Mesoscale analyses of size effect in brittle materials using DEM
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Abstract
The paper describes numerical mesoscale results of a size effect on strength, brittleness and fracture in brittle materials like concrete. The discrete element method (DEM) was used to simulate the size effect during quasistatic splitting tension with the experimentalbased mesostructure. The twodimensional (2D) calculations were carried out on concrete cylindrical specimens with two diameters wherein two different failure modes occurred (quasibrittle and very brittle with the snapback instability). Concrete was modelled as a random heterogeneous 4phase material composed of aggregate particles, cement matrix, interfacial transitional zones and macrovoids, based on xray microCTimages of the real concrete mesostructure. Attention was paid to the effect of the different specimen diameter on both the strength, brittleness and fracture pattern. Each internal energy component was analyzed in the fracture process zone and beyond it, and compared for the different postpeak behaviour of concrete. The evolutions of the number of broken contacts, coordination number, crack displacements and normal contact forces were also shown. Of specific interest was the fracture initiation and formation of two different failure modes. Next, the 2D DEM results of a size effect for 4 different specimen diameters were directly compared with corresponding experiments from the research literature. The experimental size effect was realistically reproduced in numerical calculations, i.e. the concrete strength and ductility decreased with increasing concrete specimen diameter. The calculated decreasing strength approached an asymptote with increasing cylindrical specimen diameter within the considered specimen size range.
Keywords
Concrete Size effect Splitting test DEM Fracture zone Absorbed energy Released energy1 Introduction
The size effect is a fundamental phenomenon in brittle materials like concrete. It denotes that both the: (1) nominal structural strength (corresponding to the maximal load value reached in the loading process) and (2) material ductility (ratio between the energy consumed during the loading process before and after the load–deflection peak) always decrease with increasing structural size [1, 2, 3]. These two deformation process properties are of major importance for the assessment of the member safety and its interaction with adjacent structural members. Brittle members exhibit a transition from quasibrittle response in the postcritical phase for small size members to the snapback response (a catastrophic drop in strength related to a positive slope in a load–displacement softening branch) for large size members [4, 5, 6]. The size effect is an important practical result of a fracture phenomenon. Usually, the size effect has been specified for geometrically similar structures, differing only by the value of the size factor [3]. In the case of reinforced concrete, the size effect solely occurs if the failure takes place in concrete and the reinforcement yielding is excluded when a high reinforcement ratio is assumed.
Two mechanical size effects are important in brittle materials under loading: energetic (or deterministic) and statistical (or stochastic) one [1, 2, 3, 7, 8]. The deterministic size effect is caused by the formation of a region of intense strain localization with a certain volume (microcrack regioncalled also fracture process zone FPZ) that always precedes discrete macrocracks. The strain localization zone size is not negligible relative to the crosssection dimensions and is large enough to cause significant stress redistribution in the structure, i.e. the energy absorption in localization failure zones and energy release in remaining unloading regions. The value of energy absorption in localized failure zones is similar but the energy release grows with increasing member size (both normalized by the specimen size) that causes the decrease of the nominal strength for larger members; that is sensitive to the ratio between the size of strain localization zones and the specimen size. Thus it cannot be appropriately estimated in laboratory tests since it is different for various specimen sizes (the size of localized zones cannot be experimentally scaled). The postpeak behaviour depends upon the ratio between the energy absorption increment and energy release increment. The global snapback instability is typical for large and slender structures, low fracture toughness and high tensile strength [9, 10, 11]. A statistical (stochastic) effect is caused by the spatial variability/randomness of the local material strength and occurs in concrete structures of a positive geometry [3, 12]. The first statistical theory has been introduced by Weibull [13] (called also the weakest link theory) which postulates that a structure fails when its strength is exceeded in the weakest spot (when the stress redistribution is not considered). A combination of the energetic theory with the Weibull statistical theory provides the general energeticstatistical theory for geometrically similar structures [8]. The deterministic size effect is important for moderate size structures. The Weibull statistical size effect is usually smaller and significantly increases as an asymptotic limit for very large size structures.
The main aim of our research works is to describe a quasistatic size effect in plain concrete with a realistic mechanical model during different failure mechanisms. The current paper numerically analyzes a quasistatic size effect in concrete at the mesolevel in splitting tension tests during two different failure mechanisms: (1) quasibrittle and (2) very brittle with a snapback instability. The discrete element method (DEM) was employed to better understand the global size effect (expressed by strength and brittleness) and related failure mode change with respect to fracture. The focus was on the initiation and formation of local fractures at the aggregate level during two different failure mechanisms. The advantage of the discrete mesoscale approach is that it is able to directly simulate a heterogeneous mesostructure of materials. Thus it may be used for a detailed study of mechanisms of the initiation, growth and propagation of both microcracks and discrete macrocracks that greatly control the macroscopic concrete behaviour [14, 15, 16, 17]. It may also be used for a better calibration of continuous models for concrete with respect to the e.g. characteristic length of microstructure, crack opening width, formation instant of a discrete macrocrack, effective elastic and inelastic parameters, damage evolution rule, fracture toughness and micro and macrocracking. Our experimental and theoretical investigation results on the coupled deterministicstatistical size effect in concrete and RC concrete at the macrolevel were described in [4, 5, 6, 11, 18].
In the current paper, the damage growth and size effect in concrete cylinders of two diameters (D = 0.05 m and D = 0.15 m) were theoretically analysed for plane stress conditions. The 2D numerical results for D = 0.15 m were verified by the own laboratory experiments (wherein a snapback instability occurred) with respect to the stressdisplacement curve, shape and width of FPZ and macrocrack [14]. The concrete geometry at the mesolevel was incorporated into DEM for D = 0.15 m from real concrete specimens with 3D xray microtomography images with a very highresolution [14] using advanced tomography system Skyscan 1173 of the new generation [19]. The combined DEM/microCTscan model emerged as a powerful tool to realistically capture concrete fracture under splitting tension [14] (2D analyses), bending [15, 16] (2D and 3D analyses) and uniaxial compression [17] (2D and 3D analyses). Therefore the DEM results for D = 0.15 m constitute a suitable reference for evaluating the performance of other specimen diameters. In DEM analyses, the smaller concrete specimen with the diameter of D = 0.05 m was intentionally cut out from the larger specimen to avoid a statistical size effect.
