Packing microstructure
In this section, effect of particle size ratio and contribution of granulometric fractions on structural properties of bidisperse mixtures subjected to a vertical pressure of 100 kPa was studied. For this purpose, the evolution of solid fraction \((\varPhi )\) of samples with various particle size ratios, defined as the fraction of sample volume filled by grains, with volume fraction of small particles was presented in Fig. 1. The mean values are plotted with the error bars indicating ± one standard deviation. Solid fraction was significantly larger in binary mixtures with size ratio of 0.4 and 0.6, as compared to monodisperse packings composed of large spheres. In these mixtures, an increase in solid fraction with an increase in fvalue up to 0.6 was observed, which was then followed by decrease in \(\varPhi \) value with increasing contribution of small spheres in sample. A maximum \(\varPhi \) value was observed in packings with volume fraction of small particles of 0.6, which corroborated findings reported earlier by inter alia McGeary [2], Rassously [3], Jalali and Li [6]. In samples with particle size ratio of 0.8, solid fraction varied slightly with increasing contribution of small particles in mixtures. No evident maximum in value of parameter was observed in these packings, which has been reported earlier by Wiącek [1]. The author indicated that solid fraction did not reach maximum in binary samples wherein the ratio of the diameter of small and large spheres was larger than certain critical value.
In this study, an effect of geometric and statistical factors on geometric anisotropy of binary mixtures was investigated through the comparison of contact normal orientation distributions. A contact angle is defined as:
$$\begin{aligned} \alpha =\arctan \frac{F_z^n }{F_x^n } , \end{aligned}$$
(1)
where \(F_z^n \) and \(F_x^n \) are the z- and x- components of the contact normal force. Figure 2 presents distributions of contact normal orientation in compressed mixtures with various size ratios. Regardless on g value, heterogeneous distribution of contact angles with a favored vertical contact direction was observed in examined packings. An increase in global anisotropy of the contact normal orientation was observed with increasing difference between the dimeters of small and large spheres in samples. Packings composed of particles with a small degree of particle size heterogeneity are arranged in a nearly crystalline formation with a favored vertical contact direction of \(90^\circ \) and \(270^\circ \). In samples with \(g=0.8\), each particle is supported by several neighboring particles and it supports the other ones, providing more homogeneous distribution of contact normal orientations, as compared to mixtures with \(g=0.4\). In packings with small g value, number of contacts directed upward is smaller than number of contacts directed downward, resulting in asymmetric distribution. In more heterogeneous packings, where small grains partially fill the pores between larger particles and they do not necessarily support particles located higher, number of contacts ranging from \(0^\circ \) to \(180^\circ \) significantly decreases. In these mixtures, contacts directed downward prevail with favored contact direction of \(270^\circ \). In the earlier study conducted by Wiącek and Molenda [17] for polydisperse granular mixtures with continuous particle size distribution, the authors observed more homogeneous distribution of contact force orientations in polydisperse packings as compared to monodisperse ones. These findings indicate that influence of polydispersity of granular packing on distribution of contact angles in sample is determined by a number of granulometric fractions. The authors suggest a presence of a certain number of granulometric fractions in particulate assembly above which anisotropy of the contact normal orientation decreases with increasing degree of particle size heterogeneity. An establishment of that number requires further study.
The partial distributions of contact normal orientation for large (ll), large and small (ls) and small (ss) particles are shown in Fig. 3. In samples with \(g=0.8\), a slight anisotropy in contacts between large spheres was observed, which was not visible in packings with smaller g values. Regardless on g value, no anisotropy was revealed by contact normal orientations between large and small spheres. In turn, for contacts between small spheres, a favored vertical contact direction was observed. In packings with \(g=0.4\), contacts directed downward prevailed. An increase in difference between diameters of particles in mixtures resulted in more asymmetric distribution of the contact normal orientation with greater spread downward in frequency than upward.
