MESO-MECHANICAL MODELING OF ULTRA-LIGHTWEIGHT CONCRETES

Transcription

1 International RILEM Conference on Material Science MATSCI, Aachen 2010 Vol. II, HetMat 83 MESO-MECHANICAL MODELING OF ULTRA-LIGHTWEIGHT CONCRETES T. Vallée 1, M. Oppel 2, T. Tannert 3, 1 College of Engineering and Architecture of Fribourg, 2 Bauhaus- University Weimar, 3 Bern University of Applied Sciences ABSTRACT: Hybrid slab systems combining fibre-reinforced polymer composites together with traditional materials such as concrete or steel are promising load-bearing structures, since each material is optimally used. A novel concept for a hybrid sandwich bridge deck system was proposed by Schaumann et al. (2006) and Keller et al. (2007). The system uses three layers of different materials: an FRP sheet for the bottom skin, lightweight concrete for the core material and ultra high performance fibre-reinforced concrete for the top skin. Different concrete mixtures, involving expanded glass and clay aggregates were analyzed. The meso-mechanical response of the investigated brittle ultra-lightweight concretes (ULWC) was predicted based on a meso-mechanical numerical analysis backed up by experimental assessment. The approach was based on explicit representation of the main concrete components, including mortar, aggregates, their respective interfaces and eventually fibres. The effort, firstly, focused on numerically reproducing the macroscopic behaviour of concrete specimens, in traction, compression and splitting strength, in order to develop optimum mixtures, secondly on implementing the later result in the context of hybrid slabs. 1 INTRODUCTION Concrete is a material composed of different phases and its mixture produces a very heterogeneous material, influencing the formation and development of cracks, and therefore the mechanical behaviour and the softening branch. According to the scale classification of Wittmann (1983), concrete may be analyzed at different levels, micro, meso or macro. In the past, fracture problems in concrete have been analyzed mainly at macro level, in which the material is considered as a continuum and the non-linear behaviour is controlled at the constitutive level. However, this level of observation has been proven insufficient to explain crack initiation, crack propagation and coalescence which are mainly driven by the internal meso-structure of concrete, i.e. bigger aggregates. At the meso level scale, there is an explicit representation of the material heterogeneities, and hence the constitutive relations may be relaxed or even exchanged by simpler physical relations without the need of using a high number of material parameters. 1.1 Lattice models In lattice models, the continuum material is modelled by a set of rod elements (free rotations at nodes) or beam elements (non-free rotations at nodes) (Schlangen & van Mier, 1992, 1993). In this approach, beam elements are connected to each other forming a triangular grid, and thus transferring forces and moments at corners. Fracture progress in lattice models is modelled by removing all those elements that have reached their maximum tensile strength (Schlangen & Garboczi, 1996). Material heterogeneity can be modelled by superimposing the lattice mesh on an image of the internal structure of concrete. Thus, different mechanical properties are assigned to the elements depending on their location: within an aggregate,

2 84 VALLEE, OPPEL, TANNERT: Meso-Mechanical Modelling of Ultra-Lightweight Concretes within the mortar matrix or crossing an interface. Although lattice models represent a simplification of the reality, this simplification does not always lead to a reduction on the number of degrees of freedom. In general terms, to obtain a correct global material response, a large number of rod/beam elements are necessary. This seems to be one of the reasons why most of the lattice models have been developed only in 2D. In spite of that, some extensions to 3D can be found in recent literature, (Lilliu & van Mier, 2003, Cusatis et al. 2003a,b). 1.2 Continuum meso-models Earliest continuum meso-models can be found in the works of Roelfstra, Stankowski and Vonk. In Roelfstra s model (Roelfstra et al. 1985) aggregates were represented as circular inclusions embedded within the matrix. In spite of its simplicity, this model was one of the first that described the physical properties and the mechanical behaviour of concrete as a composite material. In the model of Stankowski, concrete is represented in terms of polygonal approximations with irregular shapes for the aggregates that were embedded in a mortar matrix. This model was proposed in 2D, and only considered interface elements in the contacts between aggregates and mortar. In addition, the continuum elements were equipped with an elasto-plastic constitutive law (Stankowski et al., 1993a,b). Vonk proposed another model in which interface elements are also disposed through the mortar. Interface elements are governed by a Mohr-Coulomb criterion which is combined with a second surface to limit the tensile strength. The evolution of the constitutive law is controlled by a single history variable which accounts for the total plastic strains. The results of this model have been very powerful at least in the average softening behaviour of concrete under uni-axial compression (Vonk et al., 1991). More recently, López (1999) proposed another 2D approach in which meso-structure is obtained following a Voronoï Tessellation procedure. In the approach, interface elements are inserted between aggregate-mortar intephases and through the matrix in a similar fashion as in the model used by Vonk. The approach considered linear elastic behaviour for the continuum elements, both aggregates and matrix, while mechanical behaviour of interface elements were described by a elasto-plastic constitutive interface, Carol et al. (1997). The above mentioned meso-mechanical continuum models were developed in 2D what impeded their applicability to those problems in which main cracks develop in parallel planes to the model s plane, like bi-axial compression, and tri-axial compression and extension. A 3D meso-mechanical model for concrete was developed by Caballero (2005). In that model the meso-structure is obtained by a 3D Voronoï algorithm. Similarly, interface elements are inserted a priori in the interphase aggregate-mortar and through the mortar. The model has been tested successfully against different loading scenarios like uni-axial tension and compression (Caballero et al. 2006a,b) and bi-axial (Caballero et al 2007) and tri-axial loading (Carol et al. 2007). 1.3 Objectives of this paper Almost all above mentioned publications involved specifically written FE code, which usually makes the corresponding modelling almost inaccessible for everyone else than their respective contributors. Owing to the increasing data processing power of more and more economical computational infrastructure, combined with the democratization of commercial finite element programs, it is tempting to use commercially available software to perform numerical modelling at the meso-mechanical level.