 1.
Size effect investigations in concrete specimens during 2D quasistatic splitting tension [14] with the experimentbased mesostructure of concrete using a reliable 4phase DEM concrete model.
 2.
Mesoscopic analyses of all energy components at a different stressdisplacement stage with the dissipated and released portions, referred to the fracture process zone and the remaining unloading specimen region during quasibrittle and very brittle failure.
 3.
Analyses of local fractures with respect to concrete strength and brittleness, based on the evolution of broken normal contacts, coordination number, compressive and tensile contact forces and macrocrack displacements.
The DEM calculation results for the larger cylindrical specimen (D = 0.15 m) were directly compared with our corresponding experiment with respect to the stress–strain evolution, a shape of FPZ and macrocrack and width of FPZ. In addition, the DEM simulation results of a size effect for 4 different concrete cylindrical specimen diameters (74–290 mm) were compared with the corresponding experimental results (Sect. 8).
Commonly, the deterministic size effect investigations in concrete have been performed at the macrolevel by using different enhanced constitutive models for concrete, equipped with a characteristic material length (e.g. integraltype nonlocal models), crack band and cohesive crack approaches. Note that in contrast to DEM, continuum mechanics solutions do not consider cracking from the beginning of deformation, the damage rules have usually a priori assumed sigmoid shape and they are switched on in the softening regime only. Moreover, the heterogeneity of material properties (like stiffness, strength, fracture energy) is reflected in a homogenized sense only. The only link to the mesostructure is the presence of a characteristic length in enhanced continuum laws with softening. The characteristic length is usually chosen as a too large value as compared to experimental results (to speed up the calculations). The realistic curved cracks cannot be obtained without taking the material mesostructure into account. DEM was also used to reproduce damage in concrete by other researchers [20, 21, 22, 23, 24]. The splitting test for concrete was simulated within continuum mechanics [25, 26, 27, 28, 29, 30] and discrete mechanics using DEM [31, 32, 33]. A simplified geometric mesostructure was however assumed in DEM calculations [31, 32, 33]. The size effect was found to increase with growing heterogeneity intensity [32].
The paper is organized as follows. After the introduction (Sect. 1), a short description of splitting tensile tests with respect to the size effect was addressed in Sect. 2. Next, the discrete element method (DEM) for concrete was described in Sect. 3. The DEM input data were presented in Sect. 4. The 2D DEM results of the size effect with respect to the strength, postpeak and fracture behaviour were discussed for two concrete specimens with the different diameter in Sects. 5–7. The DEM results were next compared with the corresponding experimental splitting tensile tests in Sect. 8. The final conclusions were stated in Sect. 9.
2 Size effect in split cylinder experiments of plain concrete
3 Formulation of discrete element method (DEM) for concrete
The DEM calculations were performed with the threedimensional spherical discrete element code YADE, that was developed at the University of Grenoble [45, 46]. DEM considers a material as consisting of particles interacting with each other through a contact law and Newton’s 2nd law via an explicit timestepping scheme. Outstanding advantages of DEM include its ability to explicitly handle the modelling of particlescale properties including size and shape which play an important role in the concrete fracture behaviour [45, 47, 48]. The disadvantage is a huge computational cost. The DEM model was successfully used for describing the behaviour of granular materials by taking shear localization into account [49, 50, 51, 52]. It demonstrated also its usefulness for both local and global fracture simulations in concrete [14, 15, 16, 17]. Our DEM calculations for concrete evidently exhibited that it was of major importance to take into account both the shape and place of aggregate particles, specimen macroporosity and strength and number of interfacial transitional zones (ITZs) for a realistic reproduction of discrete macrocracks [14, 15, 16]. ITZs due to a porous structure acted as an attractor for macrocracks (created by bridging interfacial microcracks) and thus controlled both the concrete strength and brittleness [14, 15, 16].
The five main local material parameters were needed for our discrete simulations: E_{c}, υ_{c}, μ, C and T. In addition, the values of the particle radius R, particle mass density ρ and damping parameters α_{d} were required. In general, the DEM material constants are calibrated with the aid of laboratory test results on concrete (e.g. uniaxial tension, uniaxial compression, simple shear, biaxial compression) due to the current data lack on mechanical properties of mortar specimens with the different initial porosity. The calibration process consists in running test simulations on a given assembly of discrete elements simulating concrete with the same material constants to reproduce the experimental mechanical behaviour [14, 16].
4 DEM input data
The section describes the input data assumed for 2D DEM calculations of a concrete splitting test. The 3D simulations are obviously more realistic than the 2D ones with respect to the fracture pattern [16, 17], however, the 2D analyses may be also useful as a means for studying several different relationships [15]. The differences between 2D and 3D DEM simulations for fractured concrete were found to be minimal at the peak load [16]. Thus the 2D results at the peak are equal to the 3D results. Those differences increased with growing postpeak deformation when assuming the same material parameters [16]. However, when the 2D DEM model is properly calibrated for concrete based on simple laboratory tests (uniaxial compression, uniaxial tension, shear, biaxial compression), it may be also used for obtaining realistic results of fracture and a postpeak stress–strain response [15]. In addition, the 2D analyses significantly cut the computation time of DEM simulations. Besides, the splitting laboratory test is usually identified as a typical 2D boundary value problem, i.e. the effect of the specimen length on the crack geometry is assumed to be negligible.
