ectc section 4 cover · from the shear force and bending moment diagrams, it is clear that there is...

UABSchoolofEngineering–MechanicalEngineering–EarlyCareerTechnicalJournal,Volume17 Page91

SECTION 4

Materials

UABSchoolofEngineering–MechanicalEngineering–EarlyCareerTechnicalJournal,Volume17 Page92

UABSchoolofEngineering-MechanicalEngineering-JournaloftheECTC,Volume17 Page93

Journal of UAB ECTC Volume 17, 2018

Department of Mechanical Engineering The University of Alabama, Birmingham

Birmingham, Alabama USA

STRUCTURAL PERFORMANCE OF WOOD-BASED BIO-COMPOSITES 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 bio-composites using the four-point-bending method. Different types of bio-composites made from Methylene diphenyl diisocyanate (MDI) and Corn Starch (CS) were manufactured using a Dieffenbacher hot press. The bio-composite made from 4% MDI with higher density and manufacturing pressure had the highest maximum flexural stress (σf) max and elastic strain among four different types of bio-composites tested. Increasing manufacturing pressure during the fabrication process did significantly affect the flexural strength and elastic behavior of the wood-based bio-composite.

KEYWORDS: Bio-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. Bio-composites are made from biocompatible and/or eco-friendly material. Researchers of the bio-composites state that the use of bio-composites improves health and safety in their production, and that they are lighter in weight, have excellent appearance, and are environmentally superior [1].

Bio-composites have been frequently used for automobiles [2], vibration damping and noise reduction applications [3], and the packaging of nuclear waste [4]. Bio-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, bio-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, bio-composites.

When bio-composites are used for industrial applications, they are subjected to flexural loading, and therefore, when 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 bio-composites, including wood composites [5-7].

The authors [8] 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 material under bending load for the development of sustainable and green material.

MATERIAL- Southern yellow pine was used to create the bio-composite

panels. The mass of the bio-composite panels was approximately 2.95 kg except for the bio-composite panel manufactured at a mat pressure of 10.5 MPa (with ram pressure of 27.58 MPa) where twice the amount of mass (5.9 kg) was used. 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


bio-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 bio-composite panel with a thickness of 6.5 mm. The varying pressure required to form the panel to the appropriate thickness was based on the ability of the composite material in the mat to be compressed to the appropriate thickness. Bio-composite samples created for the analysis of the high strain rate test were made from corn starch (CS) and methylene diphenyl diisocyanate (MDI) resin with different mass fractions (listed in Table 1). 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. Corn starch (CS) is the starch derived from the corn (maize) grain or wheat. The sampling size for each material was five (5).

Table 1: Types of bio-composites

Material Approx. pressure (MPa)

Curing Time

(Seconds)

Density kg.m-3

4%MDI 8.9 140 827 4%MDI 10.5 MPa

(2x material) 10.5 140 1389

2%CS2%MDI 8.7 140 855

2%CS4%MDI 9.2 140 822

EXPERIMENTAL PROCEDURE AND SETUP The wood-based bio-composite specimens of size 151 mm

in length x 25 mm in width x 6.5 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 D7264 standard was used to study the flexural properties of Wood-based Composites [9].

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. Thus, it gives four-point bending. Hence, this method is called four-point bending. Figure 2 shows the schematic, shear force (SFD) and bending moment diagrams (BMD) to four point

bending loading regimes and Figure 3 shows 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 shear force (Fs) and interlaminar shear stress are zero between the loading points. Thus, it leads to the pure bending loading. Such a state of stress is desirable in testing [9]. Flexural stress (σf) is defined as the stress on the surface of the specimen at the failure. It is evident from the Figure 3 that normal stress varies linearly across the thickness. The maximum in compression (σC) is on one side and an equal maximum in tension (σT) is on 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 are given by [10]

(σ#)%&'. = +,-./01

(1)

ε = ..+340-1

(2) 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 the experimental

results for the four different types of wood-based bio-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 CS, a change occurred in the average maximum flexural stress (σf)max excluding 4%MDI10.5MPa. The wood-based bio-composite material that was made with 2%CS2% MDI had an average maximum flexural stress (σf) max. of 12.07 MPa, and by increasing the mass fraction of the MDI by 2% (solids basis), the observed average maximum flexural stress (σf) max.was 16.19 MPa for 2% CS 4% MDI. A ~ 34% increase in the maximum flexural stress (σf) max.) was achieved by the addition of more MDI. This could be because of the higher volume fraction of MDI caused higher maximum flexural stress (σf) max. Furthermore, results show that increasing the volume fraction of MDI with CS adversely affected the max strain of the bio-composite. This may be due to the material properties of CS and MDI.

Figure 2. Shear force and bending moment diagrams for

four pint bending test

Figure 3. Bending and Shear stress in the thickness direction

Table 2: Flexural properties of bio-composites

Material Max Stress

(σf) max. (MPa)

Max. Strain

4%MDI 8.89 0.014 4%MDI 10.5 MPa

(2x material was used) 39.56 0.017

2%CS2%MDI 12.07 0.015 2%CS4%MDI 16.19 0.0095

Figure 4. Flexural stress-strain behavior

In addition, results show that 4%MDI 10.5 MPa had the

highest maximum flexural stress (σf) max. ~ 39.56 MPa and the maximum strain of 0.017. This could be due to the increased manufacturing pressure during fabrication causing highest maximum flexural stress (σf) max. and maximum strain. Furthermore, Bio-composite 4%MDI140s had the lowest maximum flexural stress (σf) max among different types of bio-composite tested. In addition, the single most striking observation to emerge from the data comparison was that wood-based bio-composites containing CS have consistently improved the maximum flexural stress (σf) max when made under similar manufacturing pressure and curing time.

CONCLUSIONS An experimental study was conducted to study the flexural

behavior of wood-based bio-composites under four-point

0

5

10

15

20

25

30

35

40

0 0.002 0.004 0.006 0.008

Stre

ss (M

Pa)

Strain4%MDI 4%MDI 10.5MPa2%CS 2%MDI 2%CS 4%MDI

P

h Si S0

L

Fs = P/2

Fs = P/2

𝑀 = 𝑃(𝑆8 − 𝑆:)

4

b

h

σC

σT τ


bending load. The bio-composite 4%MDI 10.5 MPa had the highest maximum flexural stress (σf) max and strain, which reveals that factrication pressure and mass density play an important role to govern the flexural properties of bio-composites. Finally, corn starch had enhanced the maximum flexural stress (σf) max, when bio-composite made under similar manufacturing condition. Regardless, future research should continue to study the effect of different volume fraction of constituents on the mechanical behavior of composites.

ACKNOWLEDGEMENTS We would like to express our gratitude to Dr. P. Raju

Mantena and Damian Stoddard for their involvement throughout this research and Matthew Lowe for the machining of the wood-based bio-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 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] Joshi, S.V., Drzal, L.T., Mohanty, A.K., and Arora, S., 2004, “Are Natural Ffiber Composites Environmentally Superior to Glass Fiber Reinforced Composites?,” Composites Part A: Applied Science and Manufacturing, 35, pp. 371-376. [2] 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. [3] Mohanty, A., and Fatima, S., 2015, “Noise Control Using Green Materials,” Sound and Vibration Magazine, 49(2), pp. 13-15. [4] Bragov, A., and Lomunov, A., 1997, “Dynamic Properties of Some Wood Species,” J Phys IV Colloq, 07(C3), pp. 487–492. [5] 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. [6] 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. [7] Hiroshi, Y., Oka. S., 2001, “Measurement of Bending Properties of Wood by Compression Bending Tests,” J Wood Sci., 47, pp. 262-268. [8] Shah, A., Pandya, T., Stoddard, D., Ukyam, S., Street, J., Wooten, J., Mitchell, B., 2017, “Dynamic Response of Wood based Bio composite under High Strain Rate Compressive Loading”, Wood and Fiber Science. 49(4), PP. 1-9.