These results were in agreement with findings reported by Sánchez et al. [9], who observed smaller partial anisotropy in the contact distribution in samples with smaller ratio between the small and large sphere. These authors also reported no relationship between anisotropy and contribution of particle size fractions in mixture. In this study, that issue was investigated in detail, providing results opposite to ones presented by Sánchez et al. [9]. Figure 4 presents global distribution of the contact normal orientation in binary packings with \(g=0.8\) and volume fraction varying from 0.2 to 0.8. An increase in contribution of small particles in binary sample resulted in more disordered packing structure and more asymmetric distribution. In each sample, anisotropy in distribution of contact normal orientation with a favored vertical contact direction was observed. Contacts directed downward prevailed with contact direction close to \(270^\circ \) . A detailed analysis of results has shown that a percentage of these contacts in all contacts increased from 3.4% in packings with volume fraction of 0.2–6.2% in mixtures with \(f=0.8\). In samples with \(g=0.4\), percentage of contacts with contact direction close to \(270^\circ \) in all contacts increased from 5.2 to 6% with f value increasing from 0.2 to 0.8. These results show an evident influence of contribution of granulometric fractions in binary granular packings on anisotropy of the contact normal orientation. However, that influence decreases with decreasing ratio of the diameter of small and large spheres. The discrepancies between findings presented in this paper and the ones reported by Sánchez et al. [9] result from larger differences between sizes of particles examined by Sánchez et al.
Force and stress distribution
The probability density functions of normal contact forces in samples with volume fraction of small spheres of 0.8 and various particle size ratios are presented in Fig. 5a. Distributions of normal contact forces are asymmetrical and left-skewed. The most homogeneous distribution of normal contact forces was obtained in sample with \(g=0.8\). Probability density functions of normal contact forces narrowed in packings with decreasing ratio between small and large sphere. As the normal contact force is a function of the effective radius of contacting particles [1], the largest forces were obtained in samples with the smallest particle size ratio. Contribution of particle size fraction in an assembly was also found to have a strong influence on distribution of normal contact forces. Figure 5b presents distributions of normal contact forces in mixtures with different f values. Evolution of the packing structure of samples from ordered to disordered with an increase in contribution of small particles in mixtures up to 40% \((f=0.4)\) resulted in more heterogeneous distributions of normal contact forces and smaller average contact forces. A further increase in number of small particles in mixture resulted in more heterogeneous distribution of contact forces; however, no effect of composition of sample on the average contact force was observed in these samples. These results corroborate numerical findings of Wiącek and Molenda [17], who reported an increase in homogeneity of distributions of normal contact forces and a decrease in average contact forces with increasing degree of heterogeneity of polydisperse granular packing.
Distribution of contacts in the granular packing strongly influences transmission of forces and spatial distribution of force chains [14, 17]. The studies of polydisperse packings, conducted by Voivret et al. [24] and Wiącek and Molenda [17] have shown that the strongest forces passed through the largest particles in system. Figure 6 presents an evolution of the average compressive force, defined as a sum of normal forces at contacts between particles, with volume fraction of small spheres in mixtures. The average compressive forces were normalized by mean compressive force in monodisperse sample comprising large particles. The forces exerted on spheres at contact points were found to be strictly related to the partial coordination numbers, presented by Wiącek [1]. The \(\bar{F}\) values and coordination numbers for different contacts followed the same paths with increasing contribution of small particles in mixtures. The average compressive forces exerted on small spheres by other particles were the largest in mixtures with g value of 0.8, which was strictly related to the largest number of contacts between these particles in mentioned samples [1]. The opposite tendency was observed for forces exerted on large spheres by small ones. The average compressive force exerted on large particles was higher in packings with smaller particle size ratio, wherein the largest average coordination number was observed [1]. Figure 6d shows that the average compressive force exerted on large spheres by the ones of the same size was slightly sensitive to the ratio of the diameter of small and large particles in binary mixture. The differences between \(\bar{F}\) values in mixtures with different particle size ratio were small and they lied within the range of scatter.
