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Effects of Particle Shape on the Shear Wave Velocity and Shear Modulus of 3D Printed Sand Analogs

Isolating the effects of individual particle properties (e.g. shape, size, mineralogy, surface roughness) on the mechanical behavior of naturally occurring coarsegrained soils is a significant challenge in experimental studies. This challenge can be addressed by recent advances in 3D printing technology which enable generation of artificial sand-sized particles with independent control over particle size and shape. In this study, bender element tests are conducted to examine the isolated effects of particle shape on the shear wave velocity and shear modulus of 3D printed sand analogs. The experimental results show that the shear wave velocity and shear modulus of the 3D printed sand specimens exhibit a relationship with mean effective stress that is in agreement to that reported for natural sands. The specimens composed of 3D printed sands with greater particle roundness and sphericity exhibit greater shear wave velocity and shear modulus for a given void ratio, relative density, and mean effective stress. The changes in shear wave velocity can be captured in terms of differences in individual particle shape parameters such as roundness and sphericity as well as combined particle shape parameters such as regularity. Regression analysis is used to develop relationships between shear wave velocity and particle shape parameters and void ratio, which are shown to be in agreement with previously-published relationships and to reliably predict the shear wave velocity of natural sands. The results presented herein highlight the usefulness of testing 3D printed soils to identify functional trends and dependencies between soil response parameters and intrinsic properties. However, this requires verification of the results against published trends and assessment of the possible effects of the differences in constituent material between the 3D printed and the natural soils.


Introduction
Inherent particle properties such as shape, gradation, surface texture and constituent material stiffness control the global mechanical behavior of granular soils [Santamarina, 2003]. Extending the understanding of the effects of individual particle properties on the behavior of soils could aid in the advancement of constitutive models as well as in the increased efficiency and robustness of geotechnical site characterization and design methodologies. A number of previous investigations have examined the effects of different particle properties on the engineering properties of coarse-grained soils, such as friction angle [e.g. Altuhafi et al., 2016, Kirkpatrick, 1965, Marschi et al., 1972, Vangla and Latha, 2015, Wang et al., 2013, Xiao et al., 2019 and shear wave velocity (V s ) and small-strain modulus (G max ) [e.g. Altuhafi et al., 2016, Bartake and Singh, 2007, Bui, 2009, Chang and Ko, 1982, Cho et al., 2006, Dutta et al., 2020, Hussien and Karray, 2015, Iwasaki and Tatsuoka, 1977, Liu and Yang, 2018, Menq, 2003, Patel et al., 2009, Payan et al., 2016a,b, Senetakis et al., 2012, Sharifipour et al., 2004, Wichtmann and Triantafyllidis, 2009, Yang and Gu, 2013. Regarding the small-strain behavior, some studies have reported increases in V s with decreases in the mean particle size (D 50 ) Singh, 2007, Patel et al., 2009]. Conversely, other studies such as Iwasaki and Tatsuoka [1977], Wichtmann and Triantafyllidis [2009], and Yang and Gu [2013] observed no significant effect of D 50 on V s , whereas Chang and Ko [1982], Menq [2003], Sharifipour et al. [2004], Bui [2009], and Hussien and Karray [2015] reported an increase in V s with increasing D 50 . Regarding the effects of particle shape, Cho et al. [2006], Bui [2009], Patel et al. [2009], and Lee et al. [2017] reported an increase in V s and G max with increasing particle roundness. However, Altuhafi et al. [2016], Liu and Yang [2018], and Liu et al. [2021] observed a decrease in small strain stiffness with increasing particle roundness.
These conflicting observations highlight the significant challenges in isolating the effects of individual particle properties and state on the response of natural soils. For instance, V s has been shown to depend on particle size, shape, surface roughness, mineralogy, and void ratio [e.g. Cho et al., 2006, Otsubo et al., 2015. Also, different parameters have been used to characterize a given particle property, and it is often unclear which one better captures the aspects of the behavior that govern the property of interest. For example, particle shape can be characterized in terms of roundness, sphericity, and regulatity, and soil state can be captured in terms of the void ratio, relative density, and state parameter, where the latter two are defined in terms of a reference (i.e. extreme void ratios and critical state line, respectively). Despite the aforementioned challenges, several studies have used synthetic soils or natural soil mixtures to isolate the effect of individual particle properties on soil response [e.g. Xiao et al., 2019].
