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Crystal Plasticity modeling of dynamic recrystallization in CP Ti
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Crystal Plasticity modeling of dynamic recrystallization in CP Ti Crystal Plasticity modeling of dynamic recrystallization in CP Ti Ritam Chatterjee, Alankar Alankar*
XV Portuguese Conference on Fracture, PCF 2016, 10-12 February 2016, Paço de Arcos, Portugal
Ritam Chatterjee, Alankar Alankar*
Department of Mechanical Engineering, Indian Institute of Technology, Bombay, Mumbai – 400076, India
Thermo-mechanical modeling of a high pressure turbine blade of an Department of Mechanical Engineering, Indian Institute of Technology, Bombay, Mumbai – 400076, India airplane gas turbine engine Abstract
P. Brandãoa, V. Infanteb, A.M. Deusc*
Abstract The present work is an attempt at modeling the phenomenon of dynamic recrystallization (DRX) in commercial purity α-titanium. a Departmentwith of Mechanical Instituto Técnico, Lisboa, Rovisco 1, 1049-001 Lisboa, DRX is associated increase Engineering, in number of grainsSuperior along with lossUniversidade of strength.de Thus it isAv. critical toPais, understand its occurrence The present work is an attempt at modeling the phenomenon of Portugal dynamic recrystallization (DRX) in commercial purity α-titanium. while b structural metal components and parts are processed. A dislocation density hardening based approach accounts for the IDMEC, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, DRX is associated increase in number of grains along with loss of strength. Thus it is critical to understand its occurrence occurrence of DRXwith subject to achieving a critical value of dislocation Portugal density for each grain. A model describing nucleation and c while structural metal components and parts are processed. A dislocation density hardening based accounts for the probability of DRX is integrated into a polycrystal plasticity framework. Only slip deformation modes areapproach considered. In the present CeFEMA, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, occurrence of DRX subjectstudies to achieving a critical of dislocation density for each grain. model describing nucleation and work, multiple parametric have been carriedvalue out. The formation new DRX grains withAdeformation has been highlighted. Portugalof probability of DRX is integrated into a polycrystal plasticity framework. Onlythis slipnumber deformation are considered. In the present Due to applying a probabilistic criterion for addition of new DRX grains, is lessmodes than the number of grains having work, multiple parametric studies havevalue beenthat carried out. Thefor formation of new DRXThe grains with deformation has been highlighted. dislocation density more than critical is required occurrence of DRX. average dislocation density over all grains Due to applying criterion for addition new DRXof grains, this number is less than the number of grainsdynamic having hasAbstract been shown atoprobabilistic decrease with deformation due toofnucleation new grains. A modified algorithm for modeling dislocation density than critical value that is required forvia occurrence The average dislocation overweights all grains recrystallization hasmore henceforth been demonstrated for CP Ti evolutionofofDRX. flow stress, dislocation densitydensity and grain of their grains operation, aircraft engine subjected demanding hasDuring been shown to decrease with deformation due components to nucleationare of new grains.to Aincreasingly modified algorithm for operating modeling conditions, dynamic ‘old’ and ‘new’ with modern deformation. especially the high pressure turbine (HPT) blades.for Such conditions causeof these to undergo different of time-dependent recrystallization has henceforth been demonstrated CP Ti via evolution flowparts stress, dislocation densitytypes and grain weights of degradation, of which is creep. A model using the finite element method (FEM) was developed, in order to be able to predict ‘old’ and ‘new’one grains with deformation. behaviourPublished of HPTby blades. Flight © the 2018creep The Authors. Elsevier B.V.data records (FDR) for a specific aircraft, provided by a commercial aviation were used to obtain thermal © company, 2019 The Authors. Published by Elsevier B.V.and mechanical data for three different flight cycles. In order to create the 3D model This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open article under thea CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) needed for access the FEM analysis, HPT blade scrap was scanned, and its chemical composition and material properties were © 2018 The Authors. Published by Elsevier B.V. Selection and peer-review under responsibility ofinto Peer-review under of the 2018 Selection and peer-review under responsibility of fed Peer-review of the SICE 2018SICE organizers. obtained. The data that was gathered was theunder FEMresponsibility modelresponsibility and different simulations were organizers. run, first with a simplified 3D This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) rectangular block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The Selection and peer-review under responsibility of Peer-review under responsibility of the the trailing SICE 2018 Keywords: Crystal plasticity, titanium, dislocation theory, ICME, overall expected behaviour in terms of displacement wasDRX observed, in particular at edgeorganizers. of the blade. Therefore such a model can be useful in the goal of predicting turbine blade life, given a set of FDR data.
Keywords: Crystal plasticity, titanium, dislocation theory, ICME, DRX * Corresponding author. Tel.: +91-22-25769356; © 2016 The Authors. Published by Elsevier B.V. E-mail address: [email protected] Peer-review under responsibility of the Scientific Committee of PCF 2016. * Corresponding author. Tel.: +91-22-25769356; E-mail address: [email protected] 1. Keywords: Introduction High Pressure Turbine Blade; Creep; Finite Element Method; 3D Model; Simulation.
1. Introduction Recrystallization is a phenomenon that involves the formation of a new set of strain free grains at heterogeneities in the microstructure of a material during hot working. These new grains subsequently grow and consume the parent Recrystallization is a phenomenon that involves the formation of a new set of strain free grains at heterogeneities in the microstructure of a material hotB.V. working. These new grains subsequently grow and consume the parent 2452-3216 © 2018 The Authors. Published during by Elsevier This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of B.V. Peer-review under responsibility of the SICE 2018 organizers. 2452-3216 © 2018 The Authors. Published by Elsevier This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under * Corresponding author. Tel.: +351responsibility 218419991. of Peer-review under responsibility of the SICE 2018 organizers. E-mail address: [email protected]
2452-3216 © 2016 The Authors. Published by Elsevier B.V.
Peer-review under responsibility of the Scientific Committee of PCF 2016.
