A numerical study on the spatial orientation of aligning fibrous particles in composites considering the wall effect

The reinforced efficiency of steel fibers in composites is closely related to their spatial orientation, which can be generally driven by the external magnetic force and restricted by the wall effect of rigid boundaries of the container. To clarify the spatial orientation of steel fibers in composites considering the effect of rigid boundaries under the electromagnetic field, a series of two-phase models consisting of fibrous particles and homogeneous matrix are generated, in which the fibers are separately simplified as spherocylindrical, cylindrical, and linear particles. Based on these models of the semi-periodic boundaries, the effect of fiber characteristics (e.g., the fiber content Vf, fiber aspect ratio ε, fiber length lsf, and fiber style) on both the spatial distribution and orientation degree of fibrous particles is studied before and after the fibers are aligned by the magnetic force. The results revealed that (1) both the effective number NA and orientation degree ξ of fibrous particles at a cross-section of the container can be greatly increased when the electromagnetic field is applied and (2) the wall effect of rigid boundaries shows an adverse impact on the amelioration of NA and ξ, and the range size of the affected region is essentially equal to the effective length of fibrous particles of different shapes (e.g., lsf +Dsf) for spherocylindrical particles and lsf for cylindrical and linear particles).


Introduction
Incorporating short fibers into matrixes is deemed to be an effective way to improve the performance of brittle materials in terms of tension, brittleness, crack resistance, conductivity, etc. [1,2]. In the field of civil engineering, concrete material is a typically brittle composite. To strengthen the macro-properties of concrete, various types of short fibers such as steel fibers, carbon fibers, polypropylene fibers, and glass fibers have been applied, and the performance of concrete material can be significantly improved due to the addition of these fibrous components [3][4][5][6][7][8]. Numerous studies suggested that the enhancement effect of fibrous components on the matrix is not only related to the geometric features of fibers, but also closely dependent on their spatial orientation. For example, by using the electromagnetic field, the distributed directions of steel fibers in cementitious composites were uniformly aligned and their fracture properties were explored in detail in the study of Ahmd et al. [9]. The results revealed that the fracture performance of aligned steel fiber-reinforced material increased up to 80% relative to the ordinary steel fiber-reinforced system. In the literature [10], the impact of straw fiber orientation on the hygrothermal property of straw bale insulation was studied, and the findings revealed that in comparison with the conventional agricultural bales, the thermal conductivity of straw bales can be reduced by 38% through the optimizing orientation of straw fibers. In other words, it can effectively make the reduction of the insulation thickness of bales for the equivalent performance. Moreover, the influence of fiber orientation on the mechanical responses of engineering cementitious composite (ECC) was also investigated by Tawfek et al. [11]. The results showed that when the fibers were oriented with the load direction of ECC specimens, improved fiber bridging strength and mechanical properties with finer crack width can be obtained. Briefly, the reinforced efficiency of short fibers in materials is closely determined by their specific orientation, which, however, would be subjected to influence from a variety of factors.
The wall effect is a ubiquitous phenomenon in the preparation and evolution of composites [12][13][14]. It generally refers to that due to the restrictive effect of rigid boundaries on the neighborhood, huge disparities of structures or properties usually exist between the adjacent area of boundaries and other areas. For instance, because of the wall effect, a particular interfacial region is commonly exited around the aggregates in concrete, which is called "interfacial transition zone" (i.e., ITZ) in the field of civil engineering [15,16]. Compared to the cement paste, the ITZ exhibits a series of special characteristics such as low strength, high permeability, and big porosity, which in turn lead to the reduction of the holistic performance of concrete materials [17,18]. Using computer simulation, Zheng and Li [19], Xu and Chen [20], and Huang et al. [21] separately studied the influence of the wall effect of sample boundaries on the structural features of granular systems, and the importance of the wall effect is clearly observed in their studies. Besides, during the preparation and aligning of steel fibers in cementitious composites [22], the spatial distribution of fibers in the neighborhood of mold also shows a difference compared to the fibers in its central region because of the effect of mold boundaries. To eliminate the interference of formwork boundaries with the tested results in the experiment, the exterior portions of specimens near the mold are generally excised, as described in the study of Mu et al. [22].
Although the fact that the spatial distribution of short fibers in composites can be restricted by the wall effect has been widely recognized by researchers and the studies about the wall effect in materials have been held all along, as described in previous studies [12][13][14][15][16][17][18][19][20][21][22], there is still no explicit exploration of the quantitative relationship of the wall effect of mold to the orientation efficiency of short fibers in composites, especially when the alignment technology of fibers is implemented. On the one hand, the studies about the structure or properties of aligning fiber-reinforced systems (AFRS) are still at a start level. On the other hand, due to the opaqueness of most materials, it is very hard to quantitatively evaluate the spatial orientation of fibrous particles in them. To compensate for the above-mentioned lacuna, computer simulation technology is adopted in this work and the quantitative effect of the wall effect of rigid boundaries on the spatial orientation of short fibers is studied based on a series of packing models of particles.
Computer simulation is a widely used research tool recently and has attracted much attention due to its repeatability and low cost [23][24][25]. Taking the steel fiberreinforced mortar in civil engineering into account, the fibrous composites are treated as a two-phase composite system, namely, fibrous particles and homogeneous solid matrix, in this article. First, the three-dimensional (3D) granular models containing different styles of short fibers (e.g., spherocylindrical, cylindrical, and linear particles) are generated, and the serial section analysis in the field of stereology is further introduced. By the combination of the generated models of semi-periodic boundary and the stereological theory, a series of the curves of the arguments including the average effective number N A and orientation degree ξ of fibrous particles at the cross-section x are simulated and the spatial orientation of the fibers in different models is discussed then. Through this relatively thorough analysis, some findings about the quantitative influence of the wall effect on the distribution of both randomly dispersed and aligning fibers are clarified in this study.