The concrete specimen was described in DEM computations as a fourphase material, composed of aggregate, cement matrix, interfacial transitional zones (ITZs) and macrovoids with the same location, shape and content of aggregates and macrovoids as in the experiment [14]. In the experiments, the minimum aggregate diameter was d_{a(min)} = 2 mm, maximum aggregate diameter was d_{a(max)} = 12 mm and mean aggregate diameter d_{a(50)} = 5 mm. The aggregate volumetric content was 47.8%. The total particle volumetric content (sand and aggregate) in concrete was 75%. The volume of all voids was p = 3.2% and the volume of voids with the diameter d_{p} < 1 mm was p = 1.6% based on microCT. The experimental width of ITZs was 20–50 μm. The experimental porosity of ITZs changed between 25% (at aggregates) down to 1.6% (cement matrix), based on the image binarization technique. The 3D xray microtomography images with a very highresolution [19] were employed to give the necessary information for the geometric statistical characterization of aggregate and macrovoid distributions in the specimens [14]. In 2D calculations, the specimen length L included one row of aggregate and mortar particles. In order to construct the real aggregate shape (2 mm ≤ d_{a} ≤ 12 mm) in 2D calculations based on images of the polished specimen surface [14], the clusters composed of spheres with the diameter of d = 1.0 mm connected to each other as rigid bodies were used. Based on experiments, all aggregate grains with the diameter in the range of 2 mm ≤ d_{a} ≤ 12 mm included ITZs. ITZs were simulated for the sake of simplicity as contacts between aggregate and cement matrix grains and thus they had not the physical width in contrast to experiments. The cement matrix was modelled however by spheres with the diameter range 0.35 mm ≤ d_{cm} < 2.0 mm without ITZs. The specimen preparation process consisted of two stages. Initially, aggregate particles and clusters simulating macrovoids were created. Later smaller particles were added until the final specimen was filled in 98.4% by particles in order to realistically model the experimental concrete microporosity of 1.6% (the micropores were assumed as the pores with the diameter d_{p} < 1 mm) [14]. Next, all contact forces due to the particle penetration U were deleted. In order to take the starting configuration into account, the initial overlap was subtracted in each calculation step when determining the normal forces (\( \vec{F}_{n} = K_{n} \left( {U_{n}  U_{0} } \right)\vec{N} \), where U_{o}the initial overlap and U_{n}—the overlap in the calculation nsteps). The grain size distribution curve was the same as in the experiment (with \( d_{cm}^{min}\) = 0.35 mm). The macrovoids (d_{p} ≥ 1 mm) were modelled as empty regions with a real shape and place (after the cement matrix was created, the particles at the place of macrovoids were removed). The initial stresses in concrete due to shrinkage were not considered since the experimental specimens were carefully prepared. The specimen was cut out from a concrete block after the seventh day. The concrete block was covered with a plastic sheet during the initial curing period to avoid the surface evaporation and autogenous shrinkage. The specimen was next kept for 28 days in water.
The effect of the different ratios of T_{ITZ}/T_{cm} and C_{ITZ}/C_{cm}, different intergranular friction angle μ in ITZs and different minimum particle diameter in the cement matrix \( d_{cm}^{min} \) was comprehensively discussed in [14, 15]. The calculations [14] showed that if \( d_{cm}^{min} \) was small enough (e.g. \( d_{cm}^{min} \) = 0.35 mm), its effect might be neglected in concrete specimens composed of a sufficiently large number of discrete elements.
5 Macroscopic DEM results: stressdisplacement diagrams and fracture geometry
6 Energy balance in process of cohesive failure and tensile contact separation
6.1 Nonfractured state (without normal contact breakage)
It was equal to the external boundary work W expended on the particle assembly by the external vertical splitting force P on the vertical specimen top displacement v (\( W = W_{prev} + \sum Pdv \)).
6.2 Fractured state (with normal contact breakage)
6.3 Energy evolution in entire specimen
The evolution of the elastic internal energy E_{e} in a normal and tangential direction was like the evolution of the mobilized specimen strength (expressed by the splitting tensile stress in Fig. 5). The elastic internal energy E_{e} was obviously higher than the plastic damping D_{p} due to cohesion. The elastic energy part due to the tangential force action E_{e(s)} was obviously smaller than that due to the normal force action E_{e(n)} in view of the lack of plastic damping in a normal direction. The kinetic energy E_{k} was insignificant due to both a quasistatic numerical test and numerical damping.
For the normalized vertical top displacement v/D = 0.35% corresponding to the peak load for D = 0.05 m, the normalized elastic internal energy was equal to \( E_{e}^{n} \) = 96% (normal energy70%, tangential energy26%, n‘normalized’), normalized plastic dissipation was \( D_{p}^{n} \) ≈ 0.0%, normalized energy of removed contacts was equal to \( E_{rc}^{n} \) = 0.5%, normalized kinetic energy was equal to \( E_{k}^{n} \) ≈ 0% and normalized numerical damping was equal to \( D_{n}^{n} \) = 3.5% with respect to the total normalized energy (Fig. 7). For v/D = 0.4 mm corresponding to the test end (Fig. 5), the normalized elastic internal energy was \( E_{e}^{n} \) = 45%, normalized plastic dissipation was \( D_{p}^{n} \) ≈ 0%, normalized energy of removed contacts was \( E_{rc}^{n} \) = 4%, normalized kinetic energy was \( E_{k}^{n} \) ≈ 0% and normalized numerical damping was \( D_{n}^{n} \) = 51% with respect to the total normalized energy (Fig. 7). The energy of removed contacts started to gradually increase for v/D = 0.2%. For v/D ≥ 0.33%, its growth was pronounced.
In the case of D = 0.15 m, for the normalized vertical top displacement v/D = 0.30% corresponding to the peak load, the normalized elastic internal energy was equal to \( E_{e}^{n} \) = 73% (normal energy54%, tangential energy19%), normalized plastic dissipation was \( D_{p}^{n} \) ≈ 0.0%, normalized energy of removed contacts was equal to \( E_{rc}^{n} \) = 2%, normalized kinetic energy was equal to \( E_{k}^{n} \) ≈ 0% and normalized numerical damping was equal to \( D_{n}^{n} \) = 25% with respect to the total normalized energy (Fig. 7). At the test end (v/D = 0.25%, Fig. 5), the normalized elastic internal energy was 64%, normalized plastic dissipation was \( D_{p}^{n} \) ≈ 0%, normalized energy of removed contacts was equal to \( E_{rc}^{n} \) = 3%, normalized kinetic energy was \( E_{k}^{n} \) ≈ 0% and normalized numerical damping was \( D_{n}^{n} \) = 33% with respect to the total normalized energy (Fig. 7). Due to the snapback instability, the total internal energy reduced by 25%, the elastic normal internal elastic energy reduced by 50% and the elastic tangential internal energy reduced by 30%. In turn, the plastic dissipation, numerical damping and elastic energy from removed contacts increased on average by the factor 2. The energy of removed contacts started to gradually increase for v/D = 0.1%. It started to strongly grow for v/D ≥ 0.28%.
