ectc section 5 cover - uab

UAB School of Engineering - Mechanical Engineering - Journal of the ECTC, Volume 18 Page 96

SECTION 5

Materials


Journal of UAB ECTC Volume 18, 2019

Department of Mechanical Engineering The University of Alabama, Birmingham

Birmingham, Alabama USA

FLEXURAL PERFORMANCE OF MICROCRYSTALLINE CELLULOSE BASED WOOD COMPOSITE UNDER FOUR- POINT BENDING

Maharshi Dave and Tejas Pandya University of Mississippi Oxford, Mississippi USA

Jason Street Mississippi State University

Mississippi State, Mississippi USA

ABSTRACT The objective of this work was to study the mechanical

behavior of newly developed wood-based composites using the four-point-bending method. Different types of wood based composites made from Methylene diphenyl diisocyanate (MDI) and Microcrystalline Cellulose (MCC) were manufactured using a Dieffenbacher hot press. The wood composite, 1%MCC4%MDI had the highest maximum flexural stress (σf) max, and 2%MCC4%MDI had the highest maximum flexural strain. MCC has significantly affected the elastic behavior of wood based composites.

KEYWORDS: Wood-composites, maximum flexural stress (σf) max, Flexural load, strain.

INTRODUCTION Composite materials (composites) are made when two or

more materials with different properties are combined to produce a new material. The mechanical and chemical properties of each of the constituent materials remain distinct in the new material. These constituent materials work synergistically to produce a composite material that has improved properties when compared with the individual constituent materials.

Wood based composites have been frequently used for automobiles (Koronis et al. 2013 [1]), vibration damping and noise reduction applications (Mohanty et al. 2015 [2]), and the packaging of nuclear waste (Bragov et al. 1997 [3]). Wood based composites have a wide range of applications due to their renewable, cheap, recyclable, and biodegradable, high strength-to-weight ratio, and customizable static and dynamic mechanical properties. Hence, wood based composites have great capability to replace man-made petroleum-based composites. This research was carried out by the Mechanical Engineering department at the University of Mississippi and the Department of Sustainable

Bioproducts at Mississippi State University to enhance the use of green materials by studying four newly developed wood-based composites.

When wood based composites are used for industrial applications, they are subjected to flexural loading, so before selecting a sustainable bio-composite for a particular application, it is necessary to investigate the flexural behavior of the material. The four-point bending test has been widely used for investigating the flexural response of wood based composites (Edgars et al. 2017[4, 5]; Gherardi et al. 2018[6]; Labsans et al. 2017[8]; Yoshihara et al. 2001[7]).

Shah et al. (2016)[9] carried out trials on wood-based, bio-composites using a Split Hopkinson Pressure Bar (SHPB). Their research showed that cornstarch improved the energy absorption capacity of the material under high strain rate loading, and the yield strength of wood-based bio-composites increased with increasing strain rate. The present work is carried out to study the flexural behavior of the Microcrystalline cellulose (MCC) based wood composite under bending load for the development of sustainable and environmentally friendly material.

MATERIAL Southern yellow pine was used to create the wood composite

panels. The amount of mass used to create the wood based composite panels was approximately 2.95 kg. The particles ranged in size from 2-3 mm, and the temperature used to form the panels was approximately 185oC. A Dieffenbacher 915 x 915 mm hot press system located at the Sustainable Bioproducts Laboratory at Mississippi State University was used to create the wood based composite panels used in this study. This hot press with steam injection capability was coupled with the Alberta Research Council's Pressman operation and monitoring software. The Dieffenbacher hot press formed the panels based on the desired thickness, so each composite material had differing pressures, which were required to produce a wood


composite panel with a thickness of 6.35 mm. The varying pressures required to form the panel to the appropriate thickness were based on the ability of the composite material in the mat to be compressed to the appropriate thickness. Wood composite samples created for analysis with the four point bending test were made with different mass fractions of microcrystalline cellulose (MCC), as listed in Table 1. A polymeric methylene diphenyl diisocyanate (MDI) resin (Rubinate 1840, Hunstman Corp., The Woodlands, TX, USA) with the release agent McLube 841 (McGee Industries Inc., Aston, PA, USA) was used to create the composites. MDI is an aromatic diisocyanate and is an efficient binder that has been used in the production of composite wood products for over 30 years. The sampling size for each material was five (5).

Table 1: Types of wood composites

EXPERIMENTAL PROCEDURE AND SETUP Wood-based composite specimens of size 151 mm in length

x 25 mm in width x 7.5 mm in height were placed in a sliding roller four-point bending fixture as shown in Figure 1, with an inner (Si) and outer span (S0) of 278.62 mm and 326.76 mm respectively. The samples were tested to failure on an EnduraTEC servo mechanical Torsion system under displacement control. Other test settings were displacement rate of 0.017 mm/s and the sampling rate of 50 Hz using WIN Test data acquisition software installed by Bose Corporation. Figure 1 shows the experimental setup located at the Structure and Dynamics Laboratory, University of Mississippi. ASTM standard D7264 was used to study the flexural properties of the Wood-based Composites.

