Model of Fracture, Friction, and Wear Phenomena of Porous Iron

Abstract

Mechanical and tribotechnical features of powdered materials are strongly influenced by pore volume, fracture character, impurities, alloying, concentration inhomogeneity, friction conditions, and other factors. Pores also have influence on acceleration of diffusion processes and reduce undercooled austenite resistance. Annealed in hydrogen, ultra pure iron powder was used to study porous iron features. Toughness fracture and tribotechnical features had nonmonotonic dependence from porosity different from all known dependences got from technical iron powders. Researches brought out the fact that in process of porosity reduction by pressing and annealing cycles, the average dimension of porous is changed. According to the analysis of porous structure were created models of friction, wear, and fracture of pure porous iron.

1. Introduction

The last few decades studies conducted in highly technologically developed countries showed that proper choice of friction pairs from the position of tribology could save approximately 1.5% of national income [1]. The usage of powdered materials best of all suits to the conditions of economic feasibility, because it allows to replace expensive and scarce alloys, increase productivity, and vacate production space. For example, the presence of residual pores in the structure of the bearing material can increase details’ durability from 1.5 to 10 times [2].

There are three groups of the main factors influencing the friction: technological (structure, mechanical, and physical properties), constructive (contact scheme, the macro- and microgeometry, surface friction), and operational (specific friction work, sliding velocity, specific load, temperature, presence of grease, medium).

The friction process causes the formation of surface and subsurface layers which are conditioned by one or another friction and wear mechanism.

The last years researches were aimed at mechanical characteristics determination, including fracture strength and its connection with tribotechnical and physical features [3–22].

The aim of this work is to investigate and to model the friction, wear, and fracture processes of ultrapure iron.

2. Fracture Toughness of Ultrapure Porous Iron

Dependence of fracture toughness 𝐾_I𝐶 and strength on porosity 𝑃 is not always strictly monotonic function [3, 4]. Defining this dependence is necessary for both theoretical concepts of crack interaction with defects and for specific practical problems solution, for example, the choice of the porosity of construction products, which can provide the highest fracture toughness. The main condition for determining reliability of 𝐾_I𝐶 is the proof of the crack propagation in plane strain. This requirement imposes limitations on sample size and crack length. Crack length should be at least 2 mm and together with a groove to be 0.45–0.55 height of the specimen (GOST 25.506-85). The height of the sample is usually taken 12 mm. According to the regulations, the most exacting requirements are imposed on the thickness, which should not be less than β(𝐾_I𝐶/𝜎_0.2)², where 𝛽=2.5. However, other factors like the flat fracture and absence of thinning edges and lips cut, even if the porosity of iron have a few percent, indicate the reliability of the results. Thus, for the iron of high purity factor of proportionality 𝛽 and the ratio between the thickness to (𝐾_I𝐶/𝜎_0.2)² was 0.3-0.4 [23] for powder Ni-Mo-Cu steel 𝛽=1.6 [24]; for high-ductility stainless steel at 𝑃=5%, 𝛽=0.04 [25]; for nickel steels 𝛽=1.4 [26].

The correctness of the definition of 𝐾_I𝐶 was examined by varying the thickness of the samples (Table 1). For sample thickness ranging from 4.3 to 9.1 mm, β varies from 0.2 to 5.5. The obtained values are within the limits of the measurement inaccuracy confidence interval, which gives grounds to take 𝐾_𝑄 for 𝐾_I𝐶. In all cases the sample thickness of 3 mm provided a satisfactory convergence of the results.

Table 1 

The dependence of powder materials fracture toughness on porosity and sample thickness.

The data (Table 1) prove the invariance of fracture strength to a sample thickness. However, this does not mean that 𝐾_I𝐶 is invariant to the sample size for all porous steels and alloys, therefore, the influence of size ratios on the 𝐾_𝑄 should be determined experimentally for different materials.

The choice of ultrapure iron is connected to a necessity to minimize the unanswered effect of impurities. Other important factors determining the fracture toughness are fracture mode, porosity, composition, and heterogeneity of alloying elements. Most of these factors which influence the 𝐾_I𝐶(P) dependence could be eliminated by the selection of the material and the sample fabrication method. The effect of porosity on fracture toughness data cited in the literature is contradictory. For example, a nonmonotonic dependence of 𝐾_I𝐶 on 𝑃 at room temperature and at −180°C was determined [27]. A monotonic dependence 𝐾_I𝐶(𝑃) has been noticed by other authors [23, 25, 26, 28–32].

The authors of [21] do not exclude that the appearance of nonmonotonic 𝐾_I𝐶(P) dependence is associated with redistribution of impurities, because the situation of high porosity it is energetically favorable for the impurities to locate at the free surface of the pores, as well as the seals they move into the matrix, concentrating at grain boundaries, thus reduce the level of fracture toughness.

Studies have shown [33] that comparing with the porosity, the more significant factor affecting the fracture toughness is the proportion of interparticle destruction; also a linear correlation between 𝐾_I𝐶 and the proportion of transgranular cleavage (test temperature minus 196°C) was established.

Carbonyl iron OSCh 6-2 was annealed 1 hour in hydrogen at 450°C before pressing to minimize the influence of impurity atoms. The oxygen content weight in sintered samples of the iron did not exceed 0.01%.

For stochastically inhomogeneous bodies fracture toughness could be accurately estimated using the effective stress intensity factor 𝐾_I𝐶 [34]. The structure of the sintered iron powder is a single-phase material containing pores of various sizes.

