On the elastic properties and mechanical damping of Ti₃SiC₂, Ti₃GeC₂, Ti₃Si_0.5Al_0.5C₂ and Ti₂AlC in the 300–1573 K temperature range

Abstract

In this paper we report on the temperature dependencies of Young’s, E, and shear moduli, μ, of polycrystalline Ti₃SiC₂, Ti₂AlC, Ti₃GeC₂ and Ti₃Si_0.5Al_0.5C₂ samples determined by resonant ultrasound spectroscopy in the 300–1573 K temperature range. For the isostructural 312 phases, both the longitudinal and shear sound velocities decrease in the following order: Ti₃SiC₂ > Ti₃Si_0.5Al_0.5C₂ > Ti₃AlC₂ > Ti₃GeC₂. Like other phases in the same family, these solids are relatively stiff and lightweight. The room temperature E values range between 340 and 277 GPa for Ti₂AlC to 340 GPa for Ti₃SiC₂; the corresponding μ values range between 119 and 144 GPa. Poisson’s ratio is around 0.19. Both E and μ decrease linearly and slowly with increasing temperature for all compositions examined. The loss factor, Q⁻¹, is found to be relatively high and a weak function of grain size and temperature up to a critical temperature, after which it increases significantly. Modest (4% strain) pre-deformation of Ti₃SiC₂ at elevated temperatures results in roughly an order of magnitude increase in Q⁻¹ as compared to as-sintered samples, which led us to the conclusion that the damping is due to the interaction of dislocation segments with the ultrasound waves. That Q⁻¹ decreases with increasing strain amplitude is consistent with such an interpretation. The loss factors of the deformed Ti₃SiC₂ sample are orders of magnitude higher than those of typical structural ceramics. The technological implications of having readily machinable solids that have stiffnesses comparable to Si₃N₄ and damping capabilities comparable to some woods are obvious and are discussed.

Introduction

The more than 50 ternary carbides and nitrides with the general formula M_n+1AX_n(MAX) – where n = 1, 2 or 3, M is an early transition metal, A is an A-group element (mostly IIIA and IVA) and X is either C or N – represent a new class of thermodynamically stable nano-laminated solids [1]. These phases with two formula units per unit cell, are layered, with M_n+1X_n layers interleaved with layers comprised of pure hexagonal nets of the A-group element and belong to the space group P63/mmc. The MAX phases have an unusual, and sometimes unique, combination of properties [1], [2]. As a class, these solids are excellent electric and thermal conductors, exceptionally thermal shock resistant and damage tolerant [1], [2], [3], [4]. Despite being elastically quite stiff, they are all readily machinable with nothing more sophisticated than a manual hacksaw.

By now it is fairly well established that these phases deform by a combination of dislocation glide and kink band formation [1], [2], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. Basal, and only basal, slip is operative; the dislocations are overwhelmingly arranged either in pileups on the same basal plane or in walls. Dislocation interactions, other than orthogonal, are difficult, unlikely to occur and have never been observed. Unlike metals the dislocations do not entangle and work hardening in the classic sense, e.g. by the formation of dislocation forests, has also not been observed. This does not imply that hardening does not occur; rather the hardening occurs as a direct result of the formation of kink boundaries, which effectively reduce the domain size [7], [14], [15]. Most importantly, because the dislocations are restricted to the basal planes and do not entangle, they can move back and forth over significant distances, which result in the dissipation of large amounts of energy per loading cycle. The energy dissipated also increases as the square of the applied stress [7], [15].

More recently, it has been postulated that the MAX phases, and any other solid with high (>1.6) c/a ratios, will deform, if at all, by the formation of incipient kink bands (IKBs) and regular kink bands (KBs) [15], [16]. The reason this is deemed a sufficient, but not necessary, condition is simple: with high c/a ratios the lengths of the Burgers vectors of non-basal dislocations render them prohibitively expensive. Furthermore, twinning, other than possibly basal, is not an option [15], [16]. These solids thus have only two mechanisms to relieve the stress: fracture and/or kinking. If the latter occurs, the solids can be best described as kinking nonlinear elastic (KNE) solids. The signature of a KNE solid is the formation of reversible hysteretic stress–strain loops during quasi-static loading–unloading cycles [2], [7], [10], [15], [16]. Macroscopic polycrystalline cylinders of Ti₃SiC₂ can be compressed, at room temperature, to stresses up to 1 GPa, and fully recover upon the removal of the load [7]. Similar behavior is also observed at the single grain level in nanoindentation experiments [13], [16], [17], [18].