[9] Mohite. P., “Composite Materials and Structures”, NPTEL Lecture notes, https:/nptel.ac.in/corces//lecture39/39_6htm. [10] ASTM International, 2015, Standard Test Methods for Flexural Properties of Polymer Matrix Composite Materials,” Annual Book of ASTM Standards, 15.03, pp. 587-597.





AN IMPROVED METHOD FOR FRAGMENT DETECTION IN EULERIAN HYDROCODES

Kenneth Walls and David Littlefield University of Alabama at Birmingham

Birmingham, Alabama, USA

ABSTRACT Dynamic fragmentation of solid materials is a subject of

great importance to the design of munitions, development of protective shielding, and a number of other defense and commercial applications. Due to their ability to develop new free surfaces, Eulerian hydrocodes are typically chosen to simulate high strain rate impact, penetration, and fragmentation problems. However, multi-material “mixed” elements make the problem of identifying individual fragments very complex. In order to systematically determine the fragment distribution within the mesh, a fragment detection algorithm was developed. To examine the effectiveness of the new fragment detection algorithm, several test cases were considered using the Eulerian hydrocode CTH.

KEY WORDS: Dynamic Fragmentation, Eulerian Hydrocodes, Fragment Detection

INTRODUCTION Dynamic fragmentation occurs when a solid material is

subjected to an intense and abrupt internal or external force that causes the material to break apart into a large number of pieces, or fragments [1]. Fragmentation events can occur over a wide range of size scales, ranging from hypervelocity impact of orbital microparticles on a Whipple shield, to fragmentation of cased munitions, to the fragmentation of the Shoemaker-Levy comet as it collided with the planet Jupiter. The complexity of the fragmentation process makes the process of computational simulation difficult, often requiring simplifications to be made which can compromise the physical accuracy of the simulation [2]. Nevertheless, many constitutive and failure models have been shown to provide a reasonable degree of accuracy and are a useful tool for performing phenomenological studies of fragmentation. A rigorous discussion of the mechanisms for dynamic fragmentation and the models used to simulate these processes is beyond the scope of this work, but an understanding of the assumptions and simplifications made by the analyst’s

model of choice is important for a proper interpretation of results.

Because of their ability to form new free surfaces, multi-material Eulerian hydrocodes are often used to simulate problems involving large deformations and high strain-rates, such as penetrations, blasts, and fragmentation events. However, because of the presence of multi-material “mixed” elements, the process of identifying the fragment distribution within the mesh is complex. While most post-processing tools, such as Paraview [3], VisIt [4], and EnSight [5], are capable of determining global and element level quantities, such as material specific mass, momentum, and kinetic energy, these tools are unable to determine the distribution of fragments within the Eulerian mesh and the properties associated with these fragments without significant effort by the analyst.

However, by taking advantage of the element level material volume fraction, a systematic approach for identifying fragments in the Eulerian mesh can be developed. Material volume fraction, fm, is used by Eulerian hydrocodes for interface reconstruction and keeping track of the contents of the computational element. The contents of the element are limited to three possibilities: (1) the element is full of the material of interest (fm = 1), (2) the element contains no material of interest (fm = 0), or (3) the element contains a mixture of the material of interest, void, and other materials (0 < fm < 1). By creating an array of material volume fraction information and performing a search of neighboring elements to form closed volumes separated by cells that contain no material of interest it is possible to build a database of individual fragments in the Eulerian mesh. Once the elements that make up the fragment are known important information, such as mass, volume, density, kinetic energy, and momentum can be determined from the element level quantities for the cells that make up the fragment. In this work a post-processing tool has been developed that automates the process of fragment detection in the Eulerian hydrocode CTH [6], providing a robust tool for the analysis of fragmentation problems.


BACKGROUND The ability to predict the natural fragmentation of materials

using numerical simulations would represent a significant, cost effective design and analysis tool for munitions and armor designers as well as those studying other fragmentation events. The primary requirement of this study was to develop a tool that could be used to automate the identification and analysis of the properties of material fragments when simulated using the Eulerian hydrocode CTH.

Since Mott’s original work during World War II [7], significant effort has gone into understanding the fragmentation of cased munitions. In recent years, important work has been done to develop models that predict the fragmentation process of ductile materials, such as steel, and brittle materials, such as ceramics. Grady [8, 9] developed an energy based approach to predict the size of fragments based on strain rate and a fragmentation toughness constant for the material. Brannon’s work on ceramics made a significant contribution to the simulation of fragmentation in brittle materials by using Weibull statistics to seed the mesh with failure sites to introduce critical flaws at which fracture can initiate [10].

The fragmentation of metal cased explosives was used as a validation problem for the fragment post-processing tool developed for this work. Information on the experimental study of fragmentation in cased munitions can be found in references [11-13].

FRAGMENT IDENTIFICATION ALGORITHM In an Eulerian framework, more than one material can

occupy a numerical cell. The method therefore requires an advanced interface tracking algorithm to describe the evolution of material boundaries with time. In the case where fracture and fragmentation are occurring, fragments may be separated by very small distances, with gaps of only one or two elements in some cases. Furthermore, some fragments may be very small in size, spanning only a few elements and may not be adequately resolved to be visualized by most post-processing software. In such situations a visual analysis of the problem domain becomes impractical, and developing an accurate count of individual fragments and their motion requires an additional fragment identification algorithm.

In this work, a post-processing tool for identifying fragments in the Eulerian hydrocode CTH was developed. CTH is a three-dimensional finite volume hydrocode developed by Sandia National Laboratories. In order to identify fragments the contents of the entire computational mesh have to be analyzed. A schematic of the fragment detection process is shown in Figure 1. This was accomplished using the post-processing tool CTHED, which is distributed as part of the CTH software suite. CTHED is used to read and edit restart dump files produced by CTH during the run. Among its options is the ability to write the contents of each element to a text file, called the oacthed.n file, where n is the processor number associated with the restart dump. There are a total of n oacthed.n files that must be read by the post-processor, corresponding to the number of cores on which the CTH problem was run. Since the post-processor is

currently serial, data from the n oacthed.n files are combined into global arrays that encompass the entire problem domain, rather than the individual processor domains. The oacthed.n files contain all pertinent element level information, such as density, mass, energy, temperature, pressure, and volume fraction for each material present in the problem.

Figure 1. The fragment detection process.

In order to determine the distribution and properties of the

fragments in the CTH mesh, a Fortran program was developed which reads through all of the oacthed.n files and stores the necessary information from each file into combined global arrays for post-processing. Because the Eulerian mesh must be large enough to ensure the problem remains inside the mesh throughout the solution process, a large portion of the elements in the computational mesh contains no material of interest. In order to limit the size of the text file, the oacthed.n file limits printing of cell information to those elements which contain material, with negative element indexes being used to signify that previous elements were either empty if no element information appears before the negative index, or have the exact same properties as the previously listed element, and thus are not printed. The oacthed.n files also used local processor I, J, K indexing to identify elements, rather than using their global I, J, K index; therefore, an I, J, K offset had to be determined for each oacthed.n file to properly position the element information in the global arrays. Furthermore, the CTHED formatting would change depending on whether the I, J, or K direction contained more elements. Therefore it was necessary to use flags in the reader program to ensure that the oacthed.n file was read using the correct formatting.

Once the reading of the oacthed.n file was completed a global four-dimensional array with dimensions for the number of materials present in the problem as well as the number of elements in the three spatial dimensions in the global computational mesh was produced. In filling this array any cell which contained a non-zero material volume fraction for the material of interest was flagged with a value of -1. Cells which contain no material of interest are given a value of 0. It is important to note that all arrays, including the four-dimensional material flagging array as well as arrays for element properties such as mass, momentum, volume fraction, pressure, and


temperature, are global arrays, rather than processor specific arrays. Since the post-processing tool is currently a serial application all data from the problem must fit into the available local memory for a single processor. This can become problematic for large problems, so future revisions of the fragment detection tool will be parallelized in order to work effectively for large data sets.

