Physical Mechanism of Concrete Damage Under Compression

materials

Article Physical Mechanism of Concrete Damage under Compression

Hankun Liu 1, Xiaodan Ren 2,*, Shixue Liang 3 and Jie Li 2

1 Sichuan Institute of Building Research, Chengdu 610081, China; [email protected] 2 School of Civil Engineering, Tongji University, 1239 Siping Road, Shanghai 200092, China; [email protected] 3 School of Civil Engineering and Architecture, Zhejiang Sci-Tech University, Hangzhou 310018, China; [email protected] * Correspondence: [email protected]; Tel.: +86-21-65981505

Received: 12 September 2019; Accepted: 9 October 2019; Published: 11 October 2019

Abstract: Although considerable effort has been taken regarding concrete damage, the physical mechanism of concrete damage under compression remains unknown. This paper presents, for the first time, the physical reality of the damage of concrete under compression in the view of statistical and probabilistic information (SPI) at the mesoscale. To investigate the mesoscale compressive fracture, the confined force chain buckling model is proposed; using which the mesoscale parameters concerned could be directly from nanoindentation by random field theory. Then, the mesoscale parameters could also be identified from macro-testing using the stochastic damage model. In addition, the link between these two mesoscale parameters could be established by the relative entropy. A good agreement between them from nano- and macro- testing when the constraint factor approaches around 33, indicates that the mesoscale parameters in the stochastic damage model could be verified through the present research. Our results suggest that concrete damage is strongly dependent on the mesoscale random failure, where meso-randomness originates from intrinsic meso-inhomogeneity and meso-fracture arises physically from the buckling of the confined force chain system. The mesoscale random buckling of the confined force chain system above tends to constitute the physical mechanism of concrete damage under compression.

Keywords: concrete; damage; compression; random ﬁeld; nanoindentation; multiscale

1. Introduction Concrete, a mixture of Portland cement, water, sand and aggregate, hydrated to form cementitious material with micro-crack, void and inhomogeneous [1], exhibits nonlinearity and randomness of mechanical properties. Under loading, concrete and its properties suffer from deterioration, which could be regarded as damage. The concept of damage is firstly developed by Kachanov [2], introduced into concrete material by Dougill soon afterwards [3]. Over decades, concrete nonlinearity, especially the characterization of properties softening, has been modeled by damage mechanics, whose branches include continuum damage mechanics [4,5] and stochastic damage mechanics [6,7]. The former one regards macroscopic homogeneity, however, the latter one pays more attention to mesoscopic inhomogeneity and the progressive transition between different levels. Mazars modeled the degradation of concrete by bi-scalar model [4], based on this, Wu and Li proposed a plastic damage model by introducing the elastoplastic damage release rate [5]. Moreover, to discover the physical mechanism of concrete damage, the latter one, idealized as a parallel fiber bundle including two levels, has been always applied due to its simplicity in reproducing randomness and nonlinearity. Actually, concrete could be always regarded as a set of parallel small concrete rods connected on two ends and deform compatibly. At the macroscale, concrete could be modeled as a fiber bundle, giving a smooth curve

Materials 2019, 12, 3295; doi:10.3390/ma12203295 www.mdpi.com/journal/materials Materials 2019, 12, 3295 2 of 17 reflecting the averaging response. While at the mesoscale, a small concrete rod could be considered as a fiber, exhibiting a kind of elastic-brittle relationship, referring to the previous researches [6–8]. Up to now, the fiber bundle model has been used for modeling concrete damage including tensile and compressive damage, under the static loading [7] and the dynamic loading [8]. In addition, Bazant and Pang also systematically investigated the size effect of concrete materials based on the bundle model [9]. It is noted that the physical mechanism for concrete tensile damage could be easily disclosed by the bundle model; however, the physical mechanism for concrete compressive damage remains a mystery. To disclose the physical mechanism, the physical experiment could always be considered as a direct and effective way. Although it is generally realized that the nonlinearity and randomness of the mechanical properties for concrete are directly related to the fracture of the meso-element and its accumulation, the direct access to the knowledge regarding the local properties has never been provided until the 1970s. Hereafter, Beaudoin and Feldman systematically studied the properties of the autoclaved calcium silicate systems including elastic modulus, micro-strength, micro-hardness and the relationships between the properties by micro-hardness testing, and additionally, suggested the linear relationship between microscale compressive strength and micro-hardness [10]. From Igarashi’s research, the microscale compressive strength and the micro-hardness both increased with increasing curing age and decreasing water to cement ratio, meanwhile, the same relationship as Beaudoin and Feldman was also verified [11]. Zhu and Bartos proposed a novel microindentation method continuously monitoring load and displacement, to assess the elastic modulus and microhardness of the interfacial zone of reinforced concrete [12]. According to the research of Buckle and Durst et al. [13,14], small indentation depths lead to mechanical phase properties, while greater indentation depths result in homogenized material properties. Georgios and Ulm proposed a novel method by means of grid indentation, using deconvolution technique to identify in situ two calcium–silicate–hydrates (C–S–H) phase (low-density C–S–H and high-density C–S–H) [15–17]. To anticipate the nature of strength, the dual-nanoindentation technique was proposed to assess the strength of bone and cohesive-frictional materials, suggesting the nanogranular friction responsible for the increased intrinsic resistance in compression [17–19]. Vandamme et al. also investigated the nanogranular origin of concrete creep by nanoindentation [20]. Liu et al. combined nanoindentation with random field modeling to study the probabilistic and statistical properties of concrete materials, suggested the multiscale SPI of concrete in a comprehensive manner and proposed a multi-scale random media model for concrete [21]. In addition, Mondal et al. studied the topological structure of cementitious materials using nanoindentation [22]. In short, microhardness testing [10,11], microindentation [12] and the nanoindentation technique [13–22] appeared successfully, using that which researchers have investigated; the fundamental knowledge including topological structure, mechanical and physical information, SPI, as well as their interrelationships. With the development of material science and technology, the materials genome could be discovered. Through the usage of this, the elementary physical properties and fundamental structural characteristics could also be predicted [23]. Usually, for concrete, a rather complex system, it is crucial and feasible to use the SPI of the intrinsic structure and properties at the nanoscale or mesoscale to investigate macroscopic properties. Unfortunately, the physical mechanism of concrete damage or constitutive relationship under compression remains unknown, despite the great achievements on continuum damage mechanics which fails to reveal the mechanism, and on stochastic damage mechanics (the bundle model, etc.) which has failed to be verified by the physical experiment until nowadays. On the basis of the recent progress on nanoindentation, stochastic mechanics and random media modeling for concrete, the paper focuses on the link between concrete damage and materials SPI at the lower scale, and on the origin of concrete damage. In this research, the damage of concrete is investigated when subjected to compressive loading. Based on the previous research achievement, it is hypothesized that the fracture of “concrete fiber” originates at the mesoscale. On the one hand, nanoindentation tests are conducted on each constituent of concrete: hardened cement paste (HCP), interfacial transition zone (ITZ) and aggregate. By application Materials 2019, 12, x FOR PEER REVIEW 3 of 17 Materials 2019, 12, 3295 3 of 17 aggregate. By application of the confined force chain buckling model proposed in this paper, ofreconstruction the confined force technique chain [21]buckling and modelrandom proposed field theory in this, paper, the SPI reconstruction of fracture behavior technique could [21] and be randomobtained field for theory, concrete the fiber SPI of at fracture the mes behavioroscale. On could the be other obtained hand, for the concrete SPI of fibermeso at-parameters the mesoscale. of Onconcrete the other could hand, also the be SPIrecognized of meso-parameters from macro- oftesting concrete employing could also the be stochastic recognized damage from macro-testing model. Thus, employingthis hypothesi thes stochastic could be damage proven, model. as long Thus, as the this SPI hypothesis of the mes couldo-fiber be proven, from the as longnanoindentation as the SPI of thecoincides meso-fiber with fromthat recognized the nanoindentation from the macro coincides-testing. with that recognized from the macro-testing.

