Wear rate of a TiAl matrix composite containing 10 wt% Ag predicted using the Newton interpolation method

Abstract

The longevity of a mechanical component was evaluated by predicting the wear rate of the material. The primary purpose of this study was to construct three prediction formulas (W₄(F), W₄(n) and W₄(t)) to predict the wear rate at different applied loads F, rotation speeds n and sliding times t based on a TiAl matrix composite containing 10 wt% Ag. Prediction formulas of wear rate, which were constructed by the Newton interpolation method, were evaluated by making a comparison between the testing error and computational error. The results showed that the prediction formulas (W₄(F), W₄(n) and W₄(t)) of wear rate were accepted for smaller computational errors. It was found that the prediction accuracy gradually increased for the higher binding force of wear debris at larger applied loads F (see formula W₄(F)), and slightly decreased for random impacting of wear debris at higher rotation speeds n (see formula W₄(n)), as well as being relatively stabilized for incessant forming of anti-friction films at different sliding times t (see formula W₄(t)).

---

1. Introduction

In recent years, TiAl alloys have been paid extensive attention in order to satisfy the urgent requirements of the aerospace and automotive industries because of their excellent mechanical and thermal behaviors.^1–4 However, TiAl alloys, due to their poor tribological behaviors, have limited further applications in the industrial field.⁵

The tribological behaviors of TiAl alloys were significantly improved by fabricating a TiAl matrix self-lubricating composite containing solid lubricants. Shi et al.⁶ studied the tribological behavior of a TiAl matrix composite containing 10 wt% Ag (TiAl-10 wt% Ag), and found that the friction coefficient and wear rate of TiAl-10 wt% Ag were lower than those of TiAl alloys at temperatures of 25–400 °C. Currently, many mechanical component failures are caused by the excessive wear of materials. Hence, the wear-resistance of materials and the longevity of mechanical components could be evaluated by investigating the wear rate.

Many tribological researchers have made a great effort to obtain the formula of wear rate. Archard⁷ thought that weight loss only happened in the contact regions of the matched surfaces during sliding wear, and defined wear volume G, as shown in formula (1):

G = KL(σ_n/H) (1)

where wear volume G is proportional to the wear coefficient K, sliding distance L and normal stress σ_n, as well as being inversely proportional to the hardness H of the softer material. Kishore et al.⁸ reported that in the initial stage, the wear rate was mainly determined by the applied load, and in the steady stage, the wear rate mainly depended on the synergetic effect of the applied load and sliding velocity. Boneh et al.⁹ found that in the steady stage, the wear rate has a linear relation with the applied load at a low velocity. Kasolang et al.¹⁰ detected that the wear rate W_s was inversely proportional to the sliding distance L, material density ρ and applied load F, and proportional to the weight loss Δm. Wear rate W_s could be defined as formula (2):

Gong et al.¹¹ thought that during dry sliding, wear debris was continually oxidized, resulting in the generation of massive oxides, lowering the weight loss Δm. Consequently, the wear volume V could be adopted to better calculate the wear rate containing the generated oxides. The wear rate W_s was optimized as formula (3):

Currently, the wear rate formula (3) is widely accepted by tribological researchers. However, the wear volume V needs to be measured after each tribological test, leading to the poor wear rate prediction of formula (3), and the waste of energy and material. The test conditions of the applied load F, rotation speed n and sliding time t needed to be chosen before the tribological tests based on the ball-on-disk tribometer. Hence, the wear rate formula (3) would be further optimized as the formula of the applied load F, rotation speed n, sliding time t and cross-section area A of the wear scar, in order to construct the prediction formula of wear rate. The Newton interpolation method was not restrained by the increasing number of interpolation nodes, and the measured values (wear rate W_s) at the different interpolation nodes (W_s, F), (W_s, n) and (W_s, t) were not influenced by the formula structure using the Newton interpolation method. Cui et al.¹² adopted the Newton interpolation method to predict the state of charge, and found that the Newton interpolation method could be successfully adopted to predict the charge state of lead batteries. Hence, the Newton interpolation method was chosen to construct the prediction formula of wear rate. To our best knowledge, few works currently report the prediction formula of wear rate in the form of a Newton interpolation polynomial.

In this study, TiAl-10 wt% Ag, for its excellent tribological behavior, was chosen as the test material. The prediction formula of wear rate was constructed using the Newton interpolation method after three steps. Step 1: formula (3) was optimized as the wear rate formula W_s of the applied load F, rotation speed n, sliding time t and cross-section area A of the wear scar. Step 2: three cross-section area formulas A₄(F), A₄(n) and A₄(t) (A₄(F) between A and F, A₄(n) between A and n, as well as A₄(t) between A and t) were constructed using the Newton interpolation method based on five equidistant interpolation nodes. Step 3: three prediction formulas (W₄(F), W₄(n) and W₄(t)) in the form of the Newton interpolation polynomial were respectively obtained by substituting the formulas A₄(F), A₄(n) or A₄(t) into the optimized formula W_s of wear rate. Three prediction formulas (W₄(F), W₄(n) and W₄(t)) of wear rate were evaluated by making a comparison between the computational error and testing error. The different computational errors were reasonably interpreted by wear mechanisms at different applied loads F, rotation speeds n or sliding times t. The prediction formulas (W₄(F), W₄(n) and W₄(t)) of wear rate could be adopted to evaluate the longevity of important mechanical components by predicting the wear rate at the different applied loads F, rotation speeds n or sliding times t.