Structural modeling of fibrous composite
As a typical man-made short fibrous substance in materials [26], a variety of steel fibers such as large-head fibers, round-straight fibers, ultra-fine fibers, and undulating fibers have been used in cementitious composites due to their great efficiency and easy fabrication. To explore the influence of fiber styles considering the wall effect, the spherocylindrical particles, cylindrical particles, and linear particles are separately applied to represent the short steel fibers of different shapes in mortar and the 3D models of the steel fiber-reinforced mortars are modeled using the random sequential packing algorithm of particles. The detailed generating procedure of models is illustrated in the form of flowchart, as shown in Figure 1. Figure 2(a)-(c) further shows the simplified models of mortar samples containing different types of fibers, in which all the systems are assumed to be a binary mixture (i.e., the randomly dispersed short fibrous particles in the homogeneous matrix).
As displayed in Figure 2(a) and (b), for the structural models containing spherocylindrical and cylindrical fibers, respectively, the relationship between the volume fraction  V f of particles and their total number N sf is separately described in the following equations: where L is the size length of the container, D sf is the width of both spherocylindrical and cylindrical particles, l sf separately denotes the length of cylindrical particle and the length of rectangular region in the center of spherocylindrical particles, and the symbol ε represents the aspect ratio of spherocylindrical and cylindrical particles, which is identical to the value of l sf /D sf as described in the study of Xu et al. [27].
Besides, by convention, the relative fraction V f of particles for the models composed of the linear fibers in Figure 2(c) is expressed by the following equation by reference to the formula in the study of Li et al. [28]: During the construction of the two-phase models, all the fibrous particles in the system are assumed to be randomly dispersed throughout the matrix (i.e., the cement mortar) and each particle cannot intersect with the others. Moreover, as illustrated in Figure 3, the origin of each fiber is assigned to be at the centroid of the particle. Then, the angle between the zenith direction and the direction of the particle is denoted by the symbol α, and its value is randomly selected in [0, π]. The signed angle measured from the azimuth reference direction to the orthogonal projection of the fibrous particle on the reference plane O-XY is symbolized by β, and its value is selected in [0, 2π]. Based on the above-mentioned description, the angle θ between the zenith direction and the direction of the particle is defined by the orientation angle of the fibrous particle here, as shown in Figure 3.
3 Aligning arrangement of fibrous particles in the electromagnetic field In the preparation of concrete materials, the rigid boundaries of molds generally present great influence on the distribution of inner granular components, e.g., gravels and fibers. To quantitatively characterize the impact of rigid walls on the spatial distribution and orientation of fibrous particles in materials, the condition of semi-periodic boundaries (i.e., the left-right planes of the model are set as rigid boundaries, and the up-down and frontbehind planes of the model are set as periodic boundaries) is applied to the structural models here, as shown in Figure 4(a). Intuitively, it can also roughly observe the major difference of fiber distribution between the neighborhood of rigid walls and the central region of the model. On the basis of magnetism, if the external electromagnetic field is adopted during the preparation of mortars, the steel fibers in them can generally tend to rotate in the same orientation as the magnetic field lines because of the strong magnetic force. As described in the study of Mu et al. [22], it is assumed that all the fibers would only be rotated around their own centroids from the initial orientation to the magnetic orientation when the external forces are acted, as shown in Figure 2. Regardless of the horizontal movement of these fibers, the steel fibers in the central region of models can be easily rotated to the magnetic direction, as shown in Figure 4(b). However, due to the wall effect of molds, the aligning movement of steel fibers near the rigid boundaries of models may be greatly limited and the final orientation of fibers is closely related to the specific locations of the fibers. In this work, to lower the complexity of the research, the rotation of steel fibers is assumed to be stopped immediately once the outline of fibrous particles is tangent to the rigid edges of models. Actually, due to the vibrating action of the external device, the vertical locations of all the fibers may be changed to a certain extent, which, in turn, maximizes the aligning degree of these fibers. Based on this, both the intersection and interaction of all these particles can also be eliminated here.

