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An experimental approach to low-cycle fatigue damage based on thermodynamic entropy
Abstract
This paper presents an experimental approach to fatigue damage in metals based on thermodynamic theory of irreversible process. Fatigue damage is an irreversible progression of cyclic plastic strain energy that reaches its critical value at the onset of fracture. In this work, irreversible cyclic plastic energy in terms of entropy generation is utilized to experimentally determine the degradation of different specimens subjected to low cyclic bending, tension-compression, and torsional fatigue. Experimental results show that the cyclic energy dissipation in the form of thermodynamic entropy can be effectively utilized to determine the fatigue damage evolution. An experimental relation between entropy generation and damage variable is developed.
... Once the dissipated energy is known, thermodynamic fundamental equations of the fatigue process can be solved to find the temperature evolution and, lastly, the entropy generation rate. Morrow's model has been used to calculate the rate of dissipation of energy in several studies with different types of loading, such as tension-compression [11][12][13], bending [8,[14][15][16], torsion [8,16,17], compact-tension [18][19][20] and rotating-bending [21], at different magnitudes of amplitude and frequency. For practical reasons and to simplify calculations in the above mentioned studies, both the entropy generation rate and the dissipated energy are considered as constants along the fatigue process. ...
... FFE can be obtained by evaluating t in Eq. 2 from the beginning of the fatigue process ( N = 0 ) until the cycles to failure N f (considering that dt = dN∕f , being f the frequency of the cyclic loading). Later, Naderi and Khonsari [17] proposed the damage model presented in Eq. 3, where D c is the critical damage value, which must be less than 1 and represents the moment of the onset of fracture [34], D 0 is the damage for a pristine material, s i is the entropy generation, s ic is the critical entropy generation value at the onset of fracture, and s g is the total fracture fatigue entropy. In Naderi and Khonsari model, it is considered that the specimens are initially free of damage, that is, D 0 = 0 . ...
... An extended evaluation of the mentioned above is presented. For the specimen evaluated according to the conditions reported in Ref. [31], the fatigue damage was assessed using the model presented in Eq. 3 by Naderi and Khonsari [17]. A scenario where the Al2024-T3 specimen is subjected to a constant stress amplitude of 276 MPa was hypothesized and the material properties from Table 4 were used to calculate the corresponding dissipated energy ΔW and consequently the entropy generation s i . ...
Morrow’s model has been widely used in recent years as part of new methodologies for assessing the fatigue life of metallic materials through the laws of thermodynamics. It is used to quantify the rate of dissipated energy in the fatigue process, represented by the hysteresis loop of the stress–strain diagram of the material. Then, the dissipated energy is used to quantify the entropy generation of the fatigue process. Morrow’s model is a function of the amplitude of the cyclic loading, and a few constants that represent material properties. However, despite its apparent simplicity, the material properties, when are experimentally obtained, usually exhibit dispersions that can lead to inaccurate calculations. In this work, the sensitivity of Morrow’s model to the variability of these parameters is evaluated. In order to assess the sensitivity, a constant amplitude value was set, and the quotient , the fatigue strength coefficient , the fatigue ductility coefficient and the cyclic strain hardening exponent were varied. A factorial design was conducted to determine the effects of the interactions of the material parameters on the response of Morrow’s model. The results showed that Morrow’s model is strongly sensitive to the above mentioned parameters, affecting its accuracy. In an extended case study, the fatigue fracture entropy and the fatigue life of Al2024-T3 were determined using material parameters from different sources in order to determine the influence of the material parameters in the assessment of the estimation of fatigue damage. The study allowed concluding that a meticulous statistical treatment is required when characterizing the material properties and the design engineer must be aware of possible inaccuracies if Morrow’s model is used for estimating fatigue life, specially in the new thermodynamic approaches.
... Similar to the grease application discussed in Section 7.2, generalized stress-and strain-based PEG and DEG models were obtained [24] for the dynamic loading of solid components and demonstrated using the experimental low-cycle bending and torsional fatigue of stainless steel SS 304 rods [8]. Here, work = = : (also termed "load" in accordance with fatigue loading), where is the stress tensor, and is the elemental strain, with elastic and plastic components: = + , = + . ...
... Similar to the grease application discussed in Section 7.2, generalized stress-and strain-based PEG and DEG models were obtained [24] for the dynamic loading of solid components and demonstrated using the experimental low-cycle bending and torsional fatigue of stainless steel SS 304 rods [8]. Here, work δW = YdX = Vσ : dε (also termed "load" in accordance with fatigue loading), where σ is the stress tensor, and ε is the elemental strain, with elastic and plastic components: σ = σ e + σ p , ε = ε e + ε p . ...
... , -shear entropy in greases, Ohmic entropy in electrochemical systems, load entropy in fatigue loading, flow and work entropies in loaded open systems-is monotonically curvilinear, confirming prior entropybased models that employ this term only [8,9,13,22,34], whereas the MST/ECT entropy (red plots) shows instantaneous transients and nonlinearities that characterize the internal fluctuations in the systems that were hitherto unobservable. Noting the order of magnitude difference between MST/ECT and work entropies, the work entropies are significantly higher. ...
Modern concepts in irreversible thermodynamics are applied to system transformation and degradation analyses. Phenomenological entropy generation (PEG) theorem is combined with the Degradation-Entropy Generation (DEG) theorem for instantaneous multi-disciplinary, multi-scale, multi-component system characterization. A transformation-PEG theorem and space materialize with system and process defining elements and dimensions. The near-100% accurate, consistent results and features in recent publications demonstrating and applying the new TPEG methods to frictional wear, grease aging, electrochemical power system cycling—including lithium-ion battery thermal runaway—metal fatigue loading and pump flow are collated herein, demonstrating the practicality of the new and universal PEG theorem and the predictive power of models that combine and utilize both theorems. The methodology is useful for design, analysis, prognostics, diagnostics, maintenance and optimization.
... Kuhn [5,6] defined a rheological energy density to describe grease transformation due to imposed shear energy. Khonsari et al. [7][8][9][10][11][12][13] have applied single-variable entropy generation to characterize degradation of materials such as metals, composites, lubricant grease; and mechanisms such as frictional wear. Sosnovskiy and Sherbakov [14,15], via various studies, characterized complex simultaneously occurring damage mechanisms by correlating a damage variable to single-variable entropy generations. ...
... Similar to the grease application discussed in Section 7.2, generalized stress-and strain-based PEG and DEG models were obtained [21] for dynamic loading of solid components, and demonstrated with experimental low-cycle bending and torsional fatigue of a stainless steel SS 304 rod [7]. Here, work = = : ...
... The third rows (step (ii)) of Tables 2-6 present phenomenological entropy generation (PEG) constituent terms for the various systems demonstrated in this study. The work entropy (singlevariable entropy generation) ′ , -shear entropy in greases, Ohmic entropy in electrochemical systems, load entropy in fatigue loading, flow and work entropies in loaded open systems-is monotonically curvilinear, confirming prior entropy-based models that employ this term only [7,8,19,32,34]; whereas the MST/ECT entropy (red plots) shows instantaneous transients and nonlinearities that characterize the internal fluctuations in the systems that were hitherto unobservable. Noting the order of magnitude difference between MST/ECT and work entropies, the work entropies are significantly higher. ...
Modern concepts in irreversible thermodynamics are applied to system transformation and degradation analyses. Phenomenological entropy generation (PEG) theorem is combined with the Degradation-Entropy Generation (DEG) theorem for instantaneous multi-disciplinary, multi-scale, multi-component system characterization. A Transformation-Phenomenological Entropy Generation (TPEG) theorem and space materialize with system and process defining elements and dimensions. The near-100% accurate, consistent results and features in recent publications demonstrating and applying the new TPEG methods to frictional wear, grease aging, electrochemical power systems cycling—including lithium-ion battery thermal runaway—metal fatigue loading and pump flow, are collated herein, demonstrating the practicality of the new and universal PEG theorem, and the predictive power of models that combine and utilize the PEG and DEG theorems. The methodology is useful for design, analysis, prognostics, diagnostics, maintenance and optimization.
... Their approach, termed mechanothermodynamics, includes damage distribution in a volume, correlating well with measurements. Entropy-based works of Khonsari et al. [11][12][13][14][15][16][17][18][19][20] have shown consistently accurate characterizations of various mechanical systems under diverse forms of loading, such as grease shearing, metal and composite laminate fatigue as well as interfacial sliding wear. Cuadras et al. [21][22][23] used entropy generation to estimate damage in resistors, capacitors and batteries, showing consistent correlations between model and measurements. ...
... Details of the DEG theorem, including statements and proof, can be found in [1]. Frictional wear [1,[24][25][26] and fatigue [11][12][13][14][15][16][17] experiments verified the theorem. A breakdown of entropy generation can be found in this article's Appendix A. Osara and Bryant [2,[27][28][29][30], via a combination of Equations (13) and (14) (see Appendix A) with Equation (1), instantaneously characterized unsteadily loaded multicomponent multi-physics systems, with verification using measured data from grease aging, lithium-ion battery and lead-acid battery degradation, and general fatigue experiments. ...
... Such a system undergoing a single predominant process is termed a single-variable system [1]. For example, steel fatigue measurements by Naderi and Khonsari [12,15] showed a pseudo-steady temperature region for most of the sample's fatigue life, prior to failure. Via temperature control, Bryant et al. [1] adequately characterized a frictional process using only the frictional work. ...