2. Materials and Methods

2.1.2.1. Materials and Preparation The material prepared herehere waswas ordinaryordinary concreteconcrete withwith water:water: cement:cement: sand: with an aggregate ratio of 0.4:1:2:5. The bars were made measuring 0.1 m 0.1 m 0.3 m, and hydrated with the humidity ratio of 0.4:1:2:5. The bars were made measuring 0.1× m × 0.1× m × 0.3 m, and hydrated with the ofhumidity 95% at the of room 95% temperature at the room for temperature three months, for which three were months, used for which uniaxial were compression used for testinguniaxial at thecompression macroscale. testing Then, at the the prisms macroscale were sliced. Then, into the small prisms specimens were sliced with approximate into small specimens dimensions with of 0.02approximate m 0.02 mdimensions0.005 m, of which 0.02 werem × 0.02 prepared m × 0.005 for nanoindentation. m, which were Afterprepared embedding for nanoindentation. into the epoxy × × resin,After grinding embedding and into polishing the epoxy with silicon resin, grcarbideinding papers, and polishing and ultrasonically with silicon cleaning, carbide the papers, samples and for nanoindentationultrasonically cleaning, (Figure 1 the) were samples prepared. for Detailsnanoindentation of sample (Figure preparation 1) were for nanoindentation prepared. Details have of beensample described preparation in the for previous nanoindentation study [21 ].have been described in the previous study [21].

(a) (b)

(c)

Figure 1.1. SamplesSamples for for testing testing in nanoindenter: in nanoindenter (a): specimen (a) specimen 1; (b) specimen 1; (b) specimen 2; and ( c2); specimens and (c) specimens installed. installed. 2.2. Nanoindentation 2.2. Nanoindentation The equipment conducting nanoindentation in the present paper is the NanoTest Vantage system (FigureThe2, from equipment Micro Materials conducting Limited nanoindentation in Wrexham, UK) in the to o presentﬀer nanomechanical paper is theand NanoTest nanotribological Vantage system (Figure 2, from Micro Materials Limited in Wrexham, UK) to offer nanomechanical and

Materials 2019, 12, x FOR PEER REVIEW 4 of 17 nanotribologicalMaterials 2019, 12, 3295 tests, with electromagnetic load application, with a maximum load of 500 4mN, of 17 load resolution of 3 nN and displacement resolution of 0.002 nN. A series of nanoindentation tests were conducted with a Berkovich tip, with a maximum depth of 300 nm, loading and unloading rate oftests, 0.2 with mN/s electromagnetic and holding time load application, of 15 s. According with a maximum to the approach load of 500of Oliver mN, load and resolution Pharr [24 of], indentation3 nN and displacement hardness H and resolution indentation of 0.002 mod nN.ular A M series were ofobtained nanoindentation from loading tests-unloading were conducted curves. Furthermorwith a Berkoviche, Young’s tip, with modulus a maximum linking depth to the of elastic 300 nm, constants loading of and specimen unloading and rateindenter of 0.2 [2 mN5], /coulds and beholding expressed time as of 15 s. According to the approach of Oliver and Pharr [24], indentation hardness H and indentation modular M were obtained from loading-unloading curves. Furthermore, Young’s 2 2 modulus linking to the elastic constants of1 specimen 1 and1 indenteri [25], could be expressed as   (1) MEEi 2 2 1 1 ν 1 νi where E and  are Young’s modulus and= Poisson’s− + ratio− of the indenter employed with a given(1) i i M E Ei value of 1140 GPa and 0.07; E and  are that of the tested materials, and the Poisson’s ratio is 0.2. whereNanoindentatiEi and νi areon Young’s on each modulus constituent and Poisson’s of concrete ratio was of theconducted indenter in employed the present with paper a given, and value the dimensionsof 1140 GPa of and the 0.07; indentE and latticeν are for that HCP, of the ITZ tested and aggregate materials, were and theall 25 Poisson’s × 20. ratio is 0.2.

(a) (b)

FigFigureure 2 2.. NanoTestNanoTest Vantage testing system:system: ( a)) a appearance;ppearance; and ( b) i internalnternal d details.etails.

2.3. MacroNanoindentation-testing on each constituent of concrete was conducted in the present paper, and the dimensions of the indent lattice for HCP, ITZ and aggregate were all 25 20. A total of seven prism specimens subjected to uniaxial compressive× loading were investigated by2.3. an Macro-Testing electro-hydraulic servo-controlled concrete testing system from MTS Systems Corporation in Eden Prairie, MN, USA (Figure 3). The hinge on the top and the scale marks on the bottom are used to guaranteeA total ofthe seven axial prismcompression specimens and subjectedthe accurate to uniaxialcentration compressive of the tested loading samples. were The investigated stiffness of 1.1by an× 10 electro-hydraulic10 N/m is enough servo-controlled to provide the closed concrete-loop testing controlled system compression from MTS Systemsand the Corporationdata collection in accuracyEden Prairie,. A pair MN, of USA extensometers (Figure3). Thewere hinge installed on the on top opposite and the sides scale of marks the specimen on the bottom shown are in used Figure to 3guarantee, which collected the axial the compression axial displacement and the data. accurate And centrationthe strain rate of the was tested 10−5 which samples. guaranteed The stiﬀ aness static of 1.1 1010 N/m is enough to provide the closed-loop controlled compression and the data collection loading.× accuracy. A pair of extensometers were installed on opposite sides of the specimen shown in Figure3, 5 which collected the axial displacement data. And the strain rate was 10− which guaranteed a static loading.

