A Damage Model for Concrete under Fatigue Loading
Shan, Zhi;Yu, Zhiwu;Li, Xiao;Lv, Xiaoyong;Liao, Zhenyu
2019-07-09 00:00:00
applied sciences Article Zhi Shan , Zhiwu Yu, Xiao Li, Xiaoyong Lv * and Zhenyu Liao School of Civil Engineering & National Engineering Laboratory for High Speed Railway Construction & Engineering Technology Research Center for Prefabricated Construction Industrialization of Hunan Province, Central South University, 68 South Shaoshan Road, Changsha 410075, China * Correspondence: lvxiaoyong@csu.edu.cn; Tel.: +86-0731-82656611 Received: 3 June 2019; Accepted: 25 June 2019; Published: 9 July 2019 Abstract: For concrete, fatigue is an essential mechanical behavior. Concrete structures subjected to fatigue loads usually experience a progressive degradation/damage process and even an abrupt failure. However, in the literature, certain essential damage behaviors are not well considered in the study of the mechanism for fatigue behaviors such as the development of irreversible/residual strains. In this work, a damage model with the concept of mode-II microcracks on the crack face and nearby areas contributing to the development of irreversible strains was proposed. By using the micromechanics method, a micro-cell-based damage model under multi-axial loading was introduced to understand the damage behaviors for concrete. By a thermodynamic interpretation of the damage behaviors, a novel fatigue damage variable (irreversible deformation fatigue damage variable) was defined. This variable is able to describe irreversible strains generated by both mode-II microcracks and irreversible frictional sliding. The proposed model considered both elastic and irreversible deformation fatigue damages. It is found that the prediction by the proposed model of cyclic creep, stiness degradation and post-fatigue stress-strain relationship of concrete agrees well with experimental results. Keywords: concrete; fatigue; damage model; mode-II microcracks; thermodynamics 1. Introduction Fatigue is an essential mechanical behavior of concrete. In real life, a large number of concrete structures are subjected to fatigue loads, e.g., o-shore structures and bridges. Although the subjected fatigue loads are lower than the relevant materials’ original strength, these structures experience a progressive degradation and subsequently an abrupt failure. In order to investigate these fatigue behaviors, several methods (e.g., fatigue life concepts [1–4] and phenomenological models [5–8]) were developed by researchers and were widely applied in structural engineering. However, during the designing and analysis of structures, these methods [1–8] are only limited to describing the fatigue behaviors at phenomenological and empirical levels without a comprehensive understanding and explanation of the internal mechanism for damage behaviors of concrete under fatigue loading. The complex constitution of concrete results in a sophisticated damage evolution process during material under loading. Specifically, in the material, the arbitrary distribution of initial defects causes the localization of stresses, which further produce the complex evolution process of damage. Experimental studies [9–14] have been conducted to understand the damage mechanism referring to concrete under fatigue loading. In detail, some experimental results showed that local stresses near the defect cause the heterogeneous crack openings perpendicular to tensile loading, i.e., mode-I cracks. The mode-I cracks were well studied in a number of research papers [9–12]. In addition, [13] applied X-ray techniques to study the microcrack mechanism of concrete, and it was found that microcracks parallel to tensile loading (mode-II cracks) can occur even under pure global axial loading. Reference [14] found the mode-II cracks are able to create irreversible strains due to local stresses. Appl. Sci. 2019, 9, 2768; doi:10.3390/app9132768 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 2768 2 of 25 Furthermore, a series of relevant comprehensive works have been conducted by researchers in the solid mechanical field for more than a decade [15–23]. Moreover, the mechanism of the development for irreversible/residual strains in concrete under fatigue loading have been studied throughout several methods [6,12,24–49]. Concretely, on one hand, based on the micro-mechanics method [6,12,24–40], it is concluded that irreversible strains are produced by a series of types of cracking, which are distinguished as follows: (I) for a compressive case, the transversely propagating crushing band [24,25], the axial wedge-splitting cracks at hard inclusions in hardened cement paste [26], the interface cracks at inclusions [6], the pore-opening axial cracks [27,28], and the inclined wing-tipped frictional cracks (i.e., wing cracks) [29–33]; (II) for a tensile case, the irreversible opening of mode-I cracks due to the locking mechanisms of crack faces [34], the irreversible sliding-like openings of mode-II crack due to the toughness of crack faces [35,36], the irreversible