THEORY OF THE CALCULATION OF STRESS AND STRAIN USING A FOUR-POINT BENDING METHOD

In this method, there are four important points (two end supports and two loading points) along the span of the beam. Hence, this method is called the four-point bending test. Figure 2 shows a schematic of the shear force diagram (SFD) and the bending moment diagram (BMD) for four point bending loading regimes. Figure 3 shows the normal stress and shear force variation through the thickness of the specimen.

Figure 1. Experimental set-up at The University of

Mississippi

From the shear force and bending moment diagrams, it is

clear that there is a stress concentration at the point of loading. However, for four-point bending there is a uniform bending moment (M) and both the shear force (Fs) and the interlaminar shear stress are zero between the loading points. Thus, this methodology leads to a pure bending loading condition. This state of stress is desirable in testing. Flexural stress (σf) is defined as the stress on the surface of the specimen at the failure. It is evident from Figure 3 that normal stress varies linearly across the thickness. The maximum stress in compression (σC) is on one side and an equal maximum stress in tension (σT) is on the other side of the thickness and passes through the zero at mid plane. The maximum flexural (σf) max. stress and maximum strain (ε) according to ASTM D7264 is given by equations 1 and 2, respectively [8].

σ .

(1)

ε . (2)

Material Approx. pressure (MPa)

Curing Time

(Seconds)

Density kg.m-3

4%MDI 8.9 140 758

1%MCC4%MDI 8.7 140 662

2%MCC4%MDI 8.4 140 684


Figure 2. Shear force and bending moment diagrams for

four pint bending test

Figure 3. Bending and Shear stress in the thickness direction

where, (σf) max and ε are the maximum stress and strain at the outer surface in the load span region, P is the applied force, L is the support span, δ is the mid-span deflection, b and h are the width and thickness of the specimen.

RESULTS AND DISCUSSION Table 2 summarizes the average values of experimental

results of three different types of wood-based composites tested using the four-point bending method. Figure 4 represents the flexural stress-strain behavior under four-point bending load.

From the analysis of Figure 4 and Table 2, it was found that by increasing the constituent of MCC, a change occurred in the average maximum flexural stress (σf)max.. The wood-based composite material that was made with 1%MCC4%MDI had an average maximum flexural stress (σf) max. of 11.47 MPa, and by increasing the mass fraction of the MCC by 2% (solids basis), the observed average maximum flexural stress (σf) max.was 10.22 MPa for 2%MCC4%MDI. This decrement in maximum flexural

stress (σf) max might be due to material properties of MCC and MDI. Furthermore, results show that increasing the volume fraction of MCC with MDI had improved the affected the maximum strain of the wood-based composite. This may be due to the enhanced volume fraction of MCC in the wood based composite.

Table 2: Flexural properties of bio-composites

Figure 4. Flexural stress-strain behavior

In addition, the single most striking observation to emerge from the data comparison was that wood-based composites containing MCC have consistently improved the maximum flexural strain among all wood based composites tested.

CONCLUSIONS An experimental study was conducted to study the flexural

behavior of wood-based composites under four-point bending. The wood based composite 1%MCC4%MDI had the highest maximum flexural stress (σf) max, and 2%MCC4%MDI had the strain, which reveals that volume fraction of MCC plays an important role to govern the flexural properties of wood-based composites. Finally, Microcrystalline cellulose (MCC) enhanced the maximum flexural strain, when the wood-based composites were otherwise made under similar manufacturing conditions.

ACKNOWLEDGEMENTS We would like to express our gratitude to Matthew Lowe for

the machining of the wood-based composite samples. The authors wish to acknowledge the support of the U.S. Department of Agriculture (USDA), Research, Education, and Economics (REE), Agriculture Research Service (ARS), Administrative and

0

2

4

6

8

10

12

0 0.001 0.002 0.003 0.004 0.005 0.006

Stress (MPa)

Strain

4%MDI

1%MCC4%MDI

2%MCC4%MDI

P

h Si S0

L

Fs = P/2

Fs = P/2

𝑀 𝑃 𝑆 𝑆

4

b

h

σC

σT τ


Financial Management (AFM), Financial Management and Accounting Division (FMAD) and Grants and Agreements Management Branch (GAMB), under Agreement No. 5B-0202-4-001. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the view of the U.S. Department of Agriculture. This material is supported by the USDA National Institute of Food and Agriculture and McIntire Stennis under grant no 1008126.

