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A closed-form expression for the dual of dissipation potential is derived within the framework of irreversible thermodynamics using the principles of dimensional analysis and self-similarity. Through this potential, a damage evolution law is proposed for concrete under fatigue loading using the concepts of damage mechanics in conjunction with fracture mechanics. The proposed law is used to compute damage in a volume element when a member is subjected to fatigue loading. The evolution of damage from microcracking to macrocracking of the entire member is captured through a series of volume elements failing one after the other. The number of loading cycles to failure of the member is obtained as the summation of number of cycles to failure for each individual volume element. A parametric study is conducted to determine the effect of the size of the volume element on the model’s prediction of fatigue life. A global damage index is also defined, and the residual moment carrying capacity of damaged beams is evaluated. Through a deterministic sensitivity analysis, it is found that the load range and maximum aggregate size are the most influencing parameters on the fatigue life of a plain concrete beam.

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... Several analytical and empirical equations have been developed in the last decades, where the fatigue behavior of the concrete is described on the fatigue lifetime scale [13,15,[23][24][25][26][27][28][29]. In this class of models the strain evolution during fatigue life i.e. fatigue creep curves are reproduced, based on experimental observations. ...

... In the present comparison the values proposed by [1] are taken asā = 1.14,b = −2.4,c = 2. 26 ...

... In the present comparison, the values specified by [1] are taken asā = 1.14,b = −2.40,c = 2. 26. For the prediction of the fatigue life under varying loading ranges, the cumulative damage rule has been proposed in the form ...

In spite of the considerable achievements that have been made in recent years in modeling and characterizing the fatigue behavior of concrete, still many open questions need to be fundamentally addressed in order to gain a deep and general insight into the fatigue phenomenology of concrete. In this thesis, a numerical, theoretical and experimental framework for the analysis and characterization of the fatigue behavior of concrete is developed to improve the understanding of the fatigue phenomenology in terms of the fundamental fatigue damage mechanisms occurring in the internal structure of the material. The presented framework aims to provide the basis for a more efficient analysis and evaluation of the fatigue behavior of structural concrete, which in the long run can contribute to the formulation of reliable and economical design concepts and codes for reinforced concrete structures under fatigue loading. The developed modeling components in this thesis are based on an enhanced fatigue modeling hypothesis that consistently represents the dissipative mechanisms associated with cumulative cyclic shear deformation at subcritical loading levels and is formulated within the thermodynamic framework. The refined fatigue hypothesis is applied within the context of bond fatigue to describe the bond deterioration under fatigue loading, and is calibrated and validated for ductile and brittle types of bond behavior. Further enhancement of the one-dimensional interface model is provided to consistently capture the 3D kinematics of zero-thickness interfaces response under monotonic and fatigue loading and providing a generic fatigue constitutive law that can be applied in a wide range of structural applications. To capture the tri-axial stress redistribution within the concrete structure under compressive fatigue loading, a novel microplane fatigue model is developed that employs the introduced fatigue hypothesis with cumulative fatigue damage due to sliding. A systematic calibration and validation procedure of the model response is provided based on accompanying performed experimental program including normal- and high-strength concretes subjected to several loading scenarios of compressive loading. Moreover, in comparison of the current fatigue characterization methods, which require a large number of expensive experiments, a combined numerical, experimental and theoretical methodology is employed to characterize the effects of the loading sequence on the fatigue behavior of concrete. As a result, an enhanced assessment rule for predicting the fatigue life of concrete under compression is proposed that takes into account the effects of the loading sequence. This rule demonstrates the potential contribution of the advanced and efficient numerical modeling approaches to the formulation of reliable design concepts related to the fatigue response of materials and structures.

... Although several fatigue models have been successful in illustrating the effect of loading frequency on fatigue life, most of them are empirical in nature and fail to explain the physical mechanisms that initiate at microscale and gradually lead to fracture. Fathima and Chandra Kishen (2015) dimensional analysis and similitude concepts. A similar approach was followed by Keerthana and Chandra Kishen (2018) to model an overload effect in concrete subjected to variable amplitude loading. ...

... However, the work done for unit increase in crack length increased with an increase in loading frequency, which can be attributed to the wide spread of microcracks. To incorporate these features into the model, a unified approach incorporating both damage and fracture mechanics concepts was used (Fathima and Chandra Kishen 2015). In the damage mechanics framework, a damage variable D is defined, which evolves within a small volume element considered at the initial crack tip a 0 . ...

... Conventionally, a volume element with a semicircular base is considered (Fathima and Chandra Kishen 2015;Keerthana and Chandra Kishen 2018). If the width of the semicircular base of the volume element is increased, its height increases proportionally. ...

