materials Article
Microscale Testing and Modelling of Cement Paste as Basis for Multi-Scale Modelling Hongzhi Zhang *, Branko Šavija, Stefan Chaves Figueiredo, Mladena Lukovic and Erik Schlangen Faculty of Civil Engineering and Geosciences, Delft 2628 CN, The Netherlands;
[email protected] (B.Š.);
[email protected] (S.C.F.);
[email protected] (M.L.);
[email protected] (E.S.) * Correspondence:
[email protected]; Tel.: +31-68-551-5349 Academic Editor: Jorge de Brito Received: 13 October 2016; Accepted: 4 November 2016; Published: 8 November 2016
Abstract: This work aims to provide a method for numerically and experimentally investigating the fracture mechanism of cement paste at the microscale. For this purpose, a new procedure was proposed to prepare micro cement paste cubes (100 × 100 × 100 µm3 ) and beams with a square cross section of 400 × 400 µm2 . By loading the cubes to failure with a Berkovich indenter, the global mechanical properties of cement paste were obtained with the aid of a nano-indenter. Simultaneously the 3D images of cement paste with a resolution of 2 µm3 /voxel were generated by applying X-ray microcomputed tomography to a micro beam. After image segmentation, a cubic volume with the same size as the experimental tested specimen was extracted from the segmented images and used as input in the lattice model to simulate the fracture process of this heterogeneous microstructure under indenter loading. The input parameters for lattice elements are local mechanical properties of different phases. These properties were calibrated from experimental measured load displacement diagrams and failure modes in which the same boundary condition as in simulation were applied. Finally, the modified lattice model was applied to predict the global performance of this microcube under uniaxial tension. The simulated Young’s modulus agrees well with the experimental data. With the method presented in this paper the framework for fitting and validation of the modelling at microscale was created, which forms a basis for multi-scale analysis of concrete. Keywords: micro-mechanics; fracture; X-ray computed tomography; lattice model
1. Introduction Cement based materials are the dominant construction materials in the world [1]. Since cement paste is the most basic and complex component of these materials, the understanding of its mechanical properties and fracture behaviour is of significant practical importance and scientific interest. Contrary to homogenous isotropic materials, the stress field inside this highly heterogeneous material is not uniform even under uniform loading, which leads to microcracking at a number of locations prior to crack localization [2]. These microcracks eventually develop and coalesce to form a critical macrocrack leading to the failure of this material at a low strain level. Since the critical scale for studying and understanding the fracture behaviour of cement paste is the microscale [2], researchers have tried various methods to simulate the fracture performance and mechanical properties of cement at this scale. In these approaches, two aspects should be included. The first one is how to obtain a realistic microstructure. This can be achieved by numerical models or experiments. Compared with experiments, the computer generated microstructural models, such as cement hydration model, are easier and faster. However, cement particles are commonly simulated as spheres [3–5], which will influence the simulated hydration of cement [6]. Therefore, although the experimentally obtained microstructure has a disadvantage in terms of spatial resolution, the realistic Materials 2016, 9, 907; doi:10.3390/ma9110907
www.mdpi.com/journal/materials
Materials 2016, 9, 907
2 of 14
particle shape and phase distribution after the onset of hydration can be captured. Over the past decades, huge advances have been made in microstructural characterisation techniques. A good resolution of the microstructure can be provided by using backscattered electron imaging (BSE) in the scanning electron microscope (SEM) [7]. However, the main shortcoming of this technique is the lack of 3D information. To overcome this, X-ray microcomputed tomography (µCT), which provides a non-destructive way of obtaining three dimensional information on the interior of materials, has been applied in the study of cement based materials over the last two decades [8–10]. By applying synchrotron radiation as the X-ray source, an improved resolution of between 0.5 and 1.0 µm for cement-based materials can be achieved [11–13], which provides very detailed information about the 3D microstructural evolution of these materials. The second aspect is how to include the local micromechanical properties of different phases in cement paste. Usually micromechanical properties of different phases are derived from standard nano-indentation measurements [14–17]. Meanwhile, an alternative way is to use molecular dynamics simulations to calculate the micromechanical properties of cement phases, e.g., Calcium-Silicate-Hydrate (C–S–H) [18,19]. Once the microstructure and the micromechanical properties are available, the micromechanical modelling approaches can be applied to simulate the fracture behaviour and mechanical properties of cement paste. Generally, the micromechanical modelling approaches can be classified in two categories: continuum approaches and discrete (lattice) approaches. Although continuum approaches are more widely used to predict the elastic properties of cement paste [20–23], they have inherent difficulties when dealing with strain localization (fracture) processes. On the other hand, lattice models show a great advantage because not only the stress-strain response, but also cracks pattern and microcracks propagation, can be simulated [24–26]. Continuum material behaviour can be, with certain limitations, reproduced by this class of models [27,28]. In lattice models, the continuum is replaced by a lattice system of beam elements and the crack growth is realized by using a sequentially-linear solution procedure [29]. This procedure implies performing a linear elastic analysis in every step; then, a single element with the highest stress/strength ratio is identified and removed from the mesh, thereby introducing a discontinuity; this procedure is then repeated until a global failure criterion is reached. Recently, the fracture process of micro cement paste cubes under uniaxial tension was simulated by Qian et al. [30] and Lukovic et al. [31] using the 3D lattice model. In spite of the similar applied simulation strategies, the obtained mechanical properties of cement paste are still not reliable because the relationship between the indentation hardness and the tensile strength (which is one of the main inputs for the model) of individual phases in hydrated cement paste is not known at present. Since inverse calculation of these local properties is possible only when an experiment on specimen size is available and uniaxial tensile testing on the micro-length scale is still impossible to achieve [2], an important question arising here is how to validate the modelling results. Recently, a new method named microcube indentation has been developed to test the global mechanical performance of micro cement cubes (100 × 100 × 100 µm3 ) using nano-indenter, which provides an unprecedented opportunity for validation of mechanical simulation results at the microscale [32]. The new method uses nano-indentation equipment to assess the fracture properties of small specimens, unlike regular nano-indentation testing that is used to assess elastic modulus and hardness. This method is further developed and presented in this paper. The method for experimental testing and numerical simulation of specimens on the same size under the same boundary conditions is addressed. The adopted mechanical properties of local phases are evaluated by comparing the simulated damage evolution and load displacement diagram with the experimental observation. In addition, the calibrated model is applied to predict the global mechanical properties of micro cement cube under uniaxial tension. The predicted results are compared with the results of the previous works.