A generalization of the Degradation-Entropy Generation (DEG) theorem to multi-disciplinary multi-physics system-process analysis via a combination with pre-existing system models is presented in this article. Existing models and the DEG methodology are reviewed, and a method for evaluating degradation coefficients Bi is proposed. These coefficients characterize the system’s transformation based on active dissipative mechanisms, including temperature effects. The consistency of entropy generation in characterizing degradation is then inherited by these often-empirical system models, thereby rendering them more robust and applicable to similar systems without the need for numerous tests and measurements for model corrections. The approach applies to all systems and can quickly analyze and predict a system’s performance and degradation, even in the absence of experimental data (using known properties and material constants). Demonstrated applications herein include mechanically loaded systems (frictional wear, grease shearing, fatigue loading), electrochemical energy systems, thermal processes, and others.
... A high-resolution infra-red camera monitored the temperature profile of an SS 304 stainless steel rod subjected to a 10 Hz displacement-controlled cyclic bending load until fatigue failure [24]. See Figure 4a. ...
... Substituting stress, strain, temperature and material property values, the integrals in Equation (24) were estimated via the methods of Section 1.2 to render the entropy generation plots in Figure 4b, where the MST entropy density ′ (red plot) and load entropy density S'W (blue plot) are the first and second integrals of Equation (24), respectively. For low-cycle fatigue, with significant plastic deformation, the modulus defined in Equations (12) and (13), substituted into entropy content density SA in Equation (24), is the hardness modulus. ...
Entropy generation, formulated by combining the first and second laws of thermodynamics with an appropriate thermodynamic potential, emerges as the difference between a phenomenological entropy function and a reversible entropy function. The phenomenological entropy function is evaluated over an irreversible path through thermodynamic state space via real-time measurements of thermodynamic states. The reversible entropy function is calculated along an ideal reversible path through the same state space. Entropy generation models for various classes of systems—thermal, externally loaded, internally reactive, open and closed—are developed via selection of suitable thermodynamic potentials. Here we simplify thermodynamic principles to specify convenient and consistently accurate system governing equations and characterization models. The formulations introduce a new and universal Phenomenological Entropy Generation (PEG) theorem. The systems and methods presented—and demonstrated on frictional wear, grease degradation, battery charging and discharging, metal fatigue and pump flow—can be used for design, analysis, and support of diagnostic monitoring and optimization.
... where T is the temperature, α is the coefficient of linear expansion, σ is the strain, C p is the specific heat capacity at constant pressure and ρ is the density of the medium. Based on the first law of thermodynamics, Eq. (6) [56]combined with the stress-strain-temperature evolution process during tensile deformation, the entropy behavior is calculated: ...
... where σ a is the stress amplitude, Δε p represents the steady plastic strain amplitude, and n ′ is the work hardening index. During the static tensile deformation process, the total entropy production can be obtained by integrating time to material fracture, which can be expressed as follows [56]: ...
... Figure 23 illustrates how the number of cycles until failure can vary as a function of applied stress levels (e.g., ranging from R of 0.4 to 0.8) [91]. From the other perspective, a PMC structure, depending on the type of applied mechanical fatigue loading (i.e., bending, tension, shear tension-compression or multiaxial loading) and the magnitude of applied loading (i.e., frequency and stress/strain amplitude), may indeed experience both mechanical and thermal loading, as the result of which various fracture mechanisms may occur [92][93][94], which are deeply discussed in Section 3. Due to the entropy generation in the irreversible thermodynamic process caused by the self-heating effect [19,54,95], a structure may face mechanical failure or thermal failure. According to this, a mechanical fracture may occur when the influence of mechanical loading is comparatively more dominant than thermal loading, while thermal failure will occur if the self-heating effect is more dominant [64,89]. ...
... On the other hand, during the second phase of the nonstationary self-heating regime, the temperature profile experiences an upward trend linearly as the number of fatigue cycles increases (see Figure 24). The primary causative factors contributing to this trend can be sought in the synergy between mechanical fatigue and entropy generation (irreversible morphological and chemical variations [3]) because of the viscoelastic nature of From the other perspective, a PMC structure, depending on the type of applied mechanical fatigue loading (i.e., bending, tension, shear tension-compression or multiaxial loading) and the magnitude of applied loading (i.e., frequency and stress/strain amplitude), may indeed experience both mechanical and thermal loading, as the result of which various fracture mechanisms may occur [92][93][94], which are deeply discussed in Section 3. Due to the entropy generation in the irreversible thermodynamic process caused by the self-heating effect [19,54,95], a structure may face mechanical failure or thermal failure. According to this, a mechanical fracture may occur when the influence of mechanical loading is comparatively more dominant than thermal loading, while thermal failure will occur if the self-heating effect is more dominant [64,89]. ...
The self-heating effect can be considered as a catastrophic phenomenon that occurs in polymers and polymer–matrix composites (PMCs) subjected to fatigue loading or vibrations. This phenomenon appears in the form of temperature growth in such structures due to their relatively low thermal conductivities. The appearance of thermal stress resulting from temperature growth and the coefficient of thermal expansion (CTE) mismatch between fibers and neighboring polymer matrix initiates and/or accelerates structural degradation and consequently provokes sudden fatigue failure in the structures. Therefore, it is of primary significance for a number of practical applications to first characterize the degradation mechanism at the nano-, micro- and macroscales caused by the self-heating phenomenon and then minimize it through the implementation of numerous approaches. One viable solution is to cool the surfaces of considered structures using various cooling scenarios, such as environmental and operational factors, linked with convection, contributing to enhancing heat removal through convection. Furthermore, if materials are appropriately selected regarding their thermomechanical properties involving thermal conductivity, structural degradation may be prevented or at least minimized. This article presents a benchmarking survey of the conducted research studies associated with the fatigue performance of cyclically loaded PMC structures and an analysis of possible solutions to avoid structural degradation caused by the self-heating effect.
... Significant accomplishments in this area for fatigue and wear are reported by Basaran and Yan (1998), Tang and Basaran (2001), Basaran and Chandaroy (2002), Basaran and Tang (2002;Tang and Basaran (2003;Basaran et al. (2003), Basaran and Nie (2004), Basaran and Nie (2007), Basaran (2012, 2013). Further research was done by Bryant (2009Bryant ( , 2010, Bryant et al. (2008), Ling et al. (2002), Doelling et al. (2000), Bryant (2009), , 2010c, , Naderi and Khonsari (2010a, 2010c Khonsari (2010, 2011), Amiri et al. ( , 2011, Khonsari and Amiri (2013), Naderi and Khonsari (2012a), Naderi et al. (2012) and Aghdam et al. (2012). The motivation of the latter study was a series of experimental and theoretical investigations that appeared in papers by Ling et al. (2002) and Bryant et al. (2008). ...
... The motivation of the latter study was a series of experimental and theoretical investigations that appeared in papers by Ling et al. (2002) and Bryant et al. (2008). Development of the thermodynamics of fatigue fracture as well as thermodynamically-based damage mechanics are reported by , , 2010c, Naderi and Khonsari (2010a, Beheshti and Khonsari (2010), Amiri et al. ( , 2011, Amiri et al. (2011, Khonsari and Amiri (2013), Khonsari (2011), Naderi et al. (2012), Aghdam et al. (2012), , Amiri et al. (2011), Khonsari (2010a, 2010b) with application to structure health monitoring (Naderi and Khonsari, 2010b). A recent introduction to the thermodynamics of mechanical fatigue provides background and application of these findings (Khonsari and Amiri, 2013;. ...
... Significant accomplishments in this area for fatigue and wear are reported by Basaran and Yan (1998), Tang and Basaran (2001), Basaran and Chandaroy (2002), Basaran and Tang (2002;Tang and Basaran (2003;Basaran et al. (2003), Basaran and Nie (2004), Basaran and Nie (2007), Basaran (2012, 2013). Further research was done by Bryant (2009Bryant ( , 2010, Bryant et al. (2008), Ling et al. (2002), Doelling et al. (2000), Bryant (2009), , 2010c, , Naderi and Khonsari (2010a, 2010c Khonsari (2010, 2011), Amiri et al. ( , 2011, Khonsari and Amiri (2013), Naderi and Khonsari (2012a), Naderi et al. (2012) and Aghdam et al. (2012). The motivation of the latter study was a series of experimental and theoretical investigations that appeared in papers by Ling et al. (2002) and Bryant et al. (2008). ...
... The motivation of the latter study was a series of experimental and theoretical investigations that appeared in papers by Ling et al. (2002) and Bryant et al. (2008). Development of the thermodynamics of fatigue fracture as well as thermodynamically-based damage mechanics are reported by , , 2010c, Naderi and Khonsari (2010a, Beheshti and Khonsari (2010), Amiri et al. ( , 2011, Amiri et al. (2011, Khonsari and Amiri (2013), Khonsari (2011), Naderi et al. (2012), Aghdam et al. (2012), , Amiri et al. (2011), Khonsari (2010a, 2010b) with application to structure health monitoring (Naderi and Khonsari, 2010b). A recent introduction to the thermodynamics of mechanical fatigue provides background and application of these findings (Khonsari and Amiri, 2013;. ...
... Bryant et al. (2008) proposed a general theorem to connect the entropy generation with the irreversible degradation via the generalized thermodynamic forces and the degradation forces. Naderi and Khonsari (2010b) presented an experimental approach to predict the low-cycle fatigue damage on the basis of the thermodynamic theory of the irreversible process, and experimental results highlight that the energy dissipation in the form of thermodynamic entropy can be effectively applied to determine the evolution process of fatigue damage. Amiri et al. (2009) proposed an experimental approach to estimate the threshold value of fatigue damage based on the concept of entropy flow. ...