The total normalized internal energy was higher by 10% at the peak load (1.35 kN/m against 1.25 kN/m) and by 70% at the failure (1.6 kN/m against 0.95 kN/m) in the smaller specimen D = 0.05 m (Fig. 7a) as the result of both the higher vertical force P_{max} and ductility (Fig. 5). The contribution of the total normalized elastic energy with respect to the total normalized energy was higher for D = 0.05 m before the peak (97% versus 73%), and for D = 0.15 m at the failure (54% versus 45%) due to fracture (see Sect. 5). The total normalized elastic energy \( E_{e}^{n} \) was thus higher by 40% at the peak load and smaller by 10% at the failure in the smaller concrete specimen of D = 0.05 m due to a different fracture intensity in both the concrete specimens (see Sect. 7.1). The stronger contribution of the total normalized elastic energy at the peak connected with smaller fracture intensity caused the higher strength of the small concrete specimen. The contribution of the numerical damping D_{n} was inverse, i.e. higher for D = 0.15 m before the peak load and for D = 0.05 m after the peak load. The normalized energy of removed contacts was slightly higher for D = 0.15 m before the peak load and for D = 0.05 m after the peak load. The removed tensile contact failure energy E_{t} was much higher than the removed shear contact failure energy E_{s} (Eq. 13).
6.4 Energy evolution in fractured and unloading specimen region
For the smaller specimen (D = 0.05 m), the total normalized internal (absorbed) energy E_{abs}^{n} in the fractured zone was higher than the total normalized internal (released) energy E_{rel}^{n} in the remaining unloaded region by the factor of 1.5 at the peak load and by the factor of 2.4 at the failure (Fig. 8A, B). For the larger specimen (D = 0.15 m), the total normalized internal (absorbed) energy E_{abs}^{n} in the fractured zone was, however, smaller than the total normalized internal (released) energy E_{rel}^{n} in the remaining region by the factor of 1.2 at the peak load and larger by the factor 1.2 at the failure.
The maximum total normalized internal energy absorbed in the fractured region was higher for D = 0.05 m (than for D = 0.15 m) by the factor 1.3 for the peak load and by the factor 2.4 at the failure (Fig. 8a, b). The maximum total normalized internal energy released in the region beyond the fracture zone was higher for D = 0.15 m (than for D = 0.05 m) by the factor 1.3 for the peak load (that contributed to the size effect on strength) and smaller by the factor 1.25% at the failure. The normalized numerical damping was obviously higher in the fractured region and was smaller for D = 0.15 m before the peak load and smaller for D = 0.05 m after the peak load. The energy of removed contacts was negligible beyond the main fractured region for both the specimens.
In the case of D = 0.05 m, the increment of the total normalized internal energy after the peak load absorbed in the fractured zone ΔE_{abs}^{n} was by far higher than the increment of the total normalized internal energy after the peak load released in the remaining unloaded region ΔE_{rel}^{n} (ΔE_{abs}^{n} > ΔE_{rel}^{n}) (Fig. 8) that caused a global quasibrittle behaviour in the postpeak region of the small concrete specimen (Fig. 5). For D = 0.15 m, the increment of the total normalized internal energy after the peak load absorbed in the fractured zone ΔE_{abs}^{n} was smaller than the increment of the total normalized internal energy after the peak load released in the remaining region ΔE_{rel}^{n} (ΔE_{abs}^{n}< ΔE_{rel}^{n}) (Fig. 8) that contributed to a global very brittle behaviour with the snapback instability in the postpeak regime (Fig. 5). The same tendency occurred for the total normalized elastic energy amounts after the peak load (Fig. 8c).
7 DEM results at mesoscale level
7.1 Evolution of broken normal contacts
The continuous microcracking process already started from v/D = 0.17 for D = 0.05 m and from v/D = 0.10 for D = 0.15 m (Fig. 9) with a moderate intensity (Fig. 9). Later microcracking process became more intense in the smaller specimen of D = 0.05 m slightly before the peak load for P = 0.9P_{max} (v/D = 0.34) and in the larger specimen of D = 0.15 m more early before the peak load for P = 0.8P_{max} (v/D = 0.25) (similarly as in the experiment based on displacement measurements on the concrete surface using the DIC technique [14]). The contact damage intensity might be divided into two linear regimes for D = 0.05 m: before P = 0.9P_{max} (moderate intensity) and after P = 0.9 P_{max} (large intensity). For D = 0.15 m, it might be divided into three linear regimes: before P = 0.8P_{max} (small intensity), between P = 0.8P_{max} and P_{max} (large intensity) and after P_{max} (again moderate intensity). The results of the damage contact evolution are in agreement with the evolution of the energy of removed contacts (Fig. 7). The total number of broken normal contacts in ITZs: n = 60 for D = 0.05 m (n/D = 1200) and n = 220 for D = 0.15 m (n/D = 1450) was about 5 times smaller than in the cement matrix (n = 300/1000) (Fig. 9). Nearly 15%/70% (cement matrix) and 33%/80% (ITZs) of damaged normal contacts were broken before the peak load for D = 0.05 m/D = 0.15 m. The rate of the normal contact damage was always smaller in ITZs than in the cement matrix.
Figure 10 confirms that for the larger concrete specimen of D = 0.15 m much more contacts (relatively to the total crosssection area) were broken up to the peak and less relative contacts were broken in the softening region as compared to the specimen with D = 0.05 m (Fig. 10). At the peak, a macrocrack already developed for D = 0.15 m with the height of about 0.5D. For the smaller specimen, there existed solely many microcracks in the specimen  a macrocrack did not evolve at the peak. The snapback behaviour occurred in the specimen of D = 0.15 m in the postpeak regime since this specimen was already strongly fractured at the peak load and a smaller %number of local contacts was needed to fully damage the specimen.