Materials 2016, 9, 907 Materials 2016, 9, 907
3 of 14 3 of 14
2. Experimental Experimental
Materials 2016, 9, 907
3 of 14
2.1. Sample Preparation 2.1. Sample Preparation 2. Experimental 3 The following section section describes describesthe theprocedure proceduretotoprepare preparemicrocubes microcubes(100 (100× × × 100 3) for) The following 100100 × 100 µmµm 2 for global mechanical performance microbeams with a cross section of 400 × 400 for 2.1. Sample Preparation global mechanical performance testtest andand microbeams with a cross section of 400 × 400 µm2µm for CT CT scan. Standard grade OPC CEM I 42.5 N cement paste with w/c ratio 0.40 was cast in a PVC 3) for scan. Standard gradesection OPC CEM I 42.5the N cement paste with w/cmicrocubes ratio 0.40 was PVC The following describes procedure to prepare (100 cast × 100in×a100 µmcylinder cylinder (diameter, 24 mm, height 39 mm) in sealed condition. After 24 h rotation and curing 28 days 2 (diameter, 24 mm, height 39 mm) in sealed condition. After 24 h rotation and curing 28 days at room global mechanical performance test and microbeams with a cross section of 400 × 400 µm for CT ◦ C), specimens were demoulded and two discs with the thickness of 2 mm at room temperature (20 temperature (20 °C), specimens and two with0.40 the was thickness 2 mm were cut scan. Standard grade OPC CEMwere I 42.5demoulded N cement paste withdiscs w/c ratio cast inof a PVC cylinder were cut middle from middle One the used tomicrocubes, create theand microcubes, while other from the part. One part. of pieces waspieces used towas create the while the 28 other was used to (diameter, 24the mm, height 39the mm) inof sealed condition. After 24 h rotation curing days atthe room was used to create the microbeams. The hydration was arrested by solvent exchange method using temperature (20 °C), specimens were demoulded and two discs with the thickness of 2 mm were cut create the microbeams. The hydration was arrested by solvent exchange method using isopropanol isopropanol [33]. In order toofenable faster water-solvent exchange, samples were immerged five times from middle part. One the pieces was used to create the microcubes, while thefive other wasand usedtaken to [33]. In the order to enable faster water-solvent exchange, samples were immerged times and taken out for aof period of one minute. Afterwards, were placed 3 daysin inisopropanol isopropanol and microbeams. hydration was arrested solvent exchange using isopropanol outcreate for athe period oneThe minute. Afterwards, theybythey were placed for for 3method days and subsequently taken out, and solvent was removed by evaporation at ambient conditions. [33]. In order to enable faster water-solvent exchange, samples were immerged five times and taken subsequently taken out, and solvent was removed by evaporation at ambient conditions. outTo a period of oneand minute. Afterwards, the they were placed for 3 is days in isopropanol and create microcubes and the microbeams, microbeams, thefollowing following procedure isfollowed: followed: The first step Tofor create microcubes the procedure The first step is subsequently taken out, and solvent was removed by evaporation at ambient conditions. is make thethickness thicknessofofthe thespecimens specimenseven evenand andequal equaltoto100 100µm µmand and 400 400 µm µm (for (for microcube to to make the microcube and and To create microcubes and theusing microbeams, the following procedure is followed: The first step isthe microbeam creation, respectively) Labopol-5 thin sectioning machine; afterwards, microbeam creation, respectively) usinga aStruers Struers Labopol-5 thin sectioning machine; afterwards, to make the thickness of the specimens even and equal to 100 µm and 400 µm (for microcube and microcubes andand microbeams areare generated byby running a aprecise the microcubes microbeams generated running precisediamond diamondsaw sawfor for semiconductor semiconductor microbeam creation, respectively) using a Struers Labopol-5 thin sectioning machine; afterwards, wafers wafers (MicroAce (MicroAce Series Series 33 Dicing Dicing Saw, Saw, with with aa 260 260 µm µm wide wide blade) blade) in in two two directions directions over over the the thin thin the microcubes and microbeams are generated by running a precise diamond saw for semiconductor section as schematically shown in Figure 1. To prevent chipping of the edges of the microcubes section as schematically shown in Figure 1. To prevent chipping of the edges of the microcubes and and wafers (MicroAcecutting, Series 3 Dicing Saw, with a 260 µmwas wide blade) on in the twosurface directions over thesection, thin microbeams applied of the the thin microbeams while while cutting, aa thin thin layer layer of of soluble soluble glue glue was applied on the surface of thin section, section as schematically shown in Figure 1. To prevent chipping of the edges of the microcubes and which which was was later later removed removed by by soaking soaking the the specimen specimen for for aa short short time time in in acetone. acetone. The The final final cubes cubes are are at at microbeams while cutting, a thin layer of soluble glue was applied on the surface of the thin2 section, 3 and 3 2 the dimension of 100 × 100 × 100 µm beams are with a cross-section of 400 × 400 µm as shown thewhich dimension of 100 × 100 × 100 µm and beams are with a cross-section of 400 × 400 µm as shown was later removed by soaking the specimen for a short time in acetone. The final cubes are at in 2. in Figure Figure 2. the dimension of 100 × 100 × 100 µm3 and beams are with a cross-section of 400 × 400 µm2 as shown in Figure 2.
(a) (a)
(b)
(b)
Figure 1. 1. Schematic view of of sample sample generation: generation: (a) (a) microcubes microcubes and and (b) (b) microbeams. microbeams. Figure Schematic view Figure 1. Schematic view of sample generation: (a) microcubes and (b) microbeams.
(a) (a)
(b) (b)
Figure Scanningelectron electronmicroscope microscope images microbeam. Figure 2. Scanning images of of single single(a) (a)microcube microcubeand and(b) (b) microbeam. Figure 2. 2. Scanning electron microscope of single (a) microcube and (b) microbeam.
Materials 2016, 9, 907
4 of 14
Materials 2016, 9, 907
4 of 14 4 of 14
2016, 9, 907 2.2. Materials Global Micro-Mechanical Performance Using Microcube Indentation
2.2. Global Micro-Mechanical Performance Using Microcube Indentation 2.2. Performance Using Microcube Indentation In Global order Micro-Mechanical to obtain the global mechanical properties, the microcubical cement paste samples In order to Agilent obtain the global mechanical properties, the microcubical cement paste samples were placed in an G200 nano-indenter and were loaded using a cement Berkovich tipsamples (Figure 3). In order to obtain the global mechanical properties, the microcubical paste were placed in an Agilent G200 nano-indenter and were loaded using a Berkovich tip (Figure 3). A A displacement controlled test wasnano-indenter performed by using CSMusing (Continuous Stiffness Method) were placed in an Agilent G200 and werethe loaded a Berkovich tip (Figure 3). A [34]. displacement controlled test was performed by using the CSM (Continuous Stiffness Method) [34]. Thisdisplacement method relies on applying a small harmonic load with frequency on theStiffness nominal load. The controlled test was performed by using the CSM (Continuous Method) [34].CSM This method relies on applying a small harmonic load with frequency on the nominal load. The CSM settings applied in this study were: 2 nm amplitude, 45 Hz frequency and a displacement rate of This method relies on applying a small harmonic load with frequency on the nominal load. The CSM settings applied in this study were: 2 nm amplitude, 45 Hz frequency and a displacement rate of 50 − 1 settings in thisabout studythe were: 2 nm amplitude, frequency and aindisplacement ratenano of 50tube 50 nm ·s . applied More detail loading procedure 45 is Hz discussed in [34] which carbon nm·s−1. More detail about the loading procedure is discussed in [34] in which carbon nano tube nm·s−1 . More detail about with the loading procedure is discussed in [34] in which carbonup nano tube bundles are loaded to failure a flat nano-indenter. The load-displacement response until failure bundles are loaded to failure with a flat nano-indenter. The load-displacement response up until bundles are loaded to failure with4.aIn flat nano-indenter. The load-displacement response up until of the microcube is shown in Figure total, 8 load-displacements are measured in the experiments failure of the microcube is shown in Figure 4. In total, 8 load-displacements are measured in the failure of the microcube isratio shown Figure 4.measurements In total, 8 load-displacements are measured in the for cement paste with a w/c 0.4.ainMultiple on different show a high degree experiments for cement paste with w/c ratio 0.4. Multiple measurements oncubes different cubes show a experiments for cement paste with a w/c ratio 0.4. Multiple measurements on different cubes show a of repeatability. regimes can distinguished from the paragraph. In regimeIn(I), the load high degree ofTwo repeatability. Twobe regimes can be distinguished from the paragraph. regime (I), on high degree of repeatability. Two regimes can be distinguished from the paragraph. In regime (I), the load on sample increases monotonically forindenting increasinguntil indenting untilthe reaches peak load. The sample increases monotonically for increasing reaches peak the load. The maximum the load on sample increases monotonically for increasing indenting until reaches the peak load. The load that canbefore be applied before the microcube is between mN and 450atmN at loadmaximum that can be applied the microcube collapses collapses is between 350 mN350 and 450 mN a critical maximum load that can be applied before the microcube collapses is between 350 mN and 450 mN at a critical displacement between 1015 µm andOnce 15 µm.this Once thisisload is exceeded, the system transitionsfrom displacement between 10 µm and µm. load exceeded, the system transitions a critical displacement between 10 µm and 15 µm. Once this load is exceeded, the system transitions from a stable regime (I) towards an unstable(II) regime (II) with rapid displacement bursts. The a stable towards unstable with (II) rapid displacement bursts. The horizontal fromregime a stable(I)regime (I) an towards an regime unstable regime with rapid displacement bursts. The horizontal line in regime (II) indicates structural collapse of the microcube, which results in an line horizontal in regime line (II) indicates collapse of thecollapse microcube, resultswhich in an overshoot of the in regimestructural (II) indicates structural of thewhich microcube, results in an overshoot of the nano-indentation tip towards the substrate. Since displacement control of the nano-indentation tip towards the substrate. Since displacement control of the nano-indenter is not overshoot of the nano-indentation tip towards the substrate. Since displacement control of the fast nano-indenter is not fast enough, it is not possible at present to capture the post-peak behaviour of enough, it is not possible presentitto the at post-peak ofpost-peak the specimen. Therefore, nano-indenter is not fastatenough, is capture not possible present tobehaviour capture the behaviour of the specimen. Therefore, the calibration of the numerical model was carried out only in regime (I). It the specimen. Therefore, the calibration of the numerical model was carried out only in regime (I). It test the calibration of the numerical model was carried out only in regime (I). It is observed that the is observed that the test results still show variability which is induced by the inherent heterogeneity is observed that the test results still show variability which is induced by the inherent heterogeneity results stillmaterial. show variability which is induced by the inherent heterogeneity of the material. of the of the material.
