Compressing a Cylindrical Shell with a Rigid Core
CCompressing a Cylindrical Shell with a Rigid Core
Hung-Chieh Fan Chiang, Hsin-Huei Li, and Tzay-Ming Hong † Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan, Republic of China (Dated: January 22, 2020)Compressed cylindrical shells are common in our daily life, such as the diamond shape in rolled-upsleeves, crumpled aluminum cans, and retreated package of now defunct drinking straws. The kindof deformation is formally called the Yoshimura pattern. However, there are many other equallyprevalent modes of deformation, depending on the relative size of radius between the shell andits inner core, the thickness and rigidity and plasticity of the shell, etc. To elucidate the phasediagram for these modes, we combine molecular dynamics simulations and experiments to study theenergetic, mechanical, and morphological responses of a compressed cylindrical shell with a hardcore.
PACS numbers:
I. INTRODUCTION
The deformation of soft material is general and canbe due to mechanical force, temperature, pH value, hu-midity, electric field, and van der Waals interactions[1–12]. Plentiful examples exist in living creatures, such asthe wrinkling of skins[13–15], differential growth of bacte-rial biofilms[16], and pattern selection in growing tubulartissues[17]. Researchers who were interested at distinctundulating topologies for fruits and vegetables[18] or softelastic cylindrical shell[19, 20] have discussed differentmodes of deformation for core-shell structures. But, inreal-life examples of rolled-up sleeves, taken-off pants,shedding skin of snake, (now extinct) wrinkled wrappingwhen removed from drinking straws, or more exotic ba-nana leaves as shown in Fig. 1(a-c), there is a gap be-tween the core and shell. Intuitively this interval lendsmore freedom to the deformation and may allow for thecreation of new modes. To clarify this conjecture wewill discuss the morphology and dynamics of a cylindri-cal shell with a coaxial core whose radii R and R in areindependently varied. Note that R is defined as the av-erage of inner and outer radii of shell. In addition, thecompression rate v and shell properties, such as initiallength L , thickness t , and hardness, will be among ourtuning parameters.We use a stepping motor to compress cylindrical shellsof silica gel and paper with a rigid steel rod at its core,as shown in Fig. 1(d). To avoid friction, we lubricatethe core with a thin layer of oil before each round ofcompression. In order to better understand the distribu-tion and ratio of bending and stretching energies, molec-ular dynamics (MD) simulation will also be employed. Incontrast to just one mode (diamond) of deformation forstand-alone shells[21] and two (wrinkle and sagging) forcore-shell without a gap[19, 20], our core-gap-shell sys-tem can exhibit as many as five different modes. Amongthem, the spiral, ladder, diamond, and sagging modesshow up in thick shells, as shown in Fig.1(e, f, g, i).Whiles, only diamond, wrinkle, and sagging modes areobserved in Fig.1(g, h, i) for thin shells. FIG. 1: (color online) Examples of compressed cylindricalshell, (a) banana leaf, (b) package of chopsticks, and (c)rolled-up sleeves. Panel (d) shows the experimental set-up.Different modes of compressed shell can be observed: (e) spi-ral, (f) ladder, (g) diamond, (h) wrinkle, and (i) sagging. Theinset of (e, f) are silica gel tube, (g) is paper roll, and (h, i) areballoon. Further detail of inset (h): left part shows wrinkle,and the right part shows ridge.[20]
II. MD SIMULATION
Our MD simulation adopts the Weeks-Chandler-Anderson potential [22] to enforce the excluded volumefor each lattice point and define the length unit σ andthe energy unit (cid:15) . We choose a hexagonal lattice withmean spacing a = 1 . σ to form a cylindrical shell. a r X i v : . [ n li n . PS ] J a n Two impenetrable walls are arranged at both ends ofthe shell. The reduction of length to L is carried outby moving one of the walls. Since k b /k s = 3 t /
32 where k b and k s are the bending and stretching modulii [23],and two values of t , 0 . σ and 2 . σ , are tested by vary-ing k b = 10 ∼ (cid:15)/σ while holding k s fixed. In themean time, different materials are simulated by chang-ing k s for the same t . The elastic energy comprises twoforms: stretching energy E s = k s ( a − a ) / E b = k b ( θ − θ ) / a is the length betweenadjacent beads and θ is the angle spanned by three con-secutive beads along a lattice direction and θ = π alongthe length and θ = [(2 πR − / (2 πR )] π in the circularcross section. Plasticity is included by halving the mag-nitude of k b beyond a yield angle | θ − θ | of 10 ◦ [24].The compression rate is set at v = − − σ/v s where v s = σ ( m/(cid:15) ) / is the time unit and m is the mass ofbead. All simulations are performed using LAMMPS ver-sion 16Mar18[25]. III. MODE MAP
