A β-phase Ti–45Nb alloy was processed by several severe plastic deformation (SPD) methods as high-pressure torsion, cold rolling and folding, and hydrostatic extrusion to enhance its strength by achieving an ultrafine grained structure without affecting the Young’s modulus being close to that of bone material. Mechanical properties during processing were monitored by direct torque and Vickers hardness measurements, while the micro-/nano-structural evolution was investigated by transmission electron microscopy and X-ray line profile analysis. Simulations of both mechanical and micro-/nano-structural data were performed on the basis of the SPD work-hardening model by Zehetbauer. The simulations not only found a good agreement with the deformation-specific evolution of strength and density of individual dislocations but also well reflected mesoscopic structural quantities such as the sizes of cell/grain interiors and walls without introducing additional fitting parameters.
Introduction
In recent years, TiNb based alloys are getting more attention in biomedical research areas because of the unique combination of their properties such as high specific strength, good corrosion resistance and excellent biocompatibility compared to other traditional alloys [1‐3]. The alloys used for implants should have Young’s moduli close to those of human bones (around 30 GPa) as this condition minimizes the stress shielding effect [4]. In binary Ti–Nb system, the Young’s modulus shows a dependence on Nb content and exhibits two local minima, at about 15 and 42.5 wt% Nb, respectively. The single β-phase alloy Ti–45 wt% Nb lies close to the second minimum showing a Young’s modulus of 65 GPa [5]. However, the mechanical strength is also much lower [6, 7] than that of other conventional biomedical materials such as Ti–6Al–4V, Ti–6Al–7Nb, Ti–5Al–2.5Fe, etc. On the other hand, it seems possible to increase the mechanical properties of metallic materials significantly by applying methods of severe plastic deformation (SPD) [8, 9], without increasing the Young’s modulus [10‐13]. Such methods, e.g. cold rolling and folding (CR&F), high-pressure torsion (HPT) or hydrostatic extrusion (HE), are very effective in grain refinement [8]; by applying a high deformation strain, an ultrafine or even nanoscaled structure can be achieved, which results in a marked increase of strength often paired with a considerable plastic deformability [8, 9].
In this investigation, the biomedical alloy Ti–45 wt% Nb was subjected to CR&F, HPT and HE to produce ultrafined or nanostructured bulk material with very high mechanical properties, that is high strength, good plasticity and low Young’s modulus. Besides, several mechanical test methods (for the results on Young’s modulus, see Ref. [14]) such as transmission electron microscopy (TEM) have been applied to study the strain-dependent evolution of grain/cell interior and grain/cell size, whereas those of X-ray line profile analyses (XPA) were carried out for obtaining the dislocation density, as well as the coherently scattering domain size (also called ‘crystallite size’ in the literature). As this latter quantity indicates the smallest size of defect-free regular crystalline area, in largely strained materials it has been repeatedly identified with the area size of the accumulated edge dislocations near the grain/cell boundaries [15‐17]. Among other quantities, this edge area size will serve as a useful parameter in the current study in comparing data from experiments with those from simulations. The latter were done using the dislocation-based constitutive large strain work-hardening model by Zehetbauer [18‐20]. In several previous publications [8, 15, 16, 21], it has been shown that the model is capable of predicting the strain-dependent evolution of the strength and also the structural features of material such as the dislocation density, the cell/cell wall size, the misorientation angle and the concentration of strain-induced vacancies of SPD-processed materials as the function of external hydrostatic pressure [15, 16, 22].
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Methodology
Experiments
The initial material used in this study was a hot-extruded bar of Ti–45Nb alloy. Its chemical composition is listed in Table 1.
Table 1
Chemical composition of hot-extruded Ti–Nb45 (wt%)
Nb
Fe
Cr
Mn
Mg
Si
K
Na
O
N
Ti
44.94
<0.03
<0.01
<0.01
<0.01
<0.1
<0.01
<0.01
0.095
0.007
bal.
The samples for CR&F processing were cut from the bar of 42-mm diameter and 10-mm thickness. The CR&F process consisted of several steps of intermediate rolling and folding at room temperature. Steps with (i) no folds, (ii) twofolds and (iii) fourfolds pertained to true strains of 1.5, 3.0 and 4.8, respectively.