Materials 2019, 12, 3295 5 of 17 Materials 2019, 12, x FOR PEER REVIEW 5 of 17 Materials 2019, 12, x FOR PEER REVIEW 5 of 17

Figure 3. MacroMacro-testing-testing system of u uniaxialniaxial loading loading.. Figure 3. Macro-testing system of uniaxial loading. 3. Theoretical Theoretical,, Experimental and Numerical Approach 3. Theoretical, Experimental and Numerical Approach 3.1. Shear Fracture Strain (SFS) from Nanoindentation 3.1. Shear Fracture Strain (SFS) from Nanoindentation 3.1. Shear Fracture Strain (SFS) from Nanoindentation 3.1.1. Force Chain Based Modeling for Hardness 3.1.1. Force Chain Based Modeling for Hardness 3.1.1.Due Force to Chain the nanogranular Based Modeling nature for Hardnes of C–S–Hs [26], the aggregative particles of C–S–H could be Due to the nanogranular nature of C–S–H [26], the aggregative particles of C–S–H could be assumedDue to to yield the nano the conﬁnedgranular three-particle nature of C force–S–H chain [26], buckling the aggregative mechanism particles which ofwas C ﬁrstly–S–H proposedcould be assumed to yield the confined three-particle force chain buckling mechanism which was firstly assumedby Tordesillas to yield and the Muthuswamy confined three [27].-particle The schematic force chain is shown buckling in Figure mechanism4. Based onwhich the force was firstlychain proposed by Tordesillas and Muthuswamy [27]. The schematic is shown in Figure 4. Based on the proposedtheory, a connectionby Tordesillas could and be Muthuswamy established from [27 nanoscale]. The schematic to mesoscale. is shown in Figure 4. Based on the force chain theory, a connection could be established from nanoscale to mesoscale. force chain theory, a connection could be established from nanoscale to mesoscale.    1 n 1 n E E s  s  1 n  s 1 n  s

1 1  n  cos  1 1  n  cos  Undeformed Deformed Geometric relationship Undeformed Deformed Geometric relationship

Figure 4. Three-particle force chain model. Figure 4. Three-particle force chain model Figure 4. Three-particle force chain model For simplicity and possibility, only the contact force between particles and the lateral supporting For simplicity and possibility, only the contact force between particles and the lateral forceFor from simplicity the surrounding and possibility, weak forceonly chainsthe contact are assumed force between for C–S–H particles in the and present the lateral paper. supporting force from the surrounding weak force chains are assumed for C–S–H in the present supportingThe assumption force couldfrom the be surroundingextended to cementweak force paste, chains ITZ are and assumed aggregate, for becauseC–S–H in of the the present similar paper. The assumption could be extended to cement paste, ITZ and aggregate, because of the paper.characteristics The assumption among the could constituents be extended of concrete. to cement Next, the paste contact, ITZ law andis aggregate, introduced because to describe of the the similar characteristics among the constituents of concrete. Next, the contact law is introduced to similarkey features characteristics in the force among chain model. the constituents of concrete. Next, the contact law is introduced to describe the key features in the force chain model. describe the key features in the force chain model. Contact Models Contact Models The homogenized normal stress and tangential stress between particles are respectively The homogenized normal stress and tangential stress between particles are respectively

Materials 2019, 12, 3295 6 of 17

Contact Models The homogenized normal stress and tangential stress between particles are respectively

σn = Enεn (2)

σt = Etεt (3) where En and Et denote the normal and tangential modulus respectively; εn and εt are the normal and tangential strain. To reflect the constraint effect of the weak force chain network, the applied confined pressure could be expressed as follows. σs = Esεs (4) where Es and εs denote the confined modulus and the corresponding stain.

Averaged Potential Energy Density Considering a certain region (at the mesoscale) of cement paste in the surrounding area of the indenter, when applying compression by a strain ε, the averaged potential energy density ep could be expressed as follows: 1 2 1 2 1 2 ep = u w = Enε + Etε + Esε σε (5) − 2 n 2 t 2 s − where u is the stored energy density; w is the work density with the stress σ. As shown in Figure4, the strain εn, εt and εs could be rewritten in terms of θ and εn as follows:

ε = (1 ε ) sin θ sin θ (6) t − n ≈ ε = (1 ε ) sin θ sin θ (7) s − n ≈ ε = 1 (1 εn) cos θ (8) − − Substituting Equations (6)–(8) into Equation (5), one obtains the potential energy density:

1 2 1 2 1 2 ep = En(εn) + Et(sin θ) + Es(sin θ) σ[1 (1 εn) cos θ] (9) 2 2 2 − − − It is evidently observed that the normal strain and the tangential strain could be decoupled.

Critical Load According to the principle of resident potential energy, the partial derivative of the averaged potential energy density with respect to each degree of freedom should be zero, we get

∂ep ∂ep = 0, = 0 (10) ∂θ ∂εn

Substituting Equation (9) into Equation (10), and solving Equation (10), the relationship between the stress and the normal strain yields Equation (11).

