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Transactions on Modelling and Simulation vol 21, © 1999 WIT Press, www.witpress.com, ISSN 1743-355X Micromechanical behaviour of concrete interpreted by computer simulation systems for material structure P. Stroeven & M. Stroeven Delft University of Technology, Delft, The Netherlands Email: p. stroeven@ct. tudelft. nl m. stroeven@ct. tudelft. nl Abstract Interest in material structure has been raised by upgrading of cementitious systems. Computer-simulation of material structure has been pursued so far by using random generators. Such systems were successfully applied for estimating global properties. Realistic estimation of structure-sensitive properties, however, requires a more appropriate simulation of material structure. The SPACE system has been developed for that purpose. It incorporates particle interaction and gravitational influences, which are typical for the 'natural' processes undelying the formation of material structure in the production stage. The capabilities are discussed of the two computer-simulation methods for generating structural information relevant for either structureinsensitive, or structure-sensitive properties in bulk and in the interfacial transition zone (ITZ). The structural information will deal with composition and configuration of the material, respectively. The use of random generator-based (KG) systems is demonstrated not justified when dealing with structure-sensitive problems. Transactions on Modelling and Simulation vol 21, © 1999 WIT Press, www.witpress.com, ISSN 1743-355X 572 Computational Methods and Experimental Measurements 1 Introduction A growing interest in material structure is resulting from efforts to upgrade concrete materials. Such efforts involve the addition of silica fume and other (super-)fine mineral admixtures to concrete, the blending of cements, or the increase in the range of particle sizes of the aggregate. A realistic structure-simulation systems would offer insight into the relevant features of the aggregation of particulate matter, ultimately even setting the scene for systematic optimization of such materials. Close to the mould, material structure on mesolevel should deviate from that in bulk. Surfaces of aggregate particles will similarly disturb the 'natural' processes of aggregation of matter on microlevel. Mostly, experimental approaches focus on compositional changes in the vicinity of interfaces. Structure-sensitive properties of the material will depend, however, on material configuration. Insight into material configuration is therefore a pre-requisite in understanding mechanical behaviour of such composites. This holds in particular for the post-peak range, where the non-uniformity of the packing is of greater significance than any regular property. In 1927, Brandzaeg* already stated: "The process of failure must necessarily be intimately connected with the properties of small parts of the material, in particular those small parts where failure first begins to develop. It would seem the individual rather than the average properties of the constituent parts of the material need to be considered." The common way to simulate particulate composites is by using so called random generator (RG) systems. This approach was proven successful for simulating structural features relevant for global pre-peak concrete behaviour. However, it still has to be established whether local structural characteristics, which are relevant for structure-sensitive properties like (meso)crack initiation, or (macro)crack concentration in the post-peak region, can be realistically simulated as well. Since major contributions to local strength in cementitous materials are due to van der Waals binding forces, the spacing of neighbouring particles will be of paramount importance. Moreover, once bond cracks have been formed at particle-matrix interfaces, the process of coalescence which ultimately will lead to fracture, will also be governed by (particle) spacing. So, for the estimation of such structure-sensitive phenomena, the nearest neighbour spacing should Transactions on Modelling and Simulation vol 21, © 1999 WIT Press, www.witpress.com, ISSN 1743-355X Computational Methods and Experimental Measurements 573 be realistically simulated. Since this is expected to be impossible by RG systems, recently a dynamic simulation system, SPACE, has been developed. The differences in output of these two systems will be discussed on the basis of implementations for structure-insensitive ('micro-hardness') and structure-sensitive properties ('bond'). The particulate material will be modelled for that purpose as a multisized aggregate of spherical particles. 