This study investigated the effect of loading frequency on the fatigue damage process in concrete using digital imaging and acoustic emission techniques. It was found that the complex fatigue damage process in heterogeneous concrete is reflected in the amplitude of acoustic energy. The distribution of acoustic energy levels was utilized to classify micro- and macro-structural activities. It was found that fatigue failure at higher frequencies is governed predominantly by microcracks, while at lower frequencies both micro- and macro-cracks contribute to failure. The applied loading frequency had a marked influence on the size of the fracture process zone (FPZ). A fatigue model encapsulating frequency effects in terms of FPZ width is proposed using a unified damage and fracture mechanics approach within the framework of dimensional analysis and similitude concepts.

... Although several fatigue models have been successful in illustrating the effect of loading frequency on fatigue life, most of them are empirical in nature and fail to explain the physical mechanisms that initiate at microscale and gradually lead to fracture. Fathima and Chandra Kishen (2015) dimensional analysis and similitude concepts. A similar approach was followed by Keerthana and Chandra Kishen (2018) to model an overload effect in concrete subjected to variable amplitude loading. ...

... However, the work done for unit increase in crack length increased with an increase in loading frequency, which can be attributed to the wide spread of microcracks. To incorporate these features into the model, a unified approach incorporating both damage and fracture mechanics concepts was used (Fathima and Chandra Kishen 2015). In the damage mechanics framework, a damage variable D is defined, which evolves within a small volume element considered at the initial crack tip a 0 . ...

... Conventionally, a volume element with a semicircular base is considered (Fathima and Chandra Kishen 2015;Keerthana and Chandra Kishen 2018). If the width of the semicircular base of the volume element is increased, its height increases proportionally. ...

... Although several fatigue models have been successful in illustrating the effect of loading frequency on fatigue life, most of them are empirical in nature and fail to explain the physical mechanisms that initiate at microscale and gradually lead to fracture. Fathima and Chandra Kishen (2015) dimensional analysis and similitude concepts. A similar approach was followed by Keerthana and Chandra Kishen (2018) to model an overload effect in concrete subjected to variable amplitude loading. ...

... However, the work done for unit increase in crack length increased with an increase in loading frequency, which can be attributed to the wide spread of microcracks. To incorporate these features into the model, a unified approach incorporating both damage and fracture mechanics concepts was used (Fathima and Chandra Kishen 2015). In the damage mechanics framework, a damage variable D is defined, which evolves within a small volume element considered at the initial crack tip a 0 . ...

... Conventionally, a volume element with a semicircular base is considered (Fathima and Chandra Kishen 2015;Keerthana and Chandra Kishen 2018). If the width of the semicircular base of the volume element is increased, its height increases proportionally. ...

... Various methodologies such as damage mechanics [1] and fracture mechanics have been used to study progressive damage in the material due to fatigue. Fathima and Kishen [2] proposed a damage evolution law for fatigue in plain concrete which is based on the concepts of irreversible thermodynamics and damage mechanics. This model captures the evolution of damage from microcracking to macro-cracking until the failure of entire member by considering the failure of a series of small volume elements at the crack tip. ...

... The damage evolution law proposed by Fathima and Kishen [2] is given by equation Eq.1. ...

... D is the damage variable, D = 0 corresponds undamaged state and D = 1, corresponds to a completely damaged state. All the parameters in the above expression can be referred from [2]. By substituting the suitable limits for N and D, the number of load cycles required for the failure of one volume element or damage incurred in a volume element for a given number of cycles can be obtained. ...

... There are very few models which are developed based on physical principles. Few researchers [11,18,19] in recent times, have developed fatigue models based on physical laws using energy principles, dimensional analysis and similitude concepts. A damage evolution law for concrete under constant amplitude loading was proposed by Fathima and Kishen [18] using the concepts of damage mechanics and fracture mechanics. ...

... Few researchers [11,18,19] in recent times, have developed fatigue models based on physical laws using energy principles, dimensional analysis and similitude concepts. A damage evolution law for concrete under constant amplitude loading was proposed by Fathima and Kishen [18] using the concepts of damage mechanics and fracture mechanics. In this work, a closed form expression for dual dissipation potential within the thermodynamic framework using the principles of dimensional analysis and self similarity has been derived, and further a damage evolution law has been proposed which is used to compute damage in the volume element when subjected to fatigue loading. ...

... The main objective of the work presented in this paper is to develop a damage model to describe the fracture behaviour of plain concrete when subjected to variable amplitude loading. The model is based on physical principles through the use of dimensional analysis and concepts of self-similarity on the lines following the constant amplitude model proposed earlier by Fathima and Kishen [18]. Experiments are conducted on geometrically similar plain concrete notched beams of three different sizes under a simulated variable amplitude loading in order to calibrate and validate the proposed model. ...

