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Irreversible quasi-surface metallurgical phase transformations are the specific response of some metallic materials—such as metals and alloys—subjected to high thermomechanical loads applied very near their surface during the manufacturing processes or after being put into operation. These solid/solid phase transformations can be observed, for example, on the tread of many rails in railroad networks frequented by freight trains. The severe thermal and mechanical loads imposed on the surface of the rails and in the immediate vicinity of the surface by the wheel/rail contact often result in highly localized irreversible metallurgical transformations. A new kinetic model based on a previous study is presented here, which accounts more realistically for the nucleation and growth of these irreversible solid/solid phase transformations resulting from high thermomechanical loads. This metallurgical behavioral model was developed in the framework of continuum thermodynamics with gradients of temperature and internal variables.

The irreversible quasi-surface solid/solid phase transformations observed in many cases in the real industrial settings such as metal forming processes and the subsequent operating phases are the first material responses to high thermomechanical loads. In metallurgical phase transformations of this kind, which often occur in the rails of straight railway sections [

The behavioural laws (or constitutive relations) established in the framework of classical continuum thermodynamics [

(i) the “principle of determinism” (which has also been called the principle of “causality”), which states that knowledge of the past state suffices to be able to determine the current state of the material—the current thermodynamic state of an element of matter is only conditioned by its history; that is, its future state cannot influence its current state;

(ii) the “principle of local action” (which has also been called the principle of “local state”), whereby the behaviour of the material at a given point depends only on the physical state variables defined at this material point and not on those of neighboring material points—the thermodynamic state of the element of matter depends only on the history of a set of relevant variables;

(iii) the “principle of objectivity” (which has also been called the principle of “material frame indifference”), which states that the behavioural law must be independent of the observer, meaning that only objective physical quantities can be explicitly taken into account in the behavioral equations in order to ensure the invariance when making a change of the referential frame—the behavioural laws must therefore be written using the objective quantities, that is, the tensorial relations.

According to the above principles, the constitutive laws can be written in the following general form:

In addition to the principles presented above, classical continuum mechanics [

(iv) the “principle of material simplicity” (which has also been called the principle of “simple material”) which stipulates that the state of a material point depends entirely on both the history of the first strain gradient and that of the temperature.

Therefore, (

In the framework of continuum thermodynamics with internal state variables [

In the case of a classical problem in continuum mechanics, (

Note that in (

In line with (

In addition, (

(ii) The kinetics of irreversible near-surface solid/solid phase transformations depend on both the temperature and the strain history of the material (see (

Schematic diagram of the metallurgical variable

(iii) The thermodynamic admissibility of models of this kind was discussed and tested in [

(iv) Some numerical results of this local model were shown in the case of a one-dimensional problem [

In order to improve the prediction of these irreversible metallurgical transformations occurring in the presence of high localized thermomechanical loads near the surface of the material, a new kinetic model is presented in Section

For several years, the nonlocal theories developed in the framework of continuum mechanics [

In the framework of gradient theory, (

In order to account for the possible existence of strong mechanical and thermal gradients in the material subjected to high thermomechanical loads, the laws of evolution can be written (assuming the presence of small perturbations) as follows:

In line with (

The nonlocal kinetic model for describing these irreversible quasi-surface solid/solid phase transformations is

Schematic diagram of the yield criterion

(ii) The nonstandard metallurgical phase transformations studied here are activated by a certain level of mechanical stresses state, such as normal and shear stresses, via

(iii) This metallurgical transformation occurs when the mechanical stresses state reaches a sufficient magnitude to trigger the dislocation motion processes in the material caused by the strong strain incompatibilities between the former phase (the ferrite/pearlite phase) and the new phase (the martensite phase). The initiation and development of these solid/solid phase transformations are strongly influenced by the thermal and mechanical state of the material. The initial yield strength associated with the phase transformation

(iv) The rate-independent plasticity is the limit case of the rate-dependent viscoplastic materials, that is, it occurs when the viscous effects vanish (the relaxation time tends to zero,

(v) The validity domain of the kinetic model is ensured by the term

(vi) This nonlocal kinetic model can reduce to a local model in cases where the internal variables

(vii) The coefficients in the kinetic law under consideration here are the viscosity exponent

(viii) In line with the first kinetic model (Section

(ix) In the first step, we can take

(x) The value of the material parameter

(xi) Figure

(a) Schematic diagram of the wheel/rail problem arising in the case of an irreversible quasi-surface metallurgical phase transformation; (b) schematic diagram of the qualitative distribution of the metallurgical variable

In this paper, a new kinetic model is presented for predicting the onset and development of the irreversible quasi-surface solid/solid phase transformations occurring on the tracks of some railways. Based on the gradient theory, this metallurgical behavioral model is extended here to include irreversible solid/solid phase transformations taking into account the possible existence of strong mechanical and thermal gradients in the material as the result of the heavy thermomechanical loads induced by the wheel/rail contacts. For this purpose, the gradients of the temperature and the metallurgical variable have been explicitly included in the model.

The author declares that there is no conflict of interests regarding the publication of this paper.

The author is indebted to Dr. Jessica Blanc for her help with this paper.