Fracture toughness of iron samples (PZh4M2) with porosity 10–16% practically coincides with that one given in [33]. The porosity reduction is accompanied by a change from interparticle to transgranular ductile fracture mode (Figure 1). Therefore, a monotonic 𝐾_I𝐶 increase is associated with a reduction of the proportion of intercrystallite fracture due to increased interparticle metallic bonds. For the iron samples, a correlation between 𝐾_I𝐶 and mechanical properties was established (Figure 2).

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Figure 1 

Fractography of the iron grade PZh4M2 specimens: (a) after pressing, ×500; (b) 𝑃=18%, ×200; (c) 𝑃=10%, ×700.

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Figure 2 

Mechanical properties and fracture toughness correlation of vacuum sintered samples from technical iron PZh4M2 grade.

The relationship between intercrystallite and intragranular fracture components depends on the sintering atmosphere. Thus, after sintering at 1200°C, 4 h in vacuum the fracture toughness of the 10% porosity samples was 10-11 MN/m^3/2 and fracture mode was intercrystallite. After the same sintering in hydrogen, the fracture toughness of the same porosity iron samples (PZh4M2) was 20-21 MN/m^3/2, and fracture mode was transgranular. Other mechanical properties increased less significantly: 𝜎_𝐵 from 180 to 320 MPa, 𝜎_0.2 from 100 to 140 MPa, and 𝛿 from 10 to 16%.

𝐾_I𝐶 values for samples of iron OSCh 2-6 were higher and the dependence of 𝐾_I𝐶 on 𝑃 (from 3 to 7%) was nonmonotonic. Fracture mode was intragranular regardless of porosity (Figures 3, 4, 5, and 6). The detailed fractographs study revealed that the highest proportion of cleavage on the fracture surface was in the samples with 3.5–4% porosity. It corresponds to the inflection point of the fracture toughness on porosity dependence. The porosity changes leads to an increase of the ductile component of the fracture surface, and 𝑃=6% of the sites of cleavage fracture were not observed. With increasing porosity, pits were less deep and surface resembled a honeycomb fracture [35].

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Figure 3 

Fractography of the iron grade OSCh 6-2 with 2% porosity specimens: (a) ×75; (b) ×400; (c) ×750.

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Figure 4 

Fractography of the iron grade OSCh 6-2 with 3.6% porosity specimens: (a)  ×75; (b) ×200; (c) ×400.

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Figure 5 

Fractography of the iron grade OSCh 6-2 with 5.9% porosity specimens: (a) ×200; (b) ×750.

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Figure 6 

Fractography of the iron grade OSCh 6-2 with 7.3% porosity (a, b) and with 20% porosity (c) specimens: (a) ×100; (b) ×750; (c) ×750.

Thus, the change in the ratio of ductile and cleavage component is a mechanism for implementing nonmonotonic 𝐾_I𝐶(P) dependence. But to understand the process of destruction of the porous body it is necessary to develop the model representations that connect fracture toughness to the characteristics of the porous structure.

Reduction of impurities concentration in the iron powder does not change𝐾_I𝐶(P) dependence type, so one can assume that nonmonotonic decrease of 𝐾_I𝐶 is the result of the interaction of cracks with the pores. This result is consistent with theoretical studies [22, 36, 37], proving the possibility of a nonmonotonic change of 𝐾_I𝐶 by varying the structural defects.

Attention is drawn to the increased size of pores at 𝑃=3.5%, which corresponds to a singular point of 𝐾_I𝐶(P) dependence (Table 2). This fact gives a reason to suggest a relationship between fracture toughness and pore size distribution and, hence, the distance between the pores. Apparently, the influence of pores on the fracture toughness is not only due to the average porosity.

Table 2 

The dependence of porosity on the mechanical properties of OSCh 6-2 iron.

To describe the interaction of crack with pores its movement can be represented as the movement of a dislocation front. Reasons to study the propagation of the crack front by analogy with the dislocation front directly for pure iron are given in [38].

We assume that the pores are spherical, uniformly distributed and separated by a distance 2 𝑅. Crack bending leads to a change in its distribution energy [39–41]: 𝑔^=𝑔+𝑇𝑅,(1)𝐾_I𝐶=𝐸⋅𝑔^1−𝜇²=𝐸1−𝜇²𝑔+𝑅𝑇,(2) where 𝑔 is the crack distribution energy; 𝑇—linear crack front tension; 𝐸—elastic modulus; μ—Poisson’s ratio.

Expanding (1) in a series, since 𝑔≫𝑇, we find 𝐾_I𝐶=𝐾⁰_I𝐶(𝑃)+121−𝜇²𝐸⋅𝐾⋅𝑇_I𝐶(𝑃)⋅𝑅.(3) Equation (4) is got in correlation assumption of specific energy of uniform deformation (𝐴) and effective surface energy (𝛾_eff) of the Griffiths-Orowan theory [42]: 𝐾⁰_I𝐶=0.8⋅𝑙_𝑑⋅𝐴⋅𝐸1−𝜇²𝐴=𝛿_𝑒,(4)−𝜎_0.2/𝐸⋅𝜎_0.2+2𝜎_𝐵3+𝜎2𝐸^20.2,(5) where 𝑙_𝑑—length dimension constant; 𝛿_𝑒—uniform deformation; 𝜎_0.2 and 𝜎_𝐵 yield strength and tensile strength, respectively.

𝑙_𝑑 is a constant in the limit 𝑃→0. Extrapolation of experimental data to zero porosity gives 𝐾_I𝐶=35 MN/m^3/2, 𝜎_0.2=250 MPa, 𝜎_𝐵=350 MPa, 𝛿=49%, and 𝛿_𝑒=26%. Similar values of mechanical properties obtained in [43–45].

