Support of the project FAST-J-23-8323 is gratefully acknowledged. The first author is Brno
Ph.D. Talent Scholarship Holder – Funded by the Brno City Municipality, Czech Republic

Eliáš, J.

Středulová, M.

DSpace at University of West Bohemia

Circular representative volume elementfor discrete model of concreteM. Středulováa, J. EliášaaInstitute of Structural Mechanics, Faculty of Civil Engineering, Brno University of Technology, Veveřı́ 331/95, Brno, Czech Republic1. IntroductionRepresentative volume element (RVE) is a concept used in homogenization schemes, whichpotentially allows the reduction of computational costs of highly detailed simulations [6]. Inessence, it is a small-scale (micro or mesoscale) model, which statistically represents a materialused for a macroscale simulation. In a computational scheme (for example the Finite ElementMethod), it acts as a constitutive function substitution: An RVE is attached to an integrationpoint, loaded by proper displacement or traction and its response is obtained. Subsequently, theresults are used in the macroscale calculations [5].Loading, whether displacement or traction, may be applied on the RVE in multiple ways,however, it has been shown that periodic boundary conditions are the most advantageous sincethe response they produce is the most realistic one [8]. RVE in combination with periodicboundary conditions is overwhelmingly used only for elastic response [6]. When applied tonon-linear scenarios, for example a strain softening material with heterogenities, whose post-peak behavior is characterized by fracture, its use is also possible, albeit its statistical repre-sentativeness is questionable. It requires extensive modification to the original equations defin-ing the boundary conditions and some transformation (rotation, translation) of the coordinatesystem so that a crack may emerge in an arbitrary direction. Apart from leading to compleximplementations, such modifications inevitably add additional computational costs [2, 8–10].One of the possible solutions proposed in literature, which builds on the concept of an RVEused in conjunction with periodic boundary conditions, is to substitute the conventionally usedsquare / cubic RVE by a circular / spherical one [7,9]. Such solution potentially offers seamlessrotation of the coordinate system so that periodic boundary conditions allow the formation of acrack in arbitrary direction.The present contribution is part of a larger effort to formulate a circular RVE formed by alattice discrete particle model (LDPM) [3, 4] which could also account for a post peak strainsoftening behavior. It is concerned with the generation of a circular RVE geometry mimickingdiscrete setting which enables periodic placement of particles. We presume that the use of adiscrete model does not allow the splitting of a particle and therefore necessarily leads to thegeneration of a irregular boundary. In the first part, the formation of a continuous circularRVE described in literature is shown. Subsequently a formation of a discrete circular RVE ismimicked by altering the boundary of a RVE. It is shown that this may lead to spurious strainsbeing introduced into the RVE response. Possible modifications to the generation procedure areproposed and tested.1942. Circular RVE geometry generationIn an RVE formulation, periodic boundary conditions are prescribed for each opposing coupleof nodes on the surface of an RVE. Each couple of points is allowed the same displacement androtation. According to [7], the only requirement to which the shape of an RVE is subjected isthat of opposing surface normals for each couple of nodes: n+ = −n−.Upholding the periodicity of a material, geometry generation is according to [9] done byplacing a periodic image of a particle with center (r, θ, ϕ) to a particle of the same diameterwith center (r − D, θ, ϕ), where D is the diameter of the RVE (Fig. 1a). In case of a circularRVE with particles created by continuous model, such requirement is naturally fulfilled, becausethe heterogeneities may be split and thus the surface of a RVE is kept regular (Fig. 1b). In thecase of a discrete model with physical discretization [1], such split of particles is not possibleby the nature of the model, hence the periodicity of the material must be upheld by allowingirregular boundary (Fig. 1c).a) b) c)Fig. 1. (a) Geometry generation of a circular RVE with periodically placed particles, (b) a schematicboundary of a continuous RVE and (c) a schematic boundary of a discrete RVE2.1 SimulationsBoth of the scenarios described above have been tested in simulations, in order to asses itsperformance in elastic regime. To see the effect of the boundary manipulation, only the positionof one node couple was altered (Figs. 1b and 1c). Elastic material of given parameters wasused. No heterogeneities were incorporated. Periodic boundary conditions are applied to theopposing node couples. An in-house Finite Element Method preprocessor and solver, namedOAS (Open Academic Solver), was used. The RVEs are loaded by uniaxial tension applied inthe x direction ε = {εxx 0 0}T . Strain fields of the loaded RVEs are shown in Fig. 2.The RVEs were both loaded by a strain εx = 0.001. The strain field of a RVE with regularboundary shows constant distribution of strains, with magnitude being equal to that of the ap-plied