... It can be obviously shown that the normalized energy dissipation E/E C monotonically increases with the normalized cycle number N/N f until it reaches to the maximum value at the onset of fatigue fracture. It is interesting to note that the similar trends for the normalized entropy flow plotted against the normalized cycles to failure in bending fatigue were reported and demonstrated by Naderi et al. (Naderi and Khonsari, 2010b) and Amiri et al. (2009), respectively. From Fig. 11, the correlation between the normalized cycle number and the normalized energy dissipation is approximately linear and can be written as: ...
The objective of this paper is to use quantitative thermography for fatigue damage assessment and life prediction of welded components. The energy dissipation during fatigue process is taken as an effective index of fatigue damage to develop a nonlinear damage accumulation model, and the model is extended to evaluate the fatigue damage and residual life of the specimen subjected to variable loading. Experiments are performed to validate its applicability, and the relationships between the fatigue damage variable to the cycle number and the accumulated energy dissipation are presented and discussed. The results show that the energy dissipation has the same trend as the temperature increment evolution, and a power law relationship between the stress range and the energy dissipation is experimentally determined. The linear relationship between the accumulated energy dissipation and the cycle number is confirmed at different constant stress levels, and the total energy dissipation to failure is shown to be a constant value. The developed nonlinear damage accumulation model is used to predict the residual fatigue life, and a good agreement is achieved. It is concluded that the developed approach is applicable for fatigue damage accumulation and residual life evaluation of materials.
... As known, the degradation of components is the consequence of irreversible thermodynamic process. According to this hypothesis, Naderi and Khonsari [15] showed an experimental approach to predict the low cycle fatigue damage, and the experimental results highlighted that the dissipated energy in the form of thermodynamic entropy can be applied to predict fatigue damage process. Fan et al. [16] took the energy dissipation as a fatigue damage indicator to qualitatively analyze the progressive microstructural movements and to quantitatively evaluate the residual fatigue life and microplasticity. ...
... According to the reported works by Ye and Wang [5] and Naderi and Khonsari [15,30], a general logarithmic expression for fatigue damage evolution has been derived as: ...
A thermography-based energetic damage model is put forward to rapidly evaluate the high-cycle fatigue behavior of welded joints subjected to fatigue loadings. The energy dissipation is calculated using the thermal signal, and then is used to create fatigue damage model. The model is extended to predict the damage accumulation of welded specimens. The predicted fatigue damage evolution is shown to be in good agreement with the results given by the existing models. Experiments are conducted to verify the applicability of the present model for residual life prediction. It is shown that the residual life under multistage loading is well predicted in comparison with the experimental life.
... Geraci et al. were the first to establish a correlation between temperature signatures, particularly the stabilized temperature of the material [9,10], and fatigue behavior. However, numerous other thermal features have also been proposed as valuable tools for this purpose by focusing on thermoelasticplastic response and dissipated energy [11,12]; self-heating effect and stabilized temperature [13][14][15]; separating mechanical dissipation and thermoelastic source terms [16]; estimating the heat dissipation [17,18]; initial temperature rise [19]; relation between stress levels, temperature, and number of cycles to failure [20]; harmonic temperature components [21,22]; and energy methods [23,24]. ...
Over the past decades, thermographic methods have become a viable substitute for conventional approaches in the analysis of material fatigue behavior, due to their efficiency, cost-effectiveness, and nondestructive nature. By examining the temperature signature generated by intrinsic heat dissipations during the fatigue loading, valuable insights into the behavior of the materials can be investigated. Substantial intrinsic dissipation—a marker of material damage—is linked to a transition from anelastic to inelastic strains. The main aim of this work is to explore heat dissipations during fatigue of materials by combining experimental techniques and numerical simulations, focusing on the fundamental temperature component in fully reversed loading, known as the second amplitude harmonic (SAH) of temperature. The hybrid method combines experimental data with numerical modeling to identify the specific volume generating heat during fatigue testing. Additionally, the effect of the mechanical loading frequency on SAH of temperature was also examined.
... Considering this background, the frequency dependence of fatigue failure has recently been investigated, assuming a single material [32]. A fatigue failure model was proposed for a resin material based on the entropy damage law [33][34][35][36][37][38], which has been increasingly applied in fatigue studies in recent years. In our previous works, fatigue failure was simulated using the entropy damage code developed for resin in conjunction with the finite element software ABAQUS, while accounting for heat generation and dissipation. ...
A numerical simulation investigating the frequency dependence of fatigue damage progression in carbon fiber-reinforced plastics (CFRPs) is conducted in this study. The initiation and propagation of transverse cracks under varying fatigue test frequencies are successfully simulated, consistent with experiments, using an enhanced degradable Hashin failure model that was originally developed by the authors in 2022. The results obtained from the numerical simulation in the present study, which employs adjusted numerical values for the purpose of damage acceleration, indicate that the number of cycles required for the formation of three transverse cracks was 174 cycles at 0.1 Hz, 209 cycles at 1 Hz, and 165 cycles at 10 Hz. Based on these results, it is demonstrated that under high-frequency cyclic loading, internal heat generation caused by dissipated energy from mechanical deformation, attributed to the viscoelastic and/or plastic behavior of the material, exceeds thermal dissipation to the environment, leading to an increase in specimen temperature. Consequently, damage progression accelerates under high-frequency fatigue. In contrast, under low-frequency fatigue, viscoelastic dissipation becomes more pronounced, reducing the number of cycles required to reach a similar damage state. The rate of damage accumulation initially increases with test frequency but subsequently decreases. This observation underscores the importance of incorporating these findings into discussions on the fatigue damage of real structural components.
... Experiments on bending, tension -compression, and torsional fatigue demonstrate the effectiveness of entropy as a measure of fatigue damage. An experimental relationship between entropy generation and a damage variable is established, enabling the quantitative evaluation of fatigue evolution [21]. Naderi et al. proposed a damage evolution assessment method based on entropy production, applicable to constant-and variable-amplitude loading. ...
In this study, micro-scale numerical simulations were performed to evaluate the residual strength of carbon fiber-reinforced polymers (CFRPs) subjected to cyclic transverse and out-of-plane shear loading fatigue. The simulations utilized a finite element method, incorporating an entropy-based damage criterion for the matrix resin. This method aimed to link entropy generation to strength degradation, with the parameter αo(s) determined as a function of entropy. Cyclic tensile and shear analyses were conducted to correlate residual strength with entropy accumulation, establishing a linear relationship for αo(s). The results demonstrated meso-scale strength degradation based on micro-scale numerical simulations. Material constants for the epoxy resin matrix were determined through creep and tensile tests, and a generalized Maxwell model with 15 elements was used to represent viscoelastic behavior. Numerical simulations employed the Abaqus/Standard 2020 software, with the epoxy resin matrix behavior implemented via a UMAT subroutine. The analysis revealed a linear relationship between entropy and residual strength for both cyclic tensile and out-of-plane shear loading. This approach enhances experimental insights with numerical predictions, offering a comprehensive understanding of CFRP strength degradation under fatigue loading. This study represents the first numerical approach to link the entropy of the matrix resin at the micro-scale with macro-scale residual strength in CFRP, providing a novel and comprehensive framework for understanding and predicting strength degradation under cyclic loading.
... This has led to the development of various fatigue damage accumulation theories. Among these, the entropybased approach has gained significant attention due to its ability to quantify irreversible degradation [14,[20][21][22][23][24][25]. Entropy generation is calculated by dividing the dissipated energy by temperature and integrating this over the time to fracture [26,27]. ...
Accurately predicting fatigue failure in CFRP laminates requires an understanding of the cyclic behavior of their resin matrix, which plays a crucial role in the materials’ overall performance. This study focuses on the temperature elevation during the cyclic loadings of the resin, driven by inelastic deformations that increase the dissipated energy. At low loading frequencies, the dissipated energy is effectively released as heat, preventing significant temperature rise and maintaining a consistent, balanced thermal state. However, at higher frequencies, the rate of energy dissipation exceeds the system’s ability to release heat, causing temperature accumulation and accelerating damage progression. To address this issue, the study incorporates non-recoverable strain into a fatigue simulation framework, enabling the accurate modeling of the temperature-dependent fatigue behavior. At 0.1 Hz, damage accumulates rapidly due to significant inelastic deformation per cycle. As the frequency increases to around 2 Hz, the number of cycles until failure rises, indicating reduced damage per cycle. Beyond 2 Hz, higher frequencies result in accelerated temperature rises and damage progression. These findings emphasize the strong link between the loading frequency, non-recoverable strain, and temperature elevation, providing a robust tool for analyzing resin behavior. This approach represents an advancement in simulating the fatigue behavior of resin across a range of frequencies, offering insights for more reliable fatigue life predictions.
... Geraci et al. were the first to establish a correlation between temperature signatures, particularly the stabilized temperature of the material [9,10], and fatigue behavior. However, numerous other thermal features have also been proposed as valuable tools for this purpose by focusing on thermoelasticplastic response and dissipated energy [11,12]; self-heating effect and stabilized temperature [13][14][15]; separating mechanical dissipation and thermoelastic source terms [16]; estimating the heat dissipation [17,18]; initial temperature rise [19]; relation between stress levels, temperature, and number of cycles to failure [20]; harmonic temperature components [21,22]; and energy methods [23,24]. ...
Fatigue is an irreversible process accompanied by the heat dissipation which is significant when the transition from anelastic to inelastic strains happens. In view of this, in the last years, the heat dissipation has been accepted as an appropriate damage indicator of the material.
