RESUME - tcsme.orgtcsme.org/Papers/Vol32/Vol32No2Paper5.pdfRESUME II est bien connu que la presence de large zone texturee het6rogene dans les alliages de titane forgee de type quasi

MICROSTRUCTURAL MODELING OF COLD CREEP/FATIGUE IN NEAR ALPHATITANIUM ALLOYS USING CELLULAR AUTOMATA METHOD

N. Boutana\ P. Bocher', M. Jahaze, D. Pioe and F. Montheillee1Ecole de technologie superieure, 1100 rue Notre-Dame Ouest, Montreal (QC) H3C lK3, Canada

2 Aerospace manufacturing technology center (CTFA), 5145 Avenue Decelles, Montreal (QC) H3S 2S4, Canada3 Ecole nationale superieure des mines de Saint-Etienne (Centre SMS), UMR CNRS 5146, 158 Cours Fauriel, 42023

St Etienne cedex 2, FranceContact: [email protected]

Received September 2007, Accepted June 2008No. 07-CSME-38, E.I.C. Accession 3007

ABSTRACTIt is well known that the presence of large heterogeneous textured regions in forged near alpha titanium alloys

could lead to large variations of mechanical properties when fatigue and creep cycles are applied at roomtemperature. On the other hand, experimental studies and microtexture investigations are complex to set up, lengthyand costly, and one cannot expect to understand the alloy behavior by relying only on empirical approaches. Hence,numerical methods are excellent alternatives for analyzing the influence of microscopic and macroscopicheterogeneities on mechanical properties in shorter times and with minimum need for experimentation. In thepresent investigation, a cellular automata (CA) method was used to simulate the effect of texture heterogeneities, onboth local and global mechanical properties. A 2D array of cells was used and the stresses and strains developed invarious heterogeneous regions were evaluated using the Eshelby theory. Using the CA method, various types ofmicrostructures were modeled and compared with each other to quantify the influence of processing parameters onmechanical properties. The results predict, and are used to explain, the experimentally phenomena observed in creepresponses during cold fatigue/creep tests of near alpha titanium samples.

APPLICATION DE LA METHODE DES AUTOMATES CELLULAIRES POUR LAMODELISATION MICROSTRUCTURALE EN FATIGUE - FLUAGE A

TEMPERATURE AMBIANTE DES ALLIAGES DE TITANE

RESUMEII est bien connu que la presence de large zone texturee het6rogene dans les alliages de titane forgee de type quasi

alpha pourrait conduire a de fortes variations des proprietes mecaniques lors de la fatigue-fluage a temperatureambiante. Les investigations microstructurale par des methodes experimentales sont tres complexes a mettre enplace, longues et couteuses, et on ne peut pas s'attendre a comprendre Ie comportement l'alliage en se fondantuniquement sur des approches empiriques. Par consequent, les methodes numeriques sont une excellente alternativepour analyser l'influence des heterogeneites microscopiques et macroscopiques sur les proprietes mecaniques avecun minimum de temps et aussi avec un minimum recours aI'experimentation. Dans la presente etude, la methode desautomates cellulaires (AC) a ete utilisee pour simuler I'effet des heterogeneites de texture sur les proprietesmecaniques, tant aux niveaux local que globale. Une grille 2D de cellules a ete utilisee pour cette simulation. Lescontraintes et deformations developpees dans diverses regions heterogenes ont ete evaluees en utilisant la theoried'Eshelby. En utilisant la methode des AC, differents types de microstructur~s ont ete modelises et compares les unsavec les autres afin de quantifier l'influence des parametres locaux sur les proprietes mecaniques. Les resultatsobtenus sont utilises pour expliquer les differents phenomenes observes experimentalement lors des essais defatigue-fluage atemperature ambiante (froid) dans les echantillons de titane type quasi alpha.

