# Mechanothermodynamic Entropy and Analysis of Damage State of Complex Systems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Thermomechanical Statement

_{ij}are stresses; $\mathsf{\rho}$ is density; f

_{i}are volumetric forces; ${v}_{i}$ are velocities.

_{ij}is the strain rate; Π is the continuum surface; the l are the direction cosines at the continuum surface; δU/dt is the power of internal forces; δA/dt is the power of internal surfaces and volumetric forces.

_{i}is the characteristic of the heat flux per unit area of continuum surface per unit time due to heat conduction; z is the constant of heat radiation per unit mass per unit time.

^{(e)}) or inside the system itself (the increment ds

^{(i)}) [6,7]:

^{(i)}is equal to zero in reversible processes and is greater than zero in irreversible processes.

_{k}is the number of mols per unit mass.

## 3. Main Principles

_{x}, Q

_{y}, Q

_{z}, longitudinal forces N

_{x}, N

_{y}, N

_{z}and also into the bending moments M

_{x}, M

_{y}, M

_{z}. Element B is pressed to element A by the loads that are transformed into the distributed normal pressure p(x, y) and the tangential tractions q(x, y). The origin of the coordinates is placed at the point of original contact O of the two elements (prior to deformation). It is easy to see that the elements A and B together form the Tribo-Fatigue system [4] which could be reduced to the friction pair [2] in the absence of internal forces (${N}_{i}=0$, ${Q}_{i}=0$, ${M}_{i}=0$, i = x, y, z). Thus, the Tribo-Fatigue system is the friction pair in which at least one of the elements perceives non-contact loads and, consequently, undergoes volumetric deformation. This representation of the MTD system has an advantage that the analysis of the states of a solid and the components of a system can adopt the appropriate solutions known in mechanics of deformable solid, in contact mechanics, in mechanics of tribo-fatigue systems (tribo-fatigue) and in tribology.

_{Λ}takes into account the irreversible kinetic interaction of particular damage phenomena. The components ${u}_{T}^{eff},{u}_{n}^{eff},{u}_{\mathsf{\tau}}^{eff}$ of the effective energy ${u}_{\sum}^{eff}$ have no property of additivity.

_{ch}≤ 1 and can be studied, for example, as electrochemical damage under the influence of temperature (D

_{T}

_{(ch)}), stress (D

_{σ(ch)}), and friction corrosion (D

_{τ(ch)}). So function (25) takes the form:

_{0}in some area of limited size—in the dangerous volume of the MTD system.

_{0}is considered to be a fundamental constant for a given material. It shouldn’t depend on testing conditions, input energy types, damage mechanisms.

_{k}, k = 1, 2, …, are some characteristic properties (hardening-softening) of contacting materials, ${\mathsf{\Lambda}}_{k\backslash l\backslash n}$ ⋛ 1 are the functions (parameters) of dialectic interactions of effective energies (irreversible damages) that are caused by loads of different nature. This means that at Λ

_{k}> 1, the damage increase is realized, at Λ

_{l}< 1—its decrease, and at Λ

_{n}= 1—its stable development.

_{0}is the time of origination of the system and ${T}_{\oplus}$ is the time of reaching the limiting state, then the failure time of its functions corresponds to the relative lifetime (longevity) $t/{T}_{\oplus}=1$. But the system lifetime ${T}_{*}$ as the material object is longer than its lifetime as the functional integrity (${T}_{*}>>{T}_{\oplus}$) since at the time moment $t>{T}_{\oplus}$ the process of degradation—disintegration is realized by forming a great number of remains, pieces, fragments, etc. This process develops under the influence of not only possible mechanical loads but mainly of the environment—up to the system death as the material object at the time moment t = ${T}_{*}$. The system death means its complete disintegration into an infinitely large number of ultimately small particles (for example, atoms). The translimiting existence of the system as a gradually disintegrating material object can then be described by the following conditions:

_{0}which is independent of the conditions of deformation, shapes and sizes of objects. Limiting energy u

_{0}is considered as the initial activation energy of the disintegration process (critical energy) which approximately corresponds both to the sublimation heat for metals and crystals with ionic bonds and to the activation energy of thermal destruction for polymers. It is independent of conditions and ways of reaching the limiting state. Longevity of a gear or a shaft in the given conditions of operation would be different but determined by fundamental parameter u

_{0}.

