1. Introduction
Analysis of trends in the modern industry development indicates that an effective solution to the problem of obtaining specific, often incompatible characteristics in materials is the development and creation of composite materials. Among the composite materials, we can distinguish functional-gradient materials.
In a functional-gradient material (FGM), both the composition and the structure gradually change in volume, which leads to corresponding changes in the properties of the material [
1,
2]. A fairly complete overview of modern trends in the creation of FGM can be found in the works [
3].
In the case of a sharp difference between the chemical compositions of the FGM phases, one can speak of functional bimetallic materials. The concept of functional bimetallic materials (FBM) was proposed in 1984 in Japan as a means of obtaining materials for a thermal barrier [
4]. FBM is an advanced material that can achieve a transition gradient or sudden transition from one material to another for various materials [
2]. In the early stages of FBM production, welding was the main technology for combining dissimilar metals [
5], explosive welding [
6,
7] and laser welding [
8] were particularly successful. Among other technologies, laser additive manufacturing is an ideal technology for producing FBM [
8].
Functional graded structures (FGS) are another type of composite materials that occupy an intermediate position between FGM and FBM. In the study [
9], based on the analysis of technologies for building FGS, two methods of their production from CrNi and Al-powders using additive technologies are compared; these are direct metal deposition (DMD) and selective laser melting (SLM), as presented in
Figure 1.
The LDMD method for FGS manufacturing is schematically presented in
Figure 1 and was proposed earlier [
10]. The layers were formed from Ni (Diamaloy) and Al based powders on a related substrate according to the following strategy: the first two layers were pure NiCr, the next two were 70% of NiCr + 30% of Al, the third pair of layers was 50% of NiCr + 50% of Al, and finally the upper 7th and 8th layers had a ratio of 30% of NiCr + 70% of Al. For the Fe-Al system, such a system was successfully tested in [
11].
FBM and FGS-materials must have strong interlayer bonds, which are preserved during further technological processing and under operating conditions. It is assumed that the material retains its macroscopic continuity up to the initiation of an interphase crack. Delamination is considered as a process of initiation and development of continuity microdefects, leading to the formation of interlayer cracks. Delamination is formed due to a combination of two or three main delamination mechanisms (modes): normal opening mode I (a), sliding shear mode II (b), and scissor shear mode III (c) [
6,
12]. There are numerous models of damage mechanics within the framework of the phenomenological approach [
13,
14,
15,
16], which are applicable for a monolithic material and separate components of layered materials; most of them do not allow assessment of the fracture in the joint zone, as they do not take into consideration the inhomogeneity of thermomechanical properties between different phases interfaces.
It should be mentioned that materials with poor thermal conductivity obtained by fused layer deposition of metal powder are prone to cracking. The first reason for the occurrence of cracks is an increased level of residual stresses, which are formed due to uneven heating during the synthesis of layers, during which the upper layers undergo significant tensile stresses during solidification [
9]. The presence of a certain number of pores and structural defects, from which the development of cracks begins, is the second reason for the tendency to crack formation. Fundamental criteria for the initiation and propagation of fracture can be obtained using the concept of energy balance at the crack front, which, for an equilibrium crack, can be expressed as the equality of the available energy and the energy required to create a unit area of the new crack surface [
17].
Failure analysis is often used during the design phase of composite structures, which requires accurate and reliable determination of material properties. For adhesive joints, these properties are strength parameters and critical energy release (CERR) rate, which is characterized by the toughness of the material. In this case, CERR is the most defining parameter [
12]. It is advisable to determine the criteria that allow one to find CERR as a function of the ratio of modes of the involved separation mechanisms (I, II, III). A number of studies have been devoted to this issue [
12,
18,
19]. It should also be noted that the cohesion law [
12,
20], which is based on the universal law of binding energy proposed by Rose et al., is applicable to the interface of the bimetallic material [
21]. Most macroscopic fracture theories are based on the principles of solid mechanics and classical thermodynamics [
1]. With regard to additive technologies, the existing energy approaches can be expanded if we consider the conditions for the consolidation of a multiphase material, in particular FBM, from the point of view of the thermodynamics of nonequilibrium processes.
