3D Printing and Mechanics

By Lawrie Virgin

Three-dimensional (3D) printing has an element of magic about it. 3D-printed objects—which one creates with high precision and relatively little expense according to mathematical descriptions—serve as simple physical manifestations that augment many facets of mathematics and mechanics.

Although sophisticated 3D graphical capabilities have advanced considerably in recent years, there is still something to be said for producing solid objects that one can appreciate in a tactile sense. Demonstration models that speakers can hand around during a presentation occupy a useful middle ground between a purely theoretical treatment of a concept and a lab-based measurement experience. In particular, flexible 3D-printed objects provide a pedagogical opportunity to teach the behavior of elastically deformable structures. Given the underlying theory of deformable mechanics, these objects can also lend some physical insight into the fundamental mathematics that describes their behavior, including linear algebra and differential equations.

Here I describe a few examples of the ways in which 3D printing has become a useful addition to the teaching of structural mechanics at the undergraduate level. In addition, I offer some brief examples of 3D printing in the (nonlinear) research realm.

Figure 1. Sets of slender cantilevers. 1a. Static deflection that is subject to end loads. 1b. Vibration used to illustrate sympathetic resonance.

Flexure

Simple Cantilever Bending

When subject to a lateral end load \(F\), a uniform cantilever of length \(L\), width \(b\), and thickness \(d\) (giving a second moment of area \(I = bd^3/12\)) that is made from a material with Young’s modulus \(E\) will deflect by amount \(\delta = FL^3/(3EI)\) at the tip. One may obtain this result by integrating \(EI d^4/dx^4=0\) together with appropriate boundary conditions to yield a static deflection shape in the form of a cubic polynomial. One can easily print sets of cantilevers in which the geometry is varied systematically; for example, students might print specimens that are based on their ID numbers, e.g., of different lengths. Making simple measurements allows users to present data relating to \(F\) vs. \(\delta\) in a nondimensional form, e.g. \(FL^2/EI\) vs. \(\delta/L\). Even in a tactile sense, this action highlights the meaning of stiffness and the role of \(EI/L^3\) in flexure [4]. 

One can make the geometry a little more interesting by including the effect of ribs, taper, and a “turnaround” that may deflect upward at the tip despite the application of a downward force (see Figure 1a). Though the governing differential equations become a little more involved, this exercise reinforces an appreciation of the relation between flexural behavior and geometry. Users can employ a similar approach for thin elastic rings, panels, and frames. For the latter, one obtains the deflection by solving a set of simultaneous algebraic equations (essentially the finite element method) with additional insight into features of the stiffness matrix (rank, symmetry, positive-definiteness, etc.) and corresponding physical behavior, for example. For a cantilever in which the load \(P\) is applied axially at the free end, buckling occurs when the magnitude reaches \(P_{cr}=\pi^2 EI/(4L^2)\)—the lowest eigenvalue obtained from \(EI d^4/dx^4 + P d^2/dx^2 = 0\)—and a (buckling) mode shape that consists of trigonometric functions (an analysis that dates back to Leonhard Euler).

Figure 2. Curved surfaces that relate to bifurcation scenarios. 2a. Examples of catastrophe of a single variable. 2b. Nonlinear resonance of a periodically forced oscillator.

Simple Cantilever Oscillations

When plucked, the cantilever will oscillate in flexure with a dominant vibration frequency (in hertz) that is given by \(\omega = \beta/(2 \pi) \sqrt{EI/\rho A L^4}\). Here, \(\rho\) is density, \(A\) is the cross-sectional area, and \(\beta\) is a dimensionless coefficient (its value is \(3.516\) for a cantilever). One can obtain this result by solving the governing equation \(EI\partial^{4} w/\partial x^{4} + \rho A \partial^{2} w/\partial t^{2}=0\) and assuming harmonic motion (in time) and appropriate boundary conditions. The vibration mode shapes consist of hyperbolic functions. Again, this scenario provides an opportunity for researchers to scrutinize frequency’s dependence on geometry — e.g., with frequency proportional to the inverse length squared but independent of width \(b\). An interesting extension of this example is based on illustrating sympathetic resonance [3]. Figure 1b depicts a set of 16 nominally identical cantilevers that are grouped into four sets of four different lengths. When one cantilever is plucked, the other three of the same length begin to vibrate in sympathy. The small amount of energy that is transferred along the base is sufficient—with light damping—to excite the neighbors with the same natural frequency. A tiny, right-angled member is printed at the end to accentuate the visual effect.