Once the material flagging array is determined the algorithm then loops over the mesh looking for elements that are flagged with a -1. When an element containing material is found the algorithm then recursively examines adjacent cells to ascertain whether they also contain the material of interest. The algorithm begins by checking the element directly above the current element, and cycles around the 8 surrounding elements in two-dimensions or 26 surrounding elements in three-dimensions to see if it has also been flagged as containing material. If an adjacent element is found to contain material of interest, the algorithm then moves to that element and checks it for connectivity to other flagged elements. All contiguous elements that are found to contain the material of interest are grouped into a single fragment. These cells then have their flag changed to a positive number representing their fragment identification number. The marching process is continued until all fragments in the mesh have been identified. This marching process is shown for a single fragment in Figure 2. The process begins in the element with the dot, then proceeds along the path indicated by the arrows. It should be noted that it is possible for this search to miss some contiguous elements, so multiple sweeps of the mesh are performed until all cells flagged with a -1 have been replaced with a positive number representing the fragment number. In Figure 2 the element containing the triangle would have been missed by the first sweep, but would have been detected on a second sweep and associated with the fragment detected on the first sweep.

One issue with using volume fraction as the indicator for a cell containing material of interest is that Eulerian meshes have the tendency to erroneously report very small amounts of material in some elements. This can result in fragments being artificially joined together where negligible amounts of the material of interest are found in mixed cells, effectively forming a “bridge” between fragments. In order to remove this extraneous material from the calculation of fragment size, the algorithm only takes into account cells with a material volume fraction greater than a specified minimum. This volume fraction limit therefore defines the surface of a fragment and as a result dictates its ultimate size and mass. Therefore, care should be taken when choosing the volume fraction cutoff value since it has the potential to greatly influence the number of fragments identified in the mesh. For all test problems shown in this work a volume fraction cutoff value of 10-6 was chosen.

The fragment detection algorithm also stores multiple variables for the material in the cells including mass, velocity vectors, pressure, temperature, and energy. This data is summed and mass averaged to calculate the overall information for that fragment. This information can then be used to compare

simulation results with fragment data from experiments, other simulations, or analytical models.

Figure 2. Example of the search path followed for

identifying contiguous elements making up a fragment. The starting point is denoted by the dot and the search algorithm follows the path denoted by the arrows. The

element with the triangle was missed in the first sweep, and would be detected on the second sweep.

VALIDATION CASES AND DISCUSSION Several test problems were considered to ensure proper

functionality of the fragment detection post-processing tool developed for this work. Two test cases, one two-dimensional and one three-dimensional, were selected from the CTH benchmark suite which is included with the distribution of CTH version 12.0. The final test case was a three-dimensional explosively loaded fragmenting cylinder, which was chosen to demonstrate the performance of the fragment detection algorithm under its intended usage condition.

The first test case under consideration was a two-dimensional meteor impact problem. In this problem three 6061-T6 aluminum spheres impacted the non-transmitting top boundary of the mesh at 20x105 cm/s. The first meteor to strike had a radius of 100m while the two remaining meteors had a radius of 60m each. Upon impact the non-transmitting boundary the meteors fragmented into multiple pieces. The initial configuration of this test problem is shown in Figure 3. The dark grey material represents the 6061-T6 aluminum meteors, while the blue background material represents the atmospheric CO2. The mesh consisted of 30 elements in the X-direction and 60 elements in the Y-direction. The mesh was made up of a total of 1800 square elements having a constant edge length of 50m. The problem was run to a stopping time of 120ms. At this point the second meteor has just impacted the non-transmitting boundary. The material plot at 120ms is shown in Figure 3. This time state was evaluated by the fragment detection algorithm, and a total of 10 fragments were identified. The mass, volume, kinetic energy, and average pressure of the detected fragments is shown in Table 1. It is important to note that the coarse resolution of the mesh used for this test problem led to an artificially low number of fragments being identified. A visual inspection of Figure 4 will


show that at least 14 fragments are present; however, only 10 fragments were detected by the algorithm. This is because the fragment detection algorithm requires at least one empty element between fragments to consider them independent. A more refined mesh would have likely resulted in an increased number of detected fragments. Future revisions of the fragment detection algorithm will take the orientation of the material interfaces into account, making it possible to have non-contiguous fragments in neighboring cells.

The second test problem was a three-dimensional benchmark meant to demonstrate the geometric capabilities of the CTH DIATOM pre-processing card. As can be seen in Figure 5, a total of 8 shapes were defined, including a helix, a box, a cylinder, a prism, a pyramid, a tetrahedron, a torus, and a sphere. When the shapes overlap the material input that occurs last in the DIATOM record replaces previously input materials. As a result some of the materials were split into multiple discontinuous pieces. The pyramid (blue) and the tetrahedron (dark purple) were split into two and three segments, respectively. To ensure proper functionality of the fragment post-processor in determining the properties of fragments, the detected volumes and masses of the 8 materials were compared to those reported in the octh output file. Overall agreement was very good, with all error in volume and mass falling below 0.737%. The results of this test case are summarized in Table 2.

Figure 3. Initial configuration of the two-dimensional fragmenting meteor test problem.

Figure 4. Final state of the two-dimensional fragmenting

meteor test problem at 120ms.

Table 1. Properties of Fragments from Meteor Test Problem

FragID Mass (g) Volume (cm3)

KE-Tot (erg)

Pres-Avg (dyn/cm2)

1 2.16E+07 2.35E+07 1.30E+20 1.53E+10

2 2.16E+07 2.35E+07 1.30E+20 1.53E+10

3 3.68E+07 4.07E+07 8.77E+19 6.11E+09

4 3.68E+07 4.07E+07 8.77E+19 6.11E+09

5 1.95E+08 1.24E+08 8.70E+19 -8.57E+08

6 1.95E+08 1.24E+08 6.93E+19 -8.57E+08

7 1.43E+08 1.54E+08 7.78E+19 -1.82E+09

8 1.43E+08 1.54E+08 7.78E+19 -1.82E+09

9 3.59E+08 1.35E+08 7.05E+20 -4.98E+09

10 4.55E+08 2.78E+08 3.10E+20 1.93E+12

The final test case, an explosively loaded thick-walled

copper cylinder as shown in Figure 6, was chosen to demonstrate the performance of the fragment detection algorithm for a typical usage condition. In this problem a 5mm thick copper cylinder is filled with PBXN-109 and ignited at a point at the top of the explosive charge. As the cylinder expands it forms long strips of material as well as smaller fragments. This behavior is consistent with what is seen experimentally; however, these long strips are likely composed of multiple smaller fragments, but due to the close proximity of the fragments and the relative coarseness of the mesh the algorithm groups them together as a single fragment. It is likely that refinement of the mesh would increase the number of fragments detected and also reduce the maximum fragment mass. However, for the purposes of this demonstration problem, the results are sufficient. The cylinder expansion test case was run using quarter symmetry and reflected about both


planes of symmetry to provide a full visualization of the fragment field.

Figure 5. Setup of the three-dimensional shapes test problem.

Table 2. Mass Properties of Bodies in the Three-

Dimensional Shapes Test Case

Shape ID FragID Mass (g) (Detected)

Mass (g) (octh)

Error (%)

Helix 1 1.10E+04 1.11E+04 7.37E-01

Box 1 1.72E+05 1.72E+05 1.68E-05

Cylinder 1 1.97E+04 1.97E+04 1.94E-01

Prism 1 4.49E+04 4.49E+04 1.04E-02

Pyramid 1 6.61E+03 -- --

Pyramid 2 1.60E+01 -- --

Pyramid Sum 6.62E+03 6.61E+03 2.42E-01

Tetrahedron 1 3.90E+02 -- --



Tetrahedron Sum 5.25E+04 5.25E+04 9.05E-06

Torus 1 3.12E+04 3.12E+04 1.92E-06

Sphere 1 6.55E+04 6.55E+04 2.90E-06

As shown in Figure 7, the simulation of the fragmentation

of the cylinder was carried out to 150µs after detonation, when the fragment cloud had expanded approximately 30 cm from the cylinder axis. The expanded PBXN-109 products were discarded at 100µs. It is noteworthy that there are erroneous geometric effects taking place at the symmetry plane, with fragments at the symmetry plane showing greater radial expansion than other fragments. A total of 89 fragments were found for the quarter symmetry problem ranging in size from 1.11x10-6g to 467.7g. The distribution of fragments with respect to the order of

magnitude of the fragment mass is shown in Figure 8. Future work will compare the detected number of fragments and their spatial distribution to experimental results.