The estimation of the heat dissipation can be obtained by detecting the surface thermal footprint of the specimen by using thermography-based techniques. However, the energy dissipation as heat is highly sensitive to the environmental and test conditions and the microstructure status. Therefore, the experimental measurement is always associated with some inaccuracies and only provides an estimation of the heat dissipated during fatigue.
This paper is mainly focused on the numerical modeling of the heat dissipation performed by COMSOL Multiphysics software in order to investigate the factors that can affect the estimation of the heat source by means of thermography. The obtained results have been compared with an analytical solution derived from the one-dimensional heat equation. This study can provide valuable insights about the shape of the heat sources produced during the cyclic loading and differences associated with thermographic measurements and actual values, which are the main goals of this work.
... A high-resolution infra-red camera monitored the temperature profile of an SS 304 stainless steel rod subjected to a 10 Hz displacement-controlled cyclic bending load until fatigue failure [31,32].See Figure 4a. Here, boundary work ̇= :, where is the stress tensor and is the elemental strain rate tensor, both having elastic and plastic components, viz = + , = + . ...
Entropy generation, formulated by combining the first and second laws of thermodynamics with an appropriate thermodynamic potential, emerges as the difference between a phenomenological entropy function and a reversible entropy function. The phenomenological entropy function is evaluated over an irreversible path through thermodynamic state space via real time measurements of thermodynamic states. The reversible entropy function is calculated along an ideal reversible path through the same state space. Entropy generation models for various classes of systems—thermal, externally loaded, internally reactive, open and closed––are developed via selection of suitable thermodynamic potentials. Here we simplify thermodynamic principles to specify convenient and consistently accurate system governing equations and characterization models. The formulations introduce a new and universal Phenomenological Entropy Generation (PEG) theorem. The systems and methods presented––and demonstrated on grease degradation, battery charging and discharging, metal fatigue and pump flow––can be used for design, analysis, and support of diagnostic monitoring and optimization.
... While FCGR is a viable indicator of hydrogen embrittlement sensitivity, realistic prediction of the LCF fatigue life for components exposed to hydrogen environment remains challenging. In previous studies [4][5][6], one of the present authors introduced the concept of fracture fatigue entropy (FFE) as a general framework based on the principles of irreversible thermodynamics and the details of that for evaluating fatigue degradation. The details of the formulation and implementation of FFE can be found elsewhere [7]. ...
The concept of hydrogen-enhanced entropy (HEENT) for hydrogen embrittlement prediction was firstly introduced and discussed through experimental validation methods, multiscale characterization, and trapping studies. The accumulated generated entropy reaches a constant value (fatigue fracture entropy (FFE)) regardless of hydrogen content. Hydrogen-enhanced localized plasticity (HELP) is the dominant mechanism for hydrogen-assisted cracking in the studied material, due to quasi-cleavage fracture pattern with serrated marking and H-enhanced planar slip. Fatigue life is reduced due to increasing hydrogen uptake with increasing current density of H charge, and the accumulated generated entropy reaches a constant FFE value. As H content increases, a fraction of weakly trapped hydrogen at interstitials and dislocations in the ferrite increases, and the possibility of reaching hydrogen to crack tip and slip planes increases. Direct evidence of HELP mechanisms with a well-developed dislocation substructure is reported for the first time in pearlitic steels with nanoscale observations near the H-assisted crack path. The contribution of microplasticity due to the HELP mechanism to the total entropy compensates for the total entropy reduction generated due to reduced fatigue life. This introduces the hydrogen-enhanced entropy (HEENT) concept for hydrogen embrittlement prediction.
... According to Morrow's theory [45][46][47], in a steady-state hysteresis loop, the strain energy density in one cycle is ...
... In addition to thermocouples, thermal imaging infrared (IR) camera systems have been already used for temperature measurement since the 1970s [11], which is now much more widely applied due to the continuous innovation in technology. Regarding the use in fatigue evaluation, damage indicators have been defined to describe the fatigue state of different kinds of materials by means of temperature variables from direct and indirect measurements, which are used to estimate the fatigue strength of metallic materials [5,10,[12][13][14][15][16][17][18][19][20][21][22][23][24] and fatigue damage analysis of composite materials [25,26]. ...
Investigations on low carbon (non- and low-alloy) steels were conducted in form of load increase tests (LIT) and constant amplitude tests (CAT) to find the correlation among material behaviour, mechanical load, and the type of NDT method. With the help of preprogrammed load-free sequences, the thermal impact on magnetic Barkhausen noise (MBN) measurement can be avoided, so that the cyclic deformation properties of material responses can be interpreted more precisely. The results indicate differences between the change in temperature and the MBN-derived variable during LITs and CATs regarding the demonstration of the incubation stage and the cyclic hardening behaviour.
... Ensuite, nous remarquons que la valeur de l'endommagement non linéaire est toujours inférieure à celle de l'endommagement calculé par la loi de Miner. Compte tenu des travaux de Naderi et Khonsari, ce résultat est plus proche de la réalité des expériences [Naderi & Khonsari 2010b, Naderi & Khonsari 2010a. Afin d'étudier le comportement de la loi d'endommagement non linéaire, au cours d'un chargement défini par la succession de blocs, considérons le cas du chargement cyclique de la figure 8.14, constitué de deux blocs. ...
Ce travail concerne l’étude théorique, numérique et expérimentale du comportement mécanique élastoplastique en chargements cycliques complexes multiaxiaux de matériaux métalliques. Le comportement élastoplastique est décrit par modèle multisurface, avec une prise en compte des grandes transformations. Ce modèle est écrit dans l’espace multidimensionnel d’Ilyushin des déformations, de dimension 5. La modélisation qui en résulte permet de décrire le comportement multiaxial des matériaux métalliques, en chargements cycliques complexes, notamment non proportionnels, avec une prise en compte des déformations finies, de l’irréversibilité indépendante du temps, des effets secondaires cumulés (effet Poynting-Swift) et des effets d’écrouissage cyclique. Le modèle ainsi développé a été implémenté dans un code commercial de calcul par éléments finis, afin de produire un outil opérationnel de calcul des structures telles que les équipements mécaniques et les composants internes des centrales hydroélectriques (turbines, alternateur…). Le modèle proposé a été validé par confrontation à des résultats d’essais biaxiaux de traction-torsion combinées, réalisés sur un acier inoxydable. Ce modèle a été complété par une analyse énergétique et thermodynamique qui permet la mise en place, à terme, d’une approche énergétique pertinente pour le suivi de l’endommagement par fatigue. Dans le cadre de ce travail, cette approche a été illustrée par la proposition d’un critère de fatigue, validé par la comparaison de ses prédictions à celles d’autres critères de fatigue classiques, proposés dans la littérature.
... In a tribo-system, degradation is accompanied by material removal [30][31][32][33][34][35]. The same physics is involved when one deals with a deterioration of performance in batteries during charging and discharging [36,37], loss of consistency in grease due to shearing action [38][39][40][41], and accumulation of damage in cyclic fatigue [42][43][44][45]. Therefore, it is hypothesized that the degradation can be characterized using the framework of irreversible thermodynamics. ...
An extensive survey of open literature reveals the need for a unifying approach for characterizing the degradation of tribo-pairs. This paper focuses on recent efforts made towards developing unified relationships for adhesive-type wear under unlubricated conditions through a thermodynamic framework. It is shown that this framework can properly characterize many complex scenarios, such as degradation problems involving unidirectional, bidirectional (oscillatory and reciprocating motions), transient operating conditions (e.g., during the running-in period), and variable loading/speed sequencing.
... La entropía ha sido útil para evaluar la evolución del daño por fatiga de materiales metálicos sometidos a cargas uniaxiales de amplitud constante (Naderi & Khonsari, 2010a;Temfack & Basaran, 2015) y de amplitud variable (Naderi & Khonsari, 2010b). Además, se ha empleado como índice de daño para estudiar la fatiga en ambientes corrosivos y con fines de monitoreo de la salud estructural (Imanian & Modarres, 2015. ...
En este trabajo se presenta una revisión de la literatura sobre modelos para la estimación del daño de materiales metálicos sometidos a fatiga basados en la generación de entropía, los cuales corresponden a investigaciones realizadas por grupos líderes a nivel internacional. Se evaluaron tres enfoques, empleando datos de entropía existentes en la literatura de aluminio 6061-T6 sometido a fatiga a flexión, a amplitud de carga constante. Se muestran y discuten las características, semejanzas y diferencias de los modelos evaluados. Se propone extender los estudios de fatiga empleando la termodinámica, ya que aporta una base teórica que parte de principios físicos, contrario a las metodologías tradicionales para generar modelos empíricos, en las que se requiere una gran cantidad de datos experimentales para ajustar curvas.
... Dattoma and Giancane [5] used a combination of digital image correlation (DIC) and thermography to investigate energy balance between heat sources and dissipation sources of notched composite laminates under fatigue loading. A noticeable work, although focusing on metals rather than composites, has been done by Naderi and Khonsari [6]. It presented an experimental approach to fatigue damage quantification based on thermodynamic entropy and showed that the cyclic energy dissipation in the form of thermodynamic entropy can be effectively utilized to determine the fatigue damage. ...
This work proposes a novel and robust thermal imaging-based method suitable to quantify damage evolution stages in fibre-reinforced composite materials under fatigue loading. The method uses adiabatic hysteretic heating of the material to calculate the change of enthalpy by thermal imaging when the material undergoes fatigue damage. This work analytically corroborates the validity of using the calculated change of enthalpy to identify the three different fundamental fatigue damage stages of the material. The method is applied to two different types of notched sandwich specimens subject to constant amplitude fatigue loading. It is practically demonstrated that the characteristic curve progressions of the mechanically measured hysteretic dissipation curves are in good agreement with those of the enthalpy change computed by the proposed method.