A fatigue model for plain concrete under variable amplitude loading is proposed by unifying the concepts of damage mechanics and fracture mechanics through an energy equivalence, in conjunction with the principles of dimensional analysis and self-similarity. The effects of stress ratio and overloads that accelerate the crack growth rate is included in the model, in order to capture the realistic behaviour under variable amplitude loading. Experiments are performed under variable amplitude fatigue loading in order to calibrate and verify the validity of the model. The model proposed in this work encapsulates the complex behaviour of concrete under fatigue and provides a more rational method for computing fatigue life of concrete structures.

... Regarding the methods for fatigue analysis, there are roughly four categories in literature, i.e. the S-N curve based on experimental results [5,22], the linear elastic fracture mechanics based Paris' law for describing the fatigue crack propagation [23], the cohesive crack model based on the non-linear stress-crack width relationship [24] and the fatigue damage evolution model based on damage mechanics [25]. It is well known that fatigue of materials can be characterized by an S-N curve (stress level versus fatigue life), also known as the Wöhler curve [26]. ...

In this study, the flexural strength and fatigue properties of interfacial transition zone (ITZ) were experimentally investigated at the micrometre length scale. The hardened cement paste cantilevers (150 μm × 150 μm × 750 μm) attached to a quartzite aggregate surface were prepared and tested under the monotonic and cyclic load using a nanoindenter. The measured flexural strength of the ITZ (10.49–14.15 MPa) is found to be one order of magnitude higher than the macroscopic strength of ITZ reported in literature. On the other hand, the fatigue strength of the ITZ is lower than that of bulk cement paste at same length scale, measured previously by the authors. The microscopic mechanical interlocking and the electrostatic interaction between aggregate surface and hydration products are thought to contribute to the bond strength of ITZ. This study provides an experimental basis for the development of multiscale analysis of concrete subjected to both static and fatigue loading.

... Classification of structures according to fatigue loads given by [45]. [50,58,[60][61][62][63][64][65][66][67]. The fatigue behavior has been described in this class of models in term of fatigue lifetime scale. ...

This paper contributes by summarizing knowledge on the fatigue behavior of concrete, discussing current standardization, and organizing previous contributions and prospecting future developments in experimental research. It describes classical laboratory fatigue tests and analysis (S-N curves, damage effects on modulus etc.), emphasizing differences expected for real loading (with variation in frequency and temperature, among other effects). Not only current standards diverge in results, but more fundamental effects were observed. Temperature has relevant impact on fatigue life, and during tests it may change due to self-heating. Specimen dimensions are relevant for this phenomenon. Loading frequency seems to influence fatigue, and since tests in laboratory need to be accelerated (high frequency) compared to the field (low frequency), this is a major concern. Other investigated effects include loading waveforms and moisture.

... Additional dynamic related works are conducted by (Grassl et al., 2011;Lotfi, 2013, 2017;. The models at different strain rates for blast loading, seismic loading, alternate tension-compression fatigue loading, and the numerical implementation for high strain rate are developed by several authors (Fathima and Kishen, 2015;Herv e-Secourgeon et al., 2005;Huang et al., 2005;Lu and Xu, 2004;Ragueneau and Gatuingt, 2003). Lu and Xu (2004) present the constitutive model for the dynamic strength and damage for concrete which originated from the fracture mechanics of microcracks evolution, propagation, and coalescence for the damage evolution, under blast loading. ...

In the past few decades, extensive research on concrete modeling to predict behavior, crack propagation, microcrack coalescence by utilizing different approaches (fracture mechanics, continuum damage mechanics) were investigated theoretically and numerically. The presented paper aims to review the theoretical work of continuum concrete damage and plasticity modeling in part I of the work. The detailed theoretical work is presented with some of the supporting work related to multiscale modeling and phase-field modeling is also part of this paper. Few other applications related to rate-dependent models and fatigue in concrete are also discussed. In part
19 II of this work, the review of numerical work limited to finite element is presented. Some open
20 issues in concrete damage modeling and future research needed are also discussed in part II.

... Shear band/loading (Cervera et al., 2004(Cervera et al., , 2018Kotronis et al., 2003;Li Piani et al., 2019;Negi and Kumar, 2019;Poh and Sun, 2017;Thai et al., 2016;Vandoren and Simone, 2018) Transient interaction domain (Geers et al., 1998;Giry et al., 2011;Negi and Kumar, 2019;Nguyen et al., 2018;Pereira et al., 2016;Pijaudier-Cabot et al., 2004;Poh and Sun, 2017;Saroukhani et al., 2013;Thai et al., 2016;Vandoren and Simone, 2018) Fatigue (Al-Gadhib et al., 2000;Baktheer and Chudoba, 2019;Chen et al., 2018;Desmorat, 2005;Desmorat et al., 2007b;Ding et al., 2019;Fathima and Kishen, 2015;Kindrachuk et al., 2015;Li et al., 2004;Liang et al., 2015;Lü et al., 2004;Lubin and Cheng-Tzu Thomas, 1998;Mai et al., 2012;Papa and Taliercio, 1996;Perera et al., 2000;Sain and Chandra Kishen, 2007;Shan et al., 2019;Yadav and Thapa, 2020;Zhang et al., 2020a;Zhaodong and Jie, 2017) Four-point bending beam (Mazars et al., 1991) ...