The samples for HPT processing were disc-shaped with 8-mm diameter and 0.8-mm thickness. The discs were cut by spark erosion machine from an extruded initial bar in a way that the normal direction (ND) of discs was parallel to the radial direction of the initially extruded bar. The HPT experiments were carried out at room temperature at 4 GPa hydrostatic pressure with 0.5, 1, 5 and 10 turns. Additionally, the in situ torque was measured during HPT processing. The σ–ε (true stress–true strain) curve of the HPT deformation curve is obtained from in situ torque/torsion angle data using the following relations and the Von Mises factor \( \sqrt 3 \):
$$ \sigma = \sqrt 3 \frac{3Q}{{2\pi R^{2} }}, $$
(1)
$$ \varepsilon = \frac{R\phi }{d\sqrt 3 }, $$
(2)
where Q is the measured torque, R is the effective radius of the sample, ϕ is the torsion angle and t is the thickness of the sample.
Concerning HE processing, rod-shaped specimens with a diameter of d0 = 30 mm were prepared from the initial material parallel to the extrusion direction, and were subjected to HE in a multi-step process of six passes, by reducing the diameter after each pass. The true strain ε was determined by the relation ε = 2ln(df/di), where di and df denote the diameter before and after a particular pass. After six passes of HE, the final diameter of each rod was 5 mm with a total accumulated true strain of 3.5.
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After each intermediate step of deformation during CR&F, HPT and HE processing, the Vickers hardness of the samples was measured. The hardness measurements were carried out using a Vickers micro-hardness tester PAAR MHT 4, and indentations were achieved by a force of 1.96 N and a dwell time of 10 s, on the rolling plane and on the shear plane normal of CR&F and HPT processed samples, respectively. For each value, an average of 10 measurements was taken. All the experimental work-hardening curves were obtained or at least checked by Vickers hardness data. Those of CR&F and HE were done after each pass, respectively (see Table 2). The relation σ = HV/2.63 was found by the best fit of Vickers hardness data of HPT-processed sample to the in situ σ–ε true stress–true strain curves obtained from the torque/torsion angle data reported before.
Table 2
Input parameters used for simulations by Zehetbauer’s model
Parameter
Notation
Value
Comments
Taylor factor
Taylor factors were calculated from experimental textures in the frame of Taylor model (see ‘Methodology’ section)
TEM has been carried out by a JEOL M 1200 instrument. The X-ray diffraction profiles of the processed samples were measured at the beamline P07-HEMS at the Synchrotron PETRA 3 HASYLAB Hamburg, Germany, operating at 50 keV (λ = 0.0248 nm).
The Convolution Multiple Whole Profile (CMWP) fitting procedure was used to determine both the dislocation density and the ‘crystallite size’. The CMWP procedure has been described in detail in Refs. [23, 24].
Simulations
Description of the Zehetbauer’s model
The Zehetbauer’s model [15, 16, 18‐21] is a composite model, where screw and edge dislocations do not interact with each other because of different strain fields around the cores of dislocations. Thus, they are arranged in separate regions of cell interiors (areas of screw dislocations) and cell walls (areas of edge dislocations). The macroscopic hardening results from a weighted sum of contribution of hardening in cell interiors and cell walls, and can be represented as
where the ‘weights’ f1 and f2 are the structural fractions of cell/grain interiors and cell walls, respectively; f2 = 1 − f1. Equation (3) allows to treat work hardening of cell interiors and cell walls separately, according to Eqs. (4) and (5):
where C1, …, C4 are constants describing particular processes during SPD processing of the material and are connected with distinct physical parameters. C1 and C2 describe the storage of screw and edge dislocations, respectively, following the expression (6):
where μ is shear modulus, \( \alpha _{i} \in \left\langle {0.3, 1}\right\rangle, \)α1 < α2 are dislocation interaction parameters, and \( \beta _{i} \in \left\langle {10,300} \right\rangle \) denote the storage rates of specific dislocations with densities ρi (i = 1, 2 for screws and edges, respectively) relative to their mean free path Li, in terms of dislocation distance \( L_{i} = 1/\sqrt {\rho_{i} } \). C3 describes the annihilation of screw dislocations through the cross-slip mechanism according to Eq. (7):
where \( \dot{\gamma } \) is the strain rate, ωd is the Debye frequency, δG represents the enthalpy of screw dislocation annihilation, k is Boltzmann constant and T is the absolute temperature. The last parameter, C4, describes the annihilation of edge dislocations through climb processes, with the aid of deformation-induced vacancies (8):
where cv is the concentration of vacancies, ν is Poisson’s ratio, Ω is the atomic volume, Dc,0 means the core diffusion coefficient, δHm stands for the vacancy migration enthalpy and p represents the hydrostatic pressure applied during the SPD processing.