3D printing technology has advanced rapidly in the last decade, offering a solution for the individual control of particle properties. The current technological capabilities can be used to generate artificial soil analogs with independent control over particle size, shape, and gradation [e.g. Adamidis et al., 2020, Hanaor et al., 2016. Recent studies used 3D printing technology to generate analog particles of different sizes and shapes, and showed that these analogs can successfully replicate the morphology of natural particles [e.g. Adamidis et al., 2020, Ahmed and Martinez, 2020, Athanassiadis et al., 2014, Hanaor et al., 2016, Miskin and Jaeger, 2013, Su et al., 2020. Results of triaxial tests on 3D printed particles show that they exhibit stress-dilatancy behavior similar to that typical of natural soils [e.g. Adamidis et al., 2020, Ahmed and Martinez, 2021, Hanaor et al., 2016, Matsumura et al., 2017, and the interpretation of their mechanical response can be captured with the critical state soil mechanics framework [Ahmed and Martinez, 2021]. However, the 3D printed sands have a greater compressibility owing to the smaller stiffness of their polymeric constituent material. The V s and G max of 3D printed particles obtained by bender element tests have also shown a dependency on mean effective stress similar to that of natural sands [Ahmed and Martinez, 2020]. Other applications of 3D printed analogs include investigation of the frictional behavior of faults [Braun et al., 2021], effect of particle shape on clogging and discharge [Hafez et al., 2021], calibration of DEM simulations [Kittu et al., 2019, Peerun et al., 2021, permeability of uniformly graded soil [Adamidis et al., 2020, Wei et al., 2021, and development of transparent soil surrogates [Li et al., 2021]. These results highlight the usefulness of 3D printed synthetic soils as soil analogs that provide a high control over their properties. Further, the conclusions of some of these studies show that despite of the smaller stiffness of the 3D printed soils, they exhibit many of the similar behaviors and functional trends as natural soils.
The goal of this investigation is to examine the isolated effects of various particle shape parameters on the V s and G max of coarse-grained soils. To do so, seven 3D printed sands with different particle shape parameters but similar particle size distributions are first generated. Then, measurements of V s and G max are obtained in specimens composed of the 3D printed particles subjected to different magnitudes of isotropic confining effective stress. The trends obtained in this investigation are then compared to published relationships to assess consistency in the results.

3D Printing Technology
Rapid advances in the 3D printing technology have developed different methods and materials in recent years. Modern 3D printers can be used to create complex objects using methods such as fused deposition modeling (FDM), stereolithography (SLA), digital light processing (DLP), and selective laser sintering (SLS) using polymeric, metallic, or ceramic materials, resulting in a wide range of precision and cost. While specialized 3D printers can mix materials on demand to achieve the desired mechanical properties and aesthetics to produce higly complex models [Jiménez Figure 1. (a) 3D mesh of synthetic particles generated, (b) X-ray CT scans of mixes 2 and 4, and (c) 3D printed particles used in this study (not to scale). Note: mix 1 is composed of spheres; mixes 3, 6, 7 and 8 were created using spherical harmonics [after Wei et al., 2018]; and mixes 2 and 4 were generated from X-ray CT scans of rounded and angular sands, respectively. et al., , Najmon et al., 2019, typical desktop 3D printers are contrained to printing polymeric materials. These more economic printers typically print layers with a thickness as low as 10 µm and have a lateral resolution in the order of 20 to 40 µm [Ngo et al., 2018]. This study uses the polyjet 3D printing technology that offers relatively economical and fast manufacturing of small parts with high accuracy [Adamidis et al., 2020, Ahmed and Martinez, 2020, Kittu et al., 2019, Wei et al., 2021. A polyjet printer has two print heads that deposit different liquid photopolymer resins. One resin generates the desired object while the other acts as the support structure, and both resins are hardened by ultraviolet laser. The layer thickness can be as low as 30 µm. Once the printing is completed, the support structure is removed by water jetting and chemical treatment using a 2% sodium hydroxide solution. Detailed description of the polyjet technology and its use to generate sand particles is provided in Martinez [2020, 2021].