frictional sliding over the crack surface [37,38], the irreversible cracking of the fracture process zone [12,39], and other cracking mechanisms [40]. On the other hand, based on the macro-mechanics method [41–49], researchers have rarely considered the comprehensive mechanism of concrete damage, since they are usually focused on the accurate characterization of macroscopic mechanical behaviors. However, among the above-mentioned literature, certain essential damage behaviors are not well considered in the study of the mechanism for the development of irreversible/residual strains in concrete. Specifically, one type of those damage behaviors is mode-II microcracks, which has attracted the attention of researchers in the field of solids mechanics for decades [15–23]. Therefore, it is necessary to develop a continuum damage model for concrete under fatigue loading with the consideration of this damage behavior. In detail, this damage model is able to be established based on the micro-mechanics and continuum damage mechanics. Micro-mechanics enables us to understand damage behavior under multi-axial loading, and the continuum damage mechanics (CDM) method (i.e., a macro-mechanics method) oers us a convenient way to characterize the macro behavior. This work develops the above-mentioned contributions [6,12,24–49] in two aspects, the description of the micro-mechanism for mode-II microcracks in multi-axial conditions and the thermodynamics-based modeling of damage behaviors in concrete under fatigue loading. 2. Microcrack Mechanism in Concrete under Multi-Axial Loading In this section, we briefly recall here the main steps of the methodology followed by the literature [22] for the micro-mechanical description of mode-II microcracks. In addition, the random distribution of initial defects in concrete under multi-axial loading was considered in this work. 2.1. The Definition of the Mode-II Microcracks Mode-II microcracks are the local shear stress-caused by microcracks on the crack face and in the nearby area of the micro-defects and the mode-I crack. This type of crack is dierent from the mode-I crack and the mode-II crack. The dierences can be concluded as follows, the mode-II microcracks are the result of local shear stresses, which is distinguished from tensile stress-caused by the mode-I crack. Additionally, unlike the mode-II crack, mode-II microcracks usually appear on the face and nearby area of the micro-defects and mode-I crack. 2.2. The Causes for the Mode-II Microcracks under Multi-Axial Loading When the concrete is subjected to a biaxial tensile load, a micro-cell within a representative volume element (RVE) was introduced and is shown in Figure 1. In detail, the stress flow curve becomes concentrated when it approaches the crack tip, and the plane stress on the plane horizontal and vertical to the direction of the crack propagation is able to be described by the normal and shear stress as follows, and , and , respectively (Figure 1c). Due to sucient normal stress or stress intensity factor v v v h h (SIF) K at the crack tip, the crack will initiate and grow through the direction where the maximum SIF exists (i.e., transverse to the direction of maximum principal tension, Figure 1d–e). Therefore, the crack Appl. Sci. 2019, 9, x FOR PEER REVIEW 4 of 26 Appl. Sci. 2019, 9, 2768 3 of 25 type, namely the mode-I crack, decreases the eective load area of the micro-cell in the maximum principal tension direction, resulting in the stiness degradation of the specimen [33–36,50]. Figure 1. Sketch of stress flow curve around the random selected micro-cell of the representative Figure 1. Sketch of stress flow curve around the random selected micro-cell of the representative volume element (RVE) in the specimen under multi-axial tension, and the related coordinates and volume element (RVE) in the specimen under multi-axial tension, and the related coordinates and dimension, where denotes the maximum principal tensile stress. dimension, where σ1 denotes the maximum principal tensile stress. However, mode-II microcracks have not been well considered in modeling concrete under 2.3. Influence of the Mode-II Microcracks on the Irreversible Strains in Concrete multi-axial stresses in the literature [6,12,22–40]. Through dierent approaches, including experimental observations, mechanical analysis and atomic simulations [15–21], it is validated that the real It is revealed that the mode-II microcracks are attributed to crack blunting and the irreversible crack (excepting some pre-existing cracks) in the material is blunt, caused by the appearance of deformation (Figure 2) of material even in brittle material such as glass [21]. In this section, we mode-II microcracks. briefly recall the methodology followed by [22] for the micro-mechanical description and further Specifically, mode-II microcracks are produced by the local shear stresses (i.e., the shear stress develop it with consideration of stochastic properties in a multi-axial tension case, as follows. in Figures 1c and 2a) on the face and nearby area of relevant cracks. In a biaxial tensile load case, the directions of local shear stresses are arbitrary due to the random location of initial defects. It is distinguished from that in a uniaxial tensile case [22]. Moreover, several researchers [34–40] observed that the mode-II microcracks, rather than the dislocation-induced plastic flow, appear in complex composite materials such as concrete. Further description and explanation can be found in the literature [6,51]. Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 26 Appl. Sci. 2019, 9, 2768 4 of 25 Figure 2. Sketch of microcracks under multi-axial tension, leading to both the stiness degrading and Figure 2. Sketch of microcracks under multi-axial tension, leading to both the stiffness degrading the irreversible strain developing, where denotes the maximum principal tensile stress. and the irreversible strain developing, where σ1 denotes the maximum principal tensile stress. Note that it is speculated that the local constraint condition around the micro defect is stable For simplicity, we introduce a micro-cell damage model (Figure 3) considering the mode-II during the crack initiation and/or propagation under multi-axial tension; otherwise, the mode-II microcracks to describe the damage behaviors in concrete under biaxial tension. In Figure 3, the dominant failure will appear. The development of the mode-II microcrack leads to certain energy region near a certain defect is firstly highlighted and further discretized by amounts of micro-cells dissipation, and it may also release the tip stress concentration of the mode-I cracks, since it causes a (micro-cell i, micro-cell i + 1, etc.). The behavior of each micro-cell is modeled by two sets of springs relatively blunter crack tip. (spring type A and B). The spring type A can be stretched vertically along the direction of the maximum principal loading, and spring type B is attached to the middle of spring type A. Unlike 2.3. Influence of the Mode-II Microcracks on the Irreversible Strains in Concrete spring type A, spring type B cannot be stretched but it can slip between two parallel sets of It is revealed that the mode-II microcracks are attributed to crack blunting and the irreversible micro-cells. In such a micro-cell damage model, the elastic behavior and elastic deformation deformation (Figure 2) of material even in brittle material such as glass [21]. In this section, we briefly damage are described by spring type A, and the irreversible deformation damage is modeled by recall the methodology followed by [22] for the micro-mechanical description and further develop it spring type B. After unloading, there is a micro deformation b and an irreversible strain εI,f left in with consideration of stochastic properties in a multi-axial tension case, as follows. the material. The micro irreversible fractural opening is caused by mode-II microcracks illustrated For simplicity, we introduce a micro-cell damage model (Figure 3) considering the mode-II in the micro-cell damage model. microcracks to describe the damage behaviors in concrete under biaxial tension. In Figure 3, the region In summary, the elastic deformation damage in the micro-cell damage model corresponds to near a certain defect is firstly highlighted and further discretized by amounts of micro-cells (micro-cell the stiffness degradation, and the irreversible deformation damage is responsible for a certain part of i, micro-cell i + 1, etc.). The behavior of each micro-cell is modeled by two sets of springs (spring type the irreversible strain. Specifically, the local shear stresses produce mode-II microcracks on the A and B). The spring type A can be stretched vertically along the direction of the maximum principal crack face and nearby areas, which generate the micro deformation b and an irreversible strain εI,f in loading, and spring type B is attached to the middle of spring type A. Unlike spring type A, spring the material (Figure 3). type B cannot be stretched but it can slip between two parallel sets of micro-cells. In such a micro-cell damage model, the elastic behavior and elastic deformation damage are described by spring type A, 2.4. Irreversible Strains in Concrete under Multi-Axial Loading and the irreversible deformation damage is modeled by spring type B. After unloading, there is a For the irreversible strains that are not induced by mode-II microcracks, a simplified frictional micro deformation b and an irreversible strain " left in the material. The micro irreversible fractural I,f sliding model is developed in this work (Figure 4) for revealing the development of the irreversible opening is caused by mode-II microcracks illustrated in the micro-cell damage model. strains in concrete under multi-axial tension based on the literature [37,38]. In detail, as illustrated in In summary, the elastic deformation damage in the micro-cell damage model corresponds to the Figure 4, according to this model, the behavior of frictional sliding generally produces a new portion stiness degradation, and the irreversible