UAB - ECTC 2016 JOURNAL- Section 01 Page w footer€¦ · STUDY OF VARIATION IN FUNDAMENTAL...

UAB School of Engineering - Mechanical Engineering - Early Career Technical Journal, Volume 16 Page113

SECTION 5

Materials

UAB School of Engineering - Mechanical Engineering - Early Career Technical Journal, Volume 16 Page114

UAB School of Engineering - Mechanical Engineering - Early Career Technical Journal, Volume 16 Page 115

Journal of UAB ECTC Volume 16, 2017

Department of Mechanical Engineering The University of Alabama, Birmingham

Birmingham, Alabama USA

STUDY OF VARIATION IN FUNDAMENTAL FREQUENCIES OF BIO-COMPOSITES DUE TO STRUCTURAL DAMAGE

ABSTRACT A shift in natural frequency has been predicted for a bio

composite when damage was induced by experimental modal analysis and finite element analysis (FEA). The behavior of the bio composite produced from southern yellow pine (SYP), 4% sweet potato (SP) particles (solids basis) and 4% MDI (solids basis) was compared with a commercial southern yellow pine composite on the basis of the natural frequency when a known damage was induced in the material. Initially the natural frequencies for the damaged and undamaged specimens were obtained through experimental modal analysis. To verify the experimental results, FEA was completed. The FEA results and the experimental results were in agreement with maximum error of 6%. When damage was induced in the SYP-SP composite, the shifts for first three natural frequencies were -0.075%, 0.6%, and 1.2%, respectively and -0.5%, -5.12%, -19% respectively for the commercial SYP composite.

KEYWORDS: Natural Frequency, Damage, Experimental modal analysis, Poisson’s ratio, FE analysis, bio composites, southern yellow pine, sweet potato

INTRODUCTION Every structure and machine has a natural frequency. If

machines and structures are designed correctly, these natural frequencies will not affect the operation or reliability of the machines. The role of composite materials increases continuously because of their high mechanical characteristics to weight ratio. Extensive research has been done in the area of bio composites because this area has many advantages including the low manufacturing costs involved in producing final products and the production of highly valued products

from low-value materials. Producing biodegradable products has become a necessity due to petroleum-based products being a finite resource and having more potential to harm the environment. Shah et al. [1] studied the behavior of 9 different bio composites under compressive loading and found how different constituents used in producing a bio composite can change the dynamic behavior of the material. Many researchers have detected shifts in the natural frequency when there was damage in the structure or material. I Negru et al. [2] studied how transverse cracks in structures affect theirstiffness as well as natural frequency. Ján Svoreň et al. [3]studied how the natural frequencies of the circular blade canbe obtained through various methods. Rangel et al. [4]analyzed the shift in natural frequencies of structures andproposed the best method to repair damaged bridges. Magaraet al. [5] proposed a theoretical method to find the naturalfrequency for centrifugal pumps which were verified withexperimental methods. Their results indicated that for high-density centrifugal compressors, some natural frequencies ofan impeller increased with a high-density gas.

The modal analysis offers the likelihood of a global approval for a whole structure. Initially experimental modal analysis was carried out using a cantilever beam vibration technique. Many composites have been studied for their vibrational behavior [6-8]. To gauge the mechanical properties and physical properties of wood and wood-based composites, vibrational methods have been successfully used for many decades in the forest product industry and scientific research [9-11]. Previously, natural frequency and the dynamic Young’s modulus have been found using vibration beam technique. Zeng Wany et al. [12] obtained the dynamic Young’s modulus and damping ratio through the same technique and compared them for three commercially available wood- based composites.

Tejas S Pandya University of Mississippi Oxford, Mississippi, USA

Aarsh ShahUniversity of Mississippi Oxford, Mississippi, USA

Brian Mitchell Mississippi State University

Mississippi State, Mississippi, USA

Jason StreetMississippi State University

Mississippi State, Mississippi, USA


Finite element analysis (FEA) has been previously used to compute M and K matrices to determine the mode shape and natural frequencies. ABAQUS is a commercially available FEA code that has been used by many researchers to compute such parameters. Bimal Bastin et al. [13] used ABAQUS to obtain the static displacement and principle stress of a wheel rim. Zhou Kaihong [14] studied modal analysis in ABAQUS on an oval gear. K. Kharate et al. [15] predicted squeal and studied natural frequency, and mode shape of a rotor brake using experimental modal analysis and finite element analysis. They concluded that the material properties of a brake disk play an important role towards vibration mode patterns. W. R.Taylor et al. [16] determined the elastic constants of an orthotropic bone using FEA and Modal Analysis.

Various studies have been carried out to predict a shift in natural frequency, change in stiffness due to damage, and other varying parameters like density of the structure using various methods. This study precisely uses experimental modal analysis to predict a shift in the natural frequency of the material due to damage and to compare the shift with that of the SYP-SP bio composite with a commercial SYP composite.

MATERIALS Specimens were created from primarily southern yellow

pine (SYP) (2-3 mm particle size), 4 wt% (solids basis) sweet potato (SP) particles (~1 mm particle size), and 4 wt% (solids basis) methylene diphenyl diisocyanate (MDI) resin. MDI is an aromatic diisocyanate and was an efficient binder that has been used within the production of composite wood products for over 30 years. The SYP was mixed in a rolling mixer as MDI resin was sprayed into the mixer with a paint sprayer. Before spraying the MDI, precautions were taken by using PPE, which covered the researcher, including a full face mask with a P-100 activated carbon filter, which was discarded after 4 hours of being opened from the original package and a new filter was used after that point.

The amount of mass used to create the panels was the same for each panel (approximately 2.95 kg). The 22” x 24” panels were manufactured by using a mold and half of the SYP/MDI was put into the mold and leveled to form a level layer. Next the SP particles were shaken onto the top of the layer, and then the remaining SYP/MDI was input into the mold. The material was leveled and pressed with approximately 50 lbf to produce a mat. Next, the mat was put on two steel platens that were sprayed with a silicone based release agent.

A Dieffenbacher 915 mm x 915 mm hot press system located at the Sustainable Bioproducts Laboratory at Mississippi State University was used to create the 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 parameters used to form the 22” x 24” panels included a temperature of approximately 185oC, a mat pressure of approximately 8.5 MPa and a total moisture content of 6 wt%. The Dieffenbacher

hot press formed the panel based on the desired thickness of 0.25 inches. The 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. Is the resin fully sealed pores in the wood?

Resin does not fully seal pores in the wood; p-MDI’s isocyanate groups can react with hydroxyl groups in wood and moisture to form irreversible urethane linkages. p-MDI penetrates into the amorphous regions of the wood that furnish cell walls, interweaving on the molecular level until it “plasticizes” the wood cell wall itself. p-MDI’s thermosetting linkage is irreversible, giving the board a degree of moisture resistance. This helps to make the MDI-bonded boards perform better when exposed to moisture.

SPECIMEN PREPARATION Specimens of both the commercial SYP composite and

the SYP-SP based bio composite were prepared. The specimens measured 226.7mm x 25.8 mm x 6.64 mm. The specimen that was damaged had the damage induced from a distance of one inch from the free end.

EXPERIMENTAL PROCEDURE AND SETUP The natural frequencies of damaged and undamaged

commercially available SYP and the laboratory produced SYP-SP bio composite were obtained through experimental modal analysis (cantilever beam vibration technique), and the dynamic young’s modulus was obtained from it. Separate tests were carried out to get the poison’s ratio of the material. All these parameters were imported into FEA software, and natural frequencies for both cases (damaged and undamaged) were obtained and thus verified.

Frequencies for the damaged and undamaged specimens were obtained through vibration beam technique. The setup is situated in the structure and dynamics lab of the University of Mississippi. Figure 1 shows the schematic of the setup.

Figure 1: Schematic of cantilever beam vibration

technique through hammer excitation

An impact hammer was used to excite the beam. The accelerometer detected the response of the beam. Time-


dependent responses from the impact hammer and accelerometer were amplified through power amplifiers and then converted into Frequency Response Function (FRF) in the Dynamic signal analyzer (Hewlett Packard, Model #35665A). This analyzer displayed the imaginary component of the FRF magnitude. The peak amplitudes of this display were plotted accordingly to determine three different mode shapes of the beam at three respective natural frequencies.

The Poisson’s ratio of the material was found in accordance with the ASTM standard E132–04 [17]. The Poisson’s ratios of the two materials were found by testing the materials in tension in a material testing system (MTS) machine (model LVDT, serial number 10252898) situated in the structures and dynamic laboratory of the University of Mississippi. Tension was applied in the longitudinal direction of the specimen. An extensometer (model 354-025M-025-ST) which had a gauge length of 25mm, was used to measure the strain in the longitudinal and transverse direction of the material.

Initiating finite element analysis in ABAQUS, the cantilever beam was drawn to the specifications outlined in the material section, and other properties were taken from the two experiments mentioned above. After applying the material properties, the following issues were addressed: boundary conditions, mesh resolution, and mesh refinement. When modeling the cantilever beam, the first assumption included the beam being fixed in an encastre condition to replicate the experimental conditions. Using this assumption, the necessary boundary conditions were met. To validate the ideal 1D conditions, an in-plane boundary condition was applied. However, the in-plane boundary condition constrained the beam and yielded no results. After yielding no results, the decision to only use the encastre condition was made.

After determining the boundary conditions, a mesh was applied. Initially, a default mesh was used, and the simulation poorly depicted the experimental results. The coarse mesh yielded uneven displacement contours and an incorrect visual representation of the problem. In the cantilever beam application, a coarse mesh caused poor resolution and artifacts in the FEA software. This error led to the decision to further increase the mesh size. After increasing the mesh size, the displacement contours improved substantially. Experimental Results and Analysis

The dynamic Young’s modulus was obtained from the cantilever vibration beam technique. They were Eundamaged =2.25 GPa and Edamaged=2.24 Gpa for the commercial SYP composite. The SYP-SP composite had a modulus of Eundamaged =3.04 GPa and Edamaged=3.02 GPa for the undamaged and damaged sample, respectively. Densities for undamaged and damaged specimens are shown in Table 1.

Table 1: Densities of Materials before and after Damage. Material Undamaged Damaged SP-Wood 910.724 kg/m3 906.33 kg/m3

Commercial Wood 773.273 kg/m3 768.7 kg/m3

Table 1 shows that the density of each material decreased with the induction of damage. The density of SYP-SP composite was much higher than the commercial SYP composite such that there was more fiber concentration in the material.