REFERENCES [1] Koronis, G., Silva, A., and Fontul. M., 2013, “Green Composites: A Review of Adequate Materials for Automotive Applications”, Composites Part B: Engineering 44 (1), pp. 120-127. [2] Mohanty, A., and Fatima, S., 2015, “Noise Control Using Green Materials,” Sound and Vibration Magazine, 49(2), pp. 13-15. [3] Bragov, A., and Lomunov, A., 1997, “Dynamic Properties of Some Wood Species,” J Phys IV Colloq, 07(C3), pp. 487–492. [4] Edgarsa, L., Zudrags Kaspars, Z., and Kasparsa, K., 2017, “Structural Performance of Wood Based Sandwich Panels in Four Point Bending,” Procedia Engineering, 172, pp. 628-663. [5] Edgars, L., Kaspars, K., and Chiara, B., 2017, “Flexural Behavior of Sandwich Panels with Cellular Wood, Plywood stiffener/foam and Thermoplastic Composite Core,” Journal of Sandwich Structures and Materials, 0(00), pp. 1-22. [6] Gherardi, PR., Hein1, and Brancheriau, L., 2018, “Comparison Between Three-Point and Four-Point Flexural Tests to Determine Wood Strength of Eucalyptus Specimens”, Maderas. Ciencia y tecnología 20(3), pp. 333-342. [7] Hiroshi, Y., Oka. S., 2001, “Measurement of Bending Properties of Wood by Compression Bending Tests,” J Wood Sci., 47, pp. 262-268. [8] ASTM International, 2015, Standard Test Methods for Flexural Properties of Polymer Matrix Composite Materials,” Annual Book of ASTM Standards, 15.03, pp. 587-597. [9] Shah, A., Street, J., Mitchell, B., Pandya, T. 2016, “Study of Variation in Fundamental Frequencies of Bio-Composites Due to Structural Damage,” Journal of UAB ECTC 16, pp. 115-121.





MACROSCOPIC EVOLUTION OF CORROSION IN AN ALUMINUM ALLOY

Tete Tevi University of South Florida

Tampa, Florida USA

Ramana M. Pidaparti University of Georgia Athens, Georgia USA

ABSTRACT Aluminum is a very common metal used across many

industries (aerospace, automotive, and consumer) worldwide for its competitive properties compared to other metals. Although aluminum corrosion does not occur very easily in nature, it could be a problem under specific environmental conditions. Consequently, the corrosion of aluminum has been the subject of many studies for decades.

In this paper, accelerated corrosion laboratory tests on an aluminum sample are presented. Corrosion experiments were conducted on aluminum alloy metal samples in a 3 molar alkaline sodium chloride solution. A potentiostatic corrosion method was used with a potential of -0.63 V versus Ag/AgCl. Macroscopic images of the corrosion evolution with time were obtained from experiments. The macroscopic images obtained depicting the corrosion growth, the current density of the corrosion tests, along with the pH of the bulk sodium chloride solution are presented and briefly discussed to illustrate the corrosion evolution.

KEY WORDS: macroscopic corrosion, aluminum, potentiostatic, image, defects

INTRODUCTION Corrosion is a deterioration mechanism of a metal due to

chemical or electrochemical processes[1],[2]. Corrosion in many sectors of industry cost the US economy more than US $276 billion and at least 3% of many nations’ around the world gross domestic product (GDP)[3] [4]. Despite the fact that aluminum and its alloys are widely used in many sectors of industry due to their relative low cost, light weight, good thermal and mechanical properties[5] [6] [7], [8], aluminum remains very prone to corrosion under specific environmental conditions. The corrosion of aluminum and its alloys has been extensively studied in literature[9], [10], [11]–[14]. There are many techniques reported to study the corrosion evolution in the metal with different levels of complexity. Here, a simple macroscopic evaluation of an accelerated corrosion in aluminum alloy is presented to demonstrate the need to quantify the macroscopic evolution of corrosion.

EXPERIMENTS A 3 molar solution of Chloride Sodium solvated in water

[15] was used in all the electrochemical experiments in order to study the macroscopic evolution of accelerated corrosion of an aluminum alloy. The corrosion was performed in a three-probe cell configuration with Ag/AgCl as reference electrode. A graphite rod was used as the counter electrode. The working electrode was a 2024 Aluminum alloy sample. The alloy composition includes 0.23% iron, 0.63% manganese, 1.45% magnesium and 0.16% Silicon [15]. An apparent surface area of 2cm X 2cm was exposed to the sodium chloride solution. The other surfaces of the Aluminum sample were coated with paint to avoid any direct contact with the sodium chloride solution (Figure 1). A direct current (DC) potentiostatic technique was applied to run the corrosion. The working electrode potential was set at -0.63 V versus Ag/AgCl [16] . A Gamry Potentiostat/Galvanostat/ZRA Interface 1010E instrument was used to conduct all the electrochemical tests. The sample to be studied was connected to the working electrode.

The tests were conducted at room temperature. The bulk pH of the chloride solution was also recorded. At intervals of 5 minutes, a macroscopic image of the sample was taken to visually monitor the evolution of the corrosion. Before taking the images, the sample was rinsed with distilled water and dried. The images were taken using Matlab image acquisition toolbox with a generic webcam camera connected. The corrosion lasted for 90 minutes in total.

Figure 1. Aluminum Sample


RESULTS AND DISCUSSION Figure 2 shows the images of the evolution of the corrosion

of the sample taken at intervals of 15 minutes, from the uncorroded original sample to the corroded sample 1h 15 minutes later. For simplicity, only the data recorded at 15 minutes of intervals up to 75 minutes are presented and discussed.

