Influence of Thermal Sensitivity of Functionally Graded Materials on Temperature during Braking

The model of the frictional heating process during single braking to determine the temperature of the functionally graded friction elements with an account of the thermal sensitivity of materials was proposed. The basis of this model is the exact solution of the one-dimensional thermal problem of friction during braking with constant deceleration. The formulas approximating the experimental data of the temperature dependencies of properties of the functionally graded materials (FGMs) were involved in the model to improve the accuracy of the achieved results. A comparative analysis was performed for data obtained for temperature-dependent FGMs and the corresponding data, calculated without consideration of thermal sensitivity. The results revealed that the assumption of thermal stability of FGMs during braking may cause a significant overestimation of temperature of the friction pair elements.


Introduction
During intensive braking, the volume temperature of the disc braking system may be higher than 450 • C [1] and the maximum temperature on the friction surfaces of the pad and the disc during single braking may even reach a level above 1000 • C [2]. In such severe conditions, the thermal and mechanical properties of materials may highly differ from the initial, reached at the ambient temperature. Therefore, in order to improve the theoretical analysis of the thermoelastic behavior of the braking systems, it is necessary to develop mathematical models taking into consideration the thermal sensitivity of friction materials. However, the introduction of the temperature-dependent properties in formulation of the thermal problems of friction leads to nonlinearity, so most of the published analyses have been performed using numerical methods, especially the finite element method [3,4]. One of the alternative techniques used to develop such nonlinear models of frictional heating is linearization by means of the Kirchhoff substitution [5]. This method relies on the reduction of the originally nonlinear heat conduction equation to the linear one. However, it works this way only for materials with simple nonlinearity, which means that their thermal conductivity and specific heat capacity are temperature-dependent, but the thermal diffusivity remains constant [6]. For materials with arbitrary nonlinearity, only the partial linearization by the Kirchhoff substitution of such a problem is possible; as a result, another nonlinear problem is obtained for which the method of solving is known [7]. The Kirchhoff transform has a similar effect in the heat conduction problems formulated for solids with simple nonlinear thermosensitivity under complex heat exchange. Some analytic-numerical methods for the solution of such problems have been proposed in the study [8]. Another technique to take into consideration the thermal sensitivity of materials is the method of successive approximations (iterations), in which the solution of the corresponding linear problem is adopted as the initial approximation, and then the solution found in the previous step is corrected. An iteration algorithm to solve the one-dimensional problem method. Additionally, the crack propagation path was predicted by introducing the fracture mechanics analysis. It was concluded that the proper selection of an FGM gradient can lead to a significant decrease in thermal stresses [16]. A transient thermoelastic behavior of the functionally graded plate with temperature-dependent properties due to a thermal shock was considered in the paper [17]. The temperature and thermal stress distributions in the Cu-W functionally graded composite were found by means of the semi-analytical micromechanical model.
The aim of this study was to investigate the influence of FGMs thermal sensitivity on the distribution of temperature in a disc brake system. This study is a continuation of our previous articles [11,18], which concern the transient thermal problem of friction under uniform sliding and during single braking with an exponential increase in the contact pressure. Due to the appearance of a high temperature level, there is a demand to improve the results by involving the variations of material properties dependent on the actual temperature, since the thermal sensitivity effect is particularly manifested in a high temperature range. In this article, the braking with constant deceleration is considered, when the nominal pressure is reached immediately at the beginning of the process, since the increase in the time of contact pressure growth causes a drop in the achieved temperature.

