## Abstract

Entropy production during the fatigue process can serve as a measure of degradation. We postulate that the thermodynamic entropy of metals undergoing repeated cyclic load reaching the point of fracture is a constant, independent of geometry, load and frequency. That is, the necessary and sufficient condition for the final fracture of a metal undergoing fatigue load corresponds to a constant irreversible entropy gain. To examine validity, we present the results of an extensive set of both experimental tests and analytical predictions that involve bending, torsion and tension-compression of aluminium 6061-T6 and stainless steel 304 specimens. The concept of tallying up the entropy generation has application in determining the fatigue life of components undergoing cyclic bending, torsion and tension-compression.

## 1. Introduction

All structures and machinery components undergoing fatigue loading are prone to crack formation (Bullen *et al.* 1953) and its subsequent growth that increases with time. When a crack is formed, the strength of the structure or the component decreases, and it can no longer function in the intended manner for which it was designed. Moreover, the residual strength of the structure decreases progressively with increasing crack size. Eventually, after a certain time, the residual strength becomes so low that the structure fails (Broek 1982). It is, therefore, of paramount importance to be able to predict the rate of decline in the component’s residual strength and the remaining life of the system.

Many researchers have attempted to quantify fatigue in order to predict the number of cycles to failure. Among them, Miner (1945) pioneered the idea of quantifying fatigue damage based on the hypothesis that, under variable amplitude loading, the life fractions of the individual amplitudes sum to unity. Later, Coffin (1971) and Manson (1964) independently proposed the well-known empirical law Δ*ϵ*_{p}/2=*ϵ*′_{f}(*N*_{f})^{c} that relates the number of cycles to failure, *N*_{f}, in the low-cycle fatigue regime to the amplitude of the applied cyclic plastic deformation, Δ*ϵ*_{p}/2, for a material with given mechanical properties, *ϵ*′_{f} and *c*. The role of energy dissipation associated with plastic deformation during fatigue loading as a criterion for fatigue damage was also investigated by Halford (1966) and Morrow (1965).

The energy approach for estimating the fatigue life of materials under cyclic loading tests has gained considerable attention by researchers (Morrow 1965; Blotny & Kalcta 1986; Atkins *et al.* 1998; Fengchun *et al*. 1999; Park & Nelson 2000; Gasiak & Pawliczek 2003; Jahed *et al*. 2007;). Morrow’s paper (1965) is representative of a pioneering work that takes into account cyclic plastic energy dissipation and fatigue of metals that undergo cyclic loading. A descriptive theory of fatigue was presented that uses the cumulative plastic strain energy as a criterion for fatigue damage and the elastic strain energy as a criterion for fracture. For fully reversed fatigue load, Morrow derived a relation for plastic strain energy per cycle *W*_{p} in terms of the cyclic stress–strain properties, applicable when plastic strain is predominant. Park & Nelson (2000) proposed an empirical correlation for the estimation of fatigue life, taking into account the elastic strain energy *W*_{e} as well as plastic strain energy *W*_{p}. In the high-cycle regime, plastic strains are usually quite small. Park & Nelson (2000) proposed that the two energy terms, *W*_{p} and *W*_{e}, must be combined into the total strain energy parameter *W*_{t},
1.1
where the constants *A*, *α*, *B* and *β* can be determined from a set of uniaxial fatigue test data that cover a sufficiently large number of cycles. The energy dissipation owing to plastic deformation during fatigue is a fundamental irreversible thermodynamic process that must be accompanied by irreversible entropy gain.

Permanent degradations are the manifestation of irreversible processes that disorder a system and generate entropy in accordance to the second law of thermodynamics. Disorder in systems that undergo degradation continues to increase until a critical stage when failure occurs. Simultaneously, with the rise in disorder, entropy monotonically increases. Thus, entropy and thermodynamic energies offer a natural measure of component degradation (Basaran & Yan 1998; Doelling *et al.* 2000; Bryant *et al.* 2008; Amiri *et al.* in press). Of interest in this paper is to quantify the entropy rise in bending, torsion and tension-compression fatigue of metallic components, and particularly the entropy at the instance when failure occurs. According to Whaley (1983), the entropy at the fracture point can be estimated by integrating the cyclic plastic energy per temperature of material. The hypothesis of this paper is that, at the instance of failure, the fracture fatigue entropy (FFE) is constant, independent of frequency, load and specimen size.