Then we calculate 𝐾⁰_I𝐶 for all porosity values: 𝐾⁰_I𝐶=⎷0.4⋅10⁻³2𝛿_𝑒(𝑃)⋅𝐸(𝑃)−𝜎_0.2(𝑃)⋅𝜎_0.2(𝑃)+2𝜎_𝐵(𝑃)+3𝜎_0.2(𝑃)31−𝜇²(𝑃).(6)

Dependence 𝜇=𝜇(𝑃) was established in [46]:𝜇(𝑃)=(1−𝑃)𝜇1−𝑃𝜇.(7)

For the iron powder [47]: 𝐸(𝑃)=𝐸(1−𝑃)^3/4.(8)

As a result of fitting the experimental data by (3) we determined the value of 𝑇=3⋅10⁻⁴ N. Average approximation accuracy (4%) indicates a good convergence of calculated and experimental results. The crack front tension can be determined directly from the expression: 𝑇=𝜏⋅𝑅,(9) where 𝜏—crack movement stress.

The general idea is that the crack movement through the area with obstacles in it can be described on the assumption of the same principles as in dislocation movement which is described in details in [48] and later in [49]. The possibility of the crack movement is proved experimentally in [50]. In this case porous were studied as inhomogeneities (second phase). The equation was got in assumption of minimal front bend of the crack in condition of relatively large porous and long distance between them, 𝜏=𝑟²⋅𝜋⋅Γ2𝑏₀⋅Ω,(10) where 𝑟—pore radius; Γ—interface energy; Γ=1000 ergs/cm² [51]; 𝑏₀—interatomic distance; 𝑏₀=0.246⋅10⁻⁹ m; Ω—distance between the inhomogeneities (pores).

Then, as the porosity, for example 𝑃=7.5%, 𝑇 is equal to 2.5−25⋅10⁻⁴ N, which is higher than, but has the same order with, a value of 𝑇, determined experimentally. The coincidence of the front crack tension value found experimentally and calculated from published data confirms the validity of expression (3).

As seen from (3), for effective crack interaction with the defects it is necessary that the distance between the defects was several times shorter than the size of the crack, the same view expressed in [51].

Thus, the nonmonotonic dependence of fracture toughness on porosity was established and when the effect of impurities is minimal. It was shown that the type of 𝐾_I𝐶(P) dependence is the result of crack interaction with pore system, the maximum percentage of cleavage on the fracture surface coincides with a singular point of 𝐾_I𝐶(P) dependence.

A model explaining the deviation from a strictly monotonic decrease of fracture toughness with increasing porosity by crack front tension and allowing to predict fracture toughness of powder materials is given.

Note that the characteristics of plasticity may also have nonmonotonic dependence on porosity [51, 52].

3. Influence of Pores on the Friction and Wear of Iron

Mechanics of contact fracture offers a solution to the problem of wear for a half-space of moving on its surface Hertz plateau in the presence of horizontal and vertical cracks [53]. However, the features of porous materials allow to use a simple functional relation to the conservation of acceptable accuracy in many cases. First of all, if we proceed from the principle of superposition, for small friction ratio the contribution of the tangential component 𝜏_𝑧=−𝑓𝑞(𝑥) is small. For example, increasing of the contour pressure on the order during sliding friction without lubrication of oil-impregnated steel PA-ZhGr1 with 20% porosity results in 𝑓 changes only by 17% and the wear rate—on the order. Secondly, the problem is solved for certain contact surfaces. At the same contact type, changes of 𝑓 are relatively small (compared to the differences between the friction ratios for different interactions types), and porous materials have the same trend. Relationship between wear rate and the variable parameters (especially pressure) within the proposed solutions were selected empirically using the coefficients that indirectly takes into account the contribution of the friction ratio.

Physico-mechanical and antifrictional properties of heterogeneous powder materials depend on several factors which are the chemical composition, heat treatment, variations in the density and distribution of alloying elements, content and distribution of related impurities, and so forth.

Some factors mentioned could be eliminated as a result of using pure iron [21, 54, 55]. But high content of impurities in industrial iron powder, insufficient porous structure description, and narrow ranges of the test pressure variation are the reasons of determination impossibility of the contribution of each measured parameters with high reliability.

The dependence of friction ratio 𝑓 and wear rate on the porosity 𝑃 and the pressure 𝑃_𝑛 were studied in wide ranges (𝑃=2-20%, 𝑃_𝑛=0.25-4.2 MPa) for samples of pure iron (OSCh 6-2). Porosity was decreased by cycles of repressing and annealing. The size distribution of wear particles (Figure 7) were studied by microscope “Neophot-21”. Distribution shows that wear fractions are considerably smaller then porous (Table 2) and can influence on friction conditions, the forming surface geometry, and it can influence especially strongly on oil layers appearance during boundary friction of infiltrated with oil samples.

Figure 7 

The size distribution of wear particles.

Friction surface was studied by scanning microscope SEM-200 (Figure 8). Tribological properties were determined by SMC-2, with hardened steel 45 (GOST 26614-85) as a friction pair. X-ray diffraction analysis was made on a DRON-2 in the 𝜆-Co radiation.

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Figure 8 

Friction surface of porous iron OSCh 6-2 specimen, ×50: (a) 2% porosity, (b) 20% porosity.

On the friction surfaces, inclusions (presumably oxides) and microcutting traces were found, Figure 8. Such surfaces are typical for powder compacts, working at hard conditions. The friction ratio has a pronounced nonmonotonic dependence on the porosity with a maximum about 𝑃=4%, which coincides with the minimum of 𝐾_I𝐶(P) dependence.

Since there is a correlation between friction ratio and wear characteristics, and wear resistance is related to pore size [56], it can be assumed that the local extremum of 𝑓(𝑃) is also associated with the peculiarities of the porous structure.