Figure 2 shows three mode shapes obtained through the cantilever beam vibration technique, and these mode shapes show how the material vibrated at resonance. Resonant vibration is mainly caused by an interaction between the inertial and elastic properties of the materials within a structure. The displacement of the SYP commercial composite for mode one was almost constant, while the SYP-SP composite varied. When these materials were excited and their excitation matched their first natural frequency, this vibration pattern occurred.

The displacement profile for the second mode was almost the same for both materials, while the third mode displacement contour for the SYP-SP composite was not as smooth as the SYP commercial composite, but there were node points at similar locations.

Figure 2: Mode shape comparison of commercial wood and SP-wood obtained through cantilever beam vibration

technique

The Poisson’s ratios of the material were 𝜇𝜇𝑆𝑆𝑆𝑆−𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 =0.104, 𝜇𝜇𝐶𝐶𝑊𝑊𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 0.04. All these parameters were imported into ABAQUS and natural frequencies for both damaged and undamaged cases were obtained


Figure 3: Comparison of natural frequency of undamaged materials.

Figure 4: Comparison of natural frequency of damaged and undamaged SP-wood.

Figure 5: Comparison of natural frequency of damaged

and undamaged commercial wood. The graphs shown in Figures 2, 3, 4, and 6 are scaled by

1:2 to give better clarity, while Figure 7 is re-scaled. Figure 2 shows that the natural frequencies of the SYP commercial composite increased as the order of the mode increased in comparison to the SYP-SP composite, although their initial frequencies were similar. This also shows that the variation in natural frequencies was seen more in the commercial composite rather than in the SYP-SP composite.

The SYP-SP composite showed more sensitivity towards damage in comparison to the SYP commercial composite when comparing the damaged v/s undamaged results shown in Figures 3, 4, and 5. The first three natural frequencies obtained for the SYP-SP composite were 24.925 Hz, 173.75 Hz and 492.5 Hz, respectively. After damage was induced, a change of -0.075%, 0.6%, and 1.2% was realized, respectively. As far as the commercial SYP composite was concerned, the first three natural frequencies obtained were 25.75 Hz, 162.38 Hz and 454.12 Hz, respectively. When damage was induced there was a change of -0.5%, -5.12%, and -19% respectively. The change was obtained by equation 1, where 𝑓𝑓𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈 is the experimental frequency of undamaged material and 𝑓𝑓𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈is the frequency of damaged materials. This shows that the commercial SYP composite had more sensitivity towards damage.

∆𝑓𝑓(%) = �𝑓𝑓𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈−𝑓𝑓𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈

𝑓𝑓𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈� ∗ 100 (1)

The results here clearly depict a frequency change which shows that damage in the material is affecting its performance.


Figure 6: Comparison of frequency of all cases

Figure 7: Comparison of frequency of all cases-scattered

0

100

200

300

400

500

600

1 2 3

Freq

uenc

y (H

z)

Order of the Mode

Comparison of Frequncy of all cases

Undamaged Sweet Potato-wood Damaged Sweet Potato Wood

Undamaged Commercial Wood Damaged Commercial Wood

0100200300400500600700800900

100011001200130014001500160017001800190020002100

1 2 3

Freq

uenc

y (H

z)

Order of the Mode

Comparison of Frequency of all cases- Scattered

Undamaged Sweet Potato-wood Damaged Sweet Potato Wood

Undamaged Commercial Wood Damaged Commercial Wood


Figure 8: FEA Analysis

Table 2: Comparison of experimental data with FEA data

Table 2 is derived from the FEA analysis shown in Figure 8. Table 2 shows the agreement between the experimental data and the FEA predictions. A maximum frequency error of 6% was found over the first three mode shapes. The error was calculated using equation 2 below:

𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸(%) = �𝑓𝑓𝑈𝑈𝐸𝐸𝐸𝐸−𝑓𝑓𝐹𝐹𝑈𝑈𝑈𝑈

𝑓𝑓𝑈𝑈𝐸𝐸𝐸𝐸� ∗ 100 (2)

where fEXP is the experimental frequency found while performing the experiment, and fFEA is the frequency found after running the FEA simulation.

The variation in the results between the FEA predictions and the experimental data may be because when slight damage occurred, the domain signal from the dynamic signal was discrete, as there were intervals among the spectrum lines so that the change of the natural frequency could not be reflected in FRF. The damaged sample’s signal could have been submerged by the error caused by the spectrum interval; therefore, when the variation of the natural frequency was small, the damage calculated deviated much more from the practical condition


CONCLUSION: This study compared the material behavior of a

commercially available southern yellow pine (SYP) particle board composite and a SYP-SP (sweet potato) composite when damage was induced in them using the parameter of natural frequency. Based on above results and discussion it can be stated that the commercial SYP composite showed more sensitivity towards damage in comparison to the SYP-SP composite. The natural frequencies of SYP-SP did not vary as much as SYP commercial composite when the order of the mode increased. When damage was induced in the SYP-SP composite, the shift for first three natural frequencies were -0.075%, 0.6%, and 1.2% respectively the shift for the SYP commercial composite were -0.5%, -5.12%, and -19% respectively.

ACKNOWLEDGEMENTS: This material is supported by the National Institute of Food and Agriculture, U.S. Department of Agriculture, and McIntire Stennis under accession number 1009735. I would like to thank Dr. P. Raju Mantena for the lab equipment, Damian Stoddard for the help with experiments and Matthew Lowe for the help with preparation/machining of the specimens. REFERENCES [1] Aarsh Shah, Tejas S Pandya, Damian Stoddard, Suman Babu Ukyam, Jason Street, James Wooten, Brian Mitchell; “Dynamic response of wood based bio-composites under high-strain rate compressive loading.” Journal of Wood and Fiber Science, 2017. [2] “Natural frequency changes due to damage in composite beams”, I Negru, G R Gillich1, Z I Praisach1, M Tufoi, N Gillich. 11th International Conference on Damage Assessment of Structures (DAMAS 2015), IOP Publishing, Journal of Physics: Conference Series 628 (2015) 012091 doi:10.1088/1742-6596/628/1/012091 [3] Ján Svoreň, Ľubomír Javorek, Adam Droba, Dušan Paulíny; “Comparison of Natural Frequencies Values of Circular Saw Blade Determined by Different Methods”. doi:10.5552/drind.2015.1316. [4] Jaime Horta-Rangel, Socorro Carmona, Victor M. Castaño, (2008) "Shift of natural frequencies in earthquake-damaged structures: an optimization approach", Structural Survey, Vol. 26 Issue: 5, pp.400-410, https://doi.org/10.1108/02630800810922748Carrera E 2004 [5] “Natural Frequency Shift in a Centrifugal Compressor Impeller for High-Density Gas Applications”; Yohei Magara, Haruo Miura, Naohiko Takahashi, Mitsuhiro Narita; DOI: 10.1115/1.4006423;http://turbomachinery.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/journals/jotuei/926696 [6] Assessment of theories for free vibration analysis of homogeneous and multilayered plates Shock Vib. 11(3-4) 261-70

[7] Hause T and Librescu L 2006 Flexural free vibration of sandwich flat panels with laminated anisotropic face sheets J. Sound Vib. 297 823–41 [8] Gillich G R, Praisach Z I, Wahab M A and Vasile O 2014 Localization of transversal cracks in sandwich beams and evaluation of their severity Shock Vib. 9(4-5) 607125 [9] Moslemi AA. Dynamic viscoelasticity in hardboard. Forest Prod J 1967;17(1):25–33. [10] Ross RJ, Pellerin RF. Nondestructive testing for assessing wood members in structures: a review. USDA Forest service Forest Products Laboratory, General tech. rep. FPL-GTR-70; 1994. [11] Ilic J. Dynamic MOE of 55 species using small wood beams. Holz Roh Werkst, 2003;61(3):167–72. [12] Measurement of dynamic modulus of elasticity and damping ratio of wood-based composites using the cantilever beam vibration technique. Zheng Wanga,, Ling Li, Meng Gong. Journal of Construction and building materials. doi:10.1016/j.conbuildmat.2011.09.001 [13] Finite Element Analysis of Wheel Rim Using Abaqus Software Bimal Bastin, Adars S, Akshay Madhu, Arun Krishnan, Betson Jose, Praveen Raj. Int. Journal of Engineering Research and Application ;ISSN : 2248-9622, Vol. 7, Issue 2, ( Part -1) February 2017, pp.31-34 [14] Non-circular gear modal analysis based on ABAQUS; Zhou Kaihong, Li Yunpeng, Wang Congyi,Li Cheng. 2015 8th International Conference on Intelligent Computation Technology and Automation [15] Nilesh K.Kharate, Dr. Sharad S.Chaudhari; “Investigation of Natural Frequency and Modal Analysis of Brake Rotor Using Fea and Ema”; International Journal of Innovative Research in Science, Engineering and Technology; Vol. 3, Issue 10, October 2014. DOI: 10.15680/IJIRSET.2014.0310013 [16] Determination of orthotropic bone elastic constants using FEA and modal analysis. W.R. Taylora, E. Rolandb, H. Ploegc, D. Hertigc, R. Klabundec, M.D. Warnera, M.C. Hobathod, Rakotomananab, S.E. Clifta, Journal of Biomechanics 35 (2002) 767–773 [17] Standard Test Method for Poisson’s Ratio at Room Temperature. ASTM Designation: E 132 – 04.





HIGH STRAIN RATE ANALISIS OF WOOD-BASED BIO-COMPOSITES USING SHPB AND DIGITAL IMAGE CORRELATION (DIC) TECHNIQUE

Oxford, Mississippi, USA

ABSTRACT In this research, the high strain-rate response and energy

absorption characteristics of five different types of wood-based bio-composites have been investigated using the Split Hopkinson Pressure Bar (SHPB) and the DIC (Digital Image Correlation) techniques. The dynamic stress-strain curve evolution of strain in the tested specimen was obtained by Digital Image Correlation (DIC), while the stress was obtained using elastic wave theory. Different types of wood-based bio-composites i.e. 4%MDI140s, 4%MDIHP140s, 2%CS2%MDI140s, 4%CS4%MDI140s, and 1%MCC4%MDI140s (all on solid basis) were manufactured using a Dieffenbacher hot press. It was found that wood-based bio-composite 4%MDIHP140s had the highest strength among all five different types of wood-based bio-composites, which shows that increasing manufacturing pressure during fabrication significantly affected the dynamic response of the wood-based bio-composite and that high strain rate has a noticeable effect on material behavior. It was found that Corn starch marginally increased the energy absorption capacity among all wood-based bio-composites. Finally, an application of DIC to study the evolution of strain in wood-based bio-composites is described. The noncontact analysis enables measurement of the relative displacement between the interfaces of the wood-based bio-composites. Based on the ability to acquire displacement/strain measurements over a large range of deformation, DIC should serve as a valuable tool for high strain-rate research.

