- Email: [email protected]
Molecular dynamics study on the mode I fracture of calcium silicate hydrate under tensile loading
Engineering Fracture Mechanics 131 (2014) 557–569
Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Molecular dynamics study on the mode I fracture of calcium silicate hydrate under tensile loading Dongshuai Hou a, Tiejun Zhao a, Penggang Wang a, Zongjin Li b, Jinrui Zhang b,⇑ a b
Qingdao Technological University (Cooperative Innovation Center of Engineering Construction and Safety in Shandong Blue Economic Zone), Qingdao, China Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong Special Administrative Region
a r t i c l e
i n f o
Article history: Received 12 February 2014 Received in revised form 6 September 2014 Accepted 9 September 2014 Available online 21 September 2014 Keywords: Calcium silicate hydrate (C–S–H) Uniaxial tension test Crack evolution SCF DSD
a b s t r a c t Calcium silicate hydrate (C–S–H), the essential binder of cement based materials, is a poor crystal phase at nano-scale. In this study, crack growth mechanism is unraveled in lights of Molecular Dynamics (MD) by simulating the uniaxial tension test on the C–S–H gel with voids ranging from 0.5 nm to 5 nm. At nano-scale, the layered C–S–H gel demonstrates dual nature of crack propagation. In xy plane, the stable ionic-covalent bonds Si–O and Ca–O are hard to break so the cracks coalesce is slowed down, implying ductile characteristic. In z direction, cracks spread in the interlayer region with high rate due to the frequently breakage H-bonds network, exhibiting brittle nature. In the tensile process, the crack develops from the central void and ‘‘strain concentration’’ near the void boundary is induced by irreversible atomic deformation. Young’s modulus and tensile strength of C–S–H gel are significantly weakened due to the presence of central void. Additionally, due to the binding constraints’ discrepancy, bending of the calcium silicate sheet can be observed, reflecting the complicated tensile behavior of heterogeneous layered C–S–H gel. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction When cement is mixed with water, the hydration reactions produce various cementitious hydrates, of which C–S–H gel is the dominant and important binding phase. External loading on the cement materials causes complicated crack distribution ranging from nano-level to macro-level in C–S–H gel. The multi-scale crack length distribution is very important microstructure characteristic, which influences the physical properties of the concrete, such as the strength and elasticity, the permeability and diffusivity, shrinkage and durability to freezing and thawing. As the most widely used construction and building materials, the ability to resist crack growth is an important criterion to estimate cements’ serviceability. And the mechanical properties influenced by the crack have been investigated since fracture mechanics was first applied to concrete by Kaplan [21]. Researchers made great efforts to determine whether linear elastic fracture mechanics (LEFM) theory proposed by Griffth [11] can be applied to rapid crack propagation in cement-based materials. However, the large fracture process zone in concrete observed by many experiments proved that the criterion of LEFM could not directly predict the fracture behavior of cementitious materials. Since the 1980s, models with more than one fracture parameter have been proposed to explain the fracture in concrete. Wecharatana and Shah [37] used a compliance based model to calculate the length of the fracture process zone. Jenq and Shah [19] introduced a two-parameter fracture model. It was demonstrated that these two parameters were size independent. In 1984, Bazant [2] developed a size effect law to model concrete fracture. Investigations ⇑ Corresponding author. Tel.: +852 67636710. E-mail addresses: [email protected] (D. Hou), [email protected] (T. Zhao), [email protected] (P. Wang), [email protected] (Z. Li), [email protected] (J. Zhang). http://dx.doi.org/10.1016/j.engfracmech.2014.09.011 0013-7944/Ó 2014 Elsevier Ltd. All rights reserved.
558
D. Hou et al. / Engineering Fracture Mechanics 131 (2014) 557–569
Nomenclature Si Os Ow Cas Caw H SCF DSD
is is is is is is is is
the the the the the the the the
silicon atom oxygen atom bridging with silicon oxygen atom in the water molecule calcium atom in the calcium silicate sheet calcium atom in the interlayer region hydrogen atom Strain correlated function displacement standard deviation
of cracks at macro-level provide guide in design of beams, anchorage and large dams. However, fracture mechanism of cementitious materials is not yet mature without studying the crack growth mechanism at nano-level. In order to understand nano-crack growth mechanism in cement-based material, it is necessary to investigate the intrinsic structure and mechanical performance of C–S–H gel. The molecular structure of C–S–H has been studied for more than half a century, particularly using a variety of experimental techniques, including NMR ([7,8]), XRD [16] and SANS [1], and C– S–H is now widely believed to be the analogue of layered minerals: tobermorite [24,12] and jennite [3]. Based on experimental data, Pellenq et al. [29] employed the Molecular Dynamic (MD) method to construct a C–S–H model that can well describe the structural, dynamical and mechanical properties of cement at the nano-scale. Meanwhile, CSHFF force field [34] was developed to describe the interaction between atoms in cement systems, and this force field demonstrated good transferability in simulating cement-based materials. Pellenq et al. [29] simulated the stress–strain relation of C–S–H gel in resisting shear force. By comparing the mechanical performance of dry and wet C–S–H gels, it was concluded that during the loading process, large displacements of water molecules in wet C–S–H gels weaken the shear strength and result in unrecovered deformations. Youssef et al. [38] investigated structural and dynamic features of water molecules in C–S–H gels. Due to the highly hydrophilic nature of calcium silicate sheets, H-bonds constructed between the non-bridging oxygen and water are very strong. Therefore, water molecules constrained in the gel demonstrate a glassy nature: the tetrahedral spatial structure is distorted and the diffusion rate is significantly reduced. Ji et al. [20] utilized five classic water models, SPC, TIP3P, TIP4P, TIP4P05, and TIP5P to simulate calcium silicate hydrate. It was concluded that SPC and TIP5P accurately describe the intrinsic properties of C–S–H gel, with SPC (flexible) being more computationally efficient. Recently, Bonnaud et al. [4] interpreted the cohesive force of C–S–H gel by analyzing the fluid pressure of the water molecules and the courter-ions in the interlayer region. Under different humidity conditions, they found that the cohesive force mainly results from negative pressure caused by the interaction between interlayer calcium atoms and the calcium silicate sheets. Previous research on C–S–H gels can be categorized into two classes: molecular structure analysis and thermodynamics investigation. Few efforts have been made to investigate the mechanical properties evolution in presence of nano-cracks. In particular, the tensile strength of C–S–H gels at the nano-scale, considered as the most essential property of construction material, has not been studied so far. The aim of this paper is to explore the fracture mechanism of C–S–H gel at molecular level. On the one hand, the structural feature evolution can be characterized by analyzing atomic deformation, chemical bonds and crack length variation. 2. Simulation method The CSHFF force field [34], developed for cement based materials, is utilized to simulate C–S–H gels with different central voids. The force field approach has been widely used in C–S–H simulations and has been proven to be able to describe the structure, energy and mechanical properties of various calcium silicate phases satisfactorily [29,38,20,4,22]. The force field parameters of Ca, Si, O and H can be obtained from the literature [35]. 2.1. C–S–H model construction The C–S–H model in the present study is constructed based on the procedures proposed by Pellenq et al. [29]. Firstly, the layered analogue mineral of C–S–H, tobermorite 11 Å without water, was taken as the initial configuration for the C– S–H model [28,33]. Silicate chains were then broken to match the Q species distribution obtained from NMR testing [6]. The major difference for the silicate skeleton between current model and Pellenq’s model is the Q species distribution. In Pellenq’s model, Q0 species, representing the monomers, remained 13% in calcium silicate sheet, which is more than 3 times larger than the results obtained from the 29Si and 17O cross-polarized NMR test [5] ([7,8]). The overestimation of the Q0 species can influence the calcium silicate skeleton. Hence, the percentage of Q0 species is controlled below 5%, matching experimental results. Secondly, Grand Canonical Monte Carlo simulation of the water adsorption is operated
D. Hou et al. / Engineering Fracture Mechanics 131 (2014) 557–569
559
on the dry disordered C–S–H structure at 300 K. The chemical formula of the saturated C–S–H structure in current simulation is (CaO)1.69(SiO2) 1.66H2O, which is quite close to (CaO)1.7(SiO2) 1.8H2O obtained by the SANS test [1]. The molecular dynamic simulations under constant pressure and temperature (NPT) for 300 ps give the structures of C–S–H gel at equilibrium states. For each case, a further 1000 ps NPT run is employed to achieve the equilibrium configuration for structural and dynamic analysis.
2.2. Mechanical properties calculation There are two methods to represent the gel pores: voids can be created by randomly packing smaller C–S–H bricks or embedded the empty region into the large layered structure [30,10,17,18]. In previous research, the former approach has been utilized to investigate the fracture behavior of different packing models [14]. Due to the packing efficiency restriction, the former approach can only study the C–S–H gels with packing fraction lower than 74% (fcc packing). Additionally, pore sizes distribution is hard to be characterized in the packing models, which brings about difficulty in analyzing the pore size effect on the mechanical properties. In this study, influence of gel pore size can be investigated by the latter approach. The C–S–H model, constructed in Section 2.1, is periodically extended 6 times and the size of the super-cell is 138 Å 138 Å 138 Å, with more than 300,000 atoms. Large number of atoms can provide statistical reliable results and more realistic evolution of the structures of C–S–H gel. Then the cylinder pore with diameter 0, 5, 10, 15, 20, 25, 30, 50 Å at the central of the C–S–H super-cell is eliminated to achieve 8 samples, as shown in Fig. 1. While the calcium silicate sheet and interlayer Ca were removed, the interlayer water molecules remained in the gel pore. Recently, NMR investigations [27] on the calcium silicate hydrate indicated that the amount of gel pore water gradually decreases between 100% and 25% relative humidity. In current simulation, the gel pore water amount is not that high so it is relevant to C–S–H gel at low humidity state. The selected pore width is in the range of gel pore size that distributes from 0.5 to 10 nm, as proposed by Mindess et al. [25]. It should be noted that this simulation is only the preliminary study that focus on the pore size effect on the C–S–H gel. A series of work along this direction will be continued to systematically investigate the gel pores in C–S–H structure, including the influence of the geometry, the poly disperse, the connectivity and the saturated state of the pores. Atoms elimination should not change the charge neutrality in the systems. Subsequently, uniaxial tension test is performed on the new created samples in x and z direction. Young’s modulus and the tensile strength in x and z direction, describing the stiffness and interlayer cohesive force respectively, are obtained by uniaxial tension testing. In order to get the stress–strain relation, the C–S–H samples were subjected to uniaxial tensile loading through gradual elongation at a constant strain rate at 0.08/ps. In the whole simulation process, NPT ensembles are set for the system. The simulation sequence along z direction is listed as follows. Firstly, the simulation box is relaxed at 300 K and coupled to zero external pressure in the x, y, z dimensions for 100 ps. Then, after the pressures in the three directions reach equilibrium, the C–S–H structure is elongated in the z direction. Meanwhile, the pressures in the x and y directions are kept at zero. After each step elongation, the atomic configuration is relaxed and pressure evolution in the z direction is taken as the internal stress rzz. The non-pressure setting in directions perpendicular to the tension direction, also considering Poisson’s ratio, can eliminate the artificial constrain for the deformation and allow the free development of tension without any restriction. In the simulated tension process, 10,000 configurations are recorded for structural analysis.