3D Printed Sands
The shape of a particle can be described by roundness and sphericity parameters, which can be defined in different ways [Guida et al., 2020, Mitchell et al., 2005. This study considers the Wadell roundness (R) [Wadell, 1932], circle ratio sphericity (S C ), perimeter sphericity (S P ), widthto-length ratio sphericity (S WL ) and convexity (C ) [Altuhafi et al., 2016, Mitchell et al., 2005, which are defined as: where r i is the radius of curvature of the edges or corners of a soil particle; n is the number of edges; R ins is the radius of the maximum inscribed circle; D ins is the diameter of the largest inscribing circle; D cir is the diameter of the minimum circumscribing circle; P C is the perimeter of a circle having the same projected area as the particle; P S is the perimeter of the particle; d W and d L are width and length of a particle; and A 1 and A 1 + A 2 are the projected area and convex hull of a soil particle.
For this investigation, four materials were generated using the method proposed by Wei et al. [2018] that uses spherical harmonics to create random 3D shapes based on the desired shape features (mixes 3, 6, 7, and 8 in Figure 1a, Tables 1 and 2). These four materials were designed to have a D 50 of 2.5 mm and a C u of 1.26 (Table 2), R between 0.48 and 0.61, and S C between 0.53 and 0.84. To extend the range of particle shape parameters considered in this investigation, the three 3D printed mixes used by Martinez [2020, 2021] were also tested. Two of these mixes (mix 2 and 4) were generated from the X-ray CT scans of randomly selected rounded and angular natural sand particles, respectively, as described by Ahmed and Martinez [2020] ( Figure  1b, Tables 1 and 2). The D 50 and C u of these materials are 3.2 mm and 1.47, respectively. The last mix (mix 1) consisted of equal-sized spheres with a D 50 of 3.2 mm and a C u 1.0. These three mixes had greater R values (0.52 to 0.90) than the four mixes generated using spherical harmonics. Figure 2 shows the grain size distributions for all the soil mixes. Mixes 3, 6, 7 and 8 have identical grain size distributions with similar D 50 , mixes 2 and 4 have a slightly larger D 50 , and and the range of particle sizes in mix 1 is narrower. According to ASTM D2487 (Unified Soil Classification System) all the soil mixes can be considered sand since more than 50% passes no. 4 sieve. Although there are slight differences in both D 50 and C u of the soil mixes considered, no significant effect of those on the small strain behavior is expected. All the particles were generated using an Objet Eden 260V printer from Stratasys with VeroWhitePlus rigid acrylatebased polymer resin with a horizontal printing resolution of 30 µm. As previously shown by Martinez [2020, 2021], the polyjet printer is able to create 3D printed particles that succesfully reproduce the shape of natural sand particles, as evidenced by the negigible differences in the shape parameters reported in their study. The hardened polymer resin has Young's modulus of 2.4 GPa, Poisson's ratio of 0.3 and specific gravity of 1.18. The polyjet 3D printing process results in a large surface roughness, which is greater than that typical of natural soil particles, as shown in the comparison of X-ray CT scans presented in Figure 3 and further descibed in Martinez [2020, 2021]. Due to the smaller Young's modulus and the greater surface roughness, the stiffness of the interparticle contacts between 3D printed particles is smaller compared to that for natural sands. Figure 4a shows the results of interparticle uniaxial compression tests performed by Ahmed and Martinez [2020] on spherical polyjet particles with diameter of 3.175 mm. The results show that initial increases in force result in a soft contact response due to plastic yielding of the particles' microasperities. As the load is increased, the contact becomes stiffer and follows the Hertzian solution more closely. For comparison, Figure 4b presents similar results for a pair of glass spheres with diameter of 3.175 mm, showing the stiffer contact response that closely follows Hertz solution. Figure 4c shows the results of a single grain crushing test on Leighton Buzzard Sand (LBS) from Cavarretta et al. [2010]. The response of LBS also shows an initial softer response due to plastic yielding of microasperities, followed by a stiffer response that conforms well to Hertz theory.
Another important difference between the polyjet particles and natural sand particles is the magnitude and anisotropy of the friction coefficient of the former. Namely, the friction coefficient measured perpendicular to the printing direction (0.11 to 0.19) was considerably smaller than that measured along the printing direction (0.38 to 0.50) [Ahmed and Martinez, 2021]. The friction coefficients measured perpendicular to the printing direction are also considerably smaller than measurements on natural sand particles, which range between 0.17 and 0.36. The reader is referenced to Martinez [2020, 2021] for a detailed description of the compressive and frictional contact response of polyjet particles.