7.2 Evolution of coordination number
The evolution of the coordination number (average number of contacts per particle) N for two specimens D = 0.05 m and D = 0.15 m is demonstrated in Fig. 11. It was related to the evolution of broken normal contacts (Sect. 7.1).
The coordination number was always slightly smaller for D = 0.15 m due to a higher number of particles in the specimen (Fig. 11). It decreased during deformation due to fracture. Up to the peak load, the coordination number decreased from N = 4.9 down to N = 4.8 for D = 0.05 m and from N = 4.8 down to N = 4.68 for D = 0.15 m (the reduction rate was higher for D = 0.15 m). From the peak load to the test end, the coordination number reduced from N = 4.8 down to N = 4.6 for D = 0.05 m and from N = 4.68 down to N = 4.63 for D = 0.15 m (the reduction rate was higher for D = 0.05 m due to more intense fracture, Fig. 11).
7.3 Evolution of interparticle contact forces
When subjected to the external load, the granular materials develops heterogeneous force networks to transmit stress at boundaries through interparticle contacts [56, 57]. Thus the interparticle contact force transmission in materials is essential for mesoscopic modelling of constitutive behaviour. The distribution of tensile forces is e.g. of particular relevance to the stress intensity factor which controls the initiation and propagation of cracks. The cracking process may be earlier predicted based on force transmission [57].
The force transmission within particulate bodies was realized via coexisting strong and weak contacts which formed the corresponding strong and weak force networks [56]. The external vertical splitting force P was transmitted mainly via a network of strong compressive contact forces that formed clear force chains parallel to P (red lines in Fig. 12). Initially, the large vertical compressive normal contact forces were created in the specimen midregion (Fig. 12). The tensile normal forces were located in a perpendicular (horizontal) direction. In the boundary regions compression obviously dominated over tension. Before the peak of the vertical force, the compression and tensile forces increased. Later, some single tensile forces started to break due to the contact damage. When a vertical macrocrack already crossed the specimen, the contact force networks appeared to be sparse and some tensile forces became located mainly along the specimen circumference caused by compression of two separated specimen’s halves. The mean width of the region with strong compressive normal contact forces was relatively larger for the peak load with D = 0.05 m (by 40%) due to its higher strength (Fig. 12). The %contribution of strong compressive/tensile normal contact forces was 31%/31% (peak load) and 26%/23% (failure load) for D = 0.15 m and was 38%/35% (peak load) and 17%/17% (failure) for D = 0.05 m, respectively. Thus, the %number of strong compressive normal contact forces was higher at the peak load for the smaller specimen (38% against 31%) due to its higher strength. The %number of strong tensile normal contact forces was also higher at the peak load for the smaller specimen (35% against 31%) due to a smaller cracking intensity. The large changes in the %contribution of all contact forces in the postpeak regime solely occurred for D = 0.05 m due to a strong microcracking process (Fig. 10). These changes were hardly distinguishable for D = 0.15 m since the specimen was subjected to a strong fracture process before the peak load.
7.4 Evolution of crack displacements
Figure 13 presents the evolution of the average normal and shear displacement along the main central macrocrack exactly at the midheight for two different specimen diameters: D = 0.05 m and D = 0.15 m.
The maximum compressive/tensile contact forces were: 32/7 N at the peak load and 19/6 N at the failure for D = 0.05 m and 38/9 N at the peak load and 31/8 N at the failure for D = 0.15 m. The failure had a clear tensile type. i.e. the normal crack displacement always dominated over the tangential crack displacement. The crack displacements evidently increased after the peak load. The normal crack displacement was slightly higher at the peak load for D = 0.15 m (0.20 μm versus 0.15 μm) and at the failure for D = 0.05 m (0.70 μm versus 0.60 μm versus). The tangential crack displacement was 57 times smaller than the normal one (Fig. 13).
8 Size effect: comparison of DEM predictions and experimental data
The numerical results were compared with the comprehensive splitting tensile tests performed by Carmona et al. [34] (Fig. 1). In the experiments, the forceCMOD curves were listed only. The laboratory tests were carried out on cylindrical concrete specimens with the diameter of D = 74 mm, 100 mm, 150 mm and 290 mm (the uniaxial compressive strength was 60 MPa). The weight proportion of the cement, crushed gravel (with the diameter d = 5–12 mm), gravelsand (d = 0–5 mm), microsilica and water in concrete was 1:1.93:1.93:0.1:0.33. The authors in [34] measured the vertical splitting force P against CMOD that was measured at the specimen midheight between two symmetric points at the distance of 65 mm. The width of the loading/supporting plates was scaled with the specimen diameter.
Concrete was described in DEM computations as a threephase material composed of aggregate, cement matrix and interfacial transitional zones (ITZs). The exact mesostructure of concrete used in experiments could not be reproduced in simulations due to the data lack. All particles were spherical and randomly distributed in specimens. The macrovoid were neglected. The total grain volume was 75% as for normal concretes. The minimum mortar sphere diameter was \( d_{cm}^{{\min} } \) = 0.1 mm. The content of grains was the following: d = 0.1–0.125 (20%), d = 0.125–2 mm (30%) and d = 2–12 mm (50%). All aggregate grains with the diameter of 2 mm ≤ d_{a}≤ 12 mm included ITZs. The cement matrix was modelled with spheres of the diameter 0.1 mm ≤ d_{cm}< 2 mm without ITZs. The width of the loading/supporting plate was b = 12.3 mm (D = 74 mm), b = 16.7 mm (D = 100 mm), b = 25 mm (D = 150 mm) and b = 48.3 mm (D = 290 mm) (as in the experiments). It was modelled by clusters composed of stiff spheres [14].