... [86].This picture is ©2009 The Royal Society. Naderi and Khonsari [87] investigated the fatigue damage in metals based on the irreversible thermodynamic process. Their work showed that the cyclic plastic strain energy is the primary source of entropy generation in the low cycle fatigue test, and it reaches a critical value at the onset of fracture. ...
Degradation, damage evolution, and fatigue models in the literature for various engineering materials, mostly metals and composites, are reviewed. For empirical models established under the framework of Newtonian mechanics, Gurson–Tvergaard–Needleman (GTN) type model, Johnson-Cook (J-C) type damage model, microplasticity model, some other micro-mechanism based damage models, and models using irreversible entropy as a metric with an empirical evolution function are thoroughly discussed. For Physics-based models, the development and applications of unified mechanics theory is reviewed.
... Amiri and Khonsari [6] presented an extensive survey of the papers pertaining to the thermodynamic approach to tribosystems, using the concept of entropy as a natural time base with a summary of the significant contributions of outstanding researchers. Naderi et al. [22][23][24] extended the results to processes involving cyclic fatigue and showed that the necessary and required conditions for the terminal fracture of a metal undergoing fatigue loading correspond to a constant accumulation of entropy. Agdam and Khonsari [8] further investigated the relationship between wear and entropy generation. ...
Running-in plays an important role in the steady-state performance of mechanical elements. It is a transient process involving a complex interaction between friction, lubrication, asperities, plastic deformation, and wear in an intertwined fashion. The running-in process involves the evolution of the key tribological parameters such as surface roughness, surface pattern, friction coefficient, and wear rate until the steady-state prevails. Importantly, the steady-state behavior of a tribo-component is dependent upon the operating conditions during the running-in process. This paper provides a comprehensive review of the literature on this subject encompassing both experimental and analytical development to date.
... Thus, researchers have investigated thermodynamic entropy methodology to characterize fatigue performance and unveil the nature of the fatigue process. For low-cycle fatigue (LCF), Naderi et al. [23][24][25] reported the irreversible entropy model to evaluate fatigue damage and structural life monitoring. Ribeiro et al. [26] experimentally calculated the threshold entropy of Al-2024 specimen during an LCF process and compared the results with empirical Park and Nelson's model. ...
In this study, we investigated the fatigue behavior of Q460 welded joints using tensile fatigue tests. Furthermore, real-time temperature profiles of the examined specimens were recorded by infrared thermography. Based on the obtained thermographic data, we calculated the entropy production rate of the specimens under different stress amplitudes. Hypothetically, the entropy production during high-cycle fatigue (HCF) could be divided into two parts. The first is induced by inelastic behavior that corresponds to damage accumulation, and the second originates from anelasticity associated with recoverable non-damaging microstructural motions. The turning point of entropy production under different stress levels represents an index for fatigue limit estimation. Then, considering the average damage threshold that exists during HCF, the entropy production related to damage accumulation (cumulative damage entropy) is obtained by testing three specimens under the same stress amplitude above the fatigue limit. Finally, a rapid three-parameter S-N curve with a survival probability rate of 50% is obtained. Then, combined with the maximum likelihood method, the 5% and 95% survival probability rate S-N curves are established. Most of experimental data are distributed in the area between S-N curves that correspond to 5% and 95% survival probability rate, indicating good accordance with the test data.
This study proposes a novel numerical approach to simulate damage accumulation and failure in polymer materials under thermal fatigue, using an entropy-based damage criterion. Unlike the many experimental studies in this area, few numerical simulations exist due to the complexity of modeling thermal fatigue. In our method, thermal and mechanical stresses arising from thermal expansion mismatches and temperature gradients are modeled through a coupled simulation approach. A viscoelastic constitutive equation is implemented in ABAQUS via a user-defined subroutine to capture damage progression. The method includes surface and internal thermal conduction, thermal deformation, and time–temperature superposition using reduced viscosity, enabling accurate simulation under varying thermal conditions. The results show that localized thermal stresses induced by temperature gradients lead to progressive damage and failure. This study demonstrates the first successful numerical simulation of thermal fatigue-induced damage in polymer materials. The proposed framework reduces the need for extensive experiments and offers insights into residual stress prediction and durability evaluation, contributing to polymer design and application in high-performance environments.
This paper aims to determine the relationship between thermodynamic entropy generation and fatigue crack propagation life of superalloy GH4169 at 300–650°C. The entire specimen was considered as the thermodynamic system. The plastic energy dissipation in the crack tip was obtained by finite element simulation utilizing the Chaboche nonlinear hardening model. Then the cyclic entropy generation rate (CEGR) and the accumulated entropy generation are calculated by combining simulation and experimental methods. Results show that the CEGR is a power function of the stress intensity factor range, and it is almost a constant at fatigue failure. The fatigue fracture entropy (FFE) increases as fatigue cycles at failure increase at constant temperature, but it first decreases and then increases when temperature increases from 300 to 650°C. A fatigue life prediction model based on the thermodynamic damage parameter is established and verified by comparison with experimental results and available data in the literature.
Dissipation represents an irreversible deformation process and is crucial for the fatigue life of materials. A precise calculation of the evolution of dissipation under cyclic external loading can guide prediction for the fatigue life of materials. The corresponding constitutive equations must be thermodynamic consistent. Based on the first and second thermodynamic laws, we derive the constitutive equations of a unified viscoplasticity model by the maximization of mechanical dissipation, where the dissipation rate is theoretically expressed. This enables one to unify the deformation description and life prediction for complex fatigue loadings. The proposed model is compared with a series of experimental data on cyclic plasticity in 42CrMo4, and it shows good agreement with the data. This suggests that the model has promising potential for evaluating the mechanical response and degradation of materials from a thermodynamic perspective.
Three fatigue damage models based on entropy, originally developed for constant amplitude loading, were assessed and compared to each other aimed at stating whether its applicability can be extended towards variable amplitude loading conditions. A variable amplitude loading history, applied on a 2024-T3 aluminum alloy reported in the literature, was processed using both rainflow cycle counting and spectral techniques to transform it into a distribution of simple processes with constant amplitudes, then the damage models were assessed under the new loading conditions. The results showed that the model by Khonsari, combined with the rainflow technique, exhibited the highest accuracy with regard to the referenced experimental results with a −0.67 standard deviation from the average data and a 16% error from the median. Therefore, it is possible to assess the fatigue damage accumulation in metallic materials under variable amplitude loading through a thermodynamic approach with models developed for constant amplitude loading.
Infrared thermography has been under review in the last 30 years due to its versatility and potential in the detection of the thermal signature associated with intrinsic energy phenomena due to dissipative processes, specifically those relying on mechanical fatigue. Nowadays, it is a well-established technique that can support mechanical and structural engineers to implement a damage-tolerant design, assess the residual life, and finally characterize the fatigue behavior of materials. The aim of this work is to review all thermography-based approaches and procedures for fatigue limit estimation by rapid tests, drawing considerations on the applicability of thermal methods in fatigue assessment of mechanical components, proposing the capabilities of different thermal indices in fatigue assessment, and discussing the pros and cons of each method as well as the open points. On one hand, this review intends to sum up what has already been done in the field; on the other hand, it provides a guideline to direct new researches toward issues that should be resolved or understood. K E Y W O R D S fatigue assessments, NDT, rapid fatigue estimation, temperature evolution, thermographic method
A ductile material, such as a polymeric material, releases energy during deformation. The dissipated energy can be evaluated as entropy generation. If the thermodynamic entropy generation can be measured, the stress state can be evaluated by the thermodynamic entropy generation.
In this study, the thermodynamic entropy generation of Polyamide 6 (PA6) was obtained using differential scanning calorimetry (DSC) for a material subjected to arbitrary strain.
Thermodynamic entropies were measured at the beginning and at each strain state of tensile tests by using DSC, and the volumetric strain was measured with Digital Image Correlation Method.
At the 25% strain just before the necking behavior, the volumetric strain of PA6 was ~4.8%, and the entropy was ~56 kJ/K∙m3. Furthermore, the thermodynamic entropy generation of PA6 in carbon fiber reinforced plastics was evaluated under tensile conditions. The results showed that the thermodynamic entropy generation just before the transverse cracking (as same as necking in matrix resin) was ~69 kJ/K∙m3 and the volumetric strain of PA6 in composite was ~ 3.56 %. As the results, the entropy generation and volumetric strain of PA6 showed almost same values in pure PA6 and PA6 in composite.
Consequently, thermodynamic entropy generation can be measured the volumetric strain of matrix resin.
The main objective of this review paper is twofold: Recall the most captivating areas of research in predicting the fatigue life of metals with the application of thermal methodology, particularly for the stress‐controlled fatigue tests. The explicit lifetime models reviewed have been developed by scholars over the past two decades owing to the advancements in infrared thermal imaging technology.
Introduce, discuss, and conclude a broad range of alternative theoretical frameworks in thermodynamics. Some investigations are made with the previous methods, including the most recent evolutions of lifetime models for structural integrity across various fields. Some future perspectives are provided in final, which could pave the way for a new realistic or potential capability.
