Effects of ThermoMechanical Coupling in Tribological Surface Transformations: A One-Dimensional Modelling Including Irreversible Solid-Solid Phase Transformations and Classical Plasticity

Abstract

Under compressive loads combined with friction, some materials undergo Tribological Surface Transformations (TSTs) on the surface of the loaded parts and in the immediately vicinity, which in the case of metals, are known as irreversible solid-solid phase transformations. During the solid-solid phase transformations occurring under mechanical loads, TRansformation Induced Plasticity (TRIP) processes are generated at much lower stress levels than those associated with the yield strength of the material in classical plasticity. In order to assess the effects of thermomechanical coupling in these TSTs, a one-dimensional modelling based on irreversible solid-solid phase transformations and classical plasticity is presented and discussed.

1. Introduction

The TSTs which have been affecting some rails of the French railroad for the last twenty years consist of irreversible solid-solid phase transformations resulting in the development of a “White Etching Layer” (WELs) on the rail tread and which grows in depths from several nanometers to more than 100 μm [1]. After the passing of some trains, the first “white” phase layers appear on the surface of the rails (which initially have a ferritic-pearlitic structure) before penetrating more deeply, and resulting in a “quasi-surfacic” martensitic transformation, that is, the material composing the rails is transformed into martensite [2]. The initiation and development of TSTs are induced by the combined thermomechanical loads in the wheel/rail contact area, where the normal and tangential stresses are combined with a temperature increase due to the friction occuring in the contact, that is, TSTs result from the thermomechanical coupling processes. During the steel phase transformations which occur under mechanical loads, an anomalous plastic flow known as TRansformation Induced Plasticity (TRIP) is generally superimposed on the classical plastic processes at stress levels which are much lower than the initial strength of the softest phase. Greenwood and Johnson [3] has explained TRIP process by suggesting that the microplasticity generated by the incompatibility between the volumes involved in the phase transformation may be oriented by the external stress applied. Many models have been presented to account for TRIP phenomena [4–8]. In order to assess the thermomechanical coupling associated with TSTs, a one-dimensional version of the model presented [9], which was developed in a thermodynamically consistent framework, is presented in Section 2. In this model, the irreversible phase transformations are taken to result from “strong” thermomechanical coupling, that is, from the combined effects of the change in temperature and the mechanical dissipation. A numerical example is illustrated in Section 3 in the case of a bar subjected to a thermomechanical loading, which is liable to undergo an irreversible solid-solid phase transformation, possibly with the classical plasticity.

2. A 1D Phenomenological Modelling Approach Applied to Tribological Surface Transformations (TSTs): Thermomechanical Coupling

Based on the previous TRIP studies, which are extended here to the thermomechanical coupling associated with TSTs [9], the 1D phenomenological model presented in this Section is developed as follows:(i)the absolute temperature 𝑇 is explicitly considered and taken to be a state variable;(ii)the mass fraction of the daughter phase (i.e., the “white” martensite phase) is: 𝜌𝑧=^𝑑𝜌∈[0,1], (1) where 𝜌_𝑑=𝜌_martensite is partial mass density of the daughter phase, 𝜌=𝜌_𝑑+𝜌_𝑝 is total mass density, and 𝜌_𝑝=𝜌_ferrite is partial mass density of the parent phase (i.e., the ferrite phase);(iii)the total infinitesimal strain 𝜖 can be written as follows: 𝜖=𝜖^e+𝜖^th+𝜖^pc+𝜖^tr+𝜖^pzwith𝜖^th=𝛼𝑇−𝑇₀,𝜖^tr𝑧=−𝜅, (2)where 𝜖^e, 𝜖^th, 𝜖^pc denote the elastic, thermal, classical (visco-) plasticity strains, respectively, 𝜖^tr is the transformation strain associated with the volume variation, 𝜖^pz is TRIP-like strain, 𝑇₀ is initial temperature, 𝛼>0 is the thermal expansion coefficient, and 𝜅>0 is a material parameter characterizing the change in the density occurring during phase transformation. Note that the solid-solid phase transformation under consideration here induces a volume change, or more specifically, a decrease in the volume, that is, 𝜖^tr=−𝑧/𝜅<0, since the density mass of ferrite (the initial material of the rail before phase transformation) 𝜌_ferrite≈7700 kg·m⁻³, and that of martensite (material after phase transformation) 𝜌_martensite≈7800 kg·m⁻³.