Figure 6. Setup of the explosively loaded cylinder

fragmentation problem, reflected about both planes of symmetry. The green dot indicates the detonation point.

Figure 7. Predicted fragment spatial distribution at 150µs.

Figure 8. Fragment size distribution at 150µs for the

quarter cylinder.

0

5

10

15

20

25

O(10^-6)

O(10^-5)

O(10^-4)

O(10^-3)

O(10^-2)

O(10^-1)

O(10^0)

O(10^1)

O(10^2)Num

ber o

f Fra

gmen

ts

Fragment Mass Order of Magnitude (g)


CONCLUSIONS This work has presented an automated method for

identifying material fragments in an Eulerian mesh. The algorithm identifies the distribution of fragments within the mesh and determines the properties of individual fragments, such as mass, volume, kinetic energy, momentum, velocity, and averaged temperature and pressure. Several validation cases have been presented to demonstrate the accuracy and capabilities of the fragment detection algorithm. The accuracy of the fragment identification is somewhat dependent on mesh resolution and the primary limitation of the algorithm is the requirement for individual fragments to be separated by at least one element containing no material of interest. Future improvements to the algorithm will implement an interface reconstruction algorithm to determine if materials in adjacent cells should be considered contiguous or separate. However, despite this limitation the fragment detection algorithm developed in this work provides analysts with a powerful tool that can greatly reduce the effort required to post-process fragmentation problems.

ACKNOWLEDGEMENTS This work was completed as part of the Department of

Defense High Performance Computing Modernization Program (HPCMP) user Productivity Enhancement, Technology Transfer, and Training (PETTT) program.

REFERENCES [1] Grady, D., 2009, “Dynamic Fragmentation of Solids,” Shock Wave Science and Technology Reference Library, Vol. 3. Springer-Verlag, Berlin, Heidelberg. [2] Shockey, D.A., 1985, “Discussion of Mechanisms of Dynamic Fragmentation: Factors Governing Fragment Size,” Mechanics of Materials 4, pp. 321-324. [3] Ayachit, U., 2018, “The ParaView Guide: Community Edition,” Kitware Inc., https://www.paraview.org/paraview-guide/ [4] “VisIt User’s Manual: Version 1.5,” October 2005, UCRL-SM-220449, https://wci.llnl.gov/simulation/computer-codes/visit/manuals [5] “EnSight User Manual for Version 10.2,” September 2016, Computational Engineering International, Inc., http://www3.ensight.com/EnSight10_Docs/UserManual.pdf [6] McGlaun, J., Thompson, S., & Elrick, M., 1990, “CTH: A Three-dimensional Shock Wave Physics Code,” International Journal of Impact Engineering, 10, pp. 351-360. [7] Mott, N.F., 1947, “Fragmentation of Shell Cases,” Proceedings of the Royal Society Section A, 189, 300-308. [8] Grady, D.E., 2006, “Fragmentation of Rings and Shells,” Springer. [9] Grady, D.E., 1982, “Local Inertial Effects in Dynamic Fragmentation,” Journal of Applied Physics, 53(1), 322-325. [10] Brannon, R.M, Wells, J.M., & Strack, O.E., 2007, “Validating Theories for Brittle Damage,” Metallurgical and Materials Transactions, 38A, pp. 2061-2068. [11] Chhabildas, L., et al., 2001, Fragmentation Properties of

Aermet 100 steel in two material conditions,” International Symposium on Ballistics, 19, pp. 663-669. [12] Hiroe, T. Fujiwara, K., Hata, H., & Takahashi, H., 2008, “Deformation and Fragmentation Behavior of Exploded Metal Cylinders and the Effects of Wall Materials, Configuration, Explosive Energy and Initiated Locations,” International Journal of Impact Engineering, 35(12), pp. 1578-1586. [13] Cullis, I.G., et al., 2014, “Numerical Simulation of the Natural Fragmentation of Explosively Loaded Thick Walled Cylinders,” Defence Technology, 10, pp. 198-210.





DYNAMIC COMPRESSIVE BEHAVIOR AND ENERGY ABSORPTION OF DDGS- PAULOWNIA WOOD COMPOSITE USING SHPB

Mr. Suman Babu Ukyam University of Mississippi Oxford, MS 38677, USA

Mr. Damian Stoddard University of Mississippi Oxford, MS 38677, USA

Dr. Brent Tisserat

U.S. Department of Agriculture Peoria, IL 61604, USA

ABSTRACT The dynamic behavior of novel Engineered wood composites made using Dried Distiller’s Grain &

Soluble (DDGS) as an adhesive subjected to different compression loads are reported. Specimens of DDGS-Paulownia wood (PW) composites made using DDGS as an adhesive with volume fractions of 10, 15, and 25 are tested at high strain-rate tests on a modified compression Split Hopkinson Pressure Bar (SHPB). Test results of DDGS-PW composites with volume fractions of 10,15, and 25 have displayed strain rate sensitivity. DDGS-PW composite with volume fraction of 25 percent has shown the highest ultimate compressive strength. Under quasi static loading conditions, cost effective DDGS-PW composite panels are found to have superior mechanical properties compared to soybean adhesive panels.

Presentation Only





DEFORMATION AND STRESS AVERAGING FOR STATISTICAL ENSEMBLES OF PARTICLE AGGREGATE MESOSCALE MODELS

Gerald Pekmezi and David Littlefield The University of Alabama at Birmingham,

Birmingham, Alabama, USA

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 is due to the geometrical complexity of translating discrete forces and displacements to equivalent continuum microstresses and microstrains.

The current work will describe geometrical and analytical methods of quantifying the stress and deformation characteristics of particle aggregates. This will be done in the context of statistical ensembles of heterogeneous particle aggregate models of the particulate media mesoscale. From statistical averaging of the ensembles, homogenized probability distributions of stress and deformation may be obtained from the micro-characteristics of random particle aggregate realizations.

KEY WORDS: DEM,mesoscale,Hill condition,microbound

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. The aspect of the framework tackled in this work, is the quantification of stress and deformation of heterogenous DE models. Such quantification is necessary to allow DEM to “speak continuum”. DEM is a discrete particle method, wherein

the concept of “stress” does not fit naturally, but where each particle pair is considered to interact through discrete forces. Additionally, DEM is an explicit method, and equivalent continuum deformation must be defined in the context of particle kinematics.

Heretofore, DEM implementations have addressed “continuum-ization” by defining virtual simulation “walls”, with the stress defined as the average force over the wall area, while the deformation has been defined as Cartesian strains measuring the movement of the walls and comparing them with their initial positions. In a recently developed approach, being prepared for publication, the virtual “walls” are replaced with a continuous surface tracking approach. The continuous surface tracking approach, briefly described in a following section, has proven remarkably robust, but it is not the focus of the current work. The current work focuses instead on the step following imposition of the natural boundary conditions, which is to quantify the Cauchy stress, as well as the deformation gradient, from the particle forces and kinematics.

The Discrete Element Method All DEM simulations performed in the current work are

carried out using the YADE code [1]. Figure 1 shows a DEM model from Yade used in the current work.

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 [2], which is valid for elastic bodies in contact. The elastic normal force in the Hertzian solution is expressed as

(1)

where and 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.

(2)

with being the equivalent shear modulus. Finally, the normal and tangential contact stiffnesses can be written as

(3)

(4)

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

(5)

where denotes the shearing force, denotes the normal

force, and denotes the inter-particle friction angle.