Narrow ranges of friction ratio variation allow to suggest prorated relationship between contour (𝑃_𝑐) and nominal pressure (𝑃_𝑛) [57, 58].

The experimental data was treated as follows. For each value of porosity, friction ratio deformation component 𝑓_𝑑 [59] was calculated (11), suggesting that the conditions of the unsaturated plastic contact would be met. The equation was got in assumption of an additive contribution of molecular and deformative component of friction force. Deformative component is caused by the deformation of a less hard body by a harder one: 𝑓_𝑑=0.5Δ^1/2H𝑃B^𝑐^0.25≈𝐶⋅𝑃^𝑛HB^0.25,(11) where Δ—microroughnesses characteristic of a rigid material of the friction pair, 𝑃_𝑐—contour pressure, and HB—hardness, 𝐶—constant. The value of Δ was in the range 10⁻¹-10⁻³ [59].

Type of mechanical treatment determines the value of the microroughnesses characteristic. The pressure 𝑃 and the hardness values are shown in Table 3.

Table 3 

Iron powder wear characteristics.

In the first approximation, the molecular component of friction ratio 𝑓_𝑚 decreases proportional to (1−𝑃), since 𝑓_𝑚 is proportional to the area 𝐴, where shear stresses 𝜏 appear. This is due to the fact that the pore size (several microns) in 1-2 orders of magnitude higher than the arithmetic mean deviation of profile Ra=0.09 micron, hence, the contact between the surfaces in the area of the pores is unlikely. For a porous material [59], assuming 𝑓_𝑚 decreasing in (1−𝑃) times and the proportionality of 𝑃_𝑛 and 𝑃_𝑐 and subsequent calculating the 𝛽 and 𝜏, we can write 𝑓_𝑚=(1−𝑃)𝛽+𝜏⁰𝑃_𝑛,(12) where 𝛽 and 𝜏₀ are friction characteristics depending on the friction conditions.

Friction characteristics 𝛽 and 𝜏₀ were calculated for each value of porosity, varying only the 𝑃_𝑛, and then by the least squares method we found expressions for 𝛽 and 𝜏₀ from 𝑃: 𝛽(𝑃)=0.00677+0.0103⋅𝑃−0.000394⋅𝑃²,(13)𝜏₀(𝑃)=0.05(5)+0.000532⋅𝑃²−0.0000267⋅𝑃³.(14)

The number of regression equations members and their significance was determined by minimizing the residual variance 𝑆_ost [60], the model adequacy was tested by the Fisher criterion 𝐹.

For the significance level 0.25, that is, the probability that the hypothesis can be rejected, the tabulated value of 𝐹 is 2.0. The values of 𝐹, equal to 2.3 and 7.7, calculated from expressions (13) and (14), respectively, exceeded tabular values, therefore, the equations are adequate.

Behavior of stressed porous materials has some special features [61]. Firstly, the pores reduce the range of physical and mechanical properties of the material, and secondly, pores are obstacles for dislocation motion and crack growth.

Assuming that the separation of the wear particles is due to crack formation and its subsequent growth, and as the pore was examined as an obstacle for the crack movement which is similar to the dislocation movement, so stress can be calculated by the formula [48]: 𝜏_𝑚=𝑎⋅𝐺⋅𝐵𝑅,(15) where 𝜏_𝑚—crack movement stress in porous material; 𝑎≈1—coefficient; 𝐺—shear modulus; 𝐵—Burgers vector; 𝑅—the distance between the pores. 𝜏_𝑚 does not act on the entire area, but only in places of contact. The bearing surface curve is usually given with 𝑡_𝑝-𝜀 coordinates. On the basis of theoretical researches, Kragelskii and Demkin proved that the Kragelskii-Demkin equation can be used for the top of the bearing surface curve [59]: 𝑡_𝑝=𝑏⋅𝜀^𝜈,(16) where 𝑡_𝑝—relative contact area; 𝑏 and 𝜈 are bearing surface curve parameters; 𝜀—relative approach.

The bearing surface curve parameters 𝑏=3.5 and 𝜈=1.8 were taken from [59]. Kragelskii and Mihin established during theoretical and empirical studies that real field of contact is proportional to applied load and doesn’t depend on roughness of the surface (during a plastic contact). It terms of it relative approach can be determined with the formula [59]: 𝜀=𝛼¹⋅𝐶1⋅𝑃_𝑛𝑏⋅HB^1/𝜈,(17) where 𝛼₁—the coefficient depending on the type of contact (it varies from 0.5 to 1) 𝛼₁≈0.75; 𝐶1—the constant that relates the contour and nominal pressure.

Since the relation of tensions acting on contour area (𝜏_𝑚⋅𝑡_𝑝) to contour pressure 𝑃_𝑐 is analogous to a corresponded forces relation [62], so dependence of inhibition of the crack in the pores on the friction ratio can be determined from the expression: Δ𝑓=𝜏^𝑚𝐶1⋅𝑃_𝑐⋅𝑡_𝑝,(18) where 𝜏_𝑚—stress; Δ𝑓—friction ratio change. Therefore, the approximation value of 𝑓_𝑚 was carried out further in view of Δ𝑓. Accounting of the possibility of crack inhibition by pores does not improve the approximation [62], as such mechanism can operate only when the crack path to fracture, significantly longer than the distance between the pores.

In the experiment discussed, the composition of wear particles corresponded to Fe₂O₃. Their size (see Figure 7) was much smaller than the distance between the pores, so a necessary condition for crack inhibition by pores (crack blunting) was not satisfied.

However, the increased size of pores at 𝑃≈4% corresponds to the largest value of 𝑓, which indicates a relationship between the inhibition of dislocation motion and crack growth. Regardless of the mechanism by which the dislocation overcomes obstacle, stress is always inversely proportional to 𝑅 [48].