2452-3216 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of Peer-review under responsibility of the SICE 2018 organizers. 10.1016/j.prostr.2019.05.032
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grains and the cycle continues. In other words, recrystallization could be described as alternate stressing and relaxation of the material. The word dynamic when added as a prefix to recrystallization refers to work input being continuously provided to the material in order to facilitate plastic deformation. Recrystallization plays a pivotal role in imparting strength and desirable mechanical properties to the material via microstructural changes. Dynamic recrystallization was first observed nearly 80 years back by Greenwood and Worner (1939) while carrying out creep studies on an alloy of lead (Pb). Titanium is one of the most important metals known to mankind. Commercially pure titanium has widespread applications across a spectrum of industries such as aerospace, medical, nuclear, food processing etc. Due to required heat transfer characteristics, it is ideal for gas turbine blades and aircraft components. Pure titanium has excellent corrosion resistance due to which it is increasingly being used in architectural artifacts to protect them from acid rain. Due to its non-toxicity and compatibility with the human body, commercial purity Ti is used in dental and bone implants, artificial heart valves and pacemaker housings. The present work focuses on simulating the phenomenological behaviour of pure titanium under physical conditions that lead to occurrence of dynamic recrystallization. Pure titanium has been chosen as the material since the present work will later be used to compare the effect of alloying additions on the flow stress response to DRX of the strategically important aerospace alloy alpha Ti-5Al-2.5Sn. This will aid in design of optimal thermo-mechanical treatment to impart the best possible physical and mechanical properties to Ti-5Al-2.5Sn by taking advantage of beneficial facets of microstructural and texture evolution during DRX. Section 2 of the present work consists of a mathematical description of the nucleation and growth model which is based on the work of Ding and Guo (2001) and Zhou et. al (2017). The dislocation density hardening model is based on the work of Beyerlein and Tome (2008) implemented in viscoplastic self-consistent (VPSC) framework as developed by Lebensohn & Tome (1993). To update slip resistance of a slip system, the sum of contributions due to initial slip resistance, forest dislocations and debris is considered for each slip system. The dislocation density in each grain is thereafter updated based on the equation formulated by Essmann and Mughrabi (1979) that was further developed by Kocks and Mecking (1981, 2003). The occurrence of DRX in each grain is subject to achieving critical dislocation density. A flow chart is shown that depicts the complete modelling process. Nucleation is modelled based on a probabilistic scheme that allows nucleation to occur in a grain subject to its nucleation probability exceeding a critical value. Section 3 consists of a detailed discussion of results obtained from simulation of plastic deformation using VPSC which demonstrates proof concept for modeling of DRX in Ti. The flow stress behaviour of CP Ti under DRX has been captured at temperature 1173K and strain rate 0.001s-1. The updating of number of grains at each deformation step due to formation of new nuclei has been shown. Initial dislocation density has been assigned to new grains and the average dislocation density variation over all grains has been shown. Random crystallographic texture is assigned to the old grains initially as well as to the new grains.
2. Model Implementation 2.1. Nucleation and Growth Model The Ding and Guo approach (2001) for predicting the microstructural evolution due to the onset of DRX is based on achieving critical dislocation density in the grain. The approach is an attempt at improving the earlier models proposed
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by Peczak (1995) (based on temperature) and by Peczak and Luton (1994) (based on strain rate). Two important assumptions are: Initial dislocation density is same for all primary grains. DRX occurs on achieving a critical value of dislocation density. For secondary grains formed due to DRX, it increases from zero to a saturating value with deformation. New nuclei are formed only on grain boundaries. This is based on grain boundary bulging mechanism which causes strain induced boundary migration (SIBM). The nucleation rate is defined as: −𝑄𝑄 𝑁𝑁̇ = 𝐶𝐶 ∗ 𝜀𝜀̇m ∗ 𝑒𝑒𝑒𝑒𝑒𝑒 ( )
(1)
𝑅𝑅𝑅𝑅
where Q is the activation energy, T is the temperature, R is the universal gas constant, C is a constant, 𝜀𝜀̇ is the true strain rate and exponent m ~ 1. The activation energy has been taken to be 61.8KJ/mol (Stringer (1960)) and the constant ‘C’ is taken as 10000 via trial and error to obtain a high nucleation rate ~ 100s -1 per grain. 2.2. Dislocation Density Hardening based model Beyerlein and Tome (2008) developed a dislocation density hardening based algorithm for multiple slip and twinning modes in hcp material. The shear increment in each slip mode ‘m’ due to deformation can be written as the sum of shear increments in all the slip systems ‘s’ contained in the mode ‘m’: 𝑑𝑑𝛾𝛾 𝑚𝑚 = ∑𝑠𝑠∈𝑚𝑚 𝛾𝛾 𝑠𝑠̇ 𝑑𝑑𝑑𝑑 where, 𝛾𝛾 𝑠𝑠̇ = 𝛾𝛾 𝑜𝑜̇ |
(2)
𝑚𝑚𝑠𝑠 :𝜎𝜎 𝑛𝑛
| 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(𝑚𝑚 𝑠𝑠 : 𝜎𝜎)
𝜏𝜏𝑐𝑐𝑠𝑠 (𝜀𝜀̇ ,𝑇𝑇)
(3)
τc refers to the slip resistance to dislocation movement caused due to local microstructural obstacles. The ‘n’ value in the above expression is taken to be quite high ~20 so that slip or twinning only occurs when the resolved shear stress approaches the slip resistance value. The slip resistance of a slip system is the sum of contributions due to initial slip resistance, forest dislocations, debris and twin boundaries: 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝜏𝜏𝑐𝑐𝑠𝑠 = 𝜏𝜏𝑜𝑜𝑚𝑚 + 𝜏𝜏𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 + 𝜏𝜏𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 + 𝜏𝜏𝐻𝐻𝐻𝐻
(4)
