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Computational packing of aggregates for the study of virtual asphalt samples

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In this paper, a proof of concept of a method is presented for the study of granular materials, such as asphalt, based on the use of a physics engine. To begin with, virtual aggregates are generated with randomized 3D shapes and a size distribution based on a chosen gradation curve. Then, the aggregates are placed in a constrained volume and subjected to a simulated vibration until satisfactory compaction is reached. Finally, the packed stone assembly obtained is saved as a 3D model, so that the virtual aggregates can be used for further studies such as the analysis of the void space in the material. All the steps in the method are described and discussed, along with the approximations made. Furthermore, an analysis of the void space is performed to determine if the method is able to generate air pores with realistic features. The analysis is performed by comparing the void space of a computationally packed aggregate assembly to that of a real asphalt core with the same aggregate gradation. The preliminary results obtained show that the modelling approach is able to represent effectively the air pores, thus, suggesting that further studies to advance this proof of concept should be conducted.
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Computational packing of aggregates for the study of virtual
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asphalt samples
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Andrea Chiarelli1,a, Andrew R. Dawson2,b, A. García3,c
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1Nottingham Transportation Engineering Centre (NTEC), Faculty of Engineering, The University
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of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
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achiarelli.andrea@gmail.com
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bandrew.dawson@nottingham.ac.uk
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calvaro.garcia@nottingham.ac.uk
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Abstract
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In this paper, a proof of concept of a method is presented for the study of granular materials, such
11
as asphalt, based on the use of a physics engine. To begin with, virtual aggregates are generated
12
with randomized 3D shapes and a size distribution based on a chosen gradation curve. Then, the
13
aggregates are placed in a constrained volume and subjected to a simulated vibration until
14
satisfactory compaction is reached. Finally, the packed stone assembly obtained is saved as a 3D
15
model, so that the virtual aggregates can be used for further studies such as the analysis of the void
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space in the material. All the steps in the method are described and discussed, along with the
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approximations made. Furthermore, an analysis of the void space is performed to determine if the
18
method is able to generate air pores with realistic features. The analysis is performed by comparing
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the void space of a computationally packed aggregate assembly to that of a real asphalt core with
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the same aggregate gradation. The preliminary results obtained show that the modelling approach
21
is able to represent effectively the air pores, thus, suggesting that further studies to advance this
22
proof of concept should be conducted.
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Keywords: packing, modelling, aggregates, asphalt
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1. Introduction
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In civil and structural engineering, hydrogeology, and petroleum engineering, many materials are
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granular materials, thus, their microscopic structure is characterized by the presence of air voids
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with various sizes [1]. For this reason, the development of computational methods that are able to
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reproduce the shape of real granular materials is of great interest.
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Numerous applications can be envisioned for realistic 3D models of granular materials. First of
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all, a 3D model of the porous space can be used for the design of novel materials with controlled
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air void content and air void size distribution. Second, virtual models can be used to study
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properties of permeable civil engineering materials such as pavements, as this kind of material
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affects e.g. the traffic conditions in rainy weather [2] and soil pollution [3]. Third, 3D models of
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the pore space in granular materials can be used to study air and vapor movement which are of
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great importance in soil science [4] and natural air conditioning.
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In the field of computational methods, the generation of domains for the analysis of materials with
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air voids is usually approached (i) with mathematical or geometrical models, (ii) with the discrete
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elements method (DEM), or (iii) with the use of X-ray CT scans.
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Mathematical models are generally used for fluid-dynamics, as their aim is the generation of the
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pore space. Examples of mathematical models are the percolation method [5, 6, 7], or the use of
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Fourier transforms coupled with Voronoi tessellation [8]. Models for the generation of the pore
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space based only on geometrical principles have been studied by the authors [9, 10].
