# Evolution of strength and structure during SPD processing of Ti–45Nb alloys: experiments and simulations

## Abstract

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, 2, 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, 11, 12, 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, 16, 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, 19, 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].

## Methodology

### Experiments

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.

*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 \( \sqrt 3 \):

*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 *d*_{0} = 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(*d*_{f}/*d*_{i}), where *d*_{i} and *d*_{f} 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.

*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.

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) | ||

\( \left\langle {M_{{{\text{CR}}\& {\text{F}}}} } \right\rangle \) | 2.76 | Rolling and folding | |

\( \left\langle {M_{\text{HPT}} } \right\rangle \) | 2.05 | High-pressure torsion | |

\( \left\langle {M_{\text{HE}} } \right\rangle \) | 2.83 | Hydro-extrusion | |

Fraction factor | f f f | 0.92 0.80 0.92 | Values optimized during simulations considering the enhanced pressures 4 GPa (HPT processing), 1 GPa (HE) and 0.1 MPa (CR&F) |

Dislocation interaction parameters | α α α α α α | 0.31 0.36 0.25 0.39 0.30 0.37 | Values derived from fits of dislocation density |

Vacancy migration enthalpy |
| 0.53 |

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

*f*

_{1}and

*f*

_{2}are the structural fractions of cell/grain interiors and cell walls, respectively;

*f*

_{2}= 1 −

*f*

_{1}. Equation (3) allows to treat work hardening of cell interiors and cell walls separately, according to Eqs. (4) and (5):

*C*

_{1}, …,

*C*

_{4}are constants describing particular processes during SPD processing of the material and are connected with distinct physical parameters.

*C*

_{1}and

*C*

_{2}describe the storage of screw and edge dislocations, respectively, following the expression (6):

*μ*is shear modulus, \( \alpha _{i} \in \left\langle {0.3, 1}\right\rangle, \)

*α*

_{1}<

*α*

_{2}are dislocation interaction parameters, and \( \beta _{i} \in \left\langle {10,300} \right\rangle \) denote the storage rates of specific dislocations with densities

*ρ*

_{i}(

*i*= 1, 2 for screws and edges, respectively) relative to their mean free path

*L*

_{i}, in terms of dislocation distance \( L_{i} = 1/\sqrt {\rho_{i} } \).

*C*

_{3}describes the annihilation of screw dislocations through the cross-slip mechanism according to Eq. (7):

*ω*

_{d}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,

*C*

_{4}, describes the annihilation of edge dislocations through climb processes, with the aid of deformation-induced vacancies (8):

*c*

_{v}is the concentration of vacancies, ν is Poisson’s ratio,

*Ω*is the atomic volume,

*D*

_{c,0}means the core diffusion coefficient,

*δH*

_{m}stands for the vacancy migration enthalpy and

*p*represents the hydrostatic pressure applied during the SPD processing.

*α*

_{i}from fits to experimentally measured dislocation densities, according to

*L*

_{1}(area of screw dislocations) and cell wall size

*L*

_{2}(area of edge dislocations) are directly derived from the stress/strain relationships and the

*α*

_{i}/

*β*

_{i}parameters, as

*µ*as the shear modulus and

*b*as the Burgers vector.

*θ*can be calculated from the expression:

*ρ*

_{2}is the density of sessile dislocations in cell walls (edge areas, with size

*L*

_{2}) 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 \( \gamma = \left\langle M \right\rangle \varepsilon \).

_{,}\( \left\langle {M_{\text{HPT}} } \right\rangle \) 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:

*N*is a number of crystalline in the aggregate and

*M*is the Taylor factor for the particular crystallite as function of its orientation

*g*

_{i}. The orientations

*g*

_{i}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]:

*γ*

_{j}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 *C*_{i} in search of the best fit to the experimental *τ*–*γ* data. The fitting iterations are finished when the relative changes of C_{1}, …, C_{4} parameters are below 10^{−5}. The C_{1}, …, C_{4} parameters are then used to calculate the evolution of the grain/cell interior size (area of screws) *L*_{1} and the cell wall size (area of edges) *L*_{2} using the equations (10) with support of fits of Eq. (9) to the measured dislocation densities providing the required values of *α*_{i} parameters. Also, the fractions of screw/edge areas *f*_{1} and *f*_{2} were optimized by iterative fitting. Table 2 lists the values of all input parameters used for the simulations.

## Results and discussion

*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).

*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.

*τ*

_{o}is in the range 92–98 MPa. The

*β*

_{i}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 >

*β*

_{1}> 6 and 300 >

*β*

_{2}> 42 (compare previous results from applications of Zehetbauer’s model [15, 16, 18, 19, 20, 21, 29]).

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 |
---|---|---|---|

C | 663.23 | 219.23 | 1058.84 |

C | 186.96 | 29.39 | 166.78 |

C | 6.05 | 1.04 | 10.61 |

C | 0.12 | 0.17 | 253.38 |

| 98.00 | 92.12 | 95.31 |

| 10.28 | 25.07 | 6.23 |

| 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 × 10^{15} and 5.5 × 10^{15} 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 *L*_{1} 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 *L*_{2}, 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 *L*_{1}) and cell wall size (edge areas *L*_{2}) 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]).

## 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.

## Notes

### 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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