Geometric imperfection, known as a detrimental effect on the buckling load of cylindrical shells, has a new role under the emerging trend of using buckling for smart purposes. Eigenshape-based geometries were designed on the shell surface with the aim of tailoring the postbuckling response. Fourteen seeded geometric imperfection (SGI) cylinders were fabricated using polymer-based 3D printing, and their postbuckling responses were numerically simulated with a general-purpose finite element program. Results on the prototyped SGI cylinders showed a tunable elastic postbuckling response in terms of initial and final stiffness, the maximum load drop from mode switching, and the number of snap-buckling events. A response contour and discrete map is presented to show how the number of waves in the axial and circumferential directions in the seeded eigenshape imperfection can control the elastic postbuckling response. SGI cylinders provide diverse design opportunities for controllable unstable response and are good candidates for use in smart and adaptive materials/structures.

## Introduction

Recognizing the positive features of elastic instabilities for smart and adaptive applications has been gaining considerable momentum [1,2]. Among the many kinds of instabilities (e.g., buckling, snapping, wrinkling, and crumpling), this paper considers the buckling and postbuckling of cylindrical shells. While buckling is typically avoided due to the loss of load-carrying capacity that ensues, its response offers several positive features, including high-rate motion, sudden energy release, and multiple equilibrium states. These features are considered as opportunities under the perspective that they can offer advantages for the design of smart or adaptive devices.

The motivation behind this research follows the increased interest in the use of instabilities for the development of energy harvesters and self-powered sensors [3,4]. The second author and his colleagues have showcased a new energy harvesting concept for pseudostatic conditions that uses the postbuckling response of a bilaterally constrained axially compressed strip as a trigger for oscillating piezoelectric harvesters. The proposed device is to be attached, or embedded, to structural components that are experiencing quasi-static structural deformations, such as those resulting from service loads and temperature fluctuations in large civil infrastructure. Once the axial deformations, or strains, are imposed on the strip, multiple snap-through buckling events can be attained due to the presence of bilateral constraining walls. As a result, localized high-rate motions can be triggered to excite oscillating piezoelectric harvesters from a low-rate global input. Controlling the noted dynamic features in the postbuckling regime of slender structures can significantly improve the possibility of using them in the development of smart devices.

In this paper, we extend the endeavor noted above by exploring an alternative structural prototype, namely, cylindrical shells. Cylindrical shells have advantages over one- and two-dimensional elements when it comes to their ability to attain an elastic (and thus recoverable) postbuckling response with multiple mode transitions. Yet, predicting and controlling the postbuckling behavior of cylindrical shells is critical to realize their potential in the design of smart devices.

The modification of geometrical features on the shell surface is one of the approaches that can lead to a tunable and controllable postbuckling behavior for cylindrical shells under axial compression. In the 1950s, Yoshimura [5] observed the changes in buckling wave number on axially compressed shells and noticed that localized buckling was different from global buckling, even though they may have an equivalent buckling load. Several recent studies have showcased the use of artificial imperfections as a positive factor in the design of cylindrical shells. Lindgaard et al. [6] used the combination of mode shapes to find an optimal shape with the lowest buckling load. Ning and Pellegrino [7] designed cylindrical shells with a wavy cross section to make the structure less sensitive to the imperfections. A recent study by Lee et al. [8] introduced a dimplelike geometric imperfection to spherical elastic shells. Despite these efforts, most of them focused only on critical buckling, or knockdown factor to the critical load, and not the far elastic postbuckling response.

