Process model to predict yield strength of AA6063 alloy

Process model to predict yield strength of AA6063 alloy

Author’s Accepted Manuscript Process model to predict yield strength of AA6063 alloy Supriya Nandy, Kalyan Kumar Ray, Debdulal Das www.elsevier.com/l...

3MB Sizes 1 Downloads 6 Views

Author’s Accepted Manuscript Process model to predict yield strength of AA6063 alloy Supriya Nandy, Kalyan Kumar Ray, Debdulal Das

www.elsevier.com/locate/msea

PII: DOI: Reference:

S0921-5093(15)30222-7 http://dx.doi.org/10.1016/j.msea.2015.07.070 MSA32608

To appear in: Materials Science & Engineering A Received date: 30 May 2015 Revised date: 11 July 2015 Accepted date: 23 July 2015 Cite this article as: Supriya Nandy, Kalyan Kumar Ray and Debdulal Das, Process model to predict yield strength of AA6063 alloy, Materials Science & Engineering A, http://dx.doi.org/10.1016/j.msea.2015.07.070 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Process model to predict yield strength of AA6063 alloy Supriya Nandya,b, Kalyan Kumar Rayb, Debdulal Dasa* a

Department of Metallurgy and Materials Engineering, Bengal Engineering and Science University, Shibpur, Howrah - 711103, India.

b

Department of Metallurgical and Materials Engineering, Indian Institute of Technology, Kharagpur - 721302, India.

* Corresponding author. Tel.: +91 33 2668 4561/62/63; Fax: +91 33 2668 2916/4564. E-mail addresses: [email protected]; [email protected] (D. Das).

Abstract In the present study, an attempt has been made to examine the appropriateness of the models for predicting yield strength of age hardenable 6063 Al-Mg-Si alloy. Two physical models have been considered: one is classical dislocation-particle interaction model which incorporates both cutting and by-pass mechanisms around spherical shaped precipitates; and the other is based on only dislocation by-pass of the rod shaped precipitates as per modified Orowan equation. The prediction of yield strength using these models has been compared with experimentally generated yield strength data of the selected Al-alloy subjected to different combinations time and temperature of ageing. The microstructural variations due to ageing have been simulated using the well established thermodynamic and kinetic relationships. Comparison of the experimental and the simulated results assists to conclude that modified Orowan mechanism predicts yield strength better than that by the classical model up to the peak aged conditions. The generated results have been successfully used to develop thermal processing maps that can be considered as a convenient tool for selection of age hardening parameters to achieve any desired level of yield strength for AA6063 alloy.

Keywords: Aluminum alloys; Al-Mg-Si alloy; Age hardening; Precipitation; Strength; Modeling.

1|Page

Abbreviations and nomenclatures Symbol

Parameters

Unit (S.I.)

D-B

Deschamps and Brechet model

-

Z-S

Zhu and Starke model

-

a

Lattice parameter of precipitate

A

Aspect ratio of precipitate

-

A0

Avogadro number

-

b

Magnitude of Burgers vector

nm

Ce

Equilibrium solute concentration at particle/matrix interface (superscript stands for specific element)

at.%

Ci

Instant solute content (superscript stands for specific element)

at.%

Cp

Composition of precipitate

at.%

C0

Nominal solute concentration in the matrix (superscript stands for specific element)

at.%

d

Grain diameter

µm

D

Diffusion coefficient

D0

Pre-exponential coefficient

term in

nm

m2 s-1 expression

of

diffusion

m2 s-1

f

Volume fraction of precipitate

-

fm

Volume fraction of precipitate at peak aged condition

-



A parameter describes mean particle strength

-

G

Shear modulus

GPa

ΔG

Difference in shear modulus of precipitate and matrix

GPa

ΔG*

Activation energy for nucleation

ΔH0

Standard enthalpy of the dissolution/precipitation reaction

kJ mol-1 reversible

Mg2Si

kJ mol-1 MN m-3/2

ki

Locking factor

Kj

Scaling factor for solid solution hardening model (superscript represents the solute)

L

Inter-particle spacing

M

Taylor factor

-

p

A factor relating growth parameter

-

Q

Activation energy for precipitation

kJ mol-1

QD

Activation barrier for diffusion

kJ mol-1

r

Radius of precipitate

MPa wt.%-2/3 nm

nm 2|Page

rc

Critical radius for transition from shearing to by-passing

nm

r0

Dislocation core radius

nm

Critical radius for nucleation

nm

r

*

R ΔS0

J mol-1 K-1

Universal gas constant Standard entropy of the dissolution/precipitation reaction

reversible

Mg2Si

J mol-1 K-1

tA

Time of ageing

h

tP

Time to peak ageing

h

TA

Temperature of ageing

K m mol-1 3

vat

Molar volume of precipitate

Z

Zeldovich’s factor

-

β

A constant

-

β’

Dimensionless growth parameter

-

γ

Particle/matrix interfacial energy

J m-2

Γ

Dislocation line tension

Δσgb σi

Increment in strengthening

strength

N due

to

Grain

boundary

Intrinsic strength

MPa MPa

Δσss

Increment in strengthening

strength

due

to

Solid

solution

σys

Total yield strength

MPa

sysD-B

Overall yield strength form Deschamps and Brechet model

MPa

sysZ-S

Overall yield strength form Zhu-Starke model

MPa

Δτms

Increment in strength due to modulus strengthening

MPa

Δτppt

Increment in strength due to precipitation strengthening

MPa

MPa

1. Introduction Aluminum and its alloys are well-known for their suitable combination of superior specific strength, higher formability and excellent atmospheric corrosion resistance; as a consequence these materials find wide applications in automobile, aerospace and domestic appliances [1,2]. Amongst the various age-hardenable Al-alloys, Al-Mg-Si (6XXX) alloys are widely used due to their relatively low cost as compared to the Al-Cu (2XXX) and Al-Zn (7XXX) alloys [3]. One of the important varieties of 6XXX alloy series is AA6063. It 3|Page

contains small amounts of Mg (0.45-0.90 wt.%) and Si (0.2-0.6 wt.%) but exhibits excellent age-hardening

characteristic

due

to

formation

of

transient

Mg-Si

intermetallic

compounds [4]. The process of age-hardening in Al-Mg-Si alloys is governed by the composition, manufacturing process and heat treatment schedule. Therefore, comprehensive understanding related to the evolution of microstructures and their influences on properties like hardness and strength is essential. A series of attempts has been made in the past to correlate yield strength of Al-alloys with microstructural features as governed by thermodynamics [5] and kinetics of age-hardening process using various models [6,7]. Unfortunately, any comprehensive guideline for engineering applications is yet to be achieved; the present investigation is focused on this issue. The precipitation sequence during ageing of Al-Mg-Si alloys is well established [8-10] and can be summarized as: Super Saturated Solid Solution (SSSS)

Clusters with varying Mg and Si contents

(FCC structure as Al)

(No stoichiometry and crystal structure)

GP zones (MgxAl5-xSi6, Primarily monoclinic)

β

β’ + B’+ U1+ U2

β”

(Mg2Si, FCC, CaF type)

(Mg1.8Si, Hexagonal + MgSi>1, Hexagonal + MgAl2Si2, Trigonal + MgAlSi, Orthorhombic)

