In this study, the mechanical spectroscopy technique was applied to measure the torsion modulus of Ti-xZr (where x = 0, 5, 10 and 15 wt%) alloys for dental applications. The technique makes it possible to measure the torsion or shear modulus (G) of the samples without the use of Poisson’s ratio, which is an advantage when compared with other dynamic methods. The G value of the Ti-Zr samples was compared with titanium samples of equal size in terms of their thermo-mechanical treatment and composition. In order to perform this comparison, an equation that expresses G as a function of the diameter and length of the commercially pure titanium (CP-Ti) samples was determined. This equation was used to evaluate the values for G of the CP-Ti so that it was possible to compare them with those for the Ti-Zr alloys. The results showed that there was a relationship between the G values of the Ti-Zr and the CP-Ti samples and that they depended on the microstructural characteristics (after undergoing thermo-mechanical treatment) and the Zr content.

Titanium (Ti) and its alloys are well-known biomedical materials due to their favourable properties, such as excellent corrosion resistance, high strength-to-density ratio, relative low Young’s modulus, and good biocompatibility and osseointegration [1,2]. For example, Ti-based alloys have found several applications in dental, orthopedical, cardiovascular and fixation devices, such as abutments and brackets, total hip and knee replacements, stents and heart valves, and plates and screws, respectively [1,3]. In this field, the development of novel Ti-based alloys with high performance properties can impact directly on healthcare issues [2,3].

Regarding physical metallurgy, Ti is an allotropic element with a crystalline structure that changes near 883 °C (1156 K), from a hexagonal, close-packed structure (hcp, α phase) to body-centred cubic structure (bcc, β phase) [2,4]. Depending on the composition, heat treatment, and mechanical processing (e.g. rolling or forging), it is possible to obtain a mixture of α and β phases and to also precipitate metastable phases, such as martensitic (α’ and α”) and ω phases [2]. As a result, the mechanical properties are directly affected by the volumetric fraction and morphology of these phases because it is possible to manipulate the properties with adequate thermo-mechanical treatments [5].

As Ti is a transition metal of the IVB group in the periodic table, it is able to form a solid solution with a large range of elements [2]. From the biomedical point of view, Zirconium (Zr) has received special attention as an alloying element in Ti-based alloys, because the element belongs to the same elemental group as Ti and has similar chemical properties [6]. In solid solution, Zr is fully soluble in both allotropic phases of Ti, decreasing the starting martensitic temperature (Ms) and the melting point of the alloy [7,8]. Furthermore, the element can improve the mechanical strength, corrosion resistance, and biocompatibility of Ti alloys, which are crucial factors when used in biomedical applications [9,10].

Recently, novel Ti-based alloys have been developed to match the mechanical properties of human bone (Young’s modulus and hardness, in particular) in order to avoid stress shielding effects and the failure of implants [2]. For this purpose, the binary system alloys have exhibited a good combination of mechanical properties, with the advantage of being easy to manufacture, in comparison to multi-component, Ti-based alloys. Non-toxic and non-allergenic elements have been preferentially chosen to form solid solutions with Ti, such as Ti-Zr [11], Ti-Nb [12], Ti-Mo [13], and Ti-Ta [14]. Together with Young’s modulus, the torsion modulus is a fundamental parameter to consider during biomechanical loading and the study of the effects of alloying elements on this parameter is an important consideration. This study aims to analyse the effect of Zr content and thermo-mechanical treatments on the torsion modulus of biomedical Ti-based alloys by using mechanical spectroscopy.

2Materials and methods2.1Sample preparationTi-xZr alloys (where x = 5, 10, and 15 wt.%) were produced from commercially pure metals (CP-Ti grade 2 and 99% pure Zr), by argon arc-melting using a water-cooled copper crucible and a non-consumable tungsten electrode. The 60 g ingots were re-melted five times to ensure good homogeneity. Then, the samples were submitted to hot-forging at 1273 K, followed by air cooling. The samples then underwent a heat treatment in a vacuum (10−6 Torr), at 1273 K for 86.4 ks, followed by slow cooling. Details of the chemical composition of the samples can be found in Correa et al. [11].

