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Modeling hybrid polymer–nanometal lightweight structures

  • Bharat BhagaEmail author
  • Craig A. Steeves
Original Paper
  • 24 Downloads

Abstract

Increasing the environmental sustainability of aviation is a key design goal for commercial aircraft for the foreseeable future. From the perspective of structural engineering, this is accomplished through reducing the mass of aircraft components and structures. Advanced manufacturing techniques offer new avenues for design, enabling more complex structures which can have highly tailored properties. One advanced manufacturing concept is the use of 3D printed polymer preforms that are coated with nanocrystalline metal through electrodeposition. This enables the use of high-performance materials in virtually any geometry. To exploit this manufacturing approach, it is incumbent to have well-established mechanical models of the behavior of such hybrid structures. In particular, hybrid polymer–nanometal structures tend to fail due to compressive instabilities. This paper describes a model of local shell buckling, a typical compressive instability, as it applies to hybrid polymer–nanometal structures. The analysis depends upon the Southwell stress function model for radially loaded solids of rotation, and couples this with the Timoshenko analysis of local shell buckling. This combination is applicable to a range of practical configurations for truss-like hybrid structures.

Keywords

Compression Buckling Filled-shell Microtruss Nanocrystalline Instability 

1 Introduction

The design of commercial airliners is currently driven strongly by environmental considerations. The International Air Transport Association policy is to maintain carbon-neutral growth in global commercial aviation from 2020, and to achieve 50% reduction in total carbon emission from commercial aviation by 2050. Given that total commercial air travel, measured by passenger kilometer, is projected to quadruple by 2050, this implies a reduction in carbon emissions per passenger kilometer of at least 80%. This unavoidably entails large reductions in the amount of fuel burnt during commercial aircraft operations. Attaining such reductions requires significant step-changes in technology, in aerodynamics, propulsion, structures, and operations: evolutionary changes in technology will not be adequate.

In several of these categories, the most promising major technological advances have been identified. In the aerodynamics field, dramatically increased use of aerodynamic shape optimization [15] and the use of unconventional aircraft configurations, such as blended wing bodies [16], will be essential. With respect to propulsion, open rotor engines [3] or boundary-layer ingestion [10] is promising options that significantly increase fuel efficiency. Electric propulsion [6], which may be distributed [5], is also attractive options. Biofuels are certain to be an important contributor to the sustainability of any propulsion system that continues to use hydrocarbons [11]. Operational changes, such as the use of continuous descent landing [17] or formation flying [23], will also contribute to the overall sustainability of aviation.

In the area of advanced structures, the most promising options for improving sustainability are not as well defined. The overwhelming majority of the environmental impact of aviation arises during aircraft operation, rather than manufacture or disposal, so, from the perspective of the structural engineer, reducing the weight of aircraft is paramount, because it reduces the amount of fuel necessary for flight. The complexity of material and structural systems will increase because of the advantages that can be achieved by increasing the parameters available for design. Clearly, the wide use of composite materials, for their exceptional combination of low mass with high strength and stiffness, will continue. Automated fiber placement to provide greater tailoring will likely become common. Computational structural design techniques, such as topology optimization [1], will also play a significant role in improving the efficiency of aircraft structures.

Hybrid materials are another option for increasing material complexity. Hybrid materials are unconventional combinations of material classes or fabrication approaches. A particular hybrid upon which this paper will focus is the combination of 3D printed polymer with electrodeposited coating of nanocrystalline metal. Nanocrystalline metals have exceptionally small crystal grain sizes, typically less than 100 nm diameter. Because of the Hall–Petch effect, this has the consequence that the yield strength of nanocrystalline metals is extremely high, often above 1 GPa [7]. The conventional fabrication techniques cannot be used to form nanocrystalline metals, because they are too strong for machining, and are unstable when subject to high temperatures. Electrodeposition is a feasible fabrication technique, and, provided that the substrate onto which the deposition is to take place can be treated to create an electrically conductive surface; virtually any shape can be coated. 3D printed polymer is, hence, an excellent option, because 3D printing enables the construction of nearly any shape.

While nearly any geometry can be fabricated using this process, an attractive configuration is to print the polymer in a truss-like configuration, followed by electrodeposition of nanocrystalline metal onto the polymer substrate [8]. This takes advantage of the structurally efficient geometry of a truss, the high second moment of area of a coated cylinder, and the exceptional strength of the nanocrystalline metal. In addition, the nanocrystalline metal can be tailored during the electrodeposition process to control the grain size and, hence, to control the yield strength of the material [7]. This additional control enables further optimization of the microtruss geometry to reduce mass without sacrificing strength or stiffness [14].