Figure 3. Scanning electron microscope images of diamond Berkovich tip.
Figure 3. 3. Scanning imagesofofdiamond diamond Berkovich Figure Scanningelectron electron microscope microscope images Berkovich tip.tip.
Load onon sample(mN) Load sample(mN)
600 600
Regime Regime
Regime Regime
500 500 400 400 300 300 200 200 100 100 0 0 0 0
5000 5000
10000 10000
15000 15000
20000 20000
25000 25000
30000 30000
Displacement(nm) Displacement(nm)
Figure 4. Measured global mechanical response of microcubes: load versus displacement response.
Figure 4. Measured global mechanical response of microcubes: load versus displacement response. Figure 4. Measured global mechanical response of microcubes: load versus displacement response.
Materials 2016, 9, 907 Materials 2016, 9, 907
5 of 14 5 of 14
loading is observed observed in in previous previous work work [32], [32], The failure mechanism of the specimen under tip loading where different loading depths were applied. Crack patterns at various depths were obtained using ESEM. As shown in Figure 5, the typical failure mechanism obtained is the crushing of the material under the tip and three main cracks running to the sides of the cubes, starting from the three edges of the Berkovich tip. Complete Complete crushing crushing of of the sample was achieved by indenting the tip further into the specimen.
(a)
(b)
(c)
Figure loading process of of microcubes observed in ESEM: (a) Figure 5. 5. Three Threestages stagesininthe thenano-indentation nano-indentation loading process microcubes observed in ESEM: initial stage of loading; (b) three main cracks running to the sides of the cubes; (c) complete crushing (a) initial stage of loading; (b) three main cracks running to the sides of the cubes; (c) complete crushing of the sample sample (adapted (adapted from from [32]). [32]). of the
2.3. Microstructure Characterization Using Using of of Micro-CT Micro-CT 2.3. Microstructure Characterization In ordertotoobtain obtain microstructure of micro-cube, the micro-cube, a generated microbeam is scanned In order thethe microstructure of the a generated microbeam is scanned using a using a Micro CT-Scanner (Phoenix Nanotom, Boston, MA, USA). The microbeam was fixed onand the Micro CT-Scanner (Phoenix Nanotom, Boston, MA, USA). The microbeam was fixed on the holder holder and then put on the rotatable stage. The X-ray source tube worked at 120 kev/60 mA. 2800 then put on the rotatable stage. The X-ray source tube worked at 120 kev/60 mA. 2800 images with an images with exposure of 6 on s were acquired on adetector digital GE DXR detector (3072 × 2400 The exposure of 6an s were acquired a digital GE DXR (3072 × 2400 pixels). The voxelpixels). resolution 3 voxel resolution underwas these conditions was 0.5 slices µm . were Reconstructed were carried with 3 . Reconstructed under these conditions 0.5 µm carried outslices with Phoenix Datos out software. Phoenix Datos software. For saving the calculation time of mechanical model, the original resolution For saving the calculation time of mechanical model, the original resolution of reconstructed slices of reconstructed slices was reduced to 2 µm3/voxel. segmentation was performed using a 3 /voxel. was reduced to 2 µm Image segmentation wasImage performed using a so-called global threshold so-called global threshold approach [7,9,13,35]. In this method, phases were isolated from the approach [7,9,13,35]. In this method, phases were isolated from the original grey-scale map by choosing original grey-scale map by choosing the corresponding threshold step by step as shown in Figure 6. the corresponding threshold step by step as shown in Figure 6. Firstly, two threshold grey values are Firstly, two threshold grey values are definedas onshown the basis of the7:grey-level histogram shown in defined on the basis of the grey-level histogram in Figure T1 , pore/solid phase as threshold, is Figure 7: T 1, pore/solid phase threshold, is assumed as the grey value at the inflection point in the assumed as the grey value at the inflection point in the cumulative fraction curve of the histogram; T2 , cumulativeproducts/anhydrous fraction curve of cement the histogram; T2, hydration products/anhydrous cementslope phase hydration phase threshold, is a critical point at which the tangent of threshold, is a critical point at which the tangent slope of the histogram changes suddenly. Three the histogram changes suddenly. Three phases can be isolated: pore, anhydrous cement and hydration phases can be details isolated: pore, cement and hydration More However, details about this product. More about thisanhydrous approach can be found in previousproduct. work [13,35]. it is well approach can found previous work [13,35]. However, it is well known that at least three types of known that at be least threeintypes of hydration product with different mechanical properties [14–17] exist: hydration product with different mechanical properties [14–17] exist: inner product C-S-H LD, outer inner product C-S-HLD , outer product C-S-HHD and C-H. In order to simplify procedure, C–H was product C-S-HHD C-H. In orderand to simplify C–H was not considered as a separate not considered as and a separate phase, thereforeprocedure, was not explicitly modelled. This simplification is phase, and therefore was not explicitly modelled. This simplification is considered not to considered not to significantly affect the results of mechanical properties simulation [30]. However, significantly affect the of mechanical properties simulation [30]. However, in further work, in further work, C-H asresults a separated phase should be considered. C-H as a separated phase should be considered. A so called J-T model determined analytically by Tennis and Jennings [36] based on specific A called J-T model determined analytically bythe Tennis andfraction Jennings [36] based on specific surface so measurements was introduced here to calculate volume of C-S-H LD and C-S-HHD . surface measurements was introduced here to calculate the volume fraction of C-S-H LD and C-S-HHD. As shown in Figure 8, the input for this model are w/c ratio and degree of hydration which can be As shown on in Figure 8, the input for this model are w/c ratio andVdegree of and hydration which can be estimated the basis of volume fraction of anhydrous cement hydration products anhydrous estimatedaccording on the basis of volume fraction of anhydrous cement Vanhydrous and hydration products V to equation: hydrated Vhydrated according to equation: Vhydrated α=
v , VhydratedVhydrated + V anhydrous v
(1)
v α , (1) Vhydrated where v is defined as volume reaction product/volume reactant ratio and assumed as 2.2 [37]. Once the Vanhydrous v volume fraction of those two products is obtained, the threshold value T3 can be determined from
Materials 2016, 9, 907 Materials 2016, 9, 907 Materials 2016, 9, 907
6 of 14 of 14 6 of614
where v is defined as volume reaction product/volume reactant ratio and assumed as 2.2 [37]. Once where v is defined asof volume product/volume ratio and assumed 2.2determined [37]. Once the volume fraction those reaction two products is obtained,reactant the threshold value T3 canasbe thethe cumulative volumeoffraction cureproducts of grey-histogram asthe shown in Figure 7.TThe voxels with a grey volume fraction those two is obtained, threshold value 3 can be determined from the cumulative volume fraction cure of grey-histogram as shown in Figure 7. The voxels with a value lower than T3 are regarded as cure outerofhydration product, whileinthe ones with higher values from the cumulative volume fraction grey-histogram as shown Figure The higher voxels with a grey value lower than T3 are regarded as outer hydration product, while the ones7.with values aregrey inner hydration product. A cubical region of interest with 100 µm in length was extracted from value lower than T 3 are regarded as outer hydration product, while the ones with higher values are inner hydration product. A cubical region of interest with 100 µm in length was extracted from theare segmented microstructure lattice fracture analysis as shown in Figurewas 9a.extracted Microstructure hydration product. Afor cubical region of interest with µm in from the inner segmented microstructure for lattice fracture analysis as 100 shown in length Figure 9a. Microstructure characterization of cement paste with w/c ratio 0.3 and 0.5 at the same curing age of 28 days were the segmented microstructure for lattice fracture analysis as shown in Figure 9a. Microstructure characterization of cement paste with w/c ratio 0.3 and 0.5 at the same curing age of 28 days were carried outout using the same in Figure 9c. characterization ofthe cement paste withand w/cshown ratio 0.3 0.59c. at the same curing age of 28 days were carried using sameprocedure procedure and shown in and Figure carried out using the same procedure and shown in Figure 9c.