Structural analyses were undertaken by XRD measurements, using a D/Max 2100/PC diffractometer, with monochromatic CuKα radiation (λ = 0.1544 nm), at 40 kV and 20 mA. The cell parameters were calculated from the XRD patterns, using Bragg’s law. For microstructural analysis, a selected part of the sample was cut by a precision cut-off machine with a diamond cutting wheel. The samples were submitted to standard metallographic preparation by sandblasting and polishing with SiC sandpapers (#150, #180, #360, #400, #600, #800, #1200, and #1500), colloidal diamond suspension (9 μm), and alumina suspension (0.25 μm), respectively. The microstructures were revealed by an etching process using H2O, HNO3, and HF (85:10:5). The micrographs were obtained with an Olympus BX51 M microscope.

2.2Acquisition method for torsion modulus (G)The mechanical spectroscopy technique to determine the value of G, has been studied in our laboratories and can be used to investigate changes in the mechanical properties of these materials. The technique for determining G is described succinctly here. It consists of one torsion pendulum that can be adjusted to set its inertia to different values (in our case, four different values were used) and register its angular position θ in relation to an axis of rotation as a function of time t. A rotational movement sensor (RMS) was used for this purpose, see Fig. 1.

Measurement system: (1) the masses of the torsion pendulum; (2) electromagnet; (3) three-step pulley in the position above the shaft of the rotational system (RS); (4) sensor of force (FS) – PASCO: CI-6537; (5) torsion pendulum; (6) RS shaft rod; (7) sample; (8) sample fixing base; (9) rotary motion sensor (RMS) – PASCO: CI-6538; (10) measurement system structure. Complete view of the system (a); Detailed view of the system: RMS and sample fixing base (b); FS and electromagnet (c) and Pendulum (d). see ref. [20].

Mechanical spectroscopy [15], that is, the torsion motion of the pendulum, is described by the following equation:

The symbols I, G0 and i represent the rotational inertia, the torsion constant of the sample fixed to the pendulum, and the imaginary number, respectively. In this case, we assume that air resistance and the friction of the bearings of the rotational motion sensor and rotational system (RS) are negligible, because they should not significantly influence the movement [16]. The solution of the above equation is of the form θ=θ0 ei ω* t, where ω*=ω0 (1+i δ2π) and tgφ= δπ. Then, G0 can be expressed as

This equation suggests that the moment of inertia of the pendulum I, the angular velocity ω0, and the term δ/2π can be used to obtain G0. It is possible to alter the masses and/or their position on the torsion rod of the pendulum, which changes I and thus modifies the value of ω0. Then, G0 can be obtained with the frequency of the oscillation system and we, therefore have a dynamic measurement. However, if it is necessary to determine I using Eq. (2), a wire will be needed with a torsion value G0, which can be experimentally determined [16], see Fig. 1 (c). To do so, the force sensors and rotation are used to find the best fit of the force depending on the angular position. For small angles of oscillation, in which the measuring system has a linear behaviour, the best fit to the force versus the angular position is a straight line. The constant G0 is determined by the slope of the force curve F, as a function of the angular position θ, for the two quantities recorded with force sensors and rotation, respectively. With the value of the slope (ΔFΔθ) multiplied by the radius of the pulley rp, G0 can be determined as

since the main objective is to determine the torsion modulus G for a solid sample diameter of a uniform circular section and length L.Therefore, when considering problems that take into account the resistance of the material [17–19], we can calculate the resistive torque to which the sample is submitted, when the sample undergoes torsion by means of an applied torque by a pendulum (see Figs. 1). This torque is calculated by the following equation:

Where JC is the polar moment of inertia of the cross section, then the equation is:

If there is static equilibrium, i.e., the applied torque is the same as the resistive torque, Eqs. (3) and (4) can be combined to give

Thus, the magnitude of G is obtained by substituting Eq. (2) into (6) and, as a result:

The validation of Eq. (7) is presented in article [20], where we show that G can change with the length (L) and the diameter (d) of the sample, yet the experimental values found for CP-Ti are practically the same as those in the literature [21–23]. Given that we used a different dynamic technique from the traditional ones used previously (ultrasound or vibration) [24], it is perfectly understandable that the values found here would be slightly different to those presented in the literature.