3 International RILEM Conference on Material Science MATSCI, Aachen 2010 Vol. II, HetMat 85 The aim of this paper is to explore the possibilities offered by the software package ANSYS, a tool widely used in research and industry. For this purpose, a simple, and concomitantly complex, system has been considered: a lightweight concrete, involving spherical expanded clay aggregates. A corresponding cement mixture has been experimentally investigated by Keller et al. (2007) and Schaumann et al. (2008); see Fig Fig Investigated lightweight hybrid slabs concrete: randomly distributed spherical aggregates (left); the lightweight concrete experimentally investigated (right). More specifically, a subsequently closer defined concrete mixture was numerically modelled, specifically focusing on three standard tests: axial tension, axial compression and splitting tests on cylindrical numerical specimens. This paper does not claim to asses in a final form the validity of the chosen approach, since (i) not all mechanical parameters needed for the modelling were experimentally gathered, and (ii) the experimental data on the modelled cylinders is still ongoing, thus fragmentary. However, first results of the numerical modelling are quite promising, thus the authors considered it worthy enough to be shared with the academic community. 2 NUMERICAL MODELLING 2.1 Modelling of the geometry The numerical model consisted of randomly distributed spherical aggregates, embedded in a cement paste, shaped in cylinders, corresponding to the experimentally investigated specimens displayed in Fig A numerical routine was thus written to randomly generate coordinates acting as centres of the spheres, the radius of the latter also being randomly distributed around R 1 = 6 mm. For purely numerical reasons, strictly related to available computational performance, it was not possible to model the aggregates used in the previously cited experimental work, cf. Keller et al. (2007) and Schaumann et al. (2008) and Fig A succeeding numerical routine ensured that the so generated spheres did not overlap, respectively by adding a slight tolerance, even avoiding contact; refer to Fig. 2.1 (left). Subsequently, cement paste was modelled around the aggregates, defining the cylinders, Fig. 2.1 (centre, right).

4 86 VALLEE, OPPEL, TANNERT: Meso-Mechanical Modelling of Ultra-Lightweight Concretes Fig The investigated concrete: (left) randomly distributed spherical aggregates; (centre) modelled concrete cylinder, diameter exaggerated for the sake of clarity; (right) the numerical model implemented in a splitting-test setup. 2.2 Mechanical description The aggregates were modelled as being constituted of an isotropic, linear-elastic material, defined subsequently. In the following, the mechanical strength of the aggregate was so set that failure of the latter was avoided. The compression behaviour of cement paste, if defined at the level of uni-axial stress-strain (σ ε) relation can roughly be described in three phases (Fig. 2.2): Firstly, a linear-elastic part, up to around one third of the compressive strength, f c ; Secondly, a non-linear part in which cracking gradually reduces stiffness, leading to a typically curved σ ε-curve, up to reaching f c, generally associated with large cracks parallel to the first principal stress; thirdly and lastly, the characteristic softening of the cement paste occurs, which in some cases allows for a residual compressive strength. It is noteworthy to remind that the slope of the softening branch of the σ ε-curve is a direct measure for the brittleness of the material. In tension, cement paste exhibits a different behaviour, which can be summarized, once again described on the level of a uni-axial σ ε-relationship, as being constituted of two branches: firstly, a linear-elastic part peaking at the strength in tension, f t ; Secondly, the softening branch, in which the fracture energy G f is dissipated. The dissipation of G f in the softening branch is assumed to follow a linear σ ε-path or an exponential one (see Fig. 2.3). The interface between aggregate and cement paste was modelled according to the Cohesive Zone Model (CZM), which was implemented to create new surfaces. At the meso-scale, CZM can be perceived as the effect of energy dissipation mechanisms, energy dissipated both in the forward and the wake regions of the crack tip; it uses fracture energy (ideally obtained from fracture tests) as a parameter and is devoid of any ad-hoc criteria for fracture initiation and propagation. Separation of aggregates from the matrix occurs if the energy release rate, G exceeds a critical energy release rate, G c. The mechanical parameters used were the one Eurocode delivers for C30/35; the E-Modulus for the matrix was set to E m = MPa; crack were set to occur at f crack = 5 MPa. The spherical aggregate was modelled using an E-Modulus of E a = MPa, in order to be lower than E m. The corresponding strengths were set to f c,a = f t,a = 3 MPa. Different characterizations of the interface properties were tested, which can be basically divided into two categories, for which the subsequent results stand: a first set of values leading to interface failure, and a set of properties leading to failure of the aggregates.