Statement to the Problem
To develop an analytical model of frictional heating process in the braking system, the following assumptions were taken into account: 1.
The braking process with constant deceleration is considered; 2.
At the initial time moment, the temperature of a brake is equal to the ambient temperature T a ; 3.
In the heat conduction equation, only the change in temperature gradient in the perpendicular direction to the disc-pad contact surfaces is taken into consideration; 4.
The thermal contact on the friction surfaces is perfect, i.e., the temperatures of its contact surfaces are equal, and the sum of frictional heat fluxes intensities, acting along the normal direction to the contact surface to the insides of the elements equal to the specific friction power; 5.
Due to the symmetry of the system with respect to the mid plane of the disc, when determining the brake temperature, the contact of one pad and a disc with half of its thickness is considered; 6.
The pads and the disc are made of two-component thermally sensitive functionally graded materials, in such a way that their friction surfaces are materials with low thermal conductivity (i.e., cermet), while the core materials are characterized by higher thermal conductivity (titanium alloys, aluminum, etc.); 7.
The thermal conductivity of the disc and pads materials increases exponentially with the distance from the contact surface; 8.
The whole initial kinetic energy of the vehicle is transformed into heat during braking, neglecting the small part of energy associated with wear on the contact surfaces of the disc and pads; Based on the assumptions (1)- (5), in order to determine the temperature of the disc-pad system, the scheme of sliding with linearly decreasing velocity of two semi-spaces z ≥ 0 (disc) and z ≤ 0 (pad) has been adopted. Initiated by the frictional heating temperature field of such a system at a given time instant t ≥ 0 depends only on the distance from the friction surface in a perpendicular direction-independent variable z: T = T(z, t).
According to the assumption (6), the thermophysical properties of a friction pair are functions of temperature T: where K l,m , c l,m and ρ l,m -thermal conductivity, specific heat capacity and density of the first (m = 1) and second (m = 2) component of the materials of the disc (l = 1) and pad (l = 2), respectively. Corresponding values at the initial system temperature T = T 0 are marked as follows: According to the mixture law, the effective specific heat capacities and densities were also determined: where V c , V ρ -volume fractions of the phases c l,m , l = 1, 2, m = 1, 2, respectively. Based on the assumption (7), the effective thermal conductivities K l , l = 1, 2 of the disc and pad materials were established from the equations: where and t s -stop time, and parameters a l , l = 1, 2 (6) are the thicknesses of the subsurface layers actively participating in heat absorption in the disc and pads, respectively (the so-called effective depth of heat transfer [19]). During braking with constant deceleration, the specific friction power decreases linearly from the nominal value q 0 to zero [20]: where A a -nominal area of the contact between the pad and the disc; f -friction coefficient; p 0 -nominal pressure; Q 0 -nominal friction power; 0 ≤ β ≤ 2π-nominal friction power; -cover angle of the pad; R i and R e -respectively, the internal and external radii of the pads; V 0 , W 0 -the initial velocity and kinetic energy of the system, respectively. The latter, according to assumption (8), is equal to the total work of friction. In order to solve the above-formulated nonlinear problem, we will use the idea of adapting an appropriate solution of the linear problem of thermal friction. This approach in the case of homogeneous materials was used in the studies [9,21].

Solution with Temperature-Independent FGMs Properties
The key element of the proposed approach is the precise solution of the linear thermal problem of friction during braking with constant deceleration. In the case of FGMs, such a solution for the above-adopted scheme of two sliding semi-spaces for the specific friction power q(t) (8) and (9) can be written in the form [18]: where µ n > 0, n = 1, 2, 3, . . ., are the real roots of the functional equation: J k (x)-are the Bessel functions of the first kind of the kth order [22]. The temperature of the friction surfaces of both elements, in accordance with the assumption (4) of their perfect thermal contact of friction, should be the same. Substituting z = 0 in Equations (10)-(12) and (14), the following were obtained: whereφ Dimensionless variables and parameters were introduced: where parameters a and q 0 were determined accordingly from Formulas (6) and (8). Taking into account the indications (22) in Formulas (11)- (14), the dimensionless temperature rise of the friction pair elements can be presented in the form: where: and the remaining functions as well as parameters are given by Formulas (15)- (18). Substituting ζ = 0 in Formulas (23)- (25), the dimensionless rise of the temperature on the friction surfaces was obtained: Based on Fourier's law, the intensity of heat fluxes directed along the normal to the contact surface z = 0 towards the insides of the friction pair elements were defined: Taking into account the indications (22) dimensionless intensities of heat fluxes q * l = q l q −1 0 , l = 1, 2 were written as: After differentiating the solution (23)-(26) with respect to the variable ζ and subsequent substitution of the found derivatives to the right side of Formula (30), the following was found: where: and functions Ψ(µ n ) and G n (τ) can be determined from Equations (15) and (26), respectively. It should be noted that in the case of homogeneous materials ( γ i → 0, i = 1, 2 ) of the disc and pads, the dimensionless temperature rise during braking with a constant deceleration has the form [23]: (33) and (34), the known solution of Fazekas was obtained [24]:

Volume Temperature
With the given input parameters, solutions (19)- (27) make it possible to find the spacetime distribution of the temperature inside and its evolution on the friction surfaces of the pad and disc, made of thermally insensitive FGMs. In order to take into account the thermal sensitivity of materials determining the temperature of the braking system using  (1) and (2) for the volume temperature of the pad and the disc during braking [2,9]: where:θ a l -the effective depths of heat penetration (6), Based on Formulas (30)-(32), the heat partition ratio was calculated from the formula:

Numerical Analysis
The calculations were performed for the friction pair, one element of which was made of aluminium oxide Al 2 O 3 (friction surface) and cooper Cu (core) [25]. The friction surface and core of the second element are manufactured of zirconium dioxide ZrO 2 and titanium alloy Ti-6Al-4V [14]. The temperature-dependent properties of these materials are as follows: Cu [17,29] K 1,2 (T) = 31.985 + 0.0099 ZrO 2 [27,30,31] Ti-6Al-4V [32,33] K 2,2 (T) = 6.6926 + 8.9177 · 10 −3 T + 6.8432 · 10 −6 T 2 ,           The calculations were performed according to the following scheme: (1) the values of the input parameters were given (Table 1), and then from Equations (8) and (9)  , specific fric- The calculations were performed according to the following scheme: (1) the values of the input parameters were given (Table 1), and then from Equations (8) and (9) the area of the nominal contact was calculated A a = 0.0022 m 2 , specific friction power q 0 = 3.87 MW m −2 , friction power Q 0 = 8510 W and stop time t s = 12.1 s; (2) using the dependencies (40)-(51) the materials properties K   l , the effective depths of heat penetration a l and the dimensionless gradient parameters of materials γ * l , l = 1, 2 were found from Equations (3) and (5)-(7). Then, the dimensionless parameters K ε and γ ε were determined from the Formulas (16) and (17), and also the weight G l and heat partition ratios α l , l = 1, 2 were calculated from the Equations (38) and (39) ( Table 3); Table 3. Calculated parameters at the initial temperature T 0 . l,m , l, m = 1, 2 corresponding to the volume temperature ϑ l were established (Table 4) and other parameters necessary to perform the calculations (Table 5); Table 4. Material properties at volume temperature ϑ l , l = 1, 2.

Element Index
Changes in the dimensionless temperature rise Θ * (ζ, τ) during braking, at few selected distances from the contact surface are presented in Figure 4. The temperature calculated with an account of the thermal sensitivity of the materials (solid lines) is significantly lower in both friction elements compared to the results achieved without taking into account the temperature dependencies of FGMs properties (dashed lines). The maximum dimensionless temperature on the contact surface ζ = 0 without and taking into account the thermal sensitivity of the materials are 0.816 and 0.277, respectively (reduction of about 2.94 times) and are reached at the time moments τ max = 0.37 and τ max = 0.29 (reduction of 21.6%).
Increasing the distance from the contact surface ζ = 0, the temperature level of both elements drops ( Figure 5). The temperature of components made of thermally sensitive materials is lower than their temperature, found for the constant material properties. The greatest difference between these results is on the contact surface.  Increasing the distance from the contact surface 0 = ζ , the temperature level of both elements drops ( Figure 5). The temperature of components made of thermally sensitive materials is lower than their temperature, found for the constant material properties. The greatest difference between these results is on the contact surface. . It can be seen that the effective depth of heat transfer is much greater in the case that material properties remain unchanged under the influence of temperature, than in the case of considering the thermally sensitive FGMs. This effect is most noticeable for the first one ( ) 1 = l , the Al2O3-Cu element. This result is also confirmed by the parameter values l a , 2 , 1 = l presented in Tables 3 and 5.  Figure 6. It shows the dimensionless temperature isotherms Θ * (ζ, τ). It can be seen that the effective depth of heat transfer is much greater in the case that material properties remain unchanged under the influence of temperature, than in the case of considering the thermally sensitive FGMs. This effect is most noticeable for the first one (l = 1), the Al 2 O 3 -Cu element. This result is also confirmed by the parameter values a l , l = 1, 2 presented in Tables 3 and 5. ) , . It can be seen that the effective depth of heat transfer is much greater in the case that material properties remain unchanged under the influence of temperature, than in the case of considering the thermally sensitive FGMs. This effect is most noticeable for the first one ( ) 1 = l , the Al2O3-Cu element. This result is also confirmed by the parameter values l a , 2 , 1 = l presented in Tables 3 and 5.  The time profiles of the dimensionless intensities of heat fluxes q * l (τ), l = 1, 2 are shown in Figure 7. They decrease linearly during the braking process from the maximum value at the initial moment to zero at the stop. Most of the frictional heat generated is absorbed by the first element (l = 1) Al 2 O 3 -Cu. The linear change in q * l (τ) is the result of the specific friction power q * (τ) (8), which decreases linearly during braking with a constant deceleration, and the requirement to meet the boundary condition q * 1 (τ) + q * 2 (τ) = q * (τ), 0 ≤ τ ≤ τ s . The influence of thermal sensitivity on the intensity of heat fluxes is much smaller than on the temperature. For thermally sensitive materials, the maximum values of the intensity of heat fluxes are q * 1,max = 0.864 and q * 2,max = 0.136, and for constant properties of the materials, we have q * 1,max = 0.895 and q * 2,max = 0.105. the specific friction power ) (τ * q (8), which decreases linearly during braking with a constant deceleration, and the requirement to meet the boundary condition ) . The influence of thermal sensitivity on the intensity of heat fluxes is much smaller than on the temperature.