The initial slip resistance depends on the temperature, strain rate, solute content and slip mode. The last term is the Hall-Petch term that is considered only when twinning effects are taken into consideration. To calculate the debris contribution to slip resistance, the debris generation must be evaluated first using the following equation: 𝜕𝜕𝜌𝜌𝑑𝑑𝑑𝑑𝑑𝑑,𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑞𝑞 ∑𝑚𝑚 𝑓𝑓 𝑚𝑚
𝑚𝑚 𝜕𝜕𝜌𝜌𝑟𝑟𝑟𝑟𝑟𝑟
𝜕𝜕𝛾𝛾𝑚𝑚
𝑑𝑑𝛾𝛾 𝑚𝑚
(5)
where q is the coefficient that represents how debris is generated from point defects via thermally activated mechanism, f is the fraction of removed dislocations (due to recovery or as defects) that are converted to debris. 𝑓𝑓 𝑚𝑚 =
𝑛𝑛𝑛𝑛 𝑚𝑚 𝑙𝑙𝑑𝑑𝑑𝑑𝑑𝑑
= 𝐴𝐴𝑚𝑚 𝑏𝑏 𝑚𝑚 √𝜌𝜌𝑑𝑑𝑑𝑑𝑑𝑑
(6)
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where A is a constant, n is a factor multiplied with the ratio of Burgers vector and average spacing between debris. The dislocation forest contribution to slip resistance is given by: 𝑚𝑚 𝜏𝜏𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 = 𝑏𝑏 𝑚𝑚 𝜒𝜒𝜒𝜒 √𝜌𝜌𝑚𝑚
(7)
where χ is dislocation interaction coefficient between 0.1 to 1 and μ is the shear modulus. The initial slip resistance is evaluated separately for each slip mode and is of the form: −𝑇𝑇
𝜏𝜏𝑜𝑜𝑚𝑚 = 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 ( )
(8)
𝐵𝐵
where A is a constant and B is a temperature. These have been calculated via curve fitting data obtained from literature. The dislocation density evolution for each slip mode is updated based on the expression provided by Essmann and Mughrabi (1979) that has been further developed by Mecking and Kocks (1981, 2003): 𝜕𝜕𝜌𝜌𝑚𝑚 𝜕𝜕𝛾𝛾𝑚𝑚
= 𝑘𝑘1𝑚𝑚 √𝜌𝜌𝑚𝑚 − 𝑘𝑘2𝑚𝑚 (𝜀𝜀̇, 𝑇𝑇)𝜌𝜌𝑚𝑚
(9)
Here, k1 is a material constant and k2 is a function of temperature and strain which describes dynamic recovery via thermally activated mechanisms. k2 is evaluated as: 𝑘𝑘2𝑚𝑚 (𝜀𝜀̇ ,𝑇𝑇) 𝑘𝑘1𝑚𝑚
=
𝜒𝜒𝑏𝑏 𝑚𝑚 𝑔𝑔𝑚𝑚
(1 −
𝑘𝑘𝑘𝑘
𝐷𝐷𝑚𝑚 𝑏𝑏 3
𝜀𝜀̇
(10)
𝑙𝑙𝑙𝑙 ̇ ) 𝜀𝜀𝑜𝑜
where, gm is the normalized activation energy. 2.3. Simulation Algorithm A few critical material properties and model parameters for Ti are shown in Table 1 (Tromans (2011)). Table 1: A few critical VPSC input parameters C11 (GPa) C33 (GPa) C44 (GPa) 160 181 46.5 b (nm) ρin (m-2) k1 (prismatic) (m-1) 0.295 1012 1*106
C12 (GPa) 90 k1 (basal) (m-1) 1*106
C13 (GPa) 66 k1 (pyramidal) (m-1) 6*107
μ (GPa) 38.5 n
χ 0.7 ρdeb (m-2)
20
12
Here, Cij are the elements of the elastic stiffness matrix, 𝜇𝜇 is the shear modulus, χ is an interaction coefficient that strongly affects initial work hardening, b is the Burgers vector, ρin is the initial dislocation density for both old and new grains, ρdeb is the initial debris density, k1 is a parameter which defines the rate of dislocation storage due to strain hardening and n is the strain hardening exponent that is set to a large value so that slip only occurs when the resolved shear stress is close to critical value. The complete modelling process is summarized in a flow chart in Fig.1. The complete process shown in Fig.1 is within one time step of numerical integration. Within one time step, nucleation occurs several times (for various grains) based on the probability criteria. The threshold value for nucleation to occur viz. Prandom has been assumed to be 0.5. The probability for each grain is calculated as:
𝑃𝑃(𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔) =
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nucleation rate in grain average nucleation rate across all grains
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 =
Total nucleation rate in grain Number of grains at particular time instant
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(11)
(12)
As soon as the probability defined in Eq. (11) exceeds critical value, event of DRX takes place. Once DRX is confirmed, random texture is explicitly assigned to new grains. Along with texture, mechanical properties for single crystal are assigned to the new grains and the cycle continues henceforth.
Update dislocation density in each grain based on Beyerlein & Tome formulation (2008)
𝜌𝜌 ≥ 𝜌𝜌𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶
NO
YES Calculate nucleation rate based on Ding and Guo formulation (2001)
Calculate nucleation probability for each grain
NO
𝑃𝑃𝑖𝑖 ≥ 𝑃𝑃𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 YES
New grain updating
Assign dislocation density, texture and mechanical properties to new grains Recompute texture weight factor wrel
Output flow stress, dislocation density evolution
Fig. 1. Flow chart depicting implementation of DRX in VPSC model
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A few important simulation parameters are shown in Table 2. Table 2: A few important simulation parameters Ngrains T (K) 𝜀𝜀̇ (s-1) wrel 1000 1173 0.001 0.001
𝜀𝜀incr 0.01
Nsteps 30
𝜌𝜌crit (m-2) 0.37*1014
Prandom 0.5
Ngrains refers to the initial number of grains in the microstructure, T is the operating temperature and 𝜀𝜀̇ is the strain rate imposed under compressive loading. The volume of grains is realised in simulation via a parameter called relative volume weight i.e. wrel which is the volume occupied by a grain in the microstructure and is represented by using crystallographic texture. The strain is incremented by 𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 in each deformation step and the total strain imposed equals 0.3 for Nsteps. The critical dislocation density 𝜌𝜌𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 has been arbitrarily chosen to be 0.37*1014 m-2. 3. Results and Discussion The major results from the simulation are shown in Fig. 2 (a) to (d): (a)
(b)
(c)
(d)
Fig. 2. (a) Flow stress variation with strain for DRX vs non-DRX (b) evolution in average dislocation density for DRX vs non-DRX (c) new DRX grains and total grains (d) percentage of DRX grains and grain size evolution with deformation
From Fig. 2 (a) and (c), the flow stress is shown to decrease as soon as new grains start forming at total strain ~ 0.2. At peak stress, rate of dislocation storage due to strain hardening exactly balances rate of dislocation annihilation due