Serial section analysis
Quantitative stereology provides an unbiased tool for estimating the structural information of a geometric-statistical nature of an object by using the sections or projections of the object [29]. It is used to quantify the spatial distribution of discrete phases in polyphase composites. By combining stereological theory with the two-phase models in Figure 4, two configuration parameters of the distribution and orientation of short fibers (i.e., the average effective number N A of fibrous particles per unit area of cross-section and the average orientation angle ξ of fibrous particles at a cross-section) are statistically estimated here by using the serial section analysis. In particular, the second parameter ξ, which can also be called "orientation efficiency factor," is generally obtained by using the following equation: where N(x) denotes the total number of the intersected fibers with the cross-section at the distance x from the left rigid plane of samples and θ i is the orientation angle of the fibrous particle i and its value must be in the range of 0 to π/2, as shown in Figure 3.

Results and discussion
On the basis of a series of two-phase models containing different styles of fibrous particles, the influence of various characteristics of fibrous particles on the changing curves of the parameters N A and ξ through the whole models is investigated in this section. In all of the following simulations, the size lengths L of the models here are set to be 150 mm. Moreover, in order to guarantee the statistical reliability of the configuration parameters N A (x) and ξ(x), the number N total of samples should be larger enough. Figure 5 illustrates the sensitivity of the coefficient (CV) of variation for both N A (x = 10 mm) and ξ (x = 10 mm) to the number N total of samples. It can be seen that when the number N total reaches 500, the values of CV for both N A and ξ tend to be constant and small enough, which indicates that the average results of N A and ξ can be used to represent the effective number and spatial distribution of fibrous particles. Based on the curves in Figure 5, 1,000 samples are used for each kind of structure in this work and the changing curves of N A (x) and ξ(x) are plotted as a function of distance x based on the statistical average of the obtained data. Besides, based on the existing literature about the wall effect [13,20], the curves of structural parameters vs distance x generally possess approximately symmetrical variation in the range between the left and right planes with the rigid boundary condition. Accordingly, only half of these curves from the left boundary edge (i.e., x = 0 mm) to the center (i.e., x = 75 mm) of models are analyzed in the following sections.

Effect of fiber content on the spatial orientation of fibrous particles
Taking the granular systems consisting of spherocylindrical fibers into account, the influences of the contents of fibers on both the spatial distribution and orientation of fibrous particles with the wall effect are illustrated first. As shown in Figure 6(a), the effective number N A (x) of fibrous particles at a fixed cross-section x in [0 and 75 mm] for both the models of randomly dispersed fibers and the models of aligning fibers is plotted as a function of the distance x. It can be seen that, with the increase of x from 0 to 75 mm, all the curves of N A (x) vs distance x with V f = 0.5, 1.0, and 1.5% in the figure can be divided into three stages (i.e., the ascending stage, descending stage, and horizontal stage), whose trend is not subjected to the content and orientation of fibrous particles. Besides, under a constant V f , the value of N A (x) at a cross-section x for the AFRS is approximately 1.9 times the corresponding N A (x) for the randomly dispersed fiber-reinforced system (RDFRS), which indicates that once the spatial orientation of fibrous particles in materials is aligned by the electromagnetic field, the effective number of fibers in the majority of areas can be greatly increased, leading in turn to the improvement of material performance. On the basis of the simulated results in Figure 6(a), the correspondingly statistical curves of ξ(x) vs distance x for both RDFRS and AFRS are further shown in Figure  6(b). It is observed that, on the one hand, all the curves of ξ(x) vs distance with the different V f here present a similar tendency, which can be broken down into the ascending stage and the horizontal stage from the section x = 0 mm to x = 75 mm. On the other hand, when the aligning technique of fibers is applied in materials, the value of ξ(x) at all of the cross-section x is obviously increased relative to the value of ξ(x) for RDFRS.  Especially in the central region of the models, the value of ξ(x) can be increased to 1.0.
In addition, due to the wall effect, both N A (x) and ξ(x) in the neighborhood of the rigid boundaries of the container for both RDFRS and AFRS are apparently lower than the values for the central regions of the corresponding container in the work. And the closer the cross-section x reaches the rigid boundaries, the worse the spatial distribution and orientation degree of fibrous particles would be. Through the relatively careful discussion of the obtained data in both Figure 6(a) and (b), it can be found that the affected range by the wall effect for all the cases in the above-mentioned figures is roughly equal to 15.5 mm, which basically corresponds to the whole length of the ascending and descending stages of the N A (x) vs x curves in Figure 6(a), and the length of the ascending stages of the ξ(x) vs x curves in Figure 6(b).

