# On the Role of Entropy Generation in Processes Involving Fatigue

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## Abstract

**:**

## 1. Introduction

## 2. Thermodynamics of Fatigue

#### 2.1. Entropy Balance Equation

**J**

_{q}is the heat flux, T is the absolute temperature,

**σ**is the stress tensor,

**ε**

_{p}is the plastic part of strain tensor, V

_{k}represents the internal variables associated with microstructure, A

_{k}are the thermodynamic forces associated with the internal variables.

#### 2.2. Entropy Generation Approach to Fatigue Failure

_{f}, undergoing repeated cyclic load as it reaches the point of fracture is a constant value, independent of load amplitude, geometry, size of specimen, frequency and stress state; see also [8,9,10,11]. The total entropy gain, or the so-called fatigue fracture entropy, can be evaluated by integrating Equation (2) from time t = 0 to t = t

_{f}, when fracture occurs:

^{3}K and about 60 MJ/m

^{3}K for Stainless Steel 304L regardless of the load amplitude, geometry, size of specimen, frequency and stress state. In what follows typical result of accumulation of entropy generation is given to illustrate the concept.

_{f}= 4.07 MJ/m

^{3}K, regardless of the displacement amplitude. At the beginning of the test, the accumulation of entropy is nil and it linearly increases until it reaches roughly 4.07 MJ/m

^{3}K, at which point fracture occurs.

**Figure 1.**Evolution of entropy accumulation during fatigue tests pertaining to bending load of Aluminum 6061-T6.

#### 2.3. Application to Fatigue Life Prediction (Coffin-Manson Equation)

_{p}, can be estimated using the following formula presented in the pioneering work of Morrow [12] on the assessment of energy generation during fatigue as:

_{p}is the plastic strain range, ${n}^{\prime}$ is the cyclic strain hardening exponent, ${\epsilon}_{f}^{\prime}$ and ${\sigma}_{f}^{\prime}$ are fatigue ductiliy and strength coefficients of the material. Morrow [12] experimentally demonstrates that in fully reveresed fatigue tests, the amount of energy generation per cycle is aproximately constant, but varies with the strain level, Δε, and the cyclic properties of the material. Considering this assumption, Equation (4) yields to the following:

_{p}, from Equation (5), into Equation (6) and rearranging the resulting equation we obtain:

#### 2.4. Application to Variable Load Amplitude (Miner’s Rule)

_{i}, i=1, 2, …, n. Let D represent the material degradation defined as the ratio of the accumulation of entropy generation divided by the fracture fatigue entropy, viz.:

_{1}, γ

_{2}, γ

_{3}, …, are the entropy generations at stress levels σ

_{1}, σ

_{2}, σ

_{3}, …, respectively. Employing Equation (6), γ

_{i}can be written as:

_{i}, and N

_{i}denotes the number of cycles elapsed at the corresponding stress level. Given that the fracture fatigue entropy, γ

_{f}, is a material property and that it is independent of the stress level [7], the following relationship can be obtained from Equation (6):

_{f,1}, N

_{f,2}, N

_{f,3}, …, are the fatigue lives from constant stress amplitude at stresses σ

_{1}, σ

_{2}, σ

_{3}, …, respectively. Substituting Equations (9) and (10) into Equation (8), yields:

_{f}. This condition corresponds to D = 1. Therefore, from Equation (11) it follows:

#### 2.5. Degradation Coefficient (DEG Theorem)

_{i}S, in terms of experimentally measureable quantities. Following the notation of Bryant et al. [15], suppose that a system is divided into j = 1, 2, …, n subsystems with dissipative processes p

_{j}, where each ${p}_{j}={p}_{j}\left({\zeta}_{j}^{k}\right)$ depends on a set of time-dependent phenomenological variables ${\zeta}_{j}^{k}={\zeta}_{j}^{k}\left(t\right)$, k = 1, 2, …, m

_{j}. The entropy production of the entire system is the summation of the entropy production in each subsystem as follows:

_{i}S. The entropy generation presented in Equation (1) consists of a group of thermodynamic forces $X=\left\{\mathit{\sigma}/T,{A}_{k}/T,-\mathrm{\nabla}T/{T}^{2}\right\}$ and thermodynamic rates or flows $J=\left\{{\dot{\epsilon}}_{p},-{\dot{V}}_{k},{\mathit{J}}_{\mathit{q}}\right\}$.

_{j}, as does the entropy generation. Considering the fact that thermodynamic flow ${J}_{j}^{k}$ is the common parameter in Equations (13) and (14), a degradation coefficient can be defined as [15]:

_{j}measures how entropy generation and degradation interact on the level of dissipative processes p

_{j}. Bryant et al. [15] refer to this model as Degradation-Entropy Generation (DEG) theorem.

#### 2.6. Application to Paris-Erdogan Law

_{j}. We define the crack length, a, as the degradation parameter, i.e., w = a in Equation (14). Therefore, degradation can be defined as a = a{W

_{p}(N)}, where dissipative process is the plastic energy dissipation, p = W

_{p}, and the time-dependent phenomenological variable is the number of cycles, ζ = N. Equations (13) and (14) yield:

_{i}S/dt = XJ and da/dt = YJ

_{i}S/dW

_{p})(dW

_{p}/dN), and Y = (da/dW

_{p})(dW

_{p}/dN).

_{p}= Td

_{i}S, the rate of irreversible entropy production due to plastic deformation can be obtained as follows:

_{p}/dN). Applying Equation (14) yields:

_{p}. Bodner et al. [16], for example, derive a correlation for the plastic energy dissipation at crack tip as follows:

_{y}is the yield stress. Substitution of Equation (18) into Equation (17) yields:

## 3. Conclusions

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Amiri, M.; Khonsari, M.M.
On the Role of Entropy Generation in Processes Involving Fatigue. *Entropy* **2012**, *14*, 24-31.
https://doi.org/10.3390/e14010024

**AMA Style**

Amiri M, Khonsari MM.
On the Role of Entropy Generation in Processes Involving Fatigue. *Entropy*. 2012; 14(1):24-31.
https://doi.org/10.3390/e14010024

**Chicago/Turabian Style**

Amiri, Mehdi, and M. M. Khonsari.
2012. "On the Role of Entropy Generation in Processes Involving Fatigue" *Entropy* 14, no. 1: 24-31.
https://doi.org/10.3390/e14010024