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Stress wave mitigation at suture interfaces
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2017 Biomed. Phys. Eng. Express 3 035025
(http://iopscience.iop.org/2057-1976/3/3/035025)
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Biomed. Phys. Eng. Express 3 (2017) 035025 https://doi.org/10.1088/2057-1976/aa777e
PAPER
Stress wavemitigation at suture interfaces
Nayeon Lee1,2, LakieshaNWilliams1,2, SungkwangMun1 , HongjooRhee1,3, R Prabhu1,2,KabindraRBhattarai4 andMFHorstemeyer1,3
1 Center for AdvancedVehicular Systems,Mississippi StateUniversity,Mississippi State,MS 39762-5405, United States of America2 Department of Agricultural and Biological Engineering,Mississippi StateUniversity,Mississippi State,MS 39762-9632, United States of
America3 Department ofMechanical Engineering,Mississippi StateUniversity,Mississippi State,MS 39762-9552, United States of America4 Department ofMechanical Engineering, Florida StateUniversity, Tallahassee, FL 32310-6046, United States of America
E-mail: [email protected]
Keywords: suture, sinusoidal interface, stressmitigation, stress wave, wave scattering, damping
AbstractThis study investigated the stresswave dissipation in sinusoidal patterned suture interfaces thatwereinspiredby sutures in biologicalmaterials. Finite element results showed that a sutured interfacedecreased the pressure 37%more than that at anunsutured interface,which arose fromwave scatteringand greater energy dissipation at sinusoidal boundaries. Stresswave scattering resulted in convertingcompressivewaves (S11) into orthogonalflexural (S22) and shearwaves (S12), whichdecreased both thepeakpressure (attenuation) andwave speed (dispersion).Higher strain energy occurring at suturedinterfaces brought energy losswithin viscoelastic gap, too. In addition,weparameterized severalvariables related to the suture interfaces for their influence in stresswavemitigation. The following sevenparameterswere examined: (1)waviness of suture (ratio of suture height to suture period), (2) ratio ofthe suture height over the entire bar thickness, (3) gap thickness, (4) elasticmodulus, (5) type of theboundary, (6) impact amplitude, and (7) impact duration.Thefinal result of the parametric studyrevealed that thehigh ratio of the suture over the entire bar thickness had the greatest influence, followedby the short impact duration, and thenby the lowelasticmodulus. Additionally, a high ratio of thesuture over the entire bar thickness and low elasticmodulus decreased the stresswave velocity aswell.Thesefindings canbe applied for designing various synthetic damping systems so thatmanmadeengineering designs can implement the optimized sutures for impact scenarios.
1. Introduction
Biological materials are remarkably designed forefficient mechanical behavior. One elegant example isa suture joint, which is a simple geometry yet multi-functional. In biological structures, suture joints arecommonly found where two stiff components inter-lock each other. For example, within the microstruc-ture of the woodpecker beak, a wavy sinusoidal-geometry was observed under the transmission elec-tron microscope (figure 1(a)). Compared to otherbirds, whose beaks’ impact resistance is less than thatof woodpeckers, the waviness of suture shown inwoodpeckers’ beaks is greater (Lee et al 2014).Figure 1(b) shows bison’s cranial suture, which hasbeen extensively researched. Researchers reported thatcranial sutures provide flexibility for growth, move-ment and strain due to masticatory and impact energy
dissipation (Hubbard et al 1971, Behrents et al 1978,Jaslow 1990,Herring and Teng 2000, Opperman 2000,Sun et al 2004, Yu et al 2004, Byron 2006, Seimetz et al2012, Curtis et al 2013). As shown in figure 1(c), theammonoid fossil also shows a wavy structure with ahierarchical fractal pattern on its shell. The suture ofthe ammonoid fossil has been studied to investigate itsmechanical role and relation between hierarchicalstructures of sutures and function (Allen 2006, 2007,Ubukata et al 2010). De Blasio (2008) reported thatcomplex suture lines dramatically diminished thestrain and the stress in the phragmocone such thatsuture fluted septum reinforced the shell againsthydrostatic pressure. The turtle shell also has suturejoints in their carapace as shown in figure 1(d). Krausset al (2009) conducted three-point bending tests on thesuture-contained turtle bony shell and reported thatthe turtle shell withstands small loads by low-stiffness
RECEIVED
7March 2017
ACCEPTED FOR PUBLICATION
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PUBLISHED
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© 2017 IOPPublishing Ltd
deformation and becomes much stiffer when theexternal load increases beyond a certain threshold.The suture of the leatherback turtle was also studiedand revealed that the suture caused the balancebetween tension and shear, and brought structuralflexibility by causing angular displacement (Chen et al2015).