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DEM simulations are used for mechanical analyses where the particles composing the materials,
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i.e., the grains, are organized together in a confined space and the relationship between them is
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defined by using physical laws considering friction or energy [11, 12]
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X-ray CT scan imaging is a more flexible means to generate virtual domains for computational
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simulations, as the virtual models can be used for both mechanical and fluid-dynamics analyses.
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In fact, from X-ray CT scans it is possible to isolate both the grains and the pore space, thus, a
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number of different analyses can be performed [1, 13, 14, 15]. In addition, X-ray CT scans can be
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used to computationally characterize a material [16]. The use of X-ray CT scans, however, comes
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with a serious limitation, namely, the availability of a CT scanner.
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Currently, only X-ray CT scan imaging is able to describe effectively the stones and the void space
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at the same time. Mathematical and geometrical methods either neglect the stones or consider
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idealized stone shapes, while DEM methods typically use spherical virtual stones that inevitably
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yield an unrealistic shape of the void space due to the regular solids considered. The authors [17]
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were able to use a discrete element simulation to pack stones with realistic shapes in 3D, however,
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only one stone shape was chosen and therefore the stone size distribution was neglected.
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For these reasons, it is important to develop a method that allows a study of the aggregates and the
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void space based on realistic (or customized) aggregate gradations and shapes. Such a method is
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expected to allow engineers and researchers to better design granular materials, because the study
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of the physical properties could be performed based on known (and, potentially, controlled)
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assemblies of stones and pores rather than on a separate analysis of either the stones or the pores.
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In this paper, a proof of concept of a method for packing aggregates with realistic shapes and
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gradation and based on the use of a physics engine is presented. In order to achieve a preliminary
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confirmation that the development of the proof of concept should be continued, a sample virtual
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model created with the approach proposed by the authors is compared to real X-ray CT scans of
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an asphalt mixture. The comparison performed here is focused only on the study of the air pores
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in the virtual material because the 3D shapes of the real stones used were not available.
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Figure 1. Flowchart of the method for packing of aggregates and for the generation of
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virtual air pores.
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2. Development of the packing method
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In Fig. 1, a flowchart of the whole method is shown. In the next subsections, every step is
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described in detail to allow a clear understanding of the whole process
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2.1. Generation of the virtual aggregate
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Before the generation of the assembly of packed virtual stones is started, a number of parameters
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need to be chosen. To begin with, since virtual stones with realistic shapes are used, it is necessary
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to choose their number, size, and aspect ratio. Each stone is created as a virtual 3D object by
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computing the convex hull of a cloud of random points (see Fig. 2).
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Figure 2. Generation of a virtual stone (cloud of points and corresponding convex hull).
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In particular, 50 random points are seeded in the 3D space shown in Fig. 2, then the most “external”
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points in the cloud are used to define the surface of the stone (this enveloping surface is the
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abovementioned convex hull). The number of randomly seeded points can vary based on the user
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needs to create simpler or more complex 3D shapes. It is relevant to mention that too high or too
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low numbers of points could lead to unrealistic stones, e.g., a cube or oddly-shaped flake,
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respectively. The generation of random points in this paper was based on the use of the uniform
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distribution, however, any other distribution (even a custom one) could be followed.
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The points can be generated in any arbitrary interval in each direction, thus, allowing the
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generation of virtual stones of any chosen shape, aspect ratio, or faces. In this paper, the cloud of
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points that originates each virtual stone is created with a 0.6:0.7:1 aspect ratio between the x,y, and
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z axes. It is important to give the virtual stones an aspect ratio that is not 1:1:1 (i.e., a sphere),
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otherwise the maximum degree of packing will be constrained to the maximum level of packing
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that can be obtained for a set of polydisperse spheres, which corresponds to a rather high air void
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content (more than 35% [18]).
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In addition, the stones are seeded into a 3D space which is sized according to a granulometric
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curve that can be chosen by the user, thus, any kind of grading can potentially be simulated (e.g.,
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open grading or dense grading). The seeding and stone construction is executed with MATLAB
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and the stones are saved in the .obj file format to allow easy handling in the next steps.