Since we already know that there are five modes, wecan assemble a mode map to characterize their corre-sponding stages during compression. Inputs from experi-ments and MD simulation tell us that plasticity promotesdiamonds while suppressing other modes, which obser-vation is consistent with our experience with a casuallyrolled-up sleeve. Readers may protest that sleeves canalso assume the neat and more stable mode of sagging.But this is created by deliberate folding and not in thescope of our discussion. The deformation mode of silicagel tube, representative of an elastic shell, is more diversi-fied than its plastic counterpart. The mode map consistsof three dimensionless parameters: reduced length
L/L , R − R in /t , and ks/v × ρ that incorporates the effectof hardness and compression rate where ρ denotes thesurface mass density of the shell in simulation.Note that all deformations originate from the movingend. The first mode that appears is spiral that consists oftwo parallel lines spiraling up the shell Further compres-sion introduces two sets of rungs that wrap around theshell and use the spirals as side rails to form the laddermode. The rungs appear crooked at first, but improvein their orientation, namely, becoming more parallel as L decreases. When the gap R − R in is increased from2 to beyond 5, a fourth mode of diamond will interposeitself between and coexist with ladder and sagging. Butif the gap is further widened to 10, there is no saggingafter diamond. A complete survey of modes can be foundin Fig. 2 which plots gap vs. L/L . Note that once thesagging shows up, the preceding mode, either ladders ordiamonds, will be quickly suppressed to resume the sur-face back to smoothness. In contrast, the transition fromladder to diamond is gradual.For an elastic thin shell, diamond is the first modethat emerges. (a) After the diamond pattern gradually FIG. 2: (color online) For panel (a)/(b) shows the deforma-tion mode of shell with R = 20, L = 200, v = − − , k s = 2 × , and t = 0 . / .
3. Different hardness in (c)/(d)with R = 20, R in = 16 L = 200, v = − − , t = 0 . / . spreads and covers the whole shell, wrinkles start to ap-pear and coexist with the diamond. (b) A second tran-sition happens when all the wrinkles are wiped out by asudden appearance of sagging. (c) Afterwards, wrinklesreemerge between and coexist with sagging and diamond.Further compression will introduce a second sagging andthe processes (b) and (c) are duplicated. As shown in Fig.2, the number of repetitions can be as frequent as four.We have checked the effect of plasticity which turns outto suppress all the deformation mode except diamond.Furthermore, plasticity renders the arrangement of dia-mond less periodic. IV. ENERGETIC RESPONSE
The energetic and mechanical responses for a thin shellfrom MD simulations are shown in Fig. 3(a, b) where E is the initial total energy and F is the resistance forceat L/L = 0 .
99. Being the first mode to appear, dia-mond increases its size with compression. It was men-tioned in Fig. 2 that there are only three modes fora thin shell. Wrinkling emerges and coexists with thediamonds at the first transition, which is characterizedby a discontinuous d ( E b /E s ) /dL . In contrast, each ofthe latter transitions introduces one more sagging andexhibits discontinuities in both E tot and E b /E s . Analo- FIG. 3: (color online) Panel (a)/(b) shows the ener-getic/mechanical response of R = 10, R in = 9, L = 200, v = − − , k s = 2 × , t = 0 .
23. Each thin straight linehighlight the pattern transition on the shell or the formationof the sagging. During the uniform diamond mode, Y and E b /E s remain a constant. For panel (c)/(d) shows the the en-ergetic/mechanical response of R = 20, R in = 18, L = 300, v = − − , k s = 2 × , t = 2 .
3. The transition is sim-pler than thinner shell, the first mode is spiral then turn intoladder and end with sagging. A constant E b /E s during lad-der mode, and decrease during sagging mode. Besides ladder,spiral and sagging remain a constant Y . gous to the latent heat when water freezes, the differenceof E tot is converted into kinetic energy before being dis-sipated as heat. In contrast to the total E b and E s inFig. 3(a), we have also measured those for individualmode. The ratio E b /E s turns out to be insensitive to L and remains at 3/2 for wrinkles and 0.8/1.4/2.1 forsmall/medium/large diamonds that appear respectivelyin L/L > .
93, 0 . < L/L < .