The Hill-Mandel Macrohomogeneity Condition The primary assumption of any effort to quantify the

behavior of a material at the macroscale, is the existence of a Representative Volume Element (RVE). Herein, the RVE is understood in the sense of the Hill-Mandel macro-homogeneity condition [3]. Identification and determination of the RVE is rather complicated when dealing with random heterogeneous media such as soils. Mathematically, an infinite number of grains would be required to attain the RVE scale, due to the complete randomness and heterogeneity of the grains [4].

In addition to the concept of the RVE, it is also beneficial at this point to introduce the related concept of the Statistical Volume Element (SVE). The SVE is sometimes also called the Stochastic Volume Element, due to the randomness observed in the behavior at this scale. The SVE represents all realizations of the microscale greater than the typical grain, but smaller than the RVE. Where the RVE material properties are the same as those of the macroscale, the SVE properties are represented by statistical distributions.

Consider a body 𝐵"(𝜔) not under the action of any body forces or inertia and with stress and strain fields σ and ε due to some arbitrary boundary conditions. The stress/strain fields may

be represented as a superposition of their means ( and ) and trivial fluctuations ( and )

(6)

The volume average of the energy density over 𝐵"(𝜔) may then be written as:

(7)

The Hill Condition [3] means the average of a scalar product of stress and strain fields is equal to the product of their averages

(8)

which requires

(9)

Rewriting (9) in indicial notation and applying the divergence theorem

(10)

The triviality of the fluctuations is clearly satisfied for the macroscale Lmacro; however, for a finite mesoscale the Hill Condition is satisfied if and when Equation (11) holds

(11)

The condition is satisfied by three different types of uniform boundary conditions for heterogeneous media:

1. uniform displacement (Dirichlet, kinematic, KUBC) boundary condition

2. uniform traction (Neumann, static, SUBC) boundary condition

3. uniform displacement-traction (orthogonal mixed, MUBC) boundary condition

Boundaries of Particle Aggregates In Continuum Mechanics, there are two classical types of

boundary conditions: (1) the displacement boundary condition, also known as a homogeneous deformation boundary condition, and (2) the traction boundary condition, also known as a uniform stress boundary. These boundaries arise from analogous boundary conditions in Differential Equations; the displacement boundary is the continuum mechanics application of the Dirichlet boundary condition, while the traction boundary is the continuum mechanics application of the Neumann boundary condition. In particle methods with discrete bodies, such as the Discrete Element Method, these boundaries have heretofore not been rigorously implemented. Instead, they have been

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approximated using particles of infinite radii as boundary “walls”.

For most applications of the Discrete Element Method in the literature thus far, these approximations of boundary conditions are quite reasonable and sufficient. This is because DEM is typically used as a macro-scale simulation method, rather than as a micro-scale method within a hierarchical multiscale homogenization framework. The latter requires that upper and lower response bounds be considered, which dictates that the Dirichlet and Neumann boundaries must be implemented.

This last point has been recognized by several other authors in recent years. Dettmar [5] and later Miehe et al. [6] implemented micro-mechanical boundaries for Discrete Elements using a penalty method, whereby those particles designated as “boundary particles” receive increasing forces and moments if the boundary conditions are “violated”. Liu et al. [7] applied micro-mechanical boundaries for 2D Discrete Elements by implementing a servo-control mechanism at each “boundary particle”, with special treatment given to “corner particles”.

This limits the utility of the approach, since particles that are interior particles at the start cannot become boundary particles, nor can single boundary particles become multi-boundary particles or vice-versa. Furthermore, and perhaps most importantly, this approach limits the range of deformation that can be modeled. Therefore, the need for a 3D implementation of micro-mechanical boundaries in the Discrete Element Method is quite clear.

Surface Reconstruction of Particle Aggregate Boundaries with Alpha Shapes

The definition of a boundary for free-floating particles in space, also known as a “point cloud”, requires some sort of surface reconstruction algorithm. Serendipitously, surface reconstruction from point clouds has been an area of intensive research and progress in recent years. This has been spurred by rapid developments in 3D scanning and photogrammetry. There are several approaches used to extract an exterior surface from a point cloud.

Discrete Element Models do pose an additional challenge for surface reconstruction algorithms; the discrete particles are actually spheres with radii, surface area, and volume. This complication can be resolved by using “weighted” alpha shapes, that account for the radii of the DEM particles. The alpha-shape associated with a set of points is a generalization of the concept of the convex hull [8].

For each real number α, define the concept of a generalized disk of radius 1/α as follows:

• If α = 0, it is a closed half-plane (convex hull) • If α > 0, it is closed disk of radius 1/α • If α < 0, it is the closure of the complement of a

disk of radius −1/α

Weighted alpha shapes on a regular triangulation of the DEM particles can be used to construct a Laguerre-Voronoi Diagram (a.k.a. Power Diagram) of the boundary [9]. The LV

Diagram essentially is a map on the DE model boundary of the boundary areas belonging to each particle determined by the weighted alpha shape algorithm to be a boundary particle. Figure 2 shows such a diagram on a sample DE model of 10000 particles.

Figure 2: Laguerre-Voronoi Diagram on a DE Model

Surface Reconstruction for Stress and Deformation Quantification

A highlight of this novel micromechanical boundary implementation is that the boundary particles may become interior particles and vice-versa. While this makes micromechanical boundaries more versatile, it does increase computation complexity for original-to-deformed mapping schemes such as the deformation gradient. This complexity is herein resolved by introducing into the computation a fixed rectangular grid which is “fitted” to the particle aggregate boundary at either configuration. The interpolated regular surface grid makes it possible to compute stress and deformation by providing matching reference points for the current as well as the reference configuration. Figure 3 illustrates the fixed regular grid on the sample DE model of Figure 2.

Figure 3: Interpolation of a Regular Grid (right) onto the

Laguerre-Voronoi Diagram of the DE Model (left) The fitting of this regular grid is done by 1) translating the

coordinates of the LV Diagram into spherical coordinates, 2) translating the coordinates of a surface grid from a tessellation of a unit cube into spherical coordinates, 3) linear interpolation of the radii of the unit cube from the LV Diagram, 4) conversion of the interpolated spherical coordinates back to Cartesian coordinates.


Deformation Gradient from Particle Kinematics The motion of a body may be described as a transformation

defined by:

(12)

where x represents the current location of a given material point at time t, while X represents the material point in the reference configuration. If the concept of a “material point” is then extended to a discrete material particle, the transformation may then be defined as a non-linear deformation map to the current configuration:

(13)

where is the second-order particle transformation tensor, P are the interior particles in the aggregate, and Q are the boundary particles in the aggregate. The current positions of the particles may then be expressed as:

(14)

where is the macroscopic deformation gradient, while represents the displacement of particle i with respect to the macroscale current or deformed configuration.

The macroscopic deformation gradient is defined by the volume average of the macro-scale deformation:

(15)

Recall that the deformation gradient is defined as:

(16)

which may be rewritten in gradient notation as: (17)

Substitution of Equation (17) into Equation (15), followed by application of the divergence theorem, gives:

(18)

where is the unit normal vector at the outer boundary for the

reference configuration. Defining as the area

vector, then for a discrete setting Equation (18) is equivalent to:

(19)

which may be described as the volume average of the tensor products of the area vectors in the reference configuration and the particle position vectors in the current configuration.

Love-Weber Average of the Cauchy Stress Tensor In continuum mechanics, instead of a discrete particle, the

fundamental analysis unit is the presumably infinitesimal “material point”. A fundamental definition of the Cauchy stress for a material point arises naturally from the balance of momentum equation:

(20)

where denotes the Cauchy stress tensor, is the velocity, is the body force density, and is the mass density.

The volume average of the Cauchy stress tensor may be defined as:

(21)

Rewriting Equation (21) in indicial notation and noting that where is the Kronecker delta:

(22)

Furthermore, since Equation (22) may be rewritten as:

(23)

Using the chain rule, Equation (23) may be written as:

(24)

Applying the divergence theorem to the first term in the integral of Equation (24) gives:

(25)

Taking Equation (20) into account gives:

(26)

Furthermore, since at all points on the boundary, where is the exterior force, then:

(27)

where it becomes obvious that the average Cauchy stress tensor is composed of two parts.