If you replace ω*=ω0 (1+i δ2π) in the solution, θ=θ0 ei ω* t, relative to the differential equation that describes the motion of the free pendulum [15] as given above, then

In Eq. (8), taking the real part gives the following equation:

The values of the parameters δ and ω0 are obtained from the experimental θ versus t curve, which is obtained using the rotational motion sensor software and the Pasco interface for given values of L, d, and I. For this purpose, a fit function similar to Eq. (9) was devised with Origin 7.0 software, by selecting a “wave form” function such as the following:

By comparing this equation with Eq. (9), we can take into account the parameters t0 and W, as obtained by fitting Eq. (10) to the experimental values, and we are then able to obtain the parameters δ and ω0:Finally, with the W and t0 parameters and Eq. (11), we calculate (δ 2π)2 and ω0=πW and use Eq. (7) to determine G.

2.3Acquisition method for the moment of inertia (I)We determined the inertia moment (MI) of the torsion pendulum I by removing the sample and the electromagnetic system, see Fig. 1. Then, the same procedure was adopted as in references [25] and [26], where the authors considered the resulting torque (τ) to be responsible for the rotational movement of the measured pendulum system:

Here, τF is the torque of the friction force acting on the rotating shaft, while I and α0 are the MI and the angular acceleration of the pendulum, respectively. It must be remembered that the torque of the traction force is τT=m (g −α0 r) r, where r = (18.75 ± 0.03)10−3 m and g = (9.79 ± 0.01) m/s2. With Eq. (12), it can be shown that the inclination is equal to I.3Results and discussion3.1Measurement of the rotational inertia (I) of the torsion pendulumTable 1 shows the results of applying torque to the torsion pendulum (τT), by means of a traction wire with one end tied to a pulley of radius 18.75 mm and the other end connected to a mass m suspended at a suitable height from the ground. Using a rotary motion sensor, we determined the angular acceleration of the torsion pendulum (α0), see references [25] and [26].

Experimensstal results of the applied torque, τT, and angular acceleration, α0.

m(g) | α0(rad/s2) | τT(10−4 N.m) | α0(rad/s2) | τT(10−4 N.m) | α0(rad/s2) | τT(10−4 N.m) | α0(rad/s2) | τT(10−4 N.m) |
---|---|---|---|---|---|---|---|---|

25.00 | 0.0717 | 45.88 | 0.0855 | 45.88 | 0.179 | 45.87 | 0.386 | 45.86 |

45.00 | 0.139 | 82.58 | 0.178 | 82.57 | 0.354 | 82.55 | 0.736 | 82.49 |

65.00 | 0.206 | 119.27 | 0.269 | 119.25 | 0.522 | 119.20 | 1.10 | 119.06 |

85.00 | 0.276 | 155.95 | 0.36 | 155.92 | 0.678 | 155.83 | 1.43 | 155.6 |

105.00 | 0.345 | 192.61 | 0.448 | 192.58 | 0.863 | 192.42 | 1.82 | 192.07 |

125.00 | 0.412 | 229.27 | 0.544 | 229.21 | 1.02 | 229.00 | 2.18 | 228.5 |

145.00 | 0.479 | 265.92 | 0.634 | 265.84 | 1.21 | 265.55 | 2.55 | 264.87 |

165.00 | 0.549 | 302.56 | 0.725 | 302.46 | 1.38 | 302.08 | 2.92 | 301.18 |

185.00 | 0.617 | 339.19 | 0.819 | 339.06 | 1.54 | 338.59 | X | X |

205.00 | X | X | 0.904 | 375.65 | X | X | X | X |

225.00 | X | X | 1.00 | 412.22 | X | X | X | X |

255.00 | X | X | 1.14 | 467.06 | X | X | X | X |

305.00 | X | X | 1.37 | 558.4 | X | X | X | X |

These results allow the construction of a figure, whose inclinations (obtained by a linear fit), provide the values of I, see Table 2.

The sets 1 to 4 refer to the best linear fit of pairs of columns from Table 1 (2 and 3, 4 and 5, 6 and 7, and 8 and 9, respectively).

3.2Torsion modulus resultsThe results obtained for G and f0 for the four values of rotational inertia for forged Ti-Zr alloy, CP-Ti, and treated Ti-Zr alloy, are presented in Tables 3–5, 6–9 and 10–12, respectively.

Titanium forged alloy, Ti-5Zr, d = 4.40 ± 0.05 mm, L = 33.45 ± 0.05 mm.