In this part II, companion article, we present the numerical review of continuum damage mechanics and plasticity in the context of finite element. The numerical advancements in local, nonlocal, and rate-dependent models are presented. The numerical algorithms, type of elements utilized in numerical analysis, the commercial software’s or in-house codes used for the analysis, iterative schemes, explicit or implicit approaches to solving finite element equations, and degree of continuity of element are discussed in this part. Lastly, some open issues in concrete damage modeling and future research needed are also discussed.

... It should be mentioned that, even though the Miner's law may not precisely reflect the actual fatigue damage evolution of cementitious material and may lead to unconservative results, this law is widely used for its simplicity [15,[57][58][59]. An alternative would be to consider the fatigue damage evolution using thermodynamic concepts under the damage mechanics framework, as was proposed in [4,60]. ...

In this study, a numerical model using a 2D lattice network is developed to investigate the fatigue behaviour of cement paste at the microscale. Images of 2D microstructures of cement pastes obtained from XCT tests are used as inputs and mapped to the lattice model. Different local mechanical and fatigue properties are assigned to different phases of the cement paste. A cyclic constitutive law is proposed for considering the fatigue damage evolution. Fatigue experiments performed at the same length scale are used to calibrate and validate the model. The proposed model can reproduce well the flexural fatigue experimental results, in terms of S-N curve, stiffness degradation and residual deformation. The validated model is then used to predict the uniaxial tensile fatigue fracture of cement paste. The effects of microstructure and stress level on the fatigue fracture are studied using the proposed model. This model forms a basis for the multiscale analysis of concrete fatigue.

... 8. Compute the density of energy dissipated e d from Eq. (8). 9. Compare the value of e d obtained in the previous step with the critical value e crit d proposed by Fathima and Chandra Kishen (2015b). 10. ...

A micromechanics-based model is developed to study the fatigue response of cementitious composites. Microcrack growth, which is the predominant mechanism responsible for fatigue damage, is explicitly modeled at the mesoscale. The damaged state at the macroscopic scale is determined by using energetic criterion. The dissipated energy associated with each stage of microcrack propagation is computed numerically based on the elastic solutions of the stress and displacement quantities at the mesoscale. The model is used to predict the fatigue life of plain concrete beams under fatigue load cycles. The influence of the various properties of the constituent phases on the fatigue life of the composite is investigated through a parametric study.

... Models defined on lifetime scale 2.1.1. Empirical approximations of fatigue strain Several analytical and empirical equations have been developed in the last decades, where the fatigue behavior of the concrete is described on the fatigue lifetime scale [9,19,20,21,22,23,24,11,25]. In this class of models the strain evolution during fatigue life i.e. fatigue creep curves are reproduced based experimental observations. ...

In this paper an evaluation of existing numerical models for concrete fatigue behavior subjected to compressive loading is presented. The brief review of available modeling approaches is accompanied with their classification based on the postulated fatigue damage hypothesis and on the modeling scale at which the damage is introduced. Based on the review, three numerical models for concrete fatigue that are considered to be relevant for the simulation of high cycle fatigue are selected as representatives of different fatigue modeling hypotheses and evaluated in view of their ability to reproduce several aspects of concrete fatigue behavior. The strengths and weaknesses of the studied numerical models are illustrated and discussed. Each model is presented by explaining its calibration procedure and showing its ability to reproduce the Wöhler curves, fatigue creep curves, energy dissipation, effect of loading sequence and computational efficiency. The discussion of the models includes also conclusions and suggestions for possible improvements of approaches to high cycle fatigue modeling.