Conclusions
A calculation scheme was proposed to determine the temperature field of the friction elements of a disc brake, taking into account the changes in the FGMs properties depending on the actual temperature. The main part of the scheme was the adaptation of a linear solution (with temperature-independent material properties) to the thermal problem of friction during braking to thermally sensitive FGMs. A numerical analysis

Conclusions
A calculation scheme was proposed to determine the temperature field of the friction elements of a disc brake, taking into account the changes in the FGMs properties depending on the actual temperature. The main part of the scheme was the adaptation of a linear solution (with temperature-independent material properties) to the thermal problem of friction during braking to thermally sensitive FGMs. A numerical analysis was performed in the case of braking with constant deceleration of elements made of two-component functionally graded materials with exponential variations in thermal conductivities in the axial direction, across the volume of the materials. It was found that: • the influence of thermal sensitivity on the temperature of FGMs may be more significant than in the case of homogeneous materials; • for the selected friction pair, taking into account the thermal sensitivity caused an almost threefold reduction in the maximum temperature in comparison to the appropriate temperature values, found with the same properties of the materials; • the influence of thermal sensitivity on the intensity of heat fluxes directed from the friction surface to the interior of the friction pair elements is insignificant. This means that to estimate the amount of heat absorbed by the individual elements of the friction pair, appropriate solutions to linear problems can be used.
A verification of the developed theoretical model based on empirical results would be advisable. However, no information on this kind of experimental data has been found in the literature. In particular, it concerns the frictional heating of braking systems with friction elements made of thermally sensitive FGMs. Therefore, the verification of the exact solution was obtained carried out by determining from it, in cases of limit parameters, known solutions of other authors for homogeneous materials, which were verified with appropriate experimental data. A new element, significantly differentiating the results of a given article from those published earlier by us, is the incorporation in the model of the possibility of changing the frictional properties of FGMs under temperature influence. This model includes many new elements, such as determining the intensity of heat fluxes to obtain the form of the heat partition ratio, finding the volume temperature of FGMs, developing a calculation algorithm that takes into account the thermal sensitivity of all materials components, etc. We have shown that taking into consideration the thermal sensitivity of materials can significantly reduce the surface temperature contact of the pad and disc. We proposed a theoretical computational model. We hope that it will be verified with the data obtained from other authors' research positions. An indirect confirmation of the correctness of our model is also the time profiles of temperature and heat fluxes obtained on its basis, characteristic for braking with a constant deceleration.
It should be noted that all three of our papers constitute a monothematic cycle of interrelated research. We also want to develop a suitable model for braking systems operating in a short-term, repetitive mode. The problem of lowering the temperature level in such systems is up to date.