(a) (a)
(b) (b)
(c) (c)
(d) (d)
Figure 6. 2D Schematic view of image segmentation process: (a) original grey-scale map; (b) pore Figure 6. 2D Schematic view of image segmentation process: (a) original grey-scale map; (b) pore Figure 6. 2D Schematic view of image segmentation (a) map; original grey-scale map; (b)(grey) pore (blue) and solid phases (yellow) are isolated from theprocess: grey-scale (c) anhydrous cement (blue) and solid phases (yellow) are isolated from the grey-scale map; (c) anhydrous cement (grey) (blue) and solidproduct phases (yellow) (yellow) are areisolated isolatedform fromsolid the grey-scale anhydrous cement and hydration phases; (d)map; outer(c)product (yellow) and(grey) inner and hydration product (yellow) are isolated form solid phases; (d) outer product (yellow) and inner and hydration product (yellow) arehydration isolated form solid phases; (d) outer product (yellow) and inner product (red) are segmented from product. product (red) areare segmented product (red) segmentedfrom fromhydration hydrationproduct. product.
Voxel numbers Voxel numbers
1200000 1200000 1000000 1000000 800000 800000 600000 600000 400000 400000 200000 200000 0 00 0
Outer product Inner product
T2 3 T1 productTInner Outer product
Volume fraction Volume fraction Voxel numbers Voxel numbers
T1
T3
T2
Anhydrated cement Anhydrated cement
1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2
Volume fraction Volume fraction
Pore Pore
0.0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 2550.0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255
Grey-level Grey-level Figure 7. Phases evolution through grey-level histogram of CT images. Figure 7. Phases evolution through grey-level histogram of CT images.
Figure 7. Phases evolution through grey-level histogram of CT images.
Materials 2016, 9, 907
7 of 14
Materials 2016, 9, 907 Materials 2016, 9, 907
7 of 14 7 of 14 0.9 0.9
C-SH/C-S-H /C-S-H (vol.) C-SH (vol.) HD HD
0.8 0.8 0.7 0.7
w/c=0.3 w/c=0.3 w/c=0.4 w/c=0.4 w/c=0.5 w/c=0.5
0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2
0.3 0.3
0.4 0.4
0.5 0.5
0.6 0.6
Hydration degree () Hydration degree ()
0.7 0.7
0.8 0.8
Figure 8. High density Calcium-Silicate-Hydrate (C–S–HHD) evolution based on the J-T model [36].
Figure 8. High density Calcium-Silicate-Hydrate evolution based model HD Figure 8. High density Calcium-Silicate-Hydrate (C–S–H (C–S–HHD ) )evolution based onon thethe J-T J-T model [36].[36].
(a) (a)
(b) (b)
(c) (c)
Figure 9. 3D segmented microstructure (100 × 100 × 100 µm3) of cement paste with w/c ratio (a) 0.3; Figure 9. 3D segmented microstructure (100 × 100 × 100 µm3) of cement paste with w/c ratio (a) 0.3;
3 ) of cement (b) 0.4; (c) 0.5 (grey-anhydrous cement; (100 red-inner yellow-outer product; Figure 9. 3D segmented microstructure × 100product; × 100 µm paste blue-pore). with w/c ratio (a) 0.3; (b) 0.4; (c) 0.5 (grey-anhydrous cement; red-inner product; yellow-outer product; blue-pore). (b) 0.4; (c) 0.5 (grey-anhydrous cement; red-inner product; yellow-outer product; blue-pore). 3. Modelling 3. Modelling 3. Modelling Herein, a microstructure-informed lattice model was calibrated using experimental results of Herein, a microstructure-informed lattice model was calibrated using experimental results of microcube indentation (failure patternslattice and load displacement diagrams). Then, the global Herein, microstructure-informed was calibrated using experimental results microcube aindentation (failure patterns and model load displacement diagrams). Then, the global mechanical performance of microcubes with different w/c ratios were predicted by incorporating the of microcube indentation (failure patterns and load displacement diagrams). Then, the global mechanical performance of microcubes with different w/c ratios were predicted by incorporating the microstructure into the lattice model. The modelling approach, calibration and prediction procedure mechanical performance microcubes with differentapproach, w/c ratios were predicted by incorporating microstructure into the of lattice model. The modelling calibration and prediction procedure the are described in detail below. microstructure the lattice are describedinto in detail below.model. The modelling approach, calibration and prediction procedure
are 3.1. described in Approach detail below. Modelling 3.1. Modelling Approach
Again Approach the aim in this study is mainly to determine the micromechanical properties of local 3.1. Modelling
Again the aim in this study is mainly to determine the micromechanical properties of local phases in lattice model by fitting the experimental results. The lattice model is applied in this study phases inthe lattice model fitting the experimental results. The lattice model is applied in this study in this by study is behaviour mainly to determine themicrocubes. micromechanical local phases toAgain model theaim observed fracture of the tested In the properties model, the of material is to model the by observed fracture behaviourresults. of the The tested microcubes. Inapplied the model, thestudy material is in lattice model fitting the experimental lattice model is in this to model schematized as a grid of beam elements connected at the ends and all individual elements are schematized as a grid of beam elements connected at the ends and all individual elements are thedefined observed fracture behaviour of the tested microcubes. Inofthe model, material is schematized having linear elastic behaviour. Due to the low ratio length andthe height of beam elements defined elastic behaviour.atDue to the low ratio of length and height of beam elements as ain grid ofhaving beamalinear elements connected theis ends all shear individual elements defined the network, Timoshenko beam element usedand to take deformation intoare account [24].having To in the network, a Timoshenko beam element is used to take shear deformation into account [24]. To achieve thebehaviour. crack growth, prescribed displacement is imposed the lattice system andnetwork, only linear elastic Dueunit to the low ratio of length and height ofonbeam elements in the achieve the crack growth, unit prescribed displacement is imposed on the lattice system and only one elementbeam with the highest ratiodeformation is removing from the mesh[24]. at every loadingthe step. a Timoshenko element is stress/strength used to take shear into account To achieve crack one element with the highest stress/strength ratio is removing from the mesh at every loading step. The calculation procedure will repeat is several timeson until pre-defined criterion reached.with growth, unit prescribed displacement imposed the alattice systemfailure and only one iselement The calculation procedure will repeat several times until a pre-defined failure criterion is reached. method can express the physical process from of fracture behaviour, so that a realistic theThis highest stress/strength ratio is removing the mesh at every loading step.crack The patterns calculation This method can express the physical process of fracture behaviour, so that a realistic crack patterns as well as the stress-strain response will be obtained [29]. The “Berkovich tip loading” test is procedure will repeat several times until a pre-defined failure criterion is reached. This method as well as the stress-strain response will be obtained [29]. The “Berkovich tip loading” test is can simulated by applying a vertical displacement in four nodes in the centre of the top surface. The express the physical process of fracture behaviour, so that realistic crack as wellThe as the simulated by applying a vertical displacement in four nodesa in the centre of patterns the top surface. lattice element can fail either in tension or compression. The procedure to generate the lattice mesh stress-strain response be obtained [29].orThe “Berkovich tipprocedure loading” test is simulated by applying lattice element can will fail either in tension compression. The to generate the lattice mesh a and assign mechanical properties of elements is as follows. vertical displacement in four nodes in centreisofasthe top surface. The lattice element can fail either in and assign mechanical properties of the elements follows.
tension or compression. The procedure to generate the lattice mesh and assign mechanical properties of elements is as follows.