## 2. Experimental procedure

A series of fatigue tests are performed to examine the validity of the proposed hypothesis. Three different stress states examined are completely reversed bending, completely reversed torsion and axial loads. Tests are conducted with aluminium (Al) 6061-T6 and stainless steel (SS) 304 specimens. The fatigue testing apparatus used is a compact, bench-mounted unit with a variable-speed motor, variable throw crank connected to the reciprocating platen, with a failure cut-off circuit in a control box, and a cycle counter. The variable throw crank is infinitely adjustable from 0 to 50.8 mm to provide different levels of stress amplitude. The same fatigue apparatus is used for applying torsion, bending and axial load using appropriate fixtures.

Figure 1 shows a schematic of the experimental setup used for torsion tests. The torsional fatigue tests are made using a round bar specimen clamped at both ends and rotationally oscillated at one of the ends via a crank with specified amplitude and frequency. Bending fatigue tests involve a plane specimen clamped at one end and oscillated at the other end, which is connected to the crank. The tension-compression fatigue tests involve clamping a plate specimen at both ends in the grips and oscillating the lower grip at a specified amplitude and frequency. All tests are conducted by installing a fresh specimen in the apparatus, specifying the operating condition and running continuously until failure occurs. All tests are run until failure, when the specimen breaks into two pieces.

High-speed, high-resolution infrared (IR) thermography is used to record the temperature evolution of the specimen during the entire experiment. Before fatigue testing, the surface of the specimen is covered with a black paint to increase the thermal emissivity of the specimen surface. Figure 2 shows the surface-temperature evolution of a series of bending fatigue tests at the clamped end where the specimen fractures. These tests pertain to subjecting an Al specimen to different stress amplitudes. It is to be noted that a persistent trend emerges from all the experiments. Initially, the surface temperature rises as the energy density associated with the hysteresis effect gives rise to the generation of heat greater than the heat loss from the specimen by convection and radiation. Thereafter, temperature tends to become relatively uniform for a period of time until it suddenly begins to rise, shortly before failure occurs. Figure 2 also shows how the temperature of the specimen varies around a mean value. The rise of the mean temperature during fatigue tests is due to the plastic deformation of the material. The oscillation of the temperature around the mean value is caused by the thermoleastic effect (Yang *et al.* 2001; Meneghetti 2007).

## 3. Theory and formulation

Description of the relevant irreversible processes requires formulating the first and second laws of thermodynamic as applicable to a system whose properties are continuous functions of space and time. According to the first law of thermodynamics, the total energy content *E* within an arbitrary control volume can change only if energy flows into (or out of) the control volume through its boundary
3.1
where *Q* and *W* are heat flow and work across the boundary of the control volume. In terms of the specific quantities, the law of conservation of energy for a control volume can be written as (de Groot & Mazur 1962)
3.2
where *ρ* is the density, *u* is the specific internal energy, *J*_{q} is the heat flux across the boundary, ** σ** is the symmetric stress tensor and

**is the symmetric rate of deformation tensor.**

*D*The second law of thermodynamics (Clausius–Duhem inequality) postulates that the rate of entropy generation is always greater than or equal to the rate of heating divided by the temperature *T* (Lemaitre & Chaboche 1990), i.e.
3.3
where *s* represents the specific entropy. The right-hand side of equation (3.3) can be written as
3.4
Substituting equation (3.4) into equation (3.3) and replacing div*J*_{q} from equation (3.2) yields
3.5
Let Ψ represent the specific free energy defined as (Lemaitre & Chaboche 1990)
3.6

Differentiating equation (3.6) with respect to time *t*, and dividing the result by temperature *T* yields
3.7

Considering equation (3.7), the inequality (3.5) reads 3.8

For small deformations, the deformation rate tensor ** D** is replaced by , which represents the total strain rate. The total strain is decomposed to plastic and elastic strain,
3.9

The specification of the potential function (free specific energy Ψ) must be concave with respect to temperature *T* and convex with respect to other variables. Also, the potential function Ψ depends on the observable state variables and internal variables (Lemaitre & Chaboche 1990),
3.10
where *V*_{k} can be any internal variable.