2. Experimental details

2.1 Materials

TiAl-10 wt% Ag (48 at% Ti-47 at% Al-2 at% Cr-2 at% Nb-1 at% B-10 wt% Ag), which was fabricated using spark plasma sintering (SPS), was composed of Ti, Al, B, Nb, Cr and Ag. Before being fabricated, the powders of Ti, Al, B, Nb, Cr and Ag were loaded into Teflon vials, and then mixed by vibration milling (45 Hz in frequency). After being mixed and dried, the powders were sealed in the cylindrical graphite molds (25 mm in inner diameter), and sintered using D.R. Sinter® SPS3.20 apparatus (made in Sumitomo Coal & Mining, now SPS Syntex Inc.) for 8 min at the temperature and pressure of 1000 °C and 30 MPa in a pure Ar atmosphere for protection. The heating rate of 100 °C min⁻¹ was chosen.

2.2 Vicker’s microhardness and density

Vicker’s hardness of TiAl-10 wt% Ag was measured by Vicker’s hardness apparatus according to the ASTM standard E92-82.¹³ To attain the mean hardness of 5.97 GPa, ten tests were repeatedly carried out at a load of 10 N. The mean density of 4.22 g cm⁻³ was acquired using Archimedes’ principle in terms of the ASTM Standard B962-08.¹⁴

2.3 Tribological test

To investigate the tribological properties of TiAl-10 wt% Ag, tribological tests were carried out on a ball-on-disk high temperature tribometer of HT-1000 (Zhong Ke Kai Hua Corporation, China) according to the ASTM Standard G99-95.¹⁵ The cylindrical sample with the diameter and height of 25 mm and 8 mm acted as the rotary disk against the stationary Si₃N₄ ball of 6 mm diameter. The districts [1.65 N, 11.65 N], [56 rpm, 504 rpm] and [900 s, 4500 s] were respectively chosen for the applied load F (N (Newton)), rotation speed n (rpm (revolutions per minute)) and sliding time t (s (second)). At 45–65% relative humidity, three tests were repeatedly carried out to obtain the mean wear rate under the different conditions of the tribological tests, as shown in Table 1.

Table 1 Tribological test at different applied load F, rotation speed n or sliding time t

Test conditions	F-392-3600 (N-rpm-s)	9.15-n-3600 (N-rpm-s)	9.15-392-t (N-rpm-s)
1	1.65-392-3600	9.15-56-3600	9.15-392-900
2	4.15-392-3600	9.15-168-3600	9.15-392-1800
3	6.65-392-3600	9.15-280-3600	9.15-392-2700
4	9.15-392-3600	9.15-392-3600	9.15-392-3600
5	11.65-392-3600	9.15-504-3600	9.15-392-4500

---

2.4 Microstructure analysis

TiAl-10 wt% Ag was examined using XRD with Cu Kα radiation at 30 kV and 40 mA at a scanning speed of 0.01° s⁻¹. The cross-section area A of the wear scar was measured using a surface profiler of ST400 (Nanovea). The wear scar morphology was analyzed using an electron probe microanalyzer JAX-8230 (EPMA).

3. Results and discussion

3.1 Compositions of TiAl-10 wt% Ag

Fig. 1 shows the typical XRD pattern of TiAl-10 wt% Ag fabricated using SPS. As is clear in this pattern, it can be concluded from the different diffraction peaks that TiAl-10 wt% Ag was mainly composed of TiAl, TiC and Ag.

Fig. 1 Typical XRD pattern of TiAl-10 wt% Ag fabricated using SPS.

Fig. 2 exhibits the microstructure and elemental distribution of TiAl-10 wt% Ag. It is clearly shown in Fig. 2 that the solid lubricant Ag was uniformly distributed in the TiAl matrix composite.

Fig. 2 Microstructure and elemental distribution of TiAl-10 wt% Ag.

3.2 Optimizing the formula of wear rate

To construct three cross-section area formulas A₄(F), A₄(n) and A₄(t) in the form of the Newton interpolation polynomial, formula (3) needed to be further optimized.

Wear volume V in formula (3) can be written as formula (4):¹⁶

V = AC (4)

Wear volume V was additionally rewritten as formula (5) for C = 2πR:

V = 2AπR (5)

where A is the cross-section area of the wear scar in mm², C is the perimeter of the wear scar in mm and R is the mean radius of the wear scar in mm. The cross-section area A of the wear scar could be measured using the surface profiler ST400. Fig. 3 shows the profile of the wear scar of TiAl-10 wt% Ag. When the measuring stylus of the surface profiler slowly moved across the wear scar along the straight line BB (see Fig. 3b), the cross-section area A of the wear scar (see Fig. 3a) was gradually obtained by recording the coordinate positions of the measuring stylus.