Materials 2016, 9, 907
8 of 14
Materials 2016, 9, 907
8 of 14
The 3D mesh generation is described in Figure 10. First, a cubic domain (100 × 100 × 1003 µm3 ) The 3D mesh generation is described in Figure 10. First, a cubic domain (100 × 100 × 100 µm ) is 3 . Then, a sub-cell was defined within each cell is divided intoaacubic cubicgrid gridwith witha acell cellsize size 2 µm 3. Then, divided into ofof 2 µm a sub-cell was defined within each cell in in which which aanode nodeisisrandomly randomlyplaced. placed. The ratio between length of sub-cell and iscell is defined The ratio between the the length of sub-cell and cell defined as as randomness. randomness.AsAs shown in preview study [38], the choice of randomness affects the simulated shown in preview study [38], the choice of randomness affects the simulated the thefracture fracturebehaviour behaviour materials, because simulated crack shape is affected byorientation the orientation of of materials, because thethe simulated crack shape is affected by the of of meshes. meshes.InInorder ordertotoavoid avoidbig big variations length of elements introduce geometry disorder variations in in length of elements andand introduce geometry disorder of of material material texture, texture,aa randomness randomnessofof0.5 0.5isisadopted. adopted.Then, Then,Delaunay Delaunaytriangulation triangulation is performed is performed to to connect thethe fourfour nodes that are to eachto other lattice elements. Afterwards, the cross-section connect nodes thatclosest are closest eachwith other with lattice elements. Afterwards, the of is lattice element by is determined altering this in alattice homogenous latticethe of cross-section lattice element determined altering thisby parameter in aparameter homogenous model until model until the Young’s simulated global Young’s matches thevalue. assumed local value. simulated global modulus matchesmodulus the assumed local
(a)
(b)
Figure 10. Schematic view of lattice model generation: (a) lattice network construction (5 × 5 × 5); (b) Figure 10. Schematic view of lattice model generation: (a) lattice network construction (5 × 5 × 5); overlay procedure for 2D lattice mesh (yellow-outer product; red-inner product; grey-anhydrous (b) overlay procedure for 2D lattice mesh (yellow-outer product; red-inner product; grey-anhydrous cement). cement).
The overlay the heterogeneity heterogeneityofofthis thismaterial. material. this procedure, The overlayprocedure procedureisisapplied applied to to realize realize the InIn this procedure, different are assigned assignedtotocorresponding corresponding phases. Forpurpose, this purpose, differentmicromechanical micromechanical properties properties are phases. For this the themicrostructure microstructureofofcement cementpaste pastewith withw/c w/c ratio obtained Section is used here. shown ratio 0.40.4 obtained in in Section 2.32.3 is used here. AsAs shown in in Figure 10b, each node is assigned a local phase based on the voxel value it belongs to and the lattice Figure 10b, each node is assigned a local phase based on the voxel value it belongs to and the lattice element is is determined Nolattice latticenode nodeisisgenerated generated voxels element determinedby bythe thelocations locations of of its its two two nodes. nodes. No inin thethe voxels which representpore porephases, phases,as asititdoes doesnot not contribute contribute to of of thethe which represent to the the global globalmechanical mechanicalperformance performance specimen. Threesolid solidphases phasesin inthe the microstructure microstructure result asas listed in in specimen. Three resultin insix sixtypes typesofoflattice latticeelements elements listed Table 1. The shear modulus and Young’s modulus of element i–j connecting phase i and phase j are Table 1. The shear modulus and Young’s modulus element i–j connecting phase i and phase j are determined [30]: determined asas [30]: 2 1 1 2= 1 + 1 (2) ,j EijE = EEi+ E E (2) ij
i
j
where Ei , Ej and Eij , are the Young’s modulus or shear modulus for phase i, phase j and element which where Ei, Ej and Eij, are the Young’s modulus or shear modulus for phase i, phase j and element connects phase i and phase j, respectively. The compressive strength and tensile strength take the which connects phase i and phase j, respectively. The compressive strength and tensile strength take lower value of the connected two phases, which can be expressed in [30]: the lower value of the connected two phases, which can be expressed in [30]: f ijfij== min(f min(fii,, ffj),j ),
(3) (3)
where fi, fj and fij, are the compressive strength or tensile strength for phase i, phase j and element where f i , f j and f ij , are the compressive strength or tensile strength for phase i, phase j and element which connects phase i and phase j, respectively. The mechanical properties of these pure phases which connects phase i and phase j, respectively. The mechanical properties of these pure phases was was preferred to be measured in laboratory test, but in the case of a lack of experimental data, preferred to be measured in laboratory test, but in the case of a lack of experimental data, properties properties of these phases are commonly derived from the nano-indentation measurements [30,31]. of these phases are commonly derived from the nano-indentation measurements [30,31]. However, However, no data is available defining the relationship between the model parameters (tensile and nocompressive data is available defining the relationship between the model parameters (tensile and compressive strength, Young’s modulus) and nano-indentation results (indentation hardness and strength, Young’s modulus) and nano-indentation results (indentation hardness and modulus modulus of elasticity). Herein, in order to work out this relationship, the simulated fracture of elasticity). Herein, in orderwith to work out this relationship, the out simulated performance performance is compared the experimental results to find the bestfracture simulation, and these is compared with the experimental results to find out the best simulation, and these values are further values are further used in Section 3.3 to predict the mechanical and fracture properties of hydrated used in Section 3.3 to predict the mechanical and fracture properties of hydrated cement pastes
Materials 2016, 9, 907
9 of 14
Materials 2016, 9, 907
9 of 14
with different w/c ratio. Ratios of tensile strength (and compressive strength) among each phase pastes with different ratio. of tensile strength (and these compressive among are cement assumed to be equal to thew/c ratios of Ratios measured hardness among phasesstrength) in this calibration. each phase are assumed to be equal to the ratios of measured hardness among these phases this For individual phases, values reported in [14] (Table 2) were used in this work. Since scanninginelectron calibration. For individual phases, values reported in [14] (Table 2) were used in this work. Since microscope was adopted to reflect phases at the tested location for the statistics analysis, this method scanning electronresults. microscope was adopted to reflect phases at the tested location for the statistics gives more reliable analysis, this method gives more reliable results.
Table 1. Classification of lattice element types. Table 1. Classification of lattice element types.
Element Type Phase Phase Element Type Phase 1 1 Phase 2 2 A–A Anhydrous cement Anhydrous Anhydrous cement A–A Anhydrous cement cement I–I Inner product product I–I Inner product InnerInner product O–O Outer product Outer product O–O Outer product OuterInner product A–I Anhydrous cement product A–I Anhydrous cement InnerOuter product I–O Inner product product A–O Anhydrous cement Outer product I–O Inner product Outer product A–O Anhydrous cement Outer product Table 2. Measured mechanical properties of individual local phases from [14].