KEY WORDS: Split Hopkinson Pressure Bar (SHPB), Wood-based bio-composites, High Strain rate, Digital Image Correlation (DIC), Specific energy.

INTRODUCTION Wood-based bio-composites have been consistently used

for their renewability, low cost, high strength to weight ratio and desirable mechanical properties for structural applications, which might include uses as Centre board of Boat hulls, as Non-structural facia panels and for acoustic applications in Automobiles. This increase in consumer preference for environmentally friendly materials has led to an increase in

development of these materials. As a part of an effort to enhance green infrastructure, five new different wood-based bio-composites were developed in collaboration with the Department of Sustainable Bioproducts at Mississippi State University. Wood-based bio-composites could be a viable ecological alternative to carbon, glass, and man-made fiber composites.

Dynamic loading can cause different types of damage and can significantly affect the impact strength / resistance of exposed components, so these must be sufficiently robust to prevent catastrophic failure of the structure. Therefore, it is necessary to investigate the dynamic and fracture behavior of materials at such high strain rates [3].

Reddy et al. studied the quasi-static energy absorption of wood. The results revealed that the macro-mechanisms of the collapse of the cellular structure of wood depend on the density and the direction of loading; the energy absorbing characteristics of wood were a function of its relative density and deformation mechanisms under the loading conditions [1]. Vural et al. investigated the dynamic response and energy dissipation characteristics of balsa wood. The results showed that the initial failure strength of balsa wood was very sensitive to the rate of loading, while plateau stress remains unaffected by the strain rate; the degree of dynamic strength enhancement was observed to be different for buckling and kink band formation [2]. Allazadeh et al. studied the response of dry maple wood under a high strain rate compressive impact load. The results showed that the damage mode was dependent on the incident energy, strain rate, geometrical dimensions and material structure; the strain of the specimen in the wood was a function of the breaking event of the fibers, and the strain rate increased fourfold if the disk’s thickness was doubled. Moreover, a higher energy event (such as an increase in the number of fragments) produced greater strain than low energy events (such as indentation); however, the stress-strain relationship was independent of incident pressure [3].

Dave et al. investigated the dynamic and damping properties of wood-based bio-composites with the hammer excitation vibration technique. The result revealed that wood composite 4%MDI10.5MPa had the highest storage modulus (E’) and 2%CS2%MDI was potential candidate for damping materials, among nine different types of wood-based

Maharshi Dave, Tejas Pandya and Damian Stoddard

University of Mississippi


composites [4]. Tagarielli et al. studied the high strain rate response of PVC foams and end-grain balsa wood. They found that the compressive response of the cellular materials under investigation was mildly strain rate sensitive in the strain rate range 10-2-4*10-3 s-1, whereas negligible strain rate sensitivity was observed for 10-4-10-2 s-1; the compressive yield strength of the H250 PVC foam and balsa wood doubled when the strain rate was increased from quasi-static rates (10-4 s-1) to rates on the order of 103 s-1 and the H100 PVC foam displayed only a small elevation in uniaxial compressive strength (about 30%) for the same increase in strain rate [4].

Widehammar et al. investigated the stress-strain relationships for spruce wood, to include the influence of strain rate, moisture content and loading direction. They found that strain rate had large influence on the behavior of the wood, especially under the condition of full saturation, where water transport in the deforming specimen was very important [5].

Bragov et al. studied the dynamic properties of some wood species. They found that for higher strain rates, stress decreased with increasing strain rate; for low strain rates the loading and unloading branches of the deformation diagrams were nonlinear [6]. Shah et al. investigated the dynamic response of wood-based bio-composites under high strain rate compressive loading. The results showed that 4%MDIHP140s had the highest stiffness among all the biocomposites tested; corn starch was shown to enhance material properties [7].

Non-contact testing techniques have been used in the measurement of strain and 3D displacement of an object in stress analysis [9]. Optical techniques for measuring parameters such as strain and displacement in experimental mechanics have evolved over the years and have gained recognition as reliable tools for strain measurement. The availability of noncontact full-field strain measurement tools DIC has opened up new ways of understanding the behavior of composite materials. This recent breakthrough in measuring macroscopic parameters can be attributed to development of ultra-high speed imaging systems.

In this study, DIC (Proanalyst) software is used. Digital Image Correlation (DIC) is a non-contact optical technique for measuring strain and displacement. DIC has some outstanding advantages over traditional strain measurement. The advantages of this technique include: 1) measurement of full-field strain on the test specimen; 2) unaffected by rigid motion; 3) high fidelity data. The previous studies of wood contribute to the body of knowledge about the behavior of these materials in response to high strain rate dynamic loading. However, there is limited documentation of the high strain-rate characterization of wood-based composites by elastic wave theory and DIC technique. In the current study, the deformation that occurs during high strain rate deformation of different wood-based bio-composites in compression is measured using the elastic wave theory and DIC. A Split Hopkinson Pressure Bar (SHPB) is used to generate high strain rate loading. Dynamic stress-strain results are obtained via elastic wave theory and DIC technique. Furthermore, the specific energy of five different types of wood-based bio-composites were compared. This

paper would be in assistance to generate high fidelity data for high strain-rate response and selection of wood-based bio-composites for industrial and military applications.

SPECIMEN DETAILS The materials investigated in this study are different types

of wood-based bio-composites as shown in Table-1. The materials were created from Southern yellow pine. A Dieffenbacher 915 mm x 915 mm hot press system located at the Sustainable Bioproducts Laboratory at Mississippi State University was used to create the 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 forms panels based on the desired thickness, so each composite material had differing pressures, which were required to produce a 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-based bio-composite samples created for the analysis of the high strain-rate loading using the SHPB were made from Corn Starch (CS), Microcrystalline Cellulose (MCC) and a Methylene Diphenyl Diisocyanate (MDI) resin with different mass fractions (listed in Table 1). MDI is an aromatic diisocyanate and was an efficient binder that has been used in the production of composite wood products for over 30 years. Corn starch is the starch derived from corn (maize) grain or wheat, and Microcrystalline Cellulose (MCC) was formed from techniques described in a previous study [8] using pure cotton.

Table 1: Constituent Materials for Wood-Based Composites

Material Approx. Pressure

(MPa)

Cooking Time

(Seconds)

4%MDI140s 8.9 140

4%MDIHP140s (high pressure) (2x material)

10.5 140

2 % CS 2 % MDI 8.7 140

4 % CS 4 % MDI 8.4 140

1 % MCC 4 % MDI 9.2 140

Note: The sampling size for each material was three (3).

EXPERIMENTAL TECHNIQUE The high strain-rate response of wood-based bio-

composites was obtained using a Split Hopkinson Pressure Bar (SHPB) made of an Aluminum bar, and the DIC technique at The Blast and Impact Dynamics Lab., the University of Mississippi. Tests were performed under three ranges of strain rate (varying from ~550 /s to ~1250 /s). An annealed copper pulse shaper was used between the striker and the incident bar

UAB School of Engineering - Mechanical Engineering - Early Career Technical Journal, Volume 16 4

to ramp up the incident pulse and slow down the rate of loading. This method allowed the samples to be dynamically loaded at a slower rate. A typical voltage-time graph generated in the dynamic loading test is shown in Fig. 1. The evolution of strain in each tested specimen was obtained by Digital Image Correlation (DIC), while the stress was obtained using elastic wave theory. The equations for calculating the stress and strain from the incidence, reflected and transmitted waves are described in detail elsewhere [3]. High speed photography was done by using a Shimadzu HPV -2 high speed video Camera at 125,000 frames per second (FPS). A sound triggering system used for triggering the camera and strobes by the impact sound of the striker and incident bars. ProAnalyst was used to obtain evolution of strain with respect to time in different wood-based bio-composites. Fig. 2 shows the flow diagram of analysis of evolution of strain in different types of wood-based bio-composites with Classical Hopkinson bar theory and Digital image correlation technique (DIC).

Figure 1. Typical Voltage-Time graph during SHPB compression test

The strain in the specimen (εS ) at time t calculated using [3] εS(t) = −2C0

L ∫ εr

t0 dt (1)

where C0 is the wave velocity in the Hopkinson bars, L is the length of the specimen and εr is the reflected strain pulses. The average stress in the specimen (σs) at time t calculated using [3]

σs(t) = E AAS𝜀𝜀𝑡𝑡(t) (2)

where E is the s the Young’s modulus of the Hopkinson bar material, A is the cross sectional area of the Hopkinson bar material, A is the Cross sectional area of specimen, and 𝜀𝜀𝑡𝑡(t) is the transmitted strain pulse.

RESULTS AND DISCUSSION Results show that the strain-rate effect is noticeable on

dynamic response of wood-based bio-composites. Fig.3 shows a dynamic stress-strain curve obtained using the 2D DIC analysis and classical SHPB theory in the bars for the dynamic loading tests on 4%CS4%MDI140s at strain rate of ~1100 /s. The stress determined from Classical SHPB theory, while strain was obtained from classical SHPB theory and 2D DIC analysis.

Fig.4 reveals that the stress and strain histories obtained using the two techniques shows similar trends. The dynamic stress-strain curve of five different type of wood-based bio-composites were obtained at strain rate of ~950 /s, excluding 4%MDIHP140s at strain rate ~550 /s. It was found that the

maximum strain was 0.3; the wood-based bio-composite 4%MDIHP140s had the highest yield strength among all five different types of wood-based bio-composites tested. This could be due to increased ram pressure during fabrication process of wood-based bio-composites causing increasing stiffness. Wood-based bio-composite 4%CS2%MDI140s had the lowest yield strength among all five different types of wood-based bio-composites tested. This could be mainly due to altering the volume fraction of binder MDI by 2%. Our results have a number of similarities with the findings Shah et al. [7].

Figure 2. Combined evolution of strain field using SHPB

and DIC

Figure 3. Dynamic stress-strain curve of

4%CS45MDI140s bio-composite

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Figure 4. Dynamic stress-strain curves of wood-based bio-composites (DIC) (Strain rate from ~550 /s to ~950 /s)

Fig.5 shows the dynamic stress-strain curve of the wood-based biocomposites at strain rate of ~1100 /s, excluding 4%MDIHP140s at strain rate of ~ 725 /s. It was observed that wood-based bio-composite 4%MDI140s had the lowest yield strength among all wood-based bio-composites tested and that the strain rate has a noticeable effect on mechanical behavior of wood-based bio-composites; hence, the dynamic stress-strain curve differs slightly as compared to the dynamic stress-strain curve from strain rate of ~550/s to ~950/s. In addition, the results from Fig.5 show that the wood-based bio-composite 4%MDIHP140s had the highest yield strength, which could be due to increasing manufacturing pressure during fabrication causing increasing stiffness; when comparing wood-based bio-composites that were formed under the same pressure and cooking time, the 2%CS4%MDI140s had the highest yield strength and the wood-based bio-composite 4%MDI140s sample had the largest strain compared to other wood-based bio-composites. These values correlate satisfactorily well with Shah et al. [7].