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2.2 Compaction and packing of the virtual aggregate
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Once virtual aggregates have been created, they can be loaded in the 3D compaction environment.
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The virtual compaction method proposed in this paper exploits the physics engine of a videogame
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development software to vibrate the mixing container where the stones are placed based on Eq. 1-
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3.
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Some input values need to be chosen for the compaction process, which is based on the vibration
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of the volume where the stones are mixed. Therefore, the size of the mixing container has to be
8
picked based on the amount of material that is simulated, while its movement needs to be defined
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by the means of some equations. In this paper, sinusoidal equations were chosen, i.e.:
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𝑝(𝑡) = 𝑝0+ 𝑎 ∙ sin⁡(𝜔𝑡) (1)
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𝑞(𝑡) = 𝑞0+ 𝑏 ∙ sin⁡(𝜔𝑡) (2)
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𝑟(𝑡) = 𝑟0+ 𝑐 ∙ sin⁡(𝜔𝑡) (3)
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where 𝑝, 𝑞, and 𝑟 are the coordinates of the mixing container on the x,y, and z axes at time 𝑡; the
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subscript 0 describes the starting position of the container; and 𝑎, 𝑏, 𝑐, and 𝜔 are arbitrary
15
constants. For the preliminary results shown in this paper, the vertical oscillation (𝑦 axis, described
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by Eq. 2) was set to 0, as it would have added a further degree of freedom that is not discussed in
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the present analysis. Finally, the values of 𝑎 and 𝑐 were set to 1 and 𝜔 was set to 5. These values
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are arbitrary and were picked after a number of trial simulations in order to achieve a level of
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compaction that showed realistic features.
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When the stones are loaded in the mixing environment they are assigned a position, a rotation, a
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mesh and a corresponding mesh collider, which is used to detect when items with complex shapes
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touch one another. The use of a mesh collider in a physics engine is an advancement from previous
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methods of aggregate packing, because it allows the use of a range of realistic shapes for
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aggregates in the place of simple spheres (see, e.g., [19]) and prevents the need of implementing
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the specific physics that rule the contact between particles as done in [17] because they are included
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in the physics engine. When the virtual stones are imported in the mixing environment, they are
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also assigned their material properties, i.e., mass and friction factors. This is because these
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parameters rule how the particles interact with each other during the virtual compaction process.
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Bitumen and filler were not explicitly included in the simulations performed as done in recent
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studies on virtual aggregate compaction [19]. In [19], each stone is considered as if it was coated
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by a very thin layer of bitumen, thus, using modified contact laws to take bitumen and filler into
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account during the compaction and mixing process. In this paper, contact laws are not manually
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implemented and the contact between particles is completely handled by the physics engine. This
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is an approximation, however, the approach proposed can be justified by the high quality of the
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results that it yields.
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In Fig. 3, a mixing volume with virtual stones is shown. It can be observed that compaction is
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achieved only by vibration and that no load is applied to the upper surface of the virtual stones. A
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surface load could be applied, however, further studies should assess whether it would be
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beneficial for the realism of the result or not.
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1
(a) Mixing container and directions of the oscillation.(b) Detail of particles being compacted.
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Figure 3. Mixing container with virtual stones during the packing process.
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4
The mixing container can be vibrated for an arbitrary amount of time based on any chosen criteria.
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In this paper, the vibration was interrupted when the thickness of the stone assembly stopped
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decreasing for at least 1000 time steps, i.e., when no further compaction could be achieved by
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vibration in a practical amount of time. Finally, it is relevant to mention that in the packing process
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the aggregates were free to move and rotate with no constraints. Therefore, 6 degrees of freedom
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are allowed for each particle during the simulation.