The present paper is devoted to the fatigue evaluation of material or structure based on thermodynamic entropy approach. The fracture occures once the cumulative entropy reaches a critical value, the fracture fatigue entropy(FFE). Therefore, in this paper, the relationship between entropy and F-N curves is studied, and an entropy-load-number of cycles(E-F-N) surface is developed. Meanwhile, the FFE-based F-N curve can be obtained at the critical entropy value. To verify the utility of proposed method, the E-F-N surface and FFE-based F-N curve are applied to the fatigue evaluation of spot welded joints during high-cycle fatigue. It is shown that the E-F-N surface can effectively represent the fatigue performance. And the accuracy of the FFE-based F-N curve is confirmed by comparing with the traditional traditional three-parameter power function F-N curve. The maximum relative error of the fatigue limits obtained by two different methods is 10.9 %.
In this paper, changes in entropy generation, entropy flow and damage are discussed under cyclic loading for AZ31 magnesium alloy. Results show that the entropy generation increases sharply at the beginning of the cycle, decreases to a certain extent and remains stable. The evolution law of entropy flow is similar to that of temperature in the magnesium alloy fatigue process. Fatigue damage rises exponentially and increases rapidly after the damage reaches 0.3. The relationship between the fatigue fracture entropy model and cycle times, which can predict the fatigue life well, is established. The relationship between the stable value of entropy generation and cyclic loading is also established. The fatigue limit is 102.25 MPa, and the error is 6.8%.
The stress relaxation behavior of metallic materials has great influence on its mechanical properties under service conditions especially with high temperatures. This paper proposed a new engineering type of stress relaxation prediction model based on continuous damage mechanics and the second law of thermodynamics. Stress relaxation experiments of nickel-based single crystal DD6 were carried out at different temperatures. The proposed model was verified by experimental data of DD6 and other metallic materials, the results show that the model has good prediction accuracy.
Fracture due to cyclic actuation occurs when the accumulated entropy generation reaches a critical threshold known as the fracture fatigue entropy (FFE). The accumulated entropy generation is directly related to the intrinsic dissipations such as the supplied work, damage energy, cold work, and internal friction. In this paper, we examine the contribution of these mechanisms to fatigue damage via an irreversible thermodynamic framework and provide experimental evidence for the validity of the formulations. The formulations properly capture the evolution of temperature and reveal that experimentally observed abrupt temperature rise just before fracture is due to the rapid increase in damage and can be predicted.
An approach is presented to predict fatigue life and energy dissipation based on the measurement of the electrical power consumption of an external heat source. In this method, the steady-state temperature profile of a specimen experiencing cyclic fatigue test is reproduced simply by externally heating the specimen with an electric coil with a DC source. The results show a direct relationship between the electric power consumption of heating coils and the fatigue plastic energy dissipation. This relationship is used as a tool to predict fatigue life.
In this paper, the fracture fatigue entropy (FFE) was used to evaluate the fatigue reliability of Q460 welded joints during high-cycle fatigue. The Q460 butt welded joint fatigue tests were conducted under different loads. Meanwhile, the real-time temperature measurement, monitored by an infrared thermal scanner, was completed. The fatigue limit can be well estimated according to the cumulative damage entropy production rate in Phase II, and compared with the fatigue limit prediction value by the traditional staircase method, a good consistency is achieved. Furthermore, the P-S-N curves of 5%, 50%, and 95% survival probability rate are well predicted, which is consistent with the experimental data. Therefore, based on this, the accurate fatigue reliability evaluation based on the proposed method has been achieved.
Based on first principles, a hypothesis was made on the potential correlation between entropy and degradation of machinery components in an earlier investigation of stochastic characterization of degradation dynamics. This paper reports on an experimental study in which degradation in the form of wear of model machinery component pairs was made on an accelerated testing basis. Concomitant measurement of entropy flow was made by means of a simple calorimeter. Results show a strong correlation between the referenced wear and the production of entropy flow.
The science base that underlies modelling and analysis of machine reliability has remained substantially unchanged for decades.
Therefore, it is not surprising that a significant gap exists between available machinery technology and science to captur degradation dynamics for prediction of failure. Further, there is a lack of a systematic technique for the development o accelerated failure testing of machinery components. This article develops a thermodynamic characterization of degradatio dynamics, which employs entropy, a measure of thermodynamic disorder, as the fundamental measure of degradation; this relate entropy generation to irreversible degradation and shows that components of material degradation can be related to the productio of corresponding thermodynamic entropy by the irreversible dissipative processes that characterize the degradation. A theore that relates entropy generation to irreversible degradation, via generalized thermodynamic forces and degradation forces is constructed. This theorem provides the basis of a structured method for formulating degradation models consistent wit the laws of thermodynamics. Applications of the theorem to problems involving sliding wear and fretting wear, caused by effect of friction and associated with tribological components, are presented.
The fatigue damage analysis is examined from a historical perspective. The analysis indicates that some of the issues concerning the basic disparities between the experiment and model/interpretations. To help understand these issues, we have developed an approach with two driving force parameters to analyze the fatigue behavior. Such an approach helps in viewing the damage in terms of an intrinsic problem rather than an extrinsic one. In the final analysis one needs to unify the overall damage processes such that the description is complete from the crack initiation stage to short crack to long crack to final failure. In order to unify the damage process, three basic parameters are introduced for describing the overall fatigue process. These are ΔK, Kmax and internal stress contribution to Kmax. In addition, there are other effects from environment and temperature that can contribute to these parameters. In particular Kmax seems to play an important role in the overall damage process. We find that the internal stress is the missing link that can bridge the gap between the four main stages of damage that lies between the crack nucleation stage to final failure. Examples are sited in support of this view of explanation. Finally, it is suggested that systematic experimental data and analytical modeling to describe the internal stress gradients is required to help in forming a reliable life prediction methodology.
Continuum elastoplastic damage models employing irreversible thermodynamics and internal state variables are developed within two alternative dual frameworks. In a strain [stress]-based formulation, damage is characterized through the effective stress [strain] concept together with the hypothesis of strain [stress] equivalence, and plastic flow is introduced by means of an additive split of the stress [strain] tensor. In a strain-based formulation we redefine the equivalent strain, usually defined as the J2-norm of the strain tensor, as the (undamaged) energy norm of the strain tensor. In a stress-based approach we employ the complementary energy norm of the stress tensor. These thermodynamically motivated definitions result, for ductile damage, in symmetric elastic-damage moduli. For brittle damage, a simple strain-based anisotropic characterization of damage is proposed that can predict crack development parallel to the axis of loading (splitting mode). The strain- and stress-based frameworks lead to dual but not equivalent formulations, neither physically nor computationally. A viscous regularization of strain-based, rate-independent damage models is also developed, with a structure analogous to viscoplasticity of the Perzyna type, which produces retardation of microcrack growth at higher strain rates. This regularization leads to well-posed initial value problems. Application is made to the cap model with an isotropic strain-based damage mechanism. Comparisons with experimental results and numerical simulations are undertaken in Part II of this work.
The crack initiation period in an originally defect-free component can be a significant portion of its total fatigue life. The initiation phase is generally believed to constitute the nucleation and growth of short cracks, but the threshold crack length at which initiation occurs lacks a uniform definition. Moreover, available methods for predicting fatigue damage growth usually require an existing flaw (e.g. Paris law) and may be difficult to apply to the initiation phase. This paper presents a continuum damage mechanics-based approach that estimates cumulative fatigue damage, and predicts crack initiation from fundamental principles of thermodynamics and mechanics. Assuming that fatigue damage prior to localization occurs close to a state of thermodynamic equilibrium, a differential equation of isotropic damage growth under uniaxial loading is derived that is amenable to closed-form solution. Damage, as a function of the number of cycles, is computed in a recursive manner using readily available material parameters. Even though most fatigue data are obtained under constant amplitude loading conditions, most engineering systems are subjected to variable amplitude loading, which can be accommodated easily by the recursive nature of the proposed method. The predictions are compared with available experimental results.
Continuum elastoplastic damage models employing irreversible thermodynamics and internal state variables are developed within two alternative dual frameworks. In a strain [stress]-based formulation, damage is characterized through the effective stress [strain] concept together with the hypothesis of strain [stress] equivalence, and plastic flow is introduced by means of an additive split of the stress [strain] tensor. In a strain-based formulation we redefine the equivalent strain, usually defined as the J2-norm of the strain tensor, as the (undamaged) energy norm of the strain tensor. In a stress-based approach we employ the complementary energy norm of the stress tensor. These thermodynamically motivated definitions result, for ductile damage, in symmetric elastic-damage moduli. For brittle damage, a simple strain-based anisotropic characterization of damage is proposed that can predict crack development parallel to the axis of loading (splitting mode). The strain- and stress-based frameworks lead to dual but not equivalent formulations, neither physically nor computationally. A viscous regularization of strain- based, rate-independent damage models is also developed, with a structure analogous to viscoplasticity of the Perzyna type, which produces retardation of microcrack growth at higher strain rates. This regularization leads to well-posed initial value problems. Application is made to the cap model with an isotropic strain-based damage mechanism. Comparisons with experimental results and numerical simulations are undertaken in Part II of this work.
Two multiaxial fatigue damage models are proposed: a shear strain model for failures that are primarily mode II crack growth and a tensile strain model for failures that are primarily mode I crack growth. The failure mode is shown to be dependent on material, strain range and hydrostatic stress state. Tests to support these models were conducted with Inconel 718, SAE 1045, and AISI Type 304 stainless steel tubular specimens in strain control. Both proportional and nonproportional loading histories were considered. It is shown that the additional cyclic hardening that accompanies out of phase loading cannot be neglected in the fatigue damage model.