Fig.6 shows dynamic stress–strain obtained for strain rate

of ~1250/s, excluding 4%MDIHP140s at strain rate of ~ 825/s. from 2D DIC analysis. The results show a trend similar to that of results obtained from classical SHPB theory. It was observed that wood-based bio-composite 4%MDIHP140s had the highest yield strength; the wood-based bio-composite containing 1%MCC had the lowest yield strength; this could be due to MCC, among all different five types of wood-based biocomposites tested. This concurs well with [5] and also confirms our previous findings [5].

Figure 5. Dynamic stress-strain curves of wood-based bio-composites (DIC) (Strain rate from ~725/s to ~1250

/s)

Figure 6. Dynamic stress-strain curves of wood-based bio-composites (DIC) (Strain rate from ~825 /s to ~1100

/s)

In addition, the single most striking observation to emerge from the data comparison was that wood-based bio-composites containing Corn starch consistently show improved yield strength over the wood-based bio-composite containing MCC. Furthermore, the typical dynamic-stress-strain curve 2D DIC analysis shows that each wood based bio-composite is more dominated by plastic region, with the exception of 4%MDIHP140s. This result reveals that each of these wood-based bio-composites is highly incompressible in the plastic range. As these samples were being plastically deformed, they experienced a shape change when loading was experienced, because plastic strain was responsible for their shape change. Moreover a typical dynamic stress-strain diagram from the 2D

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~ 1100 /s


DIC analysis reveals that the wood-based bio-composite 4%MDIHP140s sample was shown to be highly elastic in behavior. High elastic strain in the 4%MDIHP140s wood-based bio-composite induced a volume change, so this sample was highly compressible.

From the dynamic stress strain-relationship, it was observed that the stress-strain curve of the 4%MDIHP140s composite was clearly differentiated from the rest of the stress-strain curves. This shows that the amount of pressure applied during the fabrication process had a large effect on the dynamic response of the wood. The ‘4%MDIHP140S’ sample had a much greater stiffness value in comparison to the other samples, and this is in good agreement with results obtained by [7]. The results illustrate that the entire specimen has not been extensively damaged at the peak stress as shown in Fig. 7. It appears that compression failure is dominated by plastic deformation at peak load, followed by deboning of the interface between constituents of wood-based bio-composites. Our results have a number of similarities with Shah et al.’s [7] findings. The specific energy absorption (energy absorbed per unit mass) under SHPB compression loading is shown in figs. 8, 9 and 10 for three different strain rates.

Figure 7. Dynamic stress-strain curves of wood-based bio-composite 4%MDIHP140s with progression of test

The results show that with increasing strain rate all wood-based bio-composites show increasing energy absorption capacity.

Furthermore, these results reveal that 1%MCC4%MDI140s had high energy absorption at high strain rates. This lends support to previous findings of Shah et al. [7]. The results from Figs.8,9 & 10, which show that Corn starch marginally increased the energy absorption capacity among all wood-based bio-composites. Moreover, 4%MDIHP140s had the lowest specific energy among all different types of wood-based bio-composites tested. This is in good agreement with Shah et al.’s results [7].

Figure 8. Specific energy absorption of wood-based bio-

composites at strain rate of ~550 /s to ~950/s

Figure 9. Specific energy absorption of wood-based bio-

composites at strain rate of ~725 /s to ~1100 /s

Figure 10. Specific energy of wood-based bio-composites absorption at strain rate ~825 /s to ~1200 /s

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

tested using SHPB and DIC. The digital image correlation (DIC), which is a non-contact measurement technique, gives the evolution of strain throughout the samples under high strain-rate tests. The results indicate that 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 where essential. Finally, Corn starch marginally increased the energy absorption capacity among all wood-based bio-composites and 1%MCC4%MDI140s had high energy absorption at high strain rates.

ACKNOWLEDGEMENTS We would like to acknowledge the department of

sustainable Bio products, Mississippi State University for providing wood-based composite samples to Mechanical engineering department, University of Mississippi, Oxford, Mississippi, USA. Also, we would like to express special thanks of gratitude to Dr. P. Raju Mantena, Suman Babu Ukyam and Aarsh Shah for his inputs throughout this research; Matthew Lowe for machining of wood-based composite samples.

REFERENCES [1] B.G.Vijayasimha Reddy, K.V.Sharma, T.Yella Reddy, 2011, “Quasi-static Energy Absorption of Wood,” International Journal of Scientific & Engineering Research, 2 (8), pp.1-5. [2] M. Vural and G. Ravichandran, 2003,” Dynamic Response and Energy Dissipation Characteristics of Balsa Wood: Experiment and Analysis,” International Journal of Solids and Structures, 40, pp.2147–2170. [3] M. R. Allazadeh and S. N. Wosu, “High Strain Rate Compressive Tests on Wood,” an International Journal of Experimental Mechanics, 48(2), pp.101-107. [4] V. L. Tagarielli, V. S. Deshpande and N.A. Fleck, 2008, “The high strain rate response of PVC foams and end-grain balsa wood”, Composites Part B: Engineering, 39 (1), pp. 83-91. [5] Svante Widehammar, 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]A. Bragov, A. Lomunov, 1997,” Dynamic Properties of Some Wood Species”, J. Phys. IV France 07, pp. C3-487-C3-492. [7] Aarsh Shah, Tejas S Pandya, Damian Stoddard, Suman Babu Ukyam, Jason Street, James Wooten and Brian Mitchell, 2017,” Dynamic Response of Wood based Bio Composite under High Strain Rate Compressive Loading,” Wood and fiber science, V, 49(4). [8] Chauhan, Y.P., Sapkal/, R., Sapkal, V., and Zamre, G., 2009,

“Microcrystalline cellulose from cotton rags (waste from garment and hosiery industries),” Int J Chem Sci. 7(2), pp 681-688. [9] A. S. Kobayash, 1987, “SEM Handbook of Experimental Mechanics,” Prentice-Hall, Inc., New Jersey, pp. 388- 429.





HIGH STRAIN-RATE COMPRESSIVE EVALUATION OF PULTRUDED GLASS-GRAPHITE/EPOXY HYBRID COMPOSITES USING SHPB & DIC TECHNIQUES

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

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

ABSTRACT The dynamic response and energy absorption

characteristics of pultruded hybrid combinations of glass and graphite fibers in an epoxy matrix subjected to different compression loads in a modified Split Hopkinson Pressure Bar (SHPB) apparatus, are presented. High strain-rate tests were performed on cylindrical specimens to evaluate compressive strength along the longitudinal fiber direction. Transverse tensile strength was determined by diametral compression of disc samples (Brazilian indirect tensile test method). Strain to failure under both loading conditions was monitored with a Shimadzu HPV-2 high-speed video camera, at a recording speed of 500,000 fps, and Digital Image Correlation (DIC). A comparison of the strength and specific energy absorption of pultruded hybrid composites demonstrates the benefits of hybridization. Among all, Hybrid GL48, with 48% low-cost glass fibers in the inner core and 12% high-cost graphite fibers in the outer shell, was found to exhibit better performance under both loading conditions, showing the benefits of hybridization.

KEY WORDS: Pultruded composites; hybrids; high strain-rate; compression loading; SHPB; dynamic energy absorption; Brazilian disc; indirect Tensile Test; Digital Image Correlation(DIC) technique.

INTRODUCTION In many applications, composite materials are subjected to

dynamic loading by impact, vibration or wave propagation. The improved characteristics of fiber-reinforced pultruded composites as well as their high specific energy absorption due to complex failure mechanisms make them good candidate materials for use in structures that are likely to experience extreme dynamic loading, e.g., aircraft, ships, and armored vehicles [1, 2]. In the design of these structures, the ability of materials to absorb and dissipate energy effectively during impact events is important. Therefore, a thorough understanding of the dynamic behavior and energy absorption

characteristics of fiber-reinforced pultruded composites is warranted.

Fiber-reinforced pultruded composites with improved structural strength, mechanical properties and lower weight are replacing traditional materials such as steel, wood, and aluminum, in several structural applications [3]. Two most common reinforcing fibers used are glass and graphite. Glass fibers have high tensile strength and low tensile modulus. On the other hand, graphite fibers possess very high tensile modulus, lower weight, and low impact resistance. By the “hybridization” of both glass and graphite fibers in various proportions within the same epoxy matrix, structures can be designed to achieve an optimal balance of the two material properties.

The Split Hopkinson Pressure Bar (SHPB) experimental technique has been used for evaluating fiber reinforced composites at high strain-rates. Yokoyama and Nakai [4] performed SHPB high strain-rate compression tests on cubic specimens of carbon/epoxy laminates in all three principal material directions and discussed the failure mechanism of these composites. They reported an increase of modulus and ultimate strength, and a decrease of energy absorbed up to failure strain with increasing strain-rate when compression is along the fiber direction. Harding et al. [5] used SHPB for measuring high strain-rate tensile response of the test specimen. Several aspects of the test, like generation of a tensile loading pulse, design of the tensile test specimen and holding fixtures, were more complex and needed additional care in data acquisition.

Tensile tests on high stiffness low strength materials usually consider compression induced tension to avoid premature failure caused from gripping under direct tensile loading. The Brazilian disc method which is an indirect tensile test technique was designed to measure the dynamic tensile stress-strain curve of low strength brittle materials [6]. A basic assumption for the determination of the tensile strength by the Brazilian test is that material is regarded as homogeneous, isotropic and linearly elastic before brittle failure occurs. But the assumption of homogeneous material in the Brazilian test


has been extended to study heterogeneous materials like anisotropic rocks and concrete by experimental and numerical modeling techniques [7]. It was found that the stress distribution near the center of a disc under diametral loading is quite uniform for both isotropic and anisotropic media [8]. Therefore, the Brazilian testing method is deemed to be a good and effective method for determining the transverse tensile strength due to its simplicity of sample preparation and test procedure. The fundamental requirement for the Brazilian disc method to be a valid test is that the initiation of specimen failure should occur at the center of the specimen and propagate through the diameter along the compressive loading axis [9]. In a traditional Brazilian disc method, strain at the center of the specimen is measured with a strain gage directly mounted on the specimen. But on-specimen strain gage testing is both cumbersome and complex for low strength brittle materials under dynamic loading. The complexities of conventional on-specimen strain gage testing for transverse tensile strain measurement are further reduced by non-contact digital image correlation techniques.