The material in question is initially nontransformed (𝑧₀=0, 𝜖₀^tr=0 and 𝜖^pz₀=0) and nonplasticized (𝑣₀=0, and 𝜖^pc₀=0 where 𝑣 is the isotropic hardening variable associated with classical plasticity) at the temperature 𝑇₀, and the Helmholtz free energy potential per unit mass is written as: 𝜓𝑇,𝜖,𝜖^pc,𝜖^pz𝐶,𝑣,𝑧=−^𝜖2𝑇₀𝑇−𝑇₀²−𝐸𝜌₀𝛼𝑇−𝑇₀𝜖+𝐸2𝜌₀𝜖−𝜖^pc−𝜖^tr−𝜖^pz2+12𝜌₀ℎ𝑣²+𝛿𝜌₀𝜅𝑇2𝑧²−𝑇−𝑇^𝑧_𝑖𝑧+𝐼_[0,1](𝑧)+𝜓₀, (3) where 𝐶_𝜖>0 is the specific heat capacity, 𝐸 is Young’s modulus, 𝑇^𝑧_𝑖 is the solid-solid phase transformation temperature, ℎ>0 is a material parameter characterizing the linear isotropic hardening associated with classical plasticity, 𝛿≥0 is a material parameter associated with the latent heat of the phase transformation, 𝐼_[0,1](𝑧) is the indicator function (𝐼_[0,1](𝑧)=0 when 𝑧∈[0,1] and 𝐼_[0,1](𝑧)=+∞ otherwise) and 𝜌₀ and 𝜓₀ are the initial density and the initial Helmholtz free energy of the material per unit mass, respectively. In a first approximation, the thermoelastic parameters, 𝐶_𝜖, 𝐸, and 𝛼, are not assumed to depend on the temperature and are taken to be identical in the both phases. Equation (3) is strongly inspired by SMAs model with a 1-D-Helmholtz free energy potential developed by [10] where the thermomechanical coupling process considered here is extended to include an irreversible solid-solid phase transformation. Note that in (3), the first two terms correspond to the thermal leaks (irreversible part), the third one corresponds to the free energy part at constant temperature which may be immediately recoverable by an elastic unloading (reversible part), and the next two terms correspond to the hardening associated with the classical plasticity (isotropic hardening) and the latent heat of the phase transformation, respectively.

Neglecting the viscoelastic effects and noting that the specific entropy 𝑆 is a state function, the state equations based on the above representation of the Helmholtz energy (3) can be written as: 𝑆=−𝜕𝜓𝜕𝑇,𝜎=𝜌𝜕𝜓𝜕𝜖,𝐴^𝑚=−𝜌𝜕𝜓,𝜕𝑚 (4) where 𝜎 is the Cauchy stress tensor, and 𝐴^𝑚 denote the “thermodynamic forces” associated with the variables 𝑚=(𝜖^pc,𝜖^tr,𝜖^pz,𝑣,𝑧).

Using (3)-(4) and assuming that 𝜌≃𝜌₀, the Cauchy stress tensor 𝜎 is: 𝜎=𝐸𝜖−𝛼𝑇−𝑇₀−𝜖^pc−𝜖^tr−𝜖^pz. (5) The local expression for the First Principle of Thermodynamics can be written (see e.g., [11]) as: 𝜌𝑒̇=𝜎𝜖̇−div(𝐪)⟺𝜌𝑇𝑆̇+div(𝐪)=𝜎𝜖̇−𝜌𝑆𝑇̇−𝜌𝜓̇, (6) where 𝑒=𝜓+𝑇𝑆 denotes the internal energy potential per unit mass, 𝐪 is the heat flux vector; the remote heat production rate has been omitted here.