Elasticity, plasticity, damage mechanics and cracking are all phenomena which determine the resistance of solids to deformation and fracture. The authors of this book discuss a modern method of mathematically modelling the behaviour of macroscopic volume elements. The book is self-contained and the first three chapters review physical mechanisms at the microstructural level, thermodynamics of irreversible processes, mechanics of continuous media, and the classification of the behaviour of solids. The rest of the book is devoted to the modelling of different types of material behaviour. In each case the authors present characteristic data for numerous materials, and discuss the physics underlying the phenomena together with methods for the numerical analysis of the resulting equations.
The fatigue of dual phase steel was examined in terms of calorimetric effects in order to match the energy manifestations of fatigue and constitutive equations drawn up in a thermomechanical framework. A simplified method, assuming a homogeneous fatigue test, is proposed to determine heat source development from a temperature field provided by an infrared camera. Thermoelastic and dissipative sources were then separately identified. Experimental results concerning thermoelastic effects are in close agreement with theoretical estimates. Dissipation depends on the loading frequency and stress amplitude applied to the fatigue specimen. However, as the marked decrease in dissipation observed when testing a block at high stress was not easily interpretable in terms of material effects, we questioned the homogeneous fatigue test assumption.
Structural deterioration often occurs without visible manifestation. Continuum damage mechanics (CDM) enables one to predict the state of damage in such situations and to estimate residual strength⧹service life of an existing structure. The accumulation of damage is modeled as a dissipative process that is governed by the laws of thermodynamics. The rate of dissipation in a deformable system, R, depends on the work done on the system and the evolution of the Helmholtz free energy, Ψ. Under certain thermodynamical conditions, the first variation of Ψ vanishes, and partial differential equations for damage growth in R prior to damage localization are obtained. This approach obviates the need of introducing arbitrary dissipation potential functions with undetermined constants in the damage growth equations. All solutions use only readily available material parameters. Assuming that damage occurs isotropically under uniaxial loading, closed-form solutions are obtained for ductile damage as a function of plastic strain, for creep damage as a function of time and for fatigue damage as function of number of cycles. The models are validated with published laboratory data.
Experimental results show that the inherent ductility in a material is deflected after cycle loading. Based on the concept of ductility deflection, a new fatigue damage variable Dψ for low-cycle fatigue is recommended in this paper. This new damage variable Dψ has definite physical meaning and can be measured with a simple experimental procedure. It can be connected directly with the usual mechanical properties of materials. Dψ increases slowly at the initial stage of cycle loading, but it increases rapidly after a definite cycle where the macroscale crack is formed. We define the turning point on the Dψ-N curve as the critical damage value Dcψ and its corresponding cycle number N is the fatigue life Nf.
Eight different methods are described to measure damage defined as the effective surfacic density of micro-cracks and cavities in any plane of a representative volume element: (i) direct measurements such as the observation of micrographic pictures and the measurement of the variation of the density; (ii) non direct measurements which are destructive such as the measurement of the variations of the elasticity modulus, of the ultrasonic waves propagation and of the cyclic plasticity or creep responses, or non destructive such as the measurement of the variation of the micro-hardness and of the electrical potential.
On the basis of experimental data on the fatigue process, it is possible ; to distinguish three fatigue periods. The first period, the incubation period, ; is characterized by the absence of slip bands visible in the optical microscope. ; The second period, the disintegration period, is associated with the appearance ; and development of submicroscopic cracks in the slip bands. The third or failure ; period sets in when the submicroscopic cracks grow to the dimensions of ; microcracks. A fatigue failure diagram, reflecting these periods, is shown ; schematically. The fatigue diagram for annealed Armco iron and commercial copper ; is given. The relative length of each fatigue period in turn is discussed. ; (J.S.R.);
A new energy-based approach for predicting constant amplitude multiaxial fatigue life is described. The approach is based on cyclic plastic and elastic strain energy densities and takes into account effects of stress-state and mean stresses. A wide range of test data is used to evaluate the approach as well as a critical plane approach based on maximum amplitude of shear strain modified by the maximum normal stress on the plane of maximum shear strain amplitude. Predictive capabilities of both approaches are found to be comparable, with some differences for certain types of multiaxial loading.
Fatigue behavior is strongly affected by the environment, materials, and loading conditions. The process of fatigue can be categorized into three stages: crack initiation, growth, and final fracture. Nondestruction evaluation (NDE) of fatigue damage is of critical importance for life assessments and structural integrity evaluations. Several NDE methods, including ultrasonics, acoustic emission, and thermography, have been used to monitor fatigue damage. However, relatively little work has been conducted to assess fatigue characteristics using thermographic infrared techniques. In this paper, a thermographic infrared imaging system was used to detect the heat conditions of fatigued pressure vessel steels at 1,000 Hz and 20 Hz. Moreover, the fatigue behavior has been investigated at 1,000 Hz using an advanced electrohydraulic machine.
A request has been made to the ASME Boiler and Pressure Vessel Committee that 6061-T6 aluminum be approved for use in the construction of Class 1 welded nuclear vessels so it can be used for the pressure vessel of the Advanced Neutron Source research reactor. Fatigue design curves wit and without mean stress effects have been proposed. A knock-down factor of 2 is applied to the design curve for evaluation of welds. The basis of the curves is explained. The fatigue design curves are compared to fatigue data from base metal and weldments.
An experimental study has been carried out to determine the critical damage parameter based on the concept of entropy flow. The fatigue damage is either a cumulative process that progresses toward a maximum tolerable damage, or is an irreversible progression of cyclic plastic strain energy that reaches its critical value at the onset of fracture. In the present study, irreversible heat dissipation in terms of entropy is utilized to experimentally determine the degradation of different specimens subjected to low cyclic bending fatigue. An experimental correlation between entropy and damage is proposed. It is shown that the cyclic energy dissipation in the form of thermodynamic entropy can be effectively utilized to determine the critical damage value.
Global damage indices based on equivalent modal parameters are defined using the vibrational parameters of an equivalent linear structure. In continuum mechanics-based damage models, effects of the growth and coalescence of microcracks are accounted for through the definition of an internal or local damage variable. Damage to engineering materials essentially results in a decrease of the free energy stored in the body with consequent degradation of the material stiffness. It is shown that parameter-based global damage indices can be related to local damage variables through operations of averaging over the body volume. The relationship obtained is applied to the case of numerical simulations of damage events as well as to the case of actual structures tested on shaking tables. Some results are also presented, relating parameter-based global damage indices to averages in space and time of local plastic strain.
This paper explores the potential of a time-domain identification procedure to detect structural changes on the basis of noise-polluted measurements. The method of approach requires the use of excitation and acceleration response records, to develop an equivalent multi-degree-of-freedom (MDOF) mathematical model whose order is compatible with the number of sensors used. Application of the identification procedure under discussion yields the optimum value of the elements of equivalent linear system matrices. By performing the identification task before and after potential structural changes (damage) in the physical system have occurred, quantifiable changes in the identified mathermatical model can be detected. The usefulness of the identification procedure under discussion for damage detection is demonstrated by means of an example three-degree-of-freedom (DOF) linear system. This system is used to conduct synthetic experiments to generate noise-polluted ''data'' set that are subsequently analyzed to determine the mean, variance, and probability density function corresponding to each element of the identified system matrices. Different versions of the model are investigated in which the location as well as the magnitude of the ''damage'' is varied. On the basis of this exploratory study, it appears that determining the probability density functions of the identified system matrices may furnish useful indices that can be conveniently extracted during an experimental test, to quantify changes in the characteristics of physical systems.
Failure can occur in many distinctly different ways depending on the material, stress and strain fields, temperature field, environmental effects, strain rate, etc. The proposed research program will focus on a single well defined class of failure modes common to many structures and machine elements. This class of problems is characterized by a gradually evolving microscale process which at a certain point triggers a discontinuous or singular (qualitative) change of macroscale response. Consideration of critical states of cooperative processes requires radical departure from the well traveled paths. Recently developing methods of statistical physics seem to be applicable to the class of problems under consideration. These models are generally formulated as discrete enabling consideration of spatial-temporal complexities as they raise from the microstructural disorder. However, in many cases these models in the limit do not converge to the traditional continuum theories.
Experimental results show that the static toughness synthesizing both strength and plasticity of a material is a mechanical property parameter sensitive to the fatigue damage process. The reduction of the static toughness in the fatigue damage process indicates the progressive exhaustion of the ability to absorb energy inherent in the material due to fatigue damage evolution, which is associated directly with the irreversible process of energy dissipation of fatigue failure. Based on the exhaustion of the static toughness and dissipation of the plastic strain energy during fatigue failure, a damage variable that is consistent with the fatigue damage mechanism, sensitive to the fatigue damage process and can be measured with a simple experimental procedure is defined in this paper. A corresponding fatigue damage evolution equation is derived by connecting the damage variable with the static toughness exhaustion during fatigue, which is further examined in both theory and experiment. A fatigue damage accumulation formula that has the capacity of predicting the load sequencing effect correctly and can be applied very conveniently in engineering practice is further deduced by using equivalent-loading postulate. A comparison of experimental data with the predictions shows excellent agreement for two-stage cyclic tests.
Monotonic and strain-controlled fatigue tests were conducted on 2014-T6, 6061-T6 and 7175-T73 hand-forgings, 5052-H32 and 6061-T6 sheet, and a 1983/84 production Chevrolet Corvette upper-control-arm-pivot shaft (UCAPS) cold-forged from 5454-H12 and 6061-T4 rolled rod (the 6061 UCAPS was artificially-aged to the -T6 temper, after forging). Various monotonic and cyclic fatigue stress-strain material properties are presented. The responses of the various alloys and product-forms in terms of cyclic hardening or softening are described.