Transactions ofthe CSME Ide la SCGM Vol. 32, No.2, 2008 195

INTRODUCTION

Near alpha titanium alloys offer a clear advantage in terms of reduced density and corrosion damage[1]. However, it is known that, components made of these alloys suffer from room temperature creepsensitivity which is hindering the widespread application of these alloys. Specifically, life reductions,over an order of magnitude large have been commonly observed when the fatigue cycle contained a dwellperiod at the peak stress [2]. These dwell fatigue tests have been associated with the observation ofsignificant creep at room temperature.Theoretical models, such as self-consistent, have been used to predict the mechanical behavior of the alloybased on the flow rules of the different constituents [3]. Most of these methods were originally developedfor linear elastic materials but were later extended to linear viscoplastic materials. Some approaches havebeen proposed describing the behavior of two phase nonlinear aggregates. The variational structureproposed by Castaneda describes the effective energy density of nonlinear composites in terms of thecorresponding energy density for linear composites with the similar microstructural distributions [4]. Thisapproach has been used to obtain bounds and estimations for the effective mechanical properties ofnonlinear composites using any bounds and estimates of linear composites. An alternative method hasalso been proposed by Suquet for bounding the overall properties of a class of composite materials interms of the properties of the individual phases and of their arrangement [5]. It applies to power lawmaterials and, as a special case, to rigid ideally plastic materials. Using this approach, a link between theoverall potential of a nonlinear composite and the overall energy of a fictitious linear composite ispresented with no assumptions on the arrangement of the phases. These methods have the advantage tolead to new nonlinear upper bounds. However, they only take into account the shapes of the phases, andsometimes whether one phase is embedded in the other one or not. No predictions can be made on theinfluence of the phase distribution, or of the evolution of local mechanical properties. Such results can, inprinciple, be obtained using finite element calculations, but require very long computational times toachieve accurate results [6]. In recent years, cellular automata (CA) method has been used as analternative to finite element method due to its simplicity of use and its relatively short calculation times[7]. The CA technique can be used to easily and rapidly localize the maximum stress and strain in themicrostructure and carry out statistical analysis. The technique was first developed to predict the behaviorof two-phase linear or nonlinear viscous aggregates submitted to plane strain [8, 9] and revealed to beefficient for analyzing the distribution of deformation within each phase [10].The objectives of the present paper are twofold: 1) To apply the CA method to simulate heterogeneousaggregates submitted to room temperature creep/fatigue test; and 2) To provide a detailed analysis for thelarge variations observed in creep/fatigue responses of forged titanium samples subjected to similarloading conditions. In the following, the basic features of the CA model are first described followed bythe analysis of the simulation results on the evolution of the global as well as local mechanical parametersunder creep/fatigue testing.

SIMULATION OF THE MECHANICAL BEHAVIOR OF MATERIALS

Cellular automaton approach

Cellular automata are algorithms that describe the discrete spatial and/or temporal evolution ofcomplex systems by applying local or global deterministic or probabilistic transformation rules to thesites of a lattice (Conventional cellular automata use local rules. Some modem variants consider alsointermediate or long-range interactions). The space vari~ble can stand for real space, momentum space,wave vector space,.etc.. The lattice is defined in terms of a fixed number of points (cells). In the space,these consist of the vertices of a regular, fmite-dimensionallattice which may extend to infinity, though in

Transactions ofthe CSME Ide fa SCGM Vol. 32, No.2, 2008 196

practice, periodic boundary conditions are often assumed. Time progresses in finite steps and is the sameat all cells (point) in space. Each cell (points) has dynamical state variables which range over a [mitenumber of values. The evolution of each variable is governed by a local, deterministic dynamical law(usually called a rule). The value of the cell at the next time step depends on the current state of a finitenumber of nearby cells called the neighbourhood. Finally, the rule acts on all cells (points) simultaneouslyand is the same throughout the space during the entire analysis [11]. For the purposes of this study, a CAalgorithm was developed and used with the following features and characteristics:(i) Cells are associated with material features (grain, subgrain or homogeneous phase domain): each cellis associated with one grain.(ii) The related characteristics (state) are defined (Young's modulus, bulk modulus, shear modulus, strainhardening, strain rate sensitivity): Young's modulus is the characteristic state.(iii) Relationships between neighbours: each cell can only interact with its first neighbors.(iv) Cell transition rules governing the evolution of the cell: the Eshelby inclusion method is adopted.Here the evolution of the parameters characterizing each cell is continuous.(v) The state of the cells changes over time.(vi)There is no continuity of strain or the traction vectors across the neighbouring grains.

Description of topology

The two-dimensional CA was built as a heterogeneous aggregate, submitted to axisymmetric tension.This axisymmetry is assumed at both macroscopic and local scales. Each cell has 6 neighbors and keepsthem throughout the deformation process. The nearest neighbor relations are displayed in the form of ahexagonal array (Figurel). The cell characteristics (in this case the Young's modulus) can be randomlydistributed, aligned or packed in clusters. The clusters can also be randomly distributed or arrangedaccording to a specific order.

Figure 1: a) Grain (G) of material; b) corresponding cell (C) of Cellular automaton [9].