2 Static simulation by RG system. The non-homogeneous, or granular, nature of the internal material structure is represented by a set of distinct elements. Each element corresponds to a distinguishable, characteristic phase in the material. For example, concrete is modelled as a set of elements representing the aggregate particles. The elements are dispersed in a presumably homogeneous mortar matrix moulded in a container. The parameters describing the static conditions of the internal structure are the locations, orientations, and shapes of the various individual elements. Since the elements represent real physical phases in the material, physical properties can be assigned to each element along with its shape and size, at least in principle. The most difficult task, however, is the derivation of the location and the orientation of these non-overlapping elements. In a static simulation system, the particles will be sequentially located inside a container. Each location is governed by randomly generated coordinates. 'Overlap' with earlier generated particles will result in rejection, whereupon the generation process is continued. The number of rejections will increase dramatically at high volume fractions, making the process of generation very time-consuming. Also, simulation of a multi-size particle composite requires starting with the largest particles in the mix. But maximum particle densities cannot be realized even under such conditions. The system obviously excludes the mechanism of particle interaction, which is so characteristic for the production stage of cementitious materials. Finally, a natural phenomenon as clustering will be under-estimated, resulting in a biased nearest neighbour distribution of particles, as will be demonstrated later. Three of the more popular systems in this category were developed by Roelfstra^, by Diekkamper^, and at NIST^. Transactions on Modelling and Simulation vol 21, © 1999 WIT Press, www.witpress.com, ISSN 1743-355X 574 Computational Methods and Experimental Measurements 3 Dynamic simulation by SPACE system. The SPACE system (Software Package for the Assessment of Compositional Evolution) is based on a dynamic concept of simulating the production process of composite material, i.e., in case of concrete: mixing of aggregates and paste, or cement and water. The details of the generation system have been published by this authors elsewhere^. To be able to simulate effects such as clustering and to reach high volume densities, element motion and inter-element collision^ are modelled, leading to effective ingredient mixing. This stage is referred to as the dynamic stage. The general simulation concept can be described as follows: • Initially, a structured or random 3-D dilute distribution of elements with pre-defined shape and size distribution (such as the grading curve) is generated within the boundaries of a container. This stage can be similar to that of a RG system. Fig. la presents an example of the starting situation of a multi-sized 'dilute' mixture of spherical particles inside a cubic container (Vy w 0.2). Fig. 1. Development of participate structure in SPACE system Transactions on Modelling and Simulation vol 21, © 1999 WIT Press, www.witpress.com, ISSN 1743-355X Computational Methods and Experimental Measurements 575 Next, random linear and rotational velocity vectors are assigned to each element. • The second step, the actual dynamic stage, is an iterative procedure where location and orientation of all elements are changed at each time step according to a Newtonian motion model. This motion model relates the element's linear and angular displacement to a set of conditions enforced on the element (e.g., gravity, friction, etc.). When elements meet during this time interval, a contact model defines the effect of contact on the motion/rotation update. Inter-particle influences - one of the requirements for achieving a more versatile system - are incorporated in the contact model. Additionally, electrical or chemical inter-particle forces may be added to the motion model. Higher densities are obtained by a gradual reduction in the size of the container (compacting the structure), or under the action of a gravity force. • Finally, the iteration stops when certain conditions are reached. As an example, Fig. Ib shows the multi-sized particle system of Fig. la , which was compacted to its maximum density (in this case, Vv ~ 0.72), and thereupon magnified for visual purposes. The final distribution state can incorporate effects due to gravity and to inter-particle influences. Simple physical laws may be used to define the inter-particle relationships, by introducing parameters such as specific mass and energy dissipation. High packing densities can be obtained, representing dense random packing states, typical of aggregate and binder particles in cementitious materials. 