Fig. 3 Profile of the wear scar of TiAl-10 wt% Ag: two-dimensional profile of the wear scar (a) and three-dimensional profile of the wear scar (b).

Sliding distance L in formula (3) was also expressed as formula (6) for v = (πRn)/30:

where v, n and t were the sliding velocity in m s⁻¹, rotation speed in rpm and sliding time in rpm, respectively.

Based on the ball-on-disk tribometer, formula (3) could be optimized as formula (7) by substituting formula (5) and formula (6) into formula (3):

where W_s is the wear rate in mm² N⁻¹ rpm⁻¹ s⁻¹. The units of the cross-section area A, applied load F, rotation speed n and sliding time t were mm², N, rpm and s, respectively. Hence, the unit of wear rate W_s could be expressed as mm² N⁻¹ rpm⁻¹ s⁻¹.

3.3 Wear rate

Fig. 4 shows the mean wear rate at different applied loads F, rotation speeds n and sliding times t. As shown in Fig. 4a, the wear rate continually decreased with the increase of the applied load F. As is clear in Fig. 4b, the wear rate exhibited a tendency to fluctuate upwards with the increase of rotation speed n. It can be seen from Fig. 4c that the wear rate shows a continuous downward tendency with the increase of sliding time t.

Fig. 4 Mean wear rate at the different applied loads F (a), rotation speeds n (b) and sliding times t (c).

In order to evaluate the longevity of the mechanical components at different applied loads F, rotation speeds n and sliding times t, the wear rate could be predicted by constructing three prediction formulas of wear rate. The three cross-section area formulas A₄(F), A₄(n) and A₄(t) needed to be firstly constructed using the Newton interpolation method.

3.4 Definition

The Newton interpolation method was adopted to construct the cross-section area formulas A₄(F), A₄(n) and A₄(t). The prediction formulas W₄(F), W₄(n) and W₄(t) of wear rate were respectively obtained by substituting formulas A₄(F), A₄(n) and A₄(t) into formula (7). Before being applied, the prediction formulas W₄(F), W₄(n) and W₄(t) in the form of the Newton interpolation polynomial needed to be evaluated by making a comparison between the testing error and computational error. Hence, the definitions of Newton interpolation, testing error, computational error and prediction accuracy were required and are given in the following sections.

3.4.1 Definition of Newton interpolation¹⁷. It was supposed that a function y = f(x) was known at the M + 1 points of (x₀, y₀), …, (x_m, y_m), where the values of x_k were spread out in the district [a, b] and satisfied y_k = f(x_k) for k = 0, 1, …, m, namely:

a ≤ x₀ < x₁ < … < x_M ≤ b and y_k = f(x_k) (8)

When p_k(x_k) was equivalent to y = f(x_k) at x = x_k, based on these M + 1 points, the polynomial p_M(x) of m degree was constructed in order to approximate y = f(x) in the district [a, b]. Hence, the polynomial p_M(x) was called the interpolation polynomial. [a, b] was called the interpolation district. (x₀, y₀), …, (x_m, y_m) were known as the interpolation nodes. This method, which was adopted to approximate the function y = f(x), was called the Newton interpolation method.

It was assumed that p_M(x) was the interpolation polynomial of m degree, which could be defined as formula (9):

p_M(x) = a₀ + a₁x + … + a_kx^m (9)

where a₀, a₁, …, a_k were the coefficients of the Newton interpolation polynomial for k = 0, 1, …, m. These M + 1 interpolation nodes of (x₀, y₀), …, (x_m, y_m) could be used to construct the interpolation polynomial p_M(x) of m degree. Formula (9) could be redefined as formula (10) to satisfy formula (11):

p_M(x) = a₀ + a₁(x − x₀) + … + a_m(x − x₀)(x − x₁)…(x − x_m−1) (10)

f(x_j) = p_N(x_j) for j = 0, 1, …, m (11)

where 1, x − x₀, …, (x − x₀), …, (x − x_m−1) were known as primary functions. Coefficient a₀ was the solution of function y = f(x) at x = x₀, namely: y₀ = f(x₀) at x = x₀. Coefficients a₁, a₂, …, a_m can be expressed by forward differences Δf₀/h, Δ²f₀/(2h), …, Δ^mf₀/(m!h^m). h was the step size of equidistant interpolation nodes of x₀, x₁, …, x_m, namely: h = x_k+1 − x_k. The forward differences Δf_k, Δ²f_k, …, Δ^mf_k, which were known as the forward difference of first, second, …, m order, could be defined as formula (12):

When variable x was supposed as x₀ + eh, namely: x = x₀ + eh. e could be written as (x − x₀)/h = (x − x₀)/(x_k+1 − x_k) for h = x_k+1 − x_k, namely: e = (x − x₀)/h = (x − x₀)/(x_k+1 − x_k). Hence, formula (10) was written as formula (13) by substituting formula (12) into formula (10):

In this study, five interpolation nodes were chosen, formula (13) was additionally rewritten as formula (14) for e = (x − x₀)/h = (x − x₀)/(x_k+1 − x_k):

where Δf₀, Δ²f₀, Δ³f₀ and Δ⁴f₀, which were respectively known as the forward differences of the first, second, third and fourth order, could be obtained in Table 2. In order to simplify the application of the Newton interpolation method, based on the software of MATLAB7, a small program was developed to construct the prediction formulas (W₄(F), W₄(n) and W₄(t)) of wear rates.