Figure 2. Corrosion Evolution at Different Time Intervals

The initial macroscopic state of the sample is presented in Figure 2 at t= 0 min. The aluminum sample was not subjected to the corrosion at that time. After three consecutive intervals (15 minutes) the metal showed signs of corrosion (dark spots). As the tests continue, more of the sample was covered with the dark spots. To confirm that the dark spots were corrosion defects, a localized image of the sample was taken with an optical microscope. The images confirmed that the dark spots were localized corrosion pits in the metal (Figure 3).

Figure 3. Localized Microscopic Image

Figures 4, 5 and 6 depict the growth of two random

corrosion defects (points A and B) in the metal. The growths images were recorded at times t=30, 45, and 60 minutes respectively. Point A defect has already started before t=30 minutes and point B defect appeared after that time. However, point B defect grew faster between 30 and 45 minutes. This could be explained with the fact that since point A was already corroded, it formed a thin film of aluminum oxide that prevented faster electron transfer, thus a faster corrosion growth at the same

spot [17] [18]. The same explanation is also valid for spot B between 45 and 60 minutes.

Figure 4. Corrosion at t=30 min



The sample was corroded with a constant potential of -0.63

V versus Ag/AgCl. The resulting potentiostatic current density I versus time that was induced was recorded for each test. Figure 7 presents the current evolution in time at 15, 30, 45, 60, and 75 minutes intervals. The minimum steady state current was recorded at 15 minutes, and the maximum at 30 minutes (Table 1).


Table 1. Steady State Current at Different Time Intervals

Figure 7. Current Density vs Time

The bulk pH values recorded for the sample ranged between

13.55 and 13.66 with an average value of 13.6 (Figure 8). The pH values recorded indicate an alkaline bulk solution of highly concentrated sodium chloride in water. Similar conditions of aluminum corrosion have been reported in literature [19] and the results obtained qualitatively compare to those from the literature.

Figure 8. Bulk pH Variation of the NaCl Solution

CONCLUSION Aluminum metal samples were corroded electrochemically

in a highly concentrated sodium chloride solution in order to estimate the macroscopic evolution of corrosion. Although the sample was progressively corroded, visual evaluations of the samples showed that the corrosion speed tends to slow down in spots already corroded due to the formation of oxide films that

limit electrons transfer. The bulk pH of the sodium chloride solution from the beginning to the end of the corrosion tests indicated that the alkalinity of the solution did not change. More experiments with varying experimental conditions including different concentrations of sodium chloride, are being carried out to further investigate the effect of the salt concentration on the macroscopic evolution of the corrosion.

REFERENCES [1] S. K. İpek, A. Ulus, H. Ekici, G. H. Ağaoğlu, and G. Orhan, “Continuous Casted Aluminum Flat Products Corrosion Characteristic According to Downstream Process,” in TMS Annual Meeting & Exhibition, 2018, pp. 943–952. [2] E. D. During, Corrosion atlas: a collection of illustrated case histories. Elsevier, 2018. [3] G. A. Jacobson, “NACE International’s IMPACT Study Breaks New Ground in Corrosion Management Research and Practice,” The Bridge, vol. 46, no. 2, 2016. [4] G. Koch, “1 - Cost of corrosion,” in Trends in Oil and Gas Corrosion Research and Technologies, A. M. El-Sherik, Ed. Boston: Woodhead Publishing, 2017, pp. 3–30. [5] P. Rodič, I. Milošev, M. Lekka, F. Andreatta, and L. Fedrizzi, “Corrosion behaviour and chemical stability of transparent hybrid sol-gel coatings deposited on aluminium in acidic and alkaline solutions,” Prog. Org. Coat., vol. 124, pp. 286–295, 2018. [6] G. Çam and G. \.Ipekoğlu, “Recent developments in joining of aluminum alloys,” Int. J. Adv. Manuf. Technol., vol. 91, no. 5, pp. 1851–1866, Jul. 2017. [7] S. T. Mavhungu, E. T. Akinlabi, M. A. Onitiri, and F. M. Varachia, “Aluminum Matrix Composites for Industrial Use: Advances and Trends,” Procedia Manuf., vol. 7, pp. 178–182, 2017. [8] J. Singh and A. Chauhan, “Characterization of hybrid aluminum matrix composites for advanced applications – A review,” J. Mater. Res. Technol., vol. 5, no. 2, pp. 159–169, 2016. [9] R. T. Foley and T. H. Nguyen, “The chemical nature of aluminum corrosion v. energy transfer in aluminum dissolution,” J. Electrochem. Soc., vol. 129, no. 3, pp. 464–467, 1982. [10] K. F. Khaled and M. M. Al-Qahtani, “The inhibitive effect of some tetrazole derivatives towards Al corrosion in acid solution: Chemical, electrochemical and theoretical studies,” Mater. Chem. Phys., vol. 113, no. 1, pp. 150–158, 2009. [11] Y. Xie, A. Leong, J. Zhang, and J. J. Leavitt, “Aluminum alloy corrosion in boron-containing alkaline solutions,” Mater. Corros., vol. 70, no. 5, pp. 810–819, 2019. [12] X. Li, S. Deng, T. Lin, X. Xie, and G. Du, “Cassava starch-sodium allylsulfonate-acryl amide graft copolymer as an effective inhibitor of aluminum corrosion in HCl solution,” J. Taiwan Inst. Chem. Eng., vol. 86, pp. 252–269, 2018. [13] A. Y. El-Etre, M. Shahera, M. A. Shohayeb, S. Elkomy, and S. Abdelhamed, “Inhibition of Acid Aluminum Corrosion in Presence of Aqueous Extract of Domiana,” J. Basic Environ. Sci., vol. 3, pp. 25–36, 2016. [14] S. Woerner, T. Kaiser, L. Beck, T. Grosskopf, and C. Kuenzel, “Corrosion resistance of brazed aluminum in