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to recovery. The drop in flow stress in (a) signifies annihilation of old grains due to nucleation. The stabilization in flow stress occurs due to increase in dislocation density of the newly formed grains due to difference in stored energy between nuclei and their neighbouring grains. The flow stress behaviour observed in case of simulated curves compares well with the trends of flow stress saturation post critical strain as observed in the work of Zhou et. al (2017) and Blaz et. al (1983). As the number of new grains with low dislocation density keep on increasing, the average dislocation density over all grains decreases as shown in Fig. 2(b). Drop in dislocation density occurs exactly at the onset of DRX as observed in Fig. 2 (a) and (c) which further corroborates the claim of occurrence of DRX. The number of new nuclei is less than the number of grains having dislocation density greater than critical value as shown in Fig. 2 (c). The reason is attributed to nucleation probability of grains being less than the threshold value required to form a new nucleus from a grain. Since nucleation occurs due to a variety of yet undetermined physical parameters, it is modelled stochastically in the present work due to which a critical probability 0.5 is assumed. In the present work, the grain size effect in simulation is realised via a parameter called grain weight ‘w’. After new grains are ‘added’ to the system post critical strain, the weights of all grains are re-normalized according to the following formulae: 𝑤𝑤𝑜𝑜𝑜𝑜𝑜𝑜 = (𝑤𝑤𝑒𝑒𝑒𝑒𝑒𝑒ℎ 𝑜𝑜𝑜𝑜𝑜𝑜 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 /𝑤𝑤𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎 𝑜𝑜𝑜𝑜𝑜𝑜 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 ) ∗ 𝑤𝑤𝑤𝑤ℎ𝑜𝑜𝑜𝑜𝑜𝑜
𝑤𝑤𝑛𝑛𝑛𝑛𝑛𝑛 = (𝑤𝑤𝑒𝑒𝑒𝑒𝑒𝑒ℎ 𝑛𝑛𝑛𝑛𝑛𝑛 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 /𝑤𝑤𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎 𝑛𝑛𝑛𝑛𝑛𝑛 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 ) ∗ 𝑤𝑤𝑤𝑤ℎ𝑛𝑛𝑛𝑛𝑛𝑛
(13) (14)
Here, 𝑤𝑤𝑤𝑤ℎ𝑜𝑜𝑜𝑜𝑜𝑜 and 𝑤𝑤𝑤𝑤ℎ𝑛𝑛𝑛𝑛𝑛𝑛 are phase fractions of non-recrystallized and recrystallized grains. Since, grain growth is not incorporated in the present model, these phase fractions remain unchanged during deformation. These have been obtained via trial and error to be 0.95 and 0.05 respectively while calibrating the non-DRX portion of the flow stress curve against the experimental results for CP Ti obtained by Xu & Zhu (2010) and Foul et. al (2018) at temperature 1173K and strain rate strain rate 0.001s-1 as shown in Fig. 3.
Fig. 3. Calibrated flow stress of Ti vs. strain for non-DRX vs experimental at 1173 K and 0.001 s-1
In Fig. 2 (d), the average grain size across all grains is found to decrease for the old grains and increase for the new grains. This is intuitive since the number of new grains increases rapidly with deformation and hence, the new grains occupy increasing volume in three dimensional space at the expense of older grains. At ~0.3 strain, both recrystallized and non-recrystallized grains have almost equal grain weights which signifies that an equiaxed microstructure has been realised via dynamic recrystallization. Almost all grains (~95%) are observed to have recrystallized at 0.3 strain as shown in Fig. 2(d).
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4. Conclusions The trend of flow stress variation with deformation for CP Ti under DRX shows good agreement with the simulated results obtained by Zhou et. al (2017) and experimental results obtained by Blaz et. al (1983). The probabilistic criteria used to compute number of new grains at each deformation step is novel in approach. A pseudo two phase method has been used while renormalizing grain size in simulation post critical strain. At critical strain, the decrease in average dislocation density coincides with the decrease in flow stress, increase in number of new grains and evolution in grain size of both ‘old’ non-recrystallized and ‘new’ recrystallized grains thus proving the occurrence of dynamic recrystallization. Acknowledgements The authors hereby acknowledge the grant provided by Indian Space Research Organization (ISRO) under project code 17ISROC005. References Greenwood, J. Neill, Worner, H.K., 1939. Types of creep curve obtained with lead and its dilute alloys. J. Inst. Met. 64, 135. Ding, R., Guo, Z.., 2001. Coupled quantitative simulation of microstructural evolution and plastic flow during dynamic recrystallization. Acta Mater. 49, 3163–3175. https://doi.org/10.1016/S1359-6454(01)00233-6 Zhou, G., Li, Z., Li, D., Peng, Y., Zurob, H.S., Wu, P., 2017. A polycrystal plasticity based discontinuous dynamic recrystallization simulation method and its application to copper. Int. J. Plast. 91, 48–76. https://doi.org/10.1016/j.ijplas.2017.01.001 Beyerlein, I.J., Tomé, C.N., 2008. A dislocation-based constitutive law for pure Zr including temperature effects. Int. J. Plast. 24, 867–895. https://doi.org/10.1016/j.ijplas.2007.07.017 Lebensohn, R.A., Tomé, C.N., 1993. A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: application to zirconium alloys. Acta Metall. Mater. 41, 2611–2624. Essmann, U., Mughrabi, H., 1979. Annihilation of dislocations during tensile and cyclic deformation and limits of dislocation densities. Philos. Mag. A 40, 731–756. Mecking, H., Kocks, U.F., 1981. Kinetics of flow and strain-hardening. Acta Metall. 29, 1865–1875. Kocks, U.F., Mecking, H., 2003. Physics and phenomenology of strain hardening: the FCC case. Prog. Mater. Sci. 48, 171–273. Peczak, P., 1995. A Monte Carlo study of influence of deformation temperature on dynamic recrystallization. Acta Metall. Mater. 43, 1279–1291. Peczak, P., Luton, M.J., 1994. The effect of nucleation models on dynamic recrystallization II. Heterogeneous stored-energy distribution. Philos. Mag. B 70, 817–849. Stringer, J., 1960. The oxidation of titanium in oxygen at high temperatures. Acta Metall. 8, 758–766. Tromans, D., 2011. Elastic anisotropy of HCP metal crystals and polycrystals. Int J Res Rev Appl Sci 6, 462–483. Blaz, L., Sakai, T., Jonas, J.J., 1983. Effect of initial grain size on dynamic recrystallization of copper. Met. Sci. 17, 609–616. Xu, C., Zhu, W., 2010. Transformation mechanism and mechanical properties of commercially pure titanium. Trans. Nonferrous Met. Soc. China 20, 2162–2167. https://doi.org/10.1016/S1003-6326(09)60436-2 Foul, A., Aranas Jr, C., Guo, B., Jonas, J.J., 2018. Dynamic transformation of α→ β titanium at temperatures below the β-transus in commercially pure titanium. Mater. Sci. Eng. A 722, 156–159.