Several non-contact methods have been employed to obtain tensile properties for various materials using Brazilian disc type specimens. John and Rajendran [10] performed SHPB indirect tensile testing on cementitious samples and established that the induced transverse strain at the specimen mid-point is equivalent to the strain in the transmission bar. Pramanik and Mantena [11] adapted a non-contact Laser Occluding Expansion Gage (LOEG) technique to monitor the global strain of graphite platelet reinforced vinyl ester nanocomposite disc specimens. In work reported here, the Brazilian indirect tensile test method has been used for determining the transverse tensile strength, by diametral compression of pultruded hybrid composite disc samples. The diametral crack initiation and strain to failure were monitored with a Shimadzu HPV-2 high-speed video camera at a recording speed of 500,000 fps and Digital Image Correlation (DIC). Adequate measures were taken to ensure that initiation of specimen failure occurred at the exact center of the disc specimen, and propagated through the diameter along the compressive loading axis, for these induced transverse tension tests to be valid. These results are compared with the high strain-rate compressive strength and specific energy absorption along the fiber direction, reported elsewhere [12].

MATERIALS AND METHODS Pultrusion is a relatively fast way to produce strong and

lightweight structural components [13, 14] of constant cross-section. Cylindrical rods of 9.6 mm diameter with combinations of glass and graphite in an epoxy matrix were produced using a single pass (Pulstar 804 machine) of the pultrusion process [15]. The graphite fibers were AS4W-12K (Hercules), the glass fibers were E-Glass (PPG 2001, #12), and the epoxy was EPON 862/W/537 (Shell Chemical Company, USA). The hybrid rods were manufactured to be symmetric about the neutral axis to avoid warping/bending effects in the final product due to temperature changes during production. Both

preform and post-die plates were employed to keep the fibers in the intended placement. The cylindrical rods had a total fiber volume of 60% (with 40% epoxy). Each hybrid is defined by the fiber volume fraction of each fiber type, and by their location relative to the neutral axis. For example, GR30, a 30% graphite core/30% glass shell specimen has 30% graphite fiber volume and 30% glass fiber volume, with the graphite fibers in the inner core region and the glass fibers in the outer shell region.

Cylindrical specimens were cut precisely to 7.2 mm (±0.1 mm) length for SHPB compression and Brazilian disc induced tension tests. Figure 1 shows cross-sections of the pultruded cylindrical hybrid combinations along with the corresponding percent fiber volume and specimen ID code. The average measured bulk densities of the pultruded cylindrical hybrid specimens are shown in Table 1. As expected the graphite fiber hybrids are lighter and have a lower density.

Table 1. Measured average bulk densities of pultruded

cylindrical hybrid specimens [12]

GL60 GL48 & GR12 GL30 & GR30 GR60

Density (kg/m3) 2084 1953 1808 1603

Figure 1. Cross-section of pultruded hybrid cylindrical specimens for longitudinal compression tests [12]. For Brazilian indirect tensile tests, the disc sample surface

was coated using a flat matte white spray paint and laser etched at the center, as shown in Figure 2. Laser etching provided greater contrast with the painted surface and decreased image blur, for a more consistent tracking of the crack initiation and propagation using ProAnalyst 2D DIC tracking software.


Figure 2. Cross-section of pultruded hybrid disc specimens, spray painted and laser etched, for

monitoring crack initiation and propagation using Digital Image Correlation (DIC).

EXPERIMENTAL SETUP SHPB, or Kolsky bar, first developed by Kolsky [16] in

1949, is a well-known test method for characterizing materials at high strain-rates. All tests were carried out on the modified SHPB in the Blast and Impact Dynamics Lab at the University of Mississippi at Oxford. Maraging steel bars of 19.02 mm diameter were used as striker, incident and transmission bars. A thin elastic-plastic disk made of copper is placed in between the incident and striker bars so that a ramp shaped incident pulse with almost constant stress rate can be achieved. The pulse shaping technique is discussed in detail by Frew et al. [17] for SHPB compressive tests of brittle materials.

A HPV-2 High-Speed Video Camera (Shimadzu Scientific Instruments) with a fixed resolution of 312 x 260 pixels and recording speed of 500,000 fps was used to capture the deformation/failure process. Due to the high recording speed of the camera and the need for high-intensity lighting, two 1000W strobes (Photogenic Power Lights, PL2500DR) were employed.

For longitudinal compressive loading, pultruded cylindrical specimens are placed between the incident and transmission bars with fibers aligned along the loading direction, as shown in Figure 3. Petroleum jelly was applied for mounting specimens in between the incident and transmission bar ends, and to lessen friction/shear effects during compression tests.

Figure 3. Longitudinal compressive loading on hybrid cylindrical specimen.

For Brazilian (or splitting tension) tests, the disc specimen is mounted as shown in Figure 4, with compressive loading applied along the horizontal diameter and transverse tensile stresses induced in the vertical direction normal to loading axis. In this loading configuration, the exact center of the disc is the only point at which the conditions for tensile failure at a value equal to the uniaxial tensile strength are met. Curved (instead of flat) loading anvils are used to prevent crack initiation at the loading points due to stress concentrations [18].

Figure 4. Induced transverse tension in circular disc specimen under diametral compressive loading.

RESULTS SHPB high strain-rate longitudinal [12] and diametral

compression tests were performed on pure glass, pure graphite, and the hybrids of glass-graphite fiber/epoxy cylindrical specimens to study their dynamic behavior and energy absorption characteristics.

The required dynamic equilibrium condition has been validated in all longitudinal compression tests by estimating the stresses developed at opposite faces of each specimen under compression. From the high-speed video images, it appears that compression failure is dominated by plastic deformation of the matrix at peak load followed by debonding of the fiber-matrix and fiber-fiber interfaces.

Five specimens of each hybrid combination of pultruded cylindrical specimens were tested at a strain-rate of ~700/s. Sample stress-strain curves for compression tests on pultruded hybrid combinations specimens are shown in Figure 5. Variation of compressive stiffness (slope of curve) among hybrid combinations is noted with a change in the type and location of fibers. For example, pure graphite/epoxy specimens (GR60) exhibit highest stiffness, and an increase in the volume fraction of glass fibers resulted in a reduction of stiffness. Also, higher stiffness is observed when graphite fibers are in the inner core region.


Figure 5. Stress-strain plots of hybrid cylindrical specimens from SHPB compression tests [12].

Brazilian tests were conducted at high strain-rate to induce transverse tension in the pure glass, pure graphite, and hybrids of glass-graphite fiber/epoxy disc specimens. Stress equilibrium was validated by comparing the voltage output from strain gages on each bar which is analogous to the applied force on each end of the bar [11]. It should be noted that the reported data is up to the center point crack initiation, as captured with the high-speed camera. An independent analysis was also performed to further validate these test results by comparing transmission bar strain [10] to the disc specimen center point strain. Strain at the disc center point was analyzed using 2D DIC with ProAnalyst software.

Careful visual inspection of the high-speed video images showed that initiation of specimen failure occurred at the exact center of the disc and propagated through the diameter along the loading axis, thus corroborating the Brazilian disc test data to be usable. Similar results were observed for all the other hybrid combinations.

Figure 6 shows typical transverse stress-strain output for all hybrid combinations at a strain-rate of ~450/s, with 3 specimens tested in each configuration. In this case pure Glass/Epoxy (GL60) shows the highest (slope) stiffness and, in general, decreases with increasing graphite content.

The average ultimate compressive strength, along with the data scatter for five specimens, is summarized in Figure 7. Ultimate compressive strength is higher, by more than 200 MPa, for pure glass/ epoxy (GL60) compared to pure graphite/epoxy (GR60). Hybrids with glass fibers located in the inner core exhibit higher compressive strength compared to those with glass fibers in the outer shell. The hybrids having 48% glass fibers in the inner core and 12% graphite fibers in the outer shell (GL48) demonstrated marginally higher compressive strength than pure glass/epoxy, and significantly higher than for other combinations.

Ultimate transverse tensile strength is summarized in Figure 8. The ultimate transverse tensile strength of pure glass/epoxy (GL60) is higher than pure graphite/epoxy (GR60) by approximately 10 MPa. Hybrids exhibited nearly identical transverse tensile strengths. GR60 exhibited the lowest

transverse tensile strength, similar to under longitudinal compressive loading.

Figure 6. Transverse tensile Stress-Strain from Brazilian tests on hybrid disc specimens.

Figure 7. Ultimate compressive strength of hybrid

cylindrical specimens under longitudinal compression [12].

Figure 8. Ultimate transverse tensile strength of hybrid specimens from Brazilian disc tests.


The specific energy absorption (energy absorbed per unit mass) under SHPB compression loading is compared in Figure 9. Among all the hybrid combinations, 60% glass/epoxy (GL60) and the hybrid with 48% glass fibers in the core region (GL48) show the highest specific energy absorption, while hybrids with 30% graphite fibers in core region (GR30) absorbed the least amount of energy.

Figure 9. Specific energy absorption of hybrid cylindrical specimens under longitudinal compression [12].

The specific energy absorption under induced transverse tensile loading is shown in Figure 10. Pure glass/epoxy (GL60) and pure graphite/epoxy (GR60) have the lowest specific energy absorption capacity, with the other hybrid combinations displaying better performance.

Figure 10. Specific energy absorption of hybrid specimens from Brazilian disc tests.

CONCLUSION In this study, direct longitudinal compression and Brazilian

indirect tensile tests were performed on pultruded hybrid composite specimens made of glass and graphite/epoxy using a modified SHPB. Pure glass/epoxy (GL60) shows the lowest longitudinal compressive stiffness, and highest transverse tensile stiffness compared to other hybrids. Placement of graphite fibers in the inner core region results in higher longitudinal compressive stiffness. Under both loading conditions, the pure glass/epoxy (GL60) exhibited higher strength than pure graphite/epoxy (GR60). Placement of glass

fibers at the inner core showed a significant increase in ultimate compressive strength, but not in transverse tensile strength. Pure glass/epoxy (GL60) has higher specific energy absorption capacity than pure graphite/epoxy (GR60) under longitudinal compression loading whereas pure graphite/epoxy (GR60) has higher specific energy absorption in transverse tension. Among all hybrids, GR30 has the lowest specific energy absorption under longitudinal compression and highest specific energy absorption under transverse tension. Hybrid GL48, with 48% low-cost glass fibers in the inner core and 12% high-cost graphite fibers in outer shell, was found to exhibit better performance under both loading conditions showing the benefit of hybridization.