2. Theoretical Foundations of the Research Method
The process of additive synthesis of FBM is high-temperature, and energy exchange in a local volume at the interface boundary of a bimetallic compound can be so intense that separation is possible. The purpose of this study is to identify the conditions for the consolidation of phases of a multiphase medium with different thermophysical properties characteristic of bimetallic materials, from the conditions of the balance of the thermal and stress-strain states, as well as phase equilibrium in the interface. Consolidation in this context means the absence of interphase separation under conditions of thermodynamic equilibrium. In this regard, to solve the key problem of finding conditions for the consolidation of a multiphase material from the point of view of thermodynamics, the heat transfer equation at the interface was considered, reflecting the interphase mechanical interaction.
Modeling of the bimetallic compound interface was carried out on the basis of the state analysis determined by the thermodynamics of irreversible processes. A similar approach at the macrolevel was used in [
22] in relation to a medium consisting of deformable grains. All macroscopic processes in a heterogeneous medium were considered by the methods of continuum mechanics using averaged or macroscopic parameters.
As a result, it was possible to obtain the criteria for the consolidation of two phases
and
, dependences (1) and (2), which can be considered as necessary conditions for the formation of a stable adhesive bond from the point of view of thermodynamics:
where
,
are the molar isochoric heat capacities of layers
v and
w,
,
are the molar isobaric heat capacity of the layers,
,
are the linear coefficients of thermal expansion, and
,
are the coefficients inverse to the polytropic indicator.
To find criteria (1) and (2) at the interface boundary, an analytical method was used to determine all thermodynamic quantities included in them. Isobaric and isochoric heat capacities of a pure substance from the composition of each phase were calculated according to Debye’s law of molar heat capacity [
23]:
where
θ is the Debye’s temperature, which is defined as
In Equations (3) and (4),
is Planck’s constant,
k is Boltzmann’s constant,
ν is the vibration frequency of atoms,
x is the parameter, determined on the basis of the solid-state theory [
23], and
T is temperature (all calculations are made for room temperature,
T = 298 K). The characteristic Debye temperatures of substances are known from literature, for example, [
24].
Equation (1) is valid when determining the Debye temperature for a pure substance, and for a substance in a compound (as part of a phase), the Debye temperature is calculated using the Koref’s equation [
24]. According to Koref’s rule, data on the melting points of a compound, and melting points and Debye temperatures of pure substances outside the compound, make it possible to obtain the melting temperatures of these substances in a compound, according to the dependence:
Here
θ*,
θ are the characteristic Debye temperatures of the element in the compound, with other elements of phase and the element outside the compound of phase, and
,
are the melting temperatures of the entire phase and the element outside the compound of the phase. The isochoric heat capacity (
) values are determined from
θ* using the Debye’s equation separately for each phase component. Then, summing them up according to the Neumann-Kopp rule, the isochoric heat capacity of the compound is determined. For the
AlBmDk compound, the isochoric heat capacity can be found from the dependence [
24,
25]:
The recalculation of the isochoric heat capacity to the isobaric heat capacity was carried out according to the Magnus-Lindemann equation [
24]:
where
n is the number of atoms in the compound (
n =
l +
m +
k), and
is the melting point of A
lB
mD
k.
The usual approach to assessing the properties of an FGM material is to apply the rule of mixtures. Although these are not really physical or mathematical rules, these relationships can be used to approximate the thermal or mechanical properties of a composite material in terms of individual properties and relative amounts of components. The simplest is the classical linear rule of mixtures (Voigt’s estimate) for two constituent materials, based on the assumption of uniform strain or stress of the composite structure [
1]. The upper Voigt bound [
26,
27] for the effective coefficient of thermal expansion α is provided by the expression:
where
,
,
are the volumetric concentration, coefficient of thermal expansion (CTE), and modulus of elasticity related to the first component (phase) of the composite substance,
,
,
, to the second component.
According to [
1], two-phase composite material dependences for calculating the CTE are more accurately and experimentally confirmed in works [
28,
29,
30,
31,
32].
Author Contributions
Conceptualization, A.K. and I.S.; methodology, V.S.; formal analysis, Y.E.; investigation, A.A. and V.R.; data curation, V.S. and Y.E.; writing—original draft preparation, A.K.; writing—review and editing, I.S.; visualization, V.R. All authors have read and agreed to the published version of the manuscript.
Funding
This research was funded by the Russian Science Foundation, grant number 20-69-46070.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
Conflicts of Interest
The authors declare no conflict of interest.
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