The roundness and sphericities of the printed particles were obtained from image analysis using the code by Zheng and Hryciw [2015], and the convexity was obtained using the solidity function in Matlab. The particle images were obtained using a white light scanner with a resolution of 0.1 µm (VR-3100, Keyence, Osaka, Japan).
The analysis of results also considered three combined shape parameters: regularity (R G ), overall regularity (OR) and shape-angularity group indicator (S AG I ). Regularity of a particle is the average of R and S C [Cho et al., 2006], overall regularity is the average of R, S P , S WL and C [Liu and Yang, 2018] and S AG I is defined by the relationship [Altuhafi et al., 2016]. The maximum and minimum void ratios of all the mixes were determined using the methods outlined in Carey et al. [2020], which have been shown to provide maximum and minimum void ratios similar to those provided by the ASTM D4254 and ASTM D4253 methods. The   Table 2. Figure 5 shows the variation of e max and e min with particle R, S P , and R G . These parameters were selected to allow for comparison with published relationships from Youd [1973] and Cho et al. [2006]. As shown, the measurements indicate a decrease in e max and e min as roundness, circle ratio sphericity and regularity are increased, which is consistent with the trends from literature. For the relationship with R, the e max and e min measurements fall in between the relationships from Youd [1973] and Cho et al. [2006] (Figure 5a); however, the trends reported by Cho et al. [2006] indicate a steeper decrease as S C and R G are increased (Figures 5b and 5c). While further research is required to explain these differences, it is possible that the smaller friction coefficient in relation to those reported for natural sands, as presented by Ahmed and Martinez [2021], may result in the smaller void ratio values reported.

Bender Element Test
Bender element (BE) tests were conducted on specimens with a diameter of 70 mm and a height between 65 and 76 mm contained in a latex membrane of 0.3 mm in thickness. The specimens were prepared inside split molds in five lifts,   (Cavarretta et al. [2010]). Note: diameter of the spheres is 3.175 mm, and the diameter of the LBS particle is 1.67 mm.
which were poured using a funnel. After pouring of each lift, the mold side was tapped with a rubber mallet to densify the specimen to the target void ratio. Specimens with initial void ratios (e 0 ) of 0.55 ± 0.02, 0.60 ± 0.02 and 0.65 ± 0.02 were prepared for the seven sand mixes, which corresponded to relative densities (D R ) between 30% and 80%. The BE tests were performed at isotropic confining pressures (p ) from 10 to 80 kPa applied by vacuum. However, it is noted that mixes 1, 2, and 4 were tested at p from 10 to 70 kPa due to a leak in the membrane that developed at greater p values. BEs attached to the specimen top and bottom caps were used to send and receive S-waves. V s values were calculated using the travel time of the S-waves and the distance between the BEs. The initial rise of the signal (i.e. the time when a signal first crosses the x-axis) was taken as the wave arrival time [Yamashita et al., 2009]. Ahmed and Martinez [2020] provide further information on the testing setup and results interpretation methodology. Figure 6 shows typical transmitter and receiver BE signals at different p .

Shear Wave Velocity and Small-Strain Modulus Measurements
The V s of the specimens composed of all seven materials increased as p was increased and e 0 was decreased, in agreement with trends for natural soils [e.g. Cho et al., 2006, Hardin andRichart Jr, 1963] as shown in Figures 7a, 7b and 7c for e 0 of 0.55 ± 0.02, 0.60 ± 0.02 and 0.65 ± 0.02, respectively. The V s for any given p and e 0 combination increased as the particle roundness and sphericity were increased. For example, the mix with the greatest roundness and sphericity (mix 1) has the greatest V s whereas the mix with the smallest roundness and sphericity (mix 7) has the smallest V s . The dependency of V s on particle shape is discussed in more detail in the following section.