The growth of pore size and the elongation of the distance between pores relieve the deformation in the active area, and it leads to growth of the plastically deformed contacts portion and the friction ratio. According to [63] the transition from elastic to plastic contact leads to a greater dependence of 𝑓_𝑑 on 𝑃_𝑐 and Δ (in the case of elastic contact 𝑓_𝑑∼𝑝^𝑐0.2⋅Δ^0.4, and plastic contact 𝑓∼𝑝^𝑐0.25⋅Δ^0.25, as to an increasing of 𝑓_𝑚 (in the case of elastic contact (𝑓_𝑚−𝛽)∼𝑝^𝑐−0.2⋅Δ^−0.4, and plastic contact (𝑓_𝑚−𝛽)∼HB⁻¹. Therefore, the most probable cause of the local extremum of the friction ratio dependence on the porosity is increased size of pores at 𝑃≈4%. An indirect confirmation of this assumption is the reduction of the difference between the maximum and average values of 𝑓 with contour pressure increasing (i.e., an increasing of the proportion of plastic contact).

Features of porous materials [55, 64, 65] allow us to relate the friction surface characteristics and the material structure with wear resistance, Table 4, Figure 9.

Table 4 

Iron powder characteristics.

Figure 9 

Response surface of 𝐼(𝑃,𝑃𝑛).

Iron wear during dry friction on steel is a fatigue wear [66, 67].

At a certain stage of the interaction surface reaches steady state characterized by a constant dislocation cluster density in the slip bands; lattice parameters deflection and of dislocation density at a depth of 10–15 mm remains constant [64, 67].

Thus, there are the necessary prerequisites for interpreting experimental results involving the major tenets of the theory of fatigue wear. Specific linear wear rate (𝑖_ℎ) was got in assumption of fatigue character of the wear. This assumption was based on material volume worn for one action, which is proportional to deformed volume and is inversely proportional to a number of acts leading to metal fracture [63]: 𝑖_ℎ=√𝜈/42(𝜈+1)𝑛_𝑘𝑟⋅√ℎ/𝑅_𝑐𝑣,(19) where ℎ—the rapprochement between the friction surfaces; 𝑅_𝑐𝑣=77 microns [68]—the radius of curvature; 𝑛_𝑘𝑟—the number of cycles, leading to the fracture.

Integral linear wear rate 𝐼_ℎ, and specific 𝑖_ℎ wear rate are related by [69]: 𝐼_ℎ≈𝑖_ℎ⋅𝑡_𝑝≈√𝜈/42(𝜈+1)𝑛_𝑘𝑟⋅√ℎ/𝑅_𝑐𝑣⋅𝑡_𝑝.(20)

Rapprochement between the two surfaces when the unsaturated plastic contact takes place [59]: ℎ=𝜀⋅𝑅_max=𝛼⋅𝑃^𝑐𝑏⋅HB^1/𝜈⋅𝑅_max,(21) where 𝑅_max—the greatest distance between the bumps and hollows within the gauge length.

Solving (19)–(21), we obtain the basic equation of wear 𝐼_ℎ=0.0156𝑛_𝑘𝑟⋅𝑃^𝑐HB^1.28.(22)

Fatigue wear theory assumes localized damage in a small volume. As follows from the fracture mechanics, the separation of particles occurs when the stress concentration reaches the critical value 𝐾_I𝐶.

This magnitude of stress concentration corresponds to a critical crack length 𝑙_𝑐. Dependence of stress intensity factor Δ𝐾 with the crack propagation rate 𝑑𝑙/𝑑𝑁 in case of endurance fracture determined by Paris formula: 𝑑𝑙𝑑𝑁=𝐴(Δ𝐾)^𝑛,(23) where 𝑁—number of cycles; 𝐴 and 𝑛—coefficients. Fatigue crack growth studies have shown that the coefficient 𝐴 depending on the strain hardening varies from 1.16⋅10⁻⁵ to 6.77⋅10⁻⁷; 𝑛 exponent reduces from 4.0 (for the degree of cold deformation of 50%) to 2.88 (at baseline) [70].

The stress intensity on the crack top determines stress intensity factor 𝐾 which is a tension and crack length 𝑙 function. For short cracks (𝑙/𝑎_𝑟<0.03, 𝑎_𝑟—contact pad radius) it is written in the paper [53] like 𝐾=2𝜎_𝑦√𝑙,(24) where 𝜎_𝑦—fracture stress; 𝑙—crack length. In the case of Hertz contact at the contact area interface [53]: 𝜎_𝑦=1−2𝜇2⋅𝑃^𝑐.(25)

The contact between the friction surfaces is carried out only in the microroughnesses contact plane. In this case, a critical crack length which is corresponded to a critical volume of stress intensity factor 𝐾_I𝐶, the real contact area is counted by introducing 𝑡_𝑝 taking into account (16), (17), (24), and (25) can be written like 𝑙=𝐾^I𝐶⋅𝑡_𝑝(1−2𝜇)⋅𝑃_𝑐²=0.75⋅𝐾^I𝐶HB(1−2𝜇)².(26)

For the initial size of the defect, by analogy with [56] a half the diameter of the pores (𝑙₀=𝑑/2) was taken. The values of 𝑑, 𝐾_I𝐶, and 𝜇 were taken from [70].

The deformed volume fracture occurs when Δ𝐾 reaches a value 𝐾_I𝐶, which corresponds to the critical defect size 𝑙 and the value of 𝑁=𝑛_𝑘𝑟. Solving expressions (22), (23), (26), and assuming a linear relationship between 𝑃_𝑛 and 𝑃_𝑐, we have a formula to determine the wear rate.