ACKNOWLEDGEMENTS The authors wish to acknowledge funding received from

US Army Research Office-DURIP Grant # W911NF-13-1-0248 for the high-speed digital cameras used in this research. The authors would also like to thank Mr. Paul Matthew Lowe for sample preparation. The SHPB compression tests were performed by graduate students: Soheil Daryadel, Mohammad Afrough, Damian Stoddard and Kiyun Kim.

REFERENCES [1] Alghamdi A.A.A., 2001, “Collapsible impact energy absorbers: an overview”. Thin-Walled Structures, 39, pp. 189–213. [2] Giambastiani, Lu and Yu T.X, 2003, "Energy absorption of structures and materials". Woodhead Publishing Limited, pp. 144–173. [3] Gibson,R., 2007, Principles of Composite Materials Mechanics. 2nd Edition, CRC Press, Boca Raton, USA. [4] Yokoyama, T., Nakai, K. and Odamura, T., 2007, High Strain-Rate Compressive Characteristics of a Unidirectional Carbon/Epoxy Composite: Effect of Loading Directions. In: Gdoutos, E.E., Ed., Experimental Analysis of Nano and Engineering Materials and Structures, Springer Netherlands, pp. 681-82. [5] Harding, J., Wood, E.O. and Campbell, J.D., 1960, Tensile Testing of Materials at Impact Rates of Strain, Journal of Mechanical Engineering Science, 2(2), pp. 88-96 [6] Chen, R., Dai, F., Lu, L., Lu, F. and Xia, K., 2010, Determination of Dynamic Tensile Properties for Low Strength Brittle Solids, Experimental and Applied Mechanics, 6, pp. 321-326. [7] Diyuan Li and Louis Ngai Yuen Wong, 2013, The Brazilian Disc Test for Rock Mechanics Applications: Review and New Insights, Rock Mech Rock Eng, 46, pp. 269-287 [8] Chao-Shi Chen, Ernian Pan and Bernard Amadei, 1998, Determination of Deformability and Tensile Strength of Anisotropic Rock Using Brazilian Tests, Int. J. Rock Mech., Pergamon, 35, 1, pp. 43-61. [9] Carneiro, F. and Barcellos, A., 1953, International Association of Testing and Research Laboratories for Materials and Structures, Rilem Bulletin, 13, pp. 99-125. [10] John, R., Antoun, T. and A.M. Rajendran, 1991, Effect of


Strain Rate and Size on Tensile Strength of Concrete, American Physical Society – Shock Compression of Condensed Matter, Elsevier Science Publishers, pp. 501-504. [11] Brahmananda Pramanik, P. Raju Mantena, Tezeswi Tadepalli, Arunachalam M. Rajendran., 2014, Indirect Tensile Characterization of Graphite Platelet Reinforced Vinyl Ester Nanocomposites at High-Strain Rate, Open Journal of Composite Materials, 4, pp. 201-214. [12] Seyed Soheil Daryadel, Cameron Ray, Tejas Pandya and P. Raju Mantena. 2015, Energy Absorption of Pultruded Hybrid Glass/Graphite Epoxy Composites under High Strain-Rate SHPB Compression Loading. Materials Sciences and Applications, 6, pp. 511-518. [13] Kumar, S.S and Mantena, P.R, 1996, Dynamic and Static Torsional Characterization of Pultruded Hybrid Cylindrical Composite Rods. Journal of Composite Materials, 30, pp. 918-932. [14] Vaughan, J.G., Roux, J.A. and Mantena, P.R., 1992, Characterization of Mechanical and Thermal Properties of Advanced Composite Pultrusions. Proceedings of the 1992 NSF Design and Manufacturing Systems Conference, 8-10 January 1992, pp. 1141-1145. [15] Nori, C.V., Mantena, P.R. and McCarty, T.A., 1996, Experimental and Finite Element Analysis of Pultruded Glass-Graphite/Epoxy Hybrids in Axial and Flexural Modes of Vibration. Journal of Composite Materials, 30, pp. 996-2018. [16] Kolsky, H., 1949, An Investigation of the Mechanical Properties of Materials at Very High Rates of Loading. Proc. Phys. Soc. B, 62, pp. 676-700. [17] Frew, D.J., Forrestal, M.J. and Chen, W., 2002, Pulse Shaping Techniques for Testing Brittle Materials with a Split Hopkinson Pressure Bar, Experimental Mechanics, 42(1), pp. 93-106. [18] Awaji H. and Sato S., 1979, Diametral compressive testing method, Journal of Engineering Materials and Technology- ASME, 101(2), pp. 139-147.





EVALUATING DISCRETIZATION ERRORS WITH NUMERICAL EXPERIMENTS

David Dodoo-Amoo, Julio Cesar Mendez Mechanical Engineering Department,

North Carolina Agricultural and Technical State University

Greensboro, NC, U.S.A

Dehau Feng, Frederick Ferguson Mechanical Engineering Department,

North Carolina Agricultural and Technical State University

Greensboro, NC, U.S.A

ABSTRACT The Integro-Differential Scheme (IDS) was applied to a

wide class of fluid dynamic problems. A literature survey showed that its ability to predict the flow physics is impressive [1-4]. However, the IDS capability to predict flow physics can only be considered credible after its capacity to manage and control numerical errors is formally analyzed. The research project developed herein created and implemented a set of numerical experiments in lieu of analytical methods to quantify the errors generated by the IDS. Selective Quasi-1D fluid dynamic problems are solved, both analytically and numerically using the IDS technique. Further, specialized spline routines were also developed that allow the IDS solutions to be used as ‘standard’ solutions in the place of 'exact' solutions. The IDS 'standard' solutions were compared to the known values at specified locations within the nozzle in efforts to validate their use. Also, the evaluation process facilitated a method for appropriately comparing the errors associated with the ‘standard’ solutions to the numerical solutions generated by a wide range of numerical solutions at prescribed grid levels. In this paper, the error analysis procedure is documented. A series of Quasi-1D convergent-divergent nozzle problems are solved, and their error behaviors analyzed. The numerical experiments demonstrated that the IDS procedure consistently delivers spatial discretization errors of the order one on course grids and greater for fine grids. KEY WORDS: Navier-Stokes equations, Computational Fluid Dynamics, finite difference method, the finite volume method, and Integro-Differential Scheme.

INTRODUCTION With the advent of modern computers, the non-existence of

analytical solutions to the partial differential equations governing fluid flows ceased to be a limitation to our understanding of the physics underpinning flow fields. Modern computers also gave birth to many numerical methods. Among these methods are the Conservation-Element Solution Element (CESE) method, Direct Numerical Simulation (DNS) and Large

Eddy Simulation (LES), Discontinuous Galerkin Methods (DGM) and the Integro-Differential Scheme (IDS) [2, 3]. These computational methods have all been applied to the task of solving the Navier Stokes Equations, and thus, they have pushed CFD as a tool that is consumed by the mainstream. So far, however, they have all provided varying degrees of success when it comes to elucidating the details associated with the various flow field physics of interest to engineers at realistic Reynolds and Mach numbers.

One important characteristic that these numerical schemes have in common is their inherent ability to introduce computational errors. However, each scheme handles the error generation and growth differently. The preferred way of measuring the accuracy of any numerical scheme is to evaluate its error as the difference between its numerical solution and its analytical counterpart. Analytical solutions to fluid dynamic problems of interest to modern engineers are scarce at best and non-existent at worst. Moreover, the design of experiments that may provide the flow field solution of interest is often expensive, and in many cases, almost impossible to replicate. The purpose of this paper is to describe an approach to the creation of ‘standard' solutions that may serve as reliable replacements to 'exact' solutions for numerical problems of interest to engineers. The implementation of this method in quantifying the numerical errors associated with the IDS scheme is also documented.

THE CONSERVATION LAWS IN INTEGRAL FORM Equations (1-3) represent collectively the system of mass,

momentum and energy conversation laws that dictate fluid flows. These equations, when coupled to the well-known viscous relations and equation of state form a closed system of five equations relative to five unknowns. In this research, this closed system of equations is referred to as the system of Navier-Stokes equations (NSE).

0=+∂∂

∫∫∫∫∫SV

sdVdvt

ρρ (1)


( ) ∫∫∫∫∫∫∫∫∫ +−=•+∂∂

SSSV

sdsPdVsdVdvVt

τρρ (2)

∫∫∫∫∫∫∫∫∫∫∫ +•+•−=•+∂∂

SSSSV

sdqsdVsdVPsdVEEdvt

τρρ (3)

Further, the NSE only provide a unique solution to a given fluid flow problem when the appropriate initial and boundary conditions are specified. As discussed earlier, when applied to equations (1-3) all computational schemes produce numerical errors. These errors, in general, can be described by

( )GridsTemporalSpatialSchemeNumericalF &,=ε (4) where the symbol, ε, represents the maximum error generated by the scheme during its implementation. The IDS error capacity has not been extensively tested. As discussed earlier, a literature survey [1-3, 5] concluded that the IDS captures the flow physics within complex flow fields, especially those associated with shock boundary layer interactions at realistic Reynolds and Mach numbers conditions [1, 6]. For example, the IDS scheme was used to solve the shock boundary layer interaction problem illustrated in Figure 1 [5]. As noted in Figure 1, the technical details within the flow field are experimentally documented with the flow features vividly captured. The IDS result is illustrated in Figure 2. By all standards of comparison, the IDS result shows all the flow field features previously documented [5]. Impressive or not, the credibility of the IDS will not bode well, unless its error handling behavior is fully analyzed.

Figure 1. Experimentally obtained Shock impingement. [5]

Figure 2. IDS Solution for shock impingement phenomenon

THE QUASI-1D IDS FORMULATION When the IDS elementary control volume model is applied

to the Quasi-1D form of the NS equations (1 – 3), the following conservation of mass, momentum and energy equations are derived,

( ) ( )( ) xA

AuAut Avg

nn

Avg δδδρδρρ −+ −=

∂∂ (5)

( )( ) ( )( )

( ) AvgAvg

nn

Avg dxdAp

xAAPuAPu

tu

+

+++−=

∂∂ −+

δδδρδρρ 22 (6)

( )( ) ( )( )

( ) xAApeuApeu

te

Avg

nTnT

Avg

T

δδδρδρρ −+ +−+

=

∂∂ (7)

However, in the case of the IDS methodology, the primitive variables are replaced by their average counterparts. Note, equations (5 – 7) can be expressed in vector form, as

0=−∂∂

+∂∂ JF

tU

ϑ (8)

where the U, F and J vectors are defined as follows;

VAvgTeuU

=

ρρρ

, ( )( ) ±

++=

nAvgT SAPeuAPu

uAF

,

2

ρρρ

,

VAvg

dxdAPJ

=

0

0

and the subscripts define the locations and types of operations used in evaluating respective average quantities. Finally, as in all explicit schemes, equation (8) is used to compute the update solution,

tt

UUUAvg

AvgUpdated δ

∂∂

+= (9)

The numerical details associated with the IDS scheme as summarized in this paper can be found in [2, 3]. The IDS scheme as derived herein was never used in solving Quasi-1D problems. As such, analyses of its implementation on a wide class of Quasi-1D problems and its error handling capabilities are warranted.