By referring to equation (3.8), strains are decomposed to *ϵ*−*ϵ*_{p}=*ϵ*_{e}, so we can rewrite equation (3.10) as
3.11

Using the chain rule, the rate of specific free energy can be written as
3.12
After the substitution of equation (3.12) into equation (3.8), we obtain
3.13
For small strains, the following expressions define the thermoelastic laws (Lemaitre & Chaboche 1990):
3.14
and
3.15
The constitutive laws of equations (3.14) and (3.15) arise from the fulfilment of the non-negative inequality of equation (3.13). The thermodynamic forces associated with the internal variables (Lemaitre & Chaboche 1990) are defined as follows:
3.16
Hence, the Clausius–Duhem inequality is reduced to express the fact that the volumetric entropy generation rate is positive,
3.17
Equation (3.17) is also interpreted as the product of generalized thermodynamic forces, *X*={** σ**/

*T*,

*A*

_{k}/

*T*,grad

*T*/

*T*

^{2}}, and generalized rates or flows, , (Prigogine 1967; Bejan 1988; Kondepudi & Prigogine 1998) 3.18 Irreversible thermodynamics consider forces

*X*as the drivers of flows

*J*. Each

*J*can depend on all forces (de Groot & Mazur 1962) and intensive quantities (e.g. temperature

*T*) associated with the dissipative process.

Equation (3.17) describes the entropy generation process that consists of the mechanical dissipation owing to plastic deformation, non-recoverable energy stored in the material and the thermal dissipation owing to heat conduction. For metals, non-recoverable energy represents only 5–10% of the entropy generation owing to mechanical dissipation and is often negligible (Clarebrouhg *et al.*1955, 1957; Halford 1966),
3.19

Therefore, equation (3.17) reduces to
3.20
The coupling of thermodynamics and continuum mechanics requires the selection of observable and internal variables (Basaran & Nie 2004). In the present study, two observable variables, temperature *T* and total strain *ϵ*, are chosen. By referring to equation (3.2) and replacing *ρ* d*u*/d*t* by the expression derived from *u*=Ψ+*T**s*,
3.21

Considering equations (3.12), (3.14) and (3.15) and small deformations, equation (3.21) yields 3.22

By applying the chain rule to equation (3.15), we can express by 3.23

Substitution of equations (3.27), (3.15) and (3.16) into equation (3.23) results in 3.24

By introducing the specific heat, *C*=*T*(∂*s*/∂*T*), using equations (3.9), (3.19) and (3.24) and taking into account Fourier’s law (*J*_{q}=−*k* grad *T*), equation (3.22) leads to (Lemaitre & Chaboche 1990)
3.25
where *k* is the thermal conductivity.

Equation (3.25) shows the energy balance between four terms: transfer of heat by conduction (*k*∇^{2}*T*), retardation effect owing to thermal inertia (, internal heat generation consisting of plastic deformation (—which is responsible for mean temperature rise—and the thermoelastic coupling term, , which takes into account the thermoelastic effect (figure 2).

The total energy generation in equation (3.21) is the combination of elastic and plastic energy, *W*_{t}=*W*_{e}+*W*_{p} for low- and high-cycle fatigue (Morrow 1965; Halford 1966; Park & Nelson 2000),
3.26

where *n*′ is the cyclic strain hardening exponent, *ϵ*′_{f} is the fatigue ductility coefficient, *σ*′_{f} denotes the fatigue strength coefficient, *σ*_{a} represents the stress amplitude and *υ* is Poisson’s ratio. The parameters *b*, *E* and *N* represent the fatigue strength coefficient, the modulus of elasticity and the number of cycles to failure, respectively.

As the temperature fluctuation caused by thermoelastic effect is small in comparison with the mean temperature rise (figure 2), the elastic part in equation (3.25) can be neglected (Meneghetti 2007). Therefore, equations (3.20) and (3.25) can be simplified to 3.27 and 3.28

The FFE can be obtained by the integration of equation (3.28) up to the time *t*_{f} when failure occurs,
3.29
where *γ*_{f} is the FFE. In low-cycle fatigue where the entropy generation owing to plastic deformation is dominant and the entropy generation owing to heat conduction is negligible, equation (3.25) reduces to
3.30

The experimental temperatures, such as those shown in figure 2, can be used to calculate the FFE.

