The Temperature Dependences of Elastic Modulus of Composites Lab6-Meb2

A quasi-binary eutectic composite is a system of two components in the absence of their mutual solubility. In eutectic systems, the boundary contact between the components may be coherent, semi coherent or amorphous depending on the types and structures ratio parameters of their crystal lattices. If you know the general principle of calculating of the theoretical strength under uniaxial tension through the energy of pure components, the problem of the same calculation for composite is rather ambiguous and complex, which is associated with the influence of the boundaries of the components.


Introduction
A quasi-binary eutectic composite is a system of two components in the absence of their mutual solubility. In eutectic systems, the boundary contact between the components may be coherent, semi coherent or amorphous depending on the types and structures ratio parameters of their crystal lattices. If you know the general principle of calculating of the theoretical strength under uniaxial tension through the energy of pure components, the problem of the same calculation for composite is rather ambiguous and complex, which is associated with the influence of the boundaries of the components. This paper discusses the mechanical properties such as strength, elastic modulus, maximum strain, as well as the surface energy of components contacts in composite as a function of temperature. All of these characteristics are determined by the energy of interaction between the elements in composite. The object of study is Quasi-binary eutectic composites LaB6-MeB2 (Me-Ti, Zr, Hf). Mutual insolubility components of these systems are confirmed by quantum-mechanical calculations and are presented in [1].

Research Methods, Numerical Simulation and Discussion of Results
The method of determining the total energy of the alloy per molecule within the pseudo potential method is based on the summation of the potentials of paired intermolecular interactions [2]. If the concentration of component A is denoted C, the energy of system from two components can be written 2 2 (1 ) 2 (1 ) Here, Where, Ф АА , Ф ВВ , Ф АВ -potentials of interactions between these components [2], and , , -the energy of interaction between the components А-А, В-В and A-B.

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The first component has a connection of A-A type, and the second component connection B-B, at the interface -only A-B connection. Equation (1) can be written as Where, Tension along the z axis is determined from the relation [3] ( ) 1 Where, ez -relative deformation, and Uii is a UAA or UBB, d -lattice parameter in the direction of the axis of deformation, Si -atomic plane area in a crystal lattice that is perpendicular to the axis of strain. For calculating the formulas based on strength (6,7) there is the following relationship: The paper considers in equal deformed state condition in which the permissible deformation of the composite in tension coincides with the maximum deformation of the reinforcing fibers. In equal deformed state, according to the rule of mixtures, excluding the impact of joining the borders of the two components, the maximum strength of the quasi-binary system will be max max ( ) Here, .
With a very small strain the curve stress -strain is close to a linear law, and then: and taking into account the phase of interaction ( Figure 1) ; ; For small values of deformation the calculated values of the stress and the elastic modulus of LaB6-TiB2 composite (at zero temperature) are shown (Table I). To identify the mechanical characteristics depending on the temperature, it is necessary to be able to calculate the energy of the electron-ion system of the materials at different temperatures. In the method of pseudo potentials, it means to find a change in the volume of the unit cells, at temperatures different from zero, ie, obtain the explicit dependence of the total energy of the lattice parameters and volume at nonzero temperature.
Note that the calculation of the energy of the electronion system in the second-order of perturbation theory in the pseudo potential means using the harmonic approximation. But application of this approaches in the lattice dynamics enough to calculate certain physical characteristics that are associated with a change in crystal lattice volume with increasing temperature. This problem can be avoided if use [5] "quasi-harmonic model" (developed by the authors), which allows to identify the temperature dependence of the unit cell volume in the harmonic approximation. The result is a dependence of the energy of the electron-ion system on temperature through the unit cell volume The final formula for calculating the theoretical strength at uniaxial tensile versus temperature are presented in the form:

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In borides with a high content of boron, the boron atoms form stable complexes structuras (B 6 ), their influence on the physico-mechanical characteristics of the material occurs at high temperatures, which leads to the appearance of peaks or abrupt changes in the physical and mechanical characteristics [6].
It was found that for uniaxial deformation in the temperature range from 0 to 2750K the characteristic exponential dependence of strength on temperature disrupted in a temperature range (1300-2200K) in the case of LaB 6 . In this interval an increase in temperature leads to higher theoretical strength. The same dependence is observed for eutectic systems LaB 6 -MeB 2 (Me-Ti, Zr, Hf), which constitutes mainly from the LaB 6 [1].
Accounting for the interaction energy between heterogeneous elements A and B at the interface of the two phases results in a change of its value and the maximum deformation and hence the maximum strength in the composite. Note that at T=0K for MeB 2 maximum deformation (Me-Ti, Zr, Hf) is approximately ԑmax ≈0, 099 [6]. ). The increase in the maximum deformation of hardener -the result of interactions of the components (phases), which entails an increase in the temporary matrix resistance, tending to its maximum value, and, in general, increases the strength of the composite.
Results of computational experiment on the calculation of the elastic modulus for composite and components, as well as their temperature dependence are presented in Table 2 & 3. The values obtained for the elastic modulus are close to their experimental values within acceptable limits (the maximum relative error of <9%), given that the calculations considered only perfect single crystals, real crystals always have a lower value.   Figure 2 shows the dependences of the relative values of the elastic modulus (E(T)/E(T=0)) and the theoretical strength (σ(T)/ σ(T=0)) on the temperature for LaB 6 . As the temperature rises, the LaB 6 single crystal loses its elastic properties faster than strength. Figure 3 shows almost identical dependences of the relative values of the elasticity modulus on the temperature for LaB 6 and TiB 2 . In both cases, the interval of temperature change is 0≤T≤2750K; E(T=0) and σ(T=0) is the value of the elastic modulus and strength at absolute zero temperature [7][8][9].    If a crystal is subjected to deformation and simultaneously the temperature rises, then in parallel with deformation thermal expansion will operate. At temperatures T>1300K expansion of B 6 complex on the plane (200) counteracts the uniaxial strain (stretch) of the complex in the perpendicular direction, as a result of which there appear anomalies in the graph of the strengthtemperature dependence. At relatively small deformations, in which the elastic modulus is determined, this effect is not observed (Figure 4 & 5) shows the dependence of the modulus of elasticity -maximum strength ratio of the composite on the temperature. In the temperature range of 1000-2500K, an area of the characteristic hardening of a composite is marked out.

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Accounting components interaction in composites at the interface connections increases the value of composite modulus of elasticity. The boundary of the two phases in the composite serves as a redistribution of property area: increasing the plasticity of refractory component and the theoretical strength and elastic modules of the composite as a whole. Up to the melting temperature composites LaB 6 -MeB 2 have high strength and elastic modulus.