Lausic et al. [14] analyzed and tested minimum mass configurations for polymer/nanocrystalline metal hybrid microtrusses. A consistent phenomenon was that failure of the microtrusses in three-point bending was associated with buckling instabilities in the compression members of the microtruss. Failure occurred in both global Euler buckling of struts, or in local shell buckling of the nanocrystalline metal coating. While the polymer substrate acted to strengthen the nanocrystalline metal coating against local shell buckling, it did not prevent the coating from failing through this mechanism. Local shell buckling of hollow tubes is well modeled by Timoshenko and Gere [22], and there have been several attempts at modeling local shell buckling of tubes filled with an elastic material.

The problem of local shell buckling of filled cylindrical shells was driven by the need to understand buckling of solid propellant rockets under axial loads [4, 12, 18, 24]. The earliest attempt at a filled-shell buckling model by Myint-U [18] emulated the hollow-shell derivation approach used by von Karman and Tsien [13]. Uniquely, the Myint-U [18] model assumed that a shear interaction between the shell and core was important. Independent studies by Seide [19] and Karam and Gibson [12] later showed that the shear interactions between the shell and the core were negligible. More recently, Karam and Gibson [12] utilized a foundation model derived by Gough et al. [9], which was originally intended for buckling of face sheets on sandwich panels subject to axial compression. Of all these theories, the Karam and Gibson’s [12] theory has been the most successful.

This paper describes an alternative approach to modeling local shell buckling of a hollow cylinder filled with an elastic material. It employs the Timoshenko and Gere [22] analysis to describe the behavior of the nanocrystalline metal shell, and the Southwell [20] stress functions for an axisymmetrically loaded elastic cylinder to model the behavior of the core material. Combining these two models with appropriate boundary conditions enables the creation of a general model for local shell buckling of filled tubes subject to axial compression. This approach is amenable to modeling the other configurations, including when the core has a hole in its center, and when the core and coating do not adhere. Additional manipulation of the model, by fitting an analytic equation to the numerical results generated by the Southwell model, enables its use for optimization by providing a differentiable expression for the calculation of sensitivities. The basic model for local shell buckling of a tube filled with elastic material will be detailed in this paper.

2 Physical considerations

For this analysis, consider a polymer cylinder with radius r and length l perfectly bonded to a shell of nanocrystalline metal of thickness t, as shown in Fig. 1. The relevant properties of the polymer are the Young’s modulus \(E_{\mathrm{{c}}}\) and Poisson’s ratio \(\nu _{\mathrm{{c}}}\), while the corresponding properties of the nanocrystalline metal are the unsubscripted E and \(\nu \). When subjected to an axial load sufficiently small that both materials remain elastic, such a cylinder may deform in one of three modes. The first mode is direct compression, where the shell and the core both compress elastically. The second mode is global Euler buckling, while the third mode is local shell buckling. The three modes are illustrated in Fig. 2.
Fig. 1

Geometry of a cylindrical shell filled with an elastic material, in a cylindrical (\(r, \theta , z\)) coordinate system

Fig. 2

Buckling modes of deformation for hybrid polymer–nanometal cylinders subject to an axial deformation \(\Delta \): a direct compression, no buckling; b global Euler buckling; c local shell buckling

Euler buckling occurs when the cylinder is relatively long and slender, while local shell buckling is dominant when the shell is thin and the core material compliant, regardless of the length of the cylinder. This paper is concerned exclusively with the local shell buckling mode of failure. In the local shell buckling mode, the outer shell, in this case a nanocrystalline metal coating, develops short-wavelength ripples. The development of the ripples is opposed by the presence of the elastic polymer core material, which, because it is bonded to the shell, must develop the corresponding ripples on its surface. This tends to reduce the propensity of the shell to buckle, and, hence, increases the axial load that the cylinder can support before local shell buckling begins.

Prior to the onset of local shell buckling, both the shell and the core compress uniformly and store elastic strain energy consistent with the external energy associated with the loading and end displacement of the cylinder. At the initiation of buckling, the shell begins to ripple, which leads to additional mechanisms of storage of elastic strain energy. First, the bending of the shell stores strain energy. Second, circumferential cross sections of the shell must stretch or compress to accommodate the formation of the bending ripples; this stores additional strain energy. Finally, the deformation of the core material to conform with the buckling of the shell stores further strain energy.