Table 2. Measured mechanical properties of individual local phases from [14]. Phases Modulus of Elasticity (GPa) Hardness (GPa)
Phases Anhydrous cement Anhydrous cement Inner product Inner Outerproduct product Outer product
Modulus of Elasticity (GPa) 99.2 99.2 31.6 31.6 25.2 25.2
Hardness (GPa) 8.24 8.24 1.14 1.14 0.75 0.75
3.2. Calibration and Discussion
3.2. Calibration and Discussion
During the calibration of mechanical properties of local phases, three sets of parameters as listed in During the calibration of mechanical properties of local phases, three sets of parameters as Table 3 were evaluated to study their influence on the simulated fracture performance. The simulated listed in Table 3 were evaluated to study their influence on the simulated fracture performance. The load displacement diagrams are compared with the experimental results as shown in Figure 11. simulated load displacement diagrams are compared with the experimental results as shown in A descending is observed regimein(II) for the results, which is missed in the Figure 11. Abranch descending branch is in observed regime (II) simulated for the simulated results, which is missed microcube indentation measurement. Their Their corresponding damaged patterns ininthe in the microcube indentation measurement. corresponding damaged patterns thefinal finalstage stage are presented in Figure 12. are presented in Figure 12. Table 3. Assumed local of individual individuallocal localphases phases cement paste. Table 3. Assumed localmechanical mechanicalproperties properties of inin cement paste.
Set
Set
S1 S1 S2S2 S3S3
Anhydrous Cement Inner Anhydrous Cement InnerProduct Product E (GPa) f t (GPa) f c (GPa) E (GPa) f t (GPa) E (GPa) f t (GPa) f c (GPa) E (GPa) f t (GPa) fcf c(GPa) (GPa) 99.2 0.683 6.830 31.6 0.092 0.92 99.2 0.683 6.830 31.6 0.092 0.92 99.2 0.683 68.3 31.6 0.092 9.2 99.2 0.683 68.3 31.6 0.092 9.2 99.2 0.683 31.6 0.092 ∞ 99.2 0.683 ∞∞ 31.6 0.092 ∞
Outer Outer Product Product EE(GPa) f t (GPa) ffc (GPa) (GPa) f t (GPa) c (GPa) 25.2 0.0583 0.58 25.2 0.0583 0.58 25.2 0.0583 5.8 25.2 0.0583 5.8 25.2 0.058 ∞ 25.2 0.058
600
S3 does not drop
Load (mN)
Experimental results 400
200
S1 S2 S3 0
0
10000
20000
30000
40000
Displacement (nm)
Figure 11. Comparison between simulated load displacement diagrams and experimental results of
Figure 11. Comparison between simulated load displacement diagrams and experimental results of cement paste with w/c ratio 0.4. cement paste with w/c ratio 0.4.
Materials 2016, 9, 907 Materials 2016, 9, 907
(a)
10 of 14 10 of 14
(b)
(c)
(d)
Figure 12. Crack patterns in the final failure state: (a) simulation of S1; (b) simulation of S2; Figure 12. Crack patterns in the final failure state: (a) simulation of S1; (b) simulation of S2; (c) simulation of S3; (d) element types (black-damaged element). (c) simulation of S3; (d) element types (black-damaged element).
The adopted compressive strength/tensile strength ratio of local elements has significant The adopted compressive strength ratio of localevolution. elements Figure has significant influence influence on simulated load strength/tensile displacement diagram and damage 12a shows the ondamage simulated load displacement diagram and damage evolution. Figure 12a shows the damage evolution of S1 in which the compressive strength is taken as 10 times higher than the evolution of S1 in which compressive strength is taken as 10microcracks times higher the tensile strength tensile strength (similarthe with the assumption in [39]). A few arethan initiated around the (similar with the assumption in [39]). A few microcracks are initiated around the loading points loading points causing local crushing of the indented area, thus inducing an unrealistically lowcausing load local crushing thecontrary, indented in area, inducing low load On simulated the contrary, response. Onofthe thethus simulation of an S3,unrealistically no softening branch is response. found in the in load-displacement the simulation of S3, no softening in the simulated load-displacement even diagram even branch when is thefound “indenter” reaches the bottom of the diagram microcube specimen. This is because nothe compressive is allowedspecimen. for local elements in this particular case. when the “indenter” reaches bottom offailure the microcube This is because no compressive In theisfracture the elements loading points out dragging surface down, apparent failure allowedmodel, for local in thisstarts particular case. Inthe theupper fracture model, theand loading points loadout that can be withstood the specimen becomes unrealistically high.be withstood by the specimen starts dragging the upperbysurface down, and apparent load that can Simulation of S2 and becomes unrealistically high.“measured” cures show a very high degree of consistency on the peak load and stiffness (slope the load displacement Due to the fact that some slip occurs at the Simulation of S2 andof“measured” cures showcure). a very high degree of consistency on the peak beginning of the experiments, the measurements are slightly shifted, but the slope is similar to the load and stiffness (slope of the load displacement cure). Due to the fact that some slip occurs at the one in the load displacement curve ofare S2.slightly In S2, the assumed tensile strength is 12 beginning of simulated the experiments, the measurements shifted, but the slope is similar totimes the one lower than the measured hardness, in accordance with [39,40], while the ratio between compressive in the simulated load displacement curve of S2. In S2, the assumed tensile strength is 12 times lower strength and tensile strength (100) is much higher than in [39]. This may be because of the different than the measured hardness, in accordance with [39,40], while the ratio between compressive strength resolutions and theories applied in these two models. and tensile strength (100) is much higher than in [39]. This may be because of the different resolutions and theories these two 3.3. Tensile applied Strength in Prediction andmodels. Discussion The Strength 3D latticePrediction fracture model is applied on the obtained voxel-based digital microstructures of 3.3. Tensile and Discussion cement paste to evaluate the mechanical performance under uniaxial tension. The same mesh The 3D lattice fracture model is applied on the obtained voxel-based digital microstructures described in Section 3.1 is applied here. Microstructures with different w/c are applied for the of overlay cementprocedure paste to evaluate the mechanical performance under uniaxial tension. The same mesh to investigate the influence of w/c ratio on the global performance of cement paste described in Section is applied here. Microstructures with displacement different w/c at areone applied forwhile the overlay at microscale. The 3.1 specimen is loaded by applying uniform surface, the procedure to investigate the influence of w/c ratio on the global performance of cement paste opposite surface is clamped. The lattice elements on the surfaces where boundary conditions are at microscale. applying uniformprocesses displacement one surface, while athe applied areThe not specimen allowed toisbeloaded brokenby during the fracture, as the at external load requires opposite surface is clamped. The lattice elements surfaces as where boundary properties conditionsofare path into the specimen. The adopted parameters in on S2 the are assumed the mechanical applied are notThe allowed to bestress-strain broken during the fracture, processes as the external requires local phases. simulated diagrams with corresponding damage pattern load are shown in a path into the specimen. The adopted parameters in S2 assumedbased as the properties Figures 13 and 14. Mechanical properties parameters areare calculated onmechanical the stress-strain curve, of andphases. are given Table 4 together with the previouswith results. For a w/c ratio 0.4, thepattern calculated results in local Theinsimulated stress-strain diagrams corresponding damage are shown show 13 a good with the results from [30,31]. are Since no experimental data tensile strength Figures and agreement 14. Mechanical properties parameters calculated based on theofstress-strain curve, is available for comparison at this scale [2], only the simulated Young’s modulus is compared with and are given in Table 4 together with the previous results. For a w/c ratio 0.4, the calculated results the aexperimental results [41] shownfrom in Figure It can be seen that the show good agreement with theasresults [30,31].15. Since no experimental datasimulated of tensile Young’s strength is modulus corresponds well with the results from experiments at each w/c ratio. available for comparison at this scale [2], only the simulated Young’s modulus is compared with the A moreresults localized observed inseen the cement with higher w/cmodulus ratio, experimental [41]microcracks as shown inpattern Figureis15. It can be that thepaste simulated Young’s which results in a decrease of Young’s modulus and tensile strength. This is because more corresponds well with the results from experiments at each w/c ratio. anhydrous particles are embedded in cement paste with lower w/c ratio and anhydrous particles, as A more localized microcracks pattern is observed in the cement paste with higher w/c ratio, local stiff inclusions, enable more overlaps and branches, which results in a more stable crack which results in a decrease of Young’s modulus and tensile strength. This is because more anhydrous development and less brittle fracture. Similar observation is presented in a higher scale of concrete, particles are embedded in cement paste with lower w/c ratio and anhydrous particles, as local stiff due to the influence of aggregate [42,43]. inclusions, enable more overlaps and branches, which results in a more stable crack development and
Materials 2016, 9, 907
11 of 14
less brittle fracture. Similar observation is presented in a higher scale of concrete, due to the influence of aggregate [42,43]. Materials 2016, 9, 907 11 of 14 Materials 2016, 9, 907
11 of 14
35
w/c=0.5 w/c=0.4 w/c=0.5 w/c=0.3 w/c=0.4 w/c=0.3
Stress(MPa) Stress(MPa)
3035 2530 2025 20 15 15 10 10 5 5 0 0.00 0 0.00
0.01 0.01
0.02
0.03
0.02
Strain0.03 Strain
0.04 0.04
0.05 0.05
Figure 13. Simulated stress-strain diagrams of cement paste with various water/cement ratios. Figure 13. Simulated stress-strain diagrams of cement paste with various water/cement ratios.