deformation damage is responsible for a certain part of the of irreversible deformation b’ in the micro-cell of RVE (Figure 4). irreversible strain. Specifically, the local shear stresses produce mode-II microcracks on the crack face Thus, in this work, the irreversible strains in concrete under multi-axial tension are produced and nearby areas, which generate the micro deformation b and an irreversible strain " in the material I ,f by two mechanisms: the mode-II microcracks and irreversible frictional sliding. It is noted that the (Figure 3). other mechanisms [6,12,24–36,39,40] are not employed in the work for the sake of simplicity. In addition, the irreversible deformation damages are assumed to consist of both mode-II microcracks and irreversible frictional sliding (see Figure 4). Macro behaviors Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 26 Appl. Sci. 2019, 9, 2768 5 of 25 Macro behaviors Micro mechanism Figure 3. Macro behaviors and micro mechanism of a concrete cube under bi-axial tension from Figure 3. Macro behaviors and micro mechanism of a concrete cube under bi-axial tension from various variouloading s loading statuses. statuseThis s. Thi figur s fig eu was re was developed developed based ba on sed on [22] (Repr [22] (Rep oduced rodu with cedpermission with permfr ission om [John from [John W Wiley & Sons iley Ltd], & Sons Ltd], 2016), ho 2016), however, the random wever, the distribution random distribution of of initial defects init wasial consider defected s wa in s this consid work. ered in this work. 2.4. Irreversible Strains in Concrete under Multi-Axial Loading For simplicity, the multi-axial stresses in the material are assumed to be classified into two stress spaces: the tension- and compression-dominant stress spaces (Figure 5). Precisely, the stress For the irreversible strains that are not induced by mode-II microcracks, a simplified frictional spaces are distinguished by the plane vertical to the stress line, which indicates the stresses on triaxis sliding model is developed in this work (Figure 4) for revealing the development of the irreversible are equal to each other (see Figure 5). Figure 5 illustrates that the tension-dominant stress space strains in concrete under multi-axial tension based on the literature [37,38]. In detail, as illustrated in consists of both the multi-axial tension space and a certain part of tension-compression space. The Figure 4, according to this model, the behavior of frictional sliding generally produces a new portion compression-dominant stress space represents the rest of the stress space. It is worth mentioning of irreversible deformation b’ in the micro-cell of RVE (Figure 4). that the micro damage mechanisms are different when related to the above two dominant stresses. Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 26 damage mechanism of concrete under compression-dominant stress is assumed to be similar to that Appl. Sci. 2019, 9, 2768 6 of 25 under multi-axial compression in [23]. Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 26 damage mechanism of concrete under compression-dominant stress is assumed to be similar to that Figure 4. Sketch of tensile irreversible deformations due to both mode-II microcracks and irreversible Figure 4. Sketch of tensile irreversible deformations due to both mode-II microcracks and irreversible under multi-axial compression in [23]. frictional sliding, where denotes the maximum principal tensile stress in multi-axial tension. This frictional sliding, where σ1 denotes the maximum principal tensile stress in multi-axial tension. This figure was developed based on [22] (Reproduced with permission from [John Wiley & Sons Ltd], 2016), figure was developed based on [22] (Reproduced with permission from [John Wiley & Sons Ltd], however, the random distribution of initial defects was considered in this work. 2016), however, the random distribution of initial defects was considered in this work. Thus, in this work, the irreversible strains in concrete under multi-axial tension are produced by two mechanisms: the mode-II microcracks and irreversible frictional sliding. It is noted that the other mechanisms [6,12,24–36,39,40] are not employed in the work for the sake of simplicity. In addition, the irreversible deformation damages are assumed to consist of both mode-II microcracks and irreversible frictional sliding (see Figure 4). For simplicity, the multi-axial stresses in the material are assumed to be classified into two stress spaces: the tension- and compression-dominant stress spaces (Figure 5). Precisely, the stress spaces are distinguished by the plane vertical to the stress line, which indicates the stresses on triaxis are equal to each other (see Figure 5). Figure 5 illustrates that the tension-dominant stress space consists of both the multi-axial tension space and a certain part of tension-compression space. The compression-dominant stress space represents the rest of the stress space. It is worth mentioning that the micro damage mechanisms are different when related to the above two dominant stresses. Concretely, for simplicity, the Figure 4. Sketch of tensile irreversible deformations due to both mode-II microcracks and irreversible micro damage mechanism of concrete under tension-dominant stress is assumed to be similar to that frictional sliding, where σ1 denotes the maximum principal tensile stress in multi-axial tension. This under multi-axial tension developed in this work, and the micro damage mechanism of concrete under figure was developed based on [22] (Reproduced with permission from [John Wiley & Sons Ltd], compression-dominant stress is assumed to be similar to that under multi-axial compression in [23]. 