_{0}. In cases of different sizes, shapes, number of cycles and stress concentrators effective energy is different. Therefore it may need different conditions and time for effective energy to reach fundamental material constant u

_{0}.

_{0}(see Section 5 “Mechanothermodynamical states”). Therefore the proposed formulation of entropy simultaneously contains acting and critical energy. The limiting state occurs when this ratio is equal to 1. Latter studies could show effectiveness of this in certain cases.

## 4. Energy Theory of Damage and Limiting States

_{∑}can be written with regard to the rule of disclosing the biscalar product of the stress and strain tensors σ and ε:

_{τ}is the tensor of friction-shear stresses, or, briefly, the shear tensor and σ

_{n}is the tensor of normal stresses (tension-compression), or, briefly, the tear tensor. So in (28), the tear part σ

_{n}and shear part σ

_{τ}of the tensor σ will be set as:

_{n}of the tensor σ is the sum of the tear parts of the tensors at the volume strain ${\mathsf{\sigma}}_{n}^{\left(V\right)}$ and the surface load (friction) ${\mathsf{\sigma}}_{n}^{\left(W\right)}$, whereas the shear part σ

_{τ}is the sum of the shear parts ${\mathsf{\sigma}}_{\mathsf{\tau}}^{\left(V\right)}$ and ${\mathsf{\sigma}}_{\mathsf{\tau}}^{\left(W\right)}$. This means the vital difference of the generalized approach to the construction of the criterion for the limiting state of the MTD system.

_{n}(V), A

_{τ}(V) and A

_{T}(V) that determine the fraction of the absorbed energy:

_{M\T}(V) and Λ

_{τ\n}(V) are the functions of interaction between energies of different nature. The subscript τ\n means that the function Λ describes the interaction between the shear (τ) and tear (σ) components of the effective energy, and the subscript М\T means that the function Λ describes the interaction between the mechanical (M) and thermal (T) parts of the effective energy. That fact that the coefficients A can be, generally speaking, different for different points of the volume V, enables one to take into account the inhomogeneity of environment.

_{0}, the limiting (or critical) state of the MTD system (of both separate elements of the system and the system as the integrity) is realized. Physically, this state is determined by many and different damages.

_{0}has been mentioned above. According to [82,83,84,85,86,87,88,89,90,91,92,93,94,95], the parameter u

_{0}will be interpreted as the initial activation energy of the disintegration process. It is shown that the quantity u

_{0}approximately corresponds both to the sublimation heat for metals and crystals with ionic bonds and to the activation energy of thermal destruction for polymers:

_{0}is determined as the activation energy for mechanical fracture:

_{0}can be considered to be the material constant:

_{k}is the reduction coefficient, σ

_{th}is the theoretical strength, E is the elasticity modulus, C

_{a}is the atom heat capacity, α

_{V}is the thermal expansion of the volume, k is the Boltzmann constant, T

_{S}is the melting point, θ

_{D}is the Debye temperature, h is the Planck constant. According to (49), it can be taken approximately [84]:

_{0}have also been developed [85].

_{0}is the activation energy of a given material, which is by the order of magnitude equal to 1–10 eV per one particle or molecule (~10

^{2}–10

^{3}kJ/mol), i.e., the value that is close to the energy of interatomic bond rupture in the solid [88]. Its level doesn’t depend on how the rupture is —mechanically, thermally or by their simultaneous action. In [85], it is possible to find the tables containing the u

_{0}values for different materials.

_{σ}characterizes the strength loss per 1 К.

_{0}, on the distribution function parameters p(σ

_{−1}) and p(σ) of the durability limit σ

_{−1}and the effective stresses σ considering both the effective stress probabilities P and γ

_{0}and gradients G

_{σ}:

_{k\l\n}for damage interactions in the MTD system is determined by the parameters ρ of the effective energy ratio (Λ

_{k\l\n}):

_{ch}≤ 1 is the parameter of corrosion-electrochemical damage of the body, then based on [2,4,92], criterion (26) with regard to its shape will be as follows:

_{e}are the coefficients responsible for corrosive erosion processes; m

_{V}

_{(•)}are the parameters responsible for the electrochemical activity of materials at force (the subscript σ), friction (the subscript τ), and thermodynamic (the subscript T) loads, wherein m

_{V}

_{(•)}= 2/A

_{ch}and the parameter A

_{ch}⋛ 1.