0.0E+00

1.0E-02

2.0E-02

3.0E-02

4.0E-02

0 100 200 300 400

I (A

/cm

2)

t (seconds)

75 min60 min45 min30 min

13.5613.5713.5813.5913.6

13.6113.6213.6313.6413.6513.66

0 15 30 45 60 75 90

pH

Time (min)

UAB School of Engineering - Mechanical Engineering - Journal of the ECTC, Volume 18 5

ethanol/water mixtures at high temperatures,” Automot. Engine Technol., vol. 3, no. 3–4, pp. 169–177, 2018. [15] S. Kumar, A. Kumar, and C. Vanitha, “Corrosion behaviour of Al 7075 /TiC composites processed through friction stir processing,” Front. Corros. Control Mater. FCCM-2018 28–29 June 2018 NIT Warangal Warangal Telangana State India, vol. 15, pp. 21–29, Jan. 2019. [16] R. M. Pidaparti and R. K. Patel, “Investigation of a single pit/defect evolution during the corrosion process,” Corros. Sci., vol. 52, no. 9, pp. 3150–3153, 2010. [17] B. G. Prakashaiah, D. V. Kumara, A. A. Pandith, A. N. Shetty, and B. E. A. Rani, “Corrosion inhibition of 2024-T3 aluminum alloy in 3.5% NaCl by thiosemicarbazone derivatives,” Corros. Sci., vol. 136, pp. 326–338, 2018. [18] F. Alvi, N. Aslam, and S. F. Shaukat, “Corrosion inhibition study of zinc oxide-polyaniline nanocomposite for aluminum and steel,” Am J Appl Chem, vol. 3, no. 2, pp. 57–64, 2015. [19] M. Verhoff and R. Alkire, “Experimental and modeling studies of single pits on pure aluminum in ph 11 nacl solutions i. Laser initiated single pits,” J. Electrochem. Soc., vol. 147, no. 4, pp. 1349–1358, 2000.





MESOSCALE MODELING OF GRAIN CRUSHUP IN EGLIN SAND THROUGH DISCRETE ELEMENT FRACTAL FRAGMENTATION

Gerald Pekmezi, David Littlefield The University of Alabama, Birmingham


ABSTRACT Particle methods, such as the Discrete Element Method,

have been used extensively to model particulate media in recent years. These methods are especially useful in modeling the mesoscale behavior of particulate media; however, their use in this context has thus far been limited. This work is part of a larger framework that aims to greatly expand the role of DEM in mesoscale modeling of particle aggregates.

Described in this paper, is a method for computational modeling of an important part of particulate aggregate constitutive behavior: Particle Size Distribution (PSD) comminution or “crushup”. This is achieved by replacing particles that meet the so-called “Brazilian” fracture criterion with an Apollonian fractal spherical pack. The ensemble modeling results of this method show good agreement with experimental data for the final PSD as well as volumetric stress-strain evolution of Eglin sand.

KEY WORDS: DEM,mesoscale,grain fracture, grain crushing,fractal packing

INTRODUCTION With the significant increase in available computational

power and resources over the last decade, geomaterial modeling has seen both evolutionary and revolutionary changes. One such “revolution” has been the dramatic rise in popularity of particle methods such as the Discrete Element Method (DEM). DEM is very closely related to molecular dynamics (MD). The inclusions of rotational degrees of freedom and nonlinear contact with friction are the main attributes that separate DEM from MD. DEM can be useful in modeling soils in dynamic applications considering the large deformations involved.

The current work is part of an ongoing effort to couple DEM, used as a mesoscale method, with Statistical Mechanics in order to construct a framework for the modeling, homogenization, and uncertainty quantification of particulate geomaterials. While inter-grain contact and frictional sliding in particulate media have been studied extensively, there has been less work carried out to study crushup at the mesoscale. Most of the work carried out in this regard has been toward a

phenomenological qualification of particle breakage. “Breakage factors” such as Marsal’s breakage factor [1], Lee and Farhoomand’s breakage factor [2], and Hardin’s breakage potential [3] have been used to quantify the amount of breakage a particulate medium undergoes, but with little consideration for the evolution of the mechanics or statistics at the grain scale. Work carried out in this area in recent years has only solidified the position of Marsal’s breakage factor and Hardin’s breakage potential as the standard-bearers for macroscale quantification of particle comminution [4-7].