Frequency,f0 (Hz) | Torsion modulus,G (GPa) | Rotational Inertia,I (10−4 kg m2) | I (f0)2(10−4 kg m2 Hz2) |
---|---|---|---|

8.61 ± 0.05 | 27 ± 1 | 100.6 ± 0.7 | 7458 |

5.90 ± 0.05 | 27 ± 1 | 214 ± 1 | 7449 |

4.28 ± 0.05 | 26 ± 1 | 399.6 ± 0.8 | 7320 |

3.72 ± 0.05 | 27 ± 1 | 537 ± 1 | 7431 |

Average:7415 ± 64 |

Titanium forged alloy, Ti-10Zr, d = 4.40 ± 0.05 mm, L = 33.45 ± 0.05 mm.

Frequency,f0 (Hz) | Torsion modulus,G(GPa) | Rotational Inertia,I (10−4 kg m2) | I (f0)2(10−4 kg m2 Hz2) |
---|---|---|---|

8.97 ± 0.05 | 29 ± 1 | 100.6 ± 0.7 | 8094 |

6.15 ± 0.05 | 29 ± 1 | 214 ± 1 | 8094 |

4.48 ± 0.05 | 29 ± 1 | 399.6 ± 0.8 | 8020 |

3.87 ± 0.05 | 29 ± 1 | 537 ± 1 | 8043 |

Average:8063 ± 37 |

Titanium forged alloy, Ti-15Zr, d = 5.00 ± 0.05 mm, L = 24.40 ± 0.05 mm.

Frequency,f0 (Hz) | Torsion modulus, G (GPa) | Rotational Inertia, I (10−4 kg m2) | I (f0)2 (10−4 kg m2 Hz2) |
---|---|---|---|

100.6 ± 0.7 | |||

9.03 ± 0.05 | 27 ± 1 | 214 ± 1 | 17450 |

6.58 ± 0.05 | 27 ± 1 | 399.6 ± 0.8 | 17301 |

5.69 ± 0.05 | 27 ± 1 | 537 ± 1 | 17386 |

Average:17379 ± 75 |

Tables 6–9 show the results for the construction of Figs. 2 and 3. It was observed that, with an increase in the diameter of the samples with inertia I and length L held constant, the oscillation frequency also increased (see Fig. 2). Fig. 3 shows that the value of G does not depend on the inertia but decreases with an increase of d when L is constant. This was an expected result based on Eq. (7). The fact that the value of G is slightly smaller for the CP-Ti sample (with a diameter of 3.80 ± 0.05 mm and a rotational inertia of 399.6 ± 0.8 10−4 kgm2), does not compromise the previous assertion that G does not depend on inertia. If we look at Table 7, the error associated with the values found for G was ±2 GPa, and so G was practically the same in the situation under discussion. The simple act of choosing an angular position on the experimental curve as a function of time may result in a slightly different G value.

Titanium, CP-Ti, d = 3.20 ± 0.05 mm, L = 33.45 ± 0.05 mm.

Frequency,f0 (Hz) | Torsion modulus, G (GPa) | Rotational Inertia, I (10−4 kg m2) | I (f0)2 (10−4 kg m2 Hz2) |
---|---|---|---|

5.63 ± 0.05 | 41 ± 3 | 100.6 ± 0.7 | 3189 |

3.86 ± 0.05 | 41 ± 3 | 214 ± 1 | 3189 |

2.82 ± 0.05 | 41 ± 3 | 399.6 ± 0.8 | 3178 |

2.44 ± 0.05 | 41 ± 3 | 537 ± 1 | 3197 |

Average:3188 ± 8 |

Titanium, CP-Ti, d = 4.70 ± 0.05 mm, L = 33.45 ± 0.05 mm.

Frequency,f0 (Hz) | Torsion modulus,G (GPa) | Rotational Inertia, I (10−4 kg m2) | I (f0)2 (10−4 kg m2 Hz2) |
---|---|---|---|

10.77 ± 0.05 | 32 ± 1 | 100.6 ± 0.7 | 11669 |

7.36 ± 0.05 | 32 ± 1 | 214 ± 1 | 11592 |

5.36 ± 0.05 | 32 ± 1 | 399.6 ± 0.8 | 11480 |

4.65 ± 0.05 | 32 ± 1 | 537 ± 1 | 11611 |

Average:11588 ± 79 |

Titanium, CP-Ti, d = 5.00 ± 0.05 mm, L = 33.45 ± 0.05 mm.