(Mg5Si6, monoclinic)

Jiang et al. [11] have studied the influence of time of ageing on tensile properties of AA6063 alloy during artificial ageing at 433 K. Munitz et al. [12] have investigated the role of isothermal (448 K, 1-72 h) and isochronal (5.5 h, 423-513 K) ageing treatments on tensile properties of this alloy. Siddiqui et al. [2] have attempted to correlate mechanical properties with limited ageing treatment of AA6063 alloy. The reported ageing response of AA6063 alloy by Jiang et al. [11] and Munitz et al. [12] are in good correspondence. However, the magnitude of yield strength reported by Siddiqui et al. [2] is, in general, lower than that obtained by Jiang et al. [11] and Munitz et al. [12] at a specific ageing condition for materials having almost identical chemistry. Recently, Panigrahi and his co-workers [3,13-15] have performed extensive investigations on the influence of cryorolling and ageing treatments on 4|Page

microstructure and mechanical properties of AA6063 Al-alloy. These investigators have shown that cryorolling followed by suitable ageing treatments can considerably improve mechanical properties, since severe plastic deformation at very low temperature results in accumulation of higher dislocation density and suppression of dynamic recovery; these phenomena, in turn, lead to the development of microstructure with ultrafine grain size and refined precipitates. Das et al. [16] have also shown that cryorolling of solutionized AA6063 alloy exhibits improved mechanical properties due to development of ultrafine grained microstructure. However, processing of Al-alloys by severe plastic deformation at very low temperature is yet to gain industrial acceptance. An overview of the pertinent literature infers that the available information related to the variation of mechanical properties with the state of ageing for AA6063 alloy is somewhat incoherent and insufficient to establish the optimized condition for industrial applications. Consequently, systematic study on the evaluation of the strength with ageing conditions is considered necessary for AA6063 alloy. Understanding of the inter-relations between processing parameters, microstructure and mechanical properties through mathematical models is important, because it provides the ability to optimize the industrial processing route of an existing material in a cost effective manner. Thus, research in this domain has received great deal of attention resulting in the development of numerous theoretical as well as empirical models dealing with the processing-microstructure and/or microstructure-property correlations [17-19]. A series of models has been introduced by several researchers for the evaluation of physical aspects of precipitation based on thermodynamics [5] and kinetics [6,7]. Irrespective of the several developments and important modifications of various quantitative approaches dealing with either structural or strengthening [20,21] aspects, the bridging between these two approaches remained ambiguous till the pioneering work by Shercliff and Ashby [22,23]. Based on the critical size of the precipitate, these investigators have demonstrated connectivity between the shearing and the by-pass mechanisms of dislocation motion. The model is found successful to predict the isothermal ageing response of several 2XXX and 6XXX grades heat treatable Al-alloys. Later, Deschamps and Brechet [24,25] have proposed a model that synergistically considered homogeneous isothermal precipitation kinetics accounting nucleation, growth and coarsening with structural hardening mechanisms, and is capable of predicting the yield strength of age hardenable systems. These investigators have introduced the smooth transition between cutting and by-passing of precipitates and have improved the model proposed earlier by Shercliff and Ashby [22,23] 5|Page

considering particle size distribution by using Friedel [26] and Kocks [27] distribution functions. Finally, Deschamps and Brechet [24,25] have applied their suggested approach to predict the strengthening aspects associated with commercial processing which involves simultaneous phase transformation, plastic deformation and artificial ageing of 7XXX series heat treatable Al-alloys. Later on, Myhr et al. [17] have employed this precipitation model for AA6061 Al-alloy to successfully predict their hardness values. Precipitates in the commercial age hardenable Al-alloys are mostly rod or plate shaped; therefore, the simplified models based on spherical shaped particles may be inadequate to describe their structure-property relations [28]. Zhu et al. [29,30] have used an ad hoc superposition to describe the effect of the bi-modal distribution of shearable and/or nonshearable precipitates on the strengthening of age hardenable Al-Cu-Li alloy. These investigators have attempted to describe a computer simulation process to examine the dislocation slip process through obstacles formed by various particles, and to evaluate the induced strengthening by using the modified Orowan equation [31]. The work of Zhu et al. provides a guideline to determine the optimal structural conditions for the best mechanical properties of an Al-alloy. In another attempt, Zhu and Starke [30] have studied the strengthening effect of finite size precipitates randomly distributed in one slip plane and have examined the interaction of dislocations with these precipitates under applied stress fields. These investigators have also established the modified Orowan expression that can take care of precipitate shape and their orientation with the matrix. Shuey et al. [32] have examined the suitability of the ageing models proposed by Shercliff and Ashby [22] and that described by the Nucleation-Growth-Coarsening mechanisms [17,18,24,25]. These researchers have reported that the N-G-C model with an ad hoc fitted modified Orowan expression describes better the situation of precipitation strengthening than that by the original Shercliff-Ashby model and satisfactorily predicts the experimental data of Nock [33]. But, no simulation work is found in the literature on AA6063 alloy either to demonstrate the yield strength as function of state of ageing or to compare the performance of the existing models for precipitation strengthening. In this study, the primary objective is to examine two of the currently popular strengthening models, in order to assess their potential to suitably predict the experimental yield strength of AA6063 alloy as a function of the state of ageing; the models considered are the ones proposed by Deschamps and Brechet [24,25] and the modified Orowan model suggested by Zhu and Starke [30]. Considering experimentally generated yield strength 6|Page

values under wide state of ageing for AA6063 alloy, it has been demonstrated that while both models reasonably predict yield strength, the accuracy of the prediction by the Zhu and Starke model [30] is better over the Deschamps and Brechet model [24,25]. Finally, the generated results are used to develop process map capable to provide guidelines for industrial selection of ageing parameters.

2. Experimental Procedures The AA6063 is an important variety of 6XXX series Al-Mg-Si alloy; in spite of that, data related to its variation of mechanical properties with time and temperature of artificial ageing is insufficient and is often incoherent in the existing literature. A systematic approach has been directed in this study to evaluate the alteration of hardness and yield strength 6063 Alalloy over a wide range time and temperature of ageing. The related experimental procedures are briefly outlined in the following.

2.1 Material Commercially available 6063 Al-alloy in the form of hot extruded rods of 16 mm diameter was selected for this study. The nominal chemical composition of the selected material as determined by atomic emission spectroscopy analyses is summarized in Table 1. The results in Table 1 indicate that the selected alloy is in conformity with the ASM specification of the 6063 Al-Mg-Si alloy.

2.2 Heat treatment Samples of approximately 15 mm height and 16 mm diameter and cylindrical tensile specimens with 6.25 mm diameter and 25 mm gauge length as per ASTM E8-00 were machined from the available stock materials for hardness and tensile tests, respectively. These specimens were age hardened prior to the mechanical tests. The ageing treatment consisted of solutionizing at 798 K for 2 h followed by quenching in ice-water mixture (273 K). The quenched specimens were immediately subjected to artificial ageing at different temperatures ranging from 398 to 523 K at intervals of 25 K for different durations. For a given temperature, the ageing durations were selected based on preliminary studies. The selected ageing times varied from 5 min to 42 days depending on the selected temperature. 7|Page

All the heat treatment cycles were carried out under air in a tube furnace coupled with a PID temperature controller with an accuracy level of ± 2 K.