In this study, a continuum-based model was proposed to characterise the flexural performance and damage evolution process of a plain concrete beam under high-cycle fatigue loads. The plain concrete beam under three-point bending load was modelled by combining the damaged constitutive model of concrete and the Euler–Bernoulli beam theory with varied stiffness. The dynamic stiffness matrix method, in conjunction with the discretization technique, was adopted to solve the nonlinear governing equations of motion. A global damage index corresponding to natural frequencies was introduced to quantify the performance degradation of the beam. To improve the analysis efficiency of the high-cycle fatigue problems, the accelerated algorithm, namely the jump-in-cycle method, was employed in this model. It is demonstrated that the numerical results agree well with the experimental data under fatigue harmonic loads. Performance degradation under fatigue bending loads only occurs at the midspan of the beam. Adopting the jump-in-cycle methods significantly improves the analysis efficiency of the high-cycle fatigue problem, and the calculation time is reduced by approximately 90% compared with the cycle-by-cycle method. This model is capable of rationally predicting damage evolutions, stiffness degradation processes, stress redistributions, and loading level-fatigue life (S–N) curves and can provide a basis for simulating the flexural behaviours of reinforced concrete (RC) beams under fatigue bending loads.

Based on the strain-kinetic fatigue fracture criterion, the kinetics of accumulated damages in structural carbon steel under low-cycle loading at an elevated temperature (150°C), at which the material begins to develop stress aging processes, is estimated. It is shown that stress aging, strengthening the material, narrows the durability area in which there is quasistatic destruction, the limiting case of which is the buckling failure of plastic strain, like single static destruction. This increases the fatigue fracture range, the limiting case of which is the formation of the primary fracture. The recoverable plastic strain causes the main damaging effect. Stress aging contributes to an increase in damage from elastic strain, although in the area of low-cycle failure, the share of damage from elastic strain remains small, growing with increased durability, and in the area of multi-cycle fatigue, this share becomes prevalent. Hardening from stress aging inhibits the development of damage from the one-way accumulated strain. The author confirms the validity of applying the strain-kinetic criterion of fatigue fracture to describe the damage kinetics and limit states during the cyclic elastic-plastic strain of carbon steel in the presence of weak stress aging.

To explore the fatigue fracture behavior of self-compacting concrete (SCC), constant amplitude loading tests and incremental amplitude loading tests were carried out. The experimental results of SCC were obtained and analyzed, including P-CMOD curve, fatigue life, effective crack length, fatigue crack growth (FCG) rate, stress intensity factor (SIF). The results show that the P-CMOD curves of SCC beams follow the unique envelope theory. The fatigue damage of SCC under constant amplitude load can be divided into three stages. In the second stage, cracks grow steadily. The critical SIF is the boundary point between the second and the third stages of fatigue damage. When SIF exceeds the critical SIF, cracks develop rapidly. The first stage of fatigue damage is not obvious for incremental amplitude loading tests. With the increase of load amplitude, there are obvious second and third stages of damage. Except for the maximum load stage, the crack opening rate keeps constant generally. Combined with damage mechanics and fracture mechanics, damage evolution equations of SCC under constant amplitude load and incremental amplitude load were established. The parameters of fatigue damage model based on continuous damage mechanics (CDM) were calibrated by test data, and the predicted results were compared with the experimental results. The results show that the model is sensitive to the first stage of damage. The fatigue damage model based on CDM is more suitable to predict fatigue life for incremental amplitude tests and constant amplitude tests with high cycle. The advantage of this model is that it can be applied to predict fatigue life for different load levels.

The aim of this study is to test the classical Lemaitre model based on continuum damage mechanics (CDM) approach in the range of low cycle and quasi‐static fatigue life. Study is carried out with the use of results of experimental tests for C45 steel (according to AISI: 1045 steel) carried out under variable‐amplitude loading. Loading programs are of two‐step character and include blocks of cycles of different lengths and R = ‐1 coefficients. Fatigue lives are calculated according to Lemaitre model from experimentally obtained stress and strain histories recorded during fatigue tests. The results are compared with experimental tests results and with fatigue lives calculated with the use of by traditional fatigue approach based on Palmgren‐Miner damage summation hypothesis. Experimental test of fatigue life calculation results for C45 steel reveals that continuum damage method, using the recorded stresses and strains, predicts fatigue life better as compared to the remaining methods. The study also contains many detailed analyses of experimental results.

A generalized damage model is presented. Built within the thermodynamics framework, it assumes a damage evolution governed by the main dissipative mechanim: plastictity for metals, internal sliding with friction for concrete and filled elastomers. The model applies to different classes of materials but also to different kinds of loadings, monotonic and fatigue loadings. From the Continuum Damage Mechanics point of view, the number of cycles to rupture in fatigue is reached when the damage D equals the critical damage D c . Examples of calculated fatigue curves are given for different materials.