Fig. 1. Configuration of the C–S–H gel with the central void for the uniaxial tension test. White thick chain is silicate bond; short thin chain is water molecule; large blue ball represents interlayer calcium atoms and; small red ball represents calcium atoms in the sheet. Same meaning in the following configurations. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
560
D. Hou et al. / Engineering Fracture Mechanics 131 (2014) 557–569
3. Results and discussion 3.1. Crack orientation influence Stress–strain curves, shown in Fig. 2, characterize the carrying loading ability of C–S–H gel with 30 Å central void along (1 0 0) and (0 0 1) directions. In x direction, stress evolution can be divided into three stages: stress increases with strain until reaches failure strength 3.3 GPa at strain 0.084 Å/Å; subsequently stress drops from failure strength to 2.1 GPa ; finally decrease slowly to zero at strain 0.4 Å/Å. In the post-failure region, the ladder-like stress–strain relation, a symbol of good plasticity, indicates some structural rearrangement in the layered C–S–H gel. Both Young’s modulus and tensile strength in x direction is larger than that in z direction. As exhibited in Fig. 2, the stress increase to less than 2 GPa at strain 0.08 Å/Å and then drops sharply to zero at complete rupture strain 0.2 Å/Å, indicating the poor ductility along z direction. It should be noted that the tensile strength (2 GPa) simulated by current method is larger than the shear strength (1 GPa). The difference between failure strengths results from many reasons such as the loading mode, the sample size and the molecular structure of C–S–H gel. In current C–S–H model, there are less than 5% of the silicate monomer (Q0) species, which is much smaller than that of Pellenq et al’s model (Q0 = 13%). It means that the connectivity of the silicate chains in current model is better than that of Pellenq’s model and the longer silicate chains lead to more mechanical contribution. Besides, the H2O/Si ratio in Pellenq’s model is higher than that of current model, the ‘‘hydrolytic weakening’’ effect further reduces the cohesive force. However, since both current model and Pellenq’s model have not considered the disordered calcium silicate sheet and the connected gel pores, the failure strengths simulated by both models are overestimated in some extent. On the other hand, in previous simulation on mechanical behavior of C–S–H gel by MD [15,28], the tensile-compressive strength ratio ranges from 0.23 to 0.66. As the morphology of silicate chains turns to more defective, the tensile-compressive ratio reduces to lower value. The trend in the stress–strain behavior of the C–S–H unit with defective silicate chains in tension and compression is reasonably comparable to that at the macroscopic level. 3.1.1. Crack development The stress–strain evolution is accompanied by the variation of structural deformation. More insights can be obtained by observing the molecular structure change during the stretching process. Initially, as shown in Fig. 3b, the layered structure is slightly stretched longer at strain 0.08 Å/Å and Si–O and Ca–O bonds extension takes up the strain during the elastic stage. As strain increases from 0.08 to 0.16, the dissociation of calcium atoms and short silicate chains result in crack growth surrounding the boundary of central voids. It should be noted that initiating calcium silicate sheet fracture, shown in Fig. 3c, corresponds to the first drop of the tensile stress in the stress–strain curve. The quick stress reducing rate indicates the calcium silicate sheet near the crack boundary is easily to be stretched broken due to small binding energy. Subsequently, small cracks begin to grow into neighboring calcium silicate sheet. Compared with the first fracture near the central void, the calcium sheet away from the void is relatively harder to be destroyed. Hence the stress decreasing rate can be slowed down. In addition, as shown in Fig. 3d, the interlayer calcium atoms can penetrate into the defective region due to stretch and help reconstruct the damaged calcium silicate sheet temporarily, which enhances deformation resistance. Finally, the cracks, initialed from the central void, coalesce with surrounding small cracks and propagate along direction around ±45° from x axis. The cup-and-cone failure surface, shown in Fig. 3f, has also been observed in the damages of many ductile materials and is characterized as the ductile failure. However, the necking phenomenon has not been observed at the late tensile stage as many ductile materials such as iron. Only the calcium silicate sheets in the middle have slight bend toward the central void. The presence of interlayer water molecules prevent the large bending deformation of calcium silicate sheet so that the
Fig. 2. The stress–strain curves of C–S–H gel with 30 Å central void tensioned along x and z direction.
D. Hou et al. / Engineering Fracture Mechanics 131 (2014) 557–569
561
Fig. 3. Crack evolution of C–S–H gel with central void 30 Å tensioned along x direction. a, b, c, d, e and f represent strain at 0, 0.08, 0.16, 0.2, 0.3, 0.4 Å/Å.
‘‘necking effect’’ is hard to happen. On one hand, water molecules screen the interflow of Ca, Si atoms between neighboring calcium silicate sheet so that weakens the plastic deformation of tensioned C–S–H gel. On the other hand, once the damaged structure forms, the water molecules diffuse fast to associate with the non-bridging oxygen site, which prevent interlayer calcium from bridging the fractured structure. Therefore, the plastic deformation, arouse by the diffusion of Ca, Si and O atoms, is significantly reduced. In z direction, the tensile failure of the C–S–H gel exhibits different mode. As demonstrated in Fig. 4b, at beginning of tensile process, the swell of the interlayer region and the extension of Si–O and Ca–O bond in the calcium silicate sheet takes up the strain at the elastic region. With gradually increasing of the strain, the neighboring calcium silicate sheets begin to separate at the boundary of the central void, as the strain reaches 0.08 Å/Å, where the tensile stress first reduces. It worth noting that in comparison with the ionic-covalent bonds in the calcium silicate sheets, the H-bonds network and Ca–Ow bond, breakage and formation quickly, have lower stability, so the cracks are initiated from the interlayer region. The catastrophic damage mode is attributed to the fast crack development in the interlayer region. The crack grows and coalesce fast to rupture the gel structure in only 1600 ps, which is one third of that in case of x direction. As shown in Fig. 4e, different from the coarse fracture surface in x direction, the rupture surface of C–S–H gel loading along z direction is quite flat and normal to direction of tensile loading.
562
D. Hou et al. / Engineering Fracture Mechanics 131 (2014) 557–569
Fig. 4. Crack evolution of C–S–H gel tensioned along z direction. a, b, c, d, e and f represent the strain at 0, 0.06, 0.08, 0.16, 0.2, 0.24 Å/Å.
D. Hou et al. / Engineering Fracture Mechanics 131 (2014) 557–569
563
3.1.2. Chemical bonds analysis A Strain correlated function (SCF) is used to describe the variation of chemical bonds comparison to the original structure during the tensile process. SCF can be interpreted as the time correlated function (TCF) in the presence of strain loading in Eq. (1):
CðeÞ ¼
hdbðeÞdbð0Þi hdbð0Þdbð0Þi
ð1Þ
where db(e) = b(e) hbi and b(e) is a binary operator that takes a value of one if the pair (e.g. Ca–O) is bonded and zero if not and hbi is the average value of b over all simulation time and pairs. The chemical bonds break and form due to tensile strain, which results in the connectivity variation in C–S–H gels. If the connectivity persists unchanged, the SCF of bonds maintain a constant value one. Otherwise, the more frequently bond breakage occurs, the lower the value of SCF. By comparing the deviation degree from one in the SCF curves, roles of different bonds in carrying load can be estimated. As proposed in previous research [29], Cas–Os, Si–Os and Caw–Os are the major bonding phase in the C–S–H gel playing a dominant role in loading resistance. C(e) of Cas–Os, Si–Os and Caw–Os bonds variation with strain in x and z directions are plotted in Fig. 5. As shown in Fig. 5, C(e) of Cas–Os and Caw–Os continuously decreases while that of Si–Os maintains constant value with slight fluctuation. This means that Ca–O bond breaks and forms frequently, while Si–O bond, occupying high binding energy, is broken slightly. In this respect, the silicate chains act as the backbone and the morphology of a silicate structure, to a great extent, determining the mechanical performance of C–S–H gel [36]. In x and z direction, C(e) of Ca–O stops reducing as strain reaches 0.4 and 0.2, respectively, which correspond to the rupture strain shown in stress–strain
Fig. 5. Strain correlated function (SCF) of C–S–H gel with 30 Å void tensioned in x (a) and z (b) direction.