Mechanically, the wavy suture can greatly enhancethe strength of materials. Jaslow (1990) experimentallystudied mechanical properties of sutures and reportedthat the suture increased bending strength. Similar
results on the tensile strength and bending strengthhave been reported as the suture plays a key role as anadditive to increase strength (Li et al 2011, 2012b,2013, 2014a, 2014b). In addition, a study of an inter-facial crack with hierarchical sinusoidal sutures foundthat sutures enhance interfacial fracture toughnessunderMode-I andMode-II loadings (Li et al2012a).
Although sutures are often found in the spot thatdynamic responses occur, mechanisms of aforemen-tioned properties of sutures during impact loadinghave not been extensively studied. Jaslow (1990)
Figure 1. Suture lines in biologicalmaterials; (a) suture line shown at the tip of woodpecker beaks (keratin); (b) cranial suture in abison skull (bone); (c)wavy line in a surface of an ammonoid fossil (calcium carbonate), and; (d) suture at a box turtle (bone).
Figure 2. (a) Idealized two-dimensional bar with a suture interface of gap thickness of b. The dimension of the bar is L=1000 mm,t=15 mm, and b=2 mm.The pressure initiated on the left side of the bar. (b) Schematic of an idealized bar with a flat interface. (c)Following an initial impact pressure applied in Region 1, the pressure datawere recorded at the eleven regions indicated by the redregions in the bar. Then, the peak pressures were connected by the red dotted line in the graph. As the pressure wave propagated in thebar, the peak pressure decreased.
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Biomed. Phys. Eng. Express 3 (2017) 035025 NLee et al
studied energy absorption using a pendulum on thecranial sutures of head-butting goats. Using finite ele-ment (FE) analysis, the role of cranial sutures wasinvestigated by Maloul et al (2014), who quantifiedhow sutures redistributed the stress. Zhang and Yang(2015) pointed out that hierarchically designed cranialsutures benefited the stress attenuation and energyabsorption.
Themain objective of the present study is to inves-tigate the geometrical effects of sinusoidal sutures onthe stress wavemitigation by using FEmodels. The fol-lowing sections detail the simulation setup, results,discussion, and conclusions.
2. Simulation set up
An idealized bar with a sutured interface (i.e., suturedbar) and an idealized bar with a flat interface (i.e.,unsutured bar) were created and analyzed from two-dimensional FE analysis in Abaqus/Explicit underdynamic conditions. As shown in figures 2(a) and (b),the dimension of the bar was 32 mm×1000 mm, inwhich one side of the bar was 15 mm×1000 mmwith a gap thickness of 2 mm. The wall was treated asan elastic and isotropic material with Young’s mod-ulus E=8 GPa, Poisson’s ratio ν=0.3, and densityρ=2000 kg m−3, and those material properties
Figure 3. Seven variables influencing the stress wavemitigation; (1) suturewaviness (ratio of suture height to suture period), (2)Rsuture, the ratio of the suture height to the bar thickness, (3) thickness of the gap, (4)material properties of the elastic wall, (5) type ofboundary, (6)Amplitude of the impact, and (7) impact duration. The default values are in bold font.