Concerning the microstructural quantities, the model firstly provides the dislocation interaction parameters αi from fits to experimentally measured dislocation densities, according to
Secondly, the grain/cell size L1 (area of screw dislocations) and cell wall size L2 (area of edge dislocations) are directly derived from the stress/strain relationships and the αi/βi parameters, as
where ρ2 is the density of sessile dislocations in cell walls (edge areas, with size L2) and k is a scaling factor [16].
Calculations of resolved shear stress/resolved shear strain data
For application of Zehetbauer’s model, datapairs of resolved shear stress–resolved shear strain (τ–γ) must be provided. These τ–γ datapairs can be calculated from stress–strain curves σ–ε by the Taylor factor \( \left\langle M \right\rangle \) according to the relations: \( \tau = \sigma /\left\langle M \right\rangle \) and \( \gamma = \left\langle M \right\rangle \varepsilon \).
The Taylor model [25, 26] was used to determine the Taylor factors for the samples processed by every SPD method used (\( \left\langle {M_{{{\text{CR}}\& {\text{F}}}} } \right\rangle \),\( \left\langle {M_{\text{HPT}} } \right\rangle \) and \( \left\langle {M_{\text{HE}} } \right\rangle \)). In the Taylor model it is assumed that all the grains experience the same strain. The average volume-weighted Taylor factor \( \left\langle M \right\rangle \) for the aggregate can be calculated from the equation:
$$ \left\langle M \right\rangle = \frac{{\sum\limits_{i}^{N} {M(g_{i} )} }}{N}, $$
(12)
where N is a number of crystalline in the aggregate and M is the Taylor factor for the particular crystallite as function of its orientation gi. The orientations gi of the crystallite can be obtained from the experimentally measured textures. The measuring procedure and the texture results will be published elsewhere [14]. The Taylor factor for each crystalline can be calculated from Eq. (13) [25]:
where γj is the shear strain of the particularly slip system (j) and dε is the magnitude of applied strain. The values of the obtained Taylor factors are included in Table 2.
Description of fit procedure
With the τ–γ data obtained as described in ‘Simulations’ section, the simulations were performed so that first the Zehetbauer’s model (Eqs. 3–5) was suitably implemented into the MATLAB and Simulink toolboxes. The Levenberg–Marquardt least-square minimization method was used since it finds reliable minima even if the first guess is very far from solution, using not only a Gauss–Newton approach for parameter values close to the optimum but also the gradient descent method. The procedure systematically changes the values of Ci in search of the best fit to the experimental τ–γ data. The fitting iterations are finished when the relative changes of C1, …, C4 parameters are below 10−5. The C1, …, C4 parameters are then used to calculate the evolution of the grain/cell interior size (area of screws) L1 and the cell wall size (area of edges) L2 using the equations (10) with support of fits of Eq. (9) to the measured dislocation densities providing the required values of αi parameters. Also, the fractions of screw/edge areas f1 and f2 were optimized by iterative fitting. Table 2 lists the values of all input parameters used for the simulations.
Results and discussion
The true strain–true stress σ–ε curves of Ti–45Nb samples processed by CR&F (dashed line and squares), HPT (full line and triangles) and HE (dotted line and diamonds) are presented in Fig. 1. It is seen that until strains up to ε = 4 CR&F and HE processes show higher true stresses than HPT. However, the much highest true strains (ε > 28) are achieved by HPT. Therefore, the final hardness was larger for HPT (2.48 GPa) in comparison to the final hardness for CR&F (2.05) and to that for HE (1.81 GPa).