(Et + Es) cos θ σ cos θ σ = , εn = (11) 1 εn En − From Equation (11), an apparently stable path for the conﬁned force chain could be achieved when the force chain structure keeps straight from the beginning of loading, which corresponds to

σ = Enεn (12) Materials 2019, 12, 3295 7 of 17

Solving Equation (11), σ, εn and ε could be obtained, which yields  s    En  (Et + Es)  σ = 1 1 4 cos θ  (13) 2 cos θ − − En 

 s    1 (Et + Es)  εn = 1 1 4 cos θ  (14) 2 − − En   s    1 (Et + Es)  ε = 1 1 + 1 4 cos θ  cos θ (15) − 2 − En 

Another stable path could be acquired when θ is zero and this critical point with the stress and strain of Equations (16) and (17) corresponds to a buckling load of the force chain system.  s    En  (Et + Es)  σcr = 1 1 4  (16) 2  − − En 

 s    1 (Et + Es)  εcr = εn = 1 1 4  (17) 2 − − En 

To conduct Taylor expansion for Equation (16) and Equation (17) at Et + Es/En = 0, the results obtained are as follows.  s    1 (Et + Es)  Et + Es εcr = 1 1 4  (18) 2 − − En  ≈ En

σcr = Enεcr Et + Es (19) ≈ As easily seen from Equation (19), a mixed-mode of shearing (Et) and constraint (Es) is included in the critical load, which means that the critical compressive load of the meso-element is attributed to the meso-element itself and the surrounding materials. With the framework of hardness theory, the hardness value H is the pressure at a limiting condition where the pressure keeps constant with increasing load [28]. Meanwhile, for indentation on cement paste, there is also a mixed fracture mode by cutting and hydrostatic pressure under the indenter at the limiting state [11]. Based on the statement above, the indentation hardness could be equivalent to the constrained compressive strength σc of the meso-element conﬁned by the surrounding material around the indenter:

σc = σcr Et + Es = H (20) ≈ 3.1.2. Constrained SFS Over decades, researchers have investigated the relationship between the strength and the micro-hardness, with the conclusion that for cementitious materials (frictional materials), the hardness and the yield stress relationship of the form H/Y were reported on the order of 20–30 [29]. Similarly, in Ref. [11], the ratio was found to be 2.7–3 more order of magnitude than that of metals, which was in the range of 30–60. As known from early on, the ratio of metal discussed above, named “the constraint factor”, was evidently less than that of cementitious materials, due to the frictional eﬀect on the hardness within the cohesive-frictional materials. Materials 2019, 12, 3295 8 of 17

Based on the previous researches [11,29] and Equation (20), the relationship between the hardness H and the compressive strength σy was followed by introducing a constraint factor C:

H C = (21) σy

With the assumption of elastic-brittle property of mesoscale element aforementioned [6–8], the SFS ∆1,s(x) from nanoindentation could be rewritten as

σy H ∆ ∆ (x) = = = con (22) 1,s E CE C where ∆con is defined as the constrained SFS. Combining Equation (1) and Equation (22), the constrained SFS could be rewritten as ! H H 1 1 ν 2 ∆ = = i (23) con 2 − E 1 ν M − Ei − 3.1.3. Random Field Modeling and Statistical Modeling Due to the attribute of a sufficient large degree of disorder, the knowledge of the probabilistic characteristics of concrete is fundamental to understanding the intrinsic random microstructure, even the nanostructure. Generally, random field theory deals effectively with the complex distributed disordered system [30]. As the random heterogeneous materials, Torquato has made considerable progress on 2D and 3D microstructure characterization, also on the relationship between mechanical properties and microstructure [31]. However, it is very difficult to use 2D or 3D modeling to investigate the physical mechanism of concrete damage, due to more parameters and complex simulation execution. To make it convenient and effective, the 1D random field is still adopted in the present paper. The random field could be defined as homogeneous, as long as the mean mX tj and the covariance RX tj, tj + τ as follows keep constant with the variance of space, in other words, these values only depend on the relative distance τ. h i mX tj = E X tj (24) h i RX tj, tj + τ = E X tj X tj + τ (25) where E[] is the expected operator, X tj is the observation on the random series with respect to tj. To investigate the statistical characteristics, Kolmogorov–Smirnov test (referred to as the K–S test) could be adopted to acquire the probabilistic density function (PDF) of each point of the random series (containing six sections). To execute the K-S test, the main procedure is outlined as follows:

(1) Choose a sample Xi from the population X and rearrange sample values xi in increasing order of magnitude.

(2) Compute the observed cumulative distribution function (CDF) Fn(xi) at each ordinal sample value. (3) Estimate the parameters of the hypothesized distribution as described below based on the observed data and determine the theoretical CDF F(xi) at the same sample value above using the hypothesized distribution.

(4) Form the diﬀerences Fn(xi) F(xi) , and calculate the statistics: − n n o D = max Fn(xi) F(xi) (26) i=1 −

(5) Select a value of α and determine the critical value Dα. (6) Accept or reject the testing hypothesis by comparing D and Dα. Materials 2019, 12, 3295 9 of 17

The procedure stated above is the classic K-S test process. However, since the critical value is approximate, the null hypothesis is usually rejected or accepted by comparing the returned P value and the signiﬁcance level α. In this study, the hypothesized PDF commonly used in civil engineering could be made including normal distribution, lognormal distribution, Weibull distribution and gamma distribution. Then, the estimated PDF could be acquired by executing the K-S test. According to Ref. [21], there is only a little diﬀerence between PDFs using the mean estimated parameters of six sections and the parameters given by a maximum possibility criterion proposed by the authors. Therefore, in this paper, the PDF with the mean estimated parameters of six points could be referred to as the best estimate. By conducting the probabilistic and statistical modeling on the constrained SFS, the 1D PDF of constrained SFS for each constituent of concrete could be obtained.

3.2. SFS from the Macro-Testing Generally, complex global behaviors could be captured on the basis of the fiber bundle model whose individual element is endowed with a simple response (elastic-brittle prosperity shown in Figure5). Under compressive loading, one of these fibers would fail when the overall strain exceeds the random SFS denoted by the random variable ∆2,s(x), which could be considered to be a homogenous lognormal random field with the mean λ and the standard deviation ζ in Refs. [6–8]. According to Refs. [7,8], the damage of the fiber bundle represented by d(ε) could be defined as follows

Z 1 d(ε) = H[ε ∆2,s(x)]dx (27) 0 − where H[ ] is the Heaviside function, ε is the elastic strain, ∆ s(x) is the 1D random ﬁeld for fracture · 2, strain, and x is the spatial coordinate of the meso-ﬁber. In the present work, the focus would be on the expected value of the damage variable d(ε) given as

E[d] = F(ε) (28) where F(ε) denotes the ﬁrst-order cumulative distribution function of ∆2,s(x). The 1D expected stress and strain relationship could be expressed as follows

E[σ] = [1 F(ε)]E ε (29) − 0 where σ denotes the stress, and E0 denotes the initial elastic modulus.

