Stress partitioning in a near-β Titanium alloy induced by elastic and plastic phase anisotropies: experimental and modeling

The load transfer induced by the elas c and plas c phase anisotropies of a Ti–10V–2Fe–3Al tanium alloy is studied. The microstructure consists in α nodules embedded in elongated β grains. EBSD performed on the alloy shows no crystallographic texture neither for α nor β phase. Tensile tests along the elonga on direc on, at a strain rate of 2 x 10 -3 s -1 give a yield stress of 830 MPa with 13% duc lity. Simula ons based on an advanced two-phase polycrystalline elasto-viscoplas c self-consistent (EVPSC) model predict that the β phase ﬁrst plas ﬁes with a sequen al onset of plas city star ng from <110> oriented β grains, then <111> and ﬁnally <100> oriented β grains. This leads to a strong load transfer from

for isotropic and kinema c hardening as well as crack ini a on. For these alloys, significant cracks have been observed a�er failure origina ng from basal slip ac vity in the α nodules [7,8].
Mean field Self-Consistent models have been successfully used in the past together with experimental techniques like X-ray diffrac on (XRD) to characterize the stresses and strains in complex materials by taking into account grain and phase interac ons [9]. Many studies have been carried out in combina on with in-situ diffrac on experiments and self-consistent schemes to characterize the plas c deforma on in tanium alloys [10 -13]. These studies employ elasto-plas c or viscoplas c models and predict the average behavior of the phases reasonably well.
The present work aims to predict the inter-granular stress and strain evolu on in the forged Ti-10-2-3 alloy during both elas c and plas c deforma ons. For this purpose, an advanced two-phase polycrystalline elasto-viscoplas c self-consistent model, with first order "affine" lineariza on of the viscoplas c flow rule, as described in [5] is applied on a simplified microstructure, i.e. α nodules embedded in a β matrix, as the first step. Alongside, an a�empt is made to illustrate the role of anisotropic elas c α/β strains on the hierarchical onset of plas city. The simulated data will be compared in the future to high energy XRD results.

II.1 Material:
Samples were cut from an α/β forged piece of the 10-2-3-tanium alloy. They were all heat treated in the α/β domain for 1 hour and water quenched to obtain a nodular microstructure with 15% of primary α nodules embedded in a β matrix. This matrix consists of millimeter large elongated β grains, which are par ally fragmented into equiaxed sub-grains with average size of 2.7 µm. The primary α nodules are present both at the β grain and sub-grain boundaries. Their average size is around 1.2 µm. Further observa on of the microstructure also confirmed that no secondary α lamellae formed during water quenching. The α volume frac on was confirmed to be around ~15% using laboratory X-ray diffrac on.

II.2 Texture and Microtexture:
Neutron texture analysis was carried out on 1 mm cube samples at Laboratoire Léon Brillouin (LLB) in France. Pole figures were generated for the β phase and the crystallographic texture was found to be random (not shown here). Addi onal EBSD measurements were performed on a JEOL JSM-6500F electron microscope equipped with the Nordlys-S camera. Several EBSD maps with 0.5µm step size were carried out along the cross-sec on (normal to the elonga on axis) to characterize the microstructure and local texture of α and β phases. The EBSD maps were post-processed with Aztec-Oxford instrumenta on so�ware.  (Figure 1c). Equiaxed β sub-grains were found throughout the microstructure with both small (>3°) and large (>15°) misorienta ons. A higher frac on of high angle grain boundaries is observed within a single β grain and this forms the basis of equiaxed morphological assump on for the β grains. Along with the high angle grain boundary sub-grains, the β grains also present a significant orienta on spread. The α nodules in the β grains are mostly equiaxed and their texture is very close to random, i.e. the orienta on rela onship with the β phase was broken.

II.3 Tensile test:
Cylindrical tensile specimens with 6.35mm diameter were cut from the α/β forged piece along the elonga on direc on (the tensile direc on TD is parallel to the elonga on direc on of the β grains). The width of the β grains is ~200µm, such that there is 100 grains in the tensile specimen's cross-sec on. The tensile tests were performed on a ZWICK Z250 machine with a force cell of 250kN and a contact extensometer. Several different strain rates were studied ( 2 x 10 -5 s -1 , 2 x 10 -4 s -1 , 2 x 10 -3 s -1 ) up to fracture. The results are more detailed in [14]. In this work, only the tensile response at 2 x 10 -3 s -1 is shown and used for the present micromechanical simula ons.