2016), however, the random distribution of initial defects was considered in this work. Figure 5. Sketch of the tension- and compression-dominant stress space in two-dimensional. The effect of the roughness and friction of crack faces on the progressive damage and irreversible strains under fatigue compression is able to be concluded as follows, the roughness and friction of the crack faces for the initial inclined frictional crack in wing cracks [23,29–33] and mode-II microcracks will influence the irreversible behaviors: higher roughness and friction results in the later initiation of the crack and further leads to a lower amount of the irreversible strains. It is worth mentioning that the new development of micro-mechanical descriptions related to mode-II microcracks in this work has been obtained in the following ways. Firstly, this work extended the description of the micro damage behaviors in concrete under multi-axial tension with the consideration of both the stochastic properties of initial cracks and the influences of mode-II microcracks. It is distinguished from the work in [22], which focused on an idealized model of the initial uniformly and horizontally distributed cracks under uniaxial tension, and from that in [23], Figure Figure 5. 5. Sketch Sketch of the t of the tension- ension- and compre and compression-dominant ssion-dominant stress stress space space in two-dimensional. in two-dimensional. The effect of the roughness and friction of crack faces on the progressive damage and irreversible strains under fatigue compression is able to be concluded as follows, the roughness and friction of the crack faces for the initial inclined frictional crack in wing cracks [23,29–33] and mode-II microcracks will influence the irreversible behaviors: higher roughness and friction results in the later initiation of the crack and further leads to a lower amount of the irreversible strains. It is worth mentioning that the new development of micro-mechanical descriptions related to mode-II microcracks in this work has been obtained in the following ways. Firstly, this work extended the description of the micro damage behaviors in concrete under multi-axial tension with the consideration of both the stochastic properties of initial cracks and the influences of mode-II microcracks. It is distinguished from the work in [22], which focused on an idealized model of the initial uniformly and horizontally distributed cracks under uniaxial tension, and from that in [23], Appl. Sci. 2019, 9, 2768 7 of 25 The eect of the roughness and friction of crack faces on the progressive damage and irreversible strains under fatigue compression is able to be concluded as follows, the roughness and friction of the crack faces for the initial inclined frictional crack in wing cracks [23,29–33] and mode-II microcracks will influence the irreversible behaviors: higher roughness and friction results in the later initiation of the crack and further leads to a lower amount of the irreversible strains. Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 26 It is worth mentioning that the new development of micro-mechanical descriptions related which involved a model of wing crack under multi-axial compression. Secondly, this work to mode-II microcracks in this work has been obtained in the following ways. Firstly, this work introduced a simplified description of damage behaviors in concrete under tension-compression, extended the description of the micro damage behaviors in concrete under multi-axial tension with which was not considered in the literature [22,23]. the consideration of both the stochastic properties of initial cracks and the influences of mode-II microcracks. It is distinguished from the work in [22], which focused on an idealized model of the 3. Thermodynamics Based Continuum Damage Mechanics Model initial uniformly and horizontally distributed cracks under uniaxial tension, and from that in [23], which involved a model of wing crack under multi-axial compression. Secondly, this work introduced Physically, the damage propagation including both the expanded crack length 2l and the a simplified description of damage behaviors in concrete under tension-compression, which was not developed crack opening b + b′ in micro-scale in Figure 6a (discretely modeled by the micro damage considered in the literature [22,23]. model in Figure 6b) is an irreversible