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generated a longitudinal wave speed of 2000 m s−1.The impact load was a Gaussian impulse and appliedon the left side of the bar as shown in figures 2(a) and(b) with the end nodes, on the same side, fullyconstrained to the y-direction. The sutured andunsutured gaps were treated as a viscoelastic materialin which the hyperelastic Ogden model was employedwith the elastic parameters, μ and a, being 15.6 KPaand 21.4 (Cheng and Gan 2007), respectively. Theviscoelastic properties were assigned by a Prony serieswith the viscoelastic parameters, d and t, being 0.549and 6.01 s, respectively, which were determined froma rat muscle study (Bosboom et al 2001). For meshing,a plane stress 4-noded element (CPS4R)was used, andthe approximate element size was 0.5 mm generatingabout 100 000 number of elements in the 2D bars.Then, a parametric study was performed to under-stand the dependence of suture geometric variablesand the external impact load. The seven variables were;(1) suture waviness, (2) Rsuture (ratio of the sutureheight to the entire bar thickness), (3) suture gapthickness, (4) elastic modulus of the wall, (5) geometryof the bar boundary walls, (6) amplitude of externalimpact load, and (7) impact duration. The detail of theexperimental case is described in figure 3. Whileexamining one variable, the other variables werefixed.
In order tomeasure the extent of dissipation in thesutured bar, pressure–time history data were recordedat eleven regions along the bar at every 100 mm, indi-cated by red regions in figure 2(c). The damping cap-ability of the sutured bar was then evaluated throughthe damping quotient, which is the ratio of the pres-sure decay from the ‘Region-1’ compared to ‘Region-11’ as the following:
Damping quotient
Pressure Pressure
Pressure. 1
region 1 region 11
region 1
=-
( )‐ ‐
‐
Further, the normalized phase velocity was alsoanalyzed to investigate the influence of sutures onwave dispersion. The following is the equation for thenormalized velocity
Normalized velocity
Phase velocity at current bar
Phase velocity at unsutured bar. 2= ( )
3. Results and discussion
FE simulations were carried out by applying externalmechanical loads to produce a stress wave thatpropagated in a continuum media. We examined thedamping capability of suture interfaces by comparingto an unsutured interface bar. Then, the variables ofthe suture interfaces such as the geometric variationsand boundary conditions were assessed by theirinfluence on the stress wave mitigation (pressurereduction of the travelingwavewithin the bar).
3.1.Dissipation of stress waves in the sutureinterfaceA sutured interface was able to reduce the stress waveeffectively compared to an unsutured interface.Figure 4 shows the peak pressure decay in the suturedcompared to an unsutured bar. The initial load was1MPa, and the peak pressure when the stress wavereached the end of the bar was 0.47MPa for theunsutured bar and 0.1 MPa at the sutured bar. While53% of the initial pressure dissipated during thepressure wave traveled the unsutured bar, 90% ofthe initial pressure dissipated at the sutured bar. Thesutured bar pressure was 37% less than that of theunsutured bar over the bar length used in this study.
There were two mechanisms associated with thesutured bar for stress wave mitigation as compared tothe unsutured bar. First, stress wave scattering occurredat the boundary of the sutured bar, in which compres-sive waves (S11) were converted into shear waves (S12)and into orthogonal flexural waves (S22). From a waveperspective, there are two basic types of wave motionfor mechanical waves: longitudinal waves and shearwaves (also called transverse waves). Displacements inlongitudinal waves occur in a parallel direction to thewave propagation, and in transverse waves, displace-ments occur in a perpendicular direction (Graff 1975).The waves related to S11 and S22 are longitudinalwaves, and thewaves related to S12 are shearwaves.
Wave scattering is an interaction of waves with aboundary or obstacles in a medium resulting in wavereflection, transmission, or refraction (Brekhovskikhand Goncharov 2012). Since the compressive inci-dence impinged the sinusoidal interfaces, wave scat-tering can be considered a reflection at a curvedsurface as described in figure 5. The reflected wavesconsist of longitudinal and shear waves with angles ofθL and θs, respectively. According to DasGupta andHagedorn (2007), wave scattering at boundaries can be
Figure 4.Comparison of pressure recorded along the suturedbar and unsutured bar. Each point represents the peakpressure at the eleven regions. The initial impact was 1 MPa,and the pressurewhen stress waves reached to the endwas0.47 MPa for the unsutured bar and 0.1 MPa for the suturedbar.