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Crystal Plasticity modeling of dynamic recrystallization in CP Ti Crystal Plasticity modeling of dynamic recrystallization in CP Ti Ritam Chatterjee, Alankar Alankar*
XV Portuguese Conference on Fracture, PCF 2016, 10-12 February 2016, Paço de Arcos, Portugal
Ritam Chatterjee, Alankar Alankar*
Department of Mechanical Engineering, Indian Institute of Technology, Bombay, Mumbai – 400076, India
Thermo-mechanical modeling of a high pressure turbine blade of an Department of Mechanical Engineering, Indian Institute of Technology, Bombay, Mumbai – 400076, India airplane gas turbine engine Abstract
P. Brandãoa, V. Infanteb, A.M. Deusc*
Abstract The present work is an attempt at modeling the phenomenon of dynamic recrystallization (DRX) in commercial purity α-titanium. a Departmentwith of Mechanical Instituto Técnico, Lisboa, Rovisco 1, 1049-001 Lisboa, DRX is associated increase Engineering, in number of grainsSuperior along with lossUniversidade of strength.de Thus it isAv. critical toPais, understand its occurrence The present work is an attempt at modeling the phenomenon of Portugal dynamic recrystallization (DRX) in commercial purity α-titanium. while b structural metal components and parts are processed. A dislocation density hardening based approach accounts for the IDMEC, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, DRX is associated increase in number of grains along with loss of strength. Thus it is critical to understand its occurrence occurrence of DRXwith subject to achieving a critical value of dislocation Portugal density for each grain. A model describing nucleation and c while structural metal components and parts are processed. A dislocation density hardening based accounts for the probability of DRX is integrated into a polycrystal plasticity framework. Only slip deformation modes areapproach considered. In the present CeFEMA, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, occurrence of DRX subjectstudies to achieving a critical of dislocation density for each grain. model describing nucleation and work, multiple parametric have been carriedvalue out. The formation new DRX grains withAdeformation has been highlighted. Portugalof probability of DRX is integrated into a polycrystal plasticity framework. Onlythis slipnumber deformation are considered. In the present Due to applying a probabilistic criterion for addition of new DRX grains, is lessmodes than the number of grains having work, multiple parametric studies havevalue beenthat carried out. Thefor formation of new DRXThe grains with deformation has been highlighted. dislocation density more than critical is required occurrence of DRX. average dislocation density over all grains Due to applying criterion for addition new DRXof grains, this number is less than the number of grainsdynamic having hasAbstract been shown atoprobabilistic decrease with deformation due toofnucleation new grains. A modified algorithm for modeling dislocation density than critical value that is required forvia occurrence The average dislocation overweights all grains recrystallization hasmore henceforth been demonstrated for CP Ti evolutionofofDRX. flow stress, dislocation densitydensity and grain of their grains operation, aircraft engine subjected demanding hasDuring been shown to decrease with deformation due components to nucleationare of new grains.to Aincreasingly modified algorithm for operating modeling conditions, dynamic ‘old’ and ‘new’ with modern deformation. especially the high pressure turbine (HPT) blades.for Such conditions causeof these to undergo different of time-dependent recrystallization has henceforth been demonstrated CP Ti via evolution flowparts stress, dislocation densitytypes and grain weights of degradation, of which is creep. A model using the finite element method (FEM) was developed, in order to be able to predict ‘old’ and ‘new’one grains with deformation. behaviourPublished of HPTby blades. Flight © the 2018creep The Authors. Elsevier B.V.data records (FDR) for a specific aircraft, provided by a commercial aviation were used to obtain thermal © company, 2019 The Authors. Published by Elsevier B.V.and mechanical data for three different flight cycles. In order to create the 3D model This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open article under thea CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) needed for access the FEM analysis, HPT blade scrap was scanned, and its chemical composition and material properties were © 2018 The Authors. Published by Elsevier B.V. Selection and peer-review under responsibility ofinto Peer-review under of the 2018 Selection and peer-review under responsibility of fed Peer-review of the SICE 2018SICE organizers. obtained. The data that was gathered was theunder FEMresponsibility modelresponsibility and different simulations were organizers. run, first with a simplified 3D This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) rectangular block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The Selection and peer-review under responsibility of Peer-review under responsibility of the the trailing SICE 2018 Keywords: Crystal plasticity, titanium, dislocation theory, ICME, overall expected behaviour in terms of displacement wasDRX observed, in particular at edgeorganizers. of the blade. Therefore such a model can be useful in the goal of predicting turbine blade life, given a set of FDR data.
Keywords: Crystal plasticity, titanium, dislocation theory, ICME, DRX * Corresponding author. Tel.: +91-22-25769356; © 2016 The Authors. Published by Elsevier B.V. E-mail address: [email protected] Peer-review under responsibility of the Scientific Committee of PCF 2016. * Corresponding author. Tel.: +91-22-25769356; E-mail address: [email protected] 1. Keywords: Introduction High Pressure Turbine Blade; Creep; Finite Element Method; 3D Model; Simulation.
1. Introduction Recrystallization is a phenomenon that involves the formation of a new set of strain free grains at heterogeneities in the microstructure of a material during hot working. These new grains subsequently grow and consume the parent Recrystallization is a phenomenon that involves the formation of a new set of strain free grains at heterogeneities in the microstructure of a material hotB.V. working. These new grains subsequently grow and consume the parent 2452-3216 © 2018 The Authors. Published during by Elsevier This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of B.V. Peer-review under responsibility of the SICE 2018 organizers. 2452-3216 © 2018 The Authors. Published by Elsevier This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under * Corresponding author. Tel.: +351responsibility 218419991. of Peer-review under responsibility of the SICE 2018 organizers. E-mail address: [email protected]
2452-3216 © 2016 The Authors. Published by Elsevier B.V.
Peer-review under responsibility of the Scientific Committee of PCF 2016.
