SIMULATING FRESH CONCRETE BEHAVIOUR ESTABLISHING A LINK BETWEEN THE BINGHAM MODEL AND PARAMETERS OF A DEM-BASED NUMERICAL MODEL

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1 International RILEM Conference on Material Science MATSCI, Aachen 2010 Vol. II, HetMat 211 SIMULATING FRESH CONCRETE BEHAVIOUR ESTABLISHING A LINK BETWEEN THE BINGHAM MODEL AND PARAMETERS OF A DEM-BASED NUMERICAL MODEL S. Shyshko, V. Mechtcherine, Institute of Construction Materials, TU Dresden, Germany ABSTRACT: The rheological behaviour of fresh concrete can be described reasonably well by the Bingham model, whose parameters can be used directly in numerical simulations by means of the Single Fluid approach. In many instances, however, it is advantageous to use the Distinct Element Method (DEM) instead of the Single Fluid approach. The difficulty is that yield stress and plastic viscosity, the parameters of models based on DEM, are not rheological constants and they cannot be obtained directly from experiments. Hence, the aim of the ongoing research project is to establish a link between the rheological properties of fresh concrete and the parameters of a DEM-based model. In this paper, an algorithm is presented to derive the model parameters connected to yield stress according to the Bingham model. For this an analytical prediction of stress-distribution at the beginning of the slump-flow test is used as reference for the corresponding numerical analysis. For the sake of verification, the analytical and numerical predictions of the final shape of the concrete cake are compared. 1 INTRODUCTION The workability of fresh concrete, i.e. its fluidity, compactability, ability to be pumped etc., all depend on its rheological properties. Knowledge of rheological properties and the ability to use them in the simulation of concrete behaviour open great opportunities in optimising concrete production and construction processes. The behaviour of fresh concrete is most often described using equation (1.1) according to Bingham: τ = τ 0 + μγ& (1.1) where τ shear stress applied to the material γ& shear strain rate τ 0 yield stress μ plastic viscosity. The last two terms are the Bingham model s rheological constants, which are assumed to characterise the rheological behaviour of fresh concrete properly. When simulating the flow behaviour of concrete using the Single Fluid approach the parameters of the Bingham model can be directly used as input for the simulation software. However, in many instances the Single Fluid approach is not an appropriate tool in simulating the behaviour of the material since many phenomena occurring during concrete work result from the concrete s heterogeneity, which cannot be reproduced adequately unless a sufficient degree of material meo-structure is modeled explicitly. The Distinct Element Method (DEM) provides indeed a better basis for simulating such important phenomena as, for example, passing among reinforced steel bars, blockage, segregation, fibre orientation, rebound of shortcrete.

2 212 SHYSHKO, MECHTCHERINE: Simulating fresh concrete behaviour Establishing a link between Previous publications by the authors [Shy06, Mec07] showed that the Particle Method, which is a variation of the Distinct Element Method, provided qualitatively sound results in principle and represented decisive phenomena correctly, as observed under experiment. A challenge in simulating concrete using DEM results from model parameters not being rheological constants and their not being directly obtainable from experiments. An algorithm for estimating model parameters was proposed in [Shy08]; however this algorithm does not work with rheological constants but with the results of slump or slump-flow tests. The goal of the ongoing research is to establish a link between the parameters of a DEMbased model and the rheological constants of fresh concrete. This paper describes the derivation of yield stress by adjusting the bond strength parameter of the DEM model in a way such that the numerical prediction corresponds to the prediction provided by the analytical solution obtained for a given yield stress. The slump-flow test is used as a reference basis for the analysis. Additionally, test geometry is considered both in the numerical and analytical predictions for the sake of verification. Firstly, the paper introduces the procedure for the analytical prediction of the concrete behaviour in slump or slump-flow test. Subsequently, the DEM model is presented and a link between the analytically predicted stress distribution at the beginning of the test and the bond strength parameter of the DEM model is established. Finally, the results of the numerical prediction with DEM model are verified by comparing them with analytical prediction of the final shape of the concrete cake. 2 ANALYSES OF STRESSES AND DEFORMATIONS IN THE SLUMP- FLOW TEST The slump-flow test is based on measurement of the deformation of self-compacting concrete due to the force of gravity on the concrete s mass. According to the assumption of the original Murata model [Mur84], the vertical stress p acting on any horiontal layer at a height of in a slump cone is: p W = (2.1) πr 2 where W weight of the material from height to the top of the slump cone r radius of the cone at height (Fig. 2.1). Applying the Tresca criterion (the shear stress acting on a plane is half of the applied normal stress), the shear stress in concrete at the height can be expressed for the given cone geometry according to equation (2.2): g ( H ) 2 2 τ = ρ ( r ) 2 + r rt + rt (2.2) 6 r where r t top radius of the cone r radius of the cone at height H total height of the cone ρ specific weight of concrete g gravity constant.