The Second Principle of Thermodynamics must be checked, whatever the local thermomechanical state 𝒮_tm=(𝑇,𝜖,𝜖^pc,𝜖^tr,𝜖^pz,𝑣,𝑧) of the material, whatever the evolution (𝑇̇,𝜖̇) locally undergone by the material, and whatever the temperature gradient ∇𝑇 acting locally on the material (see [10]) as follows: −1𝑇𝐪⋅∇𝑇+𝜌𝑇𝑆̇+div(𝐪)≥0∀𝒮_tm,∀𝑇̇,𝜖̇,∀∇𝑇. (7) Adopting Fourier’s law (𝐪=−𝑘∇𝑇, where 𝑘>0 denotes the thermal conductivity), and combining (3)-(4) and (7), the Second Principle of Thermodynamics must satisfy: 𝐴^𝑚𝑚̇≥0∀𝒮_tm,∀𝑇̇,𝜖̇,∀∇𝑇. (8) The laws of evolution of the internal state variables are.

(i) In the irreversible solid-solid phase transformation (TST) process: 𝜖̇^pz=𝑝̇𝜎|𝜎|,𝜖̇^tr𝑧̇=−𝜅,𝑧̇=𝜅𝑝̇ (9) with ̇⟨⟩𝑓𝑝=1−𝑧^pz(𝜎,𝑧;𝑇)𝜎₀𝜂𝐻|𝜎|+𝛿(1−𝑧)𝑇−𝑇^𝑧_𝑖≥0, (10) and 𝑓^pz(𝜎,𝑧;𝑇)=|𝜎|−𝜒𝜎₀𝑇exp⁰−𝑇𝑇₀𝑧+𝛽𝜅, (11) where 𝜎₀ denotes the yield strength of the softest phase (parent phase) in the TST when 𝑇=𝑇₀, 𝜒 and 𝛽 are a material parameter associated with TST’s yield strength and the hardening process, respectively, |⋅| is the absolute value function (|𝑥|=𝑥 when 𝑥≥0 and |𝑥|=−𝑥 when 𝑥<0), ⟨⋅⟩ denotes the Macaulay brackets (⟨𝑥⟩=𝑥 when 𝑥≥0 and ⟨𝑥⟩=0 when 𝑥<0),  𝜂  is the characteristic time of the viscous effects associated with the TST process, and 𝐻(⋅) denotes the Heaviside step function (𝐻(𝑥)=1 when 𝑥≥0 and 𝐻(𝑥)=0 when 𝑥<0).

It must be stressed that: (i) TSTs (or WELs), which correspond to the ferrite-pearlite → martensite transformations, do not fit the framework of conventional metallurgical transformations. According to these metallurgical theories, the white layer (martensite) may result from the martensitic transformation (high velocity cooling) after a sufficiently large temperature increase to produce an austenitic transformation. Dang Van et al. [12] have established that in the case of a steadily moving train, the temperature will increase at the surface of the rails by 150^∘C or so at most. Therefore, this temperature increase alone, which is caused by friction involved in the contact, is obviously too small to generate solid-solid phase transformations. The assumption adopted here is therefore that the thermomechanical coupling, where the thermal stresses superimposed on the purely mechanical (isothermal) stresses, can be taken to generate TSTs. (ii) Equation (11) means that the irreversible solid-solid phase transformation is activated by the stress 𝜎 which acts as “driving force,” and accelerated by temperature 𝑇 (see [9] for more details). TST occurs at stress levels which trigger of dislocation motions due to the strain incompatibilities between the ferrite-pearlite and martensite phases. The triggering of these dislocation motions is only facilitated by the increase in the temperature. In agreement with (i), (10) reflects the fact that the validity field of the law of evolution governing the TST process requires the presence of both the stress and the temperature.