The characteristic, length of a heterogeneous brittle material such as concrete represents a material property that governs the minimum possible width of a zone of strain-softening damage in nonlocal continuum formulations or the minimum possible spacing of cracks in discrete fracture models. This length is determined experimentally. The basic idea is to compare the response of two types of specimens, one in which the tensile softening damage remains distributed and one in which it localizes. The latter type of specimen is an edge-notched tensile fracture specimen, and the former type of specimen is of the same shape but without notches. Localization of softening damage is prevented by gluing to the specimen surface a layer of parallel thin-steel rods and using a cross section of a minimum possible thickness that can be cast with a given aggregate. The characteristic length/is the ratio of the fracture energy (i.e., the energy dissipated per unit area, dimension N/m) to the energy dissipated per unit volume (dimension N/m2). Evaluation of these energies from the present tests of concrete yields l = 2.7 times the maximum aggregate size.

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.

The Legendre transform is an important tool in theoretical physics, playing a critical role in classical mechanics, statistical mechanics, and thermodynamics. Yet, in typical undergraduate or graduate courses, the power of motivation and elegance of the method are often missing, unlike the treatments frequently enjoyed by Fourier transforms. We review and modify the presentation of Legendre transforms in a way that explicates the formal mathematics, resulting in manifestly symmetric equations, thereby clarifying the structure of the transform algebraically and geometrically. Then we bring in the physics to motivate the transform as a way of choosing independent variables that are more easily controlled. We demonstrate how the Legendre transform arises naturally from statistical mechanics and show how the use of dimensionless thermodynamic potentials leads to more natural and symmetric relations.

Scaling laws reveal the fundamental property of phenomena, namely self-similarity - repeating in time and/or space - which substantially simplifies the mathematical modelling of the phenomena themselves. This book begins from a non-traditional exposition of dimensional analysis, physical similarity theory, and general theory of scaling phenomena, using classical examples to demonstrate that the onset of scaling is not until the influence of initial and/or boundary conditions has disappeared but when the system is still far from equilibrium. Numerous examples from a diverse range of fields, including theoretical biology, fracture mechanics, atmospheric and oceanic phenomena, and flame propagation, are presented for which the ideas of scaling, intermediate asymptotics, self-similarity, and renormalisation were of decisive value in modelling.

This well-established textbook teaches macroscopic modeling for design, processing, testing, and control of mechanical components in engineering. The first chapter deals with the phenomenology of damage; the second couples damage to strains and covers the three-dimensional situation; the third is devoted to kinetic laws of damage evolution used by the author to unify many models; the fourth gives several methods for predicting crack initiation. Detailed calculations and many exercises help students to apply the powerful techniques to practical problems in engineering.
This second, corrected and enlarged edition also includes the damage of interfaces and statistical damage analysis with microdefects.

Crack growth caused by load repetitions in geometrically similar notched concrete specimens of various sizes is measured by means of the compliance method. It is found that the Paris law, which states that the crack length increment per cycle is a power function of the stress intensity factor amplitude, is valid only for one specimen size (the law parameters being adjusted for that size) or asymptotically, for very large specimens. To obtain a general law, the Paris law is combined with the size-effect law for fracture under monotonic loading, proposed previously by Bazant. This leads to a size-adjusted Paris law, which gives the crack length increment per cycle as a power function of the amplitude of a size-adjusted stress intensity factor. The size adjustment is based on the brittleness number of the structure, representing the ratio of the structure size d to the transistional size d0, which separates the responses governed by nominal stress and stress intensity factor. Experiments show that d0 for cyclic loading is much larger than d0 for monotonic loading, which means that the brittleness number for cyclic loading is much less than that for monotonic loading. The crack growth is alternatively also characterized in terms of the nominal stress amplitude.

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 problem of long time strength (evaluation of time to rupture) is of obvious relevance for various machine parts working under high temperature conditions. The existing data on specimens tested under uniaxial tension are insufficient for general loading conditions and inhomogeneous stress states: intensive experimental investigations are being conducted in this direction. Theoretical modelling of long time strength in the framework of continuum mechanics appears to be important. In recently published work of Hoff (1953), the moment of failure of a rod under tension is defined as the one at which the cross-sectional area becomes zero as a result of quasiviscous flow. His result is in satisfactory agreement with experimental data. A similar concept was used by Katz (1957) and by Rosenblum (1957) in their studies of times to rupture of thick-walled pipes and of a rotating disk with a hole. We note, however, that the experimental data points typically lie below the theoretical predictions and that rupture occurs at elongations not exceeding several tens per cent. The concept of Hoff has certain limitations. It predicts, for example, that creep under torsion does not result in rupture, contrary to observations. Also, his scheme does not explain fractures at small strains (`brittle' ruptures) and the change of character of rupture (from `viscous' to `brittle') if the material is not sufficiently stable. Here, we suggest a theoretical model for the time to rupture with the account of embrittlement. We emphasize that the physics of such phenomena is very complex and has not been studied sufficiently. Although we discuss microcracking, the results can be interpreted in a more general way, in terms of development of damage.