Effect of fiber aspect ratio on the spatial orientation of fibrous particles
Under the condition of V f = 1.0% and D sf = 0.5 mm for the spherocylindrical fibers, the spatial distribution and orientation of fibrous particles with the different aspect ratio ε are displayed in Figure 7. It can be seen from both Figure  7(a) and (b) that when the distribution orientation of these fibers is aligned, both N A (x) and ξ(x) at a cross-section x for AFRS are also much larger than the corresponding values of N A (x) and ξ(x) for RDFRS. The aspect ratio ε of fibrous particles here has no significant influence on the special values of N A (x) and ξ(x) in the central region of the models. However, due to the effect of the rigid boundaries of the container, the ascending stages of the curves of both N A (x) vs distance and ξ(x) vs distance with the different aspect ratio ε present a large difference in the figures. Generally, no matter for RDFRS and AFRS, the larger the aspect ratio ε of spherocylindrical fibers is, the smaller the values of N A (x) and ξ(x) at a cross-section x in the neighborhood of the rigid boundaries of the container are. Actually, the affected range by the wall effect for all the cases here can also be regarded to correspond to the length of the ascending stage of the ξ(x) vs x curves or the total length of the ascending and descending stages of the N A (x) vs x curves, which are separately equal to 10.5, 15.5, and 25.5 mm for the cases of ε = 20, 30, and 50 in the figures.

Effect of fiber length on the spatial orientation of fibrous particles
In Figure 8, we further investigate the influence of the variation of fiber length l sf on the spatial distribution and orientation of spherocylindrical particles when the wall effect of rigid boundaries of the containers is considered. On the one hand, with the increase in the length l sf , the effective number N A (x) of fibrous particles at a cross-section x for both RDFRS and AFRS is significantly reduced, as plotted in Figure 8(a), which indicates that the special number of the fibers having a reinforced effect at a fixed cross-section is decreased. This is largely due to that the total number of fibrous particles in the system would be dramatically reduced with the increase in l sf . On the other hand, no matter what the length l sf is, the value of N A (x) at a cross-section x in the central region of AFRS is always much larger than the corresponding value for RDFRS. Moreover, the effect of the length l sf on the curves of N A (x) vs distance x can also be reflected as the difference between the ascending and descending stage of these curves. On the whole, the longer the length l sf of fibrous particles is, the larger the corresponding total distance x of the ascending and descending stage in these curves would be, which means the range size of structures influenced by the wall effect would also be wider.
As shown in Figure 8(b), on the one hand, as the orientation of fibrous particles is aligned by the electromagnetic field, the value of ξ(x) with x in [0 and 75 mm] can be dramatically increased under all the conditions of l sf = 9, 15, and 24 mm. This indicates that the reinforced efficiency of the fibrous particles at all the cross-sections can be greatly enhanced. On the other hand, the influence of fiber length l sf on the values of ξ(x) here is reflected by the variation and effective length of the ascending stage of the curves of ξ(x) vs distance in the figure. In the neighborhood of rigid boundaries (i.e., x = 0 mm), the value of ξ(x) shows a declined trend as the  length l sf increases. Besides, based on the simulated results in Figure 8(b), it can further find that when the value of l sf is separately equal to 9, 15, and 24 mm, the effective horizontal length of the ascending stage of the curves in Figure 8(b) is 9.5, 15.5, and 24.5 mm, respectively, which corresponds to the range size of the regions subjected to the wall effect in the system.