The mode of deformation, whether direct compression or local shell buckling, is determined by the relative rates of energy storage of the two modes of deformation. At the outset of loading, when the deformation is near zero, the direct compression mode (Fig. 2a) is more energy efficient than the local shell buckling mode (Fig. 2c). This means that the rate of energy storage per unit axial deformation is smaller for the direct compression mode than for the local shell buckling mode. As the shell and core deform, the amount of energy stored per unit deformation increases for the direct compression mode. Eventually, the energy storage per unit deformation for the local shell buckling mode is smaller than that for the direct compression mode, and the cylinder experiences local shell buckling. The point at which this happens is called the bifurcation point, and can be expressed mathematically as follows:
$$\begin{aligned} \frac{\partial U_{\text {ic}}}{\partial \Delta } = \frac{\partial U_{\text {ib}}}{\partial \Delta } = \frac{\partial U_{\text {e}}}{\partial \Delta }, \end{aligned}$$
(1)
where \(\Delta \) is the unit of axial deformation, \(U_{\text {ic}}\) is the internal energy under continued axial compression (Fig. 2a), \(U_{\text {ib}}\) is the internal energy of the filled cylinder as it undergoes buckling (Fig. 2b, c), and \(U_\text {e}\) is the external energy due to the applied displacement.
After the formation of buckling wrinkles, the axial displacement \(\Delta \) and the radial displacement amplitude A in Fig. 2c are related geometrically through:
$$\begin{aligned} \Delta = \frac{A^2 m^2 \pi ^2}{4l}. \end{aligned}$$
(2)

3 Mathematical model

To analyze the physical process described above, all contributors to energy storage must be identified and modeled accurately. For hollow shells without the polymer core, Timoshenko and Gere [22] developed an expression for the stress \(\sigma \) at which local shell buckles would appear:
$$\begin{aligned} \sigma = \frac{E}{\sqrt{3 \left( 1 - \nu ^2 \right) }} \frac{t}{r}. \end{aligned}$$
(3)
This model accounts for both the bending and stretching of the shell. Tennyson [21] demonstrated that this model is fundamentally correct, but actual structures are highly sensitive to imperfections and tend to have strengths significantly below the prediction. Nonetheless, this model is adequate to capture the behavior of the coating during local shell buckling of a filled cylinder.
As noted above, there have been several attempts to model the local shell buckling of a filled cylinder. Karam and Gibson [12] were most successful; their model for the critical load in an axially loaded filled shell is:
$$\begin{aligned} F= & {} 2\pi Et^2 \left( 1 + \frac{r}{2t}\frac{E_{\mathrm{{c}}}}{E}\right) \left[ \frac{1}{12(1-\nu ^2)}\frac{r/t}{( \lambda _{\mathrm{cr}}/t)^2}\right. \nonumber \\&+\left. \frac{(\lambda _{\mathrm{cr}}/t)^2}{r/t} + \frac{2(E_{\mathrm{{c}}}/E)}{(3-\nu _{\mathrm{{c}}})(1+\nu _{\mathrm{{c}}})} \frac{\lambda _{\mathrm{cr}}}{t} \frac{r}{t} \right] . \end{aligned}$$
(4)
The critical buckling wavelength, \(\lambda _{\mathrm{cr}}\), is found through solution for the first positive real root of the quadratic equation arising from:
$$\begin{aligned} \frac{\partial F}{\partial \lambda } = 0. \end{aligned}$$
(5)
The Karam and Gibson model relies on the Gough et al. [9] model for the behavior of the core material. This model is derived for a planar sandwich panel, and is, hence, not axisymmetric, and has other drawbacks for the filled cylinder configuration.
An alternate approach is to employ the Southwell [20] model, which is appropriate for solids of revolution that are loaded radially. This enables the determination of the stress field within the core material and, hence, the stored elastic strain energy. The Southwell model employs a pair of stress functions, \(\phi \) and \(\psi \), that, when solved, enable the determination of the stress field by differentiating the stress functions and combining the derivatives appropriately. The differential equations to be solved for \(\phi \) and \(\psi \) are:
$$\begin{aligned}&\displaystyle \frac{\partial ^2 \phi }{\partial r^2} - \frac{1}{r} \frac{\partial \phi }{\partial r} + \frac{\partial ^2 \phi }{\partial z^2} = 0,\quad \text {and} \end{aligned}$$
(6)
$$\begin{aligned}&\displaystyle \frac{\partial ^2 \psi }{\partial r^2} - \frac{1}{r} \frac{\partial \psi }{\partial r} + \frac{\partial ^2 \psi }{\partial z^2} = \frac{\partial ^2 \phi }{\partial z^2}. \end{aligned}$$
(7)
A solution must be found for this pair of equations that is consistent with the required boundary conditions. For the local shell buckling of coated cylinders, the boundary conditions are the radial displacements associated with the rippling of the coating. This solution can be found numerically by discretizing the domain of interest and solving through finite differences. Once the solutions for the stress functions are found, they can be numerically differentiated to get stress fields, through:
$$\begin{aligned}&\displaystyle \sigma _r = \frac{1}{r} \left( \frac{\partial \phi }{\partial r} + \frac{\partial \psi }{\partial r} \right) - \frac{1}{r^2} [\psi + (1-\nu )\phi ], \end{aligned}$$
(8)
$$\begin{aligned}&\displaystyle \sigma _\theta = \frac{\nu }{r}\frac{\partial \phi }{\partial r} + \frac{1}{r^2} [\psi + (1-\nu )\phi ], \end{aligned}$$
(9)
$$\begin{aligned}&\displaystyle \sigma _z = -\frac{\nu }{r}\frac{\partial \psi }{\partial r},\quad \text {and} \end{aligned}$$
(10)
$$\begin{aligned}&\displaystyle \tau _{zr} = \frac{1}{r}\frac{\partial \phi }{\partial z}. \end{aligned}$$
(11)
These also provide the strain fields through the application of the theory of elasticity. The strain energy within the core, \(U_{\mathrm{{c}}}\), for a given amplitude, A, of radial displacements due to local shell buckling is found through:
$$\begin{aligned} U_{\mathrm{{c}}} = \int \limits _V \int \limits _\epsilon \sigma \mathrm {d}\epsilon \mathrm {d}V = \frac{m}{2} \pi K A^2 r, \end{aligned}$$
(12)
where V is the volume of the polymer cylinder and the integral involves triaxial states of stress, \(\sigma \), and strain, \(\epsilon \). The second equality, where m is the number of half wavelengths of the buckle pattern over the length of the cylinder, can be calibrated for K from the numerical calculations; K is a constant for a given set of polymer material properties, \(E_{\mathrm{{c}}}\) and \(\nu _{\mathrm{{c}}}\). The advantage of using the fitted form is that it is easily differentiable, allowing the use of Eq. 5 in the final solution for the critical local shell buckling load.
Fig. 3