2.3 Microstructural examination In order to measure average grain size of the selected material a specimen which was aged at 523 K for 8 h is chosen. That specimen was polished metallographically up to 0.25 µm diamond paste followed by etching for 10 sec using Poltoun’s reagent (12 ml HCl + 6 ml HNO3 + 1 ml HF of 48% +1ml H2O). Digital micrographs were then recorded using optical (Model: DM6000M, Germany) and scanning electron microscope (Model: JSM5800, JEOL, Japan). These were then analyzed using ImageJ 1.47V in order to estimate average grain size of the selected material using line intercept method as described in ASTM E 112-13. In order to understand the size and morphology of the precipitates suitable samples of under aged, peak aged and over aged AA6063 alloy were examined using a Tecnai 20 T analytical TEM. The TEM samples were prepared in the following stages: (i) thin foils of around 200 µm were first cut using an Isomet low speed saw, (ii) these were mechanically polished on silicon carbide abrasive papers down to 100 µm, (iii) discs of 3 mm diameter were punched out from these foils, (iv) finally electro-polishing was done using twin jet electro polisher prior to TEM examinations; electro-polishing was done using 30% HNO3 solution in methanol as the electrolyte at 243 K.

2.4 Measurement of mechanical properties All heat treated specimens were mechanically polished to remove the oxide layer and to achieve good surface finish (0.25 µm diamond polish). Vickers hardness of differently aged specimens was determined under 2 kgf load using indentation time of 10 sec with the help of a Vickers hardness tester (VMHT: Leica, Germany). A minimum of fifteen readings were taken to estimate the mean hardness and the associated standard error following the guidelines of ASTM E92. Tensile tests were carried out using a close-loop servo-hydraulic universal testing machine (8801: Instron, Norwood, MA, USA) of ±100 kN load capacity. The test control, data acquisition and data analyses were done by using the Bluehill®software (Version 2, Instron, Norwood, MA, USA). All these tests were carried out at ambient temperature of approximately 300 K using a crosshead speed of 1.92 mm/min that

8|Page

corresponds to nominal strain rate of 1 × 10−3 s−1. The reported tensile properties are average of at least two tests results.

3. Experimental results 3.1 Microstructure The average grain size of the selected AA6063 alloy after solution treatment (798 K for 2 h) and stabilization at 523 K for 8 h is found to be 52 + 5 μm. The characteristics of the precipitates formed typically at the peak aged and the over aged conditions of the AA6063 alloy are illustrated in the TEM BF images in Fig. 1. The precipitates are rod shaped and coherent in nature. No contrast or distortion free regions are noticed at the center of the particles. The observed precipitates in Fig. 1 are in general coffee bean shaped in nature. The average dimensions of the precipitates in the microstructures of the peak aged and the over aged conditions are estimated as 62.8 nm x 5.77 nm and 500 nm x 35.7 nm, respectively. These estimations indicate that the precipitates have coarsened with increase in their aspect ratio during over ageing, in agreement with natural expectation. Typical selected area diffraction pattern of the matrix in the peak aged specimen with <011> zone axis is shown as an insert in Fig. 1(a); no diffraction spot from the precipitates could be observed in the SADP, as the precipitates are fully coherent with the matrix. The precipitates are found to lie either on the {022} or on the {200} planes of the matrix.

3.2 Ageing response Solution treated specimens were subjected to age hardening at temperatures of 398, 448 and 498 K for different durations ranging from 5 min to 42 days in order to examine the effect of the state of ageing on hardness and yield strength. The variations of hardness and yield strength as function of time of ageing (tA) for various temperature of ageing (TA) are shown in Fig. 2(a) and (b), respectively. The peak hardness at 448 K (Fig. 2(a)) is found to be 89.46 (±0.252) HV2 which is in good agreement with the reported hardness of similar alloys when aged at 433 K [11]. The results obtained from tensile tests of the aged alloys, as shown in the Fig. 2(b), corroborate well with the data reported by Munitz et al. [12] at the temperature of ageing of 448 K. The present results indicate considerable deviation from that reported by Siddiqui et al. [2]. The observed difference is considered to have originated from the fact that the earlier 9|Page

researchers [2] have considered different homogenization condition. The solutionizing condition adopted by Siddiqui et al. is debated [34] because it may lead to undissolved b phase in the microstructures. The results in Fig. 2 follow the characteristic pattern for artificial ageing; no double peak has been detected at 448 K ageing unlike the results reported by Roven et al. [35]. One can note from Fig. 2, that hardness values show a plateau for considerable ageing duration unlike the ageing curves for modified 6XXX alloy [9] where relatively rapid drop of hardness from its values at the peak aged condition has been observed. The observed plateau at the peak aged condition in Fig. 2 indicates stable formation of the precipitates. The stability of precipitation is known to be strong function of alloy composition as it gets specifically perturbed by excess content of Si or other elements which leads to faster coarsening of the precipitates [8]. The variation of peak hardness and yield strength as function of temperature of ageing is depicted in Fig. 3. A linear relation between hardness and yield strength is obvious. It can be noted that the magnitude of yield strength is about three times that of hardness.

4. Models for strengthening of Al-alloys 4.1 Strength models The major strengthening mechanism for age hardening of Al-alloys is precipitation strengthening apart from solid solution strengthening, grain boundary strengthening and modulus strengthening. The classical attempts to predict the precipitation strengthening is based on interaction between spherical shaped particles and dislocations. In AA6063 Al-MgSi alloy, the precipitates are of rod shaped [36-38] and models incorporating this particle shape are also proposed by a few earlier researchers [31,39]. These models consider various modifications of Orowan equation, thermodynamics for precipitation and diffusional transformation with appropriate consideration of temperature and time. The latter considerations can provide estimation of the volume fraction of the transformed precipitates and their radius [40,41]. A brief account of these models is summarized below preceded by an outline of the conventional strengthening mechanisms which remain inherently associated with the former ones.

10 | P a g e

4.1.1 Solid solution strengthening Solute atoms, whether interstitial or substitutional, strengthen the matrix primarily due to the variations in their size, modulus and valency [19]. Irrespective of their interaction with the dislocations, the degree of strengthening depends on the concentration of the solute atoms which can be expressed as [22]. '

%(

Δσ## = ∑) k % C&

(1)

where, Δσss is the solid solution strengthening in MPa, kj is the scaling factor for the jth )

element and *& is the concentration in weight of the jth solute in matrix.

4.1.2 Grain boundary strengthening The grain boundary strengthening of a polycrystalline material is popularly evaluated using the Hall-Petch relation [42,43] as given below: 0

∆σ,- = σ& + k & d/'

(2)

where, Δσgb is the increase in yield strength due to grain boundary strengthening, σi is the intrinsic strength at infinite grain size, ki is the locking factor and d is the grain size in micron. Typical value of σi and ki for Al-alloys are 16 MPa [44] and 0.065 MPa m-1/2 [45], respectively.