Prior studies by the authors [9–12] have shown that the elastic postbuckling response of cylindrical shells can be modified by providing large and strategically designed imperfections, namely, a seeded geometric imperfection (SGI). Specifically, the first paper [10] only presented the SGI concept as one of three approaches to modify the postbuckling behavior. Experimental results on three SGI designs showed that the type of postbuckling response could be modified. In the next paper [11], the main focus was to further address some key design issues including the reliability of the test protocol, the sensitivity to initial imperfections, the efficiency of an optimization framework, and the dynamic features in the postbuckling response. Only three SGI designs were evaluated. After addressing these general issues, this paper presents a combined numerical and experimental study to expand the SGI strategy by comprehensively exploring a large number of design cases that represent the entire design domain, and the design parameters that modify and control features of interest in the elastic postbuckling regime were evaluated. The presented design insight has direct implications on the development of smart devices relying on the kinetics of large deformation (i.e., buckling and postbuckling response). Management of the extracted energy is an important issue for implementation of the concept but it is outside the scope of this paper and thus not dealt with here.

## Seeded Geometric Imperfection (SGI) Design

The concept of a seeded imperfections to design cylindrical shells with a tailored elastic postbuckling response is inspired by the well-known procedure for capturing postbuckling behavior in numerical analysis procedures, namely, the use of linearly superposed buckling eigenshapes onto the cylinder's perfect geometry. The biggest challenge in simulating the postbuckling response of thin-walled shells is typically the selection of a suitable seeding imperfection. A numerical study by Burgueño et al. [12] has shown that a discrepancies between simulations employing the noted method and experimental results still exist in the far elastic postbuckling regime. Most of the recent studies on imperfection sensitivity have used stochastic methods to develop less conservative guidelines for the design of axially compressed cylindrical shells [13]. For the proposed SGI cylinders, instead of opening the design domain to arbitrary shapes and combinations, a single mode shape with a large amplitude was selected to provide a governing role in the seeded imperfection. Clearly, introducing large geometric imperfections to the system will further reduce the critical buckling load. However, as shown in Fig. 1, and demonstrated later in this paper, the approach leads to advantages on other aspects. Further details on the SGI design concept can be found in our previous work [11].

The numerical and experimental studies presented in this paper are based on a reference/baseline cylindrical shell design with an effective length of 80 mm, an inner radius of 40 mm, and a thickness of 0.5 mm. This geometry was chosen based on an identified range (from the literature [12,14–18]) of geometrical ratios (*L*/*R* and *R*/*t*) that can lead to an elastic postbuckling response, on the size constraint of the available 3D printer used for fabrication of test units, and on pilot testing that showed that the desired features of the elastic postbuckling response could be attained. It should be noted that a perfect cylindrical shell using the following parameters (*t* = 0.5 mm, *E* = 1285 MPa, and *ν* = 0.2) would buckle at 1189 N based on the classic buckling critical load equation: $Pcr=2\pi Et2/3(1\u2212\nu 2)$. In this study, the seeded imperfection was obtained from buckling modes that combine a different number of axial and circumferential buckling waves. These buckling modes can be described by sine functions with different wave number combinations and can be illustrated along a circle (i.e., the Koiter circle) in a dimensionless wavenumber space. Figure 2 shows the Koiter circle for the noted baseline shell (obtained from Eq. (1) in Ref. [11]) along with representative buckling modes in terms of a dimensionless axial and circumferential wavenumber. The significance of *m–n* points in the Koiter circle is that a large number of possible equilibria coexist in the postbuckling regime [19].

Pilot test on three SGI cylinders by Hu and Burgueño [10] supported the hypothesis that SGI cylindrical shells can be designed for tailorable elastic postbuckling behavior. In specific, SGI designs can have superior features over uniform shells in terms their ability to attain three different response types in their postbuckling regime: stiffening, sustaining, and softening. By contrast, the typical postbuckling response of uniform cylindrical shells is only of the softening type, that is, a progressive reduction of load bearing capacity. The explored SGI cylinders also had distinct features in their postbuckling response, such as the number of buckling events, the magnitude of released strain energy at mode jumping events, and the initial stiffness. The research presented in this paper expands this initial knowledge with a quantitative evaluation on the entire design domain by selecting 14 mode shapes (as indicated by the dashed circle in Fig. 2) to study the effect of axial and circumferential buckling wavenumbers on controlling, and thus designing, the elastic postbuckling response of cylindrical shells.