3 International RILEM Conference on Material Science MATSCI, Aachen 2010 Vol. II, HetMat 213 r t d r b r H 0 τ 0 τ max τ S r t r sp h 0 h 1 τ 0 II I τ H Fig (a) Stress distributions in concrete (a) at the beginning and (b) at the end of the slump (or slump-flow) test. (b) The maximal shear stress τ max acts at the bottom of the cone (r = r b, where r b is the bottom radius of the cone), while the shear stress is 0 at the top surface. It is assumed that the lifting and removal of the cone as such does not deform concrete in any way. The initial shape of concrete is therefore assumed to be a perfect truncated cone. At some point along the height of the undeformed cylinder, the material experiences a stress that is larger than the yield stress τ 0, and the material below this point flows until the stress acting at that point is reduced to the yield stress. In the material above the yielded region, the vertical stress does not exceed the yield stress and the region remains undeformed. It is further assumed that all horiontal sections remain horiontal, i.e. deformation process occurs only due to radial flow. Therefore, the interface layer between the yielding and unyielding material is a horiontal surface that moves down as the material beneath it flows. Using the simplifications assumed, there is no friction between concrete and the horiontal surface. The final shape of the concrete cake consists of deformed (up to height h 1 ) and undeformed (the remainder, up to height h 0 ) parts. Fig. 2.1 shows schematically the stress distributions in concrete at the beginning (a) and at the end (b) of the slump (or slump-flow) test. 3 DEM MODEL The Particle Flow Code ITASCA [Ita04] was used in this investigation as the basic program enabling the modelling of the movement (translation and rotation) of distinct particles. This includes their interactions, separation, and automatic contact detection [Kon06]. Only two basic elements were used: spherical particles to see the concrete as a matrix of small, discrete bodies, and walls to simulate the boundaries. For more detailed information on the particle method used may be found in the earlier publications by the authors [Shy06, Mec07, Shy08]. Constitutive relations related in their meaning to the Bingham formula were developed and implemented into the Particle Flow Code in order to describe the interactions between two neighbouring particles in simulating fresh concrete. Fig. 3.1 shows schematically the corresponding rheological models for the normal and tangential direction. They consist of the basic rheological elements spring, dashpot, and slider, which represent respectively the elastic, viscous, and frictional components of the particle interaction.

4 214 SHYSHKO, MECHTCHERINE: Simulating fresh concrete behaviour Establishing a link between (a) (b) Spring (stiffness) Dashpot (viscosity) Slider (friction) Contact (force-displacement relation) Fig Model for particle interaction: (a) normal direction and (b) tangential direction. The interaction model includes also the element contact, positioned serially in line with the basic rheological elements. The contact element enables the definition of the strength of the contact, the simulation of the loss of an old interaction due to the reaching of a certain distance between two particles, and the formation of a new interaction. Fig. 3.2 shows schematically the force-displacement relations as introduced by the authors for the contact elements in the normal direction and subsequently used in the numerical investigations. Force Softening regime Bond strength Yield force Overlap Tension mode Loss of contact Distance between particles Compression mode Fig The force-displacement relation for the contact element (normal direction). The force-displacement curve in tension mode is defined for small deformations by a very steep ascending branch, i.e., there is practically no deformation until a given force value (here yield force ) is reached. After reaching this force level there is only a slight increase in tensile force up to a defined ultimate force (bond strength) and then a linear decrease to ero in a kind of softening regime. When the tensile force becomes ero, the particles lose contact. The cohesive force holding the aggregates together arises out of the action of fresh, fine mortar. In this study the discrete particles simulate coarser aggregate grains, while fresh cement paste or fine mortar is represented by an adequate definition of the properties of the interaction among the individual particles. An algorithm for the choice of the model parameters and for fine tuning the model was presented in previous publications by the authors [Shy08]. It recommends starting by choosing an appropriate particle sie or particle sie distribution, in consideration of the representative aggregate sies serving as reference measures. The bond strength (for the contact element) should be chosen in the second step since it is the main parameter for characterising the interaction of the neighbouring particles. The corresponding bond value was proposed to be set empirically according to the type of concrete to be simulated (stiff ordinary concrete, SCC etc.). In the following a link between bond strength and the yield stress of the concrete according to the Bingham model is established.