564
D. Hou et al. / Engineering Fracture Mechanics 131 (2014) 557–569
curve. It means that after fracture, most of chemical bonds in the separated C–S–H gel no longer break or form frequently, with only slight evolution of bonds in the rupture surface. When the sample is subject to loading along x direction, C(e) of Cas–Os and Caw–Os reduce by 0.25 and 0.2 respectively. The variation of Ca–O bond mainly result in the fracture of calcium silicate sheet while the breakage of Caw–Os bond reflects the shear damage between the interlayer region and calcium silicate skeleton, which can both be observed in Fig. 5. Both Cas–Os and Caw–Os bond play critical role in loading resistance. Considering that the number of Cas atoms is 2 times larger than that of Caw atoms, calcium silicate sheet acts major role in carrying loading. On the other hand, the chemical bonds variation demonstrates different case as the C–S–H is tensioned in z direction. The deviation degree of Cas–Os bond is only half of the Cas–Ow bond, implying that interlayer region is the main phase in loading resistance. The breakage of Cas–Os bond mainly result in the damage of calcium silicate sheet near the crack boundary. In the location far away from the crack, the calcium silicate sheet maintains integrity. So the Cas–O bonds change slightly. In addition, Caw–Os bonds’ breakage is attributed to the crack propagation along interlayer region. In comparison with Caw–Os, Cas–Os bond is more stable in loading resistance. On one hand, Cas atoms have more coordinated Os atoms so the Cas–Os bond can frequently be reconstructed after breakage. On the other hand, the Caw atoms have many neighboring water molecules in the interlayer region, which disjoins the cohesive force between Caw atoms and Os atoms in the silicate chains. 3.1.3. Displacement evolution In order to evaluate the deformation of C–S–H gels with nano-cracks quantitatively, the difference between deformation derived by MD simulation and that predicted by means of Cauchy–Born method is calculated. The Cauchy–Born hypothesis assumes that the displacement of an atom in the molecular theory of elasticity is according to the deformation gradient F. In the uniaxial tension test along x direction, F is defined in Eq. (2):
0
@x=@X
B F ¼ @0 0
0
0
0
1
1
C 1 0A
ð2Þ
In which ox/oX is corresponding to the magnitude of uniaxial elongation divided by the initial length of the specimen. The displacement standard deviation (DSD) from Cauchy–Born method is computed as Eq. (3):
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u 1 X MD 2 DSD ¼ t r r CB i 3N 1 i¼1 i
ð3Þ
where N is the total number of atoms in the simulation, r MD represents the ith atom’s spatial vector derived by MD and rCB i i is the ith atom’s spatial vector computed by the CB hypothesis. Since CB hypothesis predicts the elastic deformation in the homogenous deformation gradient, DSD can denote the irreversible deformation state induced by the atomic perturbation in the simulation system. Recently, the plastic deformation in the calcium silicate aluminate hydrate (CASH) under shear strain has been successfully predicted by the DSD [32]. For cases of C–S–H gel stretched along x and z direction, DSD of the Cas atoms versus strain is drawn in Fig. 6. In the elastic stage (strain <0.08 Å/Å), the deviation between MD and CB method is smaller than 1, indicating that the deformation of calcium silicate sheets obeys the CB rules. The large nonlinear deformation is accompanied by the development of the cracks. It should be note that the strain 0.08 Å/Å is coincide with the strain where stress begins to decrease in Fig. 2. Comparing the DSD values in x and z direction, the irreversible deformation in z direction is larger than that in x direction. Along x direction,
Fig. 6. Displacement standard deviation (DSD) of Cas atoms as a function of strain in C–S–H stretched along x direction and z direction.
D. Hou et al. / Engineering Fracture Mechanics 131 (2014) 557–569
565
breakage of Ca–O bonds results in the fracture of calcium silicate sheet. However, once the Ca atom dissociates with one O atom, it frequently reconstructs the chemical bond with other neighboring O atoms, which slows down the irreversible deformation. Considering frequently breakage and formation of H-bonds and Ca–Ow bonds in the interlayer region, the deformation develops very fast. The displacement deviation of the individual Cas atom (jr MD rCB i i j) is also drawn in Fig. 7. The large deformation begins to develop at the boundary of the cracks at strain 0.08 Å/Å, which is consistent with the crack initiation in Fig. 3. In view of chemical bonds, the Cas atoms near the void boundary, with small number of coordinated oxygen atoms, are less constrained and are easily to move in the presence of strain. As shown in Fig. 7, with progressively increasing distance from crack center along x axle, the irreversible displacement gradient from high to low value can be observed. This can be taken as the ‘‘strain concentration’’ near the void at nano-scale. In addition, the crack first develops parallel with the loading and subsequently propagates perpendicular to the loading direction. So the large irreversible deformation can be observed along 45° deviated from x axle. When the structure is completely stretched fracture at strain 0.4 Å/Å, the deformation is widely distributed surrounding the circular crack. As shown in Fig. 7, the calcium silicate sheets below the crack bend upward, which contributes to the large deformation of C–S–H gel. Fig. 8 demonstrates the displacement deviation of Cas atoms in C–S–H gel stretched along z direction. ‘‘Strain concentration’’ can also be observed at the circular region of the central void. From the crack growth (Fig. 8c) to the structure fracture (Fig. 8d) at strain less than 0.2 Å/Å, the deformation of the stretched C–S–H gel develops in relative high speed parallel to the interlayer region, which is consistent with the brittle failure mode observed in Fig. 4. Different from the gradually structure deformation in x direction, the C–S–H gel failure in z direction is in a catastrophic mode: in the rupture state, large deformation only occurs near interlayer region weak in carrying loading rather than propagate through C–S–H gel. 3.2. Void size influence As expected, the central voids have detrimental influence on the mechanical properties of C–S–H gel. As exhibited in Fig. 9b and c, in x direction, the Young’ s modulus reduces from 65 GPa to less than 40 GPa as the central void size increases from 0 to 50 Å. Tensile strength also decreases from 4.2 GPa to 2.6 GPa. It should be noted that the weakening trend is not obvious until the central void’s length reaches 20 Å. When central void’s diameter extends 20 Å, the sharply reduction of modulus and tensile strength can be clearly observed. As many layered crystal such as tobermorite and jennite [3,23,24], the interlayer distance of C–S–H gel ranges from 10 Å to 14 Å. When the pore size is smaller than 20 Å, the interaction between atoms near the crack boundary is as strong as interlayer cohesive force. In addition, it can be observed from Fig. 9a that increasing crack length results in the reduction of the fracture strain. Especially as the central void size reaches as large as 50 Å, the ladder-like stress–strain relation in the post-failure region disappears, implying significant weakening effect on the ductility of C–S–H gel. Fig. 10 demonstrates different morphologies of damaged C–S–H gels at strain 0.24 Å/Å. When the diameter of central void is around 5 Å, the crack propagates along a narrow channel toward the bottom of C–S–H structure. Nucleation of small cracks is widely distributed but only cracks developed from the central void boundary coalesce. With increasing void size, the size of crack channel becomes wider and wider. The cracks, initiated from the boundary of central void, both develop along the
Fig. 7. Individual displacement deviation of Cas atoms in C–S–H gel stretched along x direction. a, b, c and d represent strain at 0, 0.16, 0.2, 0.3 Å/Å.
566
D. Hou et al. / Engineering Fracture Mechanics 131 (2014) 557–569
Fig. 8. Individual displacement deviation of Cas atoms in C–S–H gel stretched along z direction. a, b, c and d represent strain at 0, 0.08, 0.16, 0.2 Å/Å.
Fig. 9. (a) Stress–strain relation of C–S–H gel with central voids with size from 0 to 50 Å. (b) Young’s modulus varies versus central voids’ size; (c) Tensile strength varies versus central voids’ size.
paths upward and downward, implying that the large void provides more crack propagation paths. In addition, the crack opening angle turns to large values, with increasing size of the central void when the central void reaches 50 Å, the fracture surface orients almost perpendicular with tension loading. It means that presence of crack changes C–S–H failure from ductile mode to brittle mode. Analogous, the central void weakens cohesive force and the Young’s modulus along z direction. As shown in Fig. 11b and c, the tensile strength decreases from 2.4 GPa to 1.54 GPa and Young’s modulus reduces from 44.5 GPa to 29.7 GPa with crack length increasing from 0 to 50 Å. The simulation results can be compared to those obtained from nano-indentation test, which is an effective method to explore the stiffness of cement paste at micro-scale level. Based on the experimental results, C–S–H gels have been proposed to contain high and low stiffness components, with Young’s modulus around 31 GPa and 21 GPa, respectively [9]. On one hand, the molecular simulation results confirm the experimental findings that the gel pores weaken the mechanical behavior of the C–S–H gel. On the other hand, the moduli of simulated C–S–H gel are higher than those obtained from nano-indentation. The reasons for such deviation from experimental results are as follow: (1) the simulated calcium silicate sheet arranges in the layered mode without disordered organization such as distortion; (2) the single gel pore located in the center of the C–S–H gel reduces the pore connectivity in x and z direction. (3) The gel pores are not saturated with water molecules so that the disjointing pressure from water is reduced. More realistic model, considering three above respects, will be proposed in future work to more reasonably understand the gel pore’s effect at molecular level. Furthermore, understanding the mechanical behavior of C–S–H gel at molecular level plays important role in multi-scale modelling of the cementitious materials. The tensile strength in molecular simulation ranges from 1.54 to 4.2 GPa, which is around 1000 times higher than that of cement at macro-level. It is necessary to bridge the huge gap between molecular and macro level. Some models based on the micro-mechanics, such as Mori–tanaka (MT) and self-consistent (SC) approach [13,26] were utilized to relate the nano-scale behavior to mechanical performance at macro-level. Recently, the lattice
D. Hou et al. / Engineering Fracture Mechanics 131 (2014) 557–569
567
Fig. 10. Snapshots of C–S–H gel with central void of diameter 5 (a), 15 (b), 30 (c), 50 Å (d) tensioned along x direction at strain 0.24 Å/Å.