The dependency of V s on p can be expressed by the following power-law equation [Lee and Stokoe, 1986]: where α is the shear wave velocity (m/s) at p of 1 kPa and β reflects the sensitivity to changes in p . Figure 7 includes power-law fits to the data using Equation 6. As shown, the quality of the fit is high, as evidenced by the high R 2 values. Values of α-coefficients and β-exponents for all the 3D printed sands were obtained from the fitted relationships ( Figure 8a). As shown, the α-coefficient generally increased and the β-exponent generally decreased as the e 0 was decreased, in agreement with trends reported by Cha et al. [2014]. The values of the α-coefficients ranged between 90 and 124 m/s whereas the values of the β-exponents ranged between 0.203 and 0.222. The α-coefficient and β-exponent obtained in this study are within the range of those for natural sands (Figure 8b) [Cha et al., 2014]. Analytical relationships for the β-exponents indicate values of 0.167 for a Hertzian contact and of 0.25 for particles experiencing contact yield [Cascante and Santamarina, 1996]. The β-exponents obtained in this investigation are within this range possibly due to local asperity yielding and particle rearrangement caused by the increases in p , as expected for natural sands [Cascante and Santamarina, 1996].
It is noted that the range of β-exponent values reported by Cho et al. [2006] and Cha et al. [2014]    is possible that that the greater compressibility of the polymeric material is responsible for the smaller range of β values, it is likely that the smaller range is due to the isolation of the particle shape effects from those of particle size and mineralogy. In fact, this is suggested by Ahmed and Martinez [2020], who show a similar range of β-exponents (i.e. 0.20 to 0.23) for rounded and angular natural sands with the same mineralogy and particle size and shape as the values (i.e. 0.20 to 0.25) for 3D printed sands obtained from X-ray CT scans of the natural sands.
Small-strain shear modulus values were calculated using the relation G max = ρV 2 s , where ρ is the specimen total density (Figures 7d, 7e and 7f ). It is noted that the G max values reported are smaller than those typical of natural sands due to the smaller specific gravity of the 3D printed sands (1.18 compared to 2.65 to 2.7 for silica sands) and the smaller stiffness of the polymeric constituent material (Young's modulus of 2.4 GPa compared to about 76 GPa for silica sands). The relationship between G max and p can be represented by the following power-law equation [Hardin and Richart Jr, 1963]: where A is a coefficient that depends on the particle arrangement and elastic properties of constituent material, F (e) is the function of e 0 as described by Hardin and Richart Jr [1963], and n describes the sensitivity to changes in p . Figure 8c shows the A and n values obtained by fitting power functions. As shown, the A-coefficient generally increased and the n-exponent generally decreased as e 0 was decreased. The n-exponents for all the specimens range between 0.424 and 0.463, which are greater than the value of 0.33 for the Hertz solution, in agreement with those reported in literature [e.g. Chung et al., 1984, Hardin andBlack, 1966].
The V s of the specimens composed of all seven mixes are also evaluated for specimens of similar relative density to account for differences in e max and e min and state, as shown in Figure 9. The results indicate the same trends as previously Figure 8. (a) Relationship between β-exponents and α-coefficients for all the specimens of 3D printed sands, (b) comparison of β-exponents and α-coefficients with database of natural sands from Cha et al. [2014] and (c) relationship between n-exponents and A-coefficients for all the specimens of 3D printed sands. described in Figures 7a, 7b, and 7c. Namely, V s increased as p and D R were increased. For any given combination of p and D R , V s increased as the particle roundness and sphericity were increased.

Effect of Particle Shape on Shear Wave Velocity and Small-Strain Modulus
The results presented in Figures 7 and 9 allow for examination of the effects of particle shape on V s for specific combinations of e 0 (or D R ) and p . Such results are presented in Figure 10 for a p of 50 kPa and e 0 of 0.55 and 0.65. It is noted that the trends reported here are true for all other combinations of p and e 0 , which are not shown here for brevity. The results show that V s increases as R (Figures 10a and 10e), S WL (Figures 10b and 10f), S C (Figures 10c and 10g) and S P (Figures 10d and 10h) are increased for any given e 0 . The correlation with V s is strongest with the R parameter (R 2 between 0.89 and 0.90), followed by that with S P and S C (R 2 between 0.75 and 0.84) and weakest with S WL (R 2 between 0.66 and 0.75). These differences imply that the R parameter is a stronger predictor of V s , likely because this parameter reflects the radii of the particle asperities which in turn determine the stiffness of the inter-particle contact. This contrasts with the S P , S C , and S WL parameters which capture the overall particle shape. Similar trends as reported for V s were observed for G max (shown in Figure A1 in Appendix), with an increase in G max with increasing shape parameters and a stronger correlation between G max and R. The shear wave velocity results can also be grouped according to their relative density, as shown in Figure A2 in Appendix. The results reveal similar trends, with V s increasing with increasing R, S WL , S C and S P and with increasing D R for all stress levels.