MODELING MECHANICAL BEHAVIOR DURING CREEPIFATIGUE TEST

Cells are assumed to be isotropic and incompressible. In this study, the material is submitted to acyclic creep/fatigue loading characterized by three stages: (i) the loading stage, (ii) creep period(application of a constant load for a fixed period of time), and (iii) unloading stage. Eshelby theory wasused to evaluate the local stresses and strains developed in the microstructure for various loadingconditions [12, 13].


Loading phase

According to Eshelby's analysis, the first step of the computation is that of localization, wherethe local strain of each grain (cell) ee is derived as a function of the remote loading strain eOO .The main assumption of the present model is to consider each grain as an inclusion embedded inits neighborhood. Hence, for the elastic loading phase, the strain associated with any cell (C) isgiven by Eq. (1).

(1)

Strain components are evaluated explicitly after reduction of indices by the following equations using theaxisymmetry of the loading case:

(2)

(3)

(4)

The localization factors fj,,~ are functions of the bulk modulus, K, and the shear modulus, Jl, of the cell

(C) and its neighbors (V) (see appendix 1). Figure 2 shows the main direction ofthe remote loading strainat infinity.

---Sl~

.$1

Figure 2: geometrical representation of the main direction of the imposed strain at infinite.

The bulk and shear moduli were calculated using the following equations:

Transactions a/the CSME Ide la SCGM

K= E3 (1- 2v)

EJl=---

2 (1+v)

Vol. 32, No.2, 2008

(5)

(6)

198

where E is the Young's modulus and v is Poisson factor and is taken equal to 0.3. It isimportant to mention that, in Eshelby's theory only isotropic materials are considered. Hence, inthe present investigation the crystallographic elastic anisotropy effects were taken into accountby assuming a distribution of the Young's modulus of an isotropic material (resulting toproportional distributions of K and fJ). The stresses in each cell are

then calculated using the following equations derived from the Hooke law and the loading

symmetry (£1~ = £;) .

C (Kc KC) C KC C0"11 = 1 + 2 £11 + 2 £33

C KC C 2Kc C0"33 = 1 £33 + 2 £11

where K C= K C+ 4,uc and K2C = KC _ 2,uc133

Creep period

(7)

(8)

(9)

For this step, it is assumed that the local stress calculated earlier in the cell is maintained constantduring the dwell period. It is also assumed that during the dwell period, creep takes place at roomtemperature and local strains can be evaluated as a function of the cell's local stresses and creep behavior.This behavior is described by the modified strain-rate sensitive Hollomon flow equation, where the flowstress (Von Mises equivalent) is a function ofboth strain and strain rate (Von Mises equivalent) Eq. (10).

(10)

From Eq. (10) the strain rate during plastic flow can be obtained by:

(11)

From Eq. (11) we obtain the following relation:

(12)

Integrating Eq. (12) for constant stress condition and considering that the strength parameter k isindependent of time, the local strains are then evaluated by Eq. (13):


_ (Cf C Jm~n (m + n)m:n ~&c = __ t m+n

KC m

In the model, the following relation for &~ was derived from the above Hollomon equation:

Sc ( CJ_t-t( )~ m.. Cf m+n m + n m+n -&~ = -'L __ t m+n

Y K C K C m

(13)

(14)

where (J"c is the stress tensor at the cell level (local stress), (jc the local equivalent stress (Von Mises),

&C is the local creep strain tensor, m is the strain rate sensitivity, n is the strain hardening exponent, t

denotes the creep period, KC is the strength parameter and SC the local deviatoric stress tensor calculated

using the following equation:

(15)

At the end of this step, each grain (cell) has undergone various amounts of strain.

Unloading stage

Theoretically, the global residual stresses vanish once the applied load is removed and the modelshould be able to predict such behavior. On the other hand, local residual stresses may persist at the end ofeach cycle and carried out to the next one. In the present analysis, it was assumed that the local plasticdeformation induced by the springback is equal to the difference between the cells plastic strain and the

average plastic strain of its neighbors. To evaluate the residual strain &: and stress (J": tensors after

unloading, the Eshelby theory, as modified by Mura [14], was used. To evaluate the residual strains, the

stress free transformation tensor pC of the equivalent homogeneous inclusion related to the cell C is

defined by Eq. (16).