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Biomed. Phys. Eng. Express 3 (2017) 035025 NLee et al
defined as the following numerical expression. Thetotal wavefield can be represented as the following:
u x y t A n
A n
A a n
, , e
e
e ,3
L Lx y C t
L Lx y C t
S Sk x y C t
0 0i sin cos
i sin cos
i sin cos
L L L L
L L L L
S S S S
0 0 0=
++ ´
k q q
k q q
q q
+ -
- -
- -
( ) ˆˆˆ ˆ
( )
{ }
{ }
{ }
where u is the displacement, t is the time, A is theamplitude, κ is the wave number, and θ is the anglebetween the waves. L0, L, and S are the incident waves,reflected longitudinal waves, and reflected shearwaves, respectively. The directions of thewaves are
n
n
n
sin , cos ,
sin , cos ,
sin , cos . 4
L L LT
L L LT
S S ST
0 q qq qq q
== -= -
ˆ ( )ˆ ( )ˆ ( ) ( )
Also, the speeds of longitudinal wave and shearwave are
CE
, 5Lr
= ( )
CE
2 1, 6S
r g=
+( )( )
where E is Young’smodulus, g is the Poisson ratio, andr is the density. For given material properties in thisstudy, C 2000 m sL
1= - and C 1240.3 m s .S1= -
With an assumption that a reflecting surface is a freesurface, then the boundary conditions are as follows:
0, 0. 7y y12 0 22 0s s= == =∣ ∣ ( )
The boundary conditions produce the followingrelationships:
Csin sin sin ,C C . 8
L L L L S S
L L L L S s
0 0
0
k q k q k q= =k = k = k ( )
Then,
C
C, . 9L L S
S
LL0 0q q q q= = ( )
For the given conditions of this study, the angles ofthe reflected longitudinal waves are the same as the
Figure 6. Stress waves shown as a function of time in the eleven regions along the sutured bar illustrating the stress components of (a)S11, (b) S22, (c) S12, and along the unsutured bar also illustrating (d) S11, (e) S22, and (f) S12. The stress waves were plotted until thewave reached the end of the bar. For the sutured bar, the stress wave reached the end at 0.78 ms, and for the unsutured bar, the stresswave reach the end at 0. 55 ms.
Figure 5.Reflections of compressive incidence waves striking a curved boundary. The incident longitudinal wavewith an angle of θL0reflected to a longitudinal wavewith an angle of θL and a shear wavewith an angle of θS.
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Biomed. Phys. Eng. Express 3 (2017) 035025 NLee et al
angles of incident longitudinal waves. On the otherhand, the angles of reflected shear waves are 0.53 timesthe angles of the incident longitudinal waves.
As a result of wave scattering at the suture inter-faces, the magnitude of S11 decreased, and S12 andS22 increased (figure 6). The maximum S22 generatedin the sutured bar was 1.58 MPa approximately threetimes greater than that of the unsutured bar with S22equaling 0.54 MPa; the maximum S12 generated inthe sutured barwas 1.59 MPa approximately five timesgreater than that of the unsutured bar with S12 equal-ing 0.29 MPa. Not only does one observe a pressuredecay but also wave dispersion from wave scattering.The wave speed determined from equation (5) is2000 m s−1 when there are no boundary effects. In theunsutured bar with boundaries, the wave speed
decreased to 1818.18 m s−1 and arrived at 0.55 ms.Alternatively, in the sutured bar with boundaries, thewave speed decreased to 1282.05 m s−1 and arrived atthe free end at 0.78 ms. Accordingly, the sutured barinduced y-direction longitudinal (LE22) and shearstrains (LE12). Figure 7 shows that the sutured barinduced strains in the y-direction and shear directionbut decreased strains in the x-direction.
