Issue 45

F. Qui et alii, Frattura ed Integrità Strutturale, 45 (2018) 1-13; DOI: 10.3221/IGF-ESIS.45.01 4 If the aggregate particle size distribution obeys Fuller’s gradation curve, then the amount of aggregate within the grading segment [ d s , d s+1 ] can be obtained as:           1 agg 1 a all max min , s s s s P d P d V d d V V P d P d      (2) where V a gg [ d s , d s+1 ] is the aggregate volume within the grading segment [ d s , d s+1 ], d min is the minimum size of aggregate particle, V a is the volume content of aggregate in concrete; V all is the total volume of concrete. After dividing the grading curve into different segments, it is possible to determine the size and number of aggregate particles by the grading segments containing the largest and smallest particles. For the grading segment [ d s , d s+1 ], the spherical aggregate particles can be generated in the following steps: Step 1. Calculate the volume of aggregate V ag g [ d s , d s+1 ] to be generated in grading segment [ d s , d s+1 ] according to Fuller’s grading curve. Step 2. Generate a random diameter d within the segment [ d s , d s +1 ] to define the size of aggregate particles. It is assumed that the size d obeys uniform distribution between d s and d s+ 1 , that is, d s ≤ d ≤ d s+1 . It may also be expressed as d = d s + η ( d s+1 − d s ), with η being a random number distributed uniformly between 0 and 1. Step 3. Calculate the volume of the generated particles and subtract it from the aggregate volume V agg [ d s , d s+1 ]. Step 4. Repeat Steps 2 and 3 until the remaining aggregate to be generated is less than πd s 3 /6. In other words, there is no more room to generate any aggregate particle in the current grading segment. Meanwhile, record the random number d ( d s ≤ d ≤ d s+1 ) in a size array Sz ( i , s ) = d ( i = 1, 2, …, j), and the number of generated aggregate particles in a number array Nb ( s , 1 ) = j . Next, transfer the volume of the remaining aggregate to the subsequent grading segment. Step 5. Repeat all the steps above for the next smaller size grading segment until the generation of the last aggregate of the minimum particle size. After obtaining the size and number arrays, select the near-spherical assemblies of elements to represent aggregates by replacing the properties of the mortar with those of the aggregates. Once the size and corresponding number arrays have been ob-tained then the assemblies of elements which approximate spheres can be selected to represent aggregates by replacing the proper-ties of the cement mortar with those of the aggregates. Generation of random aggregates and the ITZ In numerical simulations, the concrete specimens are either cylindrical or cubic in shape, depending on the type of concrete materials. Here, the randomly generated aggregate particles are confined to cubic specimens. Two constraints were imposed to select assemblies of elements as aggregate particles at a free position within each concrete specimen: each assembly of elements must be completely within the boundary of the specimen, and no overlap is allowed with the previously selected assembly. The random aggregates and the ITZs were generated in the following procedure. Step 1. Mesh the mortar of the specimen into regular solid hexahedral grids. The grid size should be determined on the accuracy requirements. Step 2. Generate random coordinates ( x , y , z ) in 3D space within the specimen with the centre of the assembly of elements as aggregate. Step 3. Check the central position against the previous selected size array Sz(i , s) to see if both constraints are completely satisfied. If one of the two constraints is violated, do not select the assembly of elements as aggregate. Subsequently, repeat the coordinate generation and position check. By this analogy, select the assemblies of elements within Nb ( s , l) one by one. Step 4. After the selection of all assemblies of elements within the grading segment [ d s , d s +1 ], change their material properties from mortar to aggregate. After that, proceed with the selection process in the subsequent grading segment. Step 5. Repeat the steps above to select the assemblies of elements until all the assemblies of elements as aggregates are successively selected in the specimen and the total volume content of aggregates is satisfied. Step 6. For each assembly of elements, select the outer surface nodes as aggregate, and split each node into two along the radial direction. Next, generate the ITZ elements between nodes and the corresponding new nodes. Moreover, reconstruct the mortar elements adjacent to aggregate by ERT without changing the material properties of these elements. Finally, check the quality of new elements and remove those failing the quality test (Fig. 2(e)). Based on the given distribution of aggregate grading, a 3D meso-mechanical model (Fig. 2) was generated for concrete using the program developed by Xu et al. [28]. The randomly distributed unit set and near-spherical shape of aggregate completely