3.3. Multiscale Approach Figure5 shows the topological structure, physical model and mechanical properties at three scales. At the nanoscale, a confined force chain buckling model was established for force-chain based materials, which could be extended to the mesoscale using the principle of resident potential energy, to investigate the relationship between the compressive strength and the hardness for the mesoscale concrete element. Meanwhile, at the mesoscale, the mesoscale parameters for concrete damage under compression could be not only from the identification combining model results with macro testing results but also from the direct experiment (nanoindentation). Then, further than that, the former one could be regarded as a traditional method: the parameters at the lower scale could be recognized provided that a reasonable model and the macroscale experiment results are given. The latter pays more attention to the verifiability of the model and the development of the microscale testing techniques. Actually, the fiber bundle model would be an optimal and convenient model to connect these three scales. Materials 2019, 12, 3295 10 of 17 Materials 2019, 12, x FOR PEER REVIEW 10 of 17

(From Ref. [22])



Es   s

Nano-particle model Meso-model Macro-model

n E n  n

t E t  t

s E s  s 1,s x   con C

Meso-property Macro-response

Nano-level Meso-level Macro-level

Figure 5. Concrete modeling at the nano-, meso- and macro-scales. Figure 5. Concrete modeling at the nano-, meso- and macro-scales. On the one hand, each concrete constituent (HCP, ITZ and aggregate) could be modeled as a randomOn the ﬁeld one using hand, the each observed concrete values constituent from nanoindentation (HCP, ITZ and tests. aggregate Performing) could “maximum be modeled possibility as a random field using the observed values from nanoindentation tests. Performing “maximum criterion” [21], the knowledge of PDF for constrained SFS ∆con could be provided, with which SFS  possibility∆1,s(x) could criterion” also be [21], obtained the knowledge using Equation of PDF (22) for (in constrained Section 3.1). SFS Then, con employing could be stochastic provided damage, with whichtheory, SFS the  statistical1,s  x could characteristics also be obtained of SFS using∆2,s(x Equation) could also (22) be (in obtained Section including 3.1). Then the, employing mean and the standard deviation of a homogenous lognormal random ﬁeld, by comparing the macroscale stochastic damage theory, the statistical characteristics of SFS  2,s  x could also be obtained experimental stress and strain curves with the model results (in Section 3.2). Finally, the constraint including the mean and the standard deviation of a homogenous lognormal random field, by factor in Equation (22) could be recognized by comparing ∆ s(x) with ∆2,s(x) through the relative comparing the macroscale experimental stress and strain curves1, with the model results (in Section entropy (in Section 4.3). 3.2). Finally, the constraint factor in Equation (22) could be recognized by comparing 1,s  x with

4.2,s Results x through and Discussion the relative entropy (in Section 4.3).

4.1. Concrete Damage SPI from Nanoindentation 4. Results and Discussion Specifically, to investigate the SPI of concrete damage, the phase SPI could be generated firstly 4.1.due Concrete to the obviousDamage boundSPI from among Nanoindentation each constituent of concrete. Combined with the phase SPI, concrete SPI could be reproduced by the reproduction technique [21]. Specifically, to investigate the SPI of concrete damage, the phase SPI could be generated firstly due4.1.1. to Phasethe obvious SPI and bound Random among Field each Theory constituent of concrete. Combined with the phase SPI, concrete SPI could be reproduced by the reproduction technique [21]. To characterize the probability distribution of properties from nanoindentation, three zones 4.1.1.including Phase HCP,SPI and ITZ Random and aggregate Field Theory were selected randomly shown in Figure6a before nanoindentation. Following random field modeling of nanoindentation results, it leads to a sample number of 100 for To characterize the probability distribution of properties from nanoindentation, three zones HCP, aggregate and ITZ, respectively, with respect to the constrained SFS ∆con. In Figure6b–d, they including HCP, ITZ and aggregate were selected randomly shown in Figure 6a before nanoindentation. Following random field modeling of nanoindentation results, it leads to a sample

Materials 2019, 12, x FOR PEER REVIEW 11 of 17

Materials 2019, 12, 3295 11 of 17 number of 100 for HCP, aggregate and ITZ, respectively, with respect to the constrained SFS con . In Figure 6b–d, they show the samples of the random field, and the corresponding mean and standard showdeviation the samples for HCP, of the ITZ random and aggregate, field, and the respectively. corresponding From mean the firstand standard and second deviation-order for statistic HCP,al ITZcharacteristics and aggregate,, the respectively. mean value From and the the standardfirst and second-order deviation calculated statistical both characteristics, keep in constant the mean. It is valueclear and that the the standard 1D random deviation filed for calculated the concrete both constituent keep in constant. herein is It belonging is clear that to the a stationary 1D random random filed forprocess. the concrete Notably, constituent concrete herein constituents is belonging could to a be stationary considered random to be process. homogeneous Notably, concrete from the constituentsprobabilistic could and be consideredstatistical tostandpoint be homogeneous, although from concrete the probabilistic is well and-known statistical as standpoint, a kind of althoughinhomogeneous concrete material. is well-known Actually, as a kindconcrete of inhomogeneous could also be regarded material. as Actually, homogenous, concrete provided could also that beone regarded consider ass homogenous,the complex materials provided as that random one considers media. the complex materials as random media.

5 x 10 3.5 Mean 3 Mean+SD Mean-SD 2.5 ) 6 - 2 (10 n

o 1.5 c  1

0.5

0 0 5 10 15 20 t (m) (a) (b) 5 5 x 10 x 10 3.5 3.5 Mean Mean 3 Mean+SD 3 Mean+SD Mean-SD Mean-SD 2.5 2.5 ) ) 6 6 - 2 - 2 (10 (10 n n

o 1.5 o 1.5 c c   1 1

0.5 0.5

0 0 0 5 10 15 20 0 5 10 15 20 t (m) t (m) (c) (d)

FigureFigure 6. 61D. 1D random random ﬁled filed observations: observations: (a ()a the) the indent inden zones;t zones; the the samples, samples, the the mean mean and and the the standard standard deviationdeviation for for the the constrained constrained SFS SFS of of (b ()b HCP,) HCP, (c )(c ITZ) ITZ and and (d ()d aggregate.) aggregate.

ByBy performing performing “maximum “maximum possibility possibility criterion” criterion” [ 21[21],], the the 1D 1D PDF PDF for for the the random random ﬁeld field could could be be obtained.obtained. It It shows shows the the optimal optimal parameters parameters in Table in Table1 for the1 for 1D the PDF 1D of thePDF random of the ﬁeldrandom by the field simplex by the methodsimplex [32 method,33]. Figure [32,337a–c]. Fig givesure the7a– histogramsc gives the ofhistograms the constrained of the SFSconstrained for each concreteSFS for each constituent concrete togetherconstituent with togetherthe theoretical with PDF the using theoretical the optimal PDF usingparameters the optimal in Table 1 parameters. The comparisons in Table between 1. The thecomparisons histograms b andetween the theoreticalthe histograms PDFs and show the better theoretical agreement. PDFs show better agreement.