III.1 Single grain cons tu ve behavior in α/β phases:
A one-site elasto-viscoplastic self-consistent (EVPSC) scheme, formulated using the translated field method and a first order affine approximation, is used. It has been described in detail elsewhere [5,15] and only a brief description of the single crystal law employed is presented. The simulations are performed under the infinitesimal small strain assumptions, where the total strain rate tensor ( ), calculated from the generalized Hooke's law, and the nonlinear viscoplastic strain rate tensor ( ).
The constitutive behavior at the single crystal level, for both α and β phases, is described by Méric-Cailletaud (MC) model [16]. The slip rate for each slip system 's' is defined as: Where = max(x,0), τ s is the resolved shear stress defined as τ s = R s :σ, with R s is the symmetric Schmid orienta on tensor and σ is the first moment of stresses (average stress inside a given grain, i.e. a given orienta on). The quan es x s and r s are the kinema c and isotropic hardening components on each slip system, respec vely. The evolu on of the kinema c hardening is described through a simplified law with a single material coefficient c: The viscoplas c strain rate at the scale of grain is defined as:

III.2 EVPSC scale transi on law:
The EVPSC scale transi on law from micro-to macroscopic fields as detailed in [5] was used in this paper. Other details and comparisons with other mean/full field approaches can be found in [17].

III.3 Applica on of the affine EVPSC to Ti-10-2-3:
Two types of simula ons have been carried out: the first one at the macroscopic scale and the second one at the scale of individual β grains. These simula ons are carried out assuming equiaxed grain morphology for both α and β phases with volume Simula on I: An isotropic crystallographic texture is considered for both the β and α phases, as observed experimentally with neutron texture analysis. Several simula ons were performed to iden fy the number of orienta ons required in each phase to obtain an isotropic elas c behavior (as observed experimentally from tensile tests in different direc ons -not shown here) [14]. It was found that at least 5000 random orienta ons are required in the α phase and 2000 random orienta ons are required for the β phase. The Representa ve Volume Element (RVE) is a polycrystalline (α+β) composite with 7000 random orienta ons.
Simula on II: The second simula on is performed at the scale of single β grains with α/β texture coming from the EBSD map. The choice of specific β grains oriented close to the <100> || TD was mo vated from the results of Simula on I. Indeed, those grains were seen to be the latest to enter to plas city with the model. They are herea�er referred as 'Red grains' (with respect to the Standard Stereographic Triangle (SST) color code). Figure 2 shows the choice of six 'Red grains' on the EBSD map (selec on of grains based on high-angle boundary with misorienta on angle of >25°.). The six red grains present local spread in orienta on; for example, red grain 2, 5 and 6 exhibit lower spread in orienta on locally, while the red grain 1, 3 and 4 presents significant spread in orienta on. As men oned previously, the texture of the α phase in these 'Red grains' is rather random.

III.4 Elas c and viscoplas c parameters:
The elas c and viscoplas c parameters have been fi�ed on experimental tensile curves at several strain rates. The parameters used for both simula ons I and II are presented in Table 1.
The elas c constants for the α phase have been selected from the literature [18], as there is considerable agreement. However, for the β phase, widely different elas c constants have been reported in the literature with an anisotropic factor ranging from 1.4 to 8.3. They have a significant influence on the predicted elas c and plas c incompa bili es at the grain scale as discussed in [5] for a 10-2-3-Ti alloy fully β. Here, we consider the same constants for b phase as used by Mar n et al. [19], as it provides reasonably good fit with the experimental data for a 100% β phase alloy. The parameter 'n' is chosen to be rela vely high (n=90 for the a phase and n=80 for the b phase) to predict the observed material's low strain rate sensi vity (see [14]). For the a phase, It should be noted that the inves gated microstructure is a simplified one with primary α nodules embedded in a β matrix (with no secondary α lamellae). The chosen SEC and viscoplas c parameters for the model are expected to change for the classical aged microstructure in which the β matrix is par ally transformed into fine secondary α lamellae. A recent experimental study [7] on the β metastable Ti-5Al-5Mo-5V-3Cr has also highlighted the role of elas c anisotropy of the transformed β phase on the elas c and plas c behavior.

III.5 Data analysis and interpreta on:
Along with the conven onal macroscopic curves, our simula ons were also applied to predict the evolu ons of the elas c strains with applied stress and their devia on from linearity to iden fy the onset of micro-plas city and load transfers between α and β phases. The interpreta on of elas c strain evolu on is taken from the XRD la ce strain analysis [21]. By following the evolu on of la ce (i.e. purely elas c) strains with applied stress in a composite material, one can iden fy the plas c events and load transfer amongst different phases/grain families. A deflec on of elas c strains from linearity, when plo�ed with respect to applied stress, indicates the nature of deforma on of the phase. A nega ve deflec on from linearity indicates plas city, whereas a posi ve devia on indicates elas c load takeover or "load-transfer" from the grain family/phase that has achieved plas city.