(ii) In the classical plasticity process: 𝜖̇^pc=𝑣̇𝜎|𝜎|̇𝑓with𝑣=^pc(𝜎,𝑣)𝜉𝜎_𝑦(𝑧)≥0, (12) with 𝑓^pc(𝜎,𝑣)=|𝜎|−𝜎_𝑦(𝑧)𝑇exp⁰−𝑇𝑇₀+ℎ𝑣 (13) and 𝜎_𝑦(𝑧)=(1−𝑧)𝜎₀+𝑧𝜎₁, (14) where 𝜎_𝑦(𝑧) denotes the classical yield strength of the material when 𝑇=𝑇₀,  𝜎₁ is the yield strength of the hardest phase (daughter phase) and 𝜉 is the characteristic time of the viscous effects associated with the classical plasticity process. Note that when 𝑧=0 (resp., 𝑧=1), the yield strength 𝜎_𝑦(𝑧) is only composed to one phase, that is, the softest phase (resp., the hardest phase) and it consists of two phases when 𝑧∈]0,1[.

Combining (3), (4), and (6) and Fourier’s law, the heat equation is given by: 𝜌𝐶_𝜖𝑇𝑇₀̇̇𝜖̇𝑇−𝑘Δ𝑇=−𝐸𝛼𝑇𝜖+𝜎^pc−ℎ𝑣𝑣̇+𝜎𝜖̇^tr+𝜖̇^pz−𝑇^𝑧_𝑖𝛿𝜅𝑧,̇ (15) where Δ denotes the Laplacian operator. The quantities on the right-hand side of (15) are the “source terms” corresponding to the thermoelasticity, the classical plasticity, and the irreversible solid-solid phase transformation, respectively.

It is worth noting the thermodynamic consistency of this model [9] is checked since the laws of evolution of the internal state variables verify systematically (8).

3. Numerical Example

Preliminary Remark. The case of the structure studied in this section is not representative of the complex processes that could be expected between wheel and rail, such as nonlinear contact and friction. This one-dimensional problem was simply intended to show that the thermomechanical coupling can generate the initiation and development of TSTs in the immediate vicinity where a sufficient thermomechanical load level is applied on the structure. Therefore, it leads on naturally to first the step to the setting up of a TSTs benchmark.

This example will be addressed in order to assess the thermomechanical coupling occurring in the TSTs using the approach presented in Section 2, subject to the assumptions stated therein. Let us consider a solid occupying a one-dimensional space Ω∈[0,𝐿]⊂ℜ, measuring 10mm of length, that is, a solid bar where the material considered here is made of steel. On the left end of the bar, a thermomechanical loading {𝜎^𝑔₀(𝑡),𝑇^𝑔₀(𝑡)}, which progresses linearly in conjunction with time 𝑡, is applied whereas both zero displacement (𝑢^𝑔_𝐿(𝑡)=0) and a set temperature (𝑇^𝑔_𝐿(𝑡)=𝑇₀) are imposed constant with time on the right end of the bar, that is, 𝑢^𝑔_𝐿(𝑡)=0,  ∀𝑡 and 𝑇^𝑔_𝐿(𝑡)=𝑇₀, ∀𝑡 (see Figure 1). The mechanical component of the loading is a compression which decreases from zero to −300MPa in 0≤𝑡≤0.1s and its thermal component, and it increases from 300K to 450K in the same time. Bearing in mind that the one-dimensional problem studied cannot take into account both the non-uniformity of normal and shear stresses in contact area including the temperature increase caused by friction, the maximum values chosen to describe the thermomechanical loading would seem to be realistic enough at a contact point under moderate load for pressure and temperature in increase. Furthermore, in a first approximation, the lateral surfaces of the bar is assumed to be adiabatic thereby both the temperature gradient ∇𝑇 and heat flux 𝐪 are one-dimensional (here, 𝑥-direction), that is, ∇𝑇=(𝜕𝑇/𝜕𝑥)𝐞_𝑥, ∀𝑡 and 𝐪=−𝑘(𝜕𝑇/𝜕𝑥)𝐞_𝑥, ∀𝑡 (where 𝐞_𝑥 is a unit vector).