thermodynamic process characterized in Figure 6c. Both microscale behaviors are able to be idealized/unified and thermodynamics modeled by considering 3. Thermodynamics Based Continuum Damage Mechanics Model the stiffness degradation Ed and the irreversible strains development ε (Figure 6), respectively. Thus, the co Physically, the mplex microscale c damage propagation rack behavior including both s (Fig the ure 6a,b) expanded are th crack ermodynam length 2l and ically interpr the developed eted int crack o a s opening imple macro b + bsca inle micr dama o-scale ge mec in h Figur anics mode e 6 (discr l (F etely igure modeled 6c), which obt by theamicr ined oa t damage hermod model ynamic in s based Figure CDM 6) is an model. In irreversible the fol thermodynamic lowing section, t process he de characteri finition of zed a in new type of da Figure 6. Both micr mage v oscale aria behaviors ble—the irrever are able sible d to be e idealized formation fatigue /unified and dam thermodynamics age variable—imodeled s firstly iby ntroduced considering and then the detai the stiness degradation ls for the model formu E and the irrleversible ation are strains given. development " (Figure 6), respectively. Figure 6. Thermodynamics interpretation of micro-scale damage behaviors. Specifically, the elastic Figure 6. Thermodynamics interpretation of micro-scale damage behaviors. Specifically, the elastic d deformation eformation damage (mod damage (mode-I e-I c crrac acks) ks) pro produces duces the the st stiiffness reduction ness reduction E Ed,, and the and the ir irrreversible eversible deform deformation ation d damage amage (both (both mode-II mode-II micr m ocracks icrocrack ands and irreversible irreversible friction sliding) friction s causes liding the)development causes the of irreversible strains " . development of irreversible strains ε . Thus, the complex microscale crack behaviors (Figure 6) are thermodynamically interpreted 3.1. Thermodynamics Interpretation into a simple macroscale damage mechanics model (Figure 6), which obtained a thermodynamics In this section, we briefly recall here the main steps of the methodology followed by [52] for the based CDM model. In the following section, the definition of a new type of damage variable—the thermodynamics interpretation of the damage variable. This work developed the method from [52] irreversible deformation fatigue damage variable—is firstly introduced and then the details for the for interpreting the damage variable under fatigue loading. model formulation are given. The infinite deformation behavior of concrete material with damage can be viewed within the framework of thermodynamics with internal state variables. The Helmholtz free energy per unit 3.1. Thermodynamics Interpretation mass, in an isothermal deformation process at the current state of the deformation and material In this section, we briefly recall here the main steps of the methodology followed by [52] for the damage, is assumed as follows: thermodynamics interpretation of the damage variable. This work developed the method from [52] for interpreting the damage variable under fatigue Ψ = loading. ψ + γ (1) n n n The infinite deformation behavior of concrete material with damage can be viewed within the where the subscript n denotes the cyclic number of the fatigue loading (n = 1, 2, 3, ..., N), ψ denotes framework of thermodynamics with internal state variables. The Helmholtz free energy per unit mass, the strain energy or a purely reversible stored energy, while γ represents the irreversible energy associated with specific micro structural changes produced by damage (i.e., elastic deformation damage induced by mode-I fractures, and irreversible deformation damage due to both mode-II micro-cracks and irreversible fictional sliding, see Figures 4 and 6). An explicit presentation of the irreversible energy and its rate is generally limited by the complexities of the internal micro structural changes discussed in the recent section; however, only one internal variable damage (contains two components) is considered in this work. The damage contains two components, that is, elastic deformation damage induced by mode-I fractures, and irreversible deformation damage due to both mode-II micro-cracks and irreversible fictional sliding, contribute to stiffness degradation and irreversible strains development, respectively (see Figure 6). For the purpose of developing a Appl. Sci. 2019, 9, 2768 8 of 25 in an isothermal deformation process at the current state of the deformation and material damage, is assumed as follows: = +
(1) n n n where the subscript n denotes the cyclic number of the fatigue loading (n = 1, 2, 3, ..., N), denotes the strain energy or a purely reversible stored energy, while