It addition to the MAX phases, graphite, mica, sapphire, BN and most probably ice [12], among many others, have been shown to be KNE solids [16], [17], [19]. Furthermore, we have shown that the geologically important class of solids – labeled nonlinear mesoscopic elastic [7], [19], [20], [21] – are also KNE solids [16]. The quasi-static response of these solids at relatively large strains (∼10⁻³) and low frequencies (∼10⁻² Hz) is also characterized by hysteresis and endpoint memory [20], [21]. To date, this response has been modeled phenomenologically by invoking the presence of hysteretic mesoscopic units, whose physical underpinnings were until recently unknown [20], [21]. However, we have shown that hysteretic mesoscopic units are nothing but dislocation-based IKBs [16].

In general, the mechanical response of KNE solids is explained by invoking the presence of three interrelated elements that manifest themselves at ever increasing loads: IKBs, mobile dislocation walls and KBs. The IKBs are comprised of two, near parallel, dislocation walls of opposite signs that are attracted to each other and are fully reversible when the load is removed. At higher stresses, the IKBs dissociate forming sets of parallel, nested, mobile dislocation walls. At even higher stresses, the mobile dislocation walls are swept into kink boundaries – normal to the basal planes – and lose their mobility. The dissociation of an IKB to a pair of mobile dislocation walls has to be associated with delaminations.

The remote shear stress, τ_κ, needed to trigger an IKB is given by [22]

$$\tau \approx \sqrt{\frac{2{\mu }^{2}b\gamma }{\alpha }}$$

where μ, γ and b are, respectively, the shear modulus, kinking angle and Burgers vector. α is the domain size available for the creation of the IKB. The formation of KBs per force reduces α, which in turn, according to Eq. (1), accounts for the hardening observed on repeat load cycling: a characteristic of all KNE solids explored to date [7], [14], [15], [16], [23]. Eq. (1) also explains why the slopes of quasi-static stress–strain curves are strong functions of grain size. Given that in polycrystalline solids 2α is equal to the thickness of the grains along the c-axis [15], [23], it follows that IKBs nucleate more easily in coarse-grained samples, which, in turn, results in lower apparent moduli. The slopes of the stress–strain curves for fine-grained samples, in contrast, approach the values of the Young’s moduli measured by ultrasound [24].

The nonlinear elastic behavior of KNE solids, in general, and the MAX phases in particular, makes it difficult to determine accurately their elastic moduli from quasi-static stress–strain curves. This is especially true at elevated temperatures, i.e. above the brittle-to-plastic transition temperature. Thus, the first reliable values for E and μ of Ti₃SiC₂ were obtained by the ultrasonic echo-pulse technique, but only in the 77–300 K temperature range [24]. At room temperature, E_RT and μ_RT of Ti₃SiC₂ were determined to be 339 ± 2 and 142 ± 2 GPa, respectively; Poisson’s ratio was 0.2. Both moduli were found to be relatively weak functions of temperature with a plateau at temperatures below ∼125 K. Radovic et al. [10] and Bao et al. [25] determined E of Ti₃SiC₂ in tensile and three-point bend tests, respectively. Both reported that the elastic moduli decreased slightly with temperature in the 300–1273 K temperature range. A dramatic decline in moduli was observed above ∼1273 K.

Most recently [19], we reported on the elastic properties of Ti₃SiC₂ determined using resonant ultrasound spectroscopy (RUS) in the 300–1573 K temperature range. RUS is a relatively novel technique for determining the complete set of elastic constants of a crystal by measuring its free-body resonances [26], [27], [28], [29]. In addition to the elastic constants, RUS allowed us to study the nonlinear elastic behavior, i.e. mechanical damping, of Ti₃SiC₂ under dynamic conditions as a function of temperature, grain size and deformation history. We found that while the elastic moduli decreased linearly with temperature up to 1573 K, the mechanical damping, Q⁻¹, was constant up to the brittle-to-plastic transition temperature, above which it increased dramatically. Most importantly, the downshift in the resonant frequencies of that is typical of nonlinear mesoscopic elastic solids [20], [21], [30], [31] was observed at room temperature when Ti₃SiC₂ was exposed to the increasing driving voltage (i.e. strain amplitude) in RUS. Under dynamic conditions Q⁻¹ was also an order of magnitude lower than that obtained from quasi-static conditions and dependent on frequency, ω, and loading history, but not on grain size [19]. The nonlinear behavior of Ti₃SiC₂ under dynamic conditions was thus attributed to the interaction of preexisting dislocation line segments with the ultrasonic waves and not to the formation and annihilation of IKBs as in quasi-static conditions [19].

In this paper we report on the elastic and damping properties of Ti₃SiC₂, Ti₂AlC, Ti₃GeC₂ and Ti₃Si_0.5Al_0.5C₂ in the 300–1573 K temperature range using RUS. The frequency range explored was 20–450 kHz. In the case of Ti₃SiC₂, the effect of grain size and preloading/deformation history on the elastic and damping properties was also examined.