Figure 13. Simulated stress-strain diagrams of cement paste with various water/cement ratios.
(a) (a)
(b) (b)
(c) (c) Figure 14. Crack patterns in the final failure state of cement paste with w/c ratio (a) 0.3; (b) 0.4; (c) 0.5 Figure 14. Crack patterns in the final failure state of cement paste with w/c ratio (a) 0.3; (b) 0.4; (c) 0.5 (left: 14. whole specimen; right: only damage). Figure Crack patterns in the final failure state of cement paste with w/c ratio (a) 0.3; (b) 0.4; (c) 0.5 (left: (left: whole specimen; right: only damage). whole specimen; right: only damage).
Materials 2016, 9, 907
12 of 14
Materials 2016, 9, 907
12 of 14
Table 4. Calculated global Table 4. Calculated global mechanical mechanical properties, properties, compared compared with with previous previous work. work.
w/c Ratio w/c Ratio
Strength (MPa) Strength (MPa)
0.30.3 0.40.4 0.50.5
32.9 (42.1 32.9 (42.1 [30])[30]) 22.3 (20.3 [30]; 22.3 (20.3 [30]; 19.319.3 [31])[31]) 17.3 (10.6 [30]) 17.3 (10.6 [30]) 40
Numerical reuslts: this paper Constantinides et at. [15] Qian et al. [30] Lukovic et al. [31] Experimental results: Haeker et al. [41]
35
Young's Modulus (GPa)
Young’s Modulus(GPa) (GPa) Young’s Modulus 33.8 (23.4 [30]) 33.8 (23.4 [30]) 25.4 [30];33.0 33.0[31]) [31]) 25.4 (12.9 (12.9 [30]; 17.0 (23.2 [15]; 7.0 [30]) 17.0 (23.2 [15]; 7.0 [30])
30 25 20 15 10 5
0.3
0.4
0.5
w/c ratio
Figure 15. 15.Comparison Comparisonbetween between predicted Young’s modulus of cement paste and experimental Figure predicted Young’s modulus of cement paste and experimental results. results.
4. Conclusions 4. Conclusions This work presents an approach for the fitting and validation of micromechanical modelling of This work presents an approach for the fitting and validation of micromechanical modelling of mechanical properties and fracture mechanism of cement paste. A procedure for micro specimen mechanical properties and fracture mechanism of cement paste. A procedure for micro specimen preparation and testing of global performance using nano-indenter was developed and employed. preparation and testing of global performance using nano-indenter was developed and employed. As a result, fracture pattern under different loading depth and load displacement diagrams were As a result, fracture pattern under different loading depth and load displacement diagrams were obtained. Two regimes were observed experimentally in the load displacement diagram. Since the obtained. Two regimes were observed experimentally in the load displacement diagram. Since the displacement control of the nano-indenter equipment is not fast enough to capture the fast decrease in displacement control of the nano-indenter equipment is not fast enough to capture the fast decrease load when the specimen fails, a horizontal line existed in regime (II). Thus, the ascending branch in in load when the specimen fails, a horizontal line existed in regime (II). Thus, the ascending branch regime (I) was applied to calibrate the microstructure-informed lattice model. The softening regime in regime (I) was applied to calibrate the microstructure-informed lattice model. The softening was observed in the modelling result. regime was observed in the modelling result. The 3D lattice model was built up based on a microstructure of cement paste obtained from X-ray The 3D lattice model was built up based on a microstructure of cement paste obtained from computed tomography. The mechanical properties of local phases were calibrated by validating the X-ray computed tomography. The mechanical properties of local phases were calibrated by simulated results with the experimental results. It is shown in this study that the adopted mechanical validating the simulated results with the experimental results. It is shown in this study that the properties of local phases are critical for the investigation in terms of load displacement response adopted mechanical properties of local phases are critical for the investigation in terms of load and failure mechanism. Therefore, it is of great importance to fit these parameters by designing displacement response and failure mechanism. Therefore, it is of great importance to fit these experiments on the specimens in the same size as well as under well-controlled boundary conditions. parameters by designing experiments on the specimens in the same size as well as under Furthermore, the calibrated lattice model was applied to predict the fracture performance of the well-controlled boundary conditions. micro cement paste cubes with various w/c ratios under uniaxial tension. Since an experimental Furthermore, the calibrated lattice model was applied to predict the fracture performance of the method was applied to calibrate the input mechanical properties of local phases, the proposed method micro cement paste cubes with various w/c ratios under uniaxial tension. Since an experimental in this paper gives more reliable prediction results. The predicted outcome can be used as an input method was applied to calibrate the input mechanical properties of local phases, the proposed in the multi-scale modelling of the failure mechanism in cement based materials in further study. method in this paper gives more reliable prediction results. The predicted outcome can be used as an The method adopted in this study also illustrates a basic investigation for future research to build input in the multi-scale modelling of the failure mechanism in cement based materials in further upon, and an integrity system for the upscaling of the modelling and validation on every scale. study. The method adopted in this study also illustrates a basic investigation for future research to build upon, and anThis integrity system for the upscaling of the modelling and validation every (CSC). scale. Acknowledgments: work is supported in part by the scholarship from China Scholarshipon Council The authors would also like to acknowledge the help of Arjan Thijssen with SEM and XCT experiments.
Acknowledgments: This work is supported in part by the scholarship from China Scholarship Council (CSC). The authors would also like to acknowledge the help of Arjan Thijssen with SEM and XCT experiments.
Materials 2016, 9, 907
13 of 14
Author Contributions: Hongzhi Zhang, Branko Šavija, Stefan Chaves Figueiredo, Mladena Lukovic and Erik Schlangen performed the analyses and wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.
References 1. 2. 3. 4. 5. 6. 7. 8.
9.
10.
11.
12.
13.
14. 15. 16. 17.
18. 19. 20.