4 Structure-insensitive approach The first example concerns a study of the packing density of multisized, spherical cement grains in the neighbourhood of a rigid boundary. Hence, this study deals with the compositional discontinuity in the packing of binder particle in the so called interfacial transition zone (ITZ). The particle size distribution of Portland cement particles is taken in accordance with the Rosin-Rammler cumulative particle size distribution function, G(d) = 1 - exp(-W). A mixture was simulated by both systems (RG and SPACE) representing a water to cement ratio of 0.5. Note that simulation by the RG system of more densely packed aggregates will be very complicated. 30,000 Transactions on Modelling and Simulation vol 21, © 1999 WIT Press, www.witpress.com, ISSN 1743-355X 576 Computational Methods and Experimental Measurements particles between 1 and 20 /mi were considered for parameter settings of n= 1.003 and 6=0.0288. Total volume fraction of the cement particles is plotted as function of the distance to the interface in Fig. 2. The extent of the discontinuity in volume fraction is slightly different for both systems. The compositional differences between the two systems are not dramatic. 0.45 __ 0.4 ^0.35 1 0-3 "So.25 • rH | 0.2 | 0.15 ^ 0.1 RG experimental SPACE 0.05 0 0 2 4 6 8 10 12 Distance to the interface in [fim] 14 Fig. 2. Total volume fraction changes in the ITZ of multi-sized cement particle mixture. A popular way to study packing density differences in the ITZ is by microhardness measurements. If it is assumed for the present purpose that 'hardness' is similar for all cement particles, then outcomes of microhardness tests can be associated with volume fraction of particles^. It should be emphasized that the Knoop indentor 'averages' over its width, b in a direction perpendicular to the interface. With the high hardness value of the aggregate particle influencing the outcomes for small distances to the interface, r < 6/2, a through-type of micro-hardness curve can be derived from the simulation data of Fig. 2. Such microhardness characteristics are also reported in the lit- Transactions on Modelling and Simulation vol 21, © 1999 WIT Press, www.witpress.com, ISSN 1743-355X Computational Methods and Experimental Measurements 577 erature for the ITZ™. So, either one of the systems can be employed for estimating structure-insensitive properties, such a micro-hardness. 5 Structure-sensitive approach The particle fractions underlying Fig. 2 have been demonstrated by Stroeven & Stroeven? to reveal a size-dependentfluctuatingbehaviour away from the interface, causing a long-range discontinuity in grading. This will also hold for material configuration underlying structure-sensitive problems. The volume fraction curve in Fig. 2, which is the common way of presenting computer-simulation data, gives as a consequence a misleading impression of structural variation relevant for structure-sensitive properties. The most relevant configuration parameter is probably the nearest neighbour distance. This parameter allows the assessment of the respective capabilities of the two structural simulation systems when studying structure-sensitive properties, such as crack initiation. Initiation of cracking will be mainly at interfaces between aggregate particles in concrete. It will be the result of stresses exceeding local (physical) bond capacity. Van der Waals bonding forces will be governed by surface-to-surface spacing of cement particles, or of the cement particles and the aggregrate grain interface. The nearest neighbour distance is intimately related to surface spacing, so it can be used for the present purpose. A common analytical approach in case of finite particle sizes is to reject particle 'overlap'**. Hence, this is the same procedure as used by RG systems. This leads to more evenly distributed particles than met in nature, and an under-representation of the natural process of particle 'clustering'. Conclusive evidence for these statements can be found in different fields of science. Supporting experimental data in the field of concrete are given in