The first integral on the right side of Equation (27) represents the static component of the stress, involving the forces applied at the boundary. The second integral is an inertial term,

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representing the acceleration of each material point. For static and quasi-static analyses, the second term may be ignored; however, it needs to be considered for transient dynamic analyses.

For the case of a particle aggregate with P interior particles and Q boundary particles, Equation (27) may be written as:

(28)

For a particle aggregate in equilibrium when subjected to external forces , the second term vanishes, giving the classical Love-Weber formula of the average Cauchy stress tensor [10, 11]:

(29)

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) does 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, sometimes fail to 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 [12]. 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, with a sufficiently large number of concurrent models of finite sizes, it 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.

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 perhaps not essential to the current effort, which is mostly geometric and analytical in nature; however, it is crucial to the evaluation of the overall framework the current effort is part of.

Figure 4 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 [13-19].

Figure 4: PSD of Eglin Sand

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

of 1x108 particles, hereafter referred to as the “source” model. For reference, Figure 1 shows a DEM packing of 1x104 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.

( ),

1 1

1 1q

Q P Qext q q

ij i j i i jVq q

dVV V

r+

= =

= - -å åòσ f x v γ x!

,ext qif

,

1

1 QLW ext q qij i j

qV =

= åσ f x


Table 1: Eglin Sand Parameters

Parameter Distribution Shape/StDev Scale/Mean

Particle Size Weibull 2.59 0.428 (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 The ensemble used here consists of 22000 DEMs. Each

realization is confined to three different mean stress targets of 5MPa, 20MPa, and 50MPa. Figure 5 shows the micromechanical stresses in a typical 1000-particle DEM at confinements of 5 MPa and 50 MPa.

Figure 5: Micromechanical Mean Stress in a DEM of 1000

Spheres at a Confinement of a) 5 MPa b) 50 MPa Figure 6 shows both the Laguerre-Voronoi diagram, as well

as the interpolated surface grid corresponding to the DE models of Figure 5. Each of these models is then loaded along three different paths using both the uniform displacement as well as the uniform traction boundary conditions. The loading paths used are:

1) Isotropic Compression – the forces/velocities on the boundary particles are increased until a volumetric strain of about 0.5% is achieved. The slope of the mean stress vs. volumetric strain curve gives the bulk modulus.

2) Uniaxial Stress – the forces/velocities on the boundaries are adjusted to achieve an axial strain of about 0.8%, with a constant lateral stress. The slope of the principal stress difference vs. axial strain gives the Young’s modulus.

3) Deviatoric Stress – the forces/velocities on the boundaries are adjusted to achieve a principal stress difference increase while maintaining a constant mean stress with a target deviatoric strain of about 0.6%. The slope of the deviatoric stress vs. deviatoric strain curve gives the shear modulus.

Figure 6: Laguerre-Voronoi Diagram (bold) and

Interpolated Surface Grid at a) 5 MPa b) 50 MPa The stresses and strains are computed from the Love-Weber

averages of the Cauchy stress and the averaged deformation gradient as indicated in the preceding sections. The deformation gradient tensor yields:

(30)

(31)

(32)

where is the volumetric strain, is the axial strain, is the deviator strain, and is the principal strain vector. Similarly, from the Cauchy stress tensor:

(33)

(34)

where is the mean (hydrostatic) stress, is the deviator strain, and is the principal stress vector.

RESULTS Shown in this section are ensemble average results. Each of

the three moduli—the bulk modulus, the Young’s modulus, and the shear modulus—were evaluated for each realization in the ensemble at confinements of 5MPa, 20 MPa, and 50 MPa.The mean modulus value is shown for the displacement and the traction boundary, along with the 5% confidence interval for the traction boundary and the 95% confidence interval for the displacement boundary. Also shown is the fit from the Eglin sand model identified by Martin and Cazacu [14] based on their triaxial testing of the sand.

Figure 7 shows the bulk modulus results for the ensemble under isotropic compression loading simulations. Figure 8 shows the Young’s modulus results for the ensemble under uniaxial stress loading simulations. Figure 9 shows the shear modulus results for the ensemble under deviatoric stress loading simulations.

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( )13

LWHyd trs = σ

1 3devs = -S S

Hyds devsS

a) b)

a) b)


Figure 7: Bulk Modulus Ensemble Statistics

Figure 8: Young’s Modulus Ensemble Statistics

Figure 9: Shear Modulus Ensemble Statistics

The means and bounds of the moduli compare well with the sand model of Martin and Cazacu [14] at lower confinement levels. At higher confinements, the ensemble simulation under-predicts the magnitudes of the moduli. This reflects the evolution of the PSD of sand at high confinement stresses due to bulk dissipative effects such as particle fracture, abrasion, and crushing. A mesoscale model of sand needs to account for these bulk dissipative effects to accurately model the evolution of sand at these higher confinement levels. This, along with quantification of deviatoric dissipation, will be addressed in ongoing work focusing on ensemble thermomechanics of sand using DE modeling.

CONCLUSIONS A method for obtaining the stress and deformation tensors

for heterogeneous particle aggregates 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 quantified through the volume averaged Cauchy stress tensor and deformation tensor.

Hundreds of thousands of simulations were carried out using random samples of spherical particles using the Discrete Element formulation. The Cauchy stress tensor and the deformation tensor were successfully obtained for the ensemble using two different surface reconstruction methods, along nine distinct loading paths, subjected to two different boundary conditions.

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.

REFERENCES [1] V. Šmilauer, E. Catalano, B. Chareyre, S. Dorofeenko, J. Duriez, A. Gladky, J. Kozicki, C. Modenese, L. Scholtès, L. Sibille, Yade reference documentation, Yade Documentation 474 (2010). [2] R. Mindlin, Compliance of elastic bodies in contact, Journal of applied mechanics 16 (1949). [3] R. Hill, Elastic properties of reinforced solids: some theoretical principles, Journal of the Mechanics and Physics of Solids 11(5) (1963) 357-372. [4] M. Ostoja-Starzewski, Microstructural randomness and scaling in mechanics of materials, CRC Press2007. [5] J.P. Dettmar, Static and dynamic homogenization analyses of discrete granular and atomistic structures on different time and length scales, 2006. [6] C. Miehe, J. Dettmar, D. Zäh, Homogenization and two-scale simulations of granular materials for different microstructural constraints, International Journal for Numerical Methods in Engineering 83(8-9) (2010) 1206-1236. [7] J. Liu, E. Bosco, A. Suiker, Formulation and numerical implementation of micro-scale boundary conditions for particle aggregates, Granular Matter 19(4) (2017) 72.


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IMPACT BEHAVIOR OF MICROCRYSTALLINE CELLULOSE AND CORN STARCH BASED WOOD BIO-COMPOSITES UNDER HIGH STRAIN RATE COMPRESSIVE

LOADING

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

Jason Street Mississippi State University

Mississippi State, Mississippi, USA

ABSTRACT Recent research led to the development of lightweight and

biodegradable wood-based bio-composite panels made from Corn Starch (CS), Microcrystalline Cellulose (MCC) and a Methylene Diphenyl Diisocyanate (MDI) resin with different mass fractions. The objective of this study was to study the high strain response of these novel bio-composites using Split Hopkinson Pressure Bar (SHPB) and Digital Image Correlation (DIC) techniques. The strain induced in the specimen under dynamic stress equilibrium was obtained using the SHPB. The dynamic event at the zone of the specimen was captured using a Shimadzu HPV-2 high-speed video Camera. DIC was performed using Proanalyst to obtain the strain in the specimen. Excellent agreement was found between the strains obtained using the SHPB and DIC analysis over three strain rates on two types of bio-composites. The bio-composite produced with cornstarch had the higher ultimate strength and specific energy absorption between two different types of bio-composites tested in this study.

KEY WORDS: Bio-composites, Split Hopkinson Pressure Bar (SHPB), Digital Image Correlation Technique (DIC), High strain rate, dynamic strength.