TableTable 1. 1.Optimal Optimal parameters parameters for for the the 1D 1D PDF. PDF.

OptimalOptimal Parameters Parameters ConstituentsConstituents PropertiesProperties Distribution Type Type MeanMean Value Value StandardStandard Deviation Deviation

HCP ∆Δconcon Lognormal distribution distribution 10.8210.82 0.430.43 ITZ ∆Δconcon Lognormal distribution distribution 10.7210.72 0.290.29 Agr ∆Δconcon Normal distribution distribution 111730111730 2391023910

Materials 2019, 12, 3295 12 of 17 Materials 2019, 12, x FOR PEER REVIEW 12 of 17

-5 -5 x 10 x 10 3 3.5

2.5 3 2.5 2 2 1.5 1.5 1 1 Probability density Probability density Probability 0.5 0.5

0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 -6 5 -6 5  (10 ) x 10  (10 ) x 10 con con (a) (b) -5 -5 x 10 x 10 2 2

1.5 1.5

1 1

Probability density Probability 0.5 density Probability 0.5

0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 -6 5 -6 5  (10 ) x 10  (10 ) x 10 con con (c) (d)

FigureFigure 7. 7Reconstruction. Reconstruction of of the the constrained constrained SFS SFS for for concrete: concrete: histograms histograms and and theoretical theoretical probability probability densitydensity curves curves for for (a ()a HCP,) HCP, (b )(b ITZ,) ITZ, (c )(c aggregate,) aggregate, and and (d )(d concrete) concrete reconstructed reconstructed with with Equation Equation (30). (30).

4.1.2.4.1.2. Synthesis Synthesis Technique Technique of of Concrete Concrete SPI SPI AsAs mentioned mentioned above, above, each eachconstituent constituent of concrete of concrete materials materials exhibits randomness, exhibits randomness, which constitutes which theconstitutes complex, the distributed complex disordered, distributed system. disordered The PDFsystem. of concrete, The PDF regarded of concrete, as arega randomrded mediaas a random [21], couldmedia be [ directly21], could obtained be directly by adding obtained the by individual adding the PDF, individual expressed PDF, as followsexpressed as follows

Xn n x f  x i  1,2,3 φ(x)= fiφi(ix) i   i = 1, 2, 3 (30)(30) i 1 i=1

where x denotes the property of concrete or its constituent; fi denotes the volume fraction of where x denotes the property of concrete or its constituent; fi denotes the volume fraction of HCP, ITZ HCP, ITZ or aggregate;   x denotes the PDF of the concrete constituent;   x denotes the or aggregate; φi(x) denotes thei PDF of the concrete constituent; φ(x) denotes the reconstructed concrete PDF.reconstructed According concrete to the previously PDF. Accordin reportedg to result the previously [21], a volume reported fraction result ratio [21 of], HCP: a volume aggregate: fraction ITZ ratio is aroundof HCP: 0.611:0.387:0.002, aggregate: ITZ and isthe around corresponding 0.611:0.387:0.002 random media, and distributions the corresponding of concrete random are displayed media indistributions Figure7d. It is of evident concrete that are the displayed histograms in and Figure the theoretical7d. It is evident PDFs of that the theconstrained histograms SFS and are in the closetheoret agreementical PDFs with of the each constrained other. SFS are in close agreement with each other. OneOne may may argue argue thatthat thethe optimaloptimal distributionsdistributions listed listed in in Table Table 1 remainremain subjective. subjective. However, However, the thetheoretical theoretical distribution distribution obtained obtained could could really really model model the the primary characteristicscharacteristics forfor each each constituent, constituent, eveneven concrete. concrete. In In the the meantime, meantime, it it is is really really a a choice choice for for engineers engineers and and researchers researchers to to statistically statistically and and probabilisticallyprobabilistically model model concrete concrete from from a practical a practical point ofpoint view. of Especially, view. Especially, the distribution the distribution of concrete of shownconcrete in Figure shown7d in provides Figure the7d provide physicals realitythe physical for concrete reality damage for concrete under damage compression under in compression Section 4.2. in Section 4.2.

4.2. Concrete Damage SPI from Macro-testing To validate the PDF of SFS from nanoindentation, the behaviors of the total seven prism specimens measuring 0.1 m × 0.1 m × 0.3 m subjected to uniaxial compressive loading were

MaterialsMaterials 2019 2019, ,12 12, ,x x FOR FOR PEER PEER REVIEW REVIEW 1313 of of 17 17 Materials 2019, 12, 3295 13 of 17 investigatedinvestigated by by MTS MTS in in this this research.research . Figure Figure 8a8a showsshows the the uniaxial uniaxial test test results results of of the the concrete concrete specimensspecimens atat thethe macroscale.macroscale. ItIt isis observedobserved fromfrom FigureFigure 8a8a thatthat allall thethe stress-strainstress-strain responsesresponses areare displayed,displayed,4.2. Concrete and and Damage they they appear ap SPIpear from randomly. random Macro-testingly .Taking Taking expectation expectation of of the the stress stress with with respect respect to to the the strain, strain, the the experimentalexperimentalTo validate mean mean the stress PDFstress of and and SFS strain strain from nanoindentation,relationshiprelationship co coulduld the be be behaviors also also plotted. plotted. of the By By total comparing comparing seven prism the the theoretical theoretical specimens meanmeanmeasuring stress-strainstress 0.1-strain m resultresult0.1 m fromfrom0.3 Equation mEquation subjected (29)(29) to withwith uniaxial thethe experimental compressiveexperimental loading result,result, thethe were mesoscalemes investigatedoscale parametersparameters by MTS × × arearein this foundfound research. toto bebe E FigureE == 37.0037.008a showsGPa,GPa, λλ the == 7.62 uniaxial7.62 andand testξξ == results0.52.0.52 . TheThe of thetheoreticaltheoretical concrete andand specimens experimentalexperimental at the macroscale. resultsresults areare displayeddisplayedIt is observed inin FigureFig fromure 8b, 8 Figureb ,which which8a indicates thatindicates all the thatthat stress-strain thethe ththeoreticaleoretical responses resultresult agreesagrees are displayed, wewellll withwith and thethe experimentalexperimental they appear one.one.randomly. FromFrom Takingthe the failure failure expectation patterns, patterns, of thethe the stress classic classic with failur failure respecte pattern pattern to the with withstrain, a a themajor major experimental diagonal diagonal crack mean crack stresscould could and be be observedobservedstrain relationship shown shown in in couldFigureFigure be 9. 9. also plotted. By comparing the theoretical mean stress-strain result from EquationInIn the the (29) present present with thepaper, paper, experimental the the focus focus result, wouldwould the be be mesoscale onon th thee mean mean parameters stress stress and andare foundstrain strain torelationship, relationship, be E = 37.00 from from GPa, whichwhichλ = 7.62 thethe and parameters parametersξ = 0.52. including Theincluding theoretical the the mean mean and experimentalvalue value and and the the results standard standard are displayeddeviation deviation inofof Figure thethe distribution distribution8b, which couldcouldindicates bebe recognized.recognized. that the theoretical However,However, result thethe agrees standardstandard well deviatdeviation withion the stress experimentalstress andand strainstrain one. relationshiprelationship From the failure couldcould patterns, alsoalso bebe plottedplottedthe classic in in the failure the figure, figure, pattern which which with would would a major leadlead diagonal to to thethe crack relative relative could length length be observed of of the the shownrandom random in field fieldFigure for for9. concrete concrete materials.materials. That That is is also also a a key key point point deserving deserving further further research. research. 5050 5050 TestTest results results Exp.Exp. Mean Mean Exp. Mean Theory Mean 4040 Exp. Mean 4040 Theory Mean