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2.3 Extraction of the aggregate assembly and analysis of the pore space
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At the end of the virtual compaction process the aggregates can be saved in a 3D file format for
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further elaboration. The most direct way to extract the pores from the stone assembly is the use of
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a Boolean operation. In fact, if a solid box is created the stone assembly can be subtracted from it
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(Boolean subtraction), thus, generating a 3D representation of the pores. An example of this can
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be seen in Fig. 4, where the pores are now a solid as opposed to the previous situation (see Fig. 3)
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where the aggregates were the solid part. As a matter of fact the volume representations of both
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the aggregates and the pores may be combined and represented at the same time, so that the
20
material could be studied as a whole. In Fig. 4, it can be appreciated that the position of the virtual
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stones was unconstrained during the mixing process. In addition, a number of stone sizes can be
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observed in the virtual sample.
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Another option to analyze the packed aggregates is the creation of virtual tomographies of the
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stone assembly, as done e.g. in [9] and [10]. This method is obviously less straightforward and
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requires further elaboration of the 3D volume, however, it allows a more precise analysis of the
26
data that is obtained. When the virtual tomographies are created, the distance between slices of the
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virtual material is conceptually similar to the vertical spacing in X-ray CT scanners. For this
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reason, a stack of virtual tomography images can be created to pursue the study of the pore space
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in terms of a number of properties.
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For the study of the aggregate packing method presented in this paper the authors developed an
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automatic slicing algorithm that is able to read the 3D files generated by the virtual compaction
32
system and then analyze them. In particular, the algorithm developed calculates the intersection
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between arbitrary horizontal slicing planes with the 3D model and returns the profile of the stones
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in the layer corresponding to the vertical position of the slicing plane.
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1
2
Figure 4. Pores as a 3D structure and cross sections in an open-graded mixture.
3
4
The implementation of the slicing algorithm was made in MATLAB and a visual explanation of
5
its working mechanism is shown in Fig. 5. The blue plane in Fig. 5 is the slicing plane at a height
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equal to 50% in the virtual sample shown, while the red profiles represent the intersection between
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the 3D geometry and the slicing plane. While the slicing plane is shifted from the bottom to the
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top of the packed virtual stone assembly the red profiles can be exported as images or image stacks.
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The algorithm developed offers a high flexibility, as it allows the automatic generation of an image
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stack very similar to those usually obtained with X-ray CT scanners quickly and with a very high
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accuracy. An alternative method for creating virtual tomographies is the use of the void tracking
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method studied by the authors in [9] and [10], however, the newly developed algorithm is several
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times faster and, more importantly, treats each stone as a set of surfaces, thus, allowing an exact
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projection of the stone profiles on the slicing planes. As a result, the new slicing method does not
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cause any concern about precision or grid spacing as seen in [10].
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Figure 5. Visual explanation of the slicing method used to generate virtual tomographies.
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3. Examples of outputs and comparison with a real asphalt sample
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In order to provide confirmation of the effectiveness of this proof-of-concept method a virtual
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porous asphalt core was built. The virtual asphalt core was built with the stone size distribution
3
shown in Table 1, which was previously used to build a porous asphalt core with and air void
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content of 25%. Note that the percentages shown in Table 1 are rescaled, as in the construction of
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the virtual model the dust used for the manufacture of the real sample was not considered.
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Table 1. Sample size distribution used to build a porous virtual asphalt sample.
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Aggregate size (mm)
Percent of total mass (%)
20 mm
9
14 mm
33
10 mm
32
6.3 mm
26
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The stone shapes (aspect ratios) were defined as explained in section 2.1 because the real aspect
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ratios of the stones used for the asphalt core were not known.
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Virtual tomographies were performed on the asphalt sample created computationally in order to
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show the high degree of realism reached with the packing method used. In Fig. 6, a virtual
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tomography is shown along with a real CT scan obtained from the above-mentioned asphalt core.
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(a) Virtual asphalt sample (~23% air voids). (b) Real asphalt sample (~25% air voids).