Frequency,f0 (Hz) | Torsion modulus,G (GPa) | Rotational Inertia,I (10−4 kg m2) | I (f0)2(10−4 kg m2 Hz2) |
---|---|---|---|

12.00 ± 0.05 | 31 ± 1 | 100.6 ± 0.7 | 14486 |

8.23 ± 0.05 | 31 ± 1 | 214 ± 1 | 14495 |

5.98 ± 0.05 | 31 ± 1 | 399.6 ± 0.8 | 14290 |

5.19 ± 0.05 | 31 ± 1 | 537 ± 1 | 14465 |

Average:14434 ± 97 |

Titanium, CP-Ti, d = 3.80 ± 0.05 mm, L = 33.45 ± 0.05 mm.

Frequency,f0 (Hz) | Torsion modulus,G (GPa) | Rotational Inertia, I (10−4 kg m2) | I (f0)2 (10−4 kg m2 Hz2) |
---|---|---|---|

7.64 ± 0.05 | 38 ± 2 | 100.6 ± 0.7 | 5872 |

5.24 ± 0.05 | 38 ± 2 | 214 ± 1 | 5876 |

3.79 ± 0.05 | 37 ± 2 | 399.6 ± 0.8 | 5740 |

3.29 ± 0.05 | 38 ± 2 | 537 ± 1 | 5813 |

Average:5825 ± 64 |

The values of the maximum shear stress or angle-specific deformation, according to [27], are:

Therefore, the value of G can be expressed as:

With regard to the value of G in Eq. (15), it should be noted that, if we fix L and θ (as we did during the experiment), for each selected d, the value of JC will change and, consequently, the value of T. As we can see, when d and L/θ are kept fixed in Eq. (15), the value of JC does not change. This means that T (resistive torque) is equal in this situation and that G must have the same value for different inertias. In fact, this is what happens experimentally, see Fig. 3. For other diameters, the same facts are observed. With increasing diameter, T increases, and this can be seen by analysing Fig. 2. When the diameter is large compared to another sample, the resistive torque increases and the oscillation frequency also increases. However, the value of G was determined using Eq. (7). In the experiment, we changed the rotational inertia I and, for the same experimental conditions described above (i.e. with a fixed d and L/θ), it was observed that the product I (f0)2 is constant. Then, if I decreases, an increase of f0 is expected, see Fig. 2. For this technique to be applied within the range of the elastic regime, we fixed a torsion angle of 1.32 rad in the sample (corresponding to 6 degrees in the rotation sensor).

As the dimensions of the sample were changed (which was assumed not to change the internal structure), this behaviour remained, as shown in Figs. 2 and 3. Therefore, it must be a typical pattern for this material.

When analysing the results for CP-Ti, it was found that it was possible to obtain a general expression for the value of G that depended on the diameter of the sample and its length, within the range used in this study. It was verified that the product of the rotational inertia I and the square oscillation frequency (f0)2 had a constant average value and this was different for distinct values of d and/or L, see the last column of Tables 3–12. The dependence of the product I (f0)2 on d2 was linear for the CP-Ti sample. Then, using the results from CP-Ti (i.e. G =41 GPa, d = 3.20 mm, and L = 33.45 mm) as a reference, the following equation was obtained based on Eq. (7):

Titanium alloy treated, Ti-10Zr, d = 4.40 ± 0.05 mm, L = 33.45 ± 0.05 mm.

Frequency,f0 (Hz) | Torsion modulus,G (GPa) | Rotational Inertia, I (10−4 kg m2) | I (f0)2 (10−4 kg m2 Hz2) |
---|---|---|---|

9.43 ± 0.05 | 32 ± 1 | 100.6 ± 0.7 | 8946 |

6.47 ± 0.05 | 32 ± 2 | 214 ± 1 | 8958 |

4.70 ± 0.05 | 32 ± 1 | 399.6 ± 0.8 | 8827 |

4.08 ± 0.05 | 32 ± 2 | 537 ± 1 | 8939 |

Average:8918 ± 61 |

Titanium alloy treated, Ti-15Zr, d = 5.00 ± 0.05 mm, L = 33.45 ± 0.05 mm.