3030 3030 (MPa) (MPa) (MPa) (MPa)   σ 20 σ 2020 20

1010 1010

00 00 00 10001000 20002000 30003000 40004000 50005000 60006000 00 10001000 20002000 30003000 40004000 50005000 60006000 -- ε (10--6) ε (10 (106)6) (10 6) (a)(a) (b)(b)

FigureFigure 8. 8 8. .Stress-strain StressStress-strain-strain curves curves under under uniaxial uniaxial compression: compression: ( (a a)) )the the the stress-strain stress stress-strain-strain curves curves of of seven seven prisms prisms andand a a a mean mean stress stress stress vs. vs. vs. strain strain strain curve; curve; curve; and and and (b ) ( comparison(bb)) comparison comparison of expected of of expected expected experimental experimental experimental and theoretical and and theoretical theoretical results resultswithresults the with with best-ﬁt the the best-fit best parameters.-fit parameters. parameters.

FigureFigure 9. 9.9 .Photos Photos of of failed failed specimens. speci specimens.mens .

Δ Δ 4.3.4.3. Constraint ConstraintIn the present FactorFactor: paper,: Linking Linking the focus 1,s 1, swith wouldwith  2, bes 2,s on the mean stress and strain relationship, from which the parameters including the mean value and the standard deviation of the distribution could be Δ Δ recognized.ForFor establishing establishing However, the the the link link standard between between deviation SFS SFS 1,1, stressss (with(with and an an strain unknown unknown relationship constraint constraint could factor factor also C) C) be and and plotted 2,2,s ins, ,the the relativerelativeﬁgure, which entropy entropy would theory the leadory tocould could the relative be be employed employed length of theto to randomstudy study on on ﬁeld the the for similarity concretesimilarity materials. between between That these these is alsotwo two a Δ Δ distributionsdistributionskey point deserving ofof 1,1,ss furtherandand 2, research.2,ss. . InIn statistics,statistics, thethe relativerelative entropyentropy couldcould bebe regardedregarded asas aa measuremeasure ofof

Materials 2019, 12, x FOR PEER REVIEW 14 of 17

Materialsthe distinguishability2019, 12, 3295 between two probability density distributions, which is also 14called of 17 Kullback–Leibler divergence [34]. For the probability distributions of two discrete random variables

P and Q , the relative entropy DPQKL () could be expressed as 4.3. Constraint Factor: Linking ∆1,s with ∆2,s Pi() For establishing the link between SFS()∆1,=s (with() an unknown constraint factor C) and ∆2,s, DPQKL  Piln (31) the relative entropy theory could be employed to studyi on theQi similarity() between these two distributions of ∆1,s and ∆2,s. In statistics, the relative entropy could be regarded as a measure of the distinguishability Assume that the random field in the stochastic damage model following a lognormal between two probability density distributions, which is also called Kullback–Leibler divergence [34]. distribution is represented by P, and that from the nanoindentation by Q. Apparently, a distinct For the probability distributions of two discrete random variables P and Q, the relative entropy DPQ() Drelative(P Q entropy) could beKL expressed would as be calculated with different constraint factors, the smallest one of KL k which means a minimum difference between P and Q. In other words, the constraint factor C () X P(i) meeting the minimum DPQKL isD an (optimalP Q) = valueP (fori) ln concrete. (31) KL k Q(i) Figure 10a shows the relative entropy with differenti constraint factors. It is clear that when the constraintAssume factor that is the very random small, field the inresult the stochasticof Equation damage (31) would model approach following zero. a lognormal In other distribution words, the isconstraint represented factor by Pless, and than that five from is thenot nanoindentationof any meaning byforQ concrete. Apparently, materials, a distinct despite relative the entropypseudo = Dsmaller(P Q relative) would entropy. be calculated While the with relative different entropy constraint corresponding factors, the to smallestthe constraint one of factor which C means33.12 a KL k minimumapproaches di thefference minimum, between givingP and theQ knowledge. In other words, that the the discre constraintpancy factor between C meeting these two the PDFs minimum is the Dminimum,(P Q) is namely, an optimal they value are in for good concrete. agreement. Moreover, the constraint factor identified agrees KL k well Figurewith the 10 apreviously shows the reported relative results entropy [11,24]. with di Figurefferent 10b constraint shows the factors. detailed It is PDFs clear of that concrete when Δ Δ theconstituent constraint together factor iswith very the small, PDFs the of result 1,s of and Equation 2,s obtained (31) would above. approach It is zero.easy to In othersee that words, the thedistribution constraint recognized factor less thanfrom five the is notmacro-test of any meaningis an ideal for concrete lognormal materials, distribution, despite thewhile pseudo the smallerdistribution relative reconstructed entropy. While from the the relative nano-test entropy is a correspondingtwo-peak distribution, to the constraint where the factor firstC =peak33.12 is approachesattributed to the HCP minimum, and the giving second the one knowledge is ascribed that to thethe discrepancyaggregate. It between is interesting these twoto note PDFs that is thethe minimum,difference between namely, theyprobability are in good density agreement. at around Moreover, ∆ = 3400 the (the constraint second peak) factor reaches identified a larger agrees value, well withwhich the is previously attributed reported to HCP results and ITZ [11, 24playing]. Figure a 10moreb shows important the detailed role PDFsthan ofthe concrete aggregate constituent during togetherdamage withevolution. the PDFs From of ∆ the1,s and failur∆2,es obtainedpatterns above.(seeing ItFigure is easy 9), to seethro thatugh the the distribution main crack recognized it is also fromobserved the macro-test that a small is amount an ideal of lognormal coarse aggregate distribution, was broken while the apart. distribution reconstructed from the nano-testFrom is the a two-peak nanoscale distribution, to mesoscale, where SFS could the first be peakgenerated is attributed based on to the HCP nanoindentation, and the second which one is ascribedshows that to thethe aggregate.meso-fracture It is arises interesting physically to note from that the the bulking difference of betweenconfined probabilityforce chain densitysystem atat aroundthe mesoscale.∆ = 3400 (theFrom second the peak)mesoscale reaches to amacroscale, larger value, SFS which could is attributed be recognized to HCPand based ITZ playingon the amacro-testing, more important which role thanprovides the aggregate the information during damagethat the evolution.damage of From concrete the failureis strongly patterns dependent (seeing Figureon the9 ),random through fracture the main of crackmesoscale it is also concrete observed element. that aIn small a word, amount the ofconcrete coarse aggregatedamage under was brokencompression apart. results from the random buckling of confined force chain system at the mesoscale.