Fig. 11. (a) Stress–strain relations of C–S–H gel with central voids of different sizes. (b) Young’s modulus varies versus central voids’ size; (c) Tensile strength varies versus central voids’ size.
fracture model [31] was utilized to multi-scale simulate the fracture behavior of the cementitious materials. In the lattice model, the basic blocking unit of C–S–H gel is represented by the ‘‘beam’’ element of 1 lm. The tensile behavior of the material at this level is hard to be obtained by experiment, so current lattice model was based on the assumption that the mechanical properties of the beam are brittle. Molecular dynamics simulation provides the stress–strain relation of the C–S–H gel, which is very important input parameters for the ‘‘beam’’ element. In future, the lattice model, starting from the molecular simulation, will be proposed to describe the fracture process of cementitious materials more accurately. Besides, the ductility of C–S–H gel is influenced by the size of the central void. Even though the weakening trend of ductility can be observed in cases of samples with pore diameter 0, 5, 10, 15, 30 and 50 Å, samples with pore diameter of 20 and 25 Å have ladder-like stress–strain relation, implying good ductility. The ductility enhancement can be explained in structural rearrangement. The morphologies of the C–S–H gels at strain 0.24 Å/Å are drawn in Fig. 12. In all the failure samples, the interlayer region is the path for crack propagation. Samples with void diameter 0, 10, 15 and 50 Å have similar failure mode as 30 Å void case discussed before: cracks are initiated from the void boundary and penetrate into same interlayer, where the structure completely fracture. On the other hand, samples with void diameter 20 and 25 Å exhibit different failure mechanism. More than two cracks are developing simultaneously at different interlayer region. As shown in Fig. 12b, the cracks propagate along three paths with different spreading rate, which result in curvature of de-bonding calcium silicate sheet. In this way, the calcium silicate sheet constructs the beam to bridge fractured structures, enhancing the ductility
568
D. Hou et al. / Engineering Fracture Mechanics 131 (2014) 557–569
Fig. 12. Snapshots of C–S–H gel with central void 10 (a), 20 (b) and 25 Å (c) tensioned along z direction at failure state.
of the C–S–H gel at the end of tensile stage. The failure mode of layer structure can be analogous to the ‘bending’ of the plate structure with complicated bonding restrictions at macro-scale. On the xy plane, constraint bonding of the individual calcium silicate sheet is not uniformly distributed because of the presence of central voids. Especially in the presence of small crack, as shown in Fig. 12a, the left boundary side and the right one of the central void occupy different concentration of interlayer atoms, which result in stronger binding at right part and weaker binding at the other side. The discrepancy between interlayer connections results in different deflections and bending of calcium silicate sheet. It should be noted that the bending failure can be taken as an intrinsic feature of C–S–H gel’s deformation at nano-scale. It has not been observed in previous small scale deformation simulation. The benefits of large scale simulation around 130 Å can consider more defects in the C–S–H gel, which is close to the realistic condition in the complicated cement environment. 4. Conclusions Considering molecular dynamics, the crack evolution mechanism in the C–S–H gel have been investigated. Several conclusions can be drawn from this study as follows. (1) At nano-scale, the heterogeneous layered C–S–H gel demonstrates dual nature of crack propagation. In xy plane, the stable ionic-covalent bonds Si–O and Ca–O are hard to break so the cracks coalesce is slowed down, implying ductile characteristic. In z direction, cracks spread in the interlayer region with high rate due to the frequently breakage Hbonds network, exhibiting brittle nature. (2) The presence of the central voids ranging from 20 Å to 50 Å weakens the stiffness and cohesive force of the C–S–H gel. It implies that the gel pore at nano-scale structure in the C–S–H gel is detrimental to the mechanical properties. Maximum values of nonlinear displacements are distributed around the central voids and the ‘‘strain concentration’’ occurs at the boundary of the local defective region. (3) Because of the binding constraints’ discrepancy, bending of the calcium silicate sheet can be observed, reflecting the complicated tensile behavior of heterogeneous layered C–S–H gel.
Acknowledgements Financially support from the China Ministry of Science and Technology under Grant 2015CB655104 and the Chinese National Natural Science Foundation (NSF) under Grant 51178230 and 51378269 , Major International Joint Research Project under Grant 51420105015 are gratefully acknowledged. References [1] [2] [3] [4] [5]
Allen AJ, Thomas JJ, Jennings HM. Composition and density of nanoscale calcium silicate hydrate in cement. Nature material 2007;6:311–6. Bazant Z. Size effect in blunt fracture: concrete, rock, metal. J Engng Mech 1984;110(4):518–35. Bonnacorsi E, Merlino S, Taylor H. The crystal structure of jennite Ca9Si6O18(OH)6 8H2O. Cem Concr Res 2004;34(9):1481–8. Bonnaud PA et al. Thermodynamics of water confined in porous calcium–silicate–hydrates. Langmuir 2012;28(31):11422–32. Brough A, Dobson C, Richardson I, Groves G. In situ solid-state NMR studies of Ca3SiO5: hydration at room temperature and at elevated temperatures using 29Si enrichment. J Mater Sci 1994;29(15):3926–40. [6] Chen JJ, Thomas JJ, Taylor HFW, Jennings HM. Solubility and structure of calcium silicate hydrate. Cem Concr Res 2004;34:1499–519.
D. Hou et al. / Engineering Fracture Mechanics 131 (2014) 557–569
569
[7] Cong X, Kirkpatrick R. 29Si and 17O NMR investigation of the structure of some crystalline calcium silicate hydrate. Adv Cem Based Mater 1996;3:133–43. [8] Cong X, Kirkpatrick R. 29Si MAS NMR study of the structure of calcium silicate hydrate. Adv Cem Based Mater 1996;3(3–4):144–56. [9] Costantinide G, Ulm F. The nanogranular nature of C–S–H. J Mech Phys Solids 2006;55(1):64–90. [10] Feldman RF, Sereda P. A model for hydrated Portland cement paste as deduced from sorption-length change and mechanical properties. Matériaux Construction 1968;1(6):509–20. [11] Griffith AA. The phenomena of rupture and flow in solids. Philos Trans R Soc Lond 1921;A(221):163–98. [12] Hamid S. The crystal structure of the 11 A natural tobermorite Ca2.25 Si3 O7.5(OH)1.5H2O. Zeitschrifit Kristallographie 1981;154(3–4):189–98. [13] Hershey A. The elasticity of an isotropic aggregate of anisotropic cubic crystals. J Appl Mech-Trans ASME 1954;21(3):236–40. [14] Hou D, Li Z. Large scale simulation of calcium silicate hydrate (C–S–H) by molecular dynamics. Adv Cem Res 2014. [15] Hou D, Ma H, Zhu Y, Li Z. Calcium silicate hydrate from dry to saturated state: structure, dynamics and mechanical properties. Acta Mater 2014;67:81–94. [16] Janika JA, Kurdowsk W, Podsiadey R, Samset J. Fractal structure of CSH and tobermorite phases. Acta Phys Polonic 2001;100:529–37. [17] Jennings HM. A model for the microstructure of calcium silicate hydrate in cement paste. Cem Concr Res 2000;30(1):101–16. [18] Jennings HM. Refinements to colloid model of C–S–H in cement: CM II. Cem Concr Res 2008;38(3):275–89. [19] Jenq YS, Shah SP. A two parameter fracture model for concrete. J Engng Mech 1985;111(4):1227–41. [20] Ji Q, Pellenq RJM, Van Vliet KJ. Comparison of computational water models for simulation of calcium silicate hydrate. Comput Mater Sci 2012;53(1):234–40. [21] Kaplan ME. Crack propagation and fracture of concrete. J ACI 1961;58(5):591–610. [22] Manzano H et al. Confined water dissociation in microporous defective silicates: mechanism, dipole distribution, and impact on substrate properties. J Am Chem Soc 2011;134(4):2208–15. [23] Merlino S, Bonaccorsi E, Armbruster T. The real structures of clinotobermorite and tobermorite 9 Å. Eur J Miner 2000;12(2):411–29. [24] Merlino S, Bonnacorsi E, Armbruster T. The real structure of tobermorite 11 A: normal and anomalous forms, OD character and polytypic modifications. Eur J Mineral 2001;13(3):577–90. [25] Mindess S, Young J, Darwin D. Concrete. 2nd ed. Prentice Hall PTR; 2003. s.l. [26] Mori T, Tanaka K. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall 1973;21(5):571–4. [27] Muller ACA, Scrivener KL, Gajewicz AM, McDonald PJ. Use of bench-top NMR to measure the density, composition and desorption isotherm of C–S–H in cement paste. Microporous Mesoporous Mater 2013;178:99–103. [28] Murray SJ, Subramani VJ, Selvam RP, Hall KD. Molecular dynamics to understand the mechanical behavior of cement paste. J Transport Res Board 2010;2142(11):75–82. [29] Pellenq RJ et al. A realistic molecular model of cement hydrates. PNAS 2009;106(38):16102–7. [30] Powers T, Brownyard T. Studies of the physical properties of hardened Portland cement paste. PCA Bull, Portland Cem Assoc 1948;22:22. [31] Qian Z, Ye G, Schlangen E, van Breugel K. 3D lattice fracture model: application to cement paste at microscale. Key Engng Mater 2011;452:65–8. [32] Qomi M, Ulm F, Pellenq R. Evidence on the dual nature of aluminum in the calcium–silicate–hydrates based on atomistic simulations. J Am Ceram Soc 2012;95(3):1128–37. [33] Selvam RP, Subramani VG, Murray S, Hall K. Potential application of nanotechnology on cement based materials, 2009; s.l.: s.n. [34] Shahsavari R, Pellenq RJM, Ulm FJ. Empirical force fileds for complex hydrated calcio-silicate layered materials. Phys Chem Chem Phys 2011;13(3):1002–11. [35] Shahsavari R. Hierarchical modeling of structure and mechanics of cement hydrate. PhD thesis of Massachusetts Institute of Technology; 2011. [36] Shahsavari R, Buechler MJ, Pellenq RJM, Ulm FJ. First-principles study of elastic constants and interlayer interactions of complex hydrated oxides: case study of tobermorite and jennite. J Am Ceram Soc 2009;92(10):2323–30. [37] Wecharatana M, Shah S. Slow crack growth in cement composites. J Struct Engng 1982;108(6):1400–13. [38] Youssef M, Pellenq RJM, Yildiz B. Glassy nature of water in an ultraconfining disordered material: the case of calcium silicate hydrate. J Am Chem Soc 2011;133(8):2499–510.
Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Molecular dynamics study on the mode I fracture of calcium silicate hydrate under tensile loading Dongshuai Hou a, Tiejun Zhao a, Penggang Wang a, Zongjin Li b, Jinrui Zhang b,⇑ a b
Qingdao Technological University (Cooperative Innovation Center of Engineering Construction and Safety in Shandong Blue Economic Zone), Qingdao, China Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong Special Administrative Region
a r t i c l e
i n f o
Article history: Received 12 February 2014 Received in revised form 6 September 2014 Accepted 9 September 2014 Available online 21 September 2014 Keywords: Calcium silicate hydrate (C–S–H) Uniaxial tension test Crack evolution SCF DSD
a b s t r a c t Calcium silicate hydrate (C–S–H), the essential binder of cement based materials, is a poor crystal phase at nano-scale. In this study, crack growth mechanism is unraveled in lights of Molecular Dynamics (MD) by simulating the uniaxial tension test on the C–S–H gel with voids ranging from 0.5 nm to 5 nm. At nano-scale, the layered C–S–H gel demonstrates dual nature of crack propagation. In xy plane, the stable ionic-covalent bonds Si–O and Ca–O are hard to break so the cracks coalesce is slowed down, implying ductile characteristic. In z direction, cracks spread in the interlayer region with high rate due to the frequently breakage H-bonds network, exhibiting brittle nature. In the tensile process, the crack develops from the central void and ‘‘strain concentration’’ near the void boundary is induced by irreversible atomic deformation. Young’s modulus and tensile strength of C–S–H gel are significantly weakened due to the presence of central void. Additionally, due to the binding constraints’ discrepancy, bending of the calcium silicate sheet can be observed, reflecting the complicated tensile behavior of heterogeneous layered C–S–H gel. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction When cement is mixed with water, the hydration reactions produce various cementitious hydrates, of which C–S–H gel is the dominant and important binding phase. External loading on the cement materials causes complicated crack distribution ranging from nano-level to macro-level in C–S–H gel. The multi-scale crack length distribution is very important microstructure characteristic, which influences the physical properties of the concrete, such as the strength and elasticity, the permeability and diffusivity, shrinkage and durability to freezing and thawing. As the most widely used construction and building materials, the ability to resist crack growth is an important criterion to estimate cements’ serviceability. And the mechanical properties influenced by the crack have been investigated since fracture mechanics was first applied to concrete by Kaplan [21]. Researchers made great efforts to determine whether linear elastic fracture mechanics (LEFM) theory proposed by Griffth [11] can be applied to rapid crack propagation in cement-based materials. However, the large fracture process zone in concrete observed by many experiments proved that the criterion of LEFM could not directly predict the fracture behavior of cementitious materials. Since the 1980s, models with more than one fracture parameter have been proposed to explain the fracture in concrete. Wecharatana and Shah [37] used a compliance based model to calculate the length of the fracture process zone. Jenq and Shah [19] introduced a two-parameter fracture model. It was demonstrated that these two parameters were size independent. In 1984, Bazant [2] developed a size effect law to model concrete fracture. Investigations ⇑ Corresponding author. Tel.: +852 67636710. E-mail addresses: [email protected] (D. Hou), [email protected] (T. Zhao), [email protected] (P. Wang), [email protected] (Z. Li), [email protected] (J. Zhang). http://dx.doi.org/10.1016/j.engfracmech.2014.09.011 0013-7944/Ó 2014 Elsevier Ltd. All rights reserved.
558
D. Hou et al. / Engineering Fracture Mechanics 131 (2014) 557–569
Nomenclature Si Os Ow Cas Caw H SCF DSD
is is is is is is is is
the the the the the the the the
silicon atom oxygen atom bridging with silicon oxygen atom in the water molecule calcium atom in the calcium silicate sheet calcium atom in the interlayer region hydrogen atom Strain correlated function displacement standard deviation
of cracks at macro-level provide guide in design of beams, anchorage and large dams. However, fracture mechanism of cementitious materials is not yet mature without studying the crack growth mechanism at nano-level. In order to understand nano-crack growth mechanism in cement-based material, it is necessary to investigate the intrinsic structure and mechanical performance of C–S–H gel. The molecular structure of C–S–H has been studied for more than half a century, particularly using a variety of experimental techniques, including NMR ([7,8]), XRD [16] and SANS [1], and C– S–H is now widely believed to be the analogue of layered minerals: tobermorite [24,12] and jennite [3]. Based on experimental data, Pellenq et al. [29] employed the Molecular Dynamic (MD) method to construct a C–S–H model that can well describe the structural, dynamical and mechanical properties of cement at the nano-scale. Meanwhile, CSHFF force field [34] was developed to describe the interaction between atoms in cement systems, and this force field demonstrated good transferability in simulating cement-based materials. Pellenq et al. [29] simulated the stress–strain relation of C–S–H gel in resisting shear force. By comparing the mechanical performance of dry and wet C–S–H gels, it was concluded that during the loading process, large displacements of water molecules in wet C–S–H gels weaken the shear strength and result in unrecovered deformations. Youssef et al. [38] investigated structural and dynamic features of water molecules in C–S–H gels. Due to the highly hydrophilic nature of calcium silicate sheets, H-bonds constructed between the non-bridging oxygen and water are very strong. Therefore, water molecules constrained in the gel demonstrate a glassy nature: the tetrahedral spatial structure is distorted and the diffusion rate is significantly reduced. Ji et al. [20] utilized five classic water models, SPC, TIP3P, TIP4P, TIP4P05, and TIP5P to simulate calcium silicate hydrate. It was concluded that SPC and TIP5P accurately describe the intrinsic properties of C–S–H gel, with SPC (flexible) being more computationally efficient. Recently, Bonnaud et al. [4] interpreted the cohesive force of C–S–H gel by analyzing the fluid pressure of the water molecules and the courter-ions in the interlayer region. Under different humidity conditions, they found that the cohesive force mainly results from negative pressure caused by the interaction between interlayer calcium atoms and the calcium silicate sheets. Previous research on C–S–H gels can be categorized into two classes: molecular structure analysis and thermodynamics investigation. Few efforts have been made to investigate the mechanical properties evolution in presence of nano-cracks. In particular, the tensile strength of C–S–H gels at the nano-scale, considered as the most essential property of construction material, has not been studied so far. The aim of this paper is to explore the fracture mechanism of C–S–H gel at molecular level. On the one hand, the structural feature evolution can be characterized by analyzing atomic deformation, chemical bonds and crack length variation. 2. Simulation method The CSHFF force field [34], developed for cement based materials, is utilized to simulate C–S–H gels with different central voids. The force field approach has been widely used in C–S–H simulations and has been proven to be able to describe the structure, energy and mechanical properties of various calcium silicate phases satisfactorily [29,38,20,4,22]. The force field parameters of Ca, Si, O and H can be obtained from the literature [35]. 2.1. C–S–H model construction The C–S–H model in the present study is constructed based on the procedures proposed by Pellenq et al. [29]. Firstly, the layered analogue mineral of C–S–H, tobermorite 11 Å without water, was taken as the initial configuration for the C– S–H model [28,33]. Silicate chains were then broken to match the Q species distribution obtained from NMR testing [6]. The major difference for the silicate skeleton between current model and Pellenq’s model is the Q species distribution. In Pellenq’s model, Q0 species, representing the monomers, remained 13% in calcium silicate sheet, which is more than 3 times larger than the results obtained from the 29Si and 17O cross-polarized NMR test [5] ([7,8]). The overestimation of the Q0 species can influence the calcium silicate skeleton. Hence, the percentage of Q0 species is controlled below 5%, matching experimental results. Secondly, Grand Canonical Monte Carlo simulation of the water adsorption is operated
D. Hou et al. / Engineering Fracture Mechanics 131 (2014) 557–569
559
on the dry disordered C–S–H structure at 300 K. The chemical formula of the saturated C–S–H structure in current simulation is (CaO)1.69(SiO2) 1.66H2O, which is quite close to (CaO)1.7(SiO2) 1.8H2O obtained by the SANS test [1]. The molecular dynamic simulations under constant pressure and temperature (NPT) for 300 ps give the structures of C–S–H gel at equilibrium states. For each case, a further 1000 ps NPT run is employed to achieve the equilibrium configuration for structural and dynamic analysis.
2.2. Mechanical properties calculation There are two methods to represent the gel pores: voids can be created by randomly packing smaller C–S–H bricks or embedded the empty region into the large layered structure [30,10,17,18]. In previous research, the former approach has been utilized to investigate the fracture behavior of different packing models [14]. Due to the packing efficiency restriction, the former approach can only study the C–S–H gels with packing fraction lower than 74% (fcc packing). Additionally, pore sizes distribution is hard to be characterized in the packing models, which brings about difficulty in analyzing the pore size effect on the mechanical properties. In this study, influence of gel pore size can be investigated by the latter approach. The C–S–H model, constructed in Section 2.1, is periodically extended 6 times and the size of the super-cell is 138 Å 138 Å 138 Å, with more than 300,000 atoms. Large number of atoms can provide statistical reliable results and more realistic evolution of the structures of C–S–H gel. Then the cylinder pore with diameter 0, 5, 10, 15, 20, 25, 30, 50 Å at the central of the C–S–H super-cell is eliminated to achieve 8 samples, as shown in Fig. 1. While the calcium silicate sheet and interlayer Ca were removed, the interlayer water molecules remained in the gel pore. Recently, NMR investigations [27] on the calcium silicate hydrate indicated that the amount of gel pore water gradually decreases between 100% and 25% relative humidity. In current simulation, the gel pore water amount is not that high so it is relevant to C–S–H gel at low humidity state. The selected pore width is in the range of gel pore size that distributes from 0.5 to 10 nm, as proposed by Mindess et al. [25]. It should be noted that this simulation is only the preliminary study that focus on the pore size effect on the C–S–H gel. A series of work along this direction will be continued to systematically investigate the gel pores in C–S–H structure, including the influence of the geometry, the poly disperse, the connectivity and the saturated state of the pores. Atoms elimination should not change the charge neutrality in the systems. Subsequently, uniaxial tension test is performed on the new created samples in x and z direction. Young’s modulus and the tensile strength in x and z direction, describing the stiffness and interlayer cohesive force respectively, are obtained by uniaxial tension testing. In order to get the stress–strain relation, the C–S–H samples were subjected to uniaxial tensile loading through gradual elongation at a constant strain rate at 0.08/ps. In the whole simulation process, NPT ensembles are set for the system. The simulation sequence along z direction is listed as follows. Firstly, the simulation box is relaxed at 300 K and coupled to zero external pressure in the x, y, z dimensions for 100 ps. Then, after the pressures in the three directions reach equilibrium, the C–S–H structure is elongated in the z direction. Meanwhile, the pressures in the x and y directions are kept at zero. After each step elongation, the atomic configuration is relaxed and pressure evolution in the z direction is taken as the internal stress rzz. The non-pressure setting in directions perpendicular to the tension direction, also considering Poisson’s ratio, can eliminate the artificial constrain for the deformation and allow the free development of tension without any restriction. In the simulated tension process, 10,000 configurations are recorded for structural analysis.