Figure 8 shows themaximum strain energy densityin the sutured and unsutured bar at Region-2 (near-front region)where the sinusoidal suture began so thatthe wave scattering started early. The peak strainenergy was 0.09 J in the sutured bar and 0.03 J in theunsutured bar. Hence, the sutured bar incurredapproximately three times greater strain energy thanthat of unsutured bar. Specifically, in the sutured bar,
Figure 7. Strain in the eleven regions along the sutured bar for the strain components of (a) LE11, (b) LE22, (c)LE12 and along theunsutured bar for the stress components of (d) LE11, (e) LE22, and (f) LE12.
Figure 8.Maximum strain energy density associatedwith different displacements in the x, y, and zdirections for the sutured andunsutured bar in Region 2 (indicated in figure 2(c))near the initial impact location. The total strain energy is greater for the suturedbar than that of the unsutured bar because the strain energy to yy and xy direction ismuch greater at the sutured bar.
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Biomed. Phys. Eng. Express 3 (2017) 035025 NLee et al
the strain energy was stored in all directions of xx(17.73%), yy (45.59), and xy (36.68%) due to reflectedstress waves, while in the unsutured bar most of thestrain energy stored was only in the xx direc-tion (86.43%).
Another mechanism that reduced the amplitudeof the traveling pressure wave was related to the strainenergy being stored in the viscoelastic suture gap. It iscommon to interleave viscoelastic layers between hardand stiff material to increase the damping of the struc-ture (Saravanos and Pereira 1992, Cupiał andNizioł 1995, Berthelot et al 2008), and biological mate-rials appear to employ the same strategy. Figure 9compares the strain energy of the gap between thesutured and unsutured bars. The viscoelastic gap mat-erial of the sutured bar allowed for energy dissipationsince the strain energy in viscoelastic material is pro-portional to the damping (Plunkett 1992).
3.2.Design variables affecting to stress wavemitigationA sinusoidal patterned interface caused a local com-plex stress redistribution, which led to wave attenua-tion and wave dispersion. In order to examine theinfluence variables regarding a sinusoidal pattern andboundary conditions, the seven variables shown infigure 3 were investigated using FE analysis. For eachcase, a pressure decay as stress waves propagated alongthe bar was observed. Also, compressive waves (S11),flexural waves (S22), and shear waves (S12) wereplotted to evaluate the transformation of longitudinalstress into the shear stress and flexural stress. Long-itudinal waves were recorded when pressure wavereached the end, and both of maximum flexural wavesand maximum shear waves were recorded whilepressure traveled the structure. Generally, as the long-itudinal wave decreased, the flexural waves and shearwaves increased. As one variable changed, the other sixvariables were fixed with the default value indicated infigure 3.
3.2.1. The effect of the suture wavinessWaviness is defined as the wave height divided by thewave period. Waviness was varied in six cases of 0.25,0.5, 0.75, 1, 1.25, and 1.5, in which the waviness heightwas fixed and the waviness width was changed. Asthe pressure wave traversed the sutured bar from theloading region to the free end, the magnitude of thepressure decreased when a suture was introduced(figure 4). However, with a suture, there was minimalrelationship between waviness and damping as shownin figure 10(a). Figure 10(b) showed that generatedshear stresses incurred the largest value at a wavinessratio of 0.5 and the generated flexural wave incurredthe largest value at a waviness ratio of 1. Hence, thegreatest conversion from a longitudinal stress to ashear stress and flexural stress were waviness ratios of0.5 and 1. We note here that the waviness ratio shownin figure 1 is 1±0.32 for the woodpecker beak;2.44±0.67 for the bison skull; 0.99±0.15 for theammonoid shell; and 0.97±0.23 for the turtle shell.