Table 2 Forward differences for constructing the Newton interpolation polynomial p₄(x)

xk	fk(xk)	Δf	Δ^2f	Δ^3f	Δ^4f
x0	f0(x0)	Δf0 = f1(x1) − f0(x0)	Δ^2f0 = Δf1 − Δf0	Δ^3f0 = Δ^2f1 − Δ^2f0	Δ^4f0 = Δ^3f1 − Δ^3f0
x1	f1(x1)	Δf1 = f2(x2) − f1(x1)	Δ^2f1 = Δf2 − Δf1	Δ^3f1 = Δ^2f2 − Δ^2f1
x2	f2(x2)	Δf2 = f3(x3) − f2(x2)	Δ^2f2 = Δf3 − Δf2
x3	f3(x3)	Δf3 = f4(x4) − f3(x3)
x4	f4(x4)

---

3.4.2 Definitions of the testing error, computational error and prediction accuracy. In order to evaluate the prediction formulas W₄(F), W₄(n) and W₄(t) of the wear rate, a comparison between the testing error and computational error needed to be conducted at different applied loads F, rotation speeds n and sliding times t. The testing error E_test and computational error E_comp were respectively expressed as formulas (15) and (16):where E_test is the testing error, W_test is the testing wear rate and W_mean-test is the mean wear rate of three tests.where E_comp is the computational error and W_comp is the computational wear rate using the wear rate formulas W₄(F), W₄(n) or W₄(t).

The prediction accuracy was improved with the decrease of the computational error. Hence, the prediction accuracy of the prediction formula of the wear rate could be defined as formula (17):

3.5 Applied load F

3.5.1 Constructing the prediction formula W₄(F) of wear rate. The cross-section area formula A₄(F) (A₄(F) between A and F) needed to be constructed using the Newton interpolation method, in order to construct the prediction formula W₄(F) of wear rate.

Five equidistant interpolation nodes of (F₀, A₀), (F₁, A₁), (F₂, A₂), (F₃, A₃) and (F₄, A₄) were chosen as (1.65, 2.07 × 10⁻²), (4.15, 4.81 × 10⁻²), (6.65, 5.58 × 10⁻²), (9.15, 7.11 × 10⁻²) and (11.65, 7.27 × 10⁻²). Table 3 shows the forward differences at different applied loads F. Hence, in the applied load district [1.65 N, 11.65 N], the cross-section area formula A₄(F) was defined as formula (18) by the Newton interpolation method:

Table 3 Forward differences at different applied loads F

Applied load F (N)	A(F)	ΔA(F)	Δ^2A(F)	Δ^3A(F)	Δ^4A(F)
1.65 (F0)	2.07 × 10^−2	2.74 × 10^−2	−1.97 × 10^−2	2.73 × 10^−2	−4.86 × 10^−2
4.15 (F1)	4.81 × 10^−2	0.77 × 10^−2	0.76 × 10^−2	−2.13 × 10^−2
6.65 (F2)	5.58 × 10^−2	1.53 × 10^−2	−1.37 × 10^−2
9.15 (F3)	7.11 × 10^−2	0.16 × 10^−2
11.65 (F4)	7.27 × 10^−2

---

Fig. 5 shows the curve graph of the cross-section area formula A₄(F) at different applied loads F. As is obvious in Fig. 5, the cross-section area A exhibited an upward tendency with an increase in the applied load from 1.65–11.65 N.

Fig. 5 Curve graph of the cross-section area formula A₄(F) at different applied loads F.

In the applied load district [1.65 N, 11.65 N], the prediction formula W₄(F) of wear rate was defined as formula (19) by substituting the cross-section area formula A₄(F) (see formula (18)) into formula (7):

Fig. 6 shows the curve chart of the prediction formula W₄(F) of wear rate at different applied loads F. It can be seen in Fig. 6 that the wear rate continuously decrease with the increase of the applied load F. It may be concluded that the pressure stress was improved with the increase of the applied load F. Bulk wear debris was crushed into many smaller wear debris pieces at higher pressure stress, resulting in the improvement of the contact areas of the wear debris, strengthening of the binding forces of the wear debris and lowering of the material loss of TiAl-10 wt% Ag. Consequently, TiAl-10 wt% Ag exhibited a smaller wear rate.

Fig. 6 Curve chart of the prediction formula W₄(F) of wear rate at different applied loads F.

