It has been well documented that solid state reactions via mechanical alloying (MA) can take place due to the negative heat of mixing and these reactions proceed via the interdiffusion of the components into thin multilayers, and a large amount of lattice defects and interfaces introduced during milling may be conducive to the solid state reactions [12]. So it has been reported that, by MA methods, metastable Ni_{3}C can be obtained in Ni–graphite powder mixtures^{24}, NbC in Cu-Nb-graphite mixtures^{25}, and TiC in Ti–n-heptane mixtures^{12}. Thus, the process of formation of TiC in the Ti–Nb alloy powders during ball milling and subsequent annealing in the present work can be described as follows: In the initial period of mechanical alloying, SA melts and evenly adsorbs at the surfaces of the Ti and Nb powders, with many defects introduced by ball milling. With ball-milling time increasing, the size of the powders decreases and Nb gradually dissolves into Ti to form a β–Ti–Nb solid solution. Most of the SA molecules enrich the grain boundary. In the subsequent annealing process, the Ti–Nb alloy powders with nano-size particles and many defects easily react with the SA molecules, and the possible reactions are as follows:

The corresponding Gibbs energy for the three possible reactions at different temperatures can be calculated according to

$$\mathrm{\Delta}{G}_{f,B}\left(T\right)={G}_{B}\left(T\right)-\left|\sum {v}_{{\epsilon}_{i}}\right|{G}_{{\epsilon}_{i}}(\mathit{T})\phantom{\rule{1em}{0ex}}\left(\mathrm{KJ}/\mathrm{mol}\right),$$

(4)

where ΔG_{f,B
}(*T*) is Gibbs energy of reaction for the formation of *B* from the elements *ε*_{i
}, G_{B
}(*T*) is Gibbs energy of *B*,

is Gibbs energy of the element *ε*_{i
}, which are listed in Table 1^{26}. It is clear that the reaction (1) yield more negative Gibbs free energy change, which implies it is easier to occur and the reaction products of TiC are more stable than NbC. So, we can only observe the presence of the TiC particles in the annealed and as-sintered samples. As for the TiH_{x} generated during low-temperature annealing, it desorbs hydrogen at further elevated temperatures under vacuum^{13}.

The TiC–β–Ti–Nb composites prepared in the present work exhibits the full dense and the ultrafine microstructures. It may be attributed to the high sintering pressure and the presence of the nano-scale TiC particles *in-situ* produced in the annealing and sintering process. It is well documented the applied high pressure during sintering could provide extra driving force for densification, promote nucleation, and reduce the overall growth rate of grains^{19}. Moreover, the fine TiC particles replace part of the grain boundaries, which may act as the barriers of the grain growth^{27}. According to the Eq. 5^{28},

where λ_{m
}, *r*_{p
} and *f* correspond to the distance apart from the reinforcements, the radius of the particles and the fractional volume of reinforcements, respectively. It can be deduced that the grain size of matrix decreases with the increase of the fractional volume of reinforcements, which is evidenced by the grain sizes of the samples with 35, and 45% TiC particles.

The TiC–β–Ti–Nb composites prepared in the present work exhibit the ultrahigh strength, which reaches 1990 MPa for the sample with 45 vol.% TiC particles, and it arises from the grain refinement and the reinforcement of the TiC particles. So, a modified rule-of-mixture (ROM)^{29} can be described as

$${\sigma}_{c}=({\sigma}_{m}+\mathrm{\Delta}{\sigma}_{\mathit{gr}}{)V}_{m}+{\sigma}_{\mathit{TiC}}{V}_{\mathit{TiC}}$$

(6)

where *σ*_{c
} is the strength of the TiC–β–Ti–Nb composites, *σ*_{m
} is the strength of β–Ti–Nb matrix, Δ*σ*_{gr
} is the strength increment arising from the grain refinement, *σ*_{TiC
} is the strength of TiC particles, *V*_{m
} is the volume fraction of β–Ti–Nb matrix, and *V*_{TiC
} is the volume fraction of TiC particles. The strength contributed by the grain refinement can be estimated by Hall-Patch relationship^{30},

where *τ* is the yield stress, *τ*_{0} is the friction stress needed to move individual dislocations, *k* is a constant, and *d* is the average grain size.

Assuming *d*_{o
} is the average grain size of β–Ti–Nb matrix without TiC particles and *d*_{gr
} is average grain size of β–Ti–Nb matrix refined by TiC particles. The yield stress of the β–Ti–Nb matrix without TiC particles, *τ*_{o
}, can be described as

and the yield stress of β–Ti–Nb matrix refined by TiC particles

Therefore, the β–Ti–Nb matrix strength increment contributed by the grain refinement can be estimated by

$$\mathrm{\Delta}{\sigma}_{\mathit{gr}}={\tau}_{\mathit{gr}}-{\tau}_{o}=\mathit{k}\left({d}_{\mathit{gr}}^{-1/2}-{d}_{o}^{-1/2}\right)$$

(10)

where *k = *0.4 MN/m^{3/2 }^{31}. Thus, as the grain size of the matrix decrease from 6 μm to 1.2 μm and 0.6 μm, the increments of the strength are estimated to be about 202 MPa and 353 MPa, according to Eq. 10.

Assuming *σ*_{TiC
} = 2600 MPa for C/Ti = 0.6~0.8^{32} (averaging the strength of TiC), we can estimated the strength for the composites with 35 vol.% and 45 vol.% TiC particles to be about 1756 MPa and 1969 MPa, according to Eqs 6 and 10, respectively. As the result, the theoretically predicted yield strength of the as-sintered samples considering the grain refinement and the second phase reinforcement can be obtained, which are listed in Table 2, in comparison with the experimental results. It is clear that the theoretical results are in a good agreement with the experimental ones.