_{ch}can be found. As seen, Equation (72) is the specification of criterion (27). According to this criterion, the limiting state of the MTD system is reached when the sum of dialectically interacting irreversible damages at force, friction, and thermodynamic loads (including electrochemical damage when acted upon by stress, friction, temperature) becomes equal to unity.

_{σ}(V) = A

_{σ}= const, A

_{τ}(V) = A

_{τ}= const, A

_{T}(V) = A

_{T}= const, ${\mathsf{\Lambda}}_{\mathsf{\tau}\backslash n}$ (V) = ${\mathsf{\Lambda}}_{\mathsf{\tau}\backslash n}$ = const, ${\mathsf{\Lambda}}_{M\backslash T}$ (V) = ${\mathsf{\Lambda}}_{M\backslash T}$ = const.

_{w}, can be neglected. Then (40) assumes the following form:

_{d}, τ

_{d}are the force and friction limiting stresses as T → 0. These are called the limits of (mechanical) destruction, T

_{d}is the destruction temperature (when σ = 0, τ

_{w}= 0) or the thermal destruction limit.

^{eff}. Then the generated energy u is equal to:

_{cons}is the non-absorbed part (it is called the conservative part) of the generated energy u.

_{cons}passed through the material strain volume V and the generated energy u is of the form:

_{ε}independent of u is the energy absorption parameter.

^{eff}= 0. If V → ∞ it appears that according to (81) u

_{cons}= u, i.e., all input energy is dissipated within such a volume.

- transition of electrons in absorbing atoms from lower to higher energy levels (quantum theory);
- generation and development of dislocation structures (dislocation theory);
- emergence of II and III order residual strains (stresses) (elasticity theory);
- formation and development of any imperfections (defects) of material composition and structure—point, planar and spatial (physical materials science);
- hardening-softening phenomena (including strain aging) developing in time (fatigue theory);
- changes in (internal) Tribo-Fatigue entropy (wear-fatigue damage mechanics [2]).

_{γ}where the subscript γ denotes the shear strain. Similarly, heat absorption in the deformable solid body can also be considered. Finally, the problem of strain energy absorption in the non-uniform (including complex) stress state can be easily solved by putting the dangerous volume $V={V}_{P\mathsf{\gamma}}$ into (81)–(83).

_{n}

_{\τ}takes into account the interaction of effective mechanical energy components caused by friction τ

_{w}and normal σ stresses, whereas Λ

_{M\}

_{T}takes into account the interaction of the thermal and mechanical components of the effective energy. The effective energy thermal component is determined by the variations of the total temperature T

_{∑}= T

_{2}− T

_{1}in the bodies contact zone caused by all heat sources, including the heat released during mechanical (spatial and surface) strain, structural changes, etc.

## 5. Mechanothermodynamical States

_{l}in formulas (41), (42), (77) [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68]:

_{l}are the specific internal energies at tear (u

_{n}), shear (u

_{τ}), thermal action (u

_{T}).

_{α}are the possible combinations of interaction of effective energies (irreversible damages).

_{l}and Λ-functions it is possible to assess energy interaction due to different-nature loads. Such interaction can cause both a sharp growth and a substantial decrease of effective energy, resulting in damages and limiting states, as compared to the one calculated by the ordinary additivity model of type (17):

_{0}is the limiting density of the internal energy interpreted as the initial activation energy of the disintegration process.

_{0}is the working volume of the solid.

^{−1}·K

^{−1}).

_{0}.

_{0}is explicitly introduced into the calculation of the specific entropy ${s}_{TF}$. Thus, ${s}_{TF}$ and ${S}_{TF}$ allow one to answer the question how much the current state of a solid or a system is dangerous in comparison with limiting states.

_{l}not yielding primary damages and, hence, the limiting states—the points of a qualitative change of the system.

_{S}, ${\mathsf{\omega}}_{S}\left(t\right)$ can grow infinitely, allowing not only the limiting states of type (93), but also different transmitting states to be described; in essence, they “provide” a quantitative description of the law of increase of entropy.