The effects of particle shape on V s are further examined using the combined shape parameters R G , OR and S AG I . The results shown in Figure 11 correspond to a p of 50 kPa. As shown, V s increases as R G and OR are increased, and decreases as S AG I is increased, in agreement with the trends in Figure 10. Stronger correlations are observed between V s and the combined shape parameters (R 2 between 0.90 and 0.98 for R G , OR and S AG I ) than between V s and the individual shape parameters (R 2 between 0.66 and 0.90 for R, S WL , and (e, f, g, h) e 0 = 0.65 ± 0.02. aspects of particle shape; the combined shape parameters seem to better capture this complexity, at least phenomenologically. The G max values show similar trends as described for V s (shown in Figure A3 in Appendix), consisting of an increase of G max with increases in R G and OR and decreases in S AG I . The trends between V s and the combined shape parameters are also true for specimens of similar D R , as shown in Fig. A4 in the Appendix, with R 2 values between 0.81 and 0.95.
The variation of α-coefficient and β-exponent values obtained from power-law fits (Equation 6) with several of the shape parameters are presented in Figures 12a to 12h. As shown, the α-coefficient increased as R, R G and OR were increased, and decreased as S AG I was increased for any given e 0 (Figures 12a to 12d). This indicates an increase in the contact stiffness at a p of 1 kPa as R, R G and OR increase and as S AG I decreases. Additionally, the α-coefficient increased as e 0 was decreased, as expected. Figures 12e, 12f, 12g and 12h indicate no clear trend between the β-exponent and the shape parameters. This suggests that for the 3D printed soils, the shape parameters are poor predictors for β and that β depends only on e 0 . Comparison of these trends with published relationships is discussed in the following section. A similar analysis for the A-coefficient and n-exponent for the power-law fits for G max (Equation 7) are presented in Figure A5 in the Appendix. These results indicate similar trends, with increases in A as R, R G and OR increase and as S AG I decreases, and no clear influence of the shape parameters on n. However, both parameters show dependence on e 0 . Several published studies show an increase in V s and α and a decrease in β as particle roundness, sphericity, and regularity are increased [Bui, 2009, Cho et al., 2006, Lee et al., 2017, Patel et al., 2009. However, other studies report a decrease in small-strain stiffness with increases in roundness, sphericity, and regularity [Altuhafi et al., 2016, Liu and Yang, 2018, Shin and Santamarina, 2013. A reason that may lead to this disagreement include differences in testing configurations and interpretation methods used to obtain shear wave velocity measurements. Another reason is the intertwined effects of different particle and soil properties and whether these were considered in the aforementioned studies. In particular, the latter may be an important consideration as D 50 , C u , e 0 , and mineralogy  have all shown to affect shear wave velocity magnitudes [e.g. Hussien and Karray, 2015, Menq, 2003, Sharifipour et al., 2004. For example, Cho et al. [2006] does not recognize the effect of e 0 in the V s , α, and β values. While Altuhafi et al. [2016] found an increase in G max with increasing S AG I , the authors also report a weak increase in G max with increasing particle surface roughness (S q ) which in turn is shown to increase with S AG I . This is however in contrast with results from analytical solutions and other experimental results, indicating a decrease in G max with increasing S q [e.g. Otsubo et al., 2015]. Indeed, Liu and Yang [2018] discuss aspects leading to uncertainty in establishing relationships between particle shape parameters and G max associated with the aforementioned interrelationships. These interdependencies among particle shape parameters, their influence on soil void ratio and density, and their aggregated effects on V s further highlight the usefulness of individually controlling particle properties, as enabled by the 3D printing technology.

