitates = α GbM ρ (7) In the ase of shearable preipitates, k is zero. As a result, the disloation ontribution to flow stre as a funtion of strain an be determined by integrating equation (6) and substituting into (7). This gives the well known Voe equation, i.e.

4 θ = o P exp ε (8) where is the saturation stre and θ o is the initial work hardening rate. These two parameters an be related to k and k as: GbMk θ = α o (9a) and GbMk s = α (9b) k The values for k and k were determined by fitting the Voe equation to the plasti 8 - behaviour of the solution treated material. This gave values of k = 7.5 m and k = 7. In subsequent alulations, the rate of dynami reovery is aumed to be independent of the preipitate population for both shearable and non-shearable preipitates. The flow stre addition problem has been treated using a generalized addition law, i.e. n n ( + ) n = + () when n is variable between and. The physial basis of this approah relates to the relative density and strength of the different obstales. For example, when the preipitates are weak obstales (or alternatively small in radius) and they are summed with strong obstales suh as forest disloations, the simulations of Foreman and Makin suggest one should sum the flow stre ontributions, i.e. n =. On the other hand, for the ase when the preipitates are strong (non-shearable), the density of obstales (preipitates and forest disloations) should add in linear manner, i.e. n =. In this work, an empirial approah has been taken by auming that n varies linearly with the preipitate radius (or alternatively preipitate strength) from the smallest preipitate radius that was experimentally measured to the transition radius, i.e. n = R () Equation () is valid for R between. and.6 nm. For R >.6 nm, it is aumed that n =..4 Non-Shearable Preipitates In this ase, it is aumed that the storage of disloations in dominated by non-shearable preipitates suh that in equation (6) k = and k desribes the storage of disloations due to preipitates [4]. The integration of equation (6) and substitution into equation (7) yields: ppt p ( exp( k ) k = αgbm ε k () where the value of k is geometrially related to the preipitate spaing as follows: M k = α (3) bl is a onstant and the preipitate spaing L is given by equation (3). In referene [4], a good fit to data was found when =.3 so that will be used here. Finally, sine by α α 94

5 definition R > alulated. R, a value of is used for n in equation (8) when the final flow stre is.5 Predition of Uniform Elongation Having developed a work hardening model, it is now poible to also predit the extent of uniform elongation using the Considére riterion, i.e. d = (4) dε d In the present ase, the derivative, was determined by numerially differentiating the d ε alulated stre-strain urve. Finally, given the magnitude of the uniform elongation, it would be straightforward to alulate the ultimate tensile stre if desired Results and isuion Figure and ompare the results for the experimental and model preditions for strestrain urves for the underaged and overaged samples, respetively. It an be observed that there is good agreement for both the yield stre and the work hardening behaviour for all ases. Figure 3 summarizes the omparison of model results and experimental results for yield stre and uniform elongation. It an be observed that over a wide range of onditions, the yield stre preditions are within ± 5 % of the experimental values while the preditions for uniform elongation also fall within this range with the exeption of the two values for highly overaged samples with non-shearable preipitates where the deviation is slightly larger. The advantage of the urrent approah is the physial basis of the model and the minimum number of adjustable parameters. The average radius, R, the volume fration, f, and width of the size distribution,, have been diretly measured. The values for kand k have a physial basis and are determined from work hardening behaviour of the solution treated sample. After the transition to non-shearable preipitates, the value of k is determined from the average preipitate spaing. The ritial adjustable parameter is the transition radius, R, whih one determined should be appliable to all alloys with similar preipitates. The weakest aspet of the model is the flow stre addition law where an empirial relation has been used, i.e. equations (8) and (9). This is an area whih, in general, requires further investigation. stre / MPa o C as solution treated.5 8 o C experiment model.5..5 strain Figure : Comparison of experimental and model true stre vs. true plasti strain urves for underaged samples. stre / MPa o C 7 5 o C.5 5 o C 6 8 o C experiment model.5..5 strain Figure : Comparison of experimental and model true stre vs. true plasti strain urves for overaged samples. Note: the samples aged for.5 hours and 7 days at 5 C are aumed to have non-shearable preipitates (see text).

6 944 Yield Stre (Model) / MPa Yield Stre (Exp.) / MPa Uniform Elongation (Model) Uniform Elongation (Exp.) Figure 3: Comparison of model preditions with experimental measurements of a) yield stre and b) uniform elongation. ashed lines represent ± 5 % deviation. Closed symbols and open symbols are for shearable and non-shearable preipitates, respetively. 4. Summary A omprehensive model framework has been presented to desribe the yield stre and work hardening behaviour for an industrially relevant aluminum alloy. Good agreement is observed between the model and experiments over a wide range of ageing onditions. The physial basis of the model offers the potential that the model an be expanded to a variety of alloys with different hemistries but where the strengthening preipitates are the same. This suggests the potential of the model for aiding alloy development. Referenes [] X. Wang,W.J. Poole,S. Esmaeili,.J. Lloyd,J.. Embury. Metall.Mater.Trans.A,34A,93-93, 3. [] S. Esmaeili, X. Wang,.J. Lloyd and W.J. Poole. Metall. Mater. Trans. A, 34A, , 3. [3] S. Esmaeili, W.J. Poole and.j. Lloyd. Ata mater., 5, 43-57, 3. [4] L.M. Cheng, W.J. Poole, J.. Embury and.j. Lloyd. Metall. Mater. Trans. A, 34A, , 3. [5] A. eshamps and Y. Brehet. Ata mater., 47, 93-35, 999. [6] Y. Estrin. Unified Constitutive Laws of Plasti eformation, ed. A.S. Krau and K. Krau, Aademi Pre, New York, NY, 69-6, 996. [7] H. Meking and U.F. Koks. Ata metall., 9, , 98. [8] U.F. Koks. Pro. 5 th Int. Conf. on Strength of Metals and Alloys, ed. P. Haasen et al., Pergamon Pre, Oxford, UK, 66-68, 979. [9] A.J. Foreman and M.J. Makin. Phil. Mag., 4, 9-94, 966.

Proposal for a new shear design method

Chapter 6 Proposal for a new shear design method The behaviour of beams failing in shear has been studied in the previous hapters, with speial attention paid to high-strength onrete beams. Some aspets