Fig. 1
True stress–true strain curves of CR&F-, HPT- and HE-processed samples. Except for the case of HPT, lines have been drawn as a guide for the eye
×
For the sake of simulations and using the relations of the Taylor model given in ‘Simulations’ section, the data presented in Fig. 1 have been recalculated to resolved shear stress τ/resolved shear strain γ ones, for direct application of the model algorithm providing the simulations. The results are shown in Fig. 2. It is seen that the τ–γ relations are—at least up to strains γ = 15—almost identical for the CR&F- and HPT-processed samples; while in case of HE-processed samples at strains beyond γ > 5, the resolved shear stress is much lower and becomes constant at much lower strains. This means that plastic deformation with CR&F and HPT processing act by different numbers and types of slip systems but essentially by the same resolved shear stress, while the deformation with HE processing follows an entirely different deformation path. This is also indicated by the individual strain-dependent evolutions of dislocation densities (Figs. 3, 5) and of grain/cell (dislocation area) dimensions (Fig. 4) of HE processing compared to the other SPD methods.
Fig. 2
The results of model simulations (lines) of the experimental τ–γ curves (symbols)
Fig. 3
Evolution of total dislocation density during CR&F, HPT and HE processes. Full/dashed/dotted lines result from the simulations by Zehetbauer’s model. The squares and triangles represent measurements by XPA
Fig. 4
Evolution of cell size (screw area L1) and of cell wall size (edge area L2) during CR&F and HPT processes. Full, dashed and dotted lines are from simulations by Zehetbauer’s model, with arrows indicating the starting values (except L1 for CR&F). Open symbols represent domain sizes measured by XPA and full symbols stand for grain/cell sizes derived from TEM observations. For measuring errors, see text
×
×
×
The simulations by Zehetbauer’s model have been carried out individually for each of the SPD processing routes. The fits to experimental stress–strain curves could be achieved within less than 1 % deviation. The output parameters resulting from the fit procedures for the cases of CR&F, HPT and HE processing are summarized in Table 3. The critical resolved shear stress τo is in the range 92–98 MPa. The βi parameters which describe the mean free path of moving dislocations in terms of the average dislocation distance are in physically reasonable ranges, i.e. 30 > β1 > 6 and 300 > β2 > 42 (compare previous results from applications of Zehetbauer’s model [15, 16, 18‐21, 29]).
Table 3
Output parameters from fits of Zehetbauer’s model to stress/strain characteristics of CR&F-, HPT- and HE-processed samples
Parameter/SPD method
CR&F
HPT
HE
C1 (MPa)
663.23
219.23
1058.84
C2 (MPa)
186.96
29.39
166.78
C3
6.05
1.04
10.61
C4 ×10−16 (MPa)
0.12
0.17
253.38
τ0 (MPa)
98.00
92.12
95.31
β1
10.28
25.07
6.23
β2
42.33
291.74
48.77
The results of simulations of the strain-dependent evolution of total dislocation density ρ during CR&F, HPT and HE processes are shown in Fig. 3. The final total dislocation densities amount to 4.8 × 1015 and 5.5 × 1015 m−2, for CR&F- and HPT-processed samples, respectively. However, during CR&F process, the total dislocation density beyond γ = 7 increases much faster than that during HPT. It is noticed that at a strain γ ≈ 10, the total dislocation density for CR&F-processed samples is almost twice as large as that for HPT-processed samples but the resolved shear stresses are almost identical, as seen in Fig. 2. This means that a significant number of the dislocations in case of the CR&F-processed samples do not—or only little—contribute to the macroscopic stress. This point will be re-discussed below in the frame of the simulation results.
Figure 4 presents a comparison of the strain-dependent simulated parameters of nanostructure with the experimental data obtained by TEM and XPA (for details see ‘Methodology’ section). There is a good agreement between the simulations and experiments. The final size of grain/cell interior after fourfolds of CR&F from TEM observations is 264 ± 70 nm, while after six passes of HE the average grain/cell interior size is 298 ± 113 nm—both values being close to the edge area sizes L1 predicted by the simulations, at least within the experimental errors. The domain size measured by XPA after fourfolds of CR&F (γ ≈ 13) is 53 ± 7 nm while after one rotation of HPT (γ ≈ 60) it is 56 ± 8 nm. Also, these experimental values are close to those predicted by the simulations for the edge dislocation areas L2, as it has been found in the previous investigations of the authors [15, 16].
The simulations suggest that—as function of resolved shear strain—at first strong decreases of both the cell interior size (screw areas L1) and cell wall size (edge areas L2) in all the processing types occur during CR&F and HE already at much lower strains as during HPT. After that, both area sizes level out to individual constant values. Thus, the increase of strengthening above γ > 3 (Fig. 2) may not be ascribed to a Hall–Petch mechanism (from decreasing grains size) but to the considerably high dislocation density (compare Fig. 4 with Fig. 3). Such behaviour has already been observed in the other as-deformed and/or as-SPD-processed materials (HPT-processed Cu [30], Nb and Ta [31]).