Frequency,f0 (Hz) | Torsion modulus,G (GPa) | Rotational Inertia, I (10−4 kg m2) | I (f0)2 (10−4 kg m2 Hz2) |
---|---|---|---|

11.03 ± 0.05 | 26 ± 1 | 100.6 ± 0.7 | 12239 |

7.56 ± 0.05 | 26 ± 1 | 214 ± 1 | 12231 |

5.49 ± 0.05 | 26 ± 1 | 399.6 ± 0.8 | 12044 |

4.76 ± 0.05 | 26 ± 1 | 537 ± 1 | 12167 |

Average:12170 ± 90 |

Thus, Eq. (16) can be used to determine the value of G for the samples of CP-Ti with diameters and/or lengths different to those of the samples used.

The next analysis considered G and the oscillation frequency relative to the experimental values of the rotational inertia used for the CP-Ti sample, with a diameter of 4.40 mm and a length L of 33.45 mm. Eq. (16) determined that G = 35 GPa. Because the product I (f0)2, which depends on d2, presents a linear behaviour and obeys Eq. (17) according to a linear fit, where each fixed value of d shows a constant value, it is possible to evaluate f0 by replacing the values of d = 4.40 mm and I = (100.6, 214, 399.6 or 537) 10−4kg m2. The oscillation frequency values were 9.88, 6.78, 4.96, and 4.28 Hz for increasing I values, respectively. It is assumed that all CP-Ti samples were homogeneous and had the same internal structure.

In a similar manner, the values G and f0 were found using Eqs. (16) and (17) for L = 24.40 mm and d = 5.00 mm, since the experimental values obtained for G for the forged Ti-15Zr sample refer to the dimensions L = 24.40 mm and d = 5.00 mm. These values were G = 22 GPa with oscillation frequencies of f0 = 11.84, 8.12, 5.93, or 5.12 Hz, for increasing I values, respectively. All of these values allow a comparison between the values of the torsion modules for Ti alloys and CP-Ti. Thus, it is possible to quantify the effect of the mass percentage of Zr and the type of thermo-mechanical treatment on the G values of Ti alloys compared to CP-Ti, see Table 13.

Cell parameters of the samples.

Sample | a (Å) | c (Å) | c/a |
---|---|---|---|

CP-Ti | 2.9501 (2) | 4.6832 (3) | 1.5875 |

Forged Ti-5Zr | 2.9616 (2) | 4.6964 (4) | 1.5856 |

Treated Ti-5Zr | 2.9618 (3) | 4.6965 (4) | 1.5858 |

Forged Ti-10Zr | 2.9697 (4) | 4.7054 (5) | 1.5847 |

Treated Ti-10Zr | 2.9701 (3) | 4.7099 (5) | 1.5858 |

Forged Ti-15Zr | 2.9808 (3) | 4.7329 (5) | 1.5844 |

Treated Ti-15Zr | 2.9812 (3) | 4.7330 (5) | 1.5876 |

Fig. 4 shows optical micrographs of forged Ti-Zr-based alloys, where the presence of small acicular structures of martensitic α’ phase can be observed. The needle-like structures were similar to those found in earlier studies [11]. These microstructures are similar to the lamellar structures of alpha in the as-received CP-Ti sample, as reported in an earlier study by the group [28]. However, the width of those lamellar structures is larger than the Ti-xZr alloys, which is typical of Ti martensitic phases. The mechanical process resulted in deformations in the size and morphological heterogeneities of the needles [4]. Furthermore, it is possible to observe a gradual increase in the quantity of needles with composition, which corresponds to a change from a lamellar morphology (α phase) to an acicular (α’ phase) morphology. The microstructure indicates that Zr content decreases the Ms temperature, according to previous reports [9,11]. These findings were confirmed by the XRD patterns (Fig. 10), which showed a characteristic diffraction peak shift toward low degrees, indicating a deformation of the hcp crystalline structure [11].

Figs. 5 and 6 were constructed based on the results of Tables 3–5. Fig. 5 shows a different behaviour than that in Fig. 2. The Ti-5Zr and Ti-10Zr samples had the same diameter (d = 4.40 mm) and length (L = 33.45 mm), but the f0 value increased for the same rotational inertia. Fig. 4 shows that the microstructures are different for these samples. The change from the lamellar α structure to acicular α’ needles and work-hardening could have led to the anisotropy of the forged alloys [2].