Fracture of the deformable volume happens when Δ𝐾 reaches 𝐾_I𝐶 value that is corresponded to a critical defect size 𝑙 and 𝑁=𝑛_𝑘𝑝 value. Substituting (23), (26) in (22) and ignoring the equation got on conditions that half of the pore diameter was taken as an initial defect size (𝑙₀=𝑑/2), on the analogy of paper [56] implying linear connection between 𝑃_𝑛 and 𝑃_𝑐, a formula for wear intensity was gotten 𝐼_ℎ=0.156⋅𝑃^𝑛/HB^0.128⋅(𝑛/2−1)⋅𝐴[(1−2𝜇)HB/0.75]^𝑛(𝑑/2)^(1−𝑛/2)−0.75𝐾_I𝐶/(1−2𝜇)HB^(2−𝑛),(27) where 𝑑—the average pore diameter.

𝐾_I𝐶,𝑑, and 𝜇 values were taken from paper [70]. Thus linear intensity of wear is determined by load, hardness, crack growth resistance, structure characteristic element size-pore size, Poisson ratio, and fatigue fracture parameters.

Fitting the experimental data by expression (27), the values of 𝐴=1.38⋅10⁻⁷ and 𝑛=2.11 were determined. As seen from Figure 9, the function 𝐼_ℎ(𝑃) is nonmonotonic over the entire range of variation of 𝑃_𝑛. A similar result for the constant values of pressure and a wide range of 𝑃 was experimentally obtained in [55]. Good agreement between the values of the coefficients 𝐴 and 𝑛 with the published data [71] and a satisfactory approximation of the experimental results confirm the correctness of the approach adopted for the study of porous iron.

4. Lubricated Friction and Wear Processes

Boundary friction conditions of powder materials are typical for most parts functioning. This mode is performed during startup and shutdown, even if the products are designed for operation in a hydrodynamic lubrication [72]. The tests differed only in that the oil in the friction zone was fed through a rotating counterbody dipping into the bowl with the lubricant, Table 5.

Table 5 

Antifriction properties of iron under limited lubrication.

Tables 4 and 5 show that for identical values of porosity and pressure the friction ratio value decreases several times with lubrication. It changes the dependence of 𝑓 on the pressure. Under dry friction an increase in pressure leads to a decrease in the 𝑓 value, but under lubricated friction pressure increasing results in 𝑓 increase. This form of the function 𝑓(𝑃) can be explained as follows. During elastic unsaturated contact friction ratio dependence on major friction unit characteristics 𝜏₀ and 𝛽, mechanical features of less hard material of friction pair (𝜇,𝐸), contour pressure and roughness of harder element of friction pair (Δ) according to [63] looks like 𝑓=2.4𝜏⁰1−𝜇²/𝐸^4/5𝑃^𝑐1/5⋅Δ^2/5+𝛽+0.23𝛼_𝑔𝑃^𝑐1/5Δ^2/51−𝜇²𝐸^1/5,(28) where 𝛼_𝑔—coefficient of hysteresis losses under friction.

Equation for elastic unsaturated contact has the same structure and is different only in indexes 𝑃_𝑐 and Δ,𝜇,𝐸 and constants values.

The presence of lubricating film dramatically reduces the frictional bond shear strength 𝜏₀, and hence the contribution of the first term of (28). Deformation component of the friction ratio is proportional to the contour pressure with a small exponent, as it observed experimentally. Features were observed only when 𝑃=8.6% (a transition from open to closed porosity occurs), and when 𝑃=4% (a local extremum of the pore size detected) [70]. The porous structure affects the surface deformation in the active area [56], so that the greatest value of the friction ratio is connected with the extreme size of the pores [62]. Low friction ratio values for high porosity samples can be explained, apparently, by the fact that in this experiment with 𝑃>10% all porosity is open, and it facilitates the circulation of oil and heat removal from the friction surfaces. The relatively low surface temperature makes it difficult to deform and promote 𝑓 decreasing.

Expression (28), Table 5, correctly reflects the trends of 𝑓(𝑃). Numerical processing of experimental data have not performed as well as the ranges of the friction ratio for most porosity values are small and the accuracy of 𝑓 at low pressures, comparable with the difference between the test results.

The thickness of oil layer was estimated according to the equation [72] which is applied in case of cylinders contact: ℎ𝑅_𝑐𝑣=1.65𝑉¹+𝑉₂⋅𝛼_𝑝⋅𝑉2𝑅_𝑛^0,73⋅𝑃^𝑛𝐸/1−𝜇²^−0.18,(29) where 𝑉₁и𝑉2—counterbodies’ velocities; 𝑉=8.7⋅10⁻⁸ m²/kg—oil dynamic viscosity; 𝛼_𝑝=4.1⋅10⁻³ kg·s/m²—oil piezocoefficient; 𝑅_𝑛—reduced radius of curvature.

Calculations showed that the porosity has a little influence on the oil layer thickness (Figure 10), which probably explains the weak dependence of the friction ratio on 𝑃 at low pressures.

Figure 10 

Dependence of the thickness ℎ of the oil film on the pressure: 1−𝑃=2%, 2−𝑃=20%.

The expression (29) tested with the pressure transducer DD-10 as follows. The sample was set and lap it dry. The sensor readings were calibrated, depending on the bias. Under dry friction the sensor readings were recorded and then counterbody was dipped into the oil. At a pressure of 0.25 MPa the rapprochement ℎ changed to 1.0–1.5 microns; this agrees well with Figure 10. At higher pressures (about 1 MPa) the accuracy of determining the change in approach is close to the calculated value of the oil film, at a pressure below 0.25 MPa tests were not carried out due to specific means for setting the sample.