2452-3216 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of Peer-review under responsibility of the SICE 2018 organizers. 10.1016/j.prostr.2019.05.032
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grains and the cycle continues. In other words, recrystallization could be described as alternate stressing and relaxation of the material. The word dynamic when added as a prefix to recrystallization refers to work input being continuously provided to the material in order to facilitate plastic deformation. Recrystallization plays a pivotal role in imparting strength and desirable mechanical properties to the material via microstructural changes. Dynamic recrystallization was first observed nearly 80 years back by Greenwood and Worner (1939) while carrying out creep studies on an alloy of lead (Pb). Titanium is one of the most important metals known to mankind. Commercially pure titanium has widespread applications across a spectrum of industries such as aerospace, medical, nuclear, food processing etc. Due to required heat transfer characteristics, it is ideal for gas turbine blades and aircraft components. Pure titanium has excellent corrosion resistance due to which it is increasingly being used in architectural artifacts to protect them from acid rain. Due to its non-toxicity and compatibility with the human body, commercial purity Ti is used in dental and bone implants, artificial heart valves and pacemaker housings. The present work focuses on simulating the phenomenological behaviour of pure titanium under physical conditions that lead to occurrence of dynamic recrystallization. Pure titanium has been chosen as the material since the present work will later be used to compare the effect of alloying additions on the flow stress response to DRX of the strategically important aerospace alloy alpha Ti-5Al-2.5Sn. This will aid in design of optimal thermo-mechanical treatment to impart the best possible physical and mechanical properties to Ti-5Al-2.5Sn by taking advantage of beneficial facets of microstructural and texture evolution during DRX. Section 2 of the present work consists of a mathematical description of the nucleation and growth model which is based on the work of Ding and Guo (2001) and Zhou et. al (2017). The dislocation density hardening model is based on the work of Beyerlein and Tome (2008) implemented in viscoplastic self-consistent (VPSC) framework as developed by Lebensohn & Tome (1993). To update slip resistance of a slip system, the sum of contributions due to initial slip resistance, forest dislocations and debris is considered for each slip system. The dislocation density in each grain is thereafter updated based on the equation formulated by Essmann and Mughrabi (1979) that was further developed by Kocks and Mecking (1981, 2003). The occurrence of DRX in each grain is subject to achieving critical dislocation density. A flow chart is shown that depicts the complete modelling process. Nucleation is modelled based on a probabilistic scheme that allows nucleation to occur in a grain subject to its nucleation probability exceeding a critical value. Section 3 consists of a detailed discussion of results obtained from simulation of plastic deformation using VPSC which demonstrates proof concept for modeling of DRX in Ti. The flow stress behaviour of CP Ti under DRX has been captured at temperature 1173K and strain rate 0.001s-1. The updating of number of grains at each deformation step due to formation of new nuclei has been shown. Initial dislocation density has been assigned to new grains and the average dislocation density variation over all grains has been shown. Random crystallographic texture is assigned to the old grains initially as well as to the new grains.
2. Model Implementation 2.1. Nucleation and Growth Model The Ding and Guo approach (2001) for predicting the microstructural evolution due to the onset of DRX is based on achieving critical dislocation density in the grain. The approach is an attempt at improving the earlier models proposed
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by Peczak (1995) (based on temperature) and by Peczak and Luton (1994) (based on strain rate). Two important assumptions are: Initial dislocation density is same for all primary grains. DRX occurs on achieving a critical value of dislocation density. For secondary grains formed due to DRX, it increases from zero to a saturating value with deformation. New nuclei are formed only on grain boundaries. This is based on grain boundary bulging mechanism which causes strain induced boundary migration (SIBM). The nucleation rate is defined as: −𝑄𝑄 𝑁𝑁̇ = 𝐶𝐶 ∗ 𝜀𝜀̇m ∗ 𝑒𝑒𝑒𝑒𝑒𝑒 ( )
(1)
𝑅𝑅𝑅𝑅
where Q is the activation energy, T is the temperature, R is the universal gas constant, C is a constant, 𝜀𝜀̇ is the true strain rate and exponent m ~ 1. The activation energy has been taken to be 61.8KJ/mol (Stringer (1960)) and the constant ‘C’ is taken as 10000 via trial and error to obtain a high nucleation rate ~ 100s -1 per grain. 2.2. Dislocation Density Hardening based model Beyerlein and Tome (2008) developed a dislocation density hardening based algorithm for multiple slip and twinning modes in hcp material. The shear increment in each slip mode ‘m’ due to deformation can be written as the sum of shear increments in all the slip systems ‘s’ contained in the mode ‘m’: 𝑑𝑑𝛾𝛾 𝑚𝑚 = ∑𝑠𝑠∈𝑚𝑚 𝛾𝛾 𝑠𝑠̇ 𝑑𝑑𝑑𝑑 where, 𝛾𝛾 𝑠𝑠̇ = 𝛾𝛾 𝑜𝑜̇ |
(2)
𝑚𝑚𝑠𝑠 :𝜎𝜎 𝑛𝑛
| 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(𝑚𝑚 𝑠𝑠 : 𝜎𝜎)
𝜏𝜏𝑐𝑐𝑠𝑠 (𝜀𝜀̇ ,𝑇𝑇)
(3)
τc refers to the slip resistance to dislocation movement caused due to local microstructural obstacles. The ‘n’ value in the above expression is taken to be quite high ~20 so that slip or twinning only occurs when the resolved shear stress approaches the slip resistance value. The slip resistance of a slip system is the sum of contributions due to initial slip resistance, forest dislocations, debris and twin boundaries: 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝜏𝜏𝑐𝑐𝑠𝑠 = 𝜏𝜏𝑜𝑜𝑚𝑚 + 𝜏𝜏𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 + 𝜏𝜏𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 + 𝜏𝜏𝐻𝐻𝐻𝐻
(4)