3.2.2. The effect of the Rsuture
Rsuture is defined as the suture height divided by the barthickness. The Rsuture was changed as 0, 0.10, 0.33,0.67, and 0.83. The height of the suture was changed as0, 1.5, 5, 10 and 12.5 mm while the bar thickness wasfixed at 15 mm. As the Rsuture increased, the pressurewhen the stress wave reached the end of the bardecreased as shown in figure 10(c). Figure 10(d)showed that a greater Rsuture increased the flexuralstress and shear stress, but the compressional stressdecreased. With respect to different animals and thehuman skull, the ‘bar thickness’ would be far greater.However, we are only concerned with the sutureheight but needed to normalize it with respect to someabsolute dimension to distinguish this feature fromthewaviness ratio.
3.2.3. The effect of the thickness of the gapThe gap thickness varies at different length scales forthe different animals. As such, we varied the suturedbar’s gap thickness: 1, 2, 4, and 6 mm. The thickness ofthe gap did not affect the amount of stress dissipation(figure 10(e)) and did not show a big difference whencomparing the shear stresses and flexural stresses(figure 10(f)) although the 2 mm, 4 mm, 6 mm of thegap thickness induced slightly more dissipation thanthe 1 mmgap.
3.2.4. The effect of thematerial propertiesFor the sutures in animals, the material comprisesmainly collagen, a structural protein that behaves likea viscoelasticmaterial. However, thematerial on eitherside of the viscoelastic collagen varied from bone tokeratin to other biological materials. Material proper-ties of the waveguide (the bar material in our study)determines the sound speed as the equations (5)and (6).
Figure 9. Strain energy occurring at the gap of the sutured bar(black) compared to the unsutured bar (red).
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Biomed. Phys. Eng. Express 3 (2017) 035025 NLee et al
In this study, five different elastic moduli weresimulated; 2, 8, 18, 32, and 50 GPa resulting in wavespeeds of 1000, 2000, 3000, 4000, and 5000 m s−1
accordingly. The dissipation occurred greater as thewave speed decreased (figure 10(g)), and also the timearriving at the end of the bar decreased. Figure 10(h)shows that the generation of a shear wave was notaffected by the wave speed while the generated flexuralwave decreased as the wave speed increased. As thewave speed increased, the longitudinal compressionstress proportionally increased when the stress wavereached the end of the bar.
3.2.5. Type of wall boundaryThe effect of the boundary was illustrated by the in-phase, out-of-phase, only center, and only outside
boundaries conditions (figures 10(i) and (j)). Stresswave dissipation was also examined with infiniteboundaries of the side walls to remove the boundaryeffect (figures 10(k) and (l)). Figure 10(i) showed thattherewas no difference in the damping andwave speedbetween the in-phase and out-of-phase boundaries.However, when changing the suture boundary to astraight boundary increased the wave speed indepen-dent of the centerline suture geometry or outsideboundary edge. Also, the results showed that aninteraction exists between the suture and the gap fordamping. Suture interfaces brought greater strainenergy to the gap compared to flat interfaces asdiscussed in figure 8. Hence, the damping of the barwith an only-outside-suture in which the suture-gapinteraction was absent was smaller than the other
Figure 10.Pressurewave decay as pressure waves traveled from the load applied region to the free end at the idealized bar with a sutureinterface, and compressional stress when the stress wave reaches to the free end,maximum shear stress andmaximum flexural stresswhile pressure wave traveling at seven variables of (a), (b)waviness, (c), (d)Rsuture which is the ratio of the suture height to the barthickness, (e), (f) thickness of the gap, (g), (h)material properties, (i)–(l) type of the wall boundary, (m), (n) amplitude of the loading,and (o), (p) impact duration. Each point in (k) is 10 mmapart while other points are 100 mmapart in the pressure decay graphs.
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Biomed. Phys. Eng. Express 3 (2017) 035025 NLee et al
Figure 10. (Continued.)