Solid Shapes

Geometrical Nonlinearity

Analysis of the cantilever and similar beam-type problems is based on Euler-Bernoulli beam theory, which is appropriate for small deflections and oscillations. In essence, a linear relation exists between force and deflection that is founded on Hooke’s law. However, there are many examples wherein a structure—typically of more complicated geometry—can exhibit a wide range of nonlinear behaviors. This fact brings us into the realm of bifurcation theory. For example, snap-through buckling may characterize the sudden transition between adjacent equilibrium configurations during which stability is lost via a saddle-node bifurcation. The classic Euler column is a prime example of a pitchfork bifurcation. Typical 3D-printer material (e.g., acrylonitrile butadiene styrene thermoplastic) and the resolution limits on slenderness mean that 3D printing is not necessarily well suited for large deflections. However, there are other applications for 3D printing in a nonlinear context.

Figure 3. Three-dimensional (3D) prints associated with co-existing solutions. 3a. A printed bifurcation diagram that evolves into numerous branches, some of which are disconnected (and artificially attached to keep their positions in space). 3b and 3c. Iso-potentials that correspond to a fixed value of the control parameter.

Equilibria and Potential Energy

Figure 2a shows three examples of elementary bifurcations: the cusp, butterfly, and swallowtail catastrophes [5]. When viewed from different angles, the curved surfaces—especially the turning points—provide clear interpretations between the behavior and control parameter. A cusp is also evident in the resonance response of a nonlinear oscillator, e.g., in Duffing’s equation. The bending-over of the response amplitude as a function of forcing frequency becomes more pronounced under greater forcing magnitude (or lighter damping). This phenomenon is evidenced in Figure 2b, where the appearance of hysteresis as the forcing frequency slowly varies.

Unfolding bifurcation scenarios sometimes lead to the branching behavior in Figure 3a. Bifurcations often emanate from a fundamental solution, losing stability and encountering new branches as a control parameter is varied. At a fixed value of the control, one thus might have many equilibrium points (both stable and unstable) that may correlate with extremum values in an associated potential energy. Figures 3b and 3c depict iso-potentials in three dimensions. The 3D prints inspire an appreciation of the connectivity between equilibria at a given energy level [1], especially in terms of the index of saddle points and possible transitions in the context of dynamics.

Figure 4. Non-simple shapes that result from highly nonlinear behavior in dynamics. 4a. The Lorenz attractor. 4b. The Rossler attractor. 4c. Mandelbrot set. 4d. Koch octahedron. The data for each element can be found using the following numerical codes from [2]: 4a: 575618, 4b: 1425939, 4c: 748, 4d: 1123004.

Chaos and Fractals

3D printing is also well-suited to the production of representations of some archetypal models in nonlinear dynamics and chaos. The Lorenz and Rössler equations produce classical chaotic attractors, which appear in Figures 4a and 4b, respectively [2].

Finally, fractal objects are an intriguing possibility that push the limit of 3D printing’s resolution. With self-similarity, the resolution game can of course never be won. Figure 4c shows the Mandelbrot set; the convergence is conventionally assigned a color coding, but here it provides the third dimension (height). Figure 4d depicts a Koch curve within a tetrahedron shape. The prints in 4c and 4d were achieved with a high-resolution 3D printer.

There is a myriad of other possibilities for 3D printing to illustrate features in mathematics, mechanics, and dynamical systems. New printer materials, higher printer resolutions, and the appeal of physical objects all point in the direction of increased use of 3D printing in pedagogical settings.

The figures in this article are courtesy of the author. 

Acknowledgments: The author acknowledges the assistance of Eli Nichols (Gilero), Yue (Susan) Guan (Duke University), and Eric Stach (Duke University)

References
[1] Guan, Y., & Virgin, L.N. (2020). Stability and bifurcation in discrete mechanical systems: An experimental and analytical study. Int. J. Bifurc. Chaos, 30(9), 2030024.
[2] MakerBot. Thingiverse. Retrieved from https://www.thingiverse.com/.
[3] Virgin, L.N. (2018). Sympathetic resonance. Am. J. Phys., 86(6), 439-442.
[4] Virgin, L.N. (2020). Comparative structural stiffness: Exploiting 3D-printing. Am. J. Phys., 88(12), 1049-1058.
[5] Zeeman, E.C. (1976, April). Catastrophe theory. Scientific American, pp. 65-70, 75-83.

Lawrie Virgin is a professor of mechanical engineering and materials science at Duke University. He is former chair of Duke’s Department of Civil and Environmental Engineering and has been a faculty member at Duke since 1988, prior to which he received his Ph.D. from the University College London. Virgin’s research interests are centered on nonlinear mechanics, especially buckling, nonlinear vibration, and their interactions. Applications of his research include ship capsize, aeroelasticity, marine risers, rocking blocks, control, sonic fatigue, solar sails, and the dynamics of slender structures.