However, as mentioned before, the value of the high dislocation density depends of the SPD processing method while the strengthening remains unaffected, see Fig. 3. In this situation, we can use the capability of Zehetbauer’s model to analyse the evolution of dislocation density in terms of screws and edges, results of which are presented in Fig. 5 for all the SPD methods applied. It is clearly seen that the particularly strong increase of dislocations during CR&F processing is mainly due to edge dislocations which obviously do not contribute to strengthening. At the same time, however, these edge dislocations—as they are stored in the wake of and within the high-angle grain boundaries—cause an increase of misorientation angle between the adjacent grains.
Fig. 5
Results of simulations by Zehetbauer’s model on the evolution of screw and edge dislocation density during CR&F, HPT and HE processes
×
Assuming for simplicity that all the edge dislocations in the boundary area contribute to misorientation (Eq. 11), thus allowing an estimation of an upper limit for the misorientation angle, the model can simulate the strain-dependent evolution of misorientation between the adjacent grains [16]. Applying Eq. (11) for the actual Ti–Nb SPD processings—CR&F, HPT and HE gives the graphs shown in Fig. 6. According to the rapid increase in edge dislocation density, the strain-dependent increase of misorientation angle of CR&F-processed samples is very fast in comparison to that of HPT- and HE-processed materials. This finding agrees with several results reported in literature [32, 33] that CR&F and/or ARB processing exhibit the most effective fragmentation and the smallest grain size among all the SPD methods. To conclude this point, it is to be noted that these dislocations described show all features of the so-called ‘geometrically necessary dislocations (GNDs)’ which accommodate the imposed shear strain but do not give a marked contribution to work hardening [34].
Fig. 6
Simulations of evolution of misorientation angle between adjacent grains in CR&F-, HPT- and HE-processed samples
×
With Zehetbauer’s model, it is also possible to estimate the production of SPD-induced vacancies and their concentrations as function of SPD parameters. Figure 7 presents their evolution with increasing strain for the different processing techniques CR&F, HPT and HE applied. The concentration of vacancies during HPT and HE is indicated to be up to four orders of magnitude larger in comparison to CR&F. Although this estimation again means an upper limit, it seems reasonable at least from a qualitative point of view because of the significantly high hydrostatic pressures in cases of HPT (4 GPa) and HE (1 GPa) in comparison to CR&F. Further studies—preferably detailed experimental investigations—are to be carried out in order to check those calculations.
Fig. 7
Simulations of evolution of strain-induced vacancies for CR&F-, HPT- and HE-processed samples
×
Summary and conclusions
1.
By applying SPD methods such like CR&F, HPT and HE, it is possible to significantly increase the mechanical properties of Ti–45Nb novel biomedical alloy.
2.
Although the nanostructure achieved by CR&F processing is finer than that resulting from HPT, the work hardening and strength are similar. From this, it is concluded that hardening from dislocations (except from those in grain boundaries) strongly dominates that from the grain boundaries. This finding is substantiated by the fact that while at strains beyond ε = 2.6, no further fragmentation is indicated either in case of CR&F-, HPT- or in HE-processed samples, although work hardening is far from zero at those strains.
3.
The simulations show that the fragmentation in CR&F and HE processing is stronger than that in HPT processing. In parallel, a much higher edge dislocation density in cell/grain wall area in case of CR&F is indicated suggesting a higher density of geometrically necessary dislocations than that with the other SPD methods. This is in accordance with literature findings, where the CR&F method has been described to be the most efficient one in fragmentation and minimization of grain/cell size.
4.
As the simulations show good agreement with the experimental data, the large strain work-hardening model by Zehetbauer can be used to predict the evolution of strength and microstructural quantities of single-phase alloys processed by different SPD methods.
Acknowledgements
This work was done within the EU 7th framework programme FP7/2007-13 under Marie-Curie project Grant No. 264635 (BioTiNet-ITN). B.S. and A.P. are grateful to Prof. Werner Skrotzki, University of Technology Dresden, for his support in texture expertise. Part of the work was done within the framework of the statutory activity (11.11.180.653) of Department of Materials Science and Non-Ferrous Metals Engineering, AGH-University of Science and Technology, Kraków, Poland.
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