3.5.2 Evaluating the prediction formula W₄(F) of wear rate. Table 4 shows the comparative analysis between the testing error and computational error 2.90-392-3600 and 10.40-392-3600 (N-rpm-s). As shown in Table 4, the wear rate shows a downward tendency with the increase of applied load F. Simultaneously, the computational error was smaller than the testing error at the different applied loads F. Hence, in the district [1.65 N, 11.65 N], the formula W₄(F) was accepted to predict wear rate with small computational errors.

Table 4 Comparative analysis between testing error and computational error at 2.90-392-3600 and 10.40-392-3600 (N-rpm-s)

Test conditions	2.90-392-3600			10.40-392-3600
Wtest	6.20	5.86	5.47	3.43	3.09	2.89
Etest	6.10%	0.29%	6.34%	9.35%	1.49%	7.86%
Wcomp		5.93			3.13
Ecomp		1.48%			0.32%

---

3.5.3 Influence of wear mechanisms on the prediction accuracy. Fig. 7 exhibits the typical electron probe morphologies of the wear scar after 80 min sliding at different applied loads F. As shown in Fig. 7a, the primary wear mechanism was abrasive wear at 2.90-392-3600 (N-rpm-s) for bulk wear debris on the smooth wear scar. As is clear in Fig. 7b, the small wear debris and discontinuous wear grooves appeared on the wear scar. It was apparent that the main wear mechanisms were abrasive wear and plough wear at 10.40-392-3600. The pressure stress was continually improved with the increase of the applied load F, resulting in cracking of the bulk wear debris into small wear debris pieces, improvement of the contact areas of the wear debris, strengthening of the binding forces of the wear debris, conversion of the wear mechanisms and lowering of the wear rate of TiAl-10 wt% Ag, as well as improvement of the prediction accuracy of the formula W₄(F) (see formula (19)). Hence, the wear rate of TiAl-10 wt% Ag was smaller at higher applied loads F. Simultaneously, the smaller wear debris pieces were retained on the wear scar by the deep wear groove at 10.40-392-3600.

Fig. 7 Typical electron probe morphologies of the wear scar after 80 min sliding at different applied loads F: 2.90-392-3600 (a), 10.40-392-3600 (b).

3.5.4 Influence of surface textures on the prediction accuracy. The excellent surface texture of the wear scar, which was determined by the height parameters of S_a, S_q and S_ku, was beneficial to improvement of the contact area, lowering of the contact stress and acquiring a smooth wear scar. Mean surface roughness was measured by the arithmetical mean height (S_a). The standard deviation of the height distribution was evaluated by the root mean square height (S_q). The flatness of the height distribution could be examined by the kurtosis of the height distribution (S_ku).

Fig. 8 shows the representative surface texture of the wear scar at different applied loads F. A smoother wear scar was obtained for smaller S_a (0.398 μm), S_q (0.328 μm) and S_ku (7.326) values at 10.40-392-3600 (N-rpm-s), if compared to that at 2.90-392-3600. It can be concluded that the wear debris was refined at the higher applied load of 10.40 N, resulting in the smoothing of the wear scar and improvement of the prediction accuracy (99.68%) of the formula W₄(F).

Fig. 8 Representative surface texture of the wear scar at different applied loads F: 2.90-392-3600 (a), 10.40-392-3600 (b).

As wear rates are mainly determined by the applied load F and rotation speed n, it was rather significant to study the wear rate at different rotation speeds n.

3.6 Rotation speed n

3.6.1 Constructing the prediction formula W₄(n) of wear rate. The Newton interpolation method could be used to construct the cross-section area formula A₄(n) (A₄(n) between A and n) to obtain the prediction formula W₄(n) of wear rate.

Five interpolation nodes were chosen as (56, 1.02 × 10⁻²), (168, 3.39 × 10⁻²), (280, 6.91 × 10⁻²), (392, 7.11 × 10⁻²) and (504, 14.83 × 10⁻²). Table 5 shows the forward differences at different rotation speeds n. Hence, in the rotation speed district [56 rpm, 504 rpm], the cross-section area formula A₄(n) was defined by the Newton interpolation method, as shown in formula (20):

Table 5 Forward differences at different rotation speeds n

Rotate speed n	A(n)	ΔA(n)	Δ^2A(n)	Δ^3A(n)	Δ^4A(n)
56 (n0)	1.02 × 10^−2	2.37 × 10^−2	1.15 × 10^−2	−4.47 × 10^−2	15.31 × 10^−2
168 (n1)	3.39 × 10^−2	3.52 × 10^−2	−3.32 × 10^−2	10.84 × 10^−2
280 (n2)	6.91 × 10^−2	0.20 × 10^−2	7.52 × 10^−2
392 (n3)	7.11 × 10^−2	7.72 × 10^−2
504 (n4)	14.83 × 10^−2

---

Fig. 9 exhibits the curve graph of the formula A₄(n) of the cross-section area of the wear scar at the different rotation speeds n. As shown in Fig. 9, the cross-section area A continuously increased with the increase of the rotation speed between 56–504 rpm.