4.1.3 Modulus strengthening Several attempts have been made by earlier researchers [46,47] to assess the modulus strengthening in alloys. The widely accepted theory to quantify the modulus strengthening with small particle, as suggested by Hanson and Morris [46] can be expressed as: ∆τ2# =

:

(

∆< ' 0.95(rf) 8-; 8 < ;

(

[email protected] /' 82bln -√B;

(3)

where, Δτms is the effect of modulus strengthening on yield strength, r and f are the radius and volume fraction of precipitates, respectively, and b is the Burger’s vector. The dislocation line tension Γ is equal to Gb2/2, where G is the shear modulus of the matrix, and ΔG is the difference in shear modulus between the precipitate and the matrix. 11 | P a g e

4.1.4 Precipitation strengthening Precipitates in a matrix hinder the movement of dislocations, and this is the underlying mechanism for precipitation strengthening. The interaction of precipitates with dislocations however is structure sensitive. The physics of this interaction lies on the attractive and repulsive stress fields associated with the precipitate and the dislocation. Smaller precipitates are found to be sheared by dislocations; whereas, dislocations easily by-pass the larger precipitates leaving behind dislocation loops around the particles. In both situations, an increase in stress is required to continue the glide of dislocation in the slip plane for further deformation. In the current investigation, two existing models have been examined to assess their appropriateness to describe the precipitation strengthening behavior of AA6063 alloy during isothermal ageing. Although both the models are based on the principle of interaction between dislocation and precipitate, their responses against the nucleation barrier for precipitation are different. A brief description of the underlying theory and the constitutive equations considered in these models are given below.

4.1.4.1

Classical dislocation-particle interaction model

Classical dislocation-precipitate interaction model deals with both shearing and bypassing of spherical shaped precipitates by dislocations. As mentioned in the previous section, Deschamps and Brechet [24] have altered the Shercliff-Ashby [22] model by inducing particle size distribution. These investigators have introduced the Friedel formalism [26] to evaluate the size distribution of the precipitates. According to the Deschamps and

Brechet [24], the mean obstacle strength has been described using a parameter F , which takes

care of statistical aspects of precipitate size and distribution. The overall macroscopic contribution towards precipitation strengthening considering the classical approach is [17,25]: ∆τ""# =

$

%&

(4)

where, L is the mean effective particle spacing in the slip plane which can be evaluated using Friedel’s proposition [26]. Replacing the value of L in equation (4) as per Friedel’s suggestion, the strengthening due to precipitation can be written as [25,48]: Δτ""#

/

0 = %) (2βGb, ).0 1,56 F 0

(

/

34

7

(5)

12 | P a g e

where, β is a constant close to 0.5. Precipitates are classified as strong and weak obstacles [17] to dislocation motion. Weak obstacles are considered to be sheared off whereas, the strong obstacles are considered to be by-passed by the dislocations.

The parameter F is a function of radius of the precipitate, r. For, weak obstacle F is

proportional to r as long as the radius of precipitate is smaller than the critical radius of shearing, rc [25] and this can be evaluated from the equation: ) F = 2βGb, )

(6a)

8

On the other hand, strong non-shearable particles are encountered when r > rc, where F is

expressed as [24]: F = 2βGb,

(6b)

For uniform distribution of either weak or strong obstacles equations (5) and (6) provide agreement with the classical models proposed by Friedel [26] and Kochs [27].

4.1.4.2

Non-shearable particle model

Larger particles are considered to be by-passed by the dislocations and thereby, the basic Orowan-Ashby equation for particle bypassing can be written as [49]: ∆τ""# = 0.13

>% &

)

ln %

(7)

where, Δτppt is the increment in yield strength due to precipitation and L is the inter-particle spacing. In the Orowan-Ashby’s model [49], the precipitates are considered as spherical in shape. Therefore, the obstacles to dislocation motion are localized, and hence it is considered as the distribution of point forces. In the commercial age hardenable Al-alloys, however, the shape of the precipitates are either of rod or plate shaped; for example, β” or β’ precipitates in AlMg-Si alloys and η’ precipitates in Al-Zn-Mg alloys [18]. The rod or plate shaped precipitates have morphology with multi directional orientation. Moreover, the precipitates are usually randomly distributed even for non-slip planes. Therefore, the precise simulation of interaction of these particles with dislocations is difficult [28]. This difficulty has been dealt with by

13 | P a g e

computer aided simulation by Zhu and Starke [30] for (1 0 0) rod-shaped precipitates using the modified version of equation (7) as: 7

>%

∆τ""# = 0.15 ,) × @√f + 1.84f + 1.84f 0 E ln

,.H3,) )I

(8)

where, Δτppt is the increment of yield strength due to precipitation strengthening and r0 is the dislocation core radius = 6J̇ [50].

4.1.5 Overall yield strength In engineering alloys, usually multiple strengthening mechanisms are simultaneously operative. Ignoring other strengthening mechanisms like texture strengthening, second phase strengthening etc. the total yield strength of the alloy can be expressed as: σMN = ∆σNN + ∆σP% + MR(∆τSN , + ∆τ""# , )

(9)

where, σys is the total yield strength of the alloy and M is the Taylor factor considered as 3.1 for Al-alloys [51]. Using equations (1), (2), (3) and (5) or (8) the different strengthening contributions to yield strength can be compiled as:

σTU

V—W

=

0

7 ∑\ k Y C[Y

/

+ σ] + k ] d.0 +

7 7 , e ∆> 0 ,) .0 M^_`0.9c(rf) 1 6 1 6 12bln 6 g % > %√4

7

>%

+ 10.15 ,) × @√f + 1.84f + 1.84f 0 E ln

, ,.H3,) 6 h )I

(10) σTU

i—j

= 0 7

7

,) .

7

,

/

,

∑\ k Y C[Y + σ] + k ] d.0 + M^_`0.9c(rf) 1 6 1 60 12bln 6 0 g + ` (2βGb , ).0 1 60 F 0 g h %) % > %√4 ,5 /

e

∆>

(

/

34

7

(11) 14 | P a g e

Equations (10) and (11) describe the overall yield strength for the Al-alloy according to the models based on the propositions by Zhu and Starke (henceforth abbreviated as Z-S model) [29,30] and Deschamps-Brechet (henceforth abbreviated as D-B model) [24,25], respectively. It may be worthy to mention here that some Al-alloys [52-54] undergo discontinuous precipitation concurrently with continuous precipitation. The occurrence of discontinuous precipitation has not been considered in the earlier reports [18,24,25,55] related to strength modeling of Al alloys during artificial ageing possibly because of simplification of the model or that discontinuous precipitation has not been evidenced in such alloys. Therefore, the strength models considered in this study are limited with the consideration of only continuous precipitation.