(a)

(b)

Figure 1 

(a) Numerical example of TSTs: bar subjected to a thermomechanical loading (where 𝐿=10mm). (b) Boundary conditions (where 𝑇0=300K): the loading function 𝜎𝑔0(𝑡) (resp., 𝑇𝑔0(𝑡)) is represented in solid lines with 𝜎0𝑔(min)=−300MPa at time 𝑡𝐷=10−1s (resp., dashed lines with 𝑇0𝑔(max)=450K). Note that 𝑡𝐴=7.3×10−2s with 𝜎𝑔0(𝑡𝐴)=−220MPa and 𝑇𝑔0(𝑡𝐴)=409.5K; 𝑡𝐵=7.5×10−2s with 𝜎𝑔0(𝑡𝐵)=−224MPa and 𝑇𝑔0(𝑡𝐵)=412K; 𝑡𝐶=9.4×10−2s  with 𝜎𝑔0(𝑡𝐶)=−283MPa and 𝑇𝑔0(𝑡𝐶)=441K.

Assuming that the convective terms of particular derivative are zero, (15) can be written in the following form: 𝜌𝐶_𝜖𝑇𝑇₀𝜕𝑇𝜕𝜕𝑡−𝑘²𝑇𝜕𝑥²=−𝐸𝛼𝑇𝜕𝜖𝜕𝑡−ℎ𝑣𝜕𝑣𝜕𝑡−𝑇^𝑧_𝑖𝛿𝜅𝜕𝑧𝜕𝑡+𝜎𝜕𝜖^tr+𝜕𝑡𝜕𝜖^pz+𝜕𝑡𝜕𝜖^pc𝜕𝑡inΩ, (16) where 𝜕(⋅)/𝜕𝑡 and 𝜕²(⋅)/𝜕𝑥² denote the first and second partial derivatives with respect to 𝑡 and 𝑥 (temporal and spatial quantities).

Neglecting the effects of both the inertia and gravity, the local expression of quasi-static equilibrium reduces to: 𝜕𝜎𝜕𝑥=0inΩ. (17) In line with (17), the stress field 𝜎(𝑥,𝑡) is uniform, that is, 𝜎(𝑥,𝑡)=𝜎^𝑔₀(𝑡),∀𝑥∈[0,𝐿]. The geometry involved and the boundary conditions are summarized in Figure 1. The constitutive thermomechanical parameters used are: 𝐶_𝜖=450J⋅kg⁻¹⋅K⁻¹; 𝐸=2×10⁵MPa; 𝜌=78×10⁻⁷ kg·mm⁻³; 𝛼=12×10⁻⁶K⁻¹;  𝜅=10²;  𝜎₀=380MPa;  𝜎₁=465MPa; 𝑘=5×10⁻²W·mm⁻¹·K⁻¹;  𝜂=10²s;  𝜉=2×10²s; 𝛽=6×10³MPa;  ℎ=2×10⁴MPa;   𝜒=4/5; 𝑇^𝑧_𝑖=1000K; bmbox?>𝛿=2.5×10⁻¹⁰J·mm⁻³·K⁻¹. Note that 𝐶_𝜖, 𝑘, 𝐸, and 𝛼 are classical thermoelastic parameters associated with a ferritic or martensitic steel, which are considered here constants in temperature range studied. As pointed out in Section 2, the material parameter 𝜅 characterizing the change in the density occurring during phase transformation (where density mass of ferrite 𝜌_ferrite≈7700 kg·m⁻³ and martensite 𝜌_martensite≈7800 kg·m⁻³). Therefore, based on these experimental data, the transformation ferrite → martensite is accompanied by a volume contraction (for a constant mass given) to be taken into account, that is, 𝜌_martensite≅1+2×10⁻²𝜌_ferrite. (18) On the other hand, the equation of conservation of mass (under small perturbations assumption) can be written as: 𝜌≃𝜌₀(1−tr(𝝐))⟺𝜌≃𝜌₀(1−𝜖)inΩ. (19) Combining (5), (9), and (12) and considering both a return to thermal and mechanical equilibrium (i.e., 𝑇=𝑇₀ and 𝜎=0), we can write that: 𝜖=𝜖^pc+𝜖^tr+𝜖^pz. (20) Assuming both that |𝜖^pc|⋘|𝜖^tr| and |𝜖^pc|⋘|𝜖^pz| and using (9) (here for a mechanical evolution 𝜎̇<0 and then a return to the mechanical state 𝜎=0), (20) can be rewritten: z𝜖=−2𝜅. (21) In the case of full phase transformation (𝑧=𝑧_max=1), (21) can be reduced to: 2−𝜖=𝜅. (22) Combining (19) and (22), the expression of (18) can be put in the following form: 𝜌_martensite≅1+2𝜅⁻¹𝜌_ferrite. (23)