The microhardness of the structural components of the friction surface was from 120 to 450 HV_0.05 (areas with high hardness (1000 HV_0.05) was not enough). According to the microhardness studies regardless of porosity, the basic structural components were sorbite and troostite.

X-ray analysis showed an increase in lattice parameter with respect to the standard on 0.03–0.06% for all samples. This confirmed the assumption about the carbon saturation of the thin surface layers under friction of iron with limited lubrication.

Thus, boundary friction could result in contact surfaces heating above the temperatures of phase transformations and the carburizing of iron. Similar results for the steels are presented in [73].

As the friction ratio, wear rate is strongly related to pore size. The greatest wear (𝑃=4%) corresponded to the minimum value of 𝐾_I𝐶. This indicates a relationship between pore size (pore size is greatest when 𝑃=4%), fracture toughness, and wear resistance.

Since the active layer thickness significantly exceeds the thickness of the oxide products of wear and the size and the depth of diffusion penetration of interstitial elements, physical and mechanical properties of the material have an effect on wear resistance. Currently accepted methodology for assessing the wear rate based on the determination of physical and mechanical properties [63, 74], but have a low accuracy of prediction under lubricated friction [74], are sufficient for engineering calculations.

Linear wear rate can be determined from the expression [59]: 𝐼_ℎ=√𝜈/42(𝜈+1)𝑛_𝑘𝑟⋅√ℎ/𝑅_𝑐𝑣⋅𝑏⋅𝜀^𝜈.(30)

Regardless of the contact type: 𝜀=ℎ𝑅_max.(31)

The film thickness, Figure 10, under the boundary friction was comparable to the rapprochement between the two surfaces, and therefore elastic contact took place.

Within the contour area under elastic unsaturated contact, the rapprochement was calculated as [59]: ℎ=5𝑃^𝑐𝑅^1/21−𝜇²𝑅^𝜈max𝑏𝜈(𝜈−1)𝐾₁𝐸^2/(2𝜈+1).(32)

The coefficient 𝐾₁=0.5 was determined from nomograms [59]. The relative rapprochement under saturated contact was assessed according to the expression [59]: 𝜀=1(𝑏𝜈)^1/(𝜈−1).(33)

In case of elastic unsaturated contact (34) for linear wear rate 𝐼_ℎ is got after substitution of (31), (32), (33) in (30). 𝐼_ℎ=√𝜈/42(𝜈+1)𝑛_𝑘𝑟⋅𝑅_max(𝑏𝜈)^1/(𝜈−1)⋅𝑅_𝑐𝑣⋅𝑏⋅1(𝑏𝜈)^1/(𝜈−1)^𝜈.(34)

In case of elastic unsaturated contact, the linear wear rate 𝐼_ℎ can be determined on the basis of the formula (35): 𝐼_ℎ=√𝜈/42(𝜈+1)𝑛_𝑘𝑟⎷5𝑃^𝑐𝑅^1/2_𝑐𝑣1−𝜇²𝑅^𝜈max𝑏𝜈(𝜈−1)𝑘₁𝐸^2/2𝑛+11𝑅_𝑐𝑣×𝑏⎛⎜⎜⎝5𝑃^𝑐𝑅^1/2_𝑐𝑣1−𝜇²𝑅^𝜈max𝑏𝜈(𝜈−1)𝐾₁𝐸^2/2𝑛+11𝑅_max⎞⎟⎟⎠^𝜈.(35) The bond between 𝐾_I𝐶, 𝑃_𝑐, and 𝑙 was found according to (36): 𝐾_I𝐶=(1−2𝜇)𝑃^𝑐√𝑙𝑡_𝑝.(36)

In the case of the elastic unsaturated contact, (35) and (23) were solved simultaneously, and according to (36), (37) was got. In the case of the elastic saturated contact, (34) and (23) were solved simultaneously and according to (36), (38) was got.

The values of 𝑅_𝑐𝑣, 𝑅_max,𝑏,𝜈 for the porous materials were taken from [59].

In the case of unsaturated elastic contact: 𝐼_ℎ=𝑃^𝑛1−𝜇²/𝐸^0.98⋅(𝑛/2−1)⋅𝐴⋅30𝑃^𝑛0.565^𝑛(𝑑/2)^(1−𝑛/2)−0.03𝐾_I𝐶/𝑃^𝑛0.565^(2−𝑛).(37)

In the case of saturated elastic contact: 𝐼_ℎ=0.0056(𝑛/2−1)⋅𝐴⋅10𝑃^𝑛^𝑛(𝑑/2)^(1−𝑛/2)−0.1𝐾_I𝐶/𝑃_𝑛^(2−𝑛).(38)

The solution was found like (39) because formulas (37) and (38) are different only in exponents 𝑃_𝑐 and in constant coefficients values, minor changes of 𝐸 and 𝜇 and 𝑃_𝑐 and 𝑃_𝑛 proportionality were also taken into account: 𝐼_ℎ=𝑎¹⋅𝑃^𝑛1_𝑛(𝑛/2−1)(𝑑/2)^(1−𝑛/2)−𝐶3⋅𝐾_I𝐶/𝑃^𝑛2_𝑛^(2−𝑛).(39)

In the area of 𝑃=8-10%, the transition from closed to open porosity occurred, so the experimental data (see Table 5) was approximated separately for 𝑃 from 2 to 8.6% (numerator) and from 20 to 8.6% (denominator). In the first case, 𝑎₁=76.2,𝐶3=0.03,𝑛₁=0.66,𝑛₂=3.77,and𝑛=2.32; in the second 𝑎₁=36.3,𝐶3=6.5,𝑛₁=0.086,𝑛₂=1.37,and𝑛=1.67.