4.2 Evaluation of radius and amount of precipitates It can be noticed from equations (3) to (8) that strengthening contributions are dependent on the radius and the volume fraction of the precipitates which are obvious functions of the state of ageing (i.e., time and temperature of ageing) and/or nucleation barrier for precipitation. The evaluation of the radius of the precipitates can be done using Ham-HorvayCahn [5,56,57] theory whereas the volume fraction of the precipitates can be calculated by Becker-Doring law [40] or by relevant experimental findings [40,41]. The expression of radius and volume fraction of precipitate can be assessed as [18]: r=

,m

f=

,5)7

3

√(β′Dt m )

mqsu

A[ Zβ t m e ∗

(12) z

{∆|∗  }~€

(13)

where, D is the diffusivity of solute atom in the solvent, A0 is Avogadro number, R is the universal gas constant and, TA and tA are the temperature and time of ageing, respectively. A and β’ are the aspect ratio and the dimensionless growth parameter, respectively. β’ can be evaluated as: β′ =

(‚I .‚ƒ )

„…†‚‡ .‚ƒ ˆ

(14)

15 | P a g e

where, Cp is the composition of the precipitate, Ce is the equilibrium solute content at the precipitate-matrix interface, Z is Zeldovich’s factor (= 0.05) [18]. ΔG* is the activation energy for nucleation of precipitates and β* is a parameter expressed as [18]: β∗ =

0

‰5)∗ Š‹I Œ

(15)

where, a is the lattice parameter and r* is the critical radius of precipitate for nucleation given as [40]: r∗ =

,Žqsu €

‹

@C" ln 1 I 6 + †1 − C" ˆln 1 ‹‘

(.‹I

(.‹‘

6E

.(

(16)

in which, γ is the interfacial energy and υat is the molar volume of the precipitate. It is clear from the above equations (12) and (13) that the critical and unknown parameters are ΔG* and A. Therefore, these two parameters should be carefully estimated prior to any simulation work. After peak ageing, the volume fraction of the precipitate particles becomes constant corresponding to its maximum value of fS =

‹I .‹‘ ‹“

[18] in which

Ce can be evaluated as [17]: C” = exp 1

-∆˜I ™€ ∆šI €

6

(17)

where, ΔH0 and ΔS0 are the enthalpy and entropy of formation or dissolution of Mg5Si3 precipitates for the Al-Mg-Si alloys [17].

4.3 Modification of the radius of precipitates for 6063 Al-Mg-Si alloy Equation (12), for evaluating the radius of precipitates in order to determine the precipitation strengthening is applicable only for spherical shaped ones. But, in AA6063 alloy the precipitates are of rod shaped morphology [10,55] and such modification has been accounted in equation (18) as [18,40]: ,

r = 3 √(β′Dt m )

(18)

4.4 Estimation of nucleation barrier and aspect ratio 16 | P a g e

An attempt has been made here to explain the effect of precipitation on the yield strength of the selected alloy as a function of tA and TA using D-B and Z-S models. In order to perform these exercises, it is a priori required to determine the variation of activation energy of nucleation (ΔG*) and that of aspect ratio (A) of precipitates with TA. These estimates have been made by multivariable regression analyses using experimentally obtained yield strength values of AA6063 alloy aged at 398, 448 and 498 K. During the regression analysis, values of ΔG* and A have been kept in the ranges of 30 to 200 kJ mol-1 and 1 to 100, respectively. The guideline for computations of ΔG* and A are followed from the earlier work of Liu et al. [18] for AA6061 alloy. The composition of the precipitate (Cp), coefficient of diffusivity (D0), activation barrier of diffusion (QD), enthalpy (H0) and entropy (S0) of precipitation and molar volume (vat) and maximum volume fraction of precipitates (fmax) used for the estimation of ΔG* and A are compiled in Table 2. The estimated values of ΔG* and A against TA obtained from both D-B and Z-S models are shown in the Figs. 4 and 5, respectively. The estimated values of ΔG* with TA appears linear within the considered range. Therefore, linear regression analyses have been carried out to establish relationship between ΔG* and TA, and this relation has been subsequently used for the purpose of prediction. The estimated value of ΔG* at 473 K (71.24 kJ mol-1) from the relation suggested by Zhu and Starke shows general agreement with the experimentally obtained value of 90.7 kJ mol-1 at TA = 473 K for AA6063 alloy by Pogatscher et al. [58]. The estimated values of A in Fig. 5 are found to follow a parabolic relation with the TA. The current estimations of the theoretical and the experimental values of A and that reported in the earlier works [3,14,15,35-37] are compiled in Fig. 5, which indicate that the nature of variation of A obtained from experiment and simulation is similar. All input parameters used for prediction of yield strength are compiled in Table 2.

5. Simulation results and discussion 5.1 Prediction of yield strength The values of yield strength estimated from Z-S and D-B models along with the experimentally measured ones at three different TA (423, 473 and 523 K) against tA are depicted in Fig. 6(a) and (b), respectively. The results in Fig. 6 indicate that yield strength values estimated from both the models are in good agreement with their corresponding experimental values in the under aged regimes, but the values in the over aged regimes show 17 | P a g e

either underestimations or overestimations. The deviations of the experimental yield strength values from the predicted ones could be as high as 25% for the Z-S model and 30% for the DB model. The observed difference between the estimated and the simulated yield strength values in the over aged regime are similar to that reported by Liu et al. [18]. These investigators have attributed this difference to the disparity in the consideration of the radius of the simulated and the observed precipitates [18]. In order to examine the proposition of Liu et al. [18], limited investigation has been carried out to measure the radius of the precipitates for a specimen aged for 8 h at 523 K. It is found that the estimated average radius of the precipitates at this ageing condition is 35 (+13) nm as compared to the computed radius of precipitate as 24.5 nm. However, detailed exercise is required by TEM examinations to measure the radius of the precipitates at differently aged conditions so as to understand the observed difference between the predicted and the experimental yield strength values. The models proposed by Deschamps-Brechet and Zhu-Starke account only the grain boundary strengthening, solid solution strengthening and precipitation strengthening; however, in this study modulus strengthening has also been incorporated, as described in section 4.1.3. Typical contributions from all the considered strengthening mechanisms at 473 K are illustrated in Fig. 7. Solute concentration depletes during the process of precipitation and thereby, solid solution strengthening decreases with progress of ageing. Beyond peak ageing, no solute atom ingresses to the precipitate phase, therefore, the extent of strengthening from solute atoms approaches a saturated plateau (Fig. 7). Variation of modulus between the precipitates and the matrix contributes to increasing modulus strengthening due to increased volume fraction of precipitates till peak ageing. Beyond peak ageing, however, precipitates are not sheared but bowed off around dislocations. As a consequence, the contribution from modulus strengthening for larger precipitates is considered as constant beyond peak ageing. It is worth mentioning at this stage that peak ageing condition is encountered earlier for the yield strength values simulated from the D-B model than that from the Z-S model.