The phenomenon of fatigue is commonly observed in majority of concrete structures and it is important to mathematically model it in order to predict their remaining life. An energy approach is adopted in this research by using the framework of thermodynamics wherein the dissipative phenomenon is described by a dissipation potential. An analytical expression is derived for the dissipation potential using the concepts of dimensional analysis and self-similarity to describe a fatigue crack propagation model for concrete. This is validated using available experimental results. Through a sensitivity analysis, the hierarchy of importance of different parameters is highlighted.

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.

The paper examines the concept of self-similarity and demonstrates how certain problems are studied with the idea of establishing self-similarity of the solution. Dimensional analysis is performed to derive some basic relations concerning similarity criteria, and these criteria are applied in some sample problems to see how similarity of the solution or of some of the variables can be established. The example of a heat source in a medium with temperature-dependent thermal conductivity is considered. Another problem illustrated is that of strong explosion. A classification of self-similar solutions is given, and the concept of incomplete self-similarity is defined. The intermediate-asymptotic nature of all self-similar solutions is demonstrated. Incomplete self-similarity in the problem of isotropic homogeneous turbulence is studied.

A damage model is proposed to characterize fatigue crack initiation in notched plates. This is accomplished by assuming a damage fracture criterion and effective instantaneous tangent moduli matrix which accounts for damage accumulation. A finite element package is used to perform the numerical analyses for an edge-notched plate specimen made of Al alloy 2024-T3 under cyclic loading of varying constant amplitudes. The estimated fatigue crack initiation life agrees satisfactorily with those determined experimentally. An important observation from the results of the proposed damage model is its ability to identify the phenomenon of stress redistribution due to material degradation in two-dimension specimens during fatigue loading.

Continuum Damage Mechanics (CDM) has bee proved to be a powerful tool in all those problems where it is difficult to use classical fracture mechanics concepts as a result of the following effects: effects of large-scale yielding plasticity (for instance, ductile rupture due to large deformations, geometry dependence of the J-resistance curve), three-dimensional effects, effects of multiple sites damage such as short cracks, multiple cracking in composites, and so on. In this paper, a new nonlinear CDM plasticity damage model is proposed. The model was developed on the experimental observations that the growth of microvoids results in a nonlinear damage accumulation with plastic deformation. Three basic possible damage evolution trends are identified and taken into account by a single damage model. The model proposed was applied successfully to seven different materials that clearly exhibit the three different damage evolution trends. The effects of triaxial state of stress on material damage parameters, such as the material strain to failure, are also discussed. Material damage constants and the procedure for their identification are presented.

The background of continuum damage mechanics is first presented in the framework of thermodynamics with some examples of constitutive equations for ductile damage, creep damage and fatigue damage. After the general scheme of structural calculations for macro-crack initiation, through non-coupled or coupled strain damage equations, some examples of “simple applications” are given: fracture limits of metal forming, surface initial damage in fatigue, creep fatigue interaction, and bifurcation of cracks.

A model of isotropic ductile plastic damage based on a continuum damage variable, on the effective stress concept and on thermodynamics is derived. The damage is linear with equivalent strain and shows a large influence of triaxiality by means of a damage equivalent stress. Identification for several metals is made by means of elasticity modulus change induced by damage. A comparison with the McClintock and Rice-Tracey models and with some experiments is presented for the influence of triaxiality on the strain to rupture.

A substantial fraction of the mysteries associated with crack extension might be eliminated if the description of fracture experiments could include some reasonable estimate of the stress conditions near the leading edge of a crack particularly at points of onset of rapid fracture and at points of fracture arrest. It is pointed out that for somewhat brittle tensile fractures in situations such that a generalized plane-stress or a plane-strain analysis is appropriate, the influence of the test configuration, loads, and crack length upon the stresses near an end of the crack may be expressed in terms of two parameters. One of these is an adjustable uniform stress parallel to the direction of a crack extension. It is shown that the other parameter, called the stress-intensity factor, is proportional to the square root of the force tending to cause crack extension. Both factors have a clear interpretation and field of usefulness in investigations of brittle-fracture mechanics.

Sensitivity analysis, supported by computer hardware and software, can easily overwhelm an analyst or decision maker with data. However, this data can be organized in a readily understandable way using well-designed graphs. Two graphical techniques, spiderplots and tornado diagrams, are commonly used respectively by engineering economists and decision analysts. Their advantages are complementary. Management scientists should often use both to convey their results to,decision makers succinctly and clearly. The simpler tornado diagram can summarize the total impact of many independent variables. An individual spiderplot displays more information about a smaller number of variables. This includes the limits for each independent variable, the impact of each on the dependent outcome, and the amount of change required to reach a break-even point.