Regardless on the ratio of the diameter of small and large spheres and contribution of particle size fractions in binary mixtures, the largest compressive forces were exerted on the large spheres which confirms that mainly these particles contribute into stress transmission in granular bedding. Figure 7 shows the chains of compressive forces in compressed mixtures with \(g=0.4\) and volume fraction of small spheres \(f=0.2\) and \(f=0.8\), in xz plane. The black and white colors indicate the maximum and minimum values of forces. The chains of the largest forces passed through the largest particles in samples, which has been already observed by Wiącek and Molenda [17] in polydisperse sphere packings.
In the binary granular packings, small spheres fill partially the voids between large grains and carry reduced effective stress. They may percolate through the primary fabric produced by immobile large particles [25, 26]. Kenney and Lau [27] reported that the presence of primary fabric and loose small particles in particulate solid was an origin of internal instability of an assembly. The transfer of externally applied loads is not homogeneous in particulate bedding and stresses are transferred through limited number of particles. A number of studies have been performed over last few decades to measure the contribution of particle size fractions into stress transfer in granular materials [14, 28].
In this study, the effect of particle size ratio and volume fraction of small spheres on global and partial stress (by size particle) in bidisperse mixtures was investigated. The mean particle stress is defined as [29]:
$$\begin{aligned} p^p =\frac{1}{3}tr(\sigma ^{p}) \end{aligned}$$
(2)
where the stress tensor components for a single particle are given by [30, 31]:
$$\begin{aligned} \sigma _{{ ij}}^p =\frac{1}{V_p }\sum _{c=1}^{N_c } {l_i^{{ pc}}} F_{{ ij}}^{n { pc}}. \end{aligned}$$
(3)
In Eq. (3), \(V_{p}\) is a particle volume,\(F_{{ ij}}^{n { pc}}\) is a normal force exerted on particle p at contact c and \(N_{c}\) is a number of contacts of particle p. The branch vector connecting the centre of the particle to its contact \((l_i^{{ pc}})\), associated with particle radius \(R_i^p\) and displacement in normal direction \(\delta _n^c\), is expressed as:
$$\begin{aligned} l_i^{{ pc}} =\left( R_i^p -\frac{\delta _n^c }{2}\right) \end{aligned}$$
(4)
The mean normal stress for the whole sample comprising N particles (global stress) is given by:
$$\begin{aligned} p=\frac{1}{V}\sum _{p=1}^N (p^{p}V^{p}) \end{aligned}$$
(5)
where V is a volume of sample. The mean normal stress for small spheres (partial stress) may be computed by:
$$\begin{aligned} p_s =\frac{\Phi }{\sum _{p=1}^{N_s } {V^{p}} }\sum _{p=1}^{N_s} {(p^{p}V^{p})} \end{aligned}$$
(6)
where \(\varPhi \) is a solid fraction of sample and \(N_{s}\) is a number of small particles.
Figure 8 shows the evolution of global stress with volume fraction of small spheres in binary mixtures with different particle size ratios subjected to compressive load of 100 kPa. As the global stress in granular material is determined by its packing density, the p(f) relationships were similar to the ones between solid fraction and volume fraction of small particles in sample. In samples with particle size ratio of 0.4, the global stress increased twofold with volume fraction of small particles increasing form 0 to 0.6 due to increase in solid fraction of samples. A further increase in f value decreased the mean normal stress. An increase in the ratio of the diameter of small and large spheres from 0.4 to 0.6 resulted in decrease in global stress by 30% in packings with volume fraction of small particles of 0.6. A slight relationship between global stress and contribution of small spheres was observed in samples with \(g=0.8\), wherein slight changes in solid fraction and total compressive force with increasing f value were also observed.