5.2 Prediction of ageing kinetics One of the important processing parameters for age-hardenable Al-alloys is the peak ageing time (tp) at a given temperature of ageing (TA). The magnitudes of tp at different TA 18 | P a g e

are identified from the simulated results of yield strength (sys) versus time of ageing (tA) (Fig. 6) for the Z-S and the D-B models. These results are depicted in Fig. 8. The experimentally obtained tp values for the employed TA are also superimposed in Fig. 8 in order to compare the appropriateness of the models. The plot of tP versus TA derived from the Z-S model is in well accord with the experimentally estimated values of tP; whereas, that estimated from the D-B model are found to be consistently less than that of the Z-S model. Shercliff and Ashby [22] have proposed a method to estimate the activation energy for the precipitation process (Q) following Arrhenius type of relationship between the tP and TA. The technique proposed by Shercliff and Ashby has been used to derive the values of Q for the selected material within the investigated range of TA. The results in Fig. 9 exhibit bi-linear variation between ln(tP/TA) and 1000/TA for the experimentally obtained as well as for the simulated results following D-B and Z-S models for the investigated Al-alloy. The nature of variations of ln(tP/TA) versus 1000/TA indicates considerable deviation in the magnitude of Q beyond a particular TA possibly due to the change in the mechanism of precipitation [59]. The magnitudes of Q for the different regimes of TA have been determined by separate linear regression analyses. The estimated Q values and the identified TA at which the transition occurs are mentioned in Fig. 9. The temperature corresponding to the transition of Q value is obtained as 450 K from analyses of the experimental results. This parameter is found to be 405 K and 432 K from the results of simulation obtained using the D-B and Z-S models, respectively. From the experimental results of ln(tP/TA) versus 1000/TA, the obtained Q values are found to be 124.6 kJ mol-1 and 62 kJ mol-1 respectively for the lower and the higher temperature regimes. The Q values obtained from the Z-S and D-B models are found to be 135.2 and 157.4 kJ mol-1 respectively in the lower temperature regime, and 70.8 and 72.1 kJ mol-1 respectively in the higher temperature regime, respectively. Doan et al. [59] have reported Q values for various regimes of TA for an Al-Mg-Si alloy using DSC results following the suggested procedures by Kissinger [60] and Ozawa [61]. Doan et al. have reported average Q value of 129.3 kJ mol-1 for TA < 465 K and 93 kJ mol-1 for TA > 465 K. The estimated results from experiments and simulations are, in good agreement with the results reported by Doan et al. [59]. In the lower regime of TA, diffusion of solute atoms (e.g. Mg and Si) in the Al matrix is found to govern the mechanism of precipitation since the estimated Q values corroborate well with the activation energies of diffusion of Si and Mg in Al [59]; the magnitudes of QDSi in Al = 124 kJ mol-1 and QDMg in Al = 131 kJ mol-1 [63]. The mechanism of precipitation reaction changes at the 19 | P a g e

relatively higher TA, where the competitive formation and dissolution of β” and β’ precipitates may be the dominating ones [59]. In addition, the observed alteration in the mechanism of precipitation with TA may be attributed to the sequence of formation of transient precipitates for age-hardenable alloys. For example, Karadeniz et al. [62] have reported absence of GP zones for Al-Mg-Si system at TA > 430 K, the GP zone solvus. Interestingly, the observed transition temperatures at which sharp change in Q values occurs (see Fig. 9) are nearly the same as the GP zone solvus of the Al-Mg-Si system, as reported by Karadeniz et al. [62]. It is worth to note that the intercept of the linear fits (between ln(tP/TA) and 1000/TA) is consistently lower compared to that from the simulated results obtained from the D-B model than that of the Z-S model. This observation infers that the pre-exponential factor in the Arrhenius type expression [22] is lower in case of the D-B model than in the Z-S model. This observation implies that the tP will be relatively lower in case of D-B model as compared to that obtained from Z-S model, in agreement with the obtained results (Fig. 7).

5.3 Utility and limitation of simulation The present study emphasizes on examining the suitability of the models proposed by Deschamps-Brechet [24] and Zhu-Starke [29] to predict the yield strength with artificial ageing parameters for the precipitation hardening Al-Mg-Si (6063) alloy. The significance of the present study centers on developing a process map (between temperature of ageing, time of ageing and yield strength) which can provide a guideline to obtain suitable heat treatment schedules for achieving desired levels of yield strength for an alloy. Process maps for the AA6063 alloy are constructed following both D-B and Z-S models and these are presented in Fig. 10. Subsequently, an attempt has been made to demonstrate the utility of the developed process maps. This has been done through comparison between spectrums of the yield strength values predicted from the developed process maps (Fig. 10) and those reported in the literature, via experimental determination, for similar Al-Mg-Si system under three widely different states of ageing [64-66] as summarized in Table 3. One can note from the results in Table 3 that the difference between the reported yield strength values obtained by earlier researchers and that predicted from the process map derived from the Z-S model (Fig. 10(a)) is less than 10%; thus the map developed using Z-S model predicts the range of yield strength 20 | P a g e

with reasonable accuracy irrespective of the ageing conditions. A similar exercise to examine the compatibility of the experimental results with the process map developed through D-B model (Fig. 10(b)), however, indicates deviations specifically for the over aged state (Table 3). The constructed process maps suffer from a couple of limitations as discussed in section 4. Amongst the assumptions made for simplifying of computation process, the major one is the consideration of the radius of precipitate as a parabolic function of time of ageing. This assumption leads to underestimation of the size of precipitates beyond peak ageing due to obvious coarsening phenomenon. This limitation can possibly be overcome to a large extent if one incorporate Lifshitz-Slyozov model [67] of precipitate coarsening. However, this model requires several additional parameters to be determined experimentally like distribution of radius of precipitates at the concerned temperatures of ageing. These results can only be achieved through time consuming and expensive experimental efforts e.g. by small angle neutron scattering [55] or HRTEM analysis. In spite of this limitation; the process map developed in this study using a simplified manner predicts yield strength with acceptable accuracy for convenient industrial utilization. In generalization, this work has demonstrated in a simple way the comparison between the predicted yield strength values from the D-B and Z-S models and an idea to develop the process maps using these models for industrial utilization.

6. Conclusions The present investigation compares the appropriateness of the classical dislocationparticle interaction model proposed by Deschamps and Brechet [24,25] and the model based on the modified Orowan mechanism for non-shearable rod shaped precipitates rejuvenated by Zhu and Starke [29,30] for predicting the yield strength of AA6063 alloy based on extensive experimental evaluations of yield strength of the material at different ageing conditions. The microstructural alterations during ageing have been simulated using well established thermodynamic and kinetic models. The aspect ratio of the precipitates is found to have parabolic dependence on the ageing time and temperature and hence, it has been considered subsequently during simulations. The analyses of the results related to the simulations and experimental assessment of yield strength lead to the following conclusions:

21 | P a g e

·

Both Zhu and Starke as well as Deschamps and Brechet models reasonably predict yield strength of 6063 Al-Mg-Si alloy. In the over aged regime, the predicted results, however, show deviation from the experimentally measured yield strength values due to the differences in the estimated and the measured precipitate sizes; overestimation is encountered at lower temperatures of ageing and underestimation at higher temperatures of ageing.

·

The yield strength predicted by the Zhu and Starke model is found to be in better agreement with the experimental results as compared to that simulated by using the Deschamps and Brechet model, specifically in the domain between the under aged and the peak aged states.

·

The Mg5Si3 precipitates in 6063 alloy is rod shaped which has been duly taken care off in the Zhu and Starke model, unlike that in the Deschamps and Brechet model where precipitates are always considered as spherical shaped. Therefore, the former model is found to be more comprehensive for predicting yield strength of the selected material.

·

A thermal processing map has been developed that provides an engineering guideline for deciding the industrial ageing treatment schedule of AA6063 alloy, hitherto non-existent in the open literature.