The objective of the present study is to summarize and recapitulate some of the methods of analyses of brittle response of solids. In general, the discussion is limited to the analyses of influence that the crack-like microdefects have on the compliance and failure of some engineering materials. The attention was focused on the perfectly brittle and semi-brittle response of materials such as concrete, rocks, ceramics and certain brittle solids. Both phenomenological and micromechanical models were discussed at some length emphasizing their relative advantages and drawbacks. A section dealing with simple, one-dimensional damage models is added to help the reader not familiar with this new branch of continuum mechanics.

A continuum isotropic ductile plastic damage model has been derived in the framework of internal variable theory of thermodynamics. The damage model is based on the concept of effective stress and the principle of strain equivalence. The damage model shows non-linear variation with respect to plastic strain, and is sensitive to stress triaxiality. The predictions using the model compare well with experimental results of certain aluminium alloys. This damage model can be used for such metals to study the effect of stress triaxiality on the strain to rupture.

In this paper, plastic damage and ductile fracture processes in mild steels are investigated based on the phenomenological theory of continuum damage mechanics and on micromechanical experimental observations. Some theoretical models are further discussed and new experimental results are reported, which contain plastic damage evolution, damage mechanics criterion, effects of internal damage on plastic deformation and ductile fracture, and macrocrack initiation.

A general continuum damage mechanics (CDM) model for ductile fracture is proposed in the framework of CDM which can be reduced to linear and/or nonlinear models and can unify many known models by introducing appropriate values for the damage coefficients of materials. Comparisons of the modelling and experimental results are presented and good agreements are found.

In many materials, including tin–lead eutectic solder, a substantial portion of their fatigue life is spent in accumulation of randomly distributed microcracks. An assembly of discrete interconnected elements (e.g. grains or phases) approximates the solid in this paper. Percolation theory is employed to derive critical microcrack density and to predict size effect. Self-similarity in microcrack nucleation leads to a simple power law for lifetime prediction. Its application to the fatigue of tin–lead eutectic solder shows a good agreement with experimental data.

The practice of attempting validation of crack-propagation laws (i.e., the laws of Head, Frost and Dugdale, McEvily and Illg, Liu, and Paris) with a small amount of data, such as a few single specimen test results, is questioned. It is shown that all the laws, though they are mutually contradictory, can be in agreement with the same small sample of data. It is suggested that agreement with a wide selection of data from many specimens and over many orders of magnitudes of crack-extension rates may be necessary to validate crack-propagation laws. For such a wide comparison of data a new simple empirical law is given which fits the broad trend of the data.

Currently, the maintenance and repair of civil engineering infrastructures (especially bridges and highways) have become increasingly important, as these structures age and deteriorate. Interface between two different mixes or strengths of concrete also appear in large concrete structures involving mass concreting such as dams, nuclear containment vessels, cooling towers etc., since joints between successive lifts are inevitable. These joints and interfaces are potential sites for crack formation, leading to weakening of mechanical strength and subsequent failure. In case of a bi-material interface, the stress singularities are oscillatory in nature and the fracture behavior of a concrete-concrete bi-material interface is much more complicated. A comprehensive experimental work has been undertaken for characterization of the behavior of different concrete-concrete interfaces under static and fatigue loading. The effect of specimen size on the concrete-concrete interfaces is studied and the non-linear fracture parameters such as fracture energy, mode I fracture toughness, critical crack tip opening displacement, critical crack length, length of process zone, brittleness number, size of process zone, crack growth resistance curve and tension softening diagram. These parameters are required for modeling the concrete-concrete interfaces in non-linear finite element analysis. Presently, the advanced non-destructive techniques namely acoustic emission, digital image correlation and micro-indentation have great capabilities to characterize the fracture behavior. The damage in plain concrete and concrete interface specimens is characterized both qualitatively and quantitatively using acoustic emission technique by measuring the width of fracture process zone and width of damage zones. The DIC technique is used to obtain the fracture parameters such as mode I and mode II fracture toughness and critical energy release rate. The micro-mechanical properties are obtained by performing depth-sensing micro-indentation tests on the concrete-concrete interfaces. Civil engineering structures such as long-span bridges, offshore structures, airport pavements and gravity dams are frequently subjected to variable-amplitude cyclic loadings in actual conditions. Hence, in order to understand the fracture behaviour under fatigue loading, the fatigue crack growth in plain concrete and concrete-concrete interface is also studied using the acoustic emission technique. An attempt is made to apply the Paris’ law, which is applicable to mechanical behaviour of metals, for acoustic emission count data. All these studies show that, as the difference in the compressive strength of concrete on either side of the interface increases, the load carrying capacity decreases and the fracture parameters indicate the increase in the brittleness of the specimens. It is concluded that the repair concrete should be selected in such a way that its elastic properties are as those of the parent concrete.