## 4. Numerical simulation

Simultaneous solution of equations (3.27) and (3.29) is necessary to determine the entropy generation. For this purpose, a commercial software package (Flexpde), which employs the finite-element method to solve partial differential equations, is used.

### (a) Computational model

Three-dimensional models with 10-node quadratic tetrahedral elements and appropriate number of meshes for the specimens undergoing bending are developed. The corresponding number of finite elements for bending is 2709. Figure 3 shows the geometry and finite-element meshes used for the specimen undergoing bending fatigue, and, because of the symmetric condition, only half of the specimen is modelled.

A mesh dependency study was carried out to investigate the effect of the number of meshes on the calculated entropy generation from equation (3.29). The results of the effect of mesh refinement for the bending test of Al 6061 at 10 Hz and 49.53 mm displacement amplitude are shown in table 1. It reveals that the calculated result for the FFE is independent of mesh refinement.

### (b) Boundary conditions

Figure 4 shows a two-dimensional sketch of the computational model used for the bending load, with the notations indicating the boundary conditions. A summary of the boundary conditions is shown in table 2. Different tip displacement amplitudes (25–50 mm) at different frequencies (6–18 Hz) are considered as the applied loads in the model. Boundary W1 exchanges heat to the surroundings by convection and radiation. Walls W2 are at room temperature, *T*_{a}. Convective heat transfer is assumed as the boundary condition on walls W3. The convective heat transfer coefficient *h* is estimated using an experimental procedure that involves measuring the cooling rate of the specimen surface temperature after a sudden interruption of the fatigue test (Amiri *et al*. in press). Surface emissivity, *ϵ*_{0}, is calculated to be 0.93 and *σ*_{0} is the Stephan–Boltzmann constant that is equal to 5.67×10^{−8} Wm^{−2} K^{−4}.

Walls W4 are associated with the glass-wool insulation used in the experiments, therefore, there is zero heat flux at this boundary. The boundary W5 is considered as a symmetric boundary condition.

Thermal and mechanical properties of the materials are summarized in table 3 (ASM 1990; Bejan 1993). Fatigue properties of the selected materials are based on the experimental studies of Wong (1984) and Lin *et al.* (1992).

## 5. Results and discussion

The evolution of entropy generation is calculated for the entire fatigue life and then integrated over time to determine the entropy generated during the fatigue process (equation (3.29)). Figure 5 shows the comparison of numerical and experimental entropy generation based on equations (3.29) and (3.30) for the bending fatigue of Al 6061-T6, where the frequency and displacement amplitude are 10 Hz and 49.53 mm, respectively. The small difference between the experimental result and the numerical simulation is due to the fact that heat conduction is neglected in equation (3.30). The final value of the entropy generation (about 4 MJ m^{−3} K^{−1} for this test) is associated with the entropy at fracture when the specimen breaks into two pieces. An uncertainty analysis is performed using the method of Kline & McClintock (1953). The maximum error in calculating entropy based on uncertainty analysis is about ±1 per cent.

Figure 6 shows the results of experimental FFE for bending fatigue tests at different frequencies. Results of different displacement amplitudes and different thicknesses of specimen, i.e. 3, 4.82 and 6.35 mm, are shown in this figure. The FFE is found to be about 4 MJ m^{−3} K^{−1}, regardless of the load, frequency and thickness of the specimen. It is to be noted that the results of seven sets of experiments presented in figure 6 correspond to the different combinations of specimen thicknesses and operating frequencies. Also, the experimental data are associated with the different displacement amplitudes ranging from 25 to 50 mm. The same concept for plotting the experimental data is followed in figures 7 and 8.

Figure 7 presents the results of experimental FFE plotted as a function of the fatigue life for bending and tension-compression tests for Al 6061-T6 specimens at 10 Hz. It is seen that the FFE is independent of the type of loading.

Figure 8 presents the results of entropy generation at failure for SS 304 undergoing bending, and torsion fatigue tests. The results show that the entropy generation at the fracture point for SS 304 is about 60 MJ m^{−3} K^{−1}, independent of frequency and geometry. It is to be noted that the fatigue life of a specimen undergoing a cyclic load is only weakly dependent on the test frequencies (Morrow 1965; Liaw *et al.* 2002) up to 200 Hz.