Fig. 1. Configuration of the C–S–H gel with the central void for the uniaxial tension test. White thick chain is silicate bond; short thin chain is water molecule; large blue ball represents interlayer calcium atoms and; small red ball represents calcium atoms in the sheet. Same meaning in the following configurations. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
560
D. Hou et al. / Engineering Fracture Mechanics 131 (2014) 557–569
3. Results and discussion 3.1. Crack orientation influence Stress–strain curves, shown in Fig. 2, characterize the carrying loading ability of C–S–H gel with 30 Å central void along (1 0 0) and (0 0 1) directions. In x direction, stress evolution can be divided into three stages: stress increases with strain until reaches failure strength 3.3 GPa at strain 0.084 Å/Å; subsequently stress drops from failure strength to 2.1 GPa ; finally decrease slowly to zero at strain 0.4 Å/Å. In the post-failure region, the ladder-like stress–strain relation, a symbol of good plasticity, indicates some structural rearrangement in the layered C–S–H gel. Both Young’s modulus and tensile strength in x direction is larger than that in z direction. As exhibited in Fig. 2, the stress increase to less than 2 GPa at strain 0.08 Å/Å and then drops sharply to zero at complete rupture strain 0.2 Å/Å, indicating the poor ductility along z direction. It should be noted that the tensile strength (2 GPa) simulated by current method is larger than the shear strength (1 GPa). The difference between failure strengths results from many reasons such as the loading mode, the sample size and the molecular structure of C–S–H gel. In current C–S–H model, there are less than 5% of the silicate monomer (Q0) species, which is much smaller than that of Pellenq et al’s model (Q0 = 13%). It means that the connectivity of the silicate chains in current model is better than that of Pellenq’s model and the longer silicate chains lead to more mechanical contribution. Besides, the H2O/Si ratio in Pellenq’s model is higher than that of current model, the ‘‘hydrolytic weakening’’ effect further reduces the cohesive force. However, since both current model and Pellenq’s model have not considered the disordered calcium silicate sheet and the connected gel pores, the failure strengths simulated by both models are overestimated in some extent. On the other hand, in previous simulation on mechanical behavior of C–S–H gel by MD [15,28], the tensile-compressive strength ratio ranges from 0.23 to 0.66. As the morphology of silicate chains turns to more defective, the tensile-compressive ratio reduces to lower value. The trend in the stress–strain behavior of the C–S–H unit with defective silicate chains in tension and compression is reasonably comparable to that at the macroscopic level. 3.1.1. Crack development The stress–strain evolution is accompanied by the variation of structural deformation. More insights can be obtained by observing the molecular structure change during the stretching process. Initially, as shown in Fig. 3b, the layered structure is slightly stretched longer at strain 0.08 Å/Å and Si–O and Ca–O bonds extension takes up the strain during the elastic stage. As strain increases from 0.08 to 0.16, the dissociation of calcium atoms and short silicate chains result in crack growth surrounding the boundary of central voids. It should be noted that initiating calcium silicate sheet fracture, shown in Fig. 3c, corresponds to the first drop of the tensile stress in the stress–strain curve. The quick stress reducing rate indicates the calcium silicate sheet near the crack boundary is easily to be stretched broken due to small binding energy. Subsequently, small cracks begin to grow into neighboring calcium silicate sheet. Compared with the first fracture near the central void, the calcium sheet away from the void is relatively harder to be destroyed. Hence the stress decreasing rate can be slowed down. In addition, as shown in Fig. 3d, the interlayer calcium atoms can penetrate into the defective region due to stretch and help reconstruct the damaged calcium silicate sheet temporarily, which enhances deformation resistance. Finally, the cracks, initialed from the central void, coalesce with surrounding small cracks and propagate along direction around ±45° from x axis. The cup-and-cone failure surface, shown in Fig. 3f, has also been observed in the damages of many ductile materials and is characterized as the ductile failure. However, the necking phenomenon has not been observed at the late tensile stage as many ductile materials such as iron. Only the calcium silicate sheets in the middle have slight bend toward the central void. The presence of interlayer water molecules prevent the large bending deformation of calcium silicate sheet so that the
Fig. 2. The stress–strain curves of C–S–H gel with 30 Å central void tensioned along x and z direction.
D. Hou et al. / Engineering Fracture Mechanics 131 (2014) 557–569
561
Fig. 3. Crack evolution of C–S–H gel with central void 30 Å tensioned along x direction. a, b, c, d, e and f represent strain at 0, 0.08, 0.16, 0.2, 0.3, 0.4 Å/Å.
‘‘necking effect’’ is hard to happen. On one hand, water molecules screen the interflow of Ca, Si atoms between neighboring calcium silicate sheet so that weakens the plastic deformation of tensioned C–S–H gel. On the other hand, once the damaged structure forms, the water molecules diffuse fast to associate with the non-bridging oxygen site, which prevent interlayer calcium from bridging the fractured structure. Therefore, the plastic deformation, arouse by the diffusion of Ca, Si and O atoms, is significantly reduced. In z direction, the tensile failure of the C–S–H gel exhibits different mode. As demonstrated in Fig. 4b, at beginning of tensile process, the swell of the interlayer region and the extension of Si–O and Ca–O bond in the calcium silicate sheet takes up the strain at the elastic region. With gradually increasing of the strain, the neighboring calcium silicate sheets begin to separate at the boundary of the central void, as the strain reaches 0.08 Å/Å, where the tensile stress first reduces. It worth noting that in comparison with the ionic-covalent bonds in the calcium silicate sheets, the H-bonds network and Ca–Ow bond, breakage and formation quickly, have lower stability, so the cracks are initiated from the interlayer region. The catastrophic damage mode is attributed to the fast crack development in the interlayer region. The crack grows and coalesce fast to rupture the gel structure in only 1600 ps, which is one third of that in case of x direction. As shown in Fig. 4e, different from the coarse fracture surface in x direction, the rupture surface of C–S–H gel loading along z direction is quite flat and normal to direction of tensile loading.
562
D. Hou et al. / Engineering Fracture Mechanics 131 (2014) 557–569
Fig. 4. Crack evolution of C–S–H gel tensioned along z direction. a, b, c, d, e and f represent the strain at 0, 0.06, 0.08, 0.16, 0.2, 0.24 Å/Å.
D. Hou et al. / Engineering Fracture Mechanics 131 (2014) 557–569
563
3.1.2. Chemical bonds analysis A Strain correlated function (SCF) is used to describe the variation of chemical bonds comparison to the original structure during the tensile process. SCF can be interpreted as the time correlated function (TCF) in the presence of strain loading in Eq. (1):
CðeÞ ¼
hdbðeÞdbð0Þi hdbð0Þdbð0Þi
ð1Þ
where db(e) = b(e) hbi and b(e) is a binary operator that takes a value of one if the pair (e.g. Ca–O) is bonded and zero if not and hbi is the average value of b over all simulation time and pairs. The chemical bonds break and form due to tensile strain, which results in the connectivity variation in C–S–H gels. If the connectivity persists unchanged, the SCF of bonds maintain a constant value one. Otherwise, the more frequently bond breakage occurs, the lower the value of SCF. By comparing the deviation degree from one in the SCF curves, roles of different bonds in carrying load can be estimated. As proposed in previous research [29], Cas–Os, Si–Os and Caw–Os are the major bonding phase in the C–S–H gel playing a dominant role in loading resistance. C(e) of Cas–Os, Si–Os and Caw–Os bonds variation with strain in x and z directions are plotted in Fig. 5. As shown in Fig. 5, C(e) of Cas–Os and Caw–Os continuously decreases while that of Si–Os maintains constant value with slight fluctuation. This means that Ca–O bond breaks and forms frequently, while Si–O bond, occupying high binding energy, is broken slightly. In this respect, the silicate chains act as the backbone and the morphology of a silicate structure, to a great extent, determining the mechanical performance of C–S–H gel [36]. In x and z direction, C(e) of Ca–O stops reducing as strain reaches 0.4 and 0.2, respectively, which correspond to the rupture strain shown in stress–strain
Fig. 5. Strain correlated function (SCF) of C–S–H gel with 30 Å void tensioned in x (a) and z (b) direction.