INTRODUCTION A composite material is a material made from two or more

constituent materials with different properties that, on combination produce a material with desired characteristics that may be at the opposite ends of the spectrum, and may not be attainable by the individual constituent materials. Well-engineered composites provide design engineers with light weight, high specific strength, modulus and long life span.

Bio-composites are composites made from materials occurring naturally, such as plants. Bio-composites have been frequently used for automobiles [2], vibration damping and noise reduction applications [3], and the packaging of nuclear waste [4]. Bio-composites have a wide range of applications due to their renewability, biodegradability, low cost, high strength-to-weight ratio, and customizable static and dynamic mechanical properties. Hence, bio-composites have great capability to replace 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 sustainable and green materials by studying five newly developed wood-based, bio-composites.

When bio-composites are used for structural and nonstructural applications, they are subjected to impact and shock loading, and therefore in selecting a potential bio-composite for a particular application, it is necessary to investigate the impact behavior of material at a high strain rate. SHPB studies have been widely used for investigating the high strain response of bio-composites, including wood composites [4-10]. The SHPB works on the principle of the elastic wave theory, in which the strain pulses of elastic waves captured by strain gauges are attached to the incident and transmission bar for determining the strains generated within the specimen during high strain rate loading.

Gilbertson et al. [6] studied the effect of strain rate on the strength of hard maple and black maple specimens, as well as eastern white spine specimens using an SHPB. They found that the strength of the wood specimens is significantly affected by strain rate.

Shah et al. [11] carried out trials on wood-based, bio-composites using an 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 an extension of the work carried out by Shah et al. [11]. This study focused on the impact behavior of bio-composites using SHPB, and validation of the strain generated using DIC technique.

The DIC technique involves a non-contact optical measurement in which strain is measured directly from the captured images during the test. The DIC technique has been widely used for measuring deformation in many areas of engineering due to advances in high-speed digital cameras and computer processing power. DIC involves using non-contact deformation measurement with digital images of the dynamic event, which are collected at predetermined time intervals. As shown in Fig. 1, the reference image represents the undeformed configuration of the specimen, and the target image represents the deformed configuration of the specimen. Processing of the digital images of the dynamic event allows the software to recognize the stochastic pattern, and the software allocates coordinates to image pixels. The deformation of the specimen is measured from the overlapping digital images at different time intervals for the test event. A subset (rather than individual pixels) is used because a subset having a broad range of gray levels can be unambiguously identified from other subsets in the deformed image of the test event.

Figure 1. Comparison of Reference and Target Images The impact behaviors of wood and wood composite were

investigated using an SHPB and DIC analysis in previous studies [4, 6, 12-14]. However, the high strain rate analysis of the bio-composite made with wood/corn starch, microcrystalline cellulose and a MDI resin has never been studied by using an SHPB and DIC.

In this study, the impact behavior of two novel bio-composites was studied using an SHPB and DIC. The high strain rate compression loading was generated by the SHPB, and strain data was captured by strain gauges mounted on the SHPB bars. Finally, an evolution of strain was validated using DIC software.

MATERIAL The material investigated in this study is a wood-based bio-

composite. The specimens, made from Southern Yellow Pine wood, Corn Starch (CS), Microcrystalline Cellulose (MCC) and a Methylene Diphenyl Diisocyanate (MDI), were prepared using a Dieffenbacher 915 x 915 mm hot press system located at the

Sustainable Bioproducts Laboratory at Mississippi State University; the detailed process of the same has described by Shah et al. [11]. Two materials were studied and compared for their mechanical properties. The first set of panels, “Material 1”, was made of 4% cornstarch and 4%MDI, and the second set, “Material 2”, was made from 2%MCC and 4%MDI. The sampling size for each material was three (3).

EXPERIMENTAL PROCEDURE AND SETUP The high strain rate response, ranging from 940 s-1 to 1455

s-1, of bio-composites was studied using a SHPB at the Blast and Impact Dynamics Laboratory at the University of Mississippi. The SHPB system consists of an incident, a transmission, and momentum trap bars shown in Fig. 2. In SHPB high strain rate testing, the characteristic impedance of the bars and specimen should be compatible for accurate results [5]. Therefore, the bars and striker bar were of an Aluminum 6061-T6 bar used with a diameter of 19 mm to avoid impedance mismatch between the specimen interfaces and bar.

The striker bar was accelerated along the barrel by compressed air. The rectangular bio-composite specimen dimensions were 12.8 mm x 10 mm x 6.35 mm, and the specimen was sandwiched between the incident and transmission bars. The impact of the striker bar at the one end of the incident bar caused a one-dimensional elastic compression wave (i.e., incident wave) to propagate along the incident bar. An annealed copper pulse shaper was inserted between the striker bar and the incident bar to achieve a constant strain rate, slow down the rate of loading, and ramp up the incident pulse. However, a nearly constant strain rate (i.e., dynamic stress equilibrium) is a prerequisite for a valid SHPB test. The one-dimensional elastic compression wave further propagated through the specimen and transmission bar. During the wave propagation, a fraction of a one-dimensional elastic compression wave was transmitted from the incident bar, specimen, and transmission bar, while the wave was reflected back along the incident bar as a tensile wave (i.e., tensile pulse), as shown in Fig. 3.

Figure 2. Schematic of Experimental set-up

Figure 1. (a) Reference Image (b) Target Image


Figure 3. Typical Voltage-Time graph during SHPB

compression

Figure 4. Typical Voltage-Time graph during SHPB

compression

SHPB set up at University of Mississippi

Figure 5. Combined evolution of strain field using SHPB

and DIC

Strain gauges of 350 Ω were mounted diametrically opposite and approximately at the middle of the incident and transmission bar and connected in series to compensate strains due to bending of the bar, and the gauges only measured the axial strain of the bars. A digital oscilloscope model TDS 3014C with preamplifier from Tektronix sampled at a rate of 1.25 GS/s and was connected to strain gauges through a wheatstone bridge. A digital signal conditioner (model 2301B) from Vishay instruments was employed to record the voltage-time history of strain pulses during the dynamic event.

A high-speed video camera (Hadland HPV-2 Shimadzu) at 25,000 frames per second (FPS) was used to capture the images of the dynamic event. The impact event was captured in 102 image-frames. Post analysis DIC was performed to obtain the strain in the specimen. Fig.5 shows the sequence of DIC analysis using the proanalyst software.

Furthermore, varying ranges of strain rates (940 s-1 to 1455 s-1) were obtained by altering the pressure of the compressed air (~ 69, ~103, and ~ 138 kPa) supplied to propel the striker bar of the SHPB.

THEORY OF THE CALCULATION OF STRESS AND STRAIN USING A SHPB

According to mass conservation, the axial displacement u (t) at time t is related to the axial strain history ε(t) by

u (t) = C0∫ ε(t) dt'( (1)

where, C0 = √(E/ρ) (E and ρ is the Young's modulus elasticity and density of Hopkinson bars) is the wave velocity in SHPB.

It follows that the axial displacement at the interface between the incident bar and specimen is given by:

u*(t) = C0∫ ε*(t) dt'( (2)

Identically, the axial displacement is at the interface between the specimen and transmission bar and is given by:

u+(t) = C0∫ ε+(t) dt'( (3)

where subscripts 1 and 2 are the two ends of the specimen. Axial displacement at the interface of the incident bar,

specimen and transmission bar in the terms of the incident (ε-), transmission (ε.), and reflected strain (ε/) pulses are given by:

u*(t) = C0∫ (ε- − ε/)dt'( (4)

u+(t) = C0∫ ε. dt'( (5)

The average compressive axial strain in a specimen of length L can be calculated as

ε1(t) =u* −u+

L (6)

Hence, the average compressive strain in the specimen in terms of strain pulses can be written as

Image Data

Computer

P1 P2

Bio-composite specimen position between the Incident and transmission bar during SHPB test


ε1(t) = C(L6 (ε- − ε/ − ε.)dt(7)'

(

From momentum of conservation, the forces at the end of the specimen are given by

F* = EA(ε- +ε/) and F+ = EAε. (8(a) & 8(b))

where E and A are Young's modulus of elasticity and cross-sectional area of Hopkinson bars.