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Figure 6. Comparison between a virtual tomography and a real X-ray CT scan (thresholded
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2D images).
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It can be observed that the shapes of the stones in the virtual tomography (Fig. 6a) resemble the
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real shapes and that the planar void fraction is, visually, very similar. In addition, the virtual stones
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seem to be arranged in a realistic way when compared to the real ones. From a first visual
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examination, also the air voids on both sides of Fig. 6 seem to have similar sizes and shapes, thus,
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suggesting that there is an acceptable degree of similarity. In order to confirm what can be visually
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assessed three analyses were run to compare the morphology of the pore space in both the virtual
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and the real samples. The analyses considered in this paper measure the fractal dimension, the
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connectivity density, and the volume fraction. In Table 2, the properties calculated with the BoneJ
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8
plugin for ImageJ (connectivity density and volume fraction) and with MATLAB (fractal
1
dimension) for the two samples are shown.
2
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Table 2. Morphological properties of the air pores in the virtual and in the real asphalt
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samples.
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Property
Pores in real sample
Pores in virtual sample
Fractal dimension (-)
1.80
1.76
Connectivity density (pixel-3)
0.000346
0.000392
Volume fraction in 3D (%)
25.5
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Tortuosity (-)
1.37
1.30
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The data in Table 2 suggests that the samples are comparable and their degree of similarity is likely
7
to be acceptable, especially considering that the shape of the real stones was unknown. At this
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preliminary stage, the desirable degree of similarity is not known.
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To begin with, it is interesting to notice that the fractal dimension (MinkowskiBouligand
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dimension or box-counting dimension [9]) is similar in the samples and falls in the interval 1.63-
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1.82 previously reported in [20] for real asphalt samples. The fractal dimension is important
12
because it was shown to provide information on the homogeneity of asphalt [20], which in turn is
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related to the quality of a pavement [21].
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Figure 7. Cross sections of a composite model made by joining virtual and real air pore
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structures.
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Furthermore, a successful comparison of the connectivity density between the samples is a rather
20
strong indication of the potential of the approach. In fact, the connectivity density is calculated by
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the means of the Euler characteristic of the pore space and it is a volume weighted measure of the
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number of connected elements in a network [22]. As a consequence, similar connectivity densities
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mean that there is a similar number of connected elements per unit volume in the samples (real
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and virtual) under analysis.
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The volume fraction of the samples can be assessed either in 2D or in 3D, thus, explaining why
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the values of air void content apparent as areas in Fig. 6 don’t match the volume fractions seen in
27
Table 2. For example, the measures of air void content shown in Figure 6 are obtained by image
28
9
analysis, while with different algorithms this parameter can be computed for a 3D volume. The
1
latter option is that used by BoneJ, and the results are very satisfactory due to the fact that the
2
volume fraction of the virtual sample is very similar to that of the real sample using the same
3
aggregate gradation. This suggests that by packing virtual stones with a specific gradation curve it
4
is possible to obtain realistic air void contents. This result could be helpful for mixture design once
5
the method is more thoroughly tested.
6
It is also important to consider the tortuosity of the voids in the samples under analysis. In this
7
paper, the tortuosity of the pores was calculated by skeletonizing the pore space and by dividing
8
the length of the maximum path found by the Euclidean distance between its start and end points.
9
The similar values of tortuosity obtained suggest that fluids should behave similarly in the virtual
10
and the real materials. The lower tortuosity obtained for the virtual sample is reasonable, because
11
in the virtual sample there are no binder and filler material, thus, the paths in the pore space are
12
expected to be less tortuous than in the real sample. Tortuosity is of interest when studying
13
phenomena such as water evaporation [4], drainage [23], or the use of air convection in asphalt
14
pavements for energy harvesting purposes [24]. Currently, it is very complex to obtain large pore
15
networks for computational studies due to the fact that the computational domains used for the
16
study of fluid dynamics are usually obtained from X-ray CT scans. As a result, the size is
17
constrained by the maximum size and discrimination allowed by the scanning equipment. On the
18
contrary, when a computational method such as the one used in this paper is used, any arbitrary
19
size and thickness can be obtained for the study of fluid flow in the pores. In principle, the
20
discrimination of particle shape can also be obtained to any desired degree of complexity. In these
21
cases, though, the available computational power is the limiting factor.