Stroeven^, in which the nearest neigbour distance distribution is derived from quantitative image measurements in serial sections. Fig. 3 convincingly demonstrates the growing gap between the two structural simulation concepts (i.e. dynamic particle interference versus overlap rejection) when estimating the nearest neighbour distance distribution at increasing volume densities up to only moderate values of volume fraction. Data confirming the capability of the SPACE system to simulate the natural process of particle clustering is presented Fig. 4. A mix- Transactions on Modelling and Simulation vol 21, © 1999 WIT Press, www.witpress.com, ISSN 1743-355X 578 Computational Methods and Experimental Measurements • SPACE RG (analytical) 3.1 3.2 33 3.5 3.6 3.73.8 Fig. 3. Comparison between simulations by analytical, random generator-based approach, and by SPACE system on the basis of the shape of the nearest neighbour distance distribution for particulate aggregates with volume fraction of 1% and 30%. ture of three different particle sizes (di=3, 6^=5 mm and d^—l mm) is generated by the SPACE system for increasing total volume fractions, W, of the aggregate. A strong tendency for particle clustering when particle density increases is reflected by the distinct peaks in the nearest neighbour distance distribution curve for the highest density case (35%). The peaks correspond to zero surface-to-surface spacings of the different combinations of particle sizes. Thus, only the SPACE system is able to simulate particle configuration in agreement with experimental findings. Even for moderate particle densities, the RG systems will erroneously simulate nearest neighbour spacing. In another example, the mean free path, A, is determined for the same mixture as used for Fig. 2. The mean free path is the average of all unobstucted distances between a particle and all of its neighbours in an aggregate of finite size particles. This parameter was proposed by Pullman*\ and employed by Stroeven for experimental assessment of particle packing characteristics in concrete^. A global notion of inter-particle 'bond capacity' might be based for the present purpose on this structural parameter. Dependence of this approach on particle configuration is only moderate. The parameter A is selected, because it can be determined by stereological (i.e. quantitative microscopy) measurements in sections parallel to the interface, i.e. A - 4(1 - Vy)/Sv, in which Vy and 6y denote Transactions on Modelling and Simulation vol 21, © 1999 WIT Press, www.witpress.com, ISSN 1743-355X Computational Methods and Experimental Measurements 579 1 0.9 R=2.5mm V: = I : 4.6 : 12.6 0.8 0.7 0.6 Vw=35% 0.5 0.4 0.3 0.2 0.1 - 0 10 12 14 16 mm Fig. 4. Nearest neighbour distance distributions for aggregate mixture composed of three particle sizes and a total volume content of 5, 15 and 35%, respectively. the volume fraction and the specific surface area of the particles, respectively. The global bond capacity is taken proportional to A~^ in conformity with a van der Waals concept**. Fig. 5 presents this normalized inter-particle bond capacity as a function of distance to the interface. Characteristic differences between the two systems are reflected, despite the only moderate volume density required by the RG system. As stated earlier, the fines concentrate near the interface^. Because of their large specific surface area, global inter-particle bond capacity is rising steeply over thefirstpart of the curve. Random generators increase, however, the degree of inter-particle 'order', because clustering is suppressed. This is reflected by the dramatic underrepresentation of close nearest neighbours among particles in Fig. 3, and between particles and interface in Fig. 2. The mean free spacing is reduced, as a result, in the zone of high particle density. Consequently, the RG system seriously under-estimates the global interparticle bond capacity in the vicinity of the interface, as depicted by Transactions on Modelling and Simulation vol 21, © 1999 WIT Press, www.witpress.com, ISSN 1743-355X 580 Computational Methods and Experimental Measurements 0.45 0.4 0.35 0.3 0.25 0.2 RG 0.15 0.1 experimental SPACE 0.05 0 0 2 4 6 8 10 12 Distance to the interface in [|itm] 14 Fig. 5. Normalized global inter-particle bond capacity in the ITZ, expressed by A~^, as function of the distance to the interface. Fig. 5. Because of the same reason, the bond between particles and interface layer will be seriously under-estimated by the RG systems (not shown in Fig. 5). The long-range fluctuating grading