(16)

where &. C is the additional free stress transformation in the equivalent homogeneous inclusion to

compensate the difference from the inhomogeneous cell and &Pc is the stress free strain corresponding to

the plastic incompatibility between the matrix and the cell (more precisely here, it is the average creepstrain of the neighboring cells minus the creep plastic strain of the central cell).This general Eshelby

problem is usually solved in pC with the linear system given by the equivalence relation between the

inhomogeneous cell and the related equivalent homogeneous inclusion:

Cv (kl Ski P mn p kl) CC (kl Ski P mn kl)ijkl &'" + mn C - C = ijkl &'" + mn C - &pC (17)

Transactions ofthe CSME /de fa SCGM Va!. 32, No.2, 2008 200

with CV and cf the elastic tensors of the neighboring and the cell, respectively.

In the present case, the problem is simplified as both pC and spc are diagonal due to symmetries, and

more prreisely pC{r -~/2 ~J~d E'c = [ - Et -E:/2 :,}

Assuming that the spherical shape of the inclusion remains the same throughout the deformation processthe Eshelby tensor 8 is:

811 822 833 7 - 5v

11 = 22 = 33 =I5(1-v)

11 22 5v-I8 22 = 8 33 = ... = ----I5(1-v)

12 23 4 - 5v8 12 = 8 23 = ... = ----

I5(1-v)

(18)

where v is the Poisson factor (common for all here). Note that shear components such as S1212 are not

used here because of the diagonal form of pC . Finally, the equivalence relation leads to a simple scalar

equation giving p from se (see Appendix II). Once this stress free transformation is calculated, the

residual strains and stresses (i.e. at the grain level) can be calculated using standard Eqs. (16) and (17).

(19)

A useful and scalar formulation is given in Appendix III.Due to the incompressible residual strain state, the Hooke law can be expressed here as:

(20)

Thus, at the end of each unloading stage, each cell has a definite amount of residual strain that should betaken into account in the next loading cycle.It is important to note that in the present work, at the end of each step (loading, creep and unloadingstage) the average strain, stress, residual strain and residual stress are evaluated.

MATERIAL BEHAVIOR

Near alpha titanium alloys such IMI 834 or Ti6242 belong to the class of High-temperature alloys usedfor aero-engine applications. This class of alloys is ideal for high temperature since it combine theexcellent creep behavior of a alloys with the high strength of a + fJ alloys. Their upper operating

temperature is limited to about 500 to 550°C. However, these materials are characterized by large spatialvariations of Young's modulus E (from 108 GPa to 145 GPa), and by strong heterogeneous distribution ofcrystallographic orientations (called macrozones) [15]. Experimental investigations have indicated that


the presence of higher primary a phase in the macrozones results in lower creep resistance at ambienttemperature. Hence, in order to better understand the impact of such heterogeneities on the overall andlocal mechanical behavior of the material, four rnicrotextures were generated using a computer program.For each microstructure, 8100 grains (90 x 90 array of cells) with specific Young's modulus value wasconsidered (Figure 3). Thus although the four microstructures have approximately the same averageYoung's modulus value, each grain could have different Young's modulus value. This leads to differentdistributions of the Young's modulus for each microstructure. Figure 4 shows an example of Young'smodulus distributions for the four selected microstructures.

microstructure 1(a)

microstructure 2(b)

•.. ..I:iIII ill IlI!I IIII. I:I1II ..

IlllillIl!lI .. I

Ii.'!iII III III•

II!IIl MIl = _.1llal lfihn II

!II5_• l1li • .,g lI'lIIlID-- • I11III .. III L!IlI~1!I-...... -- .. III ..microstructure 3

(c)microstructure 4

(d)Figure 3: Spatial distribution of the grains in the four microstructures. The different intensity

corresponds to different ranges of Young's modulus values (GPa).

The power law exponents m and n in Eqs. (10) and (11) were taken equal to m = 0.01 and n = 0.025for all grains. To allow only soft grains (i.e. with minimum Young's modulus) to creep significantly, thestrength parameter K was chosen to be a linear function of the Young's modulus:

(21)


C Kmax - K min b Kmax - K min E .where E is the Young's modulus of the cell, a = and = K minE

max- E

minE

max- E min nnn

The coefficients a and b in the equation were determined using regression analysis on K max and

K min values. The best approximation for the average value of K (case of Ti6242 where

K = 1430MPa [16]) was obtained for K max = 2000MPa and K max = lOOOMPa.

mlcrostructrure 1

ITi crostructure 3

~ 0,045 ..--------::---------...,Co ~ 0,041::; 0,035'fi ~ 0,03: E 0,025.... ca 0,02~ § 0,015o 0 0,01ti >- 0,005

~ o~

8.

young modli us (M Pal

(C)

microstructure 2

microstructure 4

young modlius (MPa)

(d)

Figure 4: Distribution of Young's modulus in the microstructures: a) Microstructure 1 (averageE=122,Ol GPa); b) Microstructure 2 (average E=121,62 GPa); c) Microstructure 3 (average

E=120,16 GPa); and d) Microstructure 4 (average E=120,46 GPa).

