MODELLING HIGH TEMPERATURE FLOW STRESS CURVES OF TITANIUM ALLOYS

MODELLING HIGH TEMPERATURE FLOW STRESS CURVES OF TITANIUM ALLOYS Z. Guo, N. Saunder, J.P. Schillé, A.P. Miodownik Sente Software Ltd, Surrey Technology Centre, Guildford, GU2 7YG, U.K. Keyword: Titanium alloy, Strength, Flow tre curve, Material data, Proceing imulation. Abtract Thi paper decribe the development of a computer model for the calculation of high temperature flow tre curve of titanium alloy. Two competing mechanim for deformation, either dominated by dilocation glide or dominated by dilocation climb, were conidered in the calculation. Validation ha been carried out for a variety of titanium alloy over a wide range of temperature and train rate. The flow tre data can be ued a replacement to the contitutive equation in computer-aided-engineering imulation. Introduction Thermo-mechanical proceing imulation require critical material data uch a trength and tre-train curve (or flow tre curve). The traditional way of obtaining uch data i through experimentation, which i expenive and time-conuming, becaue mechanical propertie are temperature and train rate dependent. It i therefore of no urprie that lack of material data ha been a common problem for computer-aided-engineering (CAE) imulation tool. CAE imulation package normally provide a range of contitutive equation that decribe tre a a function of temperature, train and train rate, e.g. the widely ued Johnon-Cook model [1]. However, the choice of the equation ued can ignificantly affect the imulation reult [2,3,4], a each ha it own limitation and can only work in a certain range [5]. Not only do uer have to decide which equation to ue, but they alo have to determine the value of the material parameter ued in the equation, where experimental flow tre data i a pre-requiite [4,6,7]. The preent paper look at material flow from the viewpoint of the actual mechanim during deformation. Two competing mechanim for deformation, i.e. either dominated by dilocation glide (DDG) or dominated by dilocation climb (DDC), were conidered in the invetigation. After a recapitulation of previou work on the calculation of high temperature trength, the paper focue on calculating tre-train curve in different temperature and train rate regime. The advantage of uing uch an analytical approach to replace ubjectively elected contitutive equation are dicued. High Temperature Strength Generally peaking, the trength of an alloy decay monotonically with increaing temperature until a critical temperature i reached, above which there i a harp fall in trength. Thi harp drop in trength i invariably due to a change of deformation mechanim from being dominated by dilocation glide (DDG) at low temperature to being dominated by dilocation climb (DDC) at higher temperature [8,9]. The latter i uually the controlling mechanim for creep. Different trength model have been employed in previou work [8,9] to account for the two different mechanim. Whichever ha the lower reitance to deformation control the final trength of

.2% proof tre (MPa) 12 1 8 6 4 2 Experimental Calculated DDG mechanim DDC mechanim 2 4 6 8 Temperature ( o C) Fig. 1. Comparion between experimental and calculated proof tre for titanium alloy IMI 318 (Ti-6Al-4V). the alloy. The two region are clearly hown in Fig. 1, uing the.2% proof tre v. temperature (σ.2 -T) plot of an IMI 318 alloy (equivalent to Ti-6Al-4V) a an example. The train rate ued in calculation i taken a 3.33x1-5 -1 (i.e..2/min). Thi i the typical value for tenile teting and hereafter will be denoted a ε. The train correponding to σ.2 i.2, hereafter denoted a ε. The calculation of trength at elevated temperature form the bai of the following calculation of tre-train curve. Stre- Curve Stre a a reult of deformation i dependent on train rate (ε ), train (ε), and temperature (T), which can be decribed by the following function: σ = f ( εε,, T ) (1) The σ.2 -T plot hown in Fig. 1 i a pecial cae of Eq. 1 where ε and ε are fixed a ε and ε, repectively. The tre-train (σ-ε) curve i another pecial cae of Eq. 1 where T and ε are fixed. With the recognition of the witch of mechanim in the σ.2 -T plot, one would expect to ee uch a witch in a σ-ε curve a well. Thi i chematically hown in Fig. 2, where deformation tart in the DDG-controlled region, and then above a critical train (ε t ) the creepcontrolled flow take over. In reality, apart from the mixed mode hown in Fig. 2, the oberved σ- ε curve can be dominated by either of the two mechanim. The procedure for the calculation of σ-ε curve in the two region are decribed below, repectively. Stre DDG region DDC region ε t Fig. 2. Change of deformation mechanim from DDG to DDC in a high temperature tre-train curve at the critical tranition train (ε t )