In this study, a computational model for grain crushing is described. This model uses the fractal Apollonian sphere packing [8, 9] to replace a grain that meets a stress-based crushing criterion, called the “Brazilian” criterion [10]. It is thus named after the eponymous test carried out to determine the fracture strength of rocks. Upon meeting the criterion, the spherical particle representing the macrograin is replaced by the Apollonian packing, which results in a “blowout” fracture of the macrograin into the smaller grains.

The Discrete Element Method Discrete Element Method (DEM) is the name given to a

collection of numerical methods used to model the motion, forces, and interactions of particles. The method was originally developed by Cundall and Strack [11]. DEM employs an explicit numerical scheme to integrate the motion of spherical particles and contact detection to model the interactions between the particles. All DEM simulations performed in the current work are carried out using a version of the open-source YADE Discrete Element Modeling code [12] adapted for the mesoscale framework that the current work is a part of. Figure 1 shows a rendering of a sample DE model generated for this work and used to represent Eglin sand at a realization size of ten thousand particles.


Figure 1: A DEM Representation of Eglin sand

The contact force used in the discrete element particle

model may follow one of two formulations: linear contact or Hertzian contact. The Hertzian contact formulation is used herein. This formulation resembles the Hertz-Mindlin solution [13], which is valid for elastic bodies in contact. The elastic normal force in the Hertzian solution is expressed as

* * 3 2,

4

3n el nF E R u (1)

where ,n elF is the elastic normal force, nu is the interpenetration

distance, *E and

*R represent the equivalent effective Young’s modulus and particle radius, respectively.

The shear force is linear with respect to the relative sliding displacement at the region of contact, assuming the no micro-slip solution of Hertzian contact.

* *8s n sF G R u u (2)

where SF is the shearing force, su represents the sliding

distance between particles, and *G represents the equivalent shear modulus. Finally, the normal and tangential contact stiffnesses can be written as

* *2n nk E R u (3)

* *8s nk G R u (4)

Sliding between particles is initiated once the shear force exceeds frictional resistance. This is expressed by a frictional Mohr-Coulomb equation

tan 0s nF F (5)

where SF is the shearing force, nF represents the normal force,

and represents the inter-particle friction angle.

Statistical Mechanics Statistical Mechanics is that branch of theoretical physics

that uses Probability Theory to quantify the uncertainties inherent to non-deterministic systems within the realm of Classical Mechanics. This outlook is usually associated with the other major realm of Mechanics, the Quantum realm; however, uncertainty is present and quite important in many classical applications as well. Indeed, the term “Statistical Mechanics” is often used to refer specifically to Statistical Thermodynamics, which aims to derive classical thermodynamics from the constituent fundamental particles, i.e. atoms and molecules. This approach is easily extended to other systems composed of particles, and the Discrete Element Method is a tool that can do for geomechanics what Molecular Dynamics methods such as Dissipative Particle Dynamics (DMD) do for thermodynamics.

There is room for the DEM research community to integrate Discrete Elements within Statistical Mechanics. Studies carried out using the Discrete Element Method have heretofore, to the best of the author’s knowledge and research, failed to properly consider the uncertainties associated with a system composed of a finite number of particles. These uncertainties, rather than being dismissed or ignored, can instead be researched and quantified using Statistical Ensembles, turning DEM into a powerful tool for Uncertainty Quantification (UQ).

Ensemble Averaging A Statistical Ensemble is a model composed of many copies

of a system in its different possible states, considered simultaneously. Another way to state this is that the Statistical Ensemble is a probability distribution for the state of the system [14]. For the concrete case of a particle model of a geomaterial such as the sand considered herein, the ensemble may be defined as a collection of particle packings of a finite size, wherein the cumulative distribution is like that of a single packing of a near-infinite number of particles.

The idea here may be distilled down to the simple fact that a Discrete Element Model containing a near-infinite number of particles is both impractical and computationally prohibitive. However, a sufficiently large number of concurrent models of finite sizes, is quite feasible, especially so with access to High Performance Computing (HPC) resources. These models of finite size are henceforth called “realizations”, wherein each is a random realization of a possible collection of particles and properties of the constituent grains.

Ensemble Averaging is the computation of the average, or possibly the statistical distribution, of a certain quantity for an ensemble of realizations. In this study, the ensemble is a collection of Discrete Element Models, each of which represents a realization of the system of grains.


Weibull Survival Probabilities of Sand Grains Weibull [15] proposed the following equation for the

survival probability sP for material blocks of volume 0V under

normalized tensile stress:

00

exp

m

sP V

(6)

where 0sP V is the probability of survival of a particle of

volume 0V under a characteristic stress of 2F d , is the

magnitude of characteristic stress where 37% of the blocks survive, and m is known as the Weibull modulus or the m-modulus. The m-modulus describes variability in strength; a higher m-modulus indicates decreasing variability.