_{ψ}:

_{ψ}given in [68] can be re-written in the following form:

## 6. Laws of Mechanothermodynamics

- Damages are the fundamental physical property (and the functional duty) of any system and all of its elements.
- Damage of each object (any existing one) inevitably grows up to its breakdown—decomposition (disintegration) into a set of particles of arbitrarily small size, i.e., it is the unidirectional process of time.
- Not only the unity and struggle of opposites but also the directivity of various and complex physical processes of hardening-softening (depending on the level of loads and time) are typical of the evolution of the system by damage. It means that the Λ-function of damage interactions (of all kinds) can take three classes of values: (a) Λ < 1 when the hardening process is dominant; (b) Λ > 1 when the softening process is dominant; (c) Λ = 1 when a stable hardening to softening process ratio is found.

## 7. Analysis of Experimental Data

_{w}= 0) or purely mechanical damage [when T

_{∑}→ 0)] will be as follows:

_{w}= 0) we have:

- The growth of loading parameters (σ, τ
_{w}, T_{∑}, D) results in the corresponding acceleration of reaching the limiting state (u_{0}). - The limiting state of the system can also be reached by increasing only one (any) of the loading parameters (when maintaining the same values of other parameters).
- If $\mathsf{\Lambda}$ > 1, the damageability of the system accordingly enhances (i.e., the processes of its softening are dominant), and if $\mathsf{\Lambda}$ < 1 it slows down (i.e., the processes of its hardening appear to be preferable) in comparison with the damage due to the joint action of loading parameters alone (with no regard to the dialectic interaction of irreversible damages).

_{T}in the double logarithmic coordinates is to be a straight line with the angular coefficient (1/2). The general regularity is as follows: the higher the value of the parameter C

_{T}, the greater is the quantity ${\mathsf{\sigma}}_{-1T}$. Figure 6 shows a satisfactory evidence of this dependence for numerous different-grade steels tested for fatigue in different conditions [93,95,103,104,105]. It is seen that the C

_{T}value varied by more than two orders, i.e., by a factor of 100 or more, and the values of the endurance limit ${\mathsf{\sigma}}_{-1T}$—by more than two orders, i.e., by a factor of 10 or more, thus the testing temperature varied in the range from the helium temperature to 0.8 T

_{s}(${T}_{S}$ is the temperature of melting). As shown in Figure 6, Equation (138) adequately describes the results of 136 experiments.

_{иT}is the stress limit). In this case, it is taken that σ

_{–1}= σ

_{иT}in Equation (138). It is obvious: the correlation coefficient is very high—not less than r = 0.722 (very occasionally), but in the most cases it exceeds r = 0.9; the analysis includes more than 300 test results. Works [103,104] and may others contain other examples of successful experimental approbation of criterion (138). This allows us to hope that even more general criteria (for example, Equations (77) and (78)) will appear to be practically acceptable. In our opinion, further studies should confirm our hope.

## 8. Conclusions

- energy theory of limiting states,
- energy theory of damage,
- foundations of the theory of electrochemical damage.

_{S}, where T

_{S}is the material melting temperature) and mechanical stresses (up to the strength limit for single static loading) while the fatigue life was of the order of 10

^{6}–10

^{8}cycles.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Typical objects of modern machines and equipment with the scheme of the elementary MTD system.

**Figure 5.**Tribo-Fatigue bridges from Mechanics (M) and Thermodynamics (T) to MTD Mechanothermodynamics (the solid lines with arrows; the dashed lines show the unrealized ways (for more than 150 years) from M or T to MTD).

**Figure 8.**Explicit temperature dependences of the fatigue limit for metal materials (based on 136 test results of various authors [1]).

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Sosnovskiy, L.A.; Sherbakov, S.S.
Mechanothermodynamic Entropy and Analysis of Damage State of Complex Systems. *Entropy* **2016**, *18*, 268.
https://doi.org/10.3390/e18070268

**AMA Style**

Sosnovskiy LA, Sherbakov SS.
Mechanothermodynamic Entropy and Analysis of Damage State of Complex Systems. *Entropy*. 2016; 18(7):268.
https://doi.org/10.3390/e18070268

**Chicago/Turabian Style**

Sosnovskiy, Leonid A., and Sergei S. Sherbakov.
2016. "Mechanothermodynamic Entropy and Analysis of Damage State of Complex Systems" *Entropy* 18, no. 7: 268.
https://doi.org/10.3390/e18070268