Comparison of models of local shell buckling. This figure includes the elastic model described in this paper, the Karam and Gibson [12] model, and the Timoshenko and Gere [22] model for an unfilled shell

Fig. 4

Comparison of models of local shell buckling to finite-element predictions for equivalent geometry and material properties

With the solution for the stresses in the polymer core material, the strain energy in the core can be combined with the strain energy in the nanocrystalline metal coating, and an energy analysis performed to determine the minimum load for local shell buckling. The resulting equation is cumbersome and will not be stated here, but can be solved numerically. Figure 3 shows the non-dimensional load index (the critical buckling load F non-dimensionalized by \(Er^2\)) at which local shell buckling occurs as a function of the ratio of shell thickness to radius (t / r). The predictions provided by this new elastic model are very similar to those given by the Karam and Gibson [12] model. Figure 4 shows both models, along with finite-element predictions from Abaqus modal analysis.

Both models agree well with the finite-element simulations. While the finite-element verification suggests that the new model may be slightly more accurate than the Karam and Gibson [12] model (depending upon the accuracy of the finite-element simulations), the Karam and Gibson [12] model is significantly easier to implement. However, the advantages of the new model lie outside the analysis of highly simplified geometry and material conditions shown here. The new model can (where the Karam and Gibson [12] model cannot) be adapted easily for cores that are not solid and instead have cylindrical holes through their axes, and for shells that are not bonded to the cores [2]. Experiments have shown that all three of these issues are significant for realistic geometries and material properties [2].

4 Concluding comments

This paper details and analysis of local shell buckling of filled cylinders based upon the Southwell [20] model for radially loaded solids of revolution. By fitting a simple expression to the numerical solutions required in the new model, ease of use is recovered and a standard variational approach can be used to determine critical buckling loads. The analysis closely reproduces finite-element verification calculations, and has the advantage that it can be adapted for situations that cannot be covered by the existing models. This approach also proves to be very useful in optimization studies, where the mass of a structure is minimized for a given stiffness or load-carrying capacity.

Notes

Acknowledgements

Financial support for the first author of this paper was provided through the National Sciences and Engineering Research Council Collaborative Research and Training Experience Program Grant 414123-2012, “Environmentally Sustainable Aviation”.

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Copyright information

© Shanghai Jiao Tong University 2019

Authors and Affiliations

  1. 1.University of TorontoTorontoCanada

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