From (18), we can deduce that 𝜅=10². Note also that the characteristic times of the viscous effects associated with the TRIP-like process 𝜂 and the classical plasticity 𝜉 are rather low which means that the viscous effects are also small (when 𝜂→0 and 𝜉→0, these are nonexistent). The reader is invited to refer to [9] the physical interpretations associated with other parameters including their chosen set value.

The results of the numerical simulations are presented in Figures 2, 3, 4, and 5. They show that due to both the nonstationary heat equation with source terms (16) and quasi-static equilibrium (17) combined with the thermomechanical loading (see Figure 1), a uniform 𝜎-field (see Figure 2(a)) and a non-uniform temperature distribution strongly localized (near where the load is applied, see Figure 2(b)) are obtained throughout of the bar. These source terms present in the heat equation will provide a maximum temperature increase of a few degrees (around 15 K) which occurs near the left end of the bar (at time 𝑡_𝐷), whereas no such increase occurred elsewhere. The increase of about 10 K, which can be observed in the close vicinity of the point where the thermomechanical loading is applied (in the same time), is due only to the thermoelastic coupling. These results indicate that a “new” phase (𝑧≠0) develops in the immediate vicinity of the point on the bar where the thermomechanical loading is applied: from 𝑥=0 to about 0.2mm at time 𝑡_𝐵, 1.6mm at time 𝑡_𝐶, and 2.8mm in depth at time 𝑡_𝐷 (see Figure 3(a)). Figure 3(b) brings out the distribution of the TRIP-like strain 𝜖^p𝑧 in the depth of the bar at time 𝑡={𝑡_𝐵,𝑡_𝐶,𝑡_𝐷}. Note that at time 𝑡_𝐴, 𝑧=𝜖^pz=0, ∀𝑥. A full phase transformation (𝑧=𝑧_max=1) develops up to about 0.1mm at time 𝑡_𝐷 (see Figure 4(a)), where there is only one phase, the daughter phase (martensite). At depths of 0.1<𝑥<2.8mm, the partial transformation (𝑧<1) persists, that is, the material is a mixture between the parent phase (ferrite-pearlite) and the daughter phase (martensite). Distribution of transformation strain 𝜖^tr in total bar length is also depicted in the same time. Figure 4(b) shows the distribution of the TRIP-like and classical plasticity strain in the depth of the bar at time 𝑡_𝐷=10⁻¹s. Near the left end of the bar, the TRIP-like strain 𝜖^pz predominates, reaching with a extremum value of −10⁻² at time 𝑡_𝐷=10⁻¹s, whereas the strain associated with classical plasticity amounted to 𝜖^pc≈−0.0085×10⁻⁶ at the same time. It can be observed that at the point 𝑥=𝐿/100 (Figure 5): (i) on the loading path 0→𝐴 (where 𝑡∈[0,𝑡_𝐴]), the response of the material is purely thermoelastic (i.e.,  𝑧=𝜖^tr=𝜖^pz=𝜖^pc=0). The thermal effects gives a non-linear stress-strain curve; (ii) on the loading path 𝐴→𝐶 (where 𝑡∈[𝑡_𝐴,𝑡_𝐶]), thermo-elasto-viscoplastic behavior occurs due to the solid-solid phase transformation and classical plasticity (i.e.,  𝑧≠0, 𝜖^tr≠0, 𝜖^pz≠0, 𝜖^pc≠0); (iii) on the loading path 𝐶→𝐷 (where 𝑡∈[𝑡_𝐶,𝑡_𝐷]), a return to the thermoelastic response of the material (i.e.,  𝑧̇=𝜖̇^tr=𝜖̇^pz=𝜖̇^pc=0) occurs.