5 International RILEM Conference on Material Science MATSCI, Aachen 2010 Vol. II, HetMat LINK BETWEEN YIELD STRESS AND BOND STRENGTH Before cone lifting, all particles are pressuried by the weight of the other particles and the reaction of the boundaries, i.e. cone walls (Fig. 4.1). The pressure induced by the particles weight is balanced by the reaction of the cone. The cone lifting disturbs this balance and the concrete cake changes its shape under action of the force of gravity until another state of equilibrium is attained. This is the case when gravity is balanced by internal forces of the material, primarily by the yield stress. r t 2 τ 0 3 H r b 1 4 Shear stress τ Average normal force Analytical solution Numerical simulation Fig Algorithm of the bond strength determination. As was shown in Chapter 3, the interaction of the two neighbouring particles in the normal direction is described by an ideal spring, a dashpot and a specific contact element. In a compressive regime the deformation of the spring is proportional to the magnitude of compressive force; some elastic energy is accumulated due to this deformation. At the beginning of cone lifting, when the undermost layer of particles loses contact with the cone, the compressive force between this layer and the cone wall vanishes. It means that particles on the surface lose the balancing contact with the cone and remain in contact with other particles (mostly in the interior of the concrete cake) only. Let us consider just two particles, one at the surface and a neighboring one in the interior, in contact with one another under unbalanced compression. If the position of the interior particle is fixed, the energy of the compressed spring will be transformed into kinetic energy of the surface particle s moving in a direction opposite to the direction of the previously acting compressive force. The assumption of the fixed interior particle is made here for the sake of easier description; in reality the interior particle is not fixed. At some moment the force of compression is completely released as the kinetic energy of the external particle. The particle continues its movement and stretches the spring. This is a reversible process since the kinetic energy is later transformed back into the potential energy of the stretched spring. In the case of an ideal spring and without considering damping effects, the potential energy of the compressed spring will be equal to the energy of the stretched spring. A system like this starts oscillating. If there is a damping force, the maximum potential energy decreases cycle by cycle to ero; the first cycle has the maximum amplitude.