93, and
L/L < . E b /E s increase is that, as the height ofdiamond perimeter increases with compression, its widthnarrows to avoid stretching the surface, which impliesmore E b is required. The ratio E b /E s for sagging is alsonot a constant. This is expected because E b is storedmainly on the bending part which remains intact as sag-ging grows. As a result, the input work is totally con-verted to E s on the lengthening overlap part.Figure 3(b) tells us that the force of resistance isdiscontinuous at the second and following transitions.The Young’s modulus Y can be measured by taking thederivative of F with respect to L . The observation that Y equals roughly a constant in Fig. 3(b) should not betaken too seriously. We believe it is due to the fact thatdiamond remains the dominant mode for L/L > . Y will change when sagging take placediamond for further compression.Now let’s study the energetic and mechanical responsesfor thick shells in Fig. 3 (c, d). As depicted in Fig. 2,the first deformation to appear is a groove at either endof the shell, that is evidenced by a sudden surge of E b and E s with E b being the dominant energy. The groovespersist after spiral and ladder take turns at occupyingthe shell. The ratio E b /E s for these two modes is de- termined to be roughly 0.4 and 0.45. Unlike thin shells,sagging does not coexist with other modes, i.e., the thickshell resumes to smoothness as soon as sagging appears,which transition is signified by a discontinuity in E tot .As further compression lengthens the sagging, the foldededge where E b mainly resides remains roughly the same,while E s increases linearly with L − L . In the meantime, the shape of folded edge flips between being roundand blunt as the edge being pushed by the fictitious wall,which adds an oscillatory part to E b /E s vs. L/L in Fig.3(c).Experimental results for silica gel tube are plotted inFig.4(a) where each dip represents the appearance ofa new sagging. A lower compression rate v exhibits ahigher F . The shorter shell, as denoted by triangles, ex-hibits a steeper slope than diamonds in Fig.4(a). This isexpected from F = ( Y A/L )( L − L ). However, if mul-tiplied by their respective L , the slope for triangles willbecome slightly smaller than that for diamonds. This im-plies that, unlike homogeneous material, Y is not pure anintrinsic property, but may depend on L for a shell withdeformations. The effect of R in is also checked. The factthat the cube data overlap with diamonds before saggingimplies that, as long as R in is nonzero, Y is insensitiveto its value. And, once entered, sagging always enhances Y besides the induction of a sudden drop in F at thetransition.Our simulation results without plasticity in Fig.4(b)show that a shorter L , larger R in , and higher v requirea stronger F to achieve the same L − L . These trends areconsistent with those of silica gel in Fig.4(a). One fur-ther parameter that we tested in simulations is the effectof thickness t . As expected, thicker samples also requirea larger force. To test the effect of plasticity, we alsoarrange to sample on paper roll to obtain Fig.4(c) andcontrast it with simulations with plasticity in Fig.4(d).Two characteristics that are unique to plastic shells arethat the mode of deformation is reduced to one, i.e., di-amonds, and rings of diamonds emerge discretely. The F is found to fluctuate whenever a new ring of diamondsappears, similar to the behavior of paper roll. But oursimulations fail to predict the overall increase of F as L − L increases. Since there was no such a trend inanother experiment[21] without a core, We believe theincrement in F must be due to the increasing number ofcontacts and frictional force between the shell and thecore. V. RELEASE
After we roll up the sleeve, the deformation can persistfor a while due to its friction with our arm. How aboutthe compression shell for a purely elastic shell in MDsimulations? In other words, is the compression processintrinsically irreversible? Or is plasticity crucial to in-duce hysteresis? This is the goal of our following simula-tions in which the compressing wall is suddenly removed.
FIG. 4: (color online) Resistance force is plotted against re-duced length for real samples. (a) Silica gel tube with different R in (mm), L (mm) and v (mm/s), while R = 4 . R = 20 isfixed. (b) Paper roll with different t (mm) while R = 15 . R in = 11mm, L = 180mm, and v =0.41mm/s are fixed. Incontrast to Fig. 3, plasticity is added to MD simulations with R = 20 , L = 300, and v = 10 − to obtain plot (c) in orderto compare with (a) and (b). The recoil speed v recoil and total energy are recorded andplotted in Fig.5(a). It can be seen that v recoil varies withmodes and is the smallest for sagging.The thick shell in MD will always stuck at sagging andnever recover to smooth again. We use E t ot/E as thedegree of deformation to plot the relation to the simula-tion time t for thin shell in MD. We find that harder andmore space can release faster, and independent of tem-perature. The temperature represent the kinetic energyof the beads, which means the fluctuation of the shellis independent of the relaxation in simulation. To avoidthe shell brittle or melt, it can not change the range oftemperature to check this property in experiment. Forelastic case, as shown in Fig. 5(a), the stepped energeticresponse represents the untying of the sagging. Note thatif and only if R − R in >
3, the sagging can be untied.Compare to the plastic case, as shown in Fig.5(b), it ishard to form sagging during compression, which meansthat there will no stepped energetic response and thereis a good agreement, with R ∼ .