22
Since the numerical results shown in Table 2 do not offer visual evidence of the goodness of the
23
approach, the real and the virtual samples studied were joined in a composite model (see Fig. 7).
24
In Fig. 7, two perpendicular cross sections of such composite model are shown to let the reader
25
see that without the labels in the figure it would be very hard, if not impossible, to tell the real and
26
virtual scans from one another. The 3D models that were joined were cut to the same size and the
27
number of faces in each of them was made equal so that the data would be compatible.
28
Finally, it is relevant to add that due to the lack of bitumen and filler in the model it is not possible
29
to simulate very dense mixtures at the moment. The main reason for this is that with no bitumen
30
and filler there will always be some relatively large (based on the mixture under analysis) portions
31
of space that cannot be filled.
32
4. Possible applications
33
The proof-of-concept method described in this paper could potentially be used for a number of
34
applications due to its apparent effectiveness and efficiency, e.g.:
35
Design and analysis of asphalt mixtures: the authors expect that further developments of
36
the approach presented in this paper could allow the study of a wide variety of stone
37
assemblies with a range of stone shapes and sizes. Once the method is refined and
38
thoroughly tested, the analysis of asphalt samples could be performed not only about the
39
air pores but also about the aggregates. This is because when the stone assemblies are saved
40
after the packing simulation each stone is exported as a separate entity with its position and
41
rotation, thus, allowing further studies on the packing process. As a result, the method
42
could assist in the creation of new and optimized asphalt mixtures or other granular
43
materials.
44
10
Computational fluid dynamics simulations: the modelling approach chosen allows the
1
creation of 3D domains for computational analyses such as simulations of water, air, or
2
multi-phase flow in the pores of granular materials.
3
Improved study of contact forces between aggregates: it is expected that the modelling
4
approach could lead to the study of contact forces between stones with realistic shapes as
5
opposed to current studies that consider ideal spherical aggregates.
6
Packing models: due to the flexibility of the approach chosen, the authors expect that the
7
method could be used to study how objects with different shapes reach a random packed
8
condition when placed inside a constrained volume.
9
5. Conclusions
10
In this paper, a proof of concept of a new method for packing aggregates was introduced and
11
discussed. The following conclusions can be drawn:
12
The method provides packed stone assemblies that visually resemble real stone
13
assemblies.
14
The morphological analysis of a virtual porous sample and of a real porous sample
15
provided a first demonstration of the feasibility of the method.
16
The use of a real gradation curve to generate a packing of virtual stones allowed the
17
generation of an air void content very similar to that obtained in a real asphalt core.
18
Stones with arbitrary shapes can be used to achieve a high degree of realism as opposed
19
to stones with idealized shapes such as spheres. It is, however, important to pursue the
20
study of real aggregate shapes to increase the verisimilitude of the results.
21
For future work, further to the preliminary observations made on a single virtual sample, a
22
possible validation of the modelling approach could be pursued by comparing a high number of
23
virtual samples with different gradations and stone shapes and sizes to a high number of real X-
24
ray CT scans.
25
26
Acknowledgements
27
The authors thank the University of Nottingham for the financial support provided for the Ph.D.
28
of Andrea Chiarelli. New Lecturer's Award to Alvaro Garcia is also acknowledged.
29
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... Chiarelli et al. created polyhedral aggregates as the convex hull of a cloud of random points and compacted them using Unity3D [27]. Thus, it can be concluded that a physics engine can be employed as a geotechnical engineering simulation tool. ...
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