characteristics with distance to the interface, presented in \ are reflected by the post-peak curve of the SPACE system. So, even for this weakly structure-sensitive approach, RG systems would yield biased information. Additional experimental data should be generated to further validate this statement. Conclusions Computer-simulation allows for accurate assessment of compositional parameters characterizing particulate structures in cementitious materials, either on mesolevel where aggregate grains are packed in a container, or on microlevel where (blended) Portland cement particles are dispersed in pockets between these aggregate grains. Bulk Transactions on Modelling and Simulation vol 21, © 1999 WIT Press, www.witpress.com, ISSN 1743-355X Computational Methods and Experimental Measurements 581 properties, or discontinuities near boundaries, can be studied in this way. Compositional parameters should be used in connection with structure-insensitive properties, such as volume percentage of particles, or porosity. The paper shows how microhardness measurements in the ITZ can be estimated properly, at least qualitatively, in this way. Such compositional problem can be approached by random generator-based systems and by the SPACE system alike. RG-based systems generate more evenly distributed particles, thereby under-estimating the natural phenomenon of particle clustering. Inter-particle distances are affected by the rejection mechanism following 'overlap' during generation. The nearest neighbour distance distribution in moderately to densely packed particulate structures (as met in cementitious materials) simulated by RG systems, will be significantly biased, as a result. Hence, material configuration in such dense random packings is poorly represented by RG-based systems. Structure-sensitive properties, like van der Waals bond between densely packed hydrated Portland cement particles, or between such particles and the interface, can only be properly simulated by the SPACE system. References 1 Brandzaeg, A., Failure of a material composed of non-isotropic elements: an analytical study with special reference to concrete, Del. Kong. Norske Vidensk. Selsk. Skr., 2, pp. 1-25, 1927 2 Roelfstra, P.E., A numerical approach to investigate the properties of numerical concrete, PhD Thesis, EPFL-Lausanne, 1989. 3 Diekkamper, R., Ein Verfahren zur numerischen Simulation des Bruch- und Verformungsverhaltens sproder Werkstoffe, Techn. Wissenschaftliche Mitteilungen der Inst. fur Konstruktiven Ingenieursbau, nr. 84-7, Ruhr Univ. Bochum, 1984. 4 Bentz, D.P., Garboczi, E.J. & Stutzman, P.E., Computer modelling of the interfacial zone in concrete, Interfaces in Cementitious Composites ed. J.C. Maso, E & FN Spon, London, pp. 107-116, 1993. Transactions on Modelling and Simulation vol 21, © 1999 WIT Press, www.witpress.com, ISSN 1743-355X 582 Computational Methods and Experimental Measurements 5 Stroeven, M. and Stroeven, P., Computer-simulated Internal Structure of Materials, Acta StereoL, 15, 3, pp. 247-252, 1996. 6 Brach, R.M., Mechanical Impact Dynamics, Rigid Body Collisions, John Wiley & Sons Inc, New York, 1991. 7 Stroeven, M. & Stroeven, P., Computer-simulation of particle packing in cementitious systems, Proc. Int. Conf. on HPC and Performance and Quality of Concrete Structures, June 14, 1999, Gramado, Brazil (to be published by ACI) 8 Stroeven, M. and Stroeven, P., Simulation of Hydration and the Formation of Microstructure, Computational Plasticity, eds. D.RJ. Owen, E. Onate & E. Hinton, Barcelona, CIMNE, pp. 981-987, 1997. 9 Kozak, A. and Modry, S., Relation between Pore Structure and Microhardness of Hardened Cement Pastes, Pore Structure and Properties of Materials, Academia, Prague, D139-D148, 1974. 10 Wang, Y., The effect of bond characteristics between steel slag fine aggregate and cement paste on mechanical properties of concrete and mortar, Bonding in Cementitious Composites, eds. S. Mindess & S.P. Shah, MRS Symp. Proc., 114, pp. 49-54, 1988. 11 Bansal, P.P., Ardell, A.J., Average nearest neighbor distances between uniformly distributed finite particles, Metallography, 5, pp. 97-111, 1972. 12 Fullman, R.L., Measurement of particle sizes in opaque bodies, Trans. Met. Soc. A.I.M.E., 197, March, pp. 447-452, 1953. 13 Stroeven P., Some Aspects of the Micromechanics of Concrete, PhD Thesis, Delft Univ. Technology, Univ. Press, Delft, 1973. 14 Wittmann, F., Experiments to determine van der Waals forces, Structure, Solid Mechanics and Engineering Design, ed. M. Te'eni, Wiley-Interscience, London, pp. 295-300, 1971