Fig. 9 Curve graph of the formula A₄(n) of the cross-section area of the wear scar at different rotation speeds n.

In the district of [56 rpm, 504 rpm], the predicted formula W₄(n) of wear rate was defined as formula (21) by substituting the cross-section area formula A₄(n) (see formula (20)) into formula (7):

Fig. 10 shows the curve chart of the prediction formula W₄(n) of wear rate at different rotation speeds n. As is clear in Fig. 10, the wear rate exhibits an upward tendency with an increase of rotation speed in the range of 56–280 rpm, a downward tendency in the region of 280–392 rpm and an ascending tendency in the district of 392–504 rpm. In the range of 56–280 rpm, the centrifugal force of the wear debris was continually improved with the increase of rotation speed n, leading to the removal of the wear debris from the wear scar and increase of the wear rate of TiAl-10 wt% Ag. When the rotation speed n was increased from 280 to 504 rpm, this resulted in the further improvement of the centrifugal force of the bulk wear debris, strengthening of the interaction forces between the bulk wear debris and small wear debris, as well as random removal of the small wear debris during the impacting process. Hence, the wear rate of TiAl-10 wt% Ag showed a fluctuating tendency with the increase of rotation speed n.

Fig. 10 Curve chart of the prediction formula W₄(n) of wear rate at different rotation speeds n.

3.6.2 Evaluating the prediction formula W₄(n) of wear rate. Table 6 shows the comparative analysis between the testing error and computational error at 9.15-112-3600 and 9.15-448-3600 (N-rpm-s). As shown in Table 6, the testing error of the wear rate shows a larger fluctuation at 9.15-112-3600, if compared to that at 9.15-448-3600, leading to the improvement in the testing error with the increase of rotation speed n. A smaller computational error was obtained than the testing error at the different rotation speeds n. Hence, the prediction formula W₄(n) of the wear rate was accepted to predict the wear rate in the district of [56 rpm, 504 rpm].

Table 6 Comparative analysis between the testing error and computational error at 9.15-112-3600 and 9.15-448-3600 (N-rpm-s)

Test conditions	9.15-112-3600			9.15-448-3600
Wtest	2.34	2.21	1.68	3.71	3.41	3.23
Etest	12.68%	6.42%	19.10%	7.54%	1.16%	6.38%
Wcomp		2.07			3.59
Ecomp		0.32%			0.41%

---

3.6.3 Influence of wear mechanisms on prediction accuracy. Fig. 11 shows the representative electron probe morphologies of the wear scar after 80 min sliding at different rotation speeds n. Massive wear debris uniformly existed on the smooth wear scar (see Fig. 11a). It was evident that the primary wear mechanism was abrasive wear at 9.15-112-3600 (N-rpm-s). As shown in Fig. 11b, a great deal of wear debris appeared on the edge of the wear scar. It was evident that the primary wear mechanism was abrasive wear at 9.15-448-3600 (N-rpm-s). The centrifugal force of the wear debris was continually improved with the increase of rotation speed n, resulting in the improvement of the centrifugal force of the bulk wear debris, strengthening of the interaction forces between the bulk wear debris and small wear debris, random removal of the small wear debris during the impacting process, acquiring of the wear mechanism of abrasive wear, fluctuation of the wear rate of TiAl-10 wt% Ag and lowering of the prediction accuracy (from 99.68% to 99.59%) of formula W₄(n) (see formula (21)).

Fig. 11 Representative electron probe morphologies of the wear scar after 80 min sliding at different rotation speeds n: 9.15-112-3600 (a), 9.15-448-3600 (b).

3.6.4 Influence of surface textures on prediction accuracy. Fig. 12 shows the typical surface texture of the wear scar at different rotation speeds n. A smooth wear scar was obtained for the excellent height parameters of S_a (0.481 μm), S_q (0.368 μm) and S_ku (7.263) at 9.15-448-3600 (N-rpm-s). The centrifugal force of the wear debris at 9.15-448-3600 was larger than at 9.15-112-3600, leading to the improvement of the removal of wear debris, strengthening of the smoothness of the wear scar, fluctuation of the wear rate of TiAl-10 wt% Ag and lowering of the prediction accuracy (from 99.68% to 99.59%) of formula W₄(n).

Fig. 12 Typical surface texture of the wear scar at different rotation speeds n: 9.15-112-3600 (a), 9.15-448-3600 (b).

In order to evaluate the longevity of the mechanical components, it was necessary to investigate the wear rates at different sliding times t.

3.7 Sliding time t

3.7.1 Constructing the prediction formula W₄(t) of wear rate. Before constructing the prediction formula W₄(t) of wear rate, the cross-section area formula A₄(t) (A₄(t) between A and t) was required to firstly be constructed by the Newton interpolation method.