The average force in the specimen is given by s

F<=>?@A> = B<+(ε- +ε/ + ε.) (9)

It is assumed that F1 = F2 (force equilibrium), that is the forces are equal at both ends of a specimen. Then

ε- +ε/ = ε. (10)

Fig. 4 shows that the dynamic strain pulses on the opposite faces of the bio-composite specimen cotton motes are nearly indistinguishable during the entire period of high strain rate loading. Equation (7) simplifies to

ε1 = −2C(L6 ε/'

(dt(11)

The strain rate and stress in the specimen is given by

εE = −2C(L ε/(12)

σE = EAAEε. (13)

where, AS is the cross-sectional area of the specimen.

RESULTS AND DISCUSSION Strain rate, ultimate stress, maximum strain and specific

energy for the bio-composites at high strain rate were measured; and are presented in Table 1. These values increased with increasing strain rate as in similar studies on wood specimens by Tagarielli et al. [8] , Gilbertson and Bulleit [6], and on wood-based bio-composite particle board by Shah et al. [11] and Dave et al. [14]. Therefore, it can be deduced that the strain rate effect is noticeable on the mechanical behavior of the bio-composites.

Figs. 6-8 show the stress-strain behavior of bio-composites with stresses obtained using elastic wave theory in SHPB experiments at striker bar pressure of ~ 69, ~103 and ~ 138 kPa respectively.

Figs. 6-8 show that the bio-composite specimen Material 1 had the highest ultimate strength, yield strength and maximum strain among all tested bio-composites at three different strain rates. This increment in mechanical properties of Material 1 could be attributed due to increasing volume fraction of CS as compared to MCC.

Table 1: Dynamic properties of bio-composite specimens

Material Strain Rate

(s-1)

Ultimate

Stress

(MPa)

Max.

Strain

Specific

Energy

(kJ/kg)

Material 1

~ 1100 44.10 0.28 14.15

~ 1245 50.89 0.37 19.87

~ 1455 59.28 0.43 25.68

Material 2

~ 940 39.50 0.28 12.53

~ 1160 41.75 0.35 16.63

~ 1420 46.52 0.43 21.75

Figure 6. Dynamic stress-strain curves of bio-composites

(Striker bar pressure of ~ 69 kPa) (SHPB)


(Striker bar pressure of ~ 103 kPa) (SHPB)



(Striker bar pressure of ~138 kPa) (SHPB) The validation of the evolution of strain in bio-composite

specimens was done using the 2D DIC commercially available software Proanalyst. Figs. 9-11 present the engineering strain versus time history using the SHPB and 2D DIC at three different strain rates. Figs.9-11 show that strain histories obtained using the SHPB and the DIC technique are consistent with each other. Furthermore, from Figs. 9-11, it is evident that strain increases linearly with time during the event of impact loading. This matches well with the strain obtained from the SHPB and confirms previous findings of Dave et al. [14].

Figure 9. Strain time history of different bio-composites

obtained using 2D DIC technique and SHPB (Striker bar pressure ~ 69 kPa)





It is evident from Table 1 that the bio-composite specimen

Material 1 had the highest average strain rate between two different specimens tested at three different strain rates. The most striking observation to emerge from the data comparison, shown in Table 1, was that the bio-composite specimen with the increased volume fraction of cornstarch (CS) (Material 1) showed an enhanced ultimate strain and average strain rate among the various bio-composites tested. In general, these results suggest that corn starch can be a potential candidate to improve the strain rate of bio-composites. Furthermore, it was observed that bio-composite specimen Material 2 had the lowest strain rate among two different bio-composites tested.


The specific energy absorption (energy absorbed per unit mass) under the SHPB compression loading is shown in Table 1 for three different striker bar loading pressures. The average strain rate had a linear relationship with the energy absorption capacity. Furthermore, results reveal that Material 1 specimens had the highest energy absorption capacity at three different strain rates among all tested bio-composites. Moreover, the Material 2 sample had the lowest specific energy two different types of bio-composites tested. This is in good agreement with Shah et al. (2017).

CONCLUSIONS Two different types of wood-based bio-composites were

tested using a SHPB and the digital image correlation (DIC) technique. The DIC technique is a non-contact measurement technique, which reveals the evolution of strain throughout the samples under high strain-rate tests. Results indicate that the material behavior is dependent on the strain-rate. The results obtained show that reliable results can be achieved by using a combined analysis procedure, consisting of classical Split-Hopkinson Bar Analysis (SHPBA) and Digital Image Correlation (DIC). DIC has the potential for quality analysis applications. Corn starch marginally increased the mechanical properties as compared to Microcrystalline cellulose.

ACKNOWLEDGEMENTS We would like to express our gratitude to Dr. P. Raju

Mantena, Damian Stoddard and Sumanbabu Ukyam for their involvement throughout this research and Matthew Lowe for the machining of the wood-based bio-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 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] Swabarna Roy, Honey Mishraand B. G. Mohapatro, 2016, “Creating Sustainable Environment using Smart Materials in Smart Structures”, Indian Journal of Science and Technology, 9(30), pp. 1-9. [2] Koronis, G., Silva, A., Fontul M. 2013, “Green composites: A review of adequate materials for automotive applications”, Composites Part B: Engineering 44 (1), pp. 120-127. [3] Mohanty, A., Fatima S., 2015, “Noise control using green materials”, Sound and Vibration Magazine 49(2), pp. 13-15. [4] Bragov, A., Lomunov, A., 1997, “ Dynamic Properties of Some Wood Species”, J. Phys. IV France 07, pp. C3-487-C3- 492.

[5] Widehammar, S., 2004, “Stress-Strain Relationships for Spruce Wood: Influence of Strain Rate”, Moisture Content and Loading Direction”, Society for Experimental Mechanics, 44 (1), pp. 44-48. [6] Gilbertson, C.G., Bulleit, W.M., 2013, “Load duration effects in wood at high strain rates. Journal of Materials in Civil Engineering”, 25(11), pp. 1647-1654. [7] Renaud, M., Rueff, M., Rocaboy, A.C., 1996, “Mechanical behaviour of saturated wood under compression”, Wood Science and Technology, 30(3), pp. 153-164. [8] Tagarielli, V.L., Deshpande, V.S., and Fleck, N.A., 2008, “The high strain rate response of PVC foams and end-grain balsa wood”, Composites Part B: Engineering, 39 (1), pp. 83- 91. [9] Allazadeh, M.R., and Wosu, S.N., “High Strain Rate Compressive Tests on Wood,” An International Journal of Experimental Mechanics, 48(2), pp.101-107. [10] Vural, M., and Ravichandran, G., 2003, “Dynamic Response and Energy Dissipation Characteristics of Balsa Wood: Experiment and Analysis,” International Journal of Solids and Structures, 40, pp.2147–2170. [11] Shah, A., Pandya, T.S., Stoddard, D., Ukyam, S., Street, J., Wooten, J., and Mitchell, B., 2017, “Dynamic Response of Wood based Bio Composite under High Strain Rate Compressive Loading,” Wood and fiber science, V - 49(4). [12] Cofaru, C., Philips, W., VanPaepegem, W., 2010, “Evaluation of digital image correlation techniques using realistic ground truth speckle images” J Meas Sci Technol. 21(5), pp. 1–17. [13] Moilanen, C.S., Saarenrinne, P., Engberg, B.A., and T Björkqvist, T., 2015, “Image-based stress and strain measurement of wood in the Split-Hopkinson pressure bar”, Measurement Science and Technology, 26(8), pp.1-12. [14] Dave, M.J., Pandya, T. S., Stoddard, D., Street, J., 2018, “Dynamic characterization of biocomposites under high strain rate compression loading with split Hopkinson pressure bar and digital image correlation technique”, International wood product journal, pp. 1-7.

ectc section 4 cover · from the shear force and bending moment diagrams, it is clear that there is...

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