IV.1 Simula on I:
The simula on is carried out on the α+β composite with 7000 random crystal orienta ons, up to a total strain of 10% (below the failure strain of the material). The predicted α+β composite response along with the experimental curve for 2 x 10 -3 s -1 strain rate is shown in Figure 3. The elas c response is well reproduced but a small discrepancy could be seen regarding the elas c-plas c transi on. The difference in yield stress predicted by the model is about 40 MPa (~5% higher than experimental data). However, these discrepancies were found to lie within the sca�er of tensile data for different tested specimens of close microstructures. The evolu on of elas c strains with applied stress as well as its devia on from linearity is given in Figures 4a and 4b, for the whole composite (in black) and for the β and α orienta ons separately (respec vely in blue and red). Figure 4b indicates the onset of plas city in β phase with a strong load transfer to the α nodules. The behavior of the vast majority of α nodules remains elas c (see Figure 4b and sec on III.4 for interpreta on of the curves). As will be shown in simula on II, the behavior of α nodules is however strongly dependent on the β texture in which they are embedded. The analysis by grouping different subsets of β fibers along TD indicates a sequen al onset of plas city star ng from <110> oriented β grains with load transfer to <100> grains, see Figure 5. At higher stresses, plas city is seen to set in for the <111> β grains family subset and finally for the <100> β grains. On the average elas c strain curve for β phase (black solid line in Figure 5c), the macro-plas city in β phase starts when all the three-grain families' subset has a�ained plas city (i.e. ~860 MPa).
This sequen al onset of plas city in the β phase predicted by our simula on agrees well with the in-situ XRD observa ons on the forged Ti-10-2-3 (similar microstructure) by Raghunathan and coworkers [10].

IV.2 Simula on II:
It was seen in Simula on I that for the β phase, the <100> || TD oriented grains enter to a plas c regime at the end. Hence, several β <100> || TD oriented grains were selected from the EBSD map to predict the average behavior of the embedded α nodules (see Figure 2). For clarity, only the results of red grains 1, 3 and 6 will be discussed due to the significant presence of local orienta on spread.
Contrary to Simula on I, an inverse trend is seen for the onset of plas city and load transfer in Simula on II, see Figure 6b. The onset of plas city is now seen in the α phase first, with a nega ve devia on of elas c strains from linearity, with load transfer to β grains. At higher applied stresses (770-800 MPa), plas city is a�ained in the β phase, resul ng in a load transfer with the α phase ( Figure 6b). Further plas city is seen in the α phase with second load transfer to β phase for red grains 1 and 6 (810-830 MPa). However, in the case of red grain 3, the second load transfer to the β phase is not seen.
The plas c behavior of extremal Young's moduli oriented α nodules has also been analyzed (see Figure 6c). For the α nodules with 'c' axis parallel to tensile direc on (unfavorable orienta on -for both basal and prisma c systems), no plas c strain could be seen. However, for the ones having the 'c' axis perpendicular to the tensile direc on (unfavorable orienta on of basal systems, favorable orienta on of prisma c systems), plas city is seen in the red grains 1 and 6. It should be noted that the onset of prisma c slip is seen only a�er plas city has been set in the β grain.
Thus, simula on II highlights two major results: (1) the plas c behavior of α nodules depends on the β-orienta on it is embedded in and on the orienta on of the nodule itself, (2) having a β grain with a large orienta on spread (red grain 3) can result in modified average plas c behavior of α phase compared to a β grain with less orienta on spread (red grain 1 and 6).

V. Conclusions and perspec ves
An EVPSC model was applied to simulate the tensile behavior and the associated stress par oning for a (α+β) polycrystalline Ti-10-2-3 alloy with a forged microstructure (primary α nodules embedded in a β matrix). The real texture and local texture of each phase was measured using neutron diffrac on and EBSD acquisi ons. The numerical results indicate that the onset of plas city in a phase could be influenced by the crystallographic texture of the other phase.
At the scale of the specimen: the β phase is predicted to plas fy first with sequen al onset of plas city star ng from <110> || TD, then <111> || TD and finally <100> || TD grains. This is followed by a strong load transfer to the α phase.
At the scale of β grains (<100> || TD oriented β grains with rather random oriented α nodules): it is predicted that plas city is ini ated in the α phase first with a load transfer to the β phase. The plas c behavior of α phase was found to depend on the β grain in which it is embedded and its orienta on spread.
High energy X-ray diffrac on experiments are planned by the end of this year to obtain la ce strains evolu on with applied stress and to verify the sequen al onset of plas city and corresponding load transfer amongst phases. It is also expected to refine the anisotropic elas c constants for the β phase and the viscoplas c/hardening parameters of the model from la ce strain evolu on for different orienta ons in both phases. In-situ SEM studies are also planned to verify slip system ac vity (at least in the α phase) to check if the ac ve slips suggested by the EVPSC model are in good agreement with experiment.