Stre-train curve in DDG region Stre-train curve can be conidered a the um of an elatic region and a platic region. In the elatic region, the tre i proportional to the train according to Hooke law. The platic region i where train hardening occur. Stre i related to train via work hardening coefficient n and contant K: σ = Kε n The value of n can be etimated from σ p via the following relationhip: n aexp( b ) = σ2. where a and b are material contant dependent of alloy type. Their value have been fitted empirically for a wide range of titanium alloy [1]. When σ.2 i known, n i known. A σ.2 i the tre at train ε, K can then be calculated a: n 2. / K = σ ε (4) (2) (3) Mot material are enitive to the rate of deformation, the o-called train rate hardening. Thi behaviour often obey a power law via train rate enitivity m. rate hardening can be combined with train hardening and decribed a below: 12 n m σ = Kε ε (5) When σ.2 correpond to ε, the.2% proof tre at train rate ε can be calculated a: σ = σ ( ε / ε ) m (6) p Therefore the tre at any given train and train rate can be calculated, which give the σ- ε curve at any train rate and temperature. Fig. 3 i the calculated tre-train curve for an Ti6242 alloy at room temperature (RT) and 482 C. The train rate ued i 3.33x1-5 -1. A thi i in the DDG-controlled region, only train hardening i oberved. Cal. 482oC.1.2.3.4.5 Fig. 3. Comparion between experimental and calculated tre-train curve for an Ti6242 alloy at RT and 482 C. The train rate ued i 3.33x1-5 -1. Flow tre curve in DDC region When in the DDC region where deformation i creep-controlled, creep curve are uually ued to decribe train a a function of tre, temperature and time t, i.e.: ε = gt (, σ, t ) (7) Since there are only two independent variable in train, train rate and time, Eq. 7 and Eq. 1 are eentially equivalent. A creep curve i therefore jut another pecial cae of Eq. 1 where T and σ are fixed. A typical creep curve contain three tage: primary, econdary and tertiary. The creep model to decribe the primary creep follow the work of Li [11]: Stre (MPa) 1 8 6 4 2 Exp. RT Exp. 482oC Cal. RT

ε ε ε ε i p = K + ε Kt ln 1 (1 e ) where ε p and ε are repectively the primary and econdary creep rate, ε i i the initial creep rate and K i an empirically evaluated material contant. ε i i et to be proportional to ε in the preent tudy [12]. To account for the tertiary tage an empirical model i ued, which relate the tertiary creep rate to the econdary rate and the creep rupture life: 4 εt = 2 Ctt d ( / Rl) ε where ε t i the tertiary creep rate, C d i a "damage contant" and R l i the rupture life, which can be readily calculated from ε via a Monkman-Grant type relationhip [9]. (8) (9) Solution to Eq. 8 and 9 relie on prior knowledge of the econdary creep rate, which can be calculated via the equation below [9,13,14]: ε ( / )( / ) n' = AD Gb RT σ E (1) where A i a material contant, D the diffuion coefficient, b the Burger vector, σ now termed a the applied tre, G and E the hear and Young' modulu of the matrix at the creep temperature, repectively. The tre exponent n i related to the mechanim of the creep and wa et a 4 for both alpha and beta phae in titanium alloy. Almot all of the parameter in Eq. 1 can be calculated [9,15]. Thi leave A a the only adjutable parameter, fitted with experimental reult. With the creep rate for all the three tage known from Eq. 8, 9 and 1 for a given tre and temperature, train ε a a function of time t, i.e. the creep curve, can be calculated a below: ε = ( ε + ε + ε ) (11) p t t At a fixed temperature, the tre-train curve σ = f ( εε, ) and the creep curve ε = f ( σ, t) are effectively forming the ame urface in the 3D pace of axe σ, ε, and ε or t. If a erie of creep curve correponding to different tree are known, the tre-train curve at a fixed train rate can be obtained. The calculated flow tre curve of grade Ti-6Al-4V (ELI) are hown in Fig. 4 Stre (MPa) 22 2 18 16 14 12 1 8 6 4 2.2.4.6.8 calc 1/ 1/ 1/.1/.1/.1/ exp 1/ 1/ 1/.1/.1/.1/ Stre (MPa) 3 28 26 24 22 2 18 16 14 12 1 8 6 4 2.2.4.6.8 calc 8C 85C 9C 95C 1C 15C exp 8C 85C 9C 95C 1C 15C Fig. 4. Comparion between experimental and calculated flow tre curve for a Ti-6Al-4V (ELI) alloy at (a) 95ºC with variou train rate, and (b) variou temperature with train rate.1/.