In terms of Eq. (15), if d is the same for the Ti-5Zr and Ti-10Zr samples, this implies that JC also has the same value. Since L/θ is fixed in the experiment, then what is being modified is the resistive torque T. As we saw in Section 3.2.1, with an increase in the value of T, the frequency f0 increases as well. This can be seen in Fig. 5. The value of the torsion modulus with a higher frequency, according to Eq. (7), will present a larger G value in this case, see Fig. 6.

According to Fig. 5, the behaviour of the Ti-15Zr sample was similar to that of CP-Ti in Fig. 2, although L was slightly smaller than that of the other two samples (Ti-5Zr and Ti-10Zr). The f0 value increased for the same value of inertia.

Fig. 6 shows that, for the same d of 4.40 mm, the Ti-5Zr and Ti-10Zr samples had increasing values of G with increasing amounts of Zr. The change from the lamellar α structure to acicular α’ needles altered the torsion modulus of these titanium alloys, as shown in Fig. 4.

For the Ti-15Zr sample, because L was slightly smaller than the other two samples but d was increased to 5.00 mm, G was expected to be smaller according to Eq. (7). However, the G value was almost the same as that of the Ti-5Zr samples. An explanation for this result relies on the much larger f0 value for the same rotational inertia, which actually occurred. It is well known that properties of Ti alloys are sensitive to their phases/crystalline structure and certain phases may be stabilised by the addition of alloying elements. Recent investigations [29–32] were aimed at providing such an increase in strength in CP-Ti. Then, the decreasing grain size in CP-Ti leads to significant increases in its hardness and/or strength [33]. In this case, Fig. 4 shows that Ti-15Zr had small needles in the microstructure, which could cause changes in the mechanical properties compared to those of Ti-5Zr and Ti-10Zr. In this case, a more detailed analysis is required, see the discussion in Section 3.2.4 and Table 13. If we look at Eq. (15), JC is larger for the Ti-15Zr sample (d = 5.00 mm; L = 24.40 mm) than the forged Ti-5Zr and Ti-10Zr samples (d = 4.40 mm; L = 33.45 mm). However, L/θ is slightly lower for the forged Ti-15Zr sample. Since the value of G is close to that of the Ti-5Zr forged sample, it is evident that the resistive torque of the Ti-15Zr sample must be much higher than that of the Ti-5Zr sample. This is evidence that the resistance of the Ti-15Zr sample increased in relation to the other two samples, see Fig. 5.

3.2.3Treated Ti-Zr alloysFig. 7 shows the optical micrographs of treated Ti-Zr alloys, while Fig. 10 shows the respective XRD patterns. The heat treatment resulted in an increase in the size of the needles, while the XRD pattern remained unchanged. This indicates a better disposition and stress-relief during mechanical deformation from forging [2,34]. This increase in needle size is reflected in the cell parameters (Table 13), which exhibited a slight increase with the heat treatment.

Figs. 8 and 9 have been constructed based on Tables 10–12. Fig. 8 shows that there is a difference in the frequency of oscillation f0 for the same diameter of 4.40 mm and the same rotational inertia, which demonstrates that the amount of Zr in the Ti-5Zr and Ti-10Zr samples must have altered their torsion modulus. The frequency f0 decreased for the same value of the rotational inertia. However, the opposite behaviour was observed for forged samples of Ti-5Zr and Ti-10Zr, because the frequency of oscillation increased with the amount of Zr. The Ti-15Zr sample presents behaviour very similar to that of the CP-Ti sample, see Fig. 2 for d = 5.00 mm. The frequency f0 increased for the same value of the rotational inertia. In terms of Eq. (15), if d is the same for Ti-5Zr and Ti-10Zr samples, this implies that JC also has the same value. Since L/θ is fixed in the experiment, then what is being modified is the resistive torque T. As seen in Section 3.2.1, with a decrease in the value of T, the frequency f0 also decreases. This fact can be seen in Fig. 8. The value of the torsion modulus (with a smaller frequency according to Eq. (7)), will have a lower G value in this case, see Fig. 9.

In Fig. 9, the quantities of Zr in the Ti-Zr alloys (Ti-5Zr and Ti-10Zr with the same diameter) increased when they were submitted to thermal treatment and the torsion modulus decreased. In these same samples, when compared with CP-Ti, there was an inversion of the values of G. While the forged samples with the same diameter exhibited increased G values with an increase of Zr content, the treated samples exhibited decreased G values with increasing Zr. Treated or forged Ti-15Zr samples exhibited different G values. The torsion modulus of the treated Ti-15Zr sample was found to be lower, as would be expected based on Eq. (7).