Mehta, P.K.; Monteiro, P.J. Concrete Structures, Properties and Materials; McGraw-Hill Education: Englewood Cliffs, NJ, USA, 1993; Volume 9. Van Mier, J.G. Concrete Fracture: A Multiscale Approach; CRC Press: Boca Raton, FL, USA, 2012. Van Breugel, K. Numerical simulation of hydration and microstructural development in hardening cement-based materials: (II) applications. Cem. Concr. Res. 1995, 25, 522–530. [CrossRef] Ye, G.; Van Breugel, K.; Fraaij, A. Three-dimensional microstructure analysis of numerically simulated cementitious materials. Cem. Concr. Res. 2003, 33, 215–222. [CrossRef] Bishnoi, S.; Scrivener, K.L. µic: A new platform for modelling the hydration of cements. Cem. Concr. Res. 2009, 39, 266–274. [CrossRef] Bullard, J.W.; Garboczi, E.J. A model investigation of the influence of particle shape on portland cement hydration. Cem. Concr. Res. 2006, 36, 1007–1015. [CrossRef] Scrivener, K.L. Backscattered electron imaging of cementitious microstructures: Understanding and quantification. Cem. Concr. Compos. 2004, 26, 935–945. [CrossRef] Hall, C.; Colston, S.L.; Jupe, A.C.; Jacques, S.D.; Livingston, R.; Ramadan, A.O.; Amde, A.W.; Barnes, P. Non-destructive tomographic energy-dispersive diffraction imaging of the interior of bulk concrete. Cem. Concr. Res. 2000, 30, 491–495. [CrossRef] Gallucci, E.; Scrivener, K.; Groso, A.; Stampanoni, M.; Margaritondo, G. 3D experimental investigation of the microstructure of cement pastes using synchrotron X-ray microtomography (µct). Cem. Concr. Res. 2007, 37, 360–368. [CrossRef] Chotard, T.; Boncoeur-Martel, M.; Smith, A.; Dupuy, J.; Gault, C. Application of X-ray computed tomography to characterise the early hydration of calcium aluminate cement. Cem. Concr. Compos. 2003, 25, 145–152. [CrossRef] Bentz, D.; Quenard, D.; Kunzel, H.; Baruchel, J.; Peyrin, F.; Martys, N.; Garboczi, E. Microstructure and transport properties of porous building materials. II: Three-dimensional X-ray tomographic studies. Mater. Struct. 2000, 33, 147–153. [CrossRef] Promentilla, M.A.B.; Sugiyama, T.; Hitomi, T.; Takeda, N. Quantification of tortuosity in hardened cement pastes using synchrotron-based x-ray computed microtomography. Cem. Concr. Res. 2009, 39, 548–557. [CrossRef] Zhang, M.; He, Y.; Ye, G.; Lange, D.A.; van Breugel, K. Computational investigation on mass diffusivity in portland cement paste based on X-ray computed microtomography (µct) image. Constr. Build. Mater. 2012, 27, 472–481. [CrossRef] Hu, C.; Li, Z. Micromechanical investigation of portland cement paste. Constr. Build. Mater. 2014, 71, 44–52. [CrossRef] Constantinides, G.; Ulm, F.J. The effect of two types of csh on the elasticity of cement-based materials: Results from nanoindentation and micromechanical modeling. Cem. Concr. Res. 2004, 34, 67–80. [CrossRef] Jennings, H.M.; Thomas, J.J.; Gevrenov, J.S.; Constantinides, G.; Ulm, F.J. A multi-technique investigation of the nanoporosity of cement paste. Cem. Concr. Res. 2007, 37, 329–336. [CrossRef] Hughes, J.J.; Trtik, P. Micro-mechanical properties of cement paste measured by depth-sensing nanoindentation: A preliminary correlation of physical properties with phase type. Mater. Charact. 2004, 53, 223–231. [CrossRef] Manzano, H.; Dolado, J.; Guerrero, A.; Ayuela, A. Mechanical properties of crystalline calcium-silicatehydrates: Comparison with cementitious c-s-h gels. Phys. Status Solidi A 2007, 204, 1775–1780. [CrossRef] Manzano, H.; Dolado, J.S.; Ayuela, A. Structural, mechanical, and reactivity properties of tricalcium aluminate using first-principles calculations. J. Am. Ceram. Soc. 2009, 92, 897–902. [CrossRef] Bentz, D.P. Three-dimensional computer simulation of portland cement hydration and microstructure development. J. Am. Ceram. Soc. 1997, 80, 3–21. [CrossRef]
Materials 2016, 9, 907
21. 22. 23.
24. 25. 26. 27. 28. 29. 30. 31. 32.
33. 34.
35. 36. 37. 38. 39. 40. 41. 42. 43.
14 of 14
Bittnar, Z. Microstructure-based micromechanical prediction of elastic properties in hydrating cement paste. Cem. Concr. Res. 2006, 36, 1708–1718. Hain, M.; Wriggers, P. Numerical homogenization of hardened cement paste. Comput. Mech. 2008, 42, 197–212. [CrossRef] Valentini, L.; Parisatto, M.; Russo, V.; Ferrari, G.; Bullard, J.W.; Angel, R.J.; Dalconi, M.C.; Artioli, G. Simulation of the hydration kinetics and elastic moduli of cement mortars by microstructural modelling. Cem. Concr. Compos. 2014, 52, 54–63. [CrossRef] Schlangen, E.; Qian, Z. 3D modeling of fracture in cement-based materials. J. Multiscale Model. 2009, 1, 245–261. [CrossRef] Asahina, D.; Landis, E.; Bolander, J. Modeling of phase interfaces during pre-critical crack growth in concrete. Cem. Concr. Compos. 2011, 33, 966–977. [CrossRef] Grassl, P.; Jirásek, M. Meso-scale approach to modelling the fracture process zone of concrete subjected to uniaxial tension. Int. J. Solids Struct. 2010, 47, 957–968. [CrossRef] Mohammadipour, A.; Willam, K. Lattice approach in continuum and fracture mechanics. J. Appl. Mech. 2016, 83, 071003. [CrossRef] Asahina, D.; Aoyagi, K.; Kim, K.; Birkholzer, J.T.; Bolander, J.E. Elastically-homogeneous lattice models of damage in geomaterials. Comput. Geotech. 2017, 81, 195–206. [CrossRef] Schlangen, E. Experimental and Numerical Analysis of Fracture Processes in Concrete. Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands, 1993. Qian, Z. Multiscale Modeling of Fracture Processes in Cementitious Materials. Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands, 2012. Lukovi´c, M.; Schlangen, E.; Ye, G. Combined experimental and numerical study of fracture behaviour of cement paste at the microlevel. Cem. Concr. Res. 2015, 73, 123–135. [CrossRef] Schlangen, E.; Lukovic, M.; Šavija, B.; Copuroglu, O. Nano-indentation testing and modelling of cement paste. In Proceedings of the 10th International Conference on Mechanics and Physics of Creep, Shrinkage, and Durability of Concrete and Concrete Structures, Vienna, Austria, 21–23 September 2015. Zhang, J.; Scherer, G.W. Comparison of methods for arresting hydration of cement. Cem. Concr. Res. 2011, 41, 1024–1036. [CrossRef] Poelma, R.H.; Morana, B.; Vollebregt, S.; Schlangen, E.; van Zeijl, H.W.; Fan, X.; Zhang, G.Q. Tailoring the mechanical properties of high-aspect-ratio carbon nanotube arrays using amorphous silicon carbide coatings. Adv. Funct. Mater. 2014, 24, 5737–5744. [CrossRef] Zhang, M.; Jivkov, A.P. Microstructure-informed modelling of damage evolution in cement paste. Constr. Build. Mater. 2014, 66, 731–742. [CrossRef] Tennis, P.D.; Jennings, H.M. A model for two types of calcium silicate hydrate in the microstructure of portland cement pastes. Cem. Concr. Res. 2000, 30, 855–863. [CrossRef] Van Breugel, K. Simulation of Hydration and Formation of Structure in Hardening Cement-Based Materials. Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands, 1991. Schlangen, E.; Garboczi, E. Fracture simulations of concrete using lattice models: Computational aspects. Eng. Fract. Mech. 1997, 57, 319–332. [CrossRef] Hlobil, M.; Šmilauer, V.; Chanvillard, G. Micromechanical multiscale fracture model for compressive strength of blended cement pastes. Cem. Concr. Res. 2016, 83, 188–202. [CrossRef] Nˇemeˇcek, J.; Králík, V.; Šmilauer, V.; Polívka, L.; Jäger, A. Tensile strength of hydrated cement paste phases assessed by micro-bending tests and nanoindentation. Cem. Concr. Compos. 2016, 73, 164–173. [CrossRef] Haecker, C.-J.; Garboczi, E.; Bullard, J.; Bohn, R.; Sun, Z.; Shah, S.; Voigt, T. Modeling the linear elastic properties of portland cement paste. Cem. Concr. Res. 2005, 35, 1948–1960. [CrossRef] Schlangen, E. Crack Development in Concrete, Part 1: Fracture Experiments and CT-Scan Observations; Key Eng. Mater.; Trans Tech Publications: Stafa-Zurich, Switzerland, 2008; pp. 69–72. Schlangen, E. Crack Development in Concrete, Part 2: Modelling of Fracture Process; Key Eng. Mater.; Trans Trans Tech Publications: Stafa-Zurich, Switzerland, 2008; pp. 73–76. © 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).