Figure 9 presents evolution of the ratio between pressure in small spheres and global pressure with contribution of small particles in samples. In packings with volume fraction of small particles of 20% \((f=0.2)\), the \(p_s /p\approx 1\), indicating that contribution of small and large spheres into stress transfer is approximately equal. A decrease in \(p_s /p\) value with increasing volume fraction of small particles in mixtures up to 0.6 was observed, which was followed by slight increase in packings with higher f values. In these samples, stress in small spheres was smaller than the global mean stress and it increased as the particle size ratio increased. At small g values, small particles do not fit completely the voids between large ones and interact with the large grains to a lesser degree. Therefore, their contribution into stress transfer in granular packings was smaller. These results were partially consistent with the findings reported by Shire et al. [14] for binary packings with particle size ratios higher than 2. The authors observed that small and large spheres contributed approximately equally into stress transfer in samples with volume fraction of small particles of 0.2 and stress in small particles decreased with decreasing ratio between diameter of small and large particles. No change in \(p_s /p\) value with decreasing particle size ratio in packings with larger number of small particles was also observed by the authors. In this study, a strong influence of the contribution of particle size fractions in mixture on stress transfer in small components was found, which was due to small differences between diameters of spheres. In samples with size ratio larger than 0.4, smaller spheres may not be tapered within the voids between larger particles which results in relationships between \(p_s /p\) and volume contribution of particle size fractions different than the ones obtained for mixtures with larger particle size ratios.
Energy dissipation
The elastic energy accumulated at contact between two interacting spheres is defined as:
$$\begin{aligned} E_{{ ij}} =\int \limits _0^\delta {F_{{ ij}}} d\delta _{{ ij}}, \end{aligned}$$
(7)
where \(F_{{ ij}}\) is a contact force in normal or tangential direction and \(\delta _{ij}\) is the normal or cumulative shear displacements [1]. For non-linear Hertz-Mindlin contact model, applied in this study, the elastic energies accumulated in the normal and tangential directions \((E_{{ ij}}^n, E_{{ ij}}^t)\) at contact between particles are given by:
$$\begin{aligned} E_{{ ij}}^n= & {} \frac{2}{5}F_{{ ij}}^n \delta _{{ ij}}^n, \end{aligned}$$
(8)
$$\begin{aligned} E_{{ ij}}^n= & {} \frac{{{{\left( {{F^t}} \right) }^2}}}{{2{k_t}}}. \end{aligned}$$
(9)
The sum of the above energies is a total elastic energy accumulated at the contact point between particles (E).
It is well-known that the mechanical response of a granular packing subjected to external load is determined by the dissipative nature of material; however, the knowledge in that field is still insufficient. Therefore, in this study, the effect of the geometric and statistical factors on dissipation of energy in binary mixtures was investigated. The energy dissipated in granular packing is defined as:
$$\begin{aligned} D=\Delta W-\Delta E \end{aligned}$$
(10)
where \(\Delta W\) is the work done on the sample by external forces and \(\Delta E \)is the change of total energy of the particulate assembly. The \(\Delta W\) is calculated from the external force using the following Eq. [32]:
$$\begin{aligned} \Delta W=\int \limits _0^{\Delta H} {{ FdH}}, \end{aligned}$$
(11)
where F is the force applied on the top platen along the deformation direction and \(\Delta \hbox {H}\) is the displacement of the top platen.
Figure 10 illustrates evolution of the energy dissipation per contact in bidisperse packings with \(g=0.8\) and different volume fractions of small spheres, when subjected to compressive loads. Dissipation of energy increased with increasing vertical pressure, which was consistent with numerical results previously reported by Wiącek and Molenda [17]. Energy loss in the contact point between two particles is determined by contact force which is a function of the radii of the interacting spheres [1]. Therefore, a decrease in dissipation of energy with an increase in the contribution of small spheres in granular system was observed. For the same reason, energy dissipated in the contact points decreased with decreasing ratio of the diameter of small and large spheres in samples. The variation of energy dissipation per contact with increasing volume fraction of small particles in mixtures is presented in Fig. 11.