The results presented in figures 6–8 demonstrate the validity of the constant entropy gain at failure for Al and SS specimens. The results reveal that the necessary and sufficient condition for final fracture of Al 6061-T6 corresponds to the entropy gain of 4 MJ m^{−3} K^{−1}, regardless of the test frequency, thickness of the specimen and the stress state. For SS 304 specimens, this condition corresponds to an entropy gain of about 60 MJ m^{−3} K^{−1}.

A possible application of the proposed hypothesis of the constant entropy gain at failure is in the development of a methodology for the prevention of the catastrophic failure of metals undergoing fatigue load. As demonstrated in this work (figure 5), the entropy generation increases during the fatigue life towards a final value of *γ*_{f}. Thus, the FFE can be used as an index of failure. As the entropy generation accumulates towards the FFE, it provides the capability of shutting down of the machinery before a catastrophic breakdown occurs.

The concept of constant entropy gain at the fracture point, *γ*_{f}, assumes that the thermodynamic condition associated with the entropy generation is identical during the fatigue process and varies only in the duration of the process, i.e. failure occurs when
5.1
Within the range of the experimental tests presented, *γ*_{f} is only dependent upon the material and is independent of load, frequency and thickness. Therefore, the duration of the fatigue process varies depending on the operating conditions in order to satisfy the condition of equation (5.1).

Based on this concept, one can conduct an accelerated failure testing scheme by increasing process rates *J* while maintaining equivalent thermodynamic forces *X* to obtain the same sequence of physical processes, in identical proportions, but at a higher rate. For example, by increasing frequency, the rate of plastic deformation increases, and subsequently the rate of degradation increases while the duration of the test is shortened in order to satisfy equation (5.1). This is in accordance with the accelerated testing procedure recently put forward by Bryant *et al.* (2008) based on the thermodynamics of degradation.

Figure 9 shows the normalized entropy generation during the bending fatigue of SS 304 and Al 6061-T6 for different thicknesses, displacement amplitudes and frequencies. The abscissa of figure 9 shows the entropy generation using equation (3.29) and normalized by dividing the entropy gain at the final fracture, *γ*_{f}. The ordinate shows the number of cycles normalized by dividing the final number of cycles when failure occurs. It can be seen that the normalized entropy generation monotonically increases until it reaches the entropy at the failure point. Interestingly, a similar trend between normalized wear plotted against the normalized entropy was reported by Doelling *et al*. (2000). Their work resulted in the prediction of flow of Archard’s wear coefficient (Archard 1953) with remarkable accuracy.

The relation between the normalized cycles to failure and the normalized entropy generation is approximately linear and can be described as
5.2
where *γ*_{f} is a property of the material. Using equation (5.2), the number of cycles to failure can be expressed as
5.3
Equation (5.3) offers a methodology for prediction of the fatigue failure of a given material based on the measurement of the thermodynamic entropy generation. By having FFE (or *γ*_{f}) and calculating entropy generation *γ* at a selected number of cycles *N*, the fatigue life *N*_{f} of the specimen can be predicted. Furthermore, an accelerated testing method can be developed whereby one calculates the entropy generation *γ* over the first few cycles and determines *N*_{f}.

## 6. Conclusions

A thermodynamic approach for the characterization of material degradation is proposed, which uses the entropy generated during the entire life of the specimens undergoing fatigue tests. Results show that the cumulative entropy generation is constant at the time of failure and is independent of geometry, load and frequency. Moreover, it is shown that the FFE is directly related to the type of material. That is, materials with different properties, such as SS and Al have a different cumulative entropy generation at the fracture point. Within the range of conditions tested, the results show that the entropy generation is approximately 4 MJ m^{−3} K^{−1} for Al 6061-T6 and 60 MJ m^{−3} K^{−1} for SS 304. The implication of this finding is that, by capturing the temperature variation of a system undergoing fatigue process, the evolution of entropy generation can be calculated during the fatigue life and then compared to the appropriate FFE for the material to assess the severity of the degradation of the specimen. Also, a methodology is offered for the prediction of the fatigue failure of a given material based on the measurement of the thermodynamic entropy generation.

## Footnotes

- Received July 4, 2009.
- Accepted September 21, 2009.

- © 2009 The Royal Society