5 International RILEM Conference on Material Science MATSCI, Aachen 2010 Vol. II, HetMat 87 Fig Numerical modelling of the matrix properties in compression. Fig Numerical modelling of the matrix properties in traction. The modelling of the interface behaviour was achieved by duplicating all nodes lying on the contact zone between aggregate and matrix, even if this made the modelling more arduous. Subsequently, the properties of the interface were applied on each of the corresponding cohesive contact zones. Figure 2.4 shows the data for the interfaces, as used in the modelling. σ max A dn = 0 B 1,0 kn/cm² Stress cm u n cm dn = 1 C c u Deformation n Fig Numerical modelling of the interface properties.

6 88 VALLEE, OPPEL, TANNERT: Meso-Mechanical Modelling of Ultra-Lightweight Concretes 3 NUMERICAL RESULTS ANSYS, in its current version (v12), does not allow for the correct modelling of material softening, since its algorithms do not take into account negative slopes of the σ εrelationship. Thus, own user defined material models were implemented into ANSYS. The numerical results are best described by Fig. 3.1 for the compression tests and Fig. 3.2 for the splitting strength test. The numerical results for the tensile tests yielded in an almost linear σ ε-relationship, so that the authors did not consider it useful for display. The numerical calculation for lightweight concrete in compression shows a result that could be expected, i.e. a first almost linear part of the σ ε-relationship, followed by a sudden drop of the sustained stresses, ending in a residual stress plateau. No significant differences were obtained for interface and failure matrix. Similarly, the numerical results for the splitting test show a good agreement with what could be expected in such a test setup; especially with regard to the stress and crack pattern. σ ε Fig Numerical modelling of cylinders under compression load. Fig Numerical modelling of cylinders under splitting load.

7 International RILEM Conference on Material Science MATSCI, Aachen 2010 Vol. II, HetMat 89 4 CONCLUSIONS A numerical model was developed at the meso-mechanical level, to investigate the behaviour of a lightweight concrete based on spherical expanded clay aggregates used in a preceding study. Due to computational limitations, the authors had to deviate from the original model, i.e. simplifying its geometry by increasing the diameter of the aggregates. The numerical modelling was performed using ANSYS in order to evaluate the applicability of meso-mechanical models derived from commercial software. The author s experience has turned out to be positive, since most of the expected behaviour could be verified on three classical experimental setups: axial compression, axial tension and splitting strength tests. All features resulting from the complex material properties, including the interface, could be implemented into their numerical model. Although the previously cited limited computer resources prevented the authors to test the accuracy of their numerical modelling on their own experimental data, the experience has shown that it is per se possible to perform complex meso-mechanical modelling using standard commercial software. Since the performed numerical modelling can at best being considered at having taken place at the level of a Representative Volume Element (RVE), the authors will in a subsequent step, using extended hardware capacities, refine their model in order to achieve the experimental proof of their approach. REFERENCES [Cab05] [Cab06a] [Cab06b] [Cab07] [Car07] [Car97] [Cus03a] [Cus03b] [Kel07] [Lil03] Caballero, A. (2005) 3D meso-mechanical numerical analysis of concrete using interface elements., PhD thesis, ETSECCPB-UPC, E Barcelona (Spain). Caballero, A., Carol, I. & López, C.M. (2006) A numerical approach to the cracking processes in concrete under uniaxial loads, Fatigue and Fracture of Engineering Materials and Structures, 29(12): 979. Caballero, A., López, C.M. & Carol, I. (2006) 3D meso-structural analysis of concrete specimens under uniaxial tension, Computer Methods in Applied Mechanics and Engineering, 195: Caballero, A., Carol, I. & López, C.M. (2007) 3D mesomechanical analysis of concrete specimens under bi-axial loading, Fatigue and Fracture of Engineering Materials and Structures, 30(9): Carol, I., Idiart, A., López, C.M. & Caballero, A. (2007) Multiaxial behavior of concrete: a meso-mechanical approach, Revue Européenne de Génie Civil, 11(7-8). Carol, I., Prat, P. & López, C. (1997). A normal/shear cracking model. Application to discrete crack analysis. ASCE J. Engrg. Mech., 123: Cusatis, G., Bazant, Z. & Cedolin, L. (2003a) Confinement-shear lattice model for concrete damage in tension and compression: I. theory. ASCE J. Engrg. Mech., 129(12): Cusatis, G., Bazant, Z. & Cedolin, L. (2003b) Confinement-shear lattice model for concrete damage in tension and compression: II. computation and validation. ASCE J. Engrg. Mech., 129(12): Keller, T., Schaumann, E. & T. Vallée (2007) Flexural behavior of a hybrid FRP and lightweight concrete sandwich bridge deck, Composites Part A: Applied Science and Manufacturing, 38(3): Lilliu, G. & van Mier, J. (2003) 3D lattice type fracture model for concrete. Engineering Fracture Mechanics, 70(7/8):

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