Five interpolation points of (900, 2.40 × 10⁻²), (1800, 4.90 × 10⁻²), (2700, 7.09 × 10⁻²), (3600, 7.11 × 10⁻²) and (4500, 6.24 × 10⁻²) were chosen to construct the cross-section area formula A₄(t) using the Newton interpolation method. Table 7 shows the forward differences at different sliding times t. Hence, the forward differences were employed to construct the cross-section area formula A₄(t), as shown in formula (22):

Table 7 Forward differences at different sliding times t

Sliding time t	A(t)	ΔA(t)	Δ^2A(t)	Δ^3A(t)	Δ^4A(t)
900 (t0)	2.40 × 10^−2	2.50 × 10^−2	−0.31 × 10^−2	−1.86 × 10^−2	3.14 × 10^−2
1800 (t1)	4.90 × 10^−2	2.19 × 10^−2	−2.17 × 10^−2	1.28 × 10^−2
2700 (t2)	7.09 × 10^−2	0.02 × 10^−2	−0.89 × 10^−2
3600 (t3)	7.11 × 10^−2	−0.87 × 10^−2
4500 (t4)	6.24 × 10^−2

---

Fig. 13 shows the curve graph of the formula A₄(t) of the cross-section area of the wear scar at different sliding times t. The cross-section area A exhibited an upward tendency, when the sliding time t continuously increased in the range of 900–4500 s.

Fig. 13 Curve graph of the formula A₄(t) of the cross-section area of wear scar at different sliding times t.

In the sliding time district [900 s, 4500 s], the predicted formula W₄(t) of the wear rate was written as formula (23) by substituting the cross-section area formula A₄(t) (see formula (22)) into formula (7):

Fig. 14 exhibits the curve chart of the prediction formula W₄(t) of wear rate at different sliding times t. As shown in Fig. 14, the wear rate constantly decreased with the increase of sliding time t. Bulk wear debris was repeatedly crushed into a great deal of smaller wear debris pieces with the increase of sliding time t, resulting in incessant forming of anti-friction films and continuous lowering of the wear rate of TiAl-10 wt% Ag.

Fig. 14 Curve chart of the prediction formula W₄(t) of the wear rate at different sliding times t.

3.7.2 Evaluating the prediction formula W₄(t) of wear rate. Table 8 shows the comparative analysis between the testing error and computational error at 9.15-392-1350 and 9.15-392-4050 (N-rpm-s). As shown in Table 8, when the sliding time t increased from 1350 to 4050 s, the testing error of the wear rate was lowered at 9.15-392-4050 compared to that at 9.15-392-1350. The computational error was smaller than the testing error at the different sliding times t. Hence, the prediction formula W₄(t) of wear rate was accepted to predict the wear rate in the sliding time district [900 s, 4500 s].

Table 8 Comparative analysis between the testing error and computational error at 9.15-392-1350 and 9.15-392-4050 (N-rpm-s)

Test conditions	9.15-392-1350			9.15-392-4050
Wtest	4.51	4.11	3.89	2.80	2.67	2.50
Etest	8.15%	1.44%	6.71%	5.40%	0.50%	5.90%
Wcomp		4.27			2.72
Ecomp		2.40%			2.38%

---

3.7.3 Influence of wear mechanisms on the prediction accuracy. Fig. 15 shows the representative electron probe morphologies of the wear scar after 80 min sliding at different sliding times t. As is clear in Fig. 15a, massive shallow peeling pits and abrasive debris existed on the wear scar. It was apparent that the primary wear mechanisms were peeling and abrasive wear at 9.15-392-1350 (N-rpm-s). As shown in Fig. 15b, shallow peeling pits and discontinuous wear grooves appeared on the smooth wear scar at 9.15-392-4050 (N-rpm-s). It was obvious that the primary wear mechanisms were peeling and plough wear. Bulk wear debris was repeatedly crushed into massive smaller wear debris pieces with the increase of sliding time t, resulting in incessant forming of anti-friction films, slight conversion of wear mechanisms, continuous lowering of the wear rate of TiAl-10 wt% Ag, and stabilization of the prediction accuracy (from 97.60% to 97.62%) of the prediction formula W₄(t) of wear rate (see formula (23)).

Fig. 15 Representative electron probe morphologies of the wear scar after 80 min sliding at different sliding times t: 9.15-392-1350 (a), 9.15-392-4050 (b).

3.7.4 Influence of surface textures on the prediction accuracy. Fig. 16 exhibits the typical surface texture of the wear scar at different sliding times t. When the sliding time t increased from 1350 to 4050 s, a smooth wear scar was obtained for the smaller height parameters of S_a (0.356 μm), S_q (0.291 μm) and S_ku (6.241). Small wear debris was constantly formed in the reiterative crush with the increase of sliding time t, resulting in the formation of anti-friction films, lowering of the wear rate of TiAl-10 wt% Ag and stabilization of the prediction accuracy (from 97.60% to 97.62%) of the prediction formula W₄(t).

Fig. 16 Typical surface texture of the wear scar at different sliding times t: 9.15-392-1350 (a), 9.15-392-4050 (b).