Figure 11.Pressure contour of (a) the sutured bar and (b) the unsutured bar with infinite boundaries on the sidewalls at the time of0.15, 0.1, 0.15, and 0.2 ms. The pressure fully dissipated at 140 mmaway from loading edge in the sutured bar and 460 mmaway in theunsutured bar.
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Biomed. Phys. Eng. Express 3 (2017) 035025 NLee et al
configurations. Figure 10(j) showed the stress trans-formation at the four types of boundaries. Althoughthe damping and wave speed were similar in in-phaseand out-of-phase suture boundaries, the maximumshear stress and the maximum flexural stress weregreater at the in-phase boundary than those of out-of-phase boundary.
Figure 10(k) shows the pressure decay for the barswith infinite boundaries on the side walls. Pressure wasrecorded every 10mm not 100mm, because the pres-sure dissipated quickly compared to the bars with finiteboundaries. Figure 10(k) demonstrated that the sutureslows down the stress wave and dissipates the wave
quicker than the unsutured bar. For the sutured bar, thepressure increased from the edge up to 10mm awayfrom the loaded edge due to the reflected S11 waves andgenerated S22 waves. The pressure then decreasedrapidly and dissipated fully after traveling 140mm awayfrom the loaded edge at 0.20ms (figure 11(a)). On theother hand, for the unsutured bar, the stress wave com-pletely dissipated after traveling 460mm away from theloaded edge at 0.23ms (figure 11(b)). The calculatedwave speed was 700m s−1 in the sutured bar and2000m s−1 in the unsutured bar. Figure 10(l) shows thatboth the generated maximum shear stress and max-imum flexural stress during stress wave propagation
Figure 12.The data points of the damping quotientwith its associated the curvefit and the data points of the normalized phasevelocity and the curvefitting at seven variables of (a), (b)waviness, (c), (d)Rsuture (ratio of the suture height to the bar thickness), (e), (f)thickness of the gap, (g), (h)material properties, (i)–(l) type of wall boundaries, (m), (n) loading amplitude, and (o), (p) impactduration.
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Biomed. Phys. Eng. Express 3 (2017) 035025 NLee et al
were greater in the sutured bar than those in the unsu-tured bar. The maximum shear stress in the sutured bar(3.02MPa) was approximately 11.2 times greater thanthat in the unsutured bar (0.27MPa), and themaximumflexural stress in the suturedbar (1.84MPa)was approxi-mately 3.2 times greater than that in the unsutured bar(0.57MPa).
3.2.6. The effect of the amplitude of the impulsive loadingThe amplitude of the impact loading was changed as0.25, 0.5, 1, 2, and 4 to investigate the damping effectscaused by an input condition of amplitudes. As theamplitude of impact increased, the pressure alsoincreased. However, the damping amounts remained
the same regardless of the amplitude of the loading asshown in figure 10(m). Figure 10(n) showed that as theamplitude of the loading increased, the stresses S11,S12, and S22 also increased.
3.2.7. The effect of the impact durationThe impact duration of the loaded pressure wave waschanged to 0.01, 0.02, 0.04, 0.08, and 0.16 ms in orderto investigate the damping effects resulting by an inputcondition of different periods (and/or frequencies).Results showed that as the impact duration increased,less dissipation occurred regarding the pressure wave(figure 10(o)), and the compressional stress convertedless to the flexural stress (figure 10(p)). Hence, we can
Figure 12. (Continued.)
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Biomed. Phys. Eng. Express 3 (2017) 035025 NLee et al
conclude that as the impact becomes faster and faster,the effect of the suture gets greater and greater in termsof dissipating the stress wave!
3.3.Damping quotient and phase velocityThe damping quotient and normalized phase velocitywere evaluated to quantify the variables’ effects onstress wavemitigation. Figure 12 shows the correlationof each variable with respect to the attenuation anddispersion of the pressure waves.