For spherical particles of varying volumes, as pointed out by Nakata [16] Equation (6) may be written as:

0 0

expm

s

VP V

V

(7)

or, in terms of particle diameters:

3

0 0

expm

s

DP D

D

(8)

For a sand grain of diameter D , Equation (8) gives the probability that the grain survives intact when subjected to a

force equal to 2F D . A more useful quantity for particle aggregate simulations

using DEM, is the probability density function (PDF) of grain strengths. This may be obtained as the derivative of the survival probability function which is the complement of the cumulative distribution function (CDF).

3 31

0 0 0 0

expmm

sm

P D Dm

D D

(9)

This PDF of grain strengths may be visualized as a surface dependent on grain stresses and diameters, as illustrated in Figure 2.

It should be noted that Equation (8) implies a sort of “size-hardening law”, whereby the larger the grain the weaker it is expected to be, and vice-versa. This may be qualified by modifying the expression so as to compare the characteristic

stresses for D and 0D , which results in the following equation.

3

0 0 00

mDD D

D

(10)

Figure 2: Grain probability distribution of normalized

strengths for normalized diameters Finally, in order to make Weibull survival usable directly in

statistical mechanics mesoscale DE models, an expression for is required. Such an expression needs to take into account both the experimental results for characteristic stress, and the stochastic nature of grain survival. This can be done by solving

Equation (8) for and substituting the survival function sP

with a randomly generated survival probability qP for particle

q

1

3

00 ln

m

q qq

DP

D

(11)

where qP is drawn from the uniform distribution over [0,1]. The

scale parameter for 2-parameter Weibull sampling in this case is simply

3

00

m

q

D

D

(12)

Upon crush-up, a spherical particle is replaced by an “Apollonian” spherical packing as shown in Figure 3. The use of the Apollonian fractal packing algorithm goes beyond convenience and aesthetic considerations. Experimental observations of the evolution of particle size distributions find a fractal distribution with an exponent of ~2.5 [17]. This agrees remarkably well with the Apollonian packing’s fractal number of ~2.47 [9].


Figure 3: Apollonian Sphere Packings at increasing

levels of refinement

Brazilian Criterion of Grain Fracture The crushing criterion used here is the so-called “Brazilian”

criterion [10, 18, 19]. This criterion derives its name from the use of the distribution of stress in a 1D compression test, also widely known as a Brazilian test. The maximum tensile stress in such a test is:

1 33

2t

(13)

where 1 is the major principal stress and 3 is the minor

principal stress. A particle that meets the crushing criterion is replaced by an

Apollonian packing which has been rotated in 3D space so that the major principal stress bears down on the assumed fracture axis of the fragments. This allows for “blowout” of the packing similar to the blowout fracture seen in a Brazilian test. Figure 4 illustrates the loading and blowout of the Apollonian packing

under 1D compression ( 3 0 ).

Figure 4: Apollonian packing blowout under 1D

compression

COMPUTATIONAL APPROACH In this work, a well-graded fine sand known in various

settings as either Eglin sand, or Quikrete sand 1961, is used to

illustrate the approach and utility of the methodology developed. Eglin sand is chosen for one important reason, it is a sand that has been tested micro-experimentally and had its micro-mechanical characteristics described statistically. This, along with triaxial test data, makes it possible to evaluate the quality of both the mesoscale as well as the macroscale model. This is crucial to the evaluation of the overall framework the current effort is part of.

Figure 5 shows the Particle Size Distribution (PSD) of Eglin sand. The PSD may also be described by a Weibull Probability Distribution Function (PDF), shown in Table 1. The rest of the rows of Table 1 summarize the parameters found in experimental micromechanical testing of the sand [20-26].

Figure 5: PSD of Eglin Sand

The Weibull PDF in Table 1 is used to build a DEM packing

of 1×108 particles, hereafter referred to as the “source” model. For reference, Figure 1 shows a DEM packing of 1×104 particles. The source model is then sampled thousands of times to build mesoscale DEMs of a set size. All simulations in the current work were carried out on DEMs of 1000 particles each, sampled from the source particle model.

Table 1: Eglin Sand Parameters

Parameter Distribution Shape/StDev

Scale/Mean

Initial Particle Size Weibull 2.59 0.428 (mm) Final Particle Size Weibull 1.11 0.25 (mm) Young’s Modulus Weibull 5.48 95.9 (GPa)

Density Scalar - 2.72 (g/cm3) Shear Stiffness Scalar - 0.17

Angle of Friction Normal 5.7 º 19.1 º Crush Strength* Scalar 36 (MPa)

Fail. Ref. Diameter Scalar 1.36 (mm) Weibull Modulus Scalar 2.5

* Crush Strength is the characteristic stress at 37% survival


The ensemble used here consists of 1000 DEMs. Each realization is confined to two different mean stress targets of 5 MPa and 200 MPa. The higher confinement matches the final mean stress carried out in the Split-Hopkinson Pressure Bar test [20] that resulted in the final particle size distribution of Table 1. Figure 6 shows the micromechanical stresses in a typical 1000-particle DEM at confinements of 5 MPa and 200 MPa prior to any crushup.