-3 -4 x 10 x 10 14 4.5 from Nanotest: Hcp 12 4 from Nanotest: ITZ 3.5 from Nanotest: Agr 10 from Nanotest: Concrete 3 from Macrotest: Concrete 8 2.5 6 2

4 1.5 Relative entropy Relative Probability density Probability 2 1

0 0.5

0 -2 0 2000 4000 6000 8000 10000 0 10 20 30 40 50 60 70 Δ (10-6) Constraint factor C s (a) (b)

FigureFigure 10. SFSSFS from from the the nano- nano- and and macro-testing: macro-testing: (a ()a the) the relative relative entropy entropy vs. vs. the theconstraint constraint factor. factor. (b) (PDFsb) PDFs from from two two different diﬀerent scales scales correspondi correspondingng to the to the smallest smallest relative relative entropy. entropy.

From the nanoscale to mesoscale, SFS could be generated based on the nanoindentation, which shows that the meso-fracture arises physically from the bulking of conﬁned force chain system at the mesoscale. From the mesoscale to macroscale, SFS could be recognized based on the macro-testing,

Materials 2019, 12, 3295 15 of 17 which provides the information that the damage of concrete is strongly dependent on the random fracture of mesoscale concrete element. In a word, the concrete damage under compression results from the random buckling of conﬁned force chain system at the mesoscale.

5. Conclusions In summary, this study sets a framework to investigate the damage of concrete under compression based on the SPI. Base on the multiscale research including experimental results, theoretical derivation and numerical analysis, the following conclusions could be drawn:

(1) The confined force chain buckling model proposed indicates the relationship between mesoscale strength and mesoscale hardness. The indentation hardness could be equivalent to the constrained compressive strength confined by the surrounding material around the indenter. The nanoindentation combined with the proposed model and random field theory provides direct access to the SPI of mesoscale fracture behavior of concrete. Meanwhile, the mesoscale fracture behavior of each constituent follows the homogeneous random field. (2) Nanoindentation combined with macro-testing under compression could lead to the constraint factor linking two scales from mesoscale to macroscale, by comparing the difference between distributions of mesoscale fracture behavior from nano- and macro-testing. This multiscale method provides an effective way to investigate concrete damage under compression, to offer the physical reality of concrete damage evolution, and to estimate the effect of concrete constituents on damage evolution. Up to now, it is interesting to see that the nature of mechanical properties, e.g., strength, creep and damage, could be anticipated based on the SPI by using nanoindentation [17–20]. (3) At the mesoscale, the meso-fracture arises physically from the bulking of the confined force chain system. At the macroscale, the concrete damage is strongly dependent on the random fracture at the mesoscale. From mesoscale to macroscale, the accumulation of mesoscopic fracture results in the macroscopic damage. Notably, the mesoscale inhomogeneity and the mesoscale confined bulking are intrinsic to concrete, which may constitute the physical mechanism controlling concrete damage subjected to compression. (4) Our investigation provides the possibility to control damage and to strengthen cementitious materials. Additionally, evaluating the macroscopic properties based on the SPI at the lower scales could be a feasible option. However, in the present paper, only one concrete mix was studied. The concrete materials with higher and smaller water to cement ratios still deserve further investigation and thus the effect of concrete constituents, especially the aggregate on damage evolution could be systematically investigated in the future. Reasonable speculation could be given that the greater the concrete strength, the smaller difference in the second peak in Figure 10b, and vice versa.

Author Contributions: Conceptualization, H.L., X.R. and J.L.; methodology, H.L., X.R. and J.L.; software, H.L. and X.R.; validation, H.L.; formal analysis, H.L., X.R. and S.L.; investigation, H.L. and S.L.; resources, H.L. and S.L.; data curation, H.L.; writing—original draft preparation, H.L.; writing—review and editing, H.L. and X.R.; visualization, H.L.; supervision, X.R. and J.L.; project administration, H.L. and X.R.; funding acquisition, H.L., X.R. and J.L. Funding: This research was funded by NATIONAL KEY R&D PROGRAM OF CHINA (Grant No. 2017YFC0702900), NATIONAL SCIENCE FOUNDATION OF CHINA FOR MAJOR PROJECT (Grant No. 51261120374, 91315301), NATIONAL SCIENCE FOUNDATION OF CHINA (Grant No. 51808499) and SCIENCE FOUNDATION OF ZHEJIANG PROVINCE OF CHINA (Grant No. LQ18E080009). Acknowledgments: The authors would like to acknowledge the kindly help of Southeast University in Nanjing, China, and the helpful suggestion of Yamei Zhang, Weiwei Zhu and Wei Huang. The authors also thank the anonymous reviewers for the constructive suggestions. Conﬂicts of Interest: The authors declare no conﬂict of interest. Materials 2019, 12, 3295 16 of 17

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