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Sandwich panel structures are widely used in many industries due to their high mechanical properties. The interlayer of these structures is a very important factor in controlling and improving their mechanical properties under various loading conditions. Concave lattice structures are outstanding candidates for use as interlayers in such sandwich structures for several reasons, namely to tune their elasticity (eg, Poisson’s ratio and elastic stiffness values) and ductility (eg, high elasticity) for simplicity. The strength-to-weight ratio properties are achieved by adjusting only the geometric elements that make up the unit cell. Here, we investigate the flexural response of a 3-layer concave core sandwich panel using analytical (i.e., zigzag theory), computational (i.e., finite element) and experimental tests. We also analyzed the effect of various geometric parameters of the concave lattice structure (e.g. angle, thickness, unit cell length to height ratio) on the overall mechanical behavior of the sandwich structure. We have found that core structures with auxetic behavior (i.e. negative Poisson’s ratio) exhibit higher flexural strength and minimal out-of-plane shear stress compared to conventional gratings. Our findings may pave the way for the development of advanced engineered multilayer structures with architectural core lattices for aerospace and biomedical applications.
Due to their high strength and low weight, sandwich structures are widely used in many industries, including mechanical and sports equipment design, marine, aerospace, and biomedical engineering. Concave lattice structures are one potential candidate being considered as core layers in such composite structures due to their superior energy absorption capacity and high strength-to-weight ratio properties1,2,3. In the past, great efforts have been made to design lightweight sandwich structures with concave lattices to further improve the mechanical properties. Examples of such designs include high pressure loads in ship hulls and shock absorbers in automobiles4,5. The reason why the concave lattice structure is very popular, unique and suitable for sandwich panel construction is its ability to independently tune its elastomechanical properties (e.g. elastic stiffness and Poisson comparison). One such interesting property is the auxetic behavior (or negative Poisson’s ratio), which refers to the lateral expansion of a lattice structure when stretched longitudinally. This unusual behavior is related to the microstructural design of its constituent elementary cells7,8,9.
Since Lakes’ initial research into the production of auxetic foams, significant efforts have been made to develop porous structures with a negative Poisson’s ratio10,11. Several geometries have been proposed to achieve this goal, such as chiral, semi-rigid, and rigid rotating unit cells,12 all of which exhibit auxetic behavior. The advent of additive manufacturing (AM, also known as 3D printing) technologies has also facilitated the implementation of these 2D or 3D auxetic structures13.
The auxetic behavior provides unique mechanical properties. For example, Lakes and Elms14 have shown that auxetic foams have higher yield strength, higher impact energy absorption capacity, and lower stiffness than conventional foams. With regard to the dynamic mechanical properties of auxetic foams, they show higher resistance under dynamic breaking loads and higher elongation under pure tension15. In addition, the use of auxetic fibers as reinforcing materials in composites will improve their mechanical properties16 and resistance to damage caused by fiber stretch17.
Research has also shown that using concave auxetic structures as the core of curved composite structures can improve their out-of-plane performance, including flexural stiffness and strength18. Using a layered model, it has also been observed that an auxetic core can increase the fracture strength of composite panels19. Composites with auxetic fibers also prevent crack propagation compared to conventional fibers20.
Zhang et al.21 modeled the dynamic collision behavior of returning cell structures. They found that voltage and energy absorption could be improved by increasing the angle of the auxetic unit cell, resulting in a grating with a more negative Poisson’s ratio. They also suggested that such auxetic sandwich panels could be used as protective structures against high strain rate impact loads. Imbalzano et al.22 also reported that auxetic composite sheets can dissipate more energy (i.e. twice as much) through plastic deformation and can reduce top speed on the reverse side by 70% compared to single ply sheets.
In recent years, much attention has been paid to numerical and experimental studies of sandwich structures with auxetic filler. These studies highlight ways to improve the mechanical properties of these sandwich structures. For example, considering a sufficiently thick auxetic layer as the core of a sandwich panel can result in a higher effective Young’s modulus than the stiffest layer23. In addition, the bending behavior of laminated beams 24 or auxetic core tubes 25 can be improved with the optimization algorithm. There are other studies on mechanical testing of expandable core sandwich structures under more complex loads. For example, compression testing of concrete composites with auxetic aggregates, sandwich panels under explosive loads27, bending tests28 and low-velocity impact tests29, as well as analysis of non-linear bending of sandwich panels with functionally differentiated auxetic aggregates30.
Because computer simulations and experimental evaluations of such designs are often time consuming and costly, there is a need to develop theoretical methods that can efficiently and accurately provide the information needed to design multilayer auxetic core structures under arbitrary loading conditions. reasonable time. However, modern analytical methods have a number of limitations. In particular, these theories are not accurate enough to predict the behavior of relatively thick composite materials and to analyze composites composed of several materials with widely differing elastic properties.
Since these analytical models depend on applied loads and boundary conditions, here we will focus on the flexural behavior of auxetic core sandwich panels. The equivalent single layer theory used for such analyzes cannot correctly predict shear and axial stresses in highly inhomogeneous laminates in moderate thickness sandwich composites. Moreover, in some theories (for example, in the layered theory), the number of kinematic variables (for example, displacement, velocity, etc.) strongly depends on the number of layers. This means that the field of motion of each layer can be described independently, while satisfying certain physical continuity constraints. Therefore, this leads to taking into account a large number of variables in the model, which makes this approach computationally expensive. To overcome these limitations, we propose an approach based on zigzag theory, a specific subclass of multilevel theory. The theory provides continuity of shear stress throughout the thickness of the laminate, assuming a zigzag pattern of in-plane displacements