Results and discussion

The simulations were carried out for 6000 cycles by imposing strains (assumed remote loading) to thematerial until a global stress of 830MPa was attained. The material was then maintained under this loadfor 30 s (creep period) before unloading. Each grain (cell) evolves in a different way according to itsspecific neighborhood, properties and microstructures; therefore different global behaviors are expected.Figure 5 compares the evolution of the strain for the four microstructures considered in the present studyobtained by creep/fatigue simulation. It can be seen that, in spite the fact that similar mechanicalproperties were assumed for the four microstructures, a different deformation response is observed.Specifically, microstructure 4 deforms faster than the three others. This variation in the behavior was alsoobserved experimentally by the authors with specimens made of a bimodal IMI 834 alloy [17]. As shownin Figure 6, two specimens prepared from the same forged part display significant differences in term of


strain response to fatigue/creep test. The experimental tests were conducted at room temperature at 830MPa (peak stress) and with 30 s holding period. The Young's modulus distribution presented in Figure 4shows that the proportion of "soft" grains is higher in microstructure 4 than in the three others indicatingthat more deformation will occur in microstructure 4 than in the three other microstructures.The simulation results indicate also that, the maximum stress was not localized on the hardest grain (i.e.with maximum Young's Modulus values (Figure 7)). Finally the results indicate that, the position of themaximum is not constant and varied during the test (Figure 8). Based on the obtained results, it can beclearly stated that the global behavior of the material is sensitive to the spatial distribution andcrystallographic orientation of the grains. Also, it was observed that at the local level, somegrains attained very high levels of stress (well above the yield stress) before the 6000 cycles ofthe test were attained.

-- Microstructure 1 ......".--- - - - Microstructure 2 .__ Microstructure 3 ..........-.-..-_._._.- Microstructure 4 ................"".--.-.-.......

.4/1----.---,~.,...

0.01

0.008

0.009

0.011

0.007 I,;;""o"......~~......-=-::i~.........-=-::Ji::-=-........~=-=-........~=-=-........~1000 2000 3000 4000 5000 6000

Cycle

Figure 5: Strain accumulation values obtained during creep/fatigue tests using the CA model on thefour microstructures.

0.014

0.013 -Specimen 1Specimen 2 --- ---

0.012 ;. --- ------ ---0.011 ------

E 0.01 - ..0.009 -, ,

I

0.008 I

0.007.....

0.006

0.005 I I I

500 1000 1500 2000 2500 3000 3500Cycle

Figure 6: Strain accumulation obtained from experimental creep/fatigue tests of near alpha forgedIMI 834 samples (applied peak stress is 830 MPa and 30 s dwell period).


These imply that, premature fracture could occur as the high level of stress at the grain levelcould go above the threshold for cleavage thereby inducing crack in the microstructure. Thecrack can eventually propagate rapidly if the environment is favorable. The major advantage ofthe CA method is the possibility to follow the evolution of the stress at the grain level duringeach cycle of the creep/fatigue test and obtain more information regarding the local mechanicalbehavior of the material. The results obtained for microstructure1 and microstructure2 show that,in spite of the similar spatial distribution and mechanical properties of the two microstructures,the strain response during creep/fatigue were significantly different. The statistical analysis withcellular automata demonstrates that the global mechanical behavior is influenced by the localbehavior where the stress and strain varied from one grain to another.

Maximum stress962 141

Stress Young's(MPa) Modulus GPa

a) after one cycle of creep/fatiguecell coordinates: (43, 45

Varied Between131-143

6 neighborsVaried Between871-904

Figure 7: Zoom on the position of maximum stress after one cycle (microstructure 4).

143

Varied Between100-137

6 neighborsVaried Between650-1008

a) after 6000 cycles of creep/fatiguecell coordinates: 85,68Stress Young's

MPa Modulus GPa

Figure 8: Zoom on the position of maximum stress after 6000 cycles (microstructure 4).