loading. On the contrary, the RVE with irregular boundary shows that by the irregularities,strain magnitude differs by as much as 50 %, ranging up to 0.0015 magnitude. The differencelowers by decreases meshing of the RVE, but it remains present.Upon closer inspection, the second RVE with irregular boundary does not abide by thecondition of opposing normals. Fig. 3a illustrates the mismatch in supposedly parallel boundarysegments. Segments of the same style (either solid or dotted line) are centrally symmetric andtherefore should be parallel to allow for opposing normals. Clearly, that is not the case. Toallow the segments to be parallel, the boundary needs to be changed in a centrally symmetricmanner as well (Fig. 3b and 3c).Results of the simulations performed in the same manner as described at the beginning ofthis subsection are shown in Fig. 4. The results show that by generating the geometry in a195a) b) c)Fig. 2. (a) Scheme of the loaded circular RVE, (b) strain field (magnitude) of a circular RVE with regularboundary, (c) strain field of a circular RVE with irregular boundary (mimicking discrete RVE)a) b) c)Fig. 3. Part of the RVE boundary which are joined to the deviated node of the boundary. Lines of thesame pattern should be parallel to satisfy the opposing normals equation: (a) not parallel lines of theRVE generated based on the procedure used in [7,9], (b) and (c) suggested modifications to the geometryto maintain constant distribution of strainsa) b) c)Fig. 4. (a) Strain distribution of the first geometry generation with not parallel boundary segments, (b)and (c) results obtained by a centrally symmetric geometry generationcentrally-symmetric manner (Fig. 3b and c), the distribution of strains is equivalent to that ofthe RVE with regular boundary.3. ConclusionIt was shown to what extent geometry generation may influence the distribution of stresses ina circular RVE upon which periodic boundary conditions are applied. Following the proceduresuggested for the geometry generation of a continuous RVE, the boundary irregularity orientedinward to the RVE behaves as a concentrator, raising the strain magnitude by as much as 50 %.196The strains are lower by the boundary distorted outside of the RVE. The phenomena may beattributed to the effect of periodic boundary conditions, which are in this case applied in acentrally symmetric manner to a non-symmetric node pair.However, by distorting the boundary in a centrally symmetric manner, that is in the samedirection with respect to the centroid of the RVE, the strain distribution is constant as in thecase of the regular RVE. Although successful for the present analysis, it is left upon furtherdiscussion whether such a geometry generation produces geometrically periodic sample of amaterial, upholding the original concept of a representative volume element.AcknowledgementsSupport of the project FAST-J-23-8323 is gratefully acknowledged. The first author is BrnoPh.D. Talent Scholarship Holder – Funded by the Brno City Municipality, Czech Republic.References[1] Bolander, J. E., Eliáš, J., Cusatis, G., Nagai, K., Discrete mechanical models of concrete fracture,Engineering Fracture Mechanics 257 (2021) No. 108030.[2] Coenen, E. W., Kouznetsova, V. G., Geers, M. G., Novel boundary conditions for strain localiza-tion analyses in microstructural volume elements, International Journal for Numerical Methods inEngineering 90 (2012) 1-21.[3] Cusatis, G., Mencarelli, A., Pelessone, D., Baylot, J., Lattice discrete particle model (LDPM)for failure behavior of concrete. II: Calibration and validation, Cement and Concrete Composites33 (9) (2011) 891-905.[4] Cusatis, G., Pelessone, D., Mencarelli, A., Lattice discrete particle model (LDPM) for failurebehavior of concrete. I: Theory, Cement and Concrete Composites 33 (9) (2011) 881-890.[5] Eliáš, J., Cusatis, G., Homogenization of discrete mesoscale model of concrete for coupled masstransport and mechanics by asymptotic expansion, Journal of the Mechanics and Physics of Solids167 (2022) No. 105010.[6] Gitman, I. M., Askes, H., Sluys, L. J., Representative volume: Existence and size determination,Engineering Fracture Mechanics 74 (2007) 2518-2534.[7] Glüge, R., Weber, M., Bertram, A., Comparison of spherical and cubical statistical volume ele-ments with respect to convergence, anisotropy, and localization behavior, Computational MaterialsScience 63 (2012) 91-104.[8] Goldmann, J., Brummund, J., Ulbricht, V., On boundary conditions for homogenization of volumeelements undergoing localization, International Journal for Numerical Methods in Engineering113 (2018) 1-21.[9] Hofman, P., Ke, L., van der Meer, F. P., Circular representative volume elements for strain localiza-tion problems, International Journal for Numerical Methods in Engineering 124 (2023) 784-807.[10] Toro, S., Sánchez, P. J., Huespe, A. E., Giusti, S. M., Blanco, P. J., Feijóo, R. A., A two-scalefailure model for heterogeneous materials: Numerical implementation based on the finite elementmethod, International Journal for Numerical Methods in Engineering 97 (2014) 313-351.197

Circular representative volume element for discrete model of concrete

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Circular representative volume element for discrete model of concrete

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