99, of the relation: E tot /E = a × exp( − t/τ ) + E min , for a is dependent onthe maximum total energy of each different case, τ isrelaxation time and E min is E tot /E at the equilibriumstate. FIG. 5: (color online) We shift the time axis to 0 when therelaxation starts, and distinguish the (a) elastic and (b) plas-tic case. (a) shows the harder material recover faster with R = 20 , R in = 16 , L = 200 , t = 0 .
23, and independentof temperature in the inset. (b) compares the deformationof simulation and experiment with plastic at the right side,and shows larger space, i.e. R − R in , recover faster with R = 20 , L = 300 , v = 10 − , t = 0 . VI. DISCUSSION
Although the morphological and mechanical responsesobtained by our simulation are consisting with exper-imental results, some discrepancies still exist. For in-stance, the diamond mode that appears in simulationsfor a thick shell with R in /R > . L/L when different modes switch.Intuitively one would imagine R in not to matter whenthe deformation is not serious and the shell never touchesthe core. However, this is limited to the early stage ofcompression when L/L > .
95. As a result, it is stillacceptable for us to use the parameter ( R − R in ) /R asa label for the mode map.We acknowledge the financial support from MoST inTaiwan under grants 105-2112-M007-008-MY3 and 108- 2112-M007-011-MY3. † [email protected][1] V. Shenoy and A. Sharma, Phys. Rev. Lett., , 119(2001).[2] W. Monch and S. Herminghaus, Europhys. Lett., , 525(2001).[3] S. C. Cowin, Annu. Rev. Biomed. Eng., , 77 (2004).[4] J. Genzer and J. Groenewold, Soft Matter, , 310 (2006).[5] S. Q. Huang, Q. Y. Li, X. Q. Feng and S. W. Yu, Mech.Mater., , 88 (2006).[6] J. Kopecek, Biomaterials, , 5185 (2007).[7] L. He and L. Qiao, Europhys. Lett., , 14003 (2007).[8] W. Yang, T. Fung, K. Chian and C. Chong, J. Biomech., , 481 (2007).[9] W. Hong, X. Zhao, J. Zhou and Z. Suo, J. Mech. Phys.Solids, , 1779 (2008).[10] I. Tokarev and S. Minko, Soft Matter, , 511 (2009).[11] B. Li, X. Q. Feng, Y. Li and G. F. Wang, Appl. Phys.Lett., , 021903 (2009).[12] K. Li and L. He, Int. J. Solids Struct., , 2784 (2010).[13] K. Efimenko, M. Rackaitis, E. Manias, A. Vaziri, L. Ma-hadevan, and J. Genzer, Nat. Mater. , 293 (2005).[14] S. Zeng, R. Li, S. G. Freire, V. M. Garbellotto, E. Y.Huang, A. T. Smith, C. Hu, W. R. Tait, Z. Bian, G.Zheng, D. Zhang, and L. Sun, Adv. Mater. , 1700828 (2017).[15] J. Yin, G. J. Gerling, and X. Chen. Acta Biomaterialia, , 1487 (2010).[16] C. Zhang, B. Li, X. Huang, Y. Ni, and X.-Q. Feng, Appl.Phys. Lett. , 143701 (2016).[17] P. Ciarletta, V. Balbi, and E. Kuhl, Phys. Rev. Lett. , 248101 (2014).[18] J. Yin, Z. Cao, C. Li, I. Sheinman, and X. Chen, Proc.Natl. Acad. Sci. U.S.A. , 19132 (2008).[19] N. Stoop and M.M. M¨uller, Int. J. Nonlinear Mech. ,115 (2015).[20] Y. Yang, H. H. Dai, F. Xu, and M. Potier-Ferry, Phys.Rev. Lett. , 215503 (2018).[21] W. Johnson, P.D. Soden, and S.T.S. Al-Hassani, J. StrainAnal. , 317 (1977).[22] J. D. Weeks, D. Chandler, and H. C. Andersen, J. Chem.Phys. , 5237 (1971).[23] S. F. Liou, C. C. Lo, M. H. Chou, P. Y. Hsiao, and T.M. Hong, Phys. Rev. E , 022404 (2014).[24] https://github.com/Lina492375qW1188/Plasticity-LAMMPS[25] S. J. Plimpton, J. Comput. Phys.117