Cyclic stress-strain curves (CSSCs) and fatigue lives were obtained from fully reversed fatigue tests in strain control on two anisotropic Al-6061-T6 rods. The experiments were conducted at room temperature under three types of loading conditions: tension-compression, torsion and combined proportional tension-torsion. Based on the CSSC data, the anisotropic constitutive relations of the rods were obtained by using Hill's anisotropic plasticity theory. Yield loci and flow behaviour were determined and compared with the theoretical predictions. Two anisotropic effective-stress-effective-strain criteria were evaluated. During the fatigue tests the fatigue cracking behaviour of the rods was observed and found to be shear dominated. Four multiaxial fatigue life prediction models representing three different concepts were used to correlate the fatigue life data. A shear cracking model incorporated with a material anisotropy constant correlated with the test data very well. The other models, however, gave poor correlations.
This paper studies the thermal effects associated with the propagation of a fatigue crack in a gigacycle fatigue regime. Ultrasonic fatigue tests were carried out on a high-strength steel. The temperature fields measured by infrared thermography show a significant and very local increase in the temperature just before fracture. In order to better understand these thermal effects and to make a connection with the initiation and the propagation of the fatigue crack, a thermomechanical model is developed. The fatigue crack is modeled by a circular ring heat source whose radius increases with time. The numerical resolution of the thermal problem allows determination of the time evolution of the temperature fields in specimens and shows a good correlation with experiment. These results provide experimental proof that in a very high cycle regime, the propagation stage of the crack constitutes a small part of the lifetime of the specimen.
High-speed, high-resolution infrared thermography, as a noncontact, full-field, and nondestructive technique, was used to
study the temperature variations of a cobalt-based ULTIMET alloy subjected to high-cycle fatigue. During each fatigue cycle,
the temperature oscillations, which were due to the thermal-elastic-plastic effects, were observed and related to stress-strain
analyses. A constitutive model was developed for predicting the thermal and mechanical responses of the ULTIMET alloy subjected
to cyclic deformation. The model was constructed in light of internal-state variables, which were developed to characterize
the inelastic strain of the material during cyclic loading. The predicted stress-strain and temperature responses were found
to be in good agreement with the experimental results. In addition, the change of temperature during fatigue was employed
to reveal the accumulation of fatigue damage, and the measured temperature was utilized as an index for fatigue-life prediction.
In this paper, a method for measuring microhardness is used to investigate the surface damage evolution of cyclically loaded annealed 16Mn steel. For this purpose, the values of Vickers microhardness on the surface of specimens for different constitutive phases, ferrite and pearlite, are measured under different stress amplitudes during the fatigue damage process and their distribution characteristics are further examined. The near-surface dislocation structures in individual phases are observed at the different stages of the fatigue damage process. The variation in Vickers microhardness during cycling is then associated with the evolution of fatigue damage by introducing the concept of continuum damage mechanics (CDM) and a probability expression for a fatigue damage variable is proposed and applied in the discussion of the surface damage evolution of annealed 16Mn steel during fatigue. The results show that the damage variable defined on the basis of the variation in microhardness during cycling can reveal some characteristics of fatigue damage evolution and distribution particularly in the stages prior to the nucleation of fatigue cracks. Thus the Vickers microhardness test potentially offers a new non-destructive inspection technique of pre-nucleation fatigue damage for fatigued components under service conditions.
Determination of fatigue limit under uniaxial tests based on the experimental measurement of material thermal increments (typically by means of infrared cameras) is well documented in the literature. Anyway the energy dissipated in a unit volume of material as heat seems to be a more promising parameter for fatigue characterisation rather than the surface temperature. In fact for a given material, loading and mechanical boundary conditions the former parameter depends only on the applied stress amplitude and load ratio in a constant amplitude fatigue test, while the latter depends also on the specimen geometry, test frequency and the thermal boundary conditions that determine the rate of heat transfer from the material to the surroundings. Then it is expected that the fatigue strength of both smooth and notched specimens can be rationalised in terms of the thermal energy dissipated in a unit volume of material per cycle. The first aim of this paper is to define a theoretical model in order to derive the specific heat loss per cycle from temperature measurements performed during the fatigue test. The model has been applied to analyse the fatigue strength of smooth and notched specimens made of AISI 304 L stainless steel. Then, it has been verified to which extent the proposed approach holds true while varying the notch tip radius. Finally, it has been analysed the material response in terms of energy released as heat in two-level fatigue tests.
The objective of this study is to highlight the accomplishments, weaknesses and trends of damage mechanics and research needed for further development. The growing interest in damage mechanics is a proof that the accomplishments are significant. However, one of the messages is that the damage mechanics, in its focus on the dilute density of micro-defects and homogeneous solids, did not address the problems that are of primary interest in applications. The list of references in this paper is restricted to the current papers that list the older works.
A unified framework for coupled elastoplastic and damage theories is developed. A rigorous thermodynamic procedure is followed that is sufficiently general to include anisotropic plasticity and anisotropic damage formulations. The concept of effective stress is the critical mechanism for coupling these theories. Yield and damage functions, constructed of homogeneous functions of degree one, are shown to satisfy thermodynamic restrictions. The principle of maximum entropy provides the evolutionary relations, the loading and unloading conditions, and the convexity of the undamaging elastic domain. The plastic and damage variables evolve normal to their respective surfaces which for plasticity corresponds to an associative flow for plastic strain. This general framework is shown to be sufficiently general to encompass several popular theories for plasticity and damage. Limitations of some existing damage theories are discussed. The performance of two specific coupled formulations are illustrated by replicating the experimental behavior of an aluminum alloy.
The residual strength of fibrous composite materials depends on the remaining fatigue life. Unlike conventional materials, in which failure occurs due to a single crack, in composite materials there is a process of accumulation of many cracks. Therefore, fatigue life is expressed by loading history and residual strength is determined subsequently by the load which will cause eventual failure. A theory is developed which predicts the residual strength based on a cumulative damage concept. Experiments were performed on multidirectional laminated composite material (T300/5208) of varying construction. The results are compared with the theory and good correlation is found. The theory predicts that the residual strength is the same as the static strength for almost all the fatigue life and it starts to degrade only within the final 10% of the fatigue life.
By analysing the temperature of the external surface during the application of cyclic loading, it is possible to evaluate the dynamic behaviour of an element and to determine the fatigue limit. The methodology (Risitano method) does not need any particular testing machine and allows reliable results to be obtained using a very limited number of specimens in a very short time. This methodology also yields information on the energy retained by the specimen and mechanical components. The results of 15 years of research into a new methodology for the determination of the fatigue limit of materials or mechanical components are reported.
This paper presents a new thermographic method based on an iteration procedure for the determination of the fatigue limit of materials and components. This method is utilised to determine the fatigue limit of a mild steel Fe 510. Standard and notched specimens with semicircular notch have been fatigue tested by means of alternate symmetric (tension–compression) loads. The thermographic results have been compared to the corresponding obtained by means of a different thermographic method available in literature. These results have also been compared to those obtained by means of the Staircase Method or calculated by means of UNI Standard.
The available methods for the rapid determination of the fatigue limit (Metall. Ital. 27 (1935) 188; Rev. Metall. XVIII (1951) 11) and for the analysis of the dynamic parameters of crack mechanics (ASTM-STP 519, 246; J. Mater. 5 (1970) 4; J. Appl. Mech. 67 (1945) A159; Exp. Mech. 5 (1965) 193; Fatigue Engng Mater. Struct. 1 (1979) 37; J. Basic Eng. Trans. ASME, Ser. D 85 (1963) 528) require the determination of parameters which are highly specialised (KI, J-integral) and often too closely linked to the micro-mechanics of the material, downgrading the engineering aspects of the problem and its design definition. Following their research into the use of the thermographic method to determine the dynamic properties of materials and components commonly used in the industrial sector (hereafter called the Risitano method [Int. J. Fatigue Mater. Struct. Components 22/1 (1999) 65]), the authors now present a procedure for the definition of the whole fatigue curve.
The classical structural life predictions are based on stabilized stress-strain analysis and some parametric relations with the number of cycles to failure. During the last ten years a different approach, initiated by the works of Kachanov and Rabotnov for creep rupture, has been developed by different laboratories. This continuous Damage Mechanics, treating the damaged material as a macroscopically homogeneous one, leads to the possibility of globally modelling the nucleation and the propagation of microdefects including their effect on the stress-strain behaviour.This paper presents the general theory and several applications to a turbine blade refractory alloy. It includes the description of sequence effects and creep-fatigue interaction. The generalization for three-dimensional conditions, where anisotropic damage effects are possible, is discussed and some new proposals are given for the determination of simple anisotropic damage equations.
Fatigue damage is closely related to plastic deformation and heat dissipation, which affect the temperature of the materials. In the current research, a state-of-the-art infrared-thermography camera has been used as a nondestructive evaluation (NDE) method to investigate the temperature evolutions in both crystalline and amorphous materials during fatigue experiments. Fatigue-damage processes, such as the Lüders band growth in reactor-pressure-vessel (RPV) steels and shear-band evolution in bulk metallic glasses (BMGs), have been observed in situ and analyzed by thermography. Theoretical models combining fracture mechanics and thermodynamics have been formulated to quantify the temperature-evolution processes during fatigue. Specifically, the plastic work in RPV steel during low-cycle fatigue has been calculated and the fatigue life has been predicted from the observed temperature. The prediction matches the experimental data quite well.
Theoretical and empirical investigation of stochastic degradation in machinery components
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