Note that 𝑛 varies from 2 to 8 under fatigue fracture, prestrain and impurity content increase caused n increase. Among other factors, the value of 𝑛 depends on the pressure and cycle asymmetry.

In [73, 75] similar values of 𝑛 for cold-plastic steel at high cycle asymmetry are represented. Apparently, 𝑚 changing from 2.32 to 1.67 caused by pores changes. Closed porosity makes it difficult to circulate the oil, and therefore the main influence has reduced the concentration of stress. In case of open porosity, pores provide constant oil circulation and heat dissipation from the friction surfaces, so the value of 𝑛<2.

In general, dependence (39) describes the experiment with sufficient accuracy for engineering calculations; the number of undetermined coefficients is much smaller than in techniques used nowadays [63, 74].

The iron powder tribological characteristics study revealed the following patterns of relationship. Under dry friction the friction ratio 𝑓 and wear rate 𝐼_ℎ were changed nonmonotonic with increasing porosity. The reason for the nonmonotonic dependence of f(P) is in pore size and distance between them changes, that determine the deformation in the active area. Depending on the elastic and plastic deformed contacts portion, the type of contact is established, and it determines the friction ratio value, along with the physical and mechanical characteristics of the material. The surface fracture of iron powder at few MPa pressure is the nature of fatigue. The symbasis of 𝐼_ℎ(𝑃) and 𝐾_I𝐶(𝑃) dependences is determined by the contact surface fracture localization in a small volume, and the material separation occurs only when the critical value of stress concentration is reached.

Limited lubricated friction under the same pressures provides elastic contact and very little shear strength of frictional bond. The surface of the porous iron is saturated with carbon. The porosity of 8–10% (carbonyl iron) provides a transition from the closed to open porosity, which naturally affects the value of the friction ratio and wear rate.

As in the case of dry friction in boundary friction, 𝑓(𝑃) and 𝐼_ℎ(𝑃) functions are nonmonotonic. With sufficient accuracy for engineering calculations the wear rate under boundary friction can be calculated on the assumption of elastic contact.

The developed model predicted the values of fracture toughness, friction ratio, and integral linear wear rate of porous steel and bronze [74–84].

5. Conclusion

Pores’ structure changes during the consolidation process, that provides blunting and bend of the crack while moving between pores, is the reason for non-monotonic dependence 𝐾_I𝐶(𝑃). During the dry sliding and during the boundary friction 𝑓(𝑃)и𝐼_ℎ(𝑃) functions are nonmonotonic. Local 𝑓(𝑃) and 𝐼_ℎ(𝑃) extremum appearance is caused by different reasons (pore sizes and structure penetrability), but is mainly the consequence of pore structure transformation.

List of Symbols

𝐾I𝐶:	Fracture toughness, MN/m3/2
𝐾𝑄:	Estimated fracture toughness, MN/m3/2
𝑃:	Porosity, %
𝜎0.2:	Yield strength, MPa
𝑔:	Crack distribution energy
𝑇:	Linear crack front tension, N
𝐸:	Elastic modulus, MPa
μ:	Poisson’s ratio
𝐴:	Specific energy of uniform deformation
𝛾eff:	Effective surface energy of the Griffiths-Orowan theory
𝑙𝑑:	Length dimension constant
𝛿𝑒:	Uniform Deformation, %
𝜎𝐵:	Tensile Strength, MPa
𝜏:	Crack movement stress, MPa
𝑟:	Pore radius, m
Γ:	Interface energy, J/m2
𝑏0:	Interatomic distance, m
Ω:	Distance between the inhomogeneities (pores), m
𝑓:	Friction ratio
𝑃𝑛:	Nominal pressure, MPa
𝑃𝑐:	Contour pressure, MPa
𝑓𝑑:	Friction ratio deformation component
Δ:	Microroughnesses characteristic of a rigid material of the friction pair
HB:	Hardness
𝑓𝑚:	Molecular component of friction ratio
Ra:	Arithmetic mean deviation of profile, m
𝛽 and 𝜏0:	Friction characteristics
𝑆ost:	Residual variance
𝐹:	Fisher criterion
𝜏𝑚:	Crack movement stress in porous material, MPa
𝑎:	Coefficient
𝐺:	Shear modulus, MPa
𝐵:	Burgers vector, m
𝑅:	Distance between the pores, m
𝑏 and 𝜈:	Bearing surface curve parameters
𝜀:	Relative approach
𝑡𝑝:	Relative contact area
𝛼1:	The coefficient depending on the type of contact
C1:	The constant that relates the contour and nominal pressure
Δ𝑓:	Friction ratio change
𝑖ℎ:	Specific linear wear rate
ℎ:	Rapprochement between the friction surfaces, m
𝑅𝑐𝑣:	Radius of curvature
𝑛𝑘𝑟:	Number of cycles, leading to the fracture
𝐼ℎ:	Integral linear wear rate
𝑅max:	The greatest distance between the bumps and hollows within the gauge length
Δ𝐾:	Stress intensity factor change (Paris formula), MN/m3/2
𝑁:	Number of cycles (Paris formula)
𝐴 and 𝑛:	Coefficients (Paris formula)
𝐾:	Stress intensity factor, MN/m3/2
𝑙:	Crack length, m
𝑎𝑟:	Radius of contact area, m
𝜎𝑦:	Fracture stress, MPa
𝑑:	Average pore diameter, m
𝛼𝑔:	Coefficient of hysteresis losses under friction
𝑉1и𝑉2:	Counterbodies’ velocities, m/s
𝑉:	Oil dynamic viscosity, m2/kg
𝛼𝑝:	Oil piezocoefficient, kg·s/m2
𝑅𝑛:	Reduced radius of curvature, m
𝐾1:	Coefficient
C3:	Coefficient.