Theoretical predictions on temperature‐dependent strength for MAX phases

Abstract

The MAX phases are a group of layered ternary compounds with the general formula M n+1 AX n (where M is an early transition metal; A is a group A element (a subset of groups 13-16); X is carbon or nitrogen; n = 1, 2 or 3). 1,2 MAX phases have also been called as "Cermets", because they combine some merits of ceramics (such as high-temperature mechanical properties, good oxidation resistance, and corrosion resistance), with those of metals (such as thermal shock resistance, damage tolerance and good electrical and thermal conductivity, machinability). 1,3 Due to the remarkable properties of some MAX phases especially high-temperature strength, thermal shock resistance, damage tolerance, and oxidation

Analytical model for strength of MAX phases considering high‐temperature oxidation and plastic deformation

“…Consequently, a relationship among the strength, Young's modulus, strain-hardening exponent, and temperature is developed. The Young modulus of the MAX phase at different temperatures can be predicted by the following theoretical model, which has been verified in our previous study 32 : where E(T 0 ) is the Young modulus at T 0 ; and α(T) and C p (T) are the linear expansion coefficient and specific heat capacity at temperature T, respectively. Thus, the temperature-dependent Young modulus of the MAX phases can be obtained from the existing literature or predicted using Equation (16).…”

“…where T is the ambient temperature, W Heat (T) denotes the heat energy density at temperature T, W d (T) denotes the strain energy density related to the onset of failure of the MAX phase, and ϕ is the coefficient between heat energy and strain energy. Moreover, our previous studies 31,32 have verified that the heat energy is equivalent to the sum of the potential energy between atoms and the kinetic energy owing to the motion of all particles. Thus, W Total can be further expressed as…”

“…Moreover, our previous studies 31,32 have verified that the heat energy is equivalent to the sum of the potential energy between atoms and the kinetic energy owing to the motion of all particles. Thus, W Total can be further expressed as $$$$\begin{equation}{W}_{{\rm{Total}}} = {W}_d\ \left( T \right) + \phi \times \left[ {{E}_k\left( T \right) + {E}_p\left( T \right)} \right]\end{equation}$$$$where E k ( T ) and E p ( T ) denote the temperature‐dependent kinetic energy and potential energy of atoms per unit volume, respectively.…”

“…where σ f (T) is the strength at temperature T. Moreover, based on our previous work, 32 the potential energy of atoms (E p (T)) and the kinetic energy of atoms (E k (T)) per unit volume are respectively expressed as follows:…”

“…Moreover, based on our previous work, 32 the potential energy of atoms ( E p ( T )) and the kinetic energy of atoms ( E k ( T )) per unit volume are respectively expressed as follows: $$$$\begin{equation}{E}_p\ \left( T \right) = \frac{3}{2}\ kNT\ \end{equation}$$$$ $$$$\begin{equation}{E}_k\ \left( T \right) = \frac{3}{2}\ kNT\ \end{equation}$$$$where T denotes the ambient temperature (in Kelvin), k is the Boltzmann constant, and N is the number of atoms per unit volume.…”