## Methods and Materials

Experimental evaluation was conducted through compressive testing of 14 SGI cylinders fabricated through 3D printing (MakerBot Replicator 2, MakerBot Industries, New York). The material used was polylactic acid (PLA). The elastic modulus in the radial and vertical directions of the shells was 3.05 GPa and 2.57 GPa, respectively (determined from tensile tests on printed coupons as per ASTM D638). The seeded geometry generated through the abaqus analyses was converted into a stereolithography file for 3D printing. The cylindrical shells had a total length of 100 mm with 10 mm connecting regions on each side, as shown in Fig. 3. The tolerance of the 3D printer is 0.1 mm.

The SGI cylinders were subjected to uniform axial compression in a universal testing frame. The loading protocol consisted of loading and unloading under displacement control for a total shortening of 1.0 mm at a rate of 1/300 mm/s. Stiff top and bottom loading fixtures were used to minimize damage to the shell edges, and loading was applied through a spherical bearing at the center of the top fixture to distribute edge forces evenly. Further details on the test setup can be found in Ref. [11].

Numerical simulations were conducted using the finite element program abaqus [20] following the procedure reported in Ref. [11]. The same geometrical parameters described in the experiments were used for the numerical model. In the previous study [11], a discrepancy between numerical and experimental result was found using the elastic moduli from tensile tests. According to a study [21] on PLA materials in 3D printing, the compressive modulus is typically 50% of the tensile modulus. Thus, orthotropic material properties were defined in the abaqus models with *E*_{1} = 1.285 GPa, *E*_{2} = 1.51 GPa, and *ν*_{12} = 0.2, where 1 and 2 represent the axial and circumferential directions, respectively. The rest of the material constants were calculated from strength-of-materials relations. The cylinders were modeled with four-node shell elements with reduced integration (S4R). Up to 500 buckling mode shapes were extracted by the Lanczos method. All cylinders were seeded from a single mode with a maximum amplitude of 1 mm from the midplane. Postbuckling response was predicted using abaqus's dynamic explicit solver.

## Results and Discussion

Figure 4 shows the postbuckling response features and parameters used to evaluate results and observations from the experimental and numerical studies. The trace shown in the figure is from a test on an SGI design (*m* = 8 and *n* = 8). The number of mode transitions (*n _{t}*) is of highest interest, since they are directly associated with localized buckling events. In this test, the SGI design had five localized buckling events. For each buckling event, the maximum load drop (

*ΔP*

_{max}) and the maximum separation of the critical points (

*δ*

_{max}) are of interest because they are related to the amount of released and accumulated strain energy, respectively. For this SGI design, the maximum load drop was equal to the first load drop (

*ΔP*

_{max}=

*ΔP*

_{1}), while the maximum separation of the critical points was observed between the third and the fourth buckling event. The change in the cylinder's axial stiffness from its initial value (

*K*) to that at the unloading point (

_{i}*K*) is used to determine the cylinder's residual capacity. It can be seen that in the present SGI design,

_{e}*K*is significantly lower than

_{e}*K*. Finally, the area (

_{i}*A*) inside the hysteretic force–displacement response is of relevance due to its correlation with the dissipated strain energy from the equilibrium path transitions.

The postbuckling behavior for all 14 SGI cylinders was, as is to be expected, very different in terms of the noted parameters of interest. A response contour based on the test data was plotted in terms of four parameters: *K _{i}*, shown in Fig. 5(a),

*ΔP*

_{max}, shown in Fig. 5(b),

*A*, shown in Fig. 5(c), and

*n*, shown in Fig. 5(d). The intent of these contour maps is to understand the effect of shape wave numbers for the entire design domain of the eigenshape seeded imperfection using test results. It can be seen from the plots in Figs. 5(a)–5(c) that the larger response values lie in the upper left corner, where the circumferential wave number (

_{t}*n*) is large and the axial half-wave number (

*m*) is small. This indicates that a seeding geometry with a lower number of axial waves can achieve a higher initial stiffness, a higher single load drop from mode transitions, and a larger hysteretic area.