6 216 SHYSHKO, MECHTCHERINE: Simulating fresh concrete behaviour Establishing a link between The qualitatively described process occurs on the whole free surface of the concrete cake. The force which keeps particles together under acting tensile force (stretched spring) is the contact bond force. If the potential energy of the surface particle is higher than the contact bond strength, the neighboring particles remain in contact, i.e. the movement of particles is limited by the contact bond; irreversible deformation of the material is not possible. However, if the tension force is higher than the contact bond, the contact bond fails (cf. Fig. 4.1). From a mechanical point of view, this means that the surface particle has more kinetic energy than is needed to stretch the spring and break the contact between the neighboring particles. The overhead of the kinetic energy is used for further movement of the particle. As a result of such particle movements, the deformation process of the material is simulated. The contact behavior of just two particles was sketched above. However, each particle has as a rule contact to more than one neighboring particle. Furthermore, the deformation occurs under the gravimetric force of the upper particle(s) wedging into the space between the neighboring particles positioned below. Generally it can be stated that the system of particles representing concrete is being plastically deformed when the acting tensile forces on the meo-level become higher than the contact bond strength. This basic principle was used first in developing the algorithm, which should provide a link between the yield stress of concrete according to the Bingham model and the bond strength parameter of the particle model. Fig. 4.1 shows this algorithm schematically. In the first step, the given input parameter yield stress τ 0 is introduced into equation (2.2) (cf. the analytical model presented in Chapter 2). The solution of the equation (Step 2) provides the height of the horiontal concrete layer in the cone (measured from the bottom of the cone), at which the shear stress level is equal to the given yield stress. One value of the yield stress τ 0 corresponds to just one height. Subsequently, the numerical simulation is performed for a chosen basic set of model parameters (particle sie and sie distribution, particle stiffness). The distribution of the normal compressive forces over the height of the cone is then calculated. The average force values for different heights, when connected to each other, form a curve which has in general a similar shape as the analytically derived stress distribution. In more detail, a special numerical procedure enables the scanning of existing contacts between particles and the sorting of these contacts according to their height in the cone into chosen height intervals (0-5mm, 5-10mm,..., mm from the bottom). Further, an average normal stress value is calculated for each height interval. As explained above, the compressive force calculated for each particular height in the cone corresponds to a tensile force which instantaneously develops in the material model at this height level when the cone is lifted. Of interest is the force acting at the level on which the yield stress is attained according to the analytical formula (Step 3). The average interaction force between the particles at this height must be equal to their bond strength in order to simulate plastic deformations; so the needed bond strength is derived (Step 4). 5 AN EXAMPLE AND VERIFICATION This chapter gives an example on the application of the developed algorithm and the first verification of its quality.

7 International RILEM Conference on Material Science MATSCI, Aachen 2010 Vol. II, HetMat 217 In the numerical simulation presented here, concrete was modelled by particles of different sies in such a way that a realistic grading curve of aggregates could be represented with good approximation. Aggregates consisted of three fractions: 3/4, 4/8 and 8/16 (it means that e.g. for the fraction 8/16 particle radius R i was between 4mm and 8mm, cf. Table 5.1). Aggregate grains with a diameter below 3 mm were not explicitly considered in the simulation in order to limit the computation time, which is directly proportional to the number of particles (increasing with decreasing particle sie). Table 5.1 gives an overview of the representative radii, the number of particles in each fraction, and the percentage of the fraction in the total volume of concrete. The absolute volume of each fraction, and hence the corresponding number of particles resulted from the volume of Abram s cone to be filled with virtual concrete. The filling procedure is described in [Shy06, Mec08]. Fig. 5.1a illustrates the distribution of the particles in the cone at the beginning of the calculation. Table 5.1. Chosen particle sie distribution representing grading curve of aggregates in SCC. Aggregate fraction [mm] Representative radius [mm] Number of particle in the slump-flow test % of total volume 3/ % 4/ % 8/ % Fig. 5.1b shows the contact forces acting between particles before cone lifting. The black colored lines represent compressive forces. The thickness of these lines is proportional to the force magnitude. Fig. 5.1c gives the distribution of the calculated contact forces over the height of the cone. Each point represents the position (height) of a contact between two particles and the corresponding value of the contact force. The grey dashed line in the diagram connects the points corresponding to averaged maximum values of the contact force (averaged contact force for 10 points having highest values). This curve is smoothed for the sake of clearer presentation; in reality it cannot be smooth because concrete is simulated in a discrete manner when using DEM τ 0 =50 Pa 200 Fig (a) (b) (c) Contact force The initial state of the numerical simulation: concrete particles in the cone (a), normal contact forces (b) and (c) contact force distribution. 0