The initial slip resistance depends on the temperature, strain rate, solute content and slip mode. The last term is the Hall-Petch term that is considered only when twinning effects are taken into consideration. To calculate the debris contribution to slip resistance, the debris generation must be evaluated first using the following equation: 𝜕𝜕𝜌𝜌𝑑𝑑𝑑𝑑𝑑𝑑,𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑞𝑞 ∑𝑚𝑚 𝑓𝑓 𝑚𝑚
𝑚𝑚 𝜕𝜕𝜌𝜌𝑟𝑟𝑟𝑟𝑟𝑟
𝜕𝜕𝛾𝛾𝑚𝑚
𝑑𝑑𝛾𝛾 𝑚𝑚
(5)
where q is the coefficient that represents how debris is generated from point defects via thermally activated mechanism, f is the fraction of removed dislocations (due to recovery or as defects) that are converted to debris. 𝑓𝑓 𝑚𝑚 =
𝑛𝑛𝑛𝑛 𝑚𝑚 𝑙𝑙𝑑𝑑𝑑𝑑𝑑𝑑
= 𝐴𝐴𝑚𝑚 𝑏𝑏 𝑚𝑚 √𝜌𝜌𝑑𝑑𝑑𝑑𝑑𝑑
(6)
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where A is a constant, n is a factor multiplied with the ratio of Burgers vector and average spacing between debris. The dislocation forest contribution to slip resistance is given by: 𝑚𝑚 𝜏𝜏𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 = 𝑏𝑏 𝑚𝑚 𝜒𝜒𝜒𝜒 √𝜌𝜌𝑚𝑚
(7)
where χ is dislocation interaction coefficient between 0.1 to 1 and μ is the shear modulus. The initial slip resistance is evaluated separately for each slip mode and is of the form: −𝑇𝑇
𝜏𝜏𝑜𝑜𝑚𝑚 = 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 ( )
(8)
𝐵𝐵
where A is a constant and B is a temperature. These have been calculated via curve fitting data obtained from literature. The dislocation density evolution for each slip mode is updated based on the expression provided by Essmann and Mughrabi (1979) that has been further developed by Mecking and Kocks (1981, 2003): 𝜕𝜕𝜌𝜌𝑚𝑚 𝜕𝜕𝛾𝛾𝑚𝑚
= 𝑘𝑘1𝑚𝑚 √𝜌𝜌𝑚𝑚 − 𝑘𝑘2𝑚𝑚 (𝜀𝜀̇, 𝑇𝑇)𝜌𝜌𝑚𝑚
(9)
Here, k1 is a material constant and k2 is a function of temperature and strain which describes dynamic recovery via thermally activated mechanisms. k2 is evaluated as: 𝑘𝑘2𝑚𝑚 (𝜀𝜀̇ ,𝑇𝑇) 𝑘𝑘1𝑚𝑚
=
𝜒𝜒𝑏𝑏 𝑚𝑚 𝑔𝑔𝑚𝑚
(1 −
𝑘𝑘𝑘𝑘
𝐷𝐷𝑚𝑚 𝑏𝑏 3
𝜀𝜀̇
(10)
𝑙𝑙𝑙𝑙 ̇ ) 𝜀𝜀𝑜𝑜
where, gm is the normalized activation energy. 2.3. Simulation Algorithm A few critical material properties and model parameters for Ti are shown in Table 1 (Tromans (2011)). Table 1: A few critical VPSC input parameters C11 (GPa) C33 (GPa) C44 (GPa) 160 181 46.5 b (nm) ρin (m-2) k1 (prismatic) (m-1) 0.295 1012 1*106
C12 (GPa) 90 k1 (basal) (m-1) 1*106
C13 (GPa) 66 k1 (pyramidal) (m-1) 6*107
μ (GPa) 38.5 n
χ 0.7 ρdeb (m-2)
20
12
Here, Cij are the elements of the elastic stiffness matrix, 𝜇𝜇 is the shear modulus, χ is an interaction coefficient that strongly affects initial work hardening, b is the Burgers vector, ρin is the initial dislocation density for both old and new grains, ρdeb is the initial debris density, k1 is a parameter which defines the rate of dislocation storage due to strain hardening and n is the strain hardening exponent that is set to a large value so that slip only occurs when the resolved shear stress is close to critical value. The complete modelling process is summarized in a flow chart in Fig.1. The complete process shown in Fig.1 is within one time step of numerical integration. Within one time step, nucleation occurs several times (for various grains) based on the probability criteria. The threshold value for nucleation to occur viz. Prandom has been assumed to be 0.5. The probability for each grain is calculated as:
𝑃𝑃(𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔) =
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nucleation rate in grain average nucleation rate across all grains
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 =
Total nucleation rate in grain Number of grains at particular time instant
255 5
(11)
(12)
As soon as the probability defined in Eq. (11) exceeds critical value, event of DRX takes place. Once DRX is confirmed, random texture is explicitly assigned to new grains. Along with texture, mechanical properties for single crystal are assigned to the new grains and the cycle continues henceforth.
Update dislocation density in each grain based on Beyerlein & Tome formulation (2008)
𝜌𝜌 ≥ 𝜌𝜌𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶
NO
YES Calculate nucleation rate based on Ding and Guo formulation (2001)
Calculate nucleation probability for each grain
NO
𝑃𝑃𝑖𝑖 ≥ 𝑃𝑃𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 YES
New grain updating
Assign dislocation density, texture and mechanical properties to new grains Recompute texture weight factor wrel
Output flow stress, dislocation density evolution
Fig. 1. Flow chart depicting implementation of DRX in VPSC model
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A few important simulation parameters are shown in Table 2. Table 2: A few important simulation parameters Ngrains T (K) 𝜀𝜀̇ (s-1) wrel 1000 1173 0.001 0.001
𝜀𝜀incr 0.01
Nsteps 30
𝜌𝜌crit (m-2) 0.37*1014
Prandom 0.5
Ngrains refers to the initial number of grains in the microstructure, T is the operating temperature and 𝜀𝜀̇ is the strain rate imposed under compressive loading. The volume of grains is realised in simulation via a parameter called relative volume weight i.e. wrel which is the volume occupied by a grain in the microstructure and is represented by using crystallographic texture. The strain is incremented by 𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 in each deformation step and the total strain imposed equals 0.3 for Nsteps. The critical dislocation density 𝜌𝜌𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 has been arbitrarily chosen to be 0.37*1014 m-2. 3. Results and Discussion The major results from the simulation are shown in Fig. 2 (a) to (d): (a)
(b)
(c)
(d)
Fig. 2. (a) Flow stress variation with strain for DRX vs non-DRX (b) evolution in average dislocation density for DRX vs non-DRX (c) new DRX grains and total grains (d) percentage of DRX grains and grain size evolution with deformation
From Fig. 2 (a) and (c), the flow stress is shown to decrease as soon as new grains start forming at total strain ~ 0.2. At peak stress, rate of dislocation storage due to strain hardening exactly balances rate of dislocation annihilation due