represents the irreversible energy associated with specific micro structural changes produced by damage (i.e., elastic deformation damage induced by mode-I fractures, and irreversible deformation damage due to both mode-II micro-cracks and irreversible fictional sliding, see Figures 4 and 6). An explicit presentation of the irreversible energy and its rate is generally limited by the complexities of the internal micro structural changes discussed in the recent section; however, only one internal variable damage (contains two components) is considered in this work. The damage contains two components, that is, elastic deformation damage induced by mode-I fractures, and irreversible deformation damage due to both mode-II micro-cracks and irreversible fictional sliding, contribute to stiness degradation and irreversible strains development, respectively (see Figure 6). For the purpose of developing a schematic description of the concepts based on the proposed micro damage model, the uniaxial stress-strain curves are used in Figure 7. In Figure 7a, E denotes the initial undamaged stiness (relates to loading line OA ). The strain and the 0 0 stress at point A are denoted by " and , respectively. 0 0, 1 max 3.1.1. Interpretation in First and Second Loading Cycle Firstly, considering the stress-strain response during the first loading cycle, the unloading curve A B is simplified by the line A B in Figure 7a,b in this work. At point A , the strain " and irreversible 1 1 1 1 1 1 di damage strain " exist in the specimen. The initial stiness changes from E to E . Even though these 1 0 1 notations are for the uniaxial case, they are able to be used in indicial tensor notation in the equations without loss of generality. The total strain (described by line OB G H in Figure 7a) is given as follows: 1 1 1 E I de di " = " + " = " + " + " (2) 1 1 1 0,1 1 where the subscript 1 denotes the cyclic number of the first fatigue loading. The strain energy is expressed as follows (see the area described by points B A H in Figure 7a) 1 1 1 1 2 1 E E = E " = E " " (3) 1 1 0 1 0,1 1 2 2 e de = + (4) 0 1 e de where denotes the initial strain energy (see the area G A H in Figure 7a), and denotes the 0 1 1 1 1 elastic deformation damage strain energy during the first cycle (see the area B A G in Figure 7a), 1 1 1 that is, = E " (5) 0 0,1 1 1 de de e E = E " " = = E " " " (6) 0 0 0,1 1 0 0,1 1 0,1 1 1 2 2 And the irreversible energy is expressed as follows (see the area OA A B in Figure 7a) 0 1 1 1 1 1 1 di de d de = " + " = E " " " (7) max max 0 1 1 1 0,1 1 1 2 2 di de =
+
(8) 1 1 1 di where
denotes the irreversible-damage irreversible energy (see the area OA I B in Figure 7a), 1 0 1 1 de and
denotes the elastic deformation damage irreversible energy (see the area B I A in Figure 7a), 1 1 1 1 that is, Appl. Sci. 2019, 9, 2768 9 of 25 1 1 di di di = " = E " " (9) max 0 1 1 0,1 1 1 1 de de de = " = E " " (10) max 0 1 1 0,1 1 2 2 In regard to stored energy (contains both the purely reversible stored energy and the irreversible energy
), one is able to obtain the formula as follows Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 of 26 = +
(11) n n n (a) (b) (c) (d) Figure 7. Sketch of the mechanical parameter definition for concrete under fatigue loading. (a) Figure 7. Sketch of the mechanical parameter definition for concrete under fatigue loading. (a) Energy Energy dissipation; (b) Energy dissipation; (c) Energy dissipation (σmin = 0); (d) Energy dissipation dissipation; (b) Energy dissipation; (c) Energy dissipation ( = 0); (d) Energy dissipation ( , 0); min min (σmin ≠ 0); When the material is assumed to be a perfect elastic material, it undergoes a strain ε1 and 0 2 0 obtains a stored energy λ1 = 1/2(E0ε1 ) = ψ1 (i.e., the perfect material’s purely reversible stored energy or strain energy) due to external loads. However, the material focused on in this work is a quasi-brittle material assumed to undergo both elastic deformation damage and irreversible deformation damage. It reduces a certain part of stored energy (denoted by the area A0F1A1 in Figure 7a) caused by the elastic deformation damage (i.e., mode-I fracture in the proposed micro-cell Appl. Sci. 2019, 9, 2768 10 of 25 When the material is assumed to be a perfect elastic material, it undergoes a strain " and obtains 0 2 0 a stored energy = 1/2(E " ) = (i.e., the perfect material’s purely reversible stored energy or 1 0 1 1 strain energy) due to external loads. However, the material focused on in this work is a quasi-brittle material assumed to undergo both elastic deformation damage and irreversible deformation damage. It reduces a certain part of stored energy (denoted by the area A F A in Figure 7a) caused by the elastic 0 1 1 deformation damage (i.e., mode-I fracture in the proposed micro-cell damage model, see Figures 4 and 6). Additionally, another part of the stored energy (described by the area A C F in Figure 7a) is 0 1 1 also decreased, as a result of the irreversible damage (due to both mode-II micro-cracks and irreversible fictional sliding, see Figures 4 and 6). Thus, the damaged material’s stored energy is derived as follows 0 d = +