Relationships Between Shear Wave Velocity and Shape Parameters
The results of bender element tests indicate a dependency of the shear wave velocity on the different particle shape parameters and the initial void ratio. To quantify these interrelationships and to compare to published equations developed based on natural sands, a linear regression analysis was performed to determine empirical relationship between V s and a given shape parameter and e 0 . Here, the R parameter is considered due to its widespread use in literature and the R G , OR and S AG I parameters are considered due to their stronger correlations with V s (i.e. Figure 11). It is noted that the regression analysis can also consider D R instead of e 0 ; the former would capture the effect of state in relation to the extreme void ratios of a given soil while the latter captures the effects of absolute density and coordination number. In the regression analysis, V s (m/s) was defined according to Equation 6; it is noted that these relationships differ from those presented in Figures 12a to 12h because they include e 0 . Based on the results presented in Figures 12a to 12h, α can be defined in terms of different particle shape parameters and e 0 , and β is a function of of e 0 only. α and β are expressed as: The equations reflect linear relationships between α and the different shape parameters and e 0 , and a linear relationship between β and e 0 . The following discussion is limited to the R and R G shape parameters due to their widespread use in the literature. Correlations between α and β with R and R G have been previously published by Cho et al. [2006]; a comparison of these relationships with those provided in Equations 8, 9 and 12 are shown in Figures 13a to 13f. While Cho et al. [2006] did not consider the effect of e 0 on α, nor provided e 0 values for their dataset, the predicted values are generally consistent with one another. Namely, the values predicted by the Cho et al. [2006] (Figures 13a and 13b). However, it is noted that this falls outside of the range of e 0 values considered in the experiments on 3D printed sands. Cha et al. [2014] indicated an increase in α with decreases in e 0 , which is in agreement with Equations 8 and 9 and the results shown in Figures 12a, 12b, 12c and 12d. While Cha et al. [2014] did not provide an equation between in α and e 0 , they suggest so by their relationship between α and the coefficient of compression (C c ), which is known to have a strong dependency on e 0 [Vesić and Clough, 1968]. Figure 13. Comparison of (a, b) α-coefficients and (c, d) β-exponents relationships with roundness and regularity, and (e, f) relationships between α-coefficients and β-exponents for different roundness and regularity.  Cho et al. [2006] also provided an equation between the β-exponent and R and R G without consideration of the effect of void ratio. However, Cha et al. [2014] report a decrease in β with decreases in void ratio, in agreement with Equation 12. Figures 13c and 13d shows that the values predicted by Equation 12 are within the range of the values predicted by the Cho et al. [2006] equation, with closer agreement at greater void ratios for small roundness values and at smaller void ratios with larger roundness. It is possible that the decrease in attainable e 0 values with increases in R and R G (i.e. as reported by Youd [1973] and Cho et al. [2006] and shown in Figure 5) causes a concomitant decrease in β-exponent. In fact, Patel et al. [2009] provided an equation for V s that explicitely considers e max and e min in addition to other particle shape and size parameters.
Equations 8, 9 and 12 can be manipulated to write the following : As shown, the equations are independent of e 0 but dependent on the particle shape parameters. Figures 13e and 13f shows inverse relationship between α and β. The figures also provide the relationships presented by Cha et al. [2014] and Lee et al. [2017]. The predictions based on Equations 8 and 9 are in general agreement with the previously-published relationships, with slightly greater predicted β values likely due to the greater compressibility of the contacts of the 3D printed particles.
To further assess the applicability of Equations 8, 9 and 12 to measurements on natural sands, values predicted using these equations are compared to those reported in the literature (Table 3) in Figure 14. As shown, there is a close agreement between the predicted values and the experimental values reported by the authors, with the bulk of the data falling within the ±15% lines. This agreement, along with the comparisons between Equations 8, 9, 12, 13, and 14 provided in Figure 13, suggests that relationships developed based on tests on 3D printed soils can capture the effects of particle shape and void ratio on the shear wave velocity of natural sands for the range of mean effective stresses considered in this investigation (i.e. 10 to 80 kPa). This agreement may be unexpected considering the differences in constituent materials between the 3D printed and natural sands. One possible reason for this agreement is that the dependence of V s on particle shape, void ratio, and effective stress is governed by the particulate nature of soils, which is properly replicated by the 3D printed soils. Another possibility is that while certain parameters or behaviors may have different effects on V s , these have an aggregated effect that is similar between the 3D printed and natural sands. For example, the effect of the smaller stiffness of the polymeric material could be offset by the effect of the larger interparticle contacts owing to its greater compressibility. Indeed, further research is required to further understand the mechanisms leading to similarities and differences in the behavior of 3D printed and natural soils.