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to recovery. The drop in flow stress in (a) signifies annihilation of old grains due to nucleation. The stabilization in flow stress occurs due to increase in dislocation density of the newly formed grains due to difference in stored energy between nuclei and their neighbouring grains. The flow stress behaviour observed in case of simulated curves compares well with the trends of flow stress saturation post critical strain as observed in the work of Zhou et. al (2017) and Blaz et. al (1983). As the number of new grains with low dislocation density keep on increasing, the average dislocation density over all grains decreases as shown in Fig. 2(b). Drop in dislocation density occurs exactly at the onset of DRX as observed in Fig. 2 (a) and (c) which further corroborates the claim of occurrence of DRX. The number of new nuclei is less than the number of grains having dislocation density greater than critical value as shown in Fig. 2 (c). The reason is attributed to nucleation probability of grains being less than the threshold value required to form a new nucleus from a grain. Since nucleation occurs due to a variety of yet undetermined physical parameters, it is modelled stochastically in the present work due to which a critical probability 0.5 is assumed. In the present work, the grain size effect in simulation is realised via a parameter called grain weight ‘w’. After new grains are ‘added’ to the system post critical strain, the weights of all grains are re-normalized according to the following formulae: 𝑤𝑤𝑜𝑜𝑜𝑜𝑜𝑜 = (𝑤𝑤𝑒𝑒𝑒𝑒𝑒𝑒ℎ 𝑜𝑜𝑜𝑜𝑜𝑜 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 /𝑤𝑤𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎 𝑜𝑜𝑜𝑜𝑜𝑜 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 ) ∗ 𝑤𝑤𝑤𝑤ℎ𝑜𝑜𝑜𝑜𝑜𝑜
𝑤𝑤𝑛𝑛𝑛𝑛𝑛𝑛 = (𝑤𝑤𝑒𝑒𝑒𝑒𝑒𝑒ℎ 𝑛𝑛𝑛𝑛𝑛𝑛 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 /𝑤𝑤𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎 𝑛𝑛𝑛𝑛𝑛𝑛 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 ) ∗ 𝑤𝑤𝑤𝑤ℎ𝑛𝑛𝑛𝑛𝑛𝑛
(13) (14)
Here, 𝑤𝑤𝑤𝑤ℎ𝑜𝑜𝑜𝑜𝑜𝑜 and 𝑤𝑤𝑤𝑤ℎ𝑛𝑛𝑛𝑛𝑛𝑛 are phase fractions of non-recrystallized and recrystallized grains. Since, grain growth is not incorporated in the present model, these phase fractions remain unchanged during deformation. These have been obtained via trial and error to be 0.95 and 0.05 respectively while calibrating the non-DRX portion of the flow stress curve against the experimental results for CP Ti obtained by Xu & Zhu (2010) and Foul et. al (2018) at temperature 1173K and strain rate strain rate 0.001s-1 as shown in Fig. 3.
Fig. 3. Calibrated flow stress of Ti vs. strain for non-DRX vs experimental at 1173 K and 0.001 s-1
In Fig. 2 (d), the average grain size across all grains is found to decrease for the old grains and increase for the new grains. This is intuitive since the number of new grains increases rapidly with deformation and hence, the new grains occupy increasing volume in three dimensional space at the expense of older grains. At ~0.3 strain, both recrystallized and non-recrystallized grains have almost equal grain weights which signifies that an equiaxed microstructure has been realised via dynamic recrystallization. Almost all grains (~95%) are observed to have recrystallized at 0.3 strain as shown in Fig. 2(d).
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4. Conclusions The trend of flow stress variation with deformation for CP Ti under DRX shows good agreement with the simulated results obtained by Zhou et. al (2017) and experimental results obtained by Blaz et. al (1983). The probabilistic criteria used to compute number of new grains at each deformation step is novel in approach. A pseudo two phase method has been used while renormalizing grain size in simulation post critical strain. At critical strain, the decrease in average dislocation density coincides with the decrease in flow stress, increase in number of new grains and evolution in grain size of both ‘old’ non-recrystallized and ‘new’ recrystallized grains thus proving the occurrence of dynamic recrystallization. Acknowledgements The authors hereby acknowledge the grant provided by Indian Space Research Organization (ISRO) under project code 17ISROC005. References Greenwood, J. Neill, Worner, H.K., 1939. Types of creep curve obtained with lead and its dilute alloys. J. Inst. Met. 64, 135. Ding, R., Guo, Z.., 2001. Coupled quantitative simulation of microstructural evolution and plastic flow during dynamic recrystallization. Acta Mater. 49, 3163–3175. https://doi.org/10.1016/S1359-6454(01)00233-6 Zhou, G., Li, Z., Li, D., Peng, Y., Zurob, H.S., Wu, P., 2017. A polycrystal plasticity based discontinuous dynamic recrystallization simulation method and its application to copper. Int. J. Plast. 91, 48–76. https://doi.org/10.1016/j.ijplas.2017.01.001 Beyerlein, I.J., Tomé, C.N., 2008. A dislocation-based constitutive law for pure Zr including temperature effects. Int. J. Plast. 24, 867–895. https://doi.org/10.1016/j.ijplas.2007.07.017 Lebensohn, R.A., Tomé, C.N., 1993. A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: application to zirconium alloys. Acta Metall. Mater. 41, 2611–2624. Essmann, U., Mughrabi, H., 1979. Annihilation of dislocations during tensile and cyclic deformation and limits of dislocation densities. Philos. Mag. A 40, 731–756. Mecking, H., Kocks, U.F., 1981. Kinetics of flow and strain-hardening. Acta Metall. 29, 1865–1875. Kocks, U.F., Mecking, H., 2003. Physics and phenomenology of strain hardening: the FCC case. Prog. Mater. Sci. 48, 171–273. Peczak, P., 1995. A Monte Carlo study of influence of deformation temperature on dynamic recrystallization. Acta Metall. Mater. 43, 1279–1291. Peczak, P., Luton, M.J., 1994. The effect of nucleation models on dynamic recrystallization II. Heterogeneous stored-energy distribution. Philos. Mag. B 70, 817–849. Stringer, J., 1960. The oxidation of titanium in oxygen at high temperatures. Acta Metall. 8, 758–766. Tromans, D., 2011. Elastic anisotropy of HCP metal crystals and polycrystals. Int J Res Rev Appl Sci 6, 462–483. Blaz, L., Sakai, T., Jonas, J.J., 1983. Effect of initial grain size on dynamic recrystallization of copper. Met. Sci. 17, 609–616. Xu, C., Zhu, W., 2010. Transformation mechanism and mechanical properties of commercially pure titanium. Trans. Nonferrous Met. Soc. China 20, 2162–2167. https://doi.org/10.1016/S1003-6326(09)60436-2 Foul, A., Aranas Jr, C., Guo, B., Jonas, J.J., 2018. Dynamic transformation of α→ β titanium at temperatures below the β-transus in commercially pure titanium. Mater. Sci. Eng. A 722, 156–159.