= (12) 1 1 1 1 1 where d de di = + (13) 1 1 1 de de d = " (14) 1 1 di di d = " (15) 1 1 1 d de where denotes the total damage caused reduction of stored energy, denotes the elastic 1 1 di deformation damage (i.e., mode-I fracture) caused reduction of stored energy, denotes the irreversible damage (due to both mode-II micro-cracks and irreversible fictional sliding) caused reduction of stored energy. Secondly, considering the stress-strain response during the second loading cycle, the unloading curve A B is also simplified by the line A B in Figure 7a,b. At point A , the strain " and irreversible 2 2 2 2 2 2 di damage strain " exist in the specimen. The stiness is changed from E to E . Even though these 2 1 2 notations are for the uniaxial case, they are able to be used in indicial tensor notation in the equations without loss of generality. The total strain (described by line OB G H and OB B G H in Figure 7b) 2 2 2 1 2 1,2 2 is given as follows: E I de di di di de de " = " + " = " + " + " = " + " + " + " + " (16) 2 2 2 0,2 2 2 1 1,2 1,2 0,2 1 where " = " (see Figure 7a). 0,2 0,1 The strain energy is expressed as follows de e de de = + = + + (17) 2 1 1,2 0 1 1,2 de where the subscript 2 denotes the cyclic number of the second fatigue loading, denotes the 1,2 elastic deformation damage strain energy due to the additional elastic deformation damage during the second cycle (see the area B A G in Figure 7b), that is, 2 2 1,2 de de = E " " (18) 1,2 0,1 1,2 Thus, the strain energy is expressed as follows (see the area described by points B A H in Figure 7a) 2 2 2 1 1 E de E = E " " + " = E " " (19) 2 0 0 0,1 1 0,1 2 1,2 2 2 The irreversible energy
is expressed as follows di de di de =
+
=
+
+
+
(20) 2 1 1,2 1 1 1,2 1,2 Appl. Sci. 2019, 9, 2768 11 of 25 di where
denotes the irreversible deformation damage irreversible energy due to the additional 1,2 de irreversible deformation damage during the second cycle (see the area B A I B in Figure 7b), and 1 1 2 2 1,2 denotes the elastic deformation damage irreversible energy due to the additional elastic deformation damage during the second cycle (see the area B A G in Figure 7b), that is, 2 2 1,2 1 1 di di di = " = E " " (21) max 0 1,2 1,2 0,1 1,2 1 1 de de de = " = E " " (22) max 0 1,2 1,2 0,1 1,2 2 2 Thus, the irreversible energy
is derived as follows (see the area OA A B in Figure 7b) 2 0 2 2 h i 1 1 1 1 1 di de di de di di de de = E " " + " + " + " = E " " + " + " + " 0 0 2 0,1 1 2 1 1,2 2 1,2 0,1 1 1,2 2 1 1,2 (23) 1 di 1 de = E " " + " 0,1 2 2 The stored energy is derived as follows 0 d = +
= (24) 2 2 2 2 2 where denotes the total damage caused reduction of stored energy, that is, d d d di de di de = + = + + + (25) 2 1 1,2 1 1,2 1 1,2 di where denotes the irreversible deformation damage caused by the reduction of stored energy 1,2 due to the additional irreversible deformation damage during the second cycle (see the composite de areas C C J F , A J J and A J A in Figure 7b), and denotes the elastic deformation damage 1 2 1 1 1 3 4 1 5 2 1,2 caused reduction of stored energy due to the additional elastic deformation damage during the second cycle (see the composite areas F J J A , A J J and A J J in Figure 7b), that is, 1 1 2 1 1 2 3 1 4 5 di 1 di 1 di 0 1 di = " " + " " + " " 2 1 2 2 1 2 2 1 1,2 1 1,2 1,2 (26) 1 di 1 di 0 1 di = " " + + 2 2 2 1 1 1,2 1,2 de 1 de 1 de0 1 de = " " + " " + " " 2 2 1,2 1 2 1 1,2 2 1 1,2 2 1 (27) 1 de 1 de0 1 de = " " + + 2 2 2 1 1 1,2 1,2 Thus, the total damage caused reduction of stored energy is derived as follows (see the area A C A in Figure 7b) 0 2 2 d di de di de di de = + + + = + (28) 2 1 1 1,2 1,2 2 2 di di d = " (29) 2 2 2 de de d = " (30) 2 2 2 di where denotes the irreversible deformation damage (due to both mode-II micro-cracks and de irreversible fictional sliding) caused by the reduction of stored energy, denotes the elastic deformation damage (i.e., mode-I fracture) caused by the reduction of stored energy (see Figure 7a). Appl. Sci. 2019, 9, 2768 12 of 25 3.1.2. Interpretation in nth Loading Cycle Based on the recent thermodynamics interpretation in this work (see Equations (3), (7), (12–15), (19), (23), (24), (28)–(30)), by comparing Equation (19) with Equation (3) and replacing the cycle number 2 by n in Equation (19), it is possible to derive the strain energy after nth loading cycle, as follows: 1 1 E E de = E " " = E " " + " (31) n 0 0 0 0 n 1 n 1,n 2 2 By comparing Equation (23) with Equation (7) and replacing the cycle number 2 by n in Equation (23), it is possible to derive the irreversible energy after nth loading cycle, as follows: 1 1 1 1 di de di di de de = E " " + " = E " " + " + " + " (32) 0 0 n n n 0 0 n 1 n 1,n n 1 n 1,n 2 2 By comparing Equations (24), (28–30) with Equations (12–15) and replacing the cycle number 2 by n in Equations (24), (28–30), it is possible to derive the damage caused by the reduction of stored energy after nth loading cycle, as follows: 0 d 0 di de = = + (33) n n n n n n di di d = " (34) n n n de de d = " (35) n n n Note that, considering both the micro structural changes (based on the proposed micro damage model, see Figures 3, 4 and 6) and the macro irreversible energy (see Equation (1)), this work is ruled by second thermodynamics law, that is, "