In fact, each grain attains different levels of stress depending on its orientation andneighborhood. For example, in microstructure 1, for those grains with the maximum stress afterthe first cycle (curve 1, Figure 9), it can be seen that after 500 cycles the stress increases

Transactions ofthe CSME Ide La SCGM Vol. 32, No.2, 2008 205

continuously during the entire creep/fatigue test with a stress level of 954 MPa. By contrast, forthe grains with minimum stress (curve 2, Figures 9) the stress level decreases until it reaches aminimum value and remains constant during the entire creep/fatigue test. Other examples forlocal stress evolution are presented in Figures 10 and 11. For grains with maximum stress, thestress increased rapidly until 4500th cycle before attaining a steady state value (curve 1, Figure10) while for the grains with lower stress (curve 1, Figure 11) the steady state was attained after3000th cycle. For grains with the maximum Young's modulus value, the stress increases continuouslyduring the creep/fatigue test (curve 2, Figure 10 and Figure 11). By contrast, for the grains with theminimum value of strength parameters (K C) in microstructures 1 and 2 the stress decreases until itreaches a steady state value and remains constant during the remaining part of the creep/fatigue test(curve 3, Figure 10 and 11). This behaviour indicates that the level of deformation of such grainsincreases continuously during the test.

955

950

945

930l..-......""':1:t.OOO;;:""""-";:20~OO~~3;:iOO:::;0~~4000

940

960~---------____,

935

cwve 1

_ _ cwve 2

------------------------------

';'~e 850

b 800

750

700

650

600 L.l............~1:-::01::-00:!-'-~2:-::0~00':!'-"-'-3!:':0~0':!"0 .........-4":':0~0':!"0 .........-5~0~0':!"0 .........~600·0

Cycle

Figure 9: Evolution of local stress for grains with maximum or minimum stress obtained duringcreep/fatigue test after 1 cycle (microstructure 1).

curve 1

....._._ cwve 3.•.•.......•...._........•..

------------------------cu;;ei-

1100r--------------------....,1050

1000

';' 950

~ 900

b 850

800

750

700

650

1000 2000 3000 5000 6000Cycle

Figure 10: Evolution oflocal stress for some grains during creep/fatigue test (microstructure 1).


The influence of texture on the mechanical behavior was studied for microstructures 3 and 4.Although, the two microstructures have identical mechanical properties they possess large differences inspatial distribution and crystallographic orientation. The analysis of the data indicated that, local stressincreased continuously during the creep/fatigue test in the grains of microstructure 3 and 4 which were atmaximum stress levels after the first cycle (curve 1, Figure 12 and Figure 13). For the grains withminimum stress after the first cycle, (curve2, Figure 12 and Figure 13) the stress decreased rapidly afterthe first cycle and remained constant during the entire creep/fatigue test. For the grains with maximumstress reached after 6000 cycles (curve3, Figure 12), it can be seen that, the stress level increases until the4000th cycle and remains constant (1080 MPa) during the rest of the test. Furthermore, as shown in curve3, Figure 13 the stress level increases until the 2800th cycle and then remains at the constant value of 1052MPa during the rest of the test.

Curvel

....-._._._._. Curve 3_._._ _... ._._._._.- _ _._... ._.. _._. .

-------------___ - - - - - - - -C;";e-2......

1100r---------------------....1050

1000

950~~ 900

~ 850

800

750

700

650

1000 2000 3000 4000Cycle

5000 6000

Figure 11: Evolution of local stress for some grains during creep/fatigue test (microstructure 2).

curve 3•.•...__..._-_.•.__.•.•.....•.

.•••.•••••••.•.••••• curve 4

curve 1

2000 3000 4000 5000 6000Cycle


The stress history of the grains with maximum Young's modulus show two different behaviours: formicrostructure3 (curve 4, Figure 12) the stress increases steadily during the creep/fatigue test while formicrostructure 4 (curve 4, Figures 13), the stress increase rapidly for the first 500 cycles and then drops to

Transactions ofthe CSME Ide La SCGM Vol. 32, No.2, 2008 207

a steady state level until the end of the test. For the grains corresponding to the minimum value ofstrength parameters (K C ) for both microstructures 3 and 4 (curve 5, Figures 12 and 13) the stressdecreases until a steady state value is reached and remains constant during the remaining part of the test.The results show that the minimum value of stress does not correspond necessarily to the minimum valueof the strength parameters (K C ).The above results reveal clearly that the strain response is sensitive to thespatial distribution and grain orientation in the microstructure and highlighted the importance of texture instrain distribution at the local level. The results also indicated that, the maximum stress at the grain levelchanges during the creep/fatigue test, some grains reaching very high levels of stress depending on theirneighborhood. Considering that very high levels oflocal stresses are attained, it is possible to assume that,some may reach the threshold stress for cleavage. Once generated, the crack could continue to propagateor not according to the nature of the neighborhood and especially their crystallographic orientation.