Acknowledgment The authors are grateful to Mr. Rajdeep Sarkar for his help in TEM studies, Mr. Tarun Kar for his assistance in heat treatment and Md. Abu Bakkar for carrying out tensile tests. The assistance received from the Centre of Excellence on Microstructurally Designed Advanced Materials Development, TEQIP-II, IIEST Shibpur to carry out a part of this work is gratefully acknowledged.

References [1]

R.C. Dorward, C. Bouvier, Mater. Sci. Eng. A 254 (1998) 33-34.

[2]

R.A. Siddiqui, H.A. Abdullah, K.R. Al-Belushi, J. Mater. Proc. Tech. 102 (2000) 234240.

[3]

S.K. Panigrahi, R. Jayaganthan, J. Mater. Sci. 45 (2010) 5624-5636.

[4]

ASM Handbook, Vol. 2, ASM Internatioal. 22 | P a g e

[5]

F.S. Ham, J. Phys. Chem. Solids 6 (1958) 335-351.

[6]

Ø. Grong, O.R. Myhr, Acta Mater. 48 (2000) 445-452.

[7]

Ø. Grong, H.R. Shercliff, Prog. Mater. Sci. 47 (2002) 163-282.

[8]

L. Zhen, W.D. Fei, S.B. Kang, H.W. Kim, J. Mater. Sci. 32 (1997) 1895-1902.

[9]

A.K. Gupta, D.J. Lloyd, S.A. Court, Mater. Sci. Eng. A 316 (2001) 11-17.

[10] R. Vissers, M.A. Van Huis, J. Jansen, H.W. Zandbergen, C.D. Marioara, S.J. Andersen, Acta Mater. 55 (2007) 3815-3823. [11] D.M. Jiang, B.D. Hong, T.C. Lei, D.A. Downham, G.W. Lorimer, Meter. Sci. Tech. 7(1991) 1010-1014. [12] A. Munitz, C. Cotler, M. Talianker, J. Mater. Sci. 35 (2000) 2529-2538. [13] S.K. Panigrahi, R. Jayaganthan, V. Chawla, Mater. Sci. Eng. A 492 (2008) 300-305. [14] S.K. Panigrahi, R. Jayaganthan, Met. Mat. Trans A 41 (2010) 2675-2690. [15] S.K. Panigrahi, R. Jayaganthan, Mater. Sci. Eng. A 528 (2011) 3147-3160. [16] M. Das, T.K. Pal, G. Das, Mater. Sci. Eng. A, 552 (2012) 31-35. [17] O.R. Myhr, Ø. Grong, S.J. Andersen, Acta Mater. 49 (2001) 65–75. [18] G. Liu, G.J. Zhang, X.D. Ding, J. Sun, K.H. Chen, Mater. Sci. Eng. A 344 (2003) 113-124. [19] M. Dixit, R.S. Mishra, K.K. Sankaran, Mater. Sci. Eng. A 478 (2008) 163-172. [20] J.W. Martin, Micromechanisms in Particle-Strengthened Alloys, Cambridge University Press, Cambridge, UK, 1980. [21] T.H. Sanders Jr., in: Proc. 1st Int. Conf. Aluminum-Lithium Alloys, Stone Mountain, May 1980. [22] H.R. Shercliff, M.F. Ashby, Acta Metall. Mater. 38 (1990) 1789-1802. [23] H.R. Shercliff, M.F. Ashby, Acta Metall. Mater. 38 (1990) 1803-1812. [24] A. Deschamps, Y. Brechet, Acta Mater. 47 (1999) 281-292. [25] A. Deschamps, Y. Brechet, Acta Mater. 47 (1999) 293-305. [26] J. Friedel, Dislocations, Pergamon, Oxford, 1964. [27] U.F. Kocks, Can. J. Phys. 45 (1967) 737-755. [28] P.M. Kelly, Scripta Metall. 6 (1972) 647-656. [29] A.W. Zhu, A. Csontos, E.A. Starke Jr., Acta Mater. 47 (1999) 1713-1721. [30] A.W. Zhu, E.A. Starke Jr., Acta Mater. 47 (1999) 3263-3269. [31] J.F. Nie, B.C. Muddle, I.J. Polmear, Mater. Sci. Forum 217-222 (1996) 1257-1262. [32] R.T. Shuey, J.P. Suni, M. Tiryakioglu, Mater. Sci. Forum 28 (2004) 118-123. [33] J.A. Nock, Heat Treatment and Aging 61S Sheet, Iron Age, 159 (1947) 48-54. 23 | P a g e

[34] L.B. Ber, Mater. Sci. Eng. A, 280 (2000) 91-96. [35] H.J. Roven, M. Liu, J.C. Werenkiold, Mater. Sci. Eng. A 483 (2008) 54-58. [36] J.L. Cavazos, R. Colas, Mater. Sci. Eng. A 363 (2003) 171-178. [37] H.Y. LI, C.T. Zeng, M.S. Han, J.J. Liu, X.C. Lu, Trans. Nonferr. Met. Soc. China 23 (2013) 38-45. [38]

B. Milkereit, N. Wanderka, C. Schick, O. Kessler, Mater. Sci.

Eng. A 550 (2012) 87-96. [39] D.L. Gilmore, E.A. Starke Jr., Metall. Mater.Trans. A 28 (1997) 1399-1415. [40] A.K. Jena, M.C. Chaturvedi, Phase Transformation in Materials, Prentice-Hall Press, New Jersey, 1992. [41] R.D. Doherty, in: R.W. Cahn, P. Haasen (Eds.), Physical Metallurgy, Elsevier, New York, 1983, pp. 1363-1505. [42] E. Hall, Proc. Phys. Soc. Lond. B 64 (1951) 747. [43] N.J. Petch, J. Iron Steel Inst. 174 (1953) 25-28. [44] E. Hornbogen, E.A. Starke Jr., Acta Metall. Mater. 41 (1993) 1-16. [45] J.D. Embury, D.J. Lloyd, T.R. Ramchandran, in: A.K. Vasudevan, R.D. Doherty (Eds.), Aluminum Alloys—Contemporary Research and Applications. Treatise on Materials Science and Technology, Vol. 31, Academic Press, Inc., Boston, USA, 1989. [46] K. Hanson, J.W. Morris Jr., J. Appl. Phys. 46 (1975) 983-990. [47] G. Knowles, P.M. Kelly, The Iron and Steel Institute, London, England (1999) 9. [48] B. Reppich, R.W. Cahn, in: P. Haasen, E.J. Krammer (Eds.), Mater. Sci. Tech. 6, 1993, p. 311. [49] M.F. Ashby, in: Proc. Second Bolton Landing Conf. on Oxide Dispersion Strengthening, Gordon and Breach, Sci. Pub. Inc. New York, USA, 1968. [50] S.D. Harkness, J.J. Hren, Met. Trans. 1 (1970) 43-49. [51] G.E. Dieter, in: Mechanical Metallurgy SI Metric Ed, McGraw Hill Book Co., 1928, pp. 188-190. [52] M. Song, K. Chen, J Mater Sci. 43 (2008) 5265-5273. [53] D.B. Williams, J.W. Edington, Acta Met. 24 (1976) 323-332. [54] F. Findik, J Mater. Sci. Letter 17 (1998) 79-83. [55] D. Bardela, M. Perez, D. Nelias, A. Deschamps, C.R. Hutchinson, D. Maisonnette, T. Chaise, J. Garnier, F. Bourlier, Acta Mater. 62 (2014) 129-140. [56] F.S. Ham, J. Appl. Phys. 17 (1959) 137. [57] G. Horvay, J.W. Cahn, Acta Metall. 9 (1961) 695-705.