In this study, in order to study the variation of wear rate at different applied loads F, rotation speeds n and sliding times t, prediction formulas W₄(F), W₄(n) and W₄(t) of wear rate were constructed based on TiAl-10 wt% Ag. Five interpolation nodes were firstly chosen to respectively construct formulas A₄(F), A₄(n) and A₄(t) (A₄(F) between A and F, A₄(n) between A and n, as well as A₄(t) between A and t) using the Newton interpolation method. And then prediction formulas W₄(F), W₄(n) and W₄(t) were obtained by substituting formulas A₄(F) (see formula (18)), A₄(n) (see formula (20)) and A₄(t) (see formula (22)) into the wear rate formula W₄ (see formula (7)). Prediction formulas W₄(F), W₄(n) and W₄(t) of wear rate, which were evaluated by making a comparison between the computational error and testing error, could be accepted to predict wear rate with small computational errors. The prediction accuracies of formulas W₄(F), W₄(n) and W₄(t) were respectively given the reasonable explanations by wear mechanisms and surface texture at different applied loads F, rotation speeds n and sliding times t. It was found that the prediction accuracy of the prediction formula of the wear rate gradually increased for the higher binding force of wear debris at larger applied loads F (see formula W₄(F)), and slightly decreased for the random impacting between the bulk wear debris and small wear debris at the higher rotation speeds n (see formula W₄(n)), as well as being relatively stabilized for the incessant forming of anti-friction films at the different sliding times t (see formula W₄(t)).

4. Conclusions

Under pure Ar atmosphere protection, TiAl-10 wt% Ag was fabricated using SPS for 8 min at the temperature and pressure of 1000 °C and 30 MPa. The formulas W₄(F), W₄(n) and W₄(t), which were constructed using the Newton interpolation method, were respectively employed to predict wear rate with the increase of applied load F, rotation speed n or sliding time t at room temperature. The conclusions were obtained as follows:

(1) Formulas W₄(F), W₄(n) and W₄(t), which were constructed using the Newton interpolation method, could be employed to predict wear rate with small computational errors.

(2) In the applied load district [1.65 N, 11.65 N], the prediction accuracy of formula W₄(F) gradually increased for the higher binding force of the wear debris at larger applied loads F.

(3) The prediction accuracy of formula W₄(n) slightly decreased for the random impacting between bulk wear debris and small wear debris at higher rotation speeds n.

(4) In the sliding time district [900 s, 4500 s], the prediction accuracy of formula W₄(t) was relatively stabilized for the incessant forming of anti-friction films.

(5) The prediction formulas W₄(F), W₄(n) and W₄(t) in the form of the Newton interpolation polynomial can be respectively adopted to predict wear rate with the increase of applied load F, rotation speed n and sliding time t, in order to evaluate the longevity of mechanical components used in industrial fields.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (51275370); the Self-determined and Innovative Research Funds of WUT (135204008); the Fundamental Research Funds for the Central Universities (2015-yb-008 and 2015-zy-051); the authors also wish to gratefully thank the Material Research and Testing Center of Wuhan University of Technology for their assistance.

References

1. S. Y. Sung and Y. J. Kim, Intermetallics, 2007, 15, 468.
2. X. H. Wu, Intermetallics, 2006, 14, 1114.
3. C. M. Austin, Curr. Opin. Solid State Mater. Sci., 1999, 4, 239–242.
4. F. H. Fores, C. Suryanarayana and D. Eliezer, J. Mater. Sci., 1992, 27, 5113.
5. T. Sun, Q. D. Wang, L. Sun, G. H. Wu and Y. Na, Wear, 2010, 268, 693.
6. X. L. Shi, Z. S. Xu, M. Wang, W. Z. Zhai, J. Yao and S. Y. Song, Wear, 2013, 303(1–2), 486.
7. J. F. Archard, J. Appl. Phys., 1953, 24, 981.
8. P. Kishore Sampathkumaran, S. Seetharamu, S. Vynatheya and A. Murali, Wear, 2000, 237, 20.
9. Y. Y. Boneh, A. Sagy and Z. Reches, Earth Planet. Sci. Lett., 2013, 381, 127.
10. S. Kasolang, M. A. Ahmad, F. A. Ghazali and A. M. Azmi, Appl. Mech. Mater., 2011, 52, 464.
11. K. Gong, H. L. Luo, D. Feng and C. H. Li, Wear, 2008, 265, 1751.
12. Q. Cui, J. Shu, Z. F. Wu and G. X. Jiang, Power Electronics, 2013, 47(7), 46,  DOI:10.3969/j.issn.1000-100X.2013.07.016.
13. ASTM E92-82, Standard test method for vickers hardness of metallic materials, ASTM International, 2003.
14. ASTM B962-08, Standard test methods for density of compacted or sintered powder metallurgy (PM) products using Archimedes’ principle, ASTM International, 2008.
15. ASTM G99-95, Standard test method for wear testing with a pin-on-disk apparatus, ASTM International, 1995.
16. C. X. Li, J. Xia and H. Dong, Wear, 2006, 261, 693.
17. J. H. Mathews, K. D. Fink and F. L. Huang, Numerical methods using MATLAB, Electronic Industry Press, 4th edn, 2009.

---