Looking at Eq. (15), the resistive torque T decreased because the diameter increased and L/θ remained the same, while G decreased (see Fig. 9). This can also be seen in Fig. 8, the oscillation frequency f0 is lower than in the forged Ti-15Zr sample. The G value is also lower in relation to the Ti-5Zr and Ti-10Zr alloys of the same diameter, as seen in Fig. 9. These results are consistent with Eq. (7).

3.2.4Effect of Zr content and thermos-mechanical treatments on the torsion modulus of Ti-Zr alloysTable 14 summarises the differences between the Ti-Zr alloys and CP-Ti when they are submitted to thermo-mechanical processes. This analysis relied upon the results presented in Section 3.2.1.

Evaluation of the quantity of Zr and the type of thermo mechanical treatment on the torsion modulus of alloys compared to CP-Ti.

Ti-xZr alloysForged | G (GPa) | d (mm) | L (mm) | GTi-xZr / GCp-Ti Forged |
---|---|---|---|---|

Ti-5Zr | 26.8 ± 1 | 4.40 | 33.45 | 0.77 |

Ti-10Zr | 29 ± 1 | 4.40 | 33.45 | 0.83 |

Ti-15Zr | 27 ± 1 | 5.00 | 24.50 | 1.23 |

Ti-xZr alloysTreated | G (GPa) | d (mm) | L (mm) | GTi-xZr / GCp-Ti Treated |
---|---|---|---|---|

Ti-5Zr | 34.8 ± 2 | 4.40 | 33.45 | 0.99 |

Ti-10Zr | 32 ± 2 | 4.40 | 33.45 | 0.91 |

Ti-15Zr | 26 ± 1 | 5.00 | 33.45 | 0.84 |

CP-Ti | G(GPa) | d(mm) | L(mm) | |
---|---|---|---|---|

CP-Ti | 35 ± 2 | 4.40 | 33.45 | Note: It was used Eq. (16) in the calculation of G for CP-Ti using the same dimensions of samples of the |

CP-Ti | 22 ± 2 | 5.00 | 24.50 | Ti-xZr alloys. |

CP-Ti | 31 ± 1 | 5.00 | 33.45 |

The forged samples showed increased G values and, in the case of Ti-15Zr, this value was greater than the value of G for CP-Ti with equal dimensions. From the micrographs in Fig. 4, Ti-5Zr and Ti-10Zr differ very little, which explains the very similar effects of Zr on these alloys. The Ti-15Zr sample showed a microstructure composed of thin, needle-like structures, justifying a greater value of G than that of CP-Ti. The G values for Ti-5Zr, Ti-10Zr, and Ti-15Zr were 77%, 83%, and 123% of the value of G for CP-Ti, respectively.

For the treated samples, G values decreased with the amount of Zr present but they were always lower than the G of CP-Ti. The micrographs in Fig. 6 show that the Ti-5Zr sample was more similar to CP-Ti, while Ti-10Zr and Ti-15Zr were very similar to each other and showed an increase in needles due to the thermal treatment, which should decrease the value of G. The G values for Ti-5Zr, Ti-10Zr, and Ti-15Zr alloys were 99%, 91%, and 84% of the G value of the CP-Ti sample, respectively. There was a very similar relationship between the G values for samples of forged Ti-10Zr and treated Ti-15Zr, and they had similar micrographs.

4ConclusionThe results presented in this paper, the techniques used and the interpretation of the torsion modulus of biomedical Ti alloys have allowed us to clarify some points and to open possibilities for studying other titanium alloys. The torsion modulus of Ti-Zr alloys depended on the composition and thermo-mechanical treatments. Forging and heat treatment affected the morphology of the microstructure, leading to deformation and anisotropy in the alloys. The forged samples showed increasing G values with increasing Zr content relative to CP-Ti, while the opposite behaviour was observed for treated samples.

The authors acknowledge the Brazilian funding agencies FAPESP (grant #2015/00851-6; #2010/20440-7 and #2007/04094-9), CNPq (grant #107808/2008-0, #115545/2009-3 and #115963/2010-3) and Capes (process number: BEX 6571/14-0).