A prior numerical study by Hu and Burgueño [9] found that there is a strong correlation between *K _{i}*,

*ΔP*

_{max}, and

*A*. This observation is confirmed by the tests reported here. Another interesting finding from the results in Figs. 5(a)–5(c) is that the “hotspot” (i.e., the region with largest parameter value) was from the data of cylinder M5N8, which lies exactly at the top of the Koiter circle. Further, the data in Fig. 5(d) show that multiple mode jumps in the postbuckling regime were more frequent when the seeded imperfection mode shapes had a relatively large number of waves in the circumferential direction (

*n*). The hotspot in this case was from cylinder M5N11. Test results for all SGI cylinders are given in Table 1, which shows that postbuckling response features can be significantly modified by the chosen seeded geometry.

Mode | m | n | K (N/mm)_{i} | P_{max} (N) | ΔP_{max} (N) | δ_{max} (mm) | A (kJ) | n_{t} |
---|---|---|---|---|---|---|---|---|

192 | 2 | 5 | 1203.0 | 377.6 | 20.6 | 0.18 | 60.6 | 1 |

200 | 2 | 8 | 852.1 | 395.2 | 24.2 | 0.32 | 84.1 | 2 |

290 | 5 | 2 | 589.9 | 333.7 | — | — | 26.9 | 0 |

161 | 5 | 5 | 609.8 | 356.7 | — | — | 21.7 | 0 |

46 | 5 | 8 | 1153.4 | 706.5 | 129.5 | 0.10 | 92.0 | 3 |

188 | 5 | 11 | 943.2 | 456.5 | 24.4 | 0.14 | 39.2 | 8 |

31 | 8 | 2 | 478.5 | 304.6 | 2.5 | — | 29.8 | 1 |

19 | 8 | 5 | 559.7 | 390.0 | 1.9 | — | 40.1 | 1 |

69 | 8 | 8 | 676.3 | 392.4 | 41.9 | 0.16 | 41.9 | 5 |

248 | 8 | 12 | 974.0 | 428.8 | 25.6 | 0.24 | 52.8 | 3 |

63 | 11 | 2 | 408.5 | 299.4 | 11.2 | — | 29.0 | 1 |

151 | 11 | 8 | 981.4 | 561.4 | 23.6 | 0.09 | 58.9 | 5 |

153 | 11 | 11 | 815.4 | 402.1 | 4.8 | 0.10 | 44.8 | 2 |

198 | 12 | 5 | 610.1 | 500.4 | 70.2 | — | 49.9 | 1 |

Mode | m | n | K (N/mm)_{i} | P_{max} (N) | ΔP_{max} (N) | δ_{max} (mm) | A (kJ) | n_{t} |
---|---|---|---|---|---|---|---|---|