8 218 SHYSHKO, MECHTCHERINE: Simulating fresh concrete behaviour Establishing a link between Furthermore, it should be noted that the upper two thirds of the force distribution curve has a shape which can be expected on the basis of the analytical solution, cf. Fig. 4.1 (the expected shape of the curve is given in Fig. 5.1 by a black dashed line). The yield stress in this interval covers self-compacting and flowable concretes. The wide spreading of force values, especially in the lower one third of the cone originates primarily from the uncompleted relaxation of the sample. Equilibrium is not reached yet at this stage of the calculation; more calculation steps would bring the result nearer to the expected force distribution, which means, however, a considerable increase in the calculation time. For the selected reference material, a self-compacting concrete with a yield stress of 50Pa, continuing the calculation would not produce any improvement since the shape of the upper two thirds of the curves, i.e. the region corresponding to relatively low values of yield stress, would not change. Following the procedure described in Chapter 4 the height can be calculated using equation 2.2 at 234.5mm for the given yield stress of 50Pa. The corresponding bond strength is approximately 2*10 4, cf. Fig. 5.1c. Other parameters of the contact model for the fine-tuning can be selected as shown in a previous publication of the authors [Shy08]. Note that up to this stage no simulation of the flowing process has been involved. If the cone is lifted, the virtual concrete begins to flow, this flowing process continues until the tensile forces at work become lower than the bonding strength of particles. Fig. 5.2a shows the shape of the concrete cake in the final state, i.e. when no further flowing occurs. The corresponding distribution of the contact forces is shown in Fig. 5.2b. In principle the final force distribution has a similar form when compared to its initial state; however, the height of the concrete cake is considerably smaller. The maximum forces in the bottom part of the cake are limited by the set bond strength of 2*10 4. The points positioned to the right of the dashed line result from some inaccuracy in the simulation process (as described above, equilibrium of the entire system require an increasing number of steps as the demand for accuracy increases. The slump-flow value obtained from the simulation was 590mm. The analytical prediction using formula by Roussel and Coussot [Rou05] provides a value of 600mm for the input yield stress of concrete equal to 50Pa. This very good correspondence of numerical and analytical results can be regarded as a first validation of the methodology developed to derive the key parameter of the particle model, bond strength Contact force Fig (a) Results of a slump-flow test simulation for a concrete with a yield stress of 50Pa; a) final shape of the concrete cake; b) contact force distribution over the height of the cake. (b)

9 International RILEM Conference on Material Science MATSCI, Aachen 2010 Vol. II, HetMat SUMMARY The paper presents an approach aimed at establishing a link between the rheological properties of fresh concrete and the parameters of DEM-based models. In particular, an algorithm for deriving the bond strength parameter of the DEM model from a given value of yield stress according to the Bingham model was developed. The keystone of the algorithm was comparison of the analytically predicted stress distributions in the Abram s cone filled with concrete and the corresponding force distribution obtained from corresponding numerical simulation by means of DEM, both calculations performed before cone lifting. Subsequently, the quality of such parameter estimation was verified by comparing the final shape of the concrete cake obtained from the numerical simulation (with the derived value of the bond strength) and the corresponding prediction by analytical formula. It was found that the numerical simulation provided quantitatively correct results. Further verification of the modelling approach is a subject of ongoing investigations which include analytical solutions for other test geometries, real experiments and corresponding numerical predictions. REFERENCES [Ita04] Itasca Consulting Group, Inc.: PFC 2D; Version 3.0, Minneapolis, ICG (2004). [Kon02] Konietky, H. (ed.), Numerical Modelling in Micromechanics via Particle Methods, Proc. of the 1 st Int. PFC Symposium, Balkema Publishers, The Netherlands (2002). [Mec07] Mechtcherine, V., Shyshko, S., Simulating the behavior of fresh concrete using distinct element method, Proc. of 5 th Int. RILEM Symp. on SCC, Ghent (2007), p [Mur84] Murata, J., Flow and deformation of fresh concrete, Mater. Constr. 17 (98) (1984), p [Rou05] Roussel, N., Coussot, P., Fifty-cent rheometer for yield stress measurements: from slump to spreading flow, Journal of Rheology, 49(3) (2005), p [Shy06] Shyshko, S., Mechtcherine, V., Continuous numerical modelling of concrete from fresh to hardened state, Proc. of 16 th Int. Building Material Congress Ibausil, Weimar, vol. 2 (2006), p [Shy08] Shyshko, S., Mechtcherine, V., Simulating the workability of fresh concrete, Proc. of the Int. RILEM Symposium of Concrete Modelling CONMOD 08, Delft, 2008, (RILEM Publications S.A.R.L., Proceedings PRO58), p