The following parameters of the cohesion and tensile strengths were mainly used in all DEM analyses: cement matrix (E_{c,cm} = 11 GPa, C_{cm} = 140 MPa and T_{cm} = 40 MPa) and ITZs (E_{c,ITZ} = 8.8 GPa, C_{ITZ} = 112 MPa and T_{ITZ} = 36 MPa). With those material constants, the uniaxial compressive strength was about 60 MPa as in the experiments. The ratios E_{c,ITZ}/E_{c,cm}, C_{ITZ}/C_{cm} and T_{ITZ}/T_{cm} were again chosen 0.8. The remaining parameters were the same as in Sect. 2: υ_{c} = 0.2, μ = 18^{o}, α_{d} = 0.08 and ρ = 2.6 g/cm^{3}. The 2D concrete specimens included in total about 10,000, 19,000, 46,000 and 173,000 spheres for the specimen diameter of D = 74, 100, 150 and 290 mm, respectively. The crack opening CMOD was measured between two spheres in the specimen’s midheight at the distance of 65 mm in a horizontal direction (as in the experiment).
Figure 15 describes the numerical size effect due to the specimen strength σ and specimen brittleness, expressed by the inclination α of the initial softening curve (α = f(CMOD/D)) to the horizontal in the counterclockwise direction versus the experimental outcomes. The splitting strength increased with decreasing specimen diameter as a result of the fracture process zone with a nonhomogeneous horizontal normal stress along the vertical direction [14]. Thus a size effect took place since the vertical macrocrack did not occur at the same time along the entire specimen diameter (Sect. 7.1, [58]). The material heterogeneity slightly augments the size effect [32]. The concrete brittleness also decreased with decreasing specimen diameter. The calculated concrete tensile strength changed by 16% between D = 74 mm and D = 100 mm, by 3% between D = 100 mm and D = 150 mm and by and 2.5% between D = 150 mm and D = 290 mm. The strength reached an asymptote with the large values of D [3]. The trends of the calculated varying σ versus D and α versus D were similar to the experimental ones except for two of the smallest specimens. The too low tensile strength in experiments for the smallest specimens was probably caused by boundary effects due to concrete drying. Note that other experimental outcomes in Fig. 1 did not show this unusual behaviour.
The numerical simulations of a size effect on splitting tension within enhanced continuum mechanics is the subject of ongoing research. The numerical predictions will be directly compared with the DEM results.
9 Summary and conclusions

The experimental size effect was realistically reproduced in calculations at the aggregate level, i.e. the concrete strength and ductility decreased with increasing concrete specimen diameter. The calculated decreasing strength approached an asymptote with increasing cylindrical specimen diameter within the considered specimen diameter range.

The continuous microcracking process started in the central vertical region at a very early deformation stage, i.e. far before the peak load. It consisted of two/three intensity regimes depending upon the specimen size. Initially, it evolved with the moderate intensity before the peak load for both the specimens and later with the pronounced intensity for the smaller specimen or with the pronounced and following moderate intensity for the larger specimen. The pronounced microcracking process mainly started in the smaller specimen slightly before the peak load and in the larger specimen clearly before the peak load.

The snapback behaviour occurred after the peak load in the larger specimen since the specimen was already strongly fractured and relatively fewer contacts were needed in the postpeak regime to fully damage the specimen in contrast to the smaller specimen. The specimen failure had a clear tensile type. i.e. the normal crack displacement always dominated strongly over the tangential crack displacement. At the peak, a clear macrocrack already developed in the larger specimen with the height equal to the half of the specimen diameter. For the smaller specimen, there existed many microcracks at the peak load. The width of FPZ was about 1/3 of the maximum aggregate diameter.

The higher strength of the smaller specimen was caused by the contribution of the normalized elastic energy that was greatly higher at the peak load than for the larger specimen due to the much lower fracture process intensity, expressed by a lower relative number of broken contacts with respect to the specimen diameter. The greatest total normalized internal energy absorbed in the fractured region was higher at the peak load by 30% in the smaller specimen. The greatest total normalized internal energy released in the region beyond the fracture zone was higher at the peak load by 30% for the larger specimen.

The load was carried in the specimens by strong compressive normal contact forces. The %number of strong compressive/tensile normal contact forces was higher at the peak load for the smaller specimen due to its higher strength and low fracture intensity. After the peak, their drop was by far higher for the smaller specimen due to the high fracture intensity.
Notes
Acknowledgements
Research was carried out within the project “Experimental and numerical analysis of coupled deterministicstatistical size effect in brittle materials” financed by the Polish National Science Centre NCN (UMO2013/09/B/ST8/03598).