192 | 2 | 5 | 1203.0 | 377.6 | 20.6 | 0.18 | 60.6 | 1 |

200 | 2 | 8 | 852.1 | 395.2 | 24.2 | 0.32 | 84.1 | 2 |

290 | 5 | 2 | 589.9 | 333.7 | — | — | 26.9 | 0 |

161 | 5 | 5 | 609.8 | 356.7 | — | — | 21.7 | 0 |

46 | 5 | 8 | 1153.4 | 706.5 | 129.5 | 0.10 | 92.0 | 3 |

188 | 5 | 11 | 943.2 | 456.5 | 24.4 | 0.14 | 39.2 | 8 |

31 | 8 | 2 | 478.5 | 304.6 | 2.5 | — | 29.8 | 1 |

19 | 8 | 5 | 559.7 | 390.0 | 1.9 | — | 40.1 | 1 |

69 | 8 | 8 | 676.3 | 392.4 | 41.9 | 0.16 | 41.9 | 5 |

248 | 8 | 12 | 974.0 | 428.8 | 25.6 | 0.24 | 52.8 | 3 |

63 | 11 | 2 | 408.5 | 299.4 | 11.2 | — | 29.0 | 1 |

151 | 11 | 8 | 981.4 | 561.4 | 23.6 | 0.09 | 58.9 | 5 |

153 | 11 | 11 | 815.4 | 402.1 | 4.8 | 0.10 | 44.8 | 2 |

198 | 12 | 5 | 610.1 | 500.4 | 70.2 | — | 49.9 | 1 |

Besides different *m–n* combinations, the amplitude of the seeded imperfection provides an additional variable to tailor the postbuckling behavior. A series of tests were thus conducted on an SGI design using the same seeding geometry (*m* = 8 and *n* = 8) but three different amplitude factors (A20 = 0.1 mm; A200 = 1 mm; and A500 = 2.5 mm). The resulting postbuckling responses are shown in Fig. 6, from which it can be seen that variation of the seeding amplitude for a fixed seeding geometry can also generate diversity in the postbuckling response within a certain range (i.e., within upper and lower bounds). In terms of mode transitions, the cylinder with smaller seeding amplitude (M8N8-A20) had a much larger number of localized buckling events (nine) compared to the other two cylinders. It can also be seen that seeding amplitude has a knockdown effect on the initial stiffness, the magnitude of the load drops, and the area of the hysteretic loop. A previous study by Hu and Burgueño [11] proved that such postbuckling response is repeatable if the system boundary conditions do not change.

Improved agreement of the numerical simulations with experimental data was found by seeding imperfections from the superposition of one governing mode shape (i.e., the seeded shape) with a large amplitude plus 20 random mode shapes with a very small total amplitude (0.01 mm for each mode or 2% of the shell thickness). Obviously, the seeded geometry was expected to have a governing role over the other small imperfections on the postbuckling response. Numerical simulations were conducted on selected baseline SGI cylindrical shells to evaluate the efficiency of the modeling approach. Figure 7 shows results for two cases (out of 14) for cylinders with a seeded imperfection amplitude of 200% of the shell thickness. The predicted postbuckling response curve for cylinder M5N2 is close to the experimental one, while there is larger discrepancy between responses for cylinder M8N8. The insets in Fig. 7 show that the seeded geometry in cylinder M8N8 has a higher number of localized inward regions than cylinder M5N2, which makes cylinder M8N8 more vulnerable to initial random imperfections. Nonetheless, it is interesting see that the numerical and experimental results for cylinder M8N8 attained the same number of mode transitions. The most important point is that the simulated responses captured the general postbuckling response features for two rather different designs (M8N8: softening response and M5N2: stiffening response), even though the simulated curves did not exactly match the experimental data, with relatively minor modeling difficulty. Although the challenge of exactly predicting the postbuckling behavior of SGI cylinders still exists, the presence of a governing mode shape as a seeded imperfection results in much better agreement between simulation and experimental results compared to predictions for the postbuckling response of uniform cylinders [11]. This was true particularly for cylinders with relatively bigger waves, e.g., M5N2.

Response contours for *K _{i}* and

*n*are shown in Figs. 8(a) and 8(b), respectively, based on the simulation results of the 14 tested cylinders. Contours for

_{t}*ΔP*

_{max}and

*A*are not presented given their direct correlation with

*K*. It can be seen that the contour maps from numerically simulated data are very similar to the experimental ones in Figs. 5(a) and 5(d). While there are some differences in the actual values, the trends and hotspots are consistent. Disagreement between experimental and simulation results still exists and is expected; but the favorable comparison indicates that the key postbuckling characteristics were successfully captured by the simulations.