Considerations on the Modeling of Sand
Behavior with 3D Printed Particle Analogs A significant advantage in using 3D printed soils is the ability to control particle shape while the remaining properties (i.e. particle size, constituent material, surface roughness) are maintained constant, which expand the experimental capabilities available to researchers. A similar procedure can be used to isolate the effects of particle size, as shown by Adamidis et al. [2020]. Despite this benefit, it is important to consider the possible effects of the differences between the 3D printed and natural sands resulting from their different constituent material properties and genesis. Namely, the polymeric material has a smaller stiffness and specific gravity than natural minerals such as quartz. The smaller stiffness of the polymer results in softer interparticle contacts which leads to a greater bulk compressibility, while the smaller density can influence behaviors in which dynamic and inertial effects are important such as tamping and pluviation used for specimen preparation. The layer deposition printing process inherently results in an anisotropic configuration. This has been shown by Ahmed and Martinez [2021] to lead to anisotropy in the interparticle friction coefficient. However, the results presented by Ahmed and Martinez [2020] suggest that the contact normal force-deformation response does not exhibit anisotropy due to the layer deposition orientation. Finally, the printing process can produce a large surface roughness which also leads to softer interparticle contacts. Because different 3D printing technologies (e.g. stereolithography, selective laser sintering, fused deposition moldeling) use different manufacturing processes and are capable of printing different materials, the possible effects of each technology on the response of soil particles should be evaluated and understood. However, it is envisioned that such differences in properties will be addressed as the additive manufacturing technology enables generating objects with a broader suite of materials and processes. Ultimately, comparisons of the measurements on 3D printed analogs with experimental data on natural soils and established relationships can help validate the conclusions drawn from such studies.

Conclusion
An investigation on the effects of particle shape on the small-strain behavior of sands using seven 3D printed soils is presented. The synthetic particles were designed based on spherical harmonics and on X-ray CT images from natural sands, allowing for careful control of the materials' particle shape parameters. The same polymeric material and additive manufacturing process was used to generate all the 3D printed materials, thus minimizing the differences in constituent material properties and particle surface roughness. Measurements based on bender element tests indicate an increase in V s and G max with increasing mean effective stress and decreasing void ratio, in agreement with trends reported for natural sands. The results also show an increase in V s and G max with increases in particle roundness, sphericity, and regularity for a given e 0 , D R , and p .
Fitting the relationship between V s and p with a power law (i.e. Eq. 6) shows an increase in the α-coefficient with increases in the particle shape partameters R, S P , S C , S WL , R G , and OR and decreases in S AG I , as well as an increase in α with decreases in e 0 . In contrast, the β-exponent was found to depend on e 0 but to be independent of the particle shape parameters. The correlations between V s and the combined shape parameters R G , OR, and S AG I was shown to be stronger than with the individual shape parameters R, S P , S C and S WL , likely because the former parameters better capture the effects of different aspects of particles (i.e. the shape of the corners as well as the overall particle shape). The results of this study are used to develop equations for the α-coefficient and β-exponent that consider their dependency on particle shape and e 0 . These equations are shown to agree with published relationships and to predict the V s values of natural sands with an error ±15%. This close agreement suggests that 3D printed soils can be reliably used to model the small-strain behavior of natural sands. However, it is important to consider differences in the inherent behaviors of 3D printed and natural sands, such as the smaller contact stiffness of the 3D printed soils owing to the smaller Young's modulus of the polymer material.

Conflicts of Interest
The authors declare that there is no conflict of interest. The complete review history is available online. Figure A2. Variation of shear wave velocities with roundness, width-to-length ratio, circle ratio sphericity and perimeter sphericity, respectively, for all the specimens at p = 50 kPa with (a, b, c, d) D R = 68-80% and (e, f, g, h) D R = 51-61%.  Figure A5. Variation of (a, b, c) A-coefficients and (d, e, f ) n-exponents with regularity, overall regularity and SAGI, respectively, for all the specimens.