564
D. Hou et al. / Engineering Fracture Mechanics 131 (2014) 557–569
curve. It means that after fracture, most of chemical bonds in the separated C–S–H gel no longer break or form frequently, with only slight evolution of bonds in the rupture surface. When the sample is subject to loading along x direction, C(e) of Cas–Os and Caw–Os reduce by 0.25 and 0.2 respectively. The variation of Ca–O bond mainly result in the fracture of calcium silicate sheet while the breakage of Caw–Os bond reflects the shear damage between the interlayer region and calcium silicate skeleton, which can both be observed in Fig. 5. Both Cas–Os and Caw–Os bond play critical role in loading resistance. Considering that the number of Cas atoms is 2 times larger than that of Caw atoms, calcium silicate sheet acts major role in carrying loading. On the other hand, the chemical bonds variation demonstrates different case as the C–S–H is tensioned in z direction. The deviation degree of Cas–Os bond is only half of the Cas–Ow bond, implying that interlayer region is the main phase in loading resistance. The breakage of Cas–Os bond mainly result in the damage of calcium silicate sheet near the crack boundary. In the location far away from the crack, the calcium silicate sheet maintains integrity. So the Cas–O bonds change slightly. In addition, Caw–Os bonds’ breakage is attributed to the crack propagation along interlayer region. In comparison with Caw–Os, Cas–Os bond is more stable in loading resistance. On one hand, Cas atoms have more coordinated Os atoms so the Cas–Os bond can frequently be reconstructed after breakage. On the other hand, the Caw atoms have many neighboring water molecules in the interlayer region, which disjoins the cohesive force between Caw atoms and Os atoms in the silicate chains. 3.1.3. Displacement evolution In order to evaluate the deformation of C–S–H gels with nano-cracks quantitatively, the difference between deformation derived by MD simulation and that predicted by means of Cauchy–Born method is calculated. The Cauchy–Born hypothesis assumes that the displacement of an atom in the molecular theory of elasticity is according to the deformation gradient F. In the uniaxial tension test along x direction, F is defined in Eq. (2):
0
@x=@X
B F ¼ @0 0
0
0
0
1
1
C 1 0A
ð2Þ
In which ox/oX is corresponding to the magnitude of uniaxial elongation divided by the initial length of the specimen. The displacement standard deviation (DSD) from Cauchy–Born method is computed as Eq. (3):
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u 1 X MD 2 DSD ¼ t r r CB i 3N 1 i¼1 i
ð3Þ
where N is the total number of atoms in the simulation, r MD represents the ith atom’s spatial vector derived by MD and rCB i i is the ith atom’s spatial vector computed by the CB hypothesis. Since CB hypothesis predicts the elastic deformation in the homogenous deformation gradient, DSD can denote the irreversible deformation state induced by the atomic perturbation in the simulation system. Recently, the plastic deformation in the calcium silicate aluminate hydrate (CASH) under shear strain has been successfully predicted by the DSD [32]. For cases of C–S–H gel stretched along x and z direction, DSD of the Cas atoms versus strain is drawn in Fig. 6. In the elastic stage (strain <0.08 Å/Å), the deviation between MD and CB method is smaller than 1, indicating that the deformation of calcium silicate sheets obeys the CB rules. The large nonlinear deformation is accompanied by the development of the cracks. It should be note that the strain 0.08 Å/Å is coincide with the strain where stress begins to decrease in Fig. 2. Comparing the DSD values in x and z direction, the irreversible deformation in z direction is larger than that in x direction. Along x direction,
Fig. 6. Displacement standard deviation (DSD) of Cas atoms as a function of strain in C–S–H stretched along x direction and z direction.
D. Hou et al. / Engineering Fracture Mechanics 131 (2014) 557–569
565
breakage of Ca–O bonds results in the fracture of calcium silicate sheet. However, once the Ca atom dissociates with one O atom, it frequently reconstructs the chemical bond with other neighboring O atoms, which slows down the irreversible deformation. Considering frequently breakage and formation of H-bonds and Ca–Ow bonds in the interlayer region, the deformation develops very fast. The displacement deviation of the individual Cas atom (jr MD rCB i i j) is also drawn in Fig. 7. The large deformation begins to develop at the boundary of the cracks at strain 0.08 Å/Å, which is consistent with the crack initiation in Fig. 3. In view of chemical bonds, the Cas atoms near the void boundary, with small number of coordinated oxygen atoms, are less constrained and are easily to move in the presence of strain. As shown in Fig. 7, with progressively increasing distance from crack center along x axle, the irreversible displacement gradient from high to low value can be observed. This can be taken as the ‘‘strain concentration’’ near the void at nano-scale. In addition, the crack first develops parallel with the loading and subsequently propagates perpendicular to the loading direction. So the large irreversible deformation can be observed along 45° deviated from x axle. When the structure is completely stretched fracture at strain 0.4 Å/Å, the deformation is widely distributed surrounding the circular crack. As shown in Fig. 7, the calcium silicate sheets below the crack bend upward, which contributes to the large deformation of C–S–H gel. Fig. 8 demonstrates the displacement deviation of Cas atoms in C–S–H gel stretched along z direction. ‘‘Strain concentration’’ can also be observed at the circular region of the central void. From the crack growth (Fig. 8c) to the structure fracture (Fig. 8d) at strain less than 0.2 Å/Å, the deformation of the stretched C–S–H gel develops in relative high speed parallel to the interlayer region, which is consistent with the brittle failure mode observed in Fig. 4. Different from the gradually structure deformation in x direction, the C–S–H gel failure in z direction is in a catastrophic mode: in the rupture state, large deformation only occurs near interlayer region weak in carrying loading rather than propagate through C–S–H gel. 3.2. Void size influence As expected, the central voids have detrimental influence on the mechanical properties of C–S–H gel. As exhibited in Fig. 9b and c, in x direction, the Young’ s modulus reduces from 65 GPa to less than 40 GPa as the central void size increases from 0 to 50 Å. Tensile strength also decreases from 4.2 GPa to 2.6 GPa. It should be noted that the weakening trend is not obvious until the central void’s length reaches 20 Å. When central void’s diameter extends 20 Å, the sharply reduction of modulus and tensile strength can be clearly observed. As many layered crystal such as tobermorite and jennite [3,23,24], the interlayer distance of C–S–H gel ranges from 10 Å to 14 Å. When the pore size is smaller than 20 Å, the interaction between atoms near the crack boundary is as strong as interlayer cohesive force. In addition, it can be observed from Fig. 9a that increasing crack length results in the reduction of the fracture strain. Especially as the central void size reaches as large as 50 Å, the ladder-like stress–strain relation in the post-failure region disappears, implying significant weakening effect on the ductility of C–S–H gel. Fig. 10 demonstrates different morphologies of damaged C–S–H gels at strain 0.24 Å/Å. When the diameter of central void is around 5 Å, the crack propagates along a narrow channel toward the bottom of C–S–H structure. Nucleation of small cracks is widely distributed but only cracks developed from the central void boundary coalesce. With increasing void size, the size of crack channel becomes wider and wider. The cracks, initiated from the boundary of central void, both develop along the
Fig. 7. Individual displacement deviation of Cas atoms in C–S–H gel stretched along x direction. a, b, c and d represent strain at 0, 0.16, 0.2, 0.3 Å/Å.
566
D. Hou et al. / Engineering Fracture Mechanics 131 (2014) 557–569
Fig. 8. Individual displacement deviation of Cas atoms in C–S–H gel stretched along z direction. a, b, c and d represent strain at 0, 0.08, 0.16, 0.2 Å/Å.
Fig. 9. (a) Stress–strain relation of C–S–H gel with central voids with size from 0 to 50 Å. (b) Young’s modulus varies versus central voids’ size; (c) Tensile strength varies versus central voids’ size.
paths upward and downward, implying that the large void provides more crack propagation paths. In addition, the crack opening angle turns to large values, with increasing size of the central void when the central void reaches 50 Å, the fracture surface orients almost perpendicular with tension loading. It means that presence of crack changes C–S–H failure from ductile mode to brittle mode. Analogous, the central void weakens cohesive force and the Young’s modulus along z direction. As shown in Fig. 11b and c, the tensile strength decreases from 2.4 GPa to 1.54 GPa and Young’s modulus reduces from 44.5 GPa to 29.7 GPa with crack length increasing from 0 to 50 Å. The simulation results can be compared to those obtained from nano-indentation test, which is an effective method to explore the stiffness of cement paste at micro-scale level. Based on the experimental results, C–S–H gels have been proposed to contain high and low stiffness components, with Young’s modulus around 31 GPa and 21 GPa, respectively [9]. On one hand, the molecular simulation results confirm the experimental findings that the gel pores weaken the mechanical behavior of the C–S–H gel. On the other hand, the moduli of simulated C–S–H gel are higher than those obtained from nano-indentation. The reasons for such deviation from experimental results are as follow: (1) the simulated calcium silicate sheet arranges in the layered mode without disordered organization such as distortion; (2) the single gel pore located in the center of the C–S–H gel reduces the pore connectivity in x and z direction. (3) The gel pores are not saturated with water molecules so that the disjointing pressure from water is reduced. More realistic model, considering three above respects, will be proposed in future work to more reasonably understand the gel pore’s effect at molecular level. Furthermore, understanding the mechanical behavior of C–S–H gel at molecular level plays important role in multi-scale modelling of the cementitious materials. The tensile strength in molecular simulation ranges from 1.54 to 4.2 GPa, which is around 1000 times higher than that of cement at macro-level. It is necessary to bridge the huge gap between molecular and macro level. Some models based on the micro-mechanics, such as Mori–tanaka (MT) and self-consistent (SC) approach [13,26] were utilized to relate the nano-scale behavior to mechanical performance at macro-level. Recently, the lattice
D. Hou et al. / Engineering Fracture Mechanics 131 (2014) 557–569
567
Fig. 10. Snapshots of C–S–H gel with central void of diameter 5 (a), 15 (b), 30 (c), 50 Å (d) tensioned along x direction at strain 0.24 Å/Å.