1100 curve 3~~~~~-_._-----_.__.__.__._.__._---

1000 :r:::.·.. curv..tL_

';' J 1I:lo< 900 curve~'-'

b 800

I

6000500020001000 3000 4000Cycle


Thus, if cleavage can propagate in the neighborhood area, the crack will progress and could quickly reacha size equivalent to the size of the region having a favorable texture orientation. In other words, if a graingenerates a cleavage crack and its local texture is close to orientations susceptible of cleavage, it isprobable that the crack progresses in the entire textured zone, i.e., within the macrozone. By contrast, ifthe neighborhood is not favorable to the propagation of the cleaved zone, the crack will not propagate.This type of model could be used to explain and predict the behavior of materials that are suffering fromsubsurface cleavage type fracture. The results obtained by the CA method presented in this paper, clearlyshow that, as the number of cycles increases, new grains can reach the cleavage stress, suggesting thepossibility of activating several sites of crack one after the other, and this at different stages of thecreep/fatigue test. Such a behavior has been observed experimentally by Sinha et al [18].

CONCLUSIONS

The Cellular Automaton method was used to study the influence of microstructure heterogeneities oncreep/fatigue (dwell fatigue) behavior of forged near alpha titanium alloys. The results of the studyindicated that the mechanical behavior of the material could be related to microstructure variations interm of elastic and creep properties. Indeed, the distribution and the local variation of the Young'smodulus generate different stress levels at the local scale. The maximum stress is not located inevitablyon the grains which have the maximum Young's modulus, but could be on the grains with high modulus


of elasticity and whose neighborhood is heterogeneous (simultaneously compound of hard and softgrains). The model shows clearly that during creep/fatigue tests, the grains reach various levels of stressaccording to their spatial distribution and their Young's modulus with the stress level changing during thecreep/fatigue cycles. This could explain why for certain orientations (i.e. some samples) a very differentstress behavior from the average is observed, and in particular, why certain premature failures occur insome manufactured parts.

ACKNOWLEDGEMENT

Financial support for this work was provided by the National Research Council Canada and theNatural Science and Engineering Research Council of Canada.

REFERENCES

[1] Bache, M. R, Copet, M., Davies, H. M., Evans, W. J. and Harrison G. "Dwell SensitiveFatigue in near alpha titanium alloy at ambient temperature", Int. J. Fatigue, Vol. 17, 1,1997, pp. 83-88.

[2] Bache, M. R., Evans, W. 1. "Dwell Sensitive Fatigue Response of Titanium Alloys forPower Plants Application", J of Engineering for Gas Turbines and Power, Vol. 125,2003,pp. 245-255.

[3] Walpole, L.J. "Overall elastic moduli of composite material", J. Mech. Phys. Solids, Vol.4,1969,pp.235-251.

[4] Castaiieda, P. P. "The effective mechanical properties of nonlinear isotropic composites",J. Mech. Phys. Solids, Vol. 39, 1991, pp.45-71.

[5] Suquet, P. "Overall potentials and external surfaces of power law or ideally plasticcomposite", Mech. Phys. Solids, Vol. 41, 1993, pp.981-991.

[6] Durand, L. and Thomas de Montpreville, "Etude du comportement mecanique desmateriaux biphases au moyen de la methode des elements finis", Res. Mechanica, Vol. 29,1990, pp. 257-285.

[7] Vichniac, G. Y. "Simulating Physics With Cellular automata", Physica 10D, 1984, pp. 96116.

[8] Montheillet, F. and Gilormini, P. "Prevision du comportement mecanique d'un agregat dedeux phases viscoplastiques lineaires al'aide de modele de type automate cellulaire", C RAcad Sci Paris II, Vol. 319, 1994, pp. 483.

[9] Montheillet, F. and Gilormini, P. "Predicting the mechanical behavior of two phasematerials with cellular automata", Int. 1. of Plasticity, Vol. 12, 1996, pp. 561.

[10] Montheillet, F. and Briottet, L. "Modelling the effect of morphology and topology onstrain inhomogeneity in two phase materials", Int Conf.( Q-MAT'97). Varsovie, 1997, pp.37.

[11] Raabe, D. "Computational Materials Science: The Simulation ofMaterials,Microstructures and Properties" Wiley-VCH Verlag GmbH ISBN: 3-527-29541, 1998.