Figure 3. Convergent-divergent nozzle with Shock


THE QUASI-1D IDS APPLICATION Consider the case of air flowing through the Quasi-1D

convergent-divergent nozzle shown in Figure 3 [1]. Further, consider the resulting flow field under two sets of boundary conditions; namely, ‘isentropic' conditions and ‘shock' conditions. In this analysis, ‘isentropic' conditions represent the coupled inlet and outlet boundary conditions that allow for the generation of isentropic flow throughout the nozzle. Whereas, ‘shock' conditions represent the coupled inlet and outlet boundary conditions that allow for the generation of a normal shock in the divergent section of the nozzle.

The Quasi-1D convergent-divergent nozzle area variation as well as the 'isentropic' and 'shock' boundary conditions of interest to this analysis are described in Table 1. In addition, all other numerical details related to this problem are tabulated. Again, refer to Figure 3. Note also, Figure 3 shows a normal shock standing in the divergent section of the nozzle. This solution is expected when the ‘shock' boundary conditions are set; again, refer to Table 1.

Table 1: Flow Conditions

( ) ( )( ) ( ) 13

Toli

2ii

10ε5001;N;N1,i;30,x;1.5x2.21.0xAA

−==∈=

−+==

Isentropic Flow Inlet Outlet 0.1: 1 =ρρ 210.2: −− −= NNN ρρρρ

321 0.2: uuuu −= 210.2: −− −= NNN uuuu 0.1: 1 =TT 210.2: −− −= NNN TTTT

Normal Shock Inlet Outlet 0.1: 1 =ρρ 1: −= NN ρρρ

321 0.2: uuuu −= 1520.0: =Nuu 0.1: 1 =TT 1: −= NN TTT

Executing the IDS formulation in accordance with the

‘isentropic conditions’ and ‘shock conditions’ described in Table 1, yields the non-dimensional density, velocity and temperature distributions within the Quasi-1D nozzle. The over-lapping plots in Figure 4 illustrate the behavior of these parameters within the convergent-divergent nozzle under both sets of conditions.

A closeup view of Figure 4 indicates the normal shock wave is generated at a unique location, as evident by the distribution of the three independent primitive variables. Besides, and as expected, the spatial distributions of the three primitive variables showed that the flow in the convergent-divergent duct is isentropic everywhere before and after the normal shock wave. Moreover, the isentropic behaviors ahead of the shock location for both cases—namely, ‘isentropic’ and ‘shock’ cases—are identical. Again, the evidence of this conclusion is clearly supported by the data presented in Figure

4. The capability of the IDS to capture the normal shock wave without smears and oscillations is noteworthy. In contrast, many existing CFD schemes fail at the task of capturing the shock wave in the convergent-divergent nozzle.

Figure 4. Density in the Convergent-Divergent Nozzle

STANDARD NUMERICAL SOLUTIONS AND ERRORS The need for developing ‘numerically exact’ solutions was

explained earlier in this paper. This need is self-evident when it comes to defining error. Consider the typical definition of the discretization error, εj;

Numericalj,Exactj SSε −= (10)

where the symbols, ExactS and Numericalj,S , represent the exact and

the numerical solutions, respectively [7, 8]. In the limited cases where the exact solutions exist, equation (10) is uniquely defined. However, in the many cases where no exact solution exists, equation (10) is not trivial, and can be challenging to evaluate [4].

This section focuses on the generation of an 'acceptable solution' when the exact solution does not exist. In the cases when the exact is no-existent, a ‘standard solution’ is defined, such that,

StandardExact SS ≈ (11)

where the symbol represents the numerical solution that best quantify the 'exact solution'.

In this paper, the ‘standard solution’ is created by numerically solving the Quasi-1D equations to the point where a grid independent solution is established. Moreover, in the case of the Quasi-1D equations, the ‘standard solution’ is a vector function and is defined by three sets of non-dimensional flow parameters; namely, density, ρ, velocity, u, and temperature, T. Of course, numerically, the grid independent solution comes only after several grid sizes are evaluated (the maximum number of grids evaluated is J). And, during each


grid level evaluation (j = 1, J), a solution represented by three sets of discrete pairs, with each pair represented as follows: ( ) j

ii ρ,x , ( ) jii u,x , and ( ) j

ii T,x are obtained. It is of interest to note that a single set consisting of three numerical pairs, obtained at a specific grid level, say j = j0, is defined as the ‘numerical solution’, such that,

( )( )( )

=∈

∈

∈

∈∈

∈∈

∈∈

J1,j

j

J1,j

j

J1,j

j

N1,iJ1,j,Gridii

N1,iJ1,j,Gridii

N1,iJ1,j,Gridii

Numericalj,

T,x

u,x

ρ,x

S (12)

Moreover, three complementary pairs of the flow parameters; ρ, u and T, are elevated to the level of a ‘numerical solution’ for the Quasi-1D equations, if and only if, they collectively solve these equations for a specific grid size at a prescribed level of tolerance. The outcome of the grid independent study is the generation of the ‘standard solution’. Therefore, the ‘standard solution’ is an element of the set of ‘numerical solutions’, such that it remains the only solution that best solves the Quasi-1D equations at the tolerance level required for it to best represent the ‘exact solution’. In this case, that criterion is met when the tolerance level is achieved for N = Nj, and j = J.

Notwithstanding the description and constraints provided above, the ‘discrete pairings’ as described in equation (12) do not best represent the ‘standard solution’, especially when it comes to practical application. Consequently, in this analysis, the ‘discrete pairs’ defined by the grid independent solution are converted to spline curves, such that,

SplineendentGrid_IndepStandard SS ≈ (13)

In the case of the Quasi-1D analysis discussed herein, the ‘standard solution’ for flow field condition, is transformed to the following three spline curves[4]:

( )

( )

( )

=

∈

∈

∈

Splinex,xxx,

Splinex,xxx,

Splinex,xxx,

SplineendentGrid_Indep

JN1

JN1

JN1

Tuρ

S (14)

where the primitive variables are defined by their respective splines starting at the inlet, the outlet and at every other point throughout the nozzle. The objective is to ensure that the ‘standard solution’ can deliver the ‘exact value’ at any location within the nozzle as needed. Further, in this analysis, the grid independent solution was obtained for N = 12801, at a maximum tolerance of 10-13, after 10 grid sets were studied.

The spline functions defined by equation (14) can be revisited and transformed into the local errors at the ith point, on the jth grid set, such that,

( ) ( ) ( )i J1,j,GridiSpline

endentGrid_Indepiij jxSxSxε ∈−= (15)

In a similar manner, the ‘maximum error’ on the jth grid can be defined as

( ) J

j

N1,ii,Gridij xεmaxε ∈= (16)

However, it is important to recall that the error at any grid level consists of three data sets; namely, ρ, u and T. As such, it must be noted that equations (12) and (14) are vector equations and their maximum coordinate values must be computed accordingly. The observed order of accuracy of the IDS method is also estimated by examining the behavior of various norms of the spatial error. For example the L1 and L2 norms are evaluated as follows:

( ) ( )∑=

∈=J

j

J1

N

1i

N1,ii,GridiL xε

N1ε (17)

( ) ( )∑=

∈=J

j

J2

N

1i

N1,ii,Gridi

2L xε

N1ε (18)

and the L∞ norm is evaluated through the use of equation (16), such that,

( ) J

j

N1,ii,GridiL xεmaxε ∈=

∞ (19)

Evaluating the spatial error at a given grid level is accomplished through the use of the illustration provided in Figure 5. As can be observed, Figure 5 consists of 7 overlapping plots showing the density distribution in a convergent-divergent duct for 7 grid levels; ranging from 201 nodes through 5001 nodes. Further, Figure 5 also qualitatively illustrates the error variation along the nozzle. In this case, when equation (19) is applied at a specified location, for example at x = 1.0, then at that point it is possible to evaluate the local error, εij, for density for the jth grid level, as per Figure 5. In a similar manner, the local errors for velocity and temperature can be obtained. Later, the maximum of the three quantities yields the maximim local error, εi, at that grid level. In a similar manner, the maximum error can be evaluated along the entire nozzle for a given grid set.

Figure 5 Numerical error at 0.1=barX for 201 grid points


PRACTIAL EVALUATION OF NUMERICAL ERRORS The errors generated by the IDS scheme are defined by the

functions described in equations (17 - 19), and in accordance with the IDS computational formulation, these functions are dependent on both time and space. However, in this analysis, it is also assumed that with the aid of the method of separation of variables, the error norms at any grid level can be further simplified and expressed as follows,

( ) ( )ΔtεΔxεε tx += (20)

In addition, the two separated, spatial and temporal error functions are predicted to have the forms,

( )px xCε ∆= (21)

αtt Aeε −= (22)

where the variable p+1 defines the order of discretization and the variable α represent the convergence and stability criteria of the numerical method, respectively.

EXPERIMENTING WITH NUMERICAL ERRORS In an effort to evaluate the entire Quasi-1D flow field

space and conclude on its spatial error performance, a set of M, M = 10, Quasi-1D fluid problems[1, 6, 9] are chosen, and each problem is evaluated on J set of grids. These convergent-divergent ducts are illustrated in Figure 6, and their functional forms are illustrated in Table 2. Moreover, the M problems are chosen from a varying set of convergent-divergent geometries that include both ‘isentropic’ and ‘shock’ boundary conditions.