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
References
 1.Bažant, Z.P.: Size effect in blunt fracture concrete, rock, metal. J. Eng. Mech. ASCE 110, 518–535 (1984)CrossRefGoogle Scholar
 2.Carpinteri, A.: Decrease of apparent tensile and bending strength with specimen size: two different explanations based on fracture mechanics. Int. J. Solids Struct. 25(4), 407–429 (1989)CrossRefGoogle Scholar
 3.Bažant, Z.P., Planas, J.: Fracture and Size Effect in Concrete and Other QuasiBrittle Materials. CRC Press, Boca Raton (1989)Google Scholar
 4.Korol, E., Tejchman, J., Mróz, Z.: FE analysis of size effects in reinforced concrete beams without shear reinforcement based on stochastic elastoplasticity with nonlocal softening. Finite Elem. Anal. Des. 1, 25–41 (2014)CrossRefGoogle Scholar
 5.SyrokaKorol, E., Tejchman, J.: Experimental investigations of size effect in reinforced concrete beams failing by shear. Eng. Struct. 58, 63–78 (2014)CrossRefGoogle Scholar
 6.Korol, E., Tejchman, J., Mróz, Z.: The effect of correlation length and material coefficient of variation on coupled energeticstatistical size effect in concrete beams under bending. Eng. Struct. 103, 239–259 (2015)CrossRefGoogle Scholar
 7.Duan, K., Hu, Z.: Specimen boundary induced size effect on quasibrittle fracture. Strength Fract. Complex. 2(2), 47–68 (2004)Google Scholar
 8.Bažant, Z.P., Pang, S.D., Vorechovsky, M., Novak, D.: Energeticstatistical size effect simulated by S6FEM with stratified sampling and crack band model. Int. J. Numer. Methods Eng. 71(11), 1297–1320 (2007)CrossRefGoogle Scholar
 9.Tanabe, T., Itoh, A., Ueda, N.: Snapback failure analysis for large scale concrete structures and its application to shear capacity study of columns. J. Adv. Concr. Technol. 2(3), 275–288 (2004)CrossRefGoogle Scholar
 10.Biolzi, L., Cangiano, S., Tognon, G., Carpintieri, A.: Snapback softening instability in high strength concrete beams. Mater. Struct. 22, 429–436 (1989)CrossRefGoogle Scholar
 11.SyrokaKorol, E., Tejchman, J., Mróz, Z.: FE calculations of a deterministic and statistical size effect in concrete under bending within stochastic elastoplasticity and nonlocal softening. Eng. Struct. 48, 205–219 (2013)CrossRefGoogle Scholar
 12.Kani, G.N.J.: How safe are our large concrete beams? ACI J. Proc. 64(3), 128–142 (1967)Google Scholar
 13.Weibull, W.: A statistical distribution function of wide applicability. J. Appl. Mech. 18(9), 293–297 (1951)zbMATHGoogle Scholar
 14.Suchorzewski, J., Tejchman, J., Nitka, M.: Experimental and numerical investigations of concrete behaviour at mesolevel during quasistatic splitting tension. Theoret. Appl. Fract. Mech. 96, 720–739 (2018)CrossRefGoogle Scholar
 15.Skarżyński, L., Nitka, M., Tejchman, J.: Modelling of concrete fracture at aggregate level using FEM and DEM based on xray µCT images of internal structure. Eng. Fract. Mech. 10(147), 13–35 (2015)CrossRefGoogle Scholar
 16.Nitka, M., Tejchman, J.: A threedimensional mesoscale approach to concrete fracture based on combined DEM with xray μCT images. Cem. Concr. Res. 107, 11–29 (2018)CrossRefGoogle Scholar
 17.Suchorzewski, J., Tejchman, J., Nitka, M.: DEM simulations of fracture in concrete under uniaxial compression based on its real internal structure. Int. J. Damage Mech. 27(4), 578–607 (2018)CrossRefGoogle Scholar
 18.Suchorzewski, J., Korol, E., Tejchman, J., Mróz, Z.: Experimental study of shear strength and failure mechanisms in RC beams scaled along height or length. Eng. Struct. 157, 203–223 (2018)CrossRefGoogle Scholar
 19.Skarzynski, L., Tejchman, J.: Experimental investigations of fracture process in concrete by means of xray microcomputed tomography. Strain 52, 26–45 (2016)CrossRefGoogle Scholar
 20.Hentz, S., Daudeville, L., Donze, F.: Identification and validation of a discrete element model for concrete. J. Eng. Mech. ASCE 130(6), 709–719 (2004)CrossRefGoogle Scholar
 21.Dupray, F., Malecot, Y., Daudeville, L., et al.: Mesoscopic model for the behaviour of concrete under high confinement. Int. J. Numer. Anal. Method Geomech. 33, 1407–1423 (2009)CrossRefGoogle Scholar
 22.Groh, U., Konietzky, H., Walter, K., et al.: Damage simulation of brittle heterogeneous materials at the grain size level. Theoret. Appl. Fract. Mech. 55, 31–38 (2011)CrossRefGoogle Scholar
 23.Chen, W., Konietzky, H.: Simulation of heterogeneity, creep, damage and lifetime for loaded brittle rocks. Tectonophysics 633, 164–175 (2014)CrossRefGoogle Scholar
 24.Poinard, C., Piotrowska, E., Malecot, Y., Daudeville, L.: Landis, E: Compression triaxial behavior of concrete: the role of the mesostructure by analysis of Xray tomographic images. Eur. J. Environ. Civil Eng. 16(S1), 115–136 (2012)CrossRefGoogle Scholar
 25.Ruiz, G., Ortiz, M., Pandolfi, A.: Threedimensional finiteelement simulation of the dynamic Brazilian tests on concrete cylinders. Int. J. Numer. Method Eng. 48, 963–994 (2000)CrossRefGoogle Scholar
 26.Ferrara, L., Gettu, R.: Size effect in splitting tests on plain and steel fiberreinforced concrete: a nonlocal damage analysis. In: Proceedings of 4th International Conference on Fracture Mechanics of Concrete and Concrete Structures, Cachan, France, pp. 677–684 (2001)Google Scholar
 27.Zhu, W.C., Tang, C.A.: Numerical simulation of Brazilian disk rock failure under static and dynamic loading. Int. J. Rock Mech. Min. Sci. 43, 236–252 (2006)CrossRefGoogle Scholar
 28.Mahabadi, O.K., Cottrell, B.E., Grasselli, G.: An example of realistic modelling of rock dynamics problems: FEM/DEM simulation of dynamic Brazilian test on Barre granite. Rock Mech. Rock Eng. 43, 707–716 (2010)ADSCrossRefGoogle Scholar
 29.Saksala, T., Hokka, M., Kuokkala, V.T., Makinen, J.: Numerical modeling and experimentation of dynamic Brazilian disc test on Kuru granite. Int. J. Rock Mech. Min. Sci. 59, 128–138 (2013)CrossRefGoogle Scholar