together with experimental data, correponding to variou train rate at 95 C and variou temperature at train rate.1-1, repectively. In all cae there i no detectable DDG region. Application of Model A computer model ha been developed to combine the calculation of tre-train curve in the two region a decribed in the previou eion. One can now calculate the tre-train curve at any given temperature and train rate. The witch of deformation mode hown in Fig. 2 i not oberved in Fig. 3 and 4. Thi i becaue the value of ε t can be quite different depending on temperature and train rate. Generally peaking, higher temperature and lower train rate reult in maller ε t. Such change in the hape of tre-train curve i clear hown in Fig. 5, uing an Ti-6Al-4V alloy a an example. The train rate ued in the calculation of Fig. 5(a) i 3.33x1-5 -1, and the temperature for Fig. 5(b) i 7 C. Switch of deformation mode i clearly oberved at 57 C in Fig. 5(a) and.1-1 in Fig. 5(b), repectively. It hould be noted that flow oftening i often uggeted to be due to recrytalliation. However, a clearly hown here, flow oftening i a natural reult of the DDC-controlled deformation whether or not recrytalliation take place. Stre (MPa) 14 25ºC 12 1 2ºC 8 4ºC 6 57ºC 4 6ºC 2 ε t 8ºC.1.2.3.4.5 Stre (MPa) 1 1-1 9 8.1-1 7.1-1 6 5 4 ε t 3 2 1.1.2.3.4.5 Fig. 5. Calculated flow tre curve for a Ti-6Al-4V alloy at (a) variou temperature with train rate 3.33x1-5 -1, and (b) 7ºC with variou train rate. Switch of deformation mechanim i clearly marked when taking place at 57 C in (a) and.1-1 in (b), repectively. The conideration of uch a witch in deformation mode allow train hardening and flow oftening to be naturally accounted for in the calculation of tre-train curve. The problem aociated with picking the right contitutive equation and aigning correct value to the material parameter involved have therefore been effectively removed. The predictive capability of the model in flow tre calculation for a wide range of Ti-alloy i demontrated in Fig. 6, which have been teted over a wide range of temperature and train rate. The model ha been implemented in computer oftware JMatPro [15]. Link with many CAE imulation package have been etablihed and the calculated flow tre data can now be organized in uch a format that can be directly read by uch package [16]. Summary The paper report the development of a computer model that i able to calculate the high temperature trength and tre-train curve in titanium alloy. Two mechanim of deformation

Calculated flow tre (MPa) 1 1 1 HT flow tre for variou Ti-Alloy 1 1 1 Experimental flow tre (MPa) Ti6Al4V 85C (.1 /) 9C (.1/) 95C (.1/) 1C (.1/) 15C (.1/) 9C (5/) 1C (5/) 11C (5/) 12C (5/) 13C (5/) 8-125C (.1/) 8-125C (1/) 8-125C (3/) 9C (.5/) 9C (.5/) 9C (.5/) 193C (.1/) 8-15C (.1/) 8-15C (.1/) 8-15C (.1/) 8-15C (1/) 8-15C (1/) 8-15C (1/) Ti-5Al-2.5Sn (ELI) 193C (.1/) Ti-B19 1C (.1/) 1C (.1/) 1C (1/) 1C (1/) Beta 21S 8C (.5/) 8C (.1/) 8C (.5/) 8C (.5/) 8C (.5/) Ti-123 8C (.42/) 8C (.42/) 8C (.42/) 8C (.42/) 75C (.42/) 75C (.42/) 75C (.42/) 75C (.42/) Ti-15333 843-126C (.1/) 843-126C (.1/) 193-126C (.1/) 927C (.1/) 927C (.1/) 927C (.1/) 193C (.1/) Ti6242 899C (.1/) 899C (.1/) 899C (.1/) 899C (1/) 927C (.1/) 927C (.1/) 927C (.1/) 927C (1/) 954C (.1/) 954C (.1/) 954C (.1/) 954C (1/) 982C (.1/) 982C (.1/) 982C (.1/) 982C (1/) 11C (.1/) 11C (.1/) 11C (.1/) 11C (1/) 25-775C (.3/) Fig. 6. Comparion between experimental and calculated flow tre for variou titanium alloy at variou temperature and train rate. were conidered in the model, dominated by either dilocation glide or dilocation climb. The model can therefore generate tre-train curve very different in hape, with train hardening, flow oftening or a mixture of the two. The problem caued by lack of material data in CAE proceing modelling ha been largely overcome. Reference [1] G.R. Johnon, W.H. Cook, in: Proceeding of the Seventh International Sympoium on Ballitic, The Hague, The Netherland, 1983, pp.541-547. [2] D. Umbrello, R. M Saoubi, J.C. Outeiro, International Journal of Machine Tool and Manufacture 47 (27) 462 47 [3] D. Umbrello, J. Mater. Proce. Tech. (27), doi:1.116/j.jmatprotec.27.5.7 [4] A. Söderberg, U. Sellgren, in: NAFEMS World Congre 25, Malta, 17-2 May 25. [5] R. Liang, A.S. Khan, International Journal of Platicity 15 (1999) 963-98. [6] Y.B. Guo, J. Mater. Proce. Tech. 142 (23) 72-81. [7] B. Langrand, P. Geoffroy, J.L. Petitniot, J. Fabi, E. Markiewicz, P. Drazetic, Aeropace Science and Technology, 1999, no. 4,215-227. [8] Z. Guo, N. Saunder, A.P. Miodownik, J.P. Schillé, Material Science Forum, 546-549 (27) 1319-1326. [9] Z. Guo, N. Saunder, A.P. Miodownik, J.P. Schillé, Rare Metal Material and Engineering, 35 (Sup. 1) (26) 18-111.

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