Figure 6. Quasi-1D Nozzle Profiles

Table 2: The Set of Quasi-1D Nozzles Evaluated

Case # Nozzle Description B-Cdts

#1 ( ) ( ) ( )3,0;5.12.20.1 2 ∈−+= xxxA

Isentropic

#2 Shock

#3 ( ) ( )3,0;5.13.50.1 ∈−+= xxxA

Isentropic

#4 Shock

#5 ( )( )

≤≤−+≤≤−+

=0.3x5.1,1.5x223.201.0

1.5x00.,1.5x0002.21.0A 2

2

Isentropic

#6 Shock

#7 ( )1.0x0

40.8x0.347tanh1.398A≤≤

−+= Isentropic

#8 Shock

#9

( )

≤<≤≤−+−<≤−

=0.15.0,0.25.05.0,sin0.15.00.1,0.2

2

xxx

xA π

Isentropic

#10 Shock

Recall that equations (17 - 19) predict the spatial errors generated at the jth grid element, and the entire set of grids varies from element j = 1 through J. So there are J grids associated with a given Quasi-1D problem, and M Quasi-1D problems to be evaluated. Also, recall, that at each of the jth grid levels, there are Nj nodes used to define the numerical grid. This is also equivalent to stating that at each node of the jth grid level there is an equivalent ∆xj that represents the grid size (uniform grids are assumed). The objective is to now develop a numerical experiment to collect the errors at each grid level for each problem, and study their behavior. Table 3 illustrates the numerical conditions under which all problems were evaluated.

Table 3: The Set of Quasi-1D Nozzles Evaluated

Nozzle Geometry ( )ii xfA = ( )3.0..0.0,...,xi ∈ ,

Gridj ( )jN1,i∈ 26, 51, 101, 201, 401, 801, 1601, 3201, 6401, 12801

Set of Nj J1,..,j;NJ = ...,12801 51, 26, NJ =

Index j Jj1 ≤≤ J = 10

Tolerance 1310−≤ Constant for All Grids

No. of Problems M M = 10, Refer to Table 2

mth Std Solution Spline Curve 12801 NJ = , for all j.


NUMERICAL ERROR DATA ANALYSIS The ten Quasi-1D problems described in Table 2 can be

classified into two major categories; those with ‘isentropic’ and those with ‘shock’ boundary conditions. The experiments that correspond to all ‘isentropic’ cases are classified by Cases #1, #3, #5, #7 and #9. On the other hand, the cases of interest to all ‘shock’ related problems are classified by Cases #2, #4, #6, #8 and #10. In addition, these Quasi-1D problems can be also be described in terms of three distinct geometric length ranges; varying from non-dimensional values of 0 to 3, 0 to 1, and -1 to 1, for both ‘isentropic’ and ‘shock’ conditions. These ‘range’ problems are described by Cases #1 - #10; with six problems in the 0 - 3 length range, two problems in the 0 to 1 range and two problems in the -1 to 1 range, respectively. Finally, these ten problems are also described in terms of their degree of smoothness of their geometric respective profiles in the throat region. The ‘geometric profile’ problems are best described by the Cases #3, #4, #5, #6, #9 and #10. In summary, the results of the experiments conducted herein will be described with the aid of the following three major characteristics: ‘boundary’ type problems, ‘range’ type problems and ‘geometric profile’ type problems. Refer to Table 2 and Figure 6 for greater details.

Also, in this section, the results from selective experiments associated with the various experimental groups described above will be analyzed and their findings summarized. The data collected from these studies are formulated and presented in the forms of log-log plots. Further, the x and y axes of the resulting log-log plots are represented by non-dimensional scaling parameters. These non-dimensional scaling parameters represent the non-dimensional grid size, GridΔx , and the non-

dimensional grid error norm, grid,L∞ , such that,

gridJ,L

gridj,Lgrid,

J

jGrid

th

th

ε

εL;

ΔxΔx

Δx∞

∞== ∞

This form of non-dimensionalization was chosen because it

allows for the comparative behavior of every grid level with respect to the finest grid. In this paper, these plots are referred to as the log-log error plots, and they are used to profile the non-dimensional grid size versus the nondimensional Quasi-1D numerical error relative to the finest grid. Also illustrated in each of the log-log error plots used in this paper are two broken lines. Each broken line indicates the classical behaviors of the first and the second order errors that are derived from numerical methods. The technical goals of these plots are to highlight the error behavior of a given set of grid in comparison to the finest grid used in the specified numerial experiment. Error Analysis of the Anderson Problems

In an effort to observe the error behavior that is associated with a fixed geometry for both the ‘isentropic’ and ‘shock’ boundary conditions, the Anderson [1] problems defined by

Cases # 1 and #2 were evaluated. The resulting log-log error plots are illustrated in Figure 7.

Figure 7. IDS Error Analysis for Anderson Problems

As seen in Figure 7, the numerical errors associated with

this problem clearly lie above the first order broken line for all grid levels equal to or greater than grid level j = 3 or higher. The major conclusion herein is that irrespective of the geometric profile or the boundary conditions, the IDS procedure delivers descritezation orders of errors greater than 1st order (p = 1) for the classical Anderson quasi-1D nozzle problem [1]. Rather, the data indicated that for very coarse grids, the order of discretization may be somewhat erratic but as the grid level is refined, the solution stays above 1st order and approaches 2nd as the IDS approaches the finest level of grids. Error Analysis of Fixed Range Problems

Consider the plots illustrated in Figure 8. They represent the results of the Quasi-1D flow fields for the fixed ‘geometric range’ but variable ‘geometric profile’ problems; namely cases #1 - #6, for both the ‘isentropic’ and ‘shock’ boundary conditions. Recall, in these cases the nozzle geometric profile varies in accordance with the plots illustrated in Figure 6 for the respective case numbers.

Figure 8. IDS Error Analysis for varying Q1D Nozzles


Again the log-log error plots indicate that the numerical errors associated with the problems with varying geometries over fixed ranges are clearly above 1st order for all grid levels equal to or greater than grid level 4. In a similar manner to the Anderson problem, for coarse grids the error behavior may be erratic, but they approach p = 2 as the grid is refined, becoming two at the finest level.

Error Analysis of ‘Isentropic and Shock’ Problems

The plots developed in Figure 9 illustrate the scaling of the non-dimensional error behavior of the five quasi-1D ‘isentropic’ problems; namely, cases #1, #3, #5, #7 and #9. Recall, these experiments were conducted under varying ranges and varying geometric profiles. Also illustrated in the log-log error plots in Figure 9, with the use of broken lines are the 1st and 2nd orders classical behaviors derived from established numerical methods. The five solid line graphs in Figure 9 qualitatively demonstrate the order of the descritization of the IDS numerical experiments. And, as noted, the results stay within the 1st and 2nd order boundaries as the grid levels are refined. In a similar manner, Figure 10 illustrates the scaling of the non-dimensional error behavour for the five Quasi-1D ‘shock’ problems; namely, cases #2, #4, #6, #8 and #10. As in the isentropic cases, these ‘shock’ experiments were also conducted under varying ranges and with varying geometric profiles. Again, as observed in Figure 10, for grid levels j = 5 or greater, the error profile stayed within the 1st and 2nd order descritization boundaries, and they aligned themseves closer to the 2nd order boundary as they approached the finest grid level.

Figure 9. IDS Error Analysis for ‘Isentropic’ Nozzles Error Analysis of ‘Varying Ranges & Profiles’ Problems

The plots illustrated in Figures 11 and 12 illustrate the scaling of the non-dimensional error behavour for Quasi-1D

Figure 10. IDS Error Analysis for ‘Shock’ Nozzles

problems with varying geometric ranges and varying geometric profiles at the throat section of the nozzle. Figure 11 illustrates the error behavior for cases #1, #2, #7, #8, #9 and #10 for both ‘isentropic’ and ‘shock’ conditions. Whereas, Figure 12 illustrates the error behavior for cases #1, #2, #3, #4, #5 and #6 for both ‘isentropic’ and ‘shock’ conditions. Once again, as observed in Figures 11 and 12, for grid levels equal to j = 5 or greater, the error profile stayed within the 1st and 2nd order descritization boundaries, and they aligned themseves closer to the 2nd order boundary they approached the finest grid level.




IDS Analysis of ‘Geometric Profiles’ Problems The order of discretization of the IDS scheme can also be

obtained from the expression [7],

( )2lnS-SS-Slnp

2hh

4h2h

= (23)

to yield, in general, a numerical scheme of (p+1)th order of accuracy[7, 8]. Applying equation (23) to Cases #1, #5 and #7 yeilds the results illustrated in Figure 13. These results indicate that the IDS consistently delivers a p value of 1 for grids below level 6. In other words, even with coarse grids, the strength of the IDS is demonstrated.


CONCLUSION In lieu of analytical methods, a set of numerical

experiments were created and implemented to analyze the error

behavior of the IDS. However, in this paper, only the spatial analysis was conducted and described. Moreover, to accommodate the experiments, the IDS was reformulated and applied to the Quasi-1D form of the NSE. It is of interest to note that the IDS was never applied to the Quasi-1D form of the NSE. As such, this analysis created a unique opportunity to independently evaluate the scheme while studying its error handling capabilities. Also note, the temporal error analysis of the IDS Quasi-1D equations will be the subject of a follow-on report, and no attempt to describe these effects is discussed herein.

The findings derived from this study indicate that the IDS as applied to Quasi-1D problems consistently delivers discretization orders of error of 1st order, and the order approaches 2nd order as the grids are refined. Also, a detailed analysis indicated that the IDS is able to capture the shock crisply between two nodes, even on coarse grids. These findings are contrary to what is observed with most established numerical schemes, where the shock is smeared across several nodes, and spurous effects dominate.

REFERENCES [1] Anderson, J. D., and Wendt, J., 1995, Computational fluid dynamics, Springer. [2] Ferguson, F., and Elamin, G., 2006, "A numerical solution of the NS equations based on the mean value theorem with applications to aerothermodynamics," Advanced Computational Methods in Heat Transfer IX, p. 97. [3] Elamin, G. A., 2008, "The Integral-differential Scheme (IDS): A New CFD Solver for the System of the Navier-Stokes Equations with Applications," North Carolina Agricultural and Technical State University. [4] Dodoo-Amoo, D. N. D., 2017, "An Error Analysis of the Integral Differential Scheme Navier-Stokes Solver " Doctoral Dissertation, North Carolina Agricultural and Technical State University (Pending). [5] Schulein, E., 2006, "Optical skin friction measurements in short-duration facilities," AIAA Journal, 44(8), pp. 1732-1742. [6] Hoffmann, K. A., and Chiang, S. T., 2000, "Computational Fluid Dynamics Volume II," Engineering Education System, Wichita, Kan, USA. [7] Roache, P. J., 1998, Verification and validation in computational science and engineering, Hermosa Albuquerque, NM. [8] Roy, C. J., 2003, "Grid convergence error analysis for mixed-order numerical schemes," AIAA journal, 41(4), pp. 595-604. [9] Pierce, N. A., 2000, "Analytic Adjoint Solutions for the Quasi-1D Euler Equations."

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