[12] Eshelby, J. D. "The determination of the elastic field of an ellipsoidal inclusion, and relatedproblems", Proc. R Soc. Lond. A241, 1975, pp. 376-396.


[13] Eshelby, 1. D. "The elastic field outside an ellipsoidal inclusion", Proc. R. Soc. Lond,A252,pp.561-569,1959,pp.561-569.

[14] Mura, T. "Micromechanics of defects in solids", Martinus Nijhoff, Dordrecht, 1987.[15] Germain, L. Gey, N. Humbert, M. Hazotte, A. Bocher, P. and Jahazi, M. "An automated

method to analyze separately the microtextures ofprimary ap grains and the secondary asinherited colonies in bimodal titanium alloys ", Materials Charaterization, Vol 54, N°3, pp.216-222,2005, pp. 216-222.

[16] Neeraj, T. Hou, D. H. Daehn, G. S. and Mills, M. 1. "Phenomenological andmicrostructural analysis ofroom temperature creep in titanium alloy", Acta Materialia,Vol. 48,pp. 1225-1238,2000,pp. 1225-1238.

[17] Germain. L, Gey. N, Humbert. M, Bocher. P, and Jahazi. M, "Beta to alpha variantselection in sharp hcp textured regions of a bimodal 1M1 834 billet" 1COTOM - Edit.Transtech publications, Van Haute, P. and Kestens, L., 2005, pp. 663-668.

[18] Sinha, V. Mills, M. J. ans Williams, J.e. "Observation on the faceted initiation site in thedwell-fatigue tested Ti6242 Alloy: cristallographic orientation and size effect"Metallurgical and Materials Transactions; 37A, 5,2006, pp. 1507-1518.

APPENDIX I

Consider an "inclusion" embedded in a homogeneous infinite matrix submitted to remotehomogeneous strain, and let Kc and K v be the bulk moduli, Pc and Pv the shear moduli of the cells

(inclusion) and the neighbors (matrix) respectively. The inclusion is assumed to be a sphere. The

localization factors d~ are then defined by:

~ = (18 Jlv+12J'c;) K/ + 60Jlv2 +( 40 + 15Kc ) Jlv Kv +48 Jlv3 +(32 f.lc +20Kc)Jlv2

I 36 /u/ +( 27 Kc + 24 f.lc) Jlv + 18 Kc f.lc Kv +32 Jlv3 +(48 f.lc + 24Kc)Jlv2 +36 Kc f.lc Jlv

(9 Jlv +6 pJ K/ +( 20 Pc -15Kc ) Jlv Kv -16 Jlv3 +(16 Pc -20KJJlv2

d~3 (36 Jlv2+( 27 Kc + 24 Pc) Jlv + 18Kc Pc)Kv +32 Jlv3 +(48 Pc + 24Kc)Jlv2 +36 Kc Pc Jlv

(18 Jlv+ 12Pc)K/ +( 40 Pc-30Kc)Jlv Kv -32 Jlv3 +(32 Pc -40Kc)Jlv2

d;\ (36 Jlv2+( 27 Kc + 24 Pc) Jlv + 18Kc Pc)Kv +32 Jlv3 +(48 Pc + 24Kc)Jlv2 +36Kc Pc Jlv

Transactions ofthe eSME Ide ta SeGM Vol. 32, No.2, 2008 210

APPENDIX II

In our context, the standard Eshelby equivalence relation between the equivalent homogeneousinclusion and the heterogeneity can be simplified as a scalar. More precisely, it becomes after projectionon the deviatoric subspace and regarding the 33 component:

v( SII 13 Sl1 13 Sl1 13 13) c( Sl1 13 SII 13 S11 )j.1 - 22 - - 22 - + 11 - = J.l - 22 - - 22 - + 11 - 82 2 2 2 p

. ... 15(1- v)j.1c8pFmally, 13 can be explIcItly gIVen: f3 = ( ) c ( ) v·

2 4 - 5v j.1 + 7 - 5v j.1APPENDIX III

Reducing the problem using the specific diagonal form of j3c here, the Eshelby relation between the

stress free transformation and the real strain of the heterogeneity (cell) can be expressed in a scalarmanner:

(

-8/2

8c = 0r

o

o-8/2

o


Transactions a/the CSME Ide la SCGM Vol. 32, No.2, 2008 212

Documents

RESUME - tcsme.orgtcsme.org/Papers/Vol32/Vol32No2Paper5.pdfRESUME II est bien connu que la presence de large zone texturee het6rogene dans les alliages de titane forgee de type quasi