_{i}The evaluation presented above, with reference to the results in Figs. 7 and 8, confirmed the approach to numerically simulate the postbuckling response and mode transitions in an axially compressed cylindrical shell with seeded imperfections. After such verification, the responses of two cylinder groups were numerically estimated according to the design domain shown in Fig. 2, and the resulting postbuckling response types (stiffening, sustaining, or softening) for each *m–n* combination were plotted as discrete response maps in Fig. 9. The first simulation group, shown in Fig. 9(a), featured SGI cylinders with a seeded imperfection amplitude of 200% of the total shell thickness. It can be seen that for a seeding amplitude of 200% the majority of designs with softening response (type 3) are outside the Koiter circle, while almost all designs with a sustaining response (type 1) are along the perimeter of the half circle or inside the circle. Further, the number of designs with sustaining response (type 2) is limited, and they are scattered inside and outside the circle. It is also interesting to note from Fig. 9(a) that softening response can be achieved by seeding a geometry with either a large number of circumferential waves (*n* > 8) or a large number of axial waves (*m* > 10).

The second simulation group considered SGI cylinders with an imperfection amplitude of 500% of the total shell thickness. The discrete response map is plotted in Fig. 9(b). It can be noted that the number of designs with sustaining type response (type 2) significantly increased, particularly in the region where the SGI cylinders were seeded with a large number of circumferential waves (*n* > 8). The reason is that changes in seeding amplitude can lead to a transition between response types. An increase in seeding amplitude has a significant knockdown effect on the first critical buckling load such that a softening-type response transforms into a sustaining response, or even a stiffening response. Further, it should be noted that some SGI cylinders did not change their response type even if the amplitude was varied, particularly in the regime where the axial half-wave is less than five. Overall, the numerical results presented in this section show the opportunities of “imperfections by design” as an approach to attain a tailorable postbuckling elastic response in cylindrical shells.

## Conclusions

Use of elastic buckling and postbuckling response in smart and adaptive materials and structures has gained increasing attention in the past decade. Such growing interest has pushed the limits on the understanding and expectations of response for many traditional structural components. Cylindrical shells, primarily used as containers, may now turn into a promising prototype for smart applications. The presented extends the knowledge from prior studies on the concept of seeded geometric imperfection by illustrating a strategy for SGI cylinder design by experimentally and numerically evaluating cases that spanned the entire design domain and covered the majority of the desired postbuckling responses.

Response contours from experimental results showed that SGI cylinders with a relatively large number of waves in the circumferential direction can attain more localized buckling events in the postbuckling regime. Further, it was shown that a seeded geometry with a lower number of axial waves can lead to a shell with higher stiffness, large load drops from postbuckling mode jumps, and larger energy dissipation from an elastic loading/unloading cycle. It follows that diverse response characteristics in the elastic postbuckling regime can be achieved, and controlled, by changing the seeding shape and its amplitude.

The difficulty of predicting the postbuckling response of SGI cylindrical shells is reduced by the presence of the governing mode shape as the seeded imperfection. A numerical modeling strategy was introduced by combining one large-amplitude mode shape (the seeded imperfection) with 20 other random higher-order modes with a smaller amplitude (0.05% of the seeded imperfection amplitude). The simulations captured reasonably well the key postbuckling characteristics as judged by comparisons with experimental data.

A discrete contour map for the postbuckling response type (stiffening, sustaining, or softening) of SGI designs was developed through numerical simulations to illustrate how such a map can be used to design cylinders with targeted elastic postbuckling behavior through the selection of axial and circumferential wave numbers in the seeded geometric imperfection. The response maps show diverse opportunities for controllable unstable response, which can be of interest for use in smart and adaptive material/structural concepts.

## Acknowledgment

The authors gratefully acknowledge the assistance of Dr. Wassim Borchani on the experimental testing, and the staff at MSU's Division of Engineering and Computing Services on specimen manufacturing.

The research described in this paper was carried out with partial funding from the U.S. National Science Foundation under Grant Nos. ECCS-1408506 and CMMI-1463164.

## Nomenclature

*A*=dissipated strain energy (enclosed area of response curve)

*K*=_{i}initial stiffness before the first bifurcation

*n*=_{t}number of mode transitions

*P*_{max}=maximum buckling load

*ΔP*_{max}=maximum load drop between first bifurcation and unloading point

*δ*_{max}=maximum spacing between localized buckling events