Figure 6: Micromechanical Mean Stress in a DEM of

1000 Spheres at a Confinement of a) 5 MPa b) 200 MPa

Figure 7: Sample realization before and after crushup

of grains at 200 MPa mean stress Figure 7 shows the original PSD at 5 MPa confinement, as

well as the final PSD after crushup at up to 200 MPa mean compressive stress. Note that Figure 7 also depicts a faithful representation of the volumetric strain in the simulation, approximately 20% at this stress for this realization.

RESULTS Shown in this section are ensemble average results of

mesoscale simulations of volumetric and crushup behavior of Eglin sand.

Figure 8 shows the evolution of the particle size distribution (PSD) of Eglin sand found from the cumulative behavior of the mesoscale model ensemble compared with that from experimental PSD evolution. The experimental data in this figure is represented by three different data series. First, the data series titled “Weibull Initial” is the Weibull distribution fit for the “as-received” PSD. The fit here is almost perfect, so the discrete data points for each sieve are not shown. The series titled “Sieves Final” shows the experimental sieve data points for the sand after

crushup [20]. The “Weibull Final” series shows the Weibull distribution fit for the sieve data points. The fit is still quite good, though not as perfect as the fit for the initial PSD. Next, the series titled “m=2.5” represents the mesoscale simulation ensemble cumulative PSD using a Weibull modulus of 2.5 for the Weibull grain survival model described in this paper.

Figure 8: Eglin Sand PSD evolution during crushup for isotropic compression up to 200 MPa

The fit here is remarkable for one very important reason: no

calibration or optimization has been carried out to obtain the computational results shown. Only experimental data has been used for the mesoscale ensemble simulations of grain crushup. So, what is remarkable is that this Discrete Element paradigm, wherein sand grains are modeled as prefect spheres, can with no calibration use lab data to approximate the complex, non-deterministic behavior of sand using fractal geometry to discretize the comminution of macrograins.

Next, Figure 9 presents a comparison of the volumetric stress-strain curve from the triaxial testing of Martin and Cazacu [21] with that obtained using the crushup model introduced in this study. The first two series shown in this figure represent the volumetric strain with no crushup, that is for a normal DEM code that does nothing to alter the number of particles or the initial PSD through crushing. The difference between these two series is that one uses the Kinematic Uniform Boundary Condition (KUBC), which yields the Voigt (stiff) bound, while the other uses the Static Uniform Boundary Condition (SUBC), which yields the Reuss (soft) bound of constitutive response. The series titled “Fractal” shows the volumetric response using the crushup model described in this paper.

0.010.11

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Sand Grain Size (mm)Percent Finer by Weight (%

)

Weibull Initial

Sieves Final

Weibull Final

M=2.5

Cutoff

a) b)


Figure 9: Eglin Sand volumetric compression response at mean stress up to 400 MPa

Again, the fit here is remarkable for the same reason as

before. Only experimental data has been used, with no calibration of optimization. The small discrepancy here can be easily attributed to the fact that no intragrain voids are assumed when macrograins are replaced by their fractal fragments.

CONCLUSIONS A method for simulating the particle size distribution (PSD)

evolution of particle aggregates during grain crushup was devised and presented in this work. It was demonstrated that through this approach, the evolution of Discrete Element Models of particle aggregates can be modeled during large stress loading and the resulting comminution approximated using the available experimental testing and data.

Thousands of simulations were carried out using random samples of spherical particles using the Discrete Element formulation. The PSD evolution and constitutive response were successfully obtained for the ensemble under loading conditions simulating lab testing, including a Split-Hopkinson Pressure bar and triaxial loading.

This is an important development, as it clears an important hurdle for utilization of DE models in the context of mesoscale modeling of particle aggregates. As such, and along with the statistical mechanics perspective herein, it represents a powerful tool in the multiscale modeling of particulate media.

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‐0.35‐0.3‐0.25‐0.2‐0.15‐0.1‐0.050

‐600

‐500

‐400

‐300

‐200

‐100

0

Volumetric Strain

Mean

Stress (MPa)

No Crush ‐ Reuss

No Crush ‐ Voigt

Exp 1

Exp 2

Fractal


individual contacts in geologic materials, Granular Matter 17(1) (2015) 95-110. [24] C. Sandeep, K. Senetakis, Grain-scale mechanics of quartz sand under normal and tangential loading, Tribology International 117 (2018) 261-271. [25] F. Wang, B. Fu, H. Luo, S. Staggs, R.A. Mirshams, W.L. Cooper, S.Y. Park, M.J. Kim, C. Hartley, H. Lu, Characterization of the Grain-Level Mechanical Behavior of Eglin Sand by Nanoindentation, Experimental Mechanics 54(5) (2014) 871-884. [26] K. Senetakis, M.R. Coop, M.C. Todisco, The inter-particle coefficient of friction at the contacts of Leighton Buzzard sand quartz minerals, Soils and Foundations 53(5) (2013) 746-755.

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