24 | P a g e

[58] S. Pogatscher, H. Antrekowitsch, H. Leitner, T. Ebner, P.J. Uggowitzer, Acta Mater. 59 (2011) 3352-3363. [59] L.C. Doan, Y. Ohmori, K. Nakai, Mater. Trans. JIM 41 (2000) 300-305. [60] H.E. Kissinger, Anal. Chem. 29 (1957) 1702-1704. [61] T. Ozawa, J. Thermal Analysis 2 (1970) 301-324. [62] E.P. Karadeniz, P. Lang, P. Warczok, A. Falahati, CALPHAD 43 (2013) 94-104. [63] JIM: Kinzoku data book 3rd Eds, Maruzen, Tokyo, 1993. [64] S.E. Urreta, F. Louchet, A. Ghilarducci, Mater. Sci. Eng. A 302 (2001) 300-307. [65] D. Jiang, C. Wang, Mater. Sci. Eng. A 352 (2003) 29-33. [66] S.S. Fettah, O.S. Mahmood, S.R.J. Rojbeyani, JZS part A 9 (2007) 31-41. [67] I.M. Lifshitz, V.V. Slyozov, J. Phys. Chem. Solids 19 (1961) 35-50.

Table 1. Composition of the investigated AA6063 alloy and its standard specification.

Si

Fe

Cu

Mn

Elements (wt.%) Mg Cr Zn

Ti

ASM Standard

0.2-0.6

<0.35

<0.1

<0.1

0.45-0.9

<0.1

<0.1

<0.1

Other elements Each Total <0.05 <0.15

Selected alloy

0.4285

0.1778

0.0010

0.0815

0.4997

0.0063

0.0665

0.0235

In conformity

25 | P a g e

Al Balance Balance

Table 2. Summary of input parameters used in the simulation processes. Para

Description of the parameter

Values

References

meters C0

Nominal solute concentration in the matrix

0.003883

Present study

4.4x10-4

[18]

130

[18]

0.59 at.%

[17]

95.9

[17]

at.% Pre-exponential term in expression of diffusion coefficient

D0

2 -1

m s

Activation barrier for diffusion

QD

kJ mol-1

Cp

Composition of precipitate Standard enthalpy of the reversible dissolution/precipitation reaction

Mg2Si

Standard entropy of the reversible dissolution/precipitation reaction

Mg2Si

ΔH0 ΔS0

Molar volume of precipitates

vat

Maximum volume fraction of precipitate

fmax

Particle/matrix interfacial energy

γ

Scaling factor for Si solid solution hardening model Scaling factor for Mg solid solution hardening model Taylor factor

KSi KMg M

kJ mol-1 112.0 J mol-1 K-1 7.62x10-5 3 -1 m mol 1.5%

[17]

-

[17]

0.26 J m

[17] [18]

2

66.3 MPa wt. %-2/3 29 MPa wt. %-2/3 3.1

[17] [17] [51]

b

0.284 nm

Grain diameter considered in simulation

d

50 µm

Lattice parameter of Mg2Si precipitate

a

0.639 nm

[36]

Zeldovich’s factor

Z

0.05

[18]

Magnitude of Burgers vector

[51] Present study

Table 3. Reported and predicted values of yield strength of AA6063 Al-alloy. State of ageing Under aged

Ageing parameters Temperature of Time of ageing, TA (K) ageing, tA (h) 430 5

Experimental yield strength (MPa) [Ref.] 93 [64]

Predicted yield strength (MPa) Z-S model D-B model (Fig. 10(a)) (Fig. 10(b)) 89-104 130-141

Peak aged

433

64

220 [65]

239-254

224-235

Over aged

433

100

208 [66]

205-220

109-120

26 | P a g e

27 | P a g e

Figure 01

Figure 01

Nandy et al., 2015

(a) 200 011

022

(b)

Fig. 1. TEM micrographs illustrating size and morphology of the precipitates typically for the (a) peak aged and (b) over aged conditions of the 6063 Al-Mg-Si alloy. The insert in (a) presents the SAD pattern of the matrix.

Figure 02

Figure 02

Nandy et al., 2015

(a)

(b)

Fig. 2. Variations of measured (a) hardness (HV2) and (b) yield strength (sys) as functions of time of ageing (tA) at different temperature of ageing (TA) of the selected Al-Mg-Si alloy.

Figure 03

Figure 03

Nandy et al., 2015

Fig. 3. Variations of measured peak hardness (HV2) and yield strength (sys) as functions of temperature of ageing (TA) of the selected 6063 Al-Mg-Si alloy.

Figure 04

Figure 04

Nandy et al., 2015

Fig. 4. Simulated values of activation energy for nucleation (DG*) as functions of temperature of ageing (TA) as determined using Zhu and Starke (Z-S) as well as Deschamps and Brechet (D-B) models.

Figure 05

Figure 05

Nandy et al., 2015

Fig. 5. Aspect ratio (A) of the precipitates estimated from the Zhu-Starke (Z-S) and Deschamps-Brechet (D-B) models. Experimentally measured aspect ratio values reported in the literature and that determined in the present study are superimposed for the selected 6063 Al-Mg-Si alloy.

Figure 06

Figure 06

Nandy et al., 2015

(a)

(b)

Fig. 6. Comparison between the observed and the predicted strength evolution as functions of time of ageing (tA) at different temperature of ageing (TA) for 6063 Al-Mg-Si alloy. Prediction using (a) Zhu-Starke (Z-S) and (b) Deschamps-Brechet (D-B) models.

Figure 07

Figure 07

Nandy et al., 2015

(a)

(b)

Fig. 7. Illustration of contribution of various strengthening mechanisms to the yield strength of AA6063 alloy aged at 473 K estimated from the (a) Zhu-Starke (Z-S) and (b) DeschampsBrechet (D-B) models.

Figure 08

Figure 08

Nandy et al., 2015

Fig. 8. Variations of the time to peak aged (tP) with temperature of ageing (TA) for AA6063 alloy as experimental determined and simulated from the Zhu-Starke (Z-S) and DeschampsBrechet (D-B) models.

Figure 09

Figure 09

Nandy et al., 2015

Fig. 9. Plots of ln(tp/TA) versus 1000/TA for AA6063 alloy, illustrating the variations of the activation energy (Q) for precipitation and its transition with respect to TA as analyzed using experimental as well as simulated results derived from the Zhu-Starke (Z-S) and DeschampsBrechet (D-B) models.

Figure 10

Figure 10

Nandy et al., 2015

(a)

(b)

Fig. 10. Process maps illustrating the correlation amongst temperature of ageing (T A), time of ageing (tA) and yield strength (sys) of AA6063 alloy constructed using (a) Zhu-Starke and (b) Deschamps-Brechet models.