Figures 12(a) and (b) show the relationships betweenwaviness and the damping quotient/phase velocity.Because of the minimal relationship between the suturewaviness and damping quotient as evinced by a slopevalue of 0.04 (R2:45), andbetween thewaviness andphasevelocitywith a slope valueof 0.07 (R2:91), the suturewavi-ness essentially did not affect the damping. Rsuture versusdamping quotient was illustrated in figure 12(c) to showthat thedampingquotientwas proportional to theRsuture.The damping quotient increased from 0.57 to 0.94 as theRsuture increased from 0 to 0.83. Hence, for every unitincrement increase of Rsuture gives a 45% increase ofdamping. Also, the normalized phase velocity pro-portionally decreased as Rsuture increased (figure 12(d)).Jaslow (1990) reported that the amount of energy absor-bed may depend on the morphology of the suture, andour findings indicated that the amount of energy
mitigated was directly related to Rsuture rather than thewaviness. Figures 12(e) and (f) show that the gap thick-ness did not substantially affect the wave dissipation anddispersion. Figures 12(g) and (h) show that the soundspeed determined by material properties gave changes indamping and phase velocity proportionally but less thanthe Rsuture effect. Regarding the type of boundary withfinite boundaries, there was no difference in the dampingquotient and phase velocity between the in-phase andout-of-phase sutures while the absence of suture lines ledto less attenuation and dispersion as shown infigures 12(i) and (j). For the barswith infinite boundaries,the damping quotients were unity in both the suturedand unsutured bars, because all the pressure waves weredissipated before reaching the end (figure 12(k)). How-ever, sutures played an important role in dispersing thepressure waves in the bar with infinite boundaries(figure 12(l)) as thewave speed velocitywas reduced 65%.Figures 12(m) and (n) show that the impact amplitudedidnot correlate to thedampingquotient andphase velo-city. Figures 12(o) and (p) indicate that the impact dura-tion affected the wave attenuation but not the wavedispersion. As a result, the three variables includingRsuture, speed of sound, and impact duration affected thedamping quotient, and two variables including Rsuture,speedof soundaffected thenormalizedphase velocity.
Figure 12. (Continued.)
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4. Conclusions
One unique characteristic of biological materials is theeffective use of elasticity and viscoelasticity formitigat-ing and dissipating energy. Although shock absorberssuch as car bumpers or guard rails are designed toabsorb impact energy through plastic deformation,biological materials cannot use this strategy forabsorbing energy, because severe plastic deformationcould cause fatal damage. To keep structural integrity,biological materials use elastic and viscoelasticresponses effectively to dampen stress waves andabsorb energy. Sutures are found in nature in whichenergy absorption and stress wave damping areimportant, and they function in two roles: (i) sutureinterfaces transform longitudinal waves into shearwaves and flexural waves so that elastic deformationarises in not only the longitudinal direction but thetransverse and shear directions as well; and (ii) theinteraction between viscoelastic material in the gapand suture geometry lead to stress wave damping.
In addition, we investigated variations of sutureinterfaces and boundary conditions to evaluate theircorrelation to damping. As a result, there were threevariables that increasedwave attenuation: (i) high ratioof the suture height to the bar thickness, (ii) a shortexternal impact duration, and (iii) low sound speeddictated by the elastic modulus. The two variablescausing wave dispersion were a high ratio of the sutureheight over the bar thickness and a low sound speed. Ifthematerial properties and impact duration cannot becontrolled in the engineering design of a structuralcomponent or system, making the suture heightgreater becomes the only controllable design variablethatmatters.
Acknowledgments
The authors would like to acknowledge the supportfrom the Department of Agricultural and BiologicalEngineering, theDepartment ofMechanical Engineer-ing, and the Center for Advanced Vehicular Systems(CAVS) at Mississippi State University. Effort spon-sored by the Engineering Research & DevelopmentCenter under Cooperative Agreement numberW912HZ-15-2-0004. The views and conclusions con-tained herein are those of the authors and should notbe interpreted as necessarily representing the officialpolicies or endorsements, either expressed or implied,of the Engineering Research&Development Center ortheUSGovernment.
ORCID
SungkwangMun https://orcid.org/0000-0002-3347-2587
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