Fig. 11. (a) Stress–strain relations of C–S–H gel with central voids of different sizes. (b) Young’s modulus varies versus central voids’ size; (c) Tensile strength varies versus central voids’ size.
fracture model [31] was utilized to multi-scale simulate the fracture behavior of the cementitious materials. In the lattice model, the basic blocking unit of C–S–H gel is represented by the ‘‘beam’’ element of 1 lm. The tensile behavior of the material at this level is hard to be obtained by experiment, so current lattice model was based on the assumption that the mechanical properties of the beam are brittle. Molecular dynamics simulation provides the stress–strain relation of the C–S–H gel, which is very important input parameters for the ‘‘beam’’ element. In future, the lattice model, starting from the molecular simulation, will be proposed to describe the fracture process of cementitious materials more accurately. Besides, the ductility of C–S–H gel is influenced by the size of the central void. Even though the weakening trend of ductility can be observed in cases of samples with pore diameter 0, 5, 10, 15, 30 and 50 Å, samples with pore diameter of 20 and 25 Å have ladder-like stress–strain relation, implying good ductility. The ductility enhancement can be explained in structural rearrangement. The morphologies of the C–S–H gels at strain 0.24 Å/Å are drawn in Fig. 12. In all the failure samples, the interlayer region is the path for crack propagation. Samples with void diameter 0, 10, 15 and 50 Å have similar failure mode as 30 Å void case discussed before: cracks are initiated from the void boundary and penetrate into same interlayer, where the structure completely fracture. On the other hand, samples with void diameter 20 and 25 Å exhibit different failure mechanism. More than two cracks are developing simultaneously at different interlayer region. As shown in Fig. 12b, the cracks propagate along three paths with different spreading rate, which result in curvature of de-bonding calcium silicate sheet. In this way, the calcium silicate sheet constructs the beam to bridge fractured structures, enhancing the ductility
568
D. Hou et al. / Engineering Fracture Mechanics 131 (2014) 557–569
Fig. 12. Snapshots of C–S–H gel with central void 10 (a), 20 (b) and 25 Å (c) tensioned along z direction at failure state.
of the C–S–H gel at the end of tensile stage. The failure mode of layer structure can be analogous to the ‘bending’ of the plate structure with complicated bonding restrictions at macro-scale. On the xy plane, constraint bonding of the individual calcium silicate sheet is not uniformly distributed because of the presence of central voids. Especially in the presence of small crack, as shown in Fig. 12a, the left boundary side and the right one of the central void occupy different concentration of interlayer atoms, which result in stronger binding at right part and weaker binding at the other side. The discrepancy between interlayer connections results in different deflections and bending of calcium silicate sheet. It should be noted that the bending failure can be taken as an intrinsic feature of C–S–H gel’s deformation at nano-scale. It has not been observed in previous small scale deformation simulation. The benefits of large scale simulation around 130 Å can consider more defects in the C–S–H gel, which is close to the realistic condition in the complicated cement environment. 4. Conclusions Considering molecular dynamics, the crack evolution mechanism in the C–S–H gel have been investigated. Several conclusions can be drawn from this study as follows. (1) At nano-scale, the heterogeneous layered C–S–H gel demonstrates dual nature of crack propagation. In xy plane, the stable ionic-covalent bonds Si–O and Ca–O are hard to break so the cracks coalesce is slowed down, implying ductile characteristic. In z direction, cracks spread in the interlayer region with high rate due to the frequently breakage Hbonds network, exhibiting brittle nature. (2) The presence of the central voids ranging from 20 Å to 50 Å weakens the stiffness and cohesive force of the C–S–H gel. It implies that the gel pore at nano-scale structure in the C–S–H gel is detrimental to the mechanical properties. Maximum values of nonlinear displacements are distributed around the central voids and the ‘‘strain concentration’’ occurs at the boundary of the local defective region. (3) Because of the binding constraints’ discrepancy, bending of the calcium silicate sheet can be observed, reflecting the complicated tensile behavior of heterogeneous layered C–S–H gel.
Acknowledgements Financially support from the China Ministry of Science and Technology under Grant 2015CB655104 and the Chinese National Natural Science Foundation (NSF) under Grant 51178230 and 51378269 , Major International Joint Research Project under Grant 51420105015 are gratefully acknowledged. References [1] [2] [3] [4] [5]
Allen AJ, Thomas JJ, Jennings HM. Composition and density of nanoscale calcium silicate hydrate in cement. Nature material 2007;6:311–6. Bazant Z. Size effect in blunt fracture: concrete, rock, metal. J Engng Mech 1984;110(4):518–35. Bonnacorsi E, Merlino S, Taylor H. The crystal structure of jennite Ca9Si6O18(OH)6 8H2O. Cem Concr Res 2004;34(9):1481–8. Bonnaud PA et al. Thermodynamics of water confined in porous calcium–silicate–hydrates. Langmuir 2012;28(31):11422–32. Brough A, Dobson C, Richardson I, Groves G. In situ solid-state NMR studies of Ca3SiO5: hydration at room temperature and at elevated temperatures using 29Si enrichment. J Mater Sci 1994;29(15):3926–40. [6] Chen JJ, Thomas JJ, Taylor HFW, Jennings HM. Solubility and structure of calcium silicate hydrate. Cem Concr Res 2004;34:1499–519.
D. Hou et al. / Engineering Fracture Mechanics 131 (2014) 557–569
569
[7] Cong X, Kirkpatrick R. 29Si and 17O NMR investigation of the structure of some crystalline calcium silicate hydrate. Adv Cem Based Mater 1996;3:133–43. [8] Cong X, Kirkpatrick R. 29Si MAS NMR study of the structure of calcium silicate hydrate. Adv Cem Based Mater 1996;3(3–4):144–56. [9] Costantinide G, Ulm F. The nanogranular nature of C–S–H. J Mech Phys Solids 2006;55(1):64–90. [10] Feldman RF, Sereda P. A model for hydrated Portland cement paste as deduced from sorption-length change and mechanical properties. Matériaux Construction 1968;1(6):509–20. [11] Griffith AA. The phenomena of rupture and flow in solids. Philos Trans R Soc Lond 1921;A(221):163–98. [12] Hamid S. The crystal structure of the 11 A natural tobermorite Ca2.25 Si3 O7.5(OH)1.5H2O. Zeitschrifit Kristallographie 1981;154(3–4):189–98. [13] Hershey A. The elasticity of an isotropic aggregate of anisotropic cubic crystals. J Appl Mech-Trans ASME 1954;21(3):236–40. [14] Hou D, Li Z. Large scale simulation of calcium silicate hydrate (C–S–H) by molecular dynamics. Adv Cem Res 2014. [15] Hou D, Ma H, Zhu Y, Li Z. Calcium silicate hydrate from dry to saturated state: structure, dynamics and mechanical properties. Acta Mater 2014;67:81–94. [16] Janika JA, Kurdowsk W, Podsiadey R, Samset J. Fractal structure of CSH and tobermorite phases. Acta Phys Polonic 2001;100:529–37. [17] Jennings HM. A model for the microstructure of calcium silicate hydrate in cement paste. Cem Concr Res 2000;30(1):101–16. [18] Jennings HM. Refinements to colloid model of C–S–H in cement: CM II. Cem Concr Res 2008;38(3):275–89. [19] Jenq YS, Shah SP. A two parameter fracture model for concrete. J Engng Mech 1985;111(4):1227–41. [20] Ji Q, Pellenq RJM, Van Vliet KJ. Comparison of computational water models for simulation of calcium silicate hydrate. Comput Mater Sci 2012;53(1):234–40. [21] Kaplan ME. Crack propagation and fracture of concrete. J ACI 1961;58(5):591–610. [22] Manzano H et al. Confined water dissociation in microporous defective silicates: mechanism, dipole distribution, and impact on substrate properties. J Am Chem Soc 2011;134(4):2208–15. [23] Merlino S, Bonaccorsi E, Armbruster T. The real structures of clinotobermorite and tobermorite 9 Å. Eur J Miner 2000;12(2):411–29. [24] Merlino S, Bonnacorsi E, Armbruster T. The real structure of tobermorite 11 A: normal and anomalous forms, OD character and polytypic modifications. Eur J Mineral 2001;13(3):577–90. [25] Mindess S, Young J, Darwin D. Concrete. 2nd ed. Prentice Hall PTR; 2003. s.l. [26] Mori T, Tanaka K. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall 1973;21(5):571–4. [27] Muller ACA, Scrivener KL, Gajewicz AM, McDonald PJ. Use of bench-top NMR to measure the density, composition and desorption isotherm of C–S–H in cement paste. Microporous Mesoporous Mater 2013;178:99–103. [28] Murray SJ, Subramani VJ, Selvam RP, Hall KD. Molecular dynamics to understand the mechanical behavior of cement paste. J Transport Res Board 2010;2142(11):75–82. [29] Pellenq RJ et al. A realistic molecular model of cement hydrates. PNAS 2009;106(38):16102–7. [30] Powers T, Brownyard T. Studies of the physical properties of hardened Portland cement paste. PCA Bull, Portland Cem Assoc 1948;22:22. [31] Qian Z, Ye G, Schlangen E, van Breugel K. 3D lattice fracture model: application to cement paste at microscale. Key Engng Mater 2011;452:65–8. [32] Qomi M, Ulm F, Pellenq R. Evidence on the dual nature of aluminum in the calcium–silicate–hydrates based on atomistic simulations. J Am Ceram Soc 2012;95(3):1128–37. [33] Selvam RP, Subramani VG, Murray S, Hall K. Potential application of nanotechnology on cement based materials, 2009; s.l.: s.n. [34] Shahsavari R, Pellenq RJM, Ulm FJ. Empirical force fileds for complex hydrated calcio-silicate layered materials. Phys Chem Chem Phys 2011;13(3):1002–11. [35] Shahsavari R. Hierarchical modeling of structure and mechanics of cement hydrate. PhD thesis of Massachusetts Institute of Technology; 2011. [36] Shahsavari R, Buechler MJ, Pellenq RJM, Ulm FJ. First-principles study of elastic constants and interlayer interactions of complex hydrated oxides: case study of tobermorite and jennite. J Am Ceram Soc 2009;92(10):2323–30. [37] Wecharatana M, Shah S. Slow crack growth in cement composites. J Struct Engng 1982;108(6):1400–13. [38] Youssef M, Pellenq RJM, Yildiz B. Glassy nature of water in an ultraconfining disordered material: the case of calcium silicate hydrate. J Am Chem Soc 2011;133(8):2499–510.