 30.Benkemoun, N., Poullain, Ph, Al Khazraji, H., Choinska, M., Khelidj, A.: Mesoscale investigation of failure in the tensile splitting test: size effect and fracture energy analysis. Eng. Fract. Mech. 168, 242–259 (2016)CrossRefGoogle Scholar
 31.Carmona, H.A., Kun, F., Andrade Jr., J.S., Herrmann, H.J.: Computer simulation of fatigue under diametrical compression. Phys. Rev. E 75, 046115 (2007)ADSCrossRefGoogle Scholar
 32.Murali, K., Deb, A.: Effect of mesostructure on strength and size effect in concrete under tension. Int. J. Numer. Anal. Methods Geomech. 42, 181–207 (2018)CrossRefGoogle Scholar
 33.AlKhazraji, H., Benkemoun, N., Choinska, M., Khelidj, A.: Mesoscale analysis of the aggregate size influence on the mechanical properties of heterogeneous materials using the Brazilian splitting test. Energy Procedia 139, 266–272 (2017)CrossRefGoogle Scholar
 34.Carmona, S., Gettu, R., Aguado, A.: Study of the postpeak behaviour of concrete in the splittingtension test, Fracture Mechanics of Concrete Structures. In: Proceedings FRAMCOS3, Aedificatio Publishers, D79104 Freiburg, pp. 111–120 (1998)Google Scholar
 35.Torrent, R.J.: A general relation between tensile strength and specimen geometry for concrete—like materials. Mater. Struct. 10, 187–196 (1977)Google Scholar
 36.Hasegawa, T., Shioya, T., Okada, T.: Size effect on splitting tensile strength of concrete. In: Proceedings of 7th Conference, Japan Institute, pp. 309–312 (1985)Google Scholar
 37.Bažant, Z., Kazemi, M.T., Hasegawa, T., Mazars, J.: Size effect in brazilian splitcylinder tests: measurements and fructure analysis. ACI Mater. J. 88(3), 325–332 (1991)Google Scholar
 38.Kadlecek Sr., V., Modry, S., Kadlecek Jr., V.: Size effect of test specimens on tensile splitting strength of concrete: general relation. Mater. Struct. 35, 28–34 (2002)CrossRefGoogle Scholar
 39.Timoshenko, S., Goodier, J.N.: Theory of Elasticity. McGrawHill Book Company, New York (1977)zbMATHGoogle Scholar
 40.Rocco, C., Guine, G.V., Planas, J., Elices, M.: Review of the splittingtest standards from a fracture mechanics point of view. Cem. Concr. Res. 31, 73–82 (2001)CrossRefGoogle Scholar
 41.Wei, X.X., Chau, K.T.: Three dimensional analytical solution for finite circular cylinders subjected to indirect tensile test. Int. J. Solids Struct. 50(14), 2395–2406 (2013)CrossRefGoogle Scholar
 42.Kuorkoulis, S.K., Markides, C.F., Chatzistergos, P.E.: The standarized Brazilian disc test as a contact problem. Int. J. Rock Mech. Min. Sci. 57, 132–141 (2013)CrossRefGoogle Scholar
 43.Kuorkoulis, S.K., Markides, C.F., Bakalis, G.: Smooth elastic contact of cylinders be caustics: the contact length in the Brazilian disc test. Arch. Mech. 65(4), 313–338 (2013)zbMATHGoogle Scholar
 44.García, V.J., Márquez, C.O., ZúñigaSuárez, A.R., ZuñigaTorres, B.C., VillaltaGranda, L.J.: Brazilian test of concrete specimens subjected to different loading geometries: review and new insights. Int. J. Concr. Struct. Mater. 11(2), 343–363 (2017)CrossRefGoogle Scholar
 45.Kozicki, J., Donze, F.V.: A new opensource software developer for numerical simulations using discrete modeling methods. Comput. Methods Appl. Mech. Eng. 197, 4429–4443 (2008)ADSCrossRefGoogle Scholar
 46.Šmilauer, V., Chareyre, B.: Yade DEM Formulation. Manual, 2011Google Scholar
 47.Donze, F.V., Magnier, S.A., Daudeville, L., et al.: Numerical study of compressive behaviour of concrete at high strain rates. J. Eng. Mech. 122(80), 1154–1163 (1999)CrossRefGoogle Scholar
 48.Nitka, M., Tejchman, J.: Modelling of concrete behaviour in uniaxial compression and tension with DEM. Granul. Matter 17(1), 145–164 (2015)CrossRefGoogle Scholar
 49.Widulinski, L., Tejchman, J., Kozicki, J., Lesniewska, D.: Discrete simulations of shear zone patterning in sand in earth pressure problems of a retaining wall. Int. J. Solids Struct. 48(7–8), 1191–1209 (2011)CrossRefGoogle Scholar
 50.Kozicki, J., Niedostatkiewicz, M., Tejchman, J., Műhlhaus, H.B.: Discrete modelling results of a direct shear test for granular materials versus FE results. Granul. Matter 15(5), 607–627 (2013)CrossRefGoogle Scholar
 51.Kozicki, J., Tejchman, J., Műhlhaus, H.B.: Discrete simulations of a triaxial compression test for sand by DEM. Int. J. Num. Anal. Methods Geom. 38, 1923–1952 (2014)CrossRefGoogle Scholar
 52.Kozicki, J., Tejchman, J.: Relationship between vortex structures and shear localization in 3D granular specimens based on combined DEM and HelmholtzHodge decomposition. Granul. Matter 20(48), 1–24 (2018)Google Scholar
 53.Ergenzinger, C., Seifried, R., Eberhard, P.A.: Discrete element model to describe failure of strong rock in uniaxial compression. Granul. Matter 12(4), 341–364 (2011)CrossRefGoogle Scholar
 54.Cundall, P.A., Hart, R.D.: Numerical modelling of discontinua. Eng. Comput. 9(2), 101–113 (1992)CrossRefGoogle Scholar
 55.Xiao, J., Wengui, L., Zhihui, S., et al.: Properties of interfacial transition zones in recycled aggregate concrete tested by nanoindentation. Cem. Concr. Compos. 37, 276–292 (2013)CrossRefGoogle Scholar
 56.Deng, X., Dave, R.N.: Properties of force networks in jammed granular media. Granul. Matter 19(27), 1–10 (2017)Google Scholar
 57.Kahagalage, S., Tordesillas, A., Nitka, M., Tejchman, J.: Of cuts and cracks: data analytics on constrained graphs for early prediction of failure in cementitious materials. In: Proc. Int. Conf. Powders and Grains, 2017, EPJ Web of Conferences 140, 08012 (2017). https://doi.org/10.1051/epjconf/201714008012
 58.Tejchman, J., Bobinski, J.: In: Wu, W., Borja, R.I. (eds.) Continuous and discontinuous modelling of fracture in concrete using FEM. Springer, Berlin (2013)CrossRefGoogle Scholar
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