Effect of fiber style on the spatial orientation of fibrous particles
Finally, by simplifying the short fibers as spherocylindrical, cylindrical, and linear particles, respectively, the influence of the styles of fibers on the spatial distribution and orientation of fibrous particles in both RDFRS and AFRS considering the wall effect is also analyzed in this section. Under the condition of V f = 1.0% and l sf = 15 mm, Figure 9(a) first displays the statistical variation trends of N A (x) as a function of distance x in [0 and 75 mm] for different kinds of fibrous systems. One can clearly see that as the distance x increases from 0 to 75 mm, the value of N A (x) in the curves for all the cases shows first an increasing trend, then decreasing and finally horizontal trend. In other words, the styles of fibrous particles have almost no effect on the geometric shapes of the curves of N A (x) vs distance. But, on the other hand, the special value of N A (x) at a cross-section is obviously related to the fiber styles, as shown in Figure 9(a). This is because of the fact that with the change of the fiber styles, the total number of fibrous particles in the whole system may be greatly changed, which in turn leads to the large difference in N A (x) for different kinds of RDFRS and AFRS. Moreover, it can find that no matter what the style of fibrous particles is, the amplification of N A (x) at a cross-section x is basically unchanged when the spatial orientation of short fibers is adjusted from the randomly dispersed form to the aligning dispersed form. Finally, through the comparison of the data in the curves, it is observed that the value of distance x corresponding to the horizontal length of the ascending and descending stage of the curves in Figure 9(a) is approximately equal to 15.5, 15, and 15 mm for the spherocylindrical, cylindrical, and linear fibers, respectively. In Figure 9(b), the statistical variation trends of ξ(x) with x in [0 and 75 mm] for different kinds of ordinary and aligning fibrous systems are further illustrated. It is observed that all of the curves of ξ(x) vs distance here can also be broken down into the ascending and horizontal stages as the value of x increases. On the one hand, no matter what the style of fibrous particles is, the corresponding value of ξ(x) can be greatly improved when the fibers in the system are aligned. On the other hand, the effect of the style of fibers on the curves of ξ(x) vs distance here is mainly reflected by the effective distance x of the ascending stage of these curves, which is separately 15.5, 15, and 15 mm for the discorectangle, rectangle, and linear particles, as suggested by the data in Figure 9(b). Based on all of the variation curves of both N A (x) and ξ(x) in Figures 6-9, it can be concluded that the range size of affected regions by the wall effect in the system would be dependent on the synergies of both the total length of fibers and the fiber shape, in which the value of total length (i.e., l sf + D sf for spherocylindrical fibers and l sf for both cylindrical and linear fibers) can be generally deemed to be the major influencing factor.

Conclusions
The orientation of steel fibers in composites can be generally driven by the external electromagnetic field, and their orientation degree may be restricted by the wall effect of rigid boundaries. In this work, the quantitative effect of the wall effect on the spatial orientation of steel fiber in mortar is studied based on the computer simulation. First, by simplifying the steel fibers as spherocylindrical, cylindrical, and linear particles, respectively, a series of 3D composite models consisting of fibrous particles and homogeneous matrix are generated by the randomly sequential algorithm of particles. By combining these models of the semi-periodic boundaries with the serial section analysis in stereology, the influence of various fiber characters (e.g., the fiber content V f , fiber aspect ratio ε, fiber length l sf , and fiber style) on the variation of the spatial distribution and orientation degree (i.e., the curve of N A (x) vs distance x and the curve of ξ(x) vs distance x) of fibrous particles in both RDFRS and AFRS is quantitatively evaluated. Through the relatively systematic analysis, the following conclusions can be highlighted. (a) Through the external electromagnetic field, both the effective number N A (x) and the orientation degree ξ(x) of fibrous particles at the cross-section of the container can be greatly increased, especially in the central regions of the container, which in turn leads to the higher enhancement of material performance. (b) The wall effect of rigid boundaries shows an adverse influence on the amelioration of both N A (x) and ξ(x). The closer the cross-section away from the rigid boundaries of the container is, the more rigorous the restriction of the outside walls of the container to the spatial distribution and orientation of fibrous particles is. (c) The range size of affected regions by the wall effect is determined by the synergies of both the total length of fibers (i.e., l sf + D sf for spherocylindrical particles and l sf for cylindrical and linear particles) and the fiber shape, in which the value of total fiber length can be deemed to be the major influencing factor.