(a)

(b)

Figure 2 

(a) 𝜎-field in the depth of the bar at time 𝑡𝐴 (dotted line), 𝑡𝐵 (dashed-dotted line), 𝑡𝐶 (dashed line), and 𝑡𝐷 (solid line). (b) Distribution of temperature 𝑇 in the depth of the bar at time 𝑡𝐴 (dotted line), 𝑡𝐵 (dashed-dotted line), 𝑡𝐶 (dashed line), and 𝑡𝐷 (solid line).

(a)

(b)

Figure 3 

(a) 𝑧-field in the depth of the bar at time 𝑡𝐵 (dashed-dotted line), 𝑡𝐶 (dashed line), and 𝑡𝐷 (solid line). (b) Distribution of the TRIP-like strain 𝜖pz in the depth of the bar at time 𝑡𝐵 (dashed-dotted line), 𝑡𝐶 (dashed line), and 𝑡𝐷 (solid line) and at time 𝑡𝐴, 𝑧=𝜖pz=0, ∀𝑥.

(a)

(b)

Figure 4 

(a) Distribution of mass fraction 𝑧 (solid line) and transformation strain 𝜖tr (dashed line) in the depth of the bar at time 𝑡=𝑡𝐷. (b) Distribution of the TRIP-like strain, 𝜖pz (solid line), and the classical plasticity strain, 𝜖pc (dashed line), in the depth of the bar at time 𝑡=𝑡𝐷.

(a)

(b)

Figure 5 

Stress/strain curve (solid line) and stress/TRIP-like strain curve (dashed line) in 𝑥=𝐿/100 at time 𝑡∈[0,𝑡𝐵] (a) and 𝑡∈[𝑡𝐵,𝑡𝐷] (b).

The results presented in Section 3 are in line with the experimental available data on TSTs [1], since a full layer transformed is strongly localized and achieved a realistic TST thickness, that is, at a maximum depth of around 100 μm (see Section 1). The thermomechanical model proposed here can describe both the initiation and development of TSTs at the surface and in the immediate vicinity where the thermomechanical loading was applied. Therefore, this model will be examined on the wheel/rail contact where both the nonlinear contact and friction can take place.

4. Conclusion

The thermodynamically consistent model presented in this study can be used to simulate the initiation and development of Tribological Surface Transformations (TSTs). The results obtained here show that the thermomechanical coupling can generate a thin layer of fully irreversible solid-solid phase transformation in the immediate vicinity of the point on the bar where the thermomechanical loading is applied. These results also highlight that the layer thickness transformed is in line with the experimental available data on TSTs since a depth of around one hundred micrometers has been reached. In the future studies, it is proposed to test the present thermomechanical model in more realistic numerical simulations involving issues relatied to wheel/rail contact problems.

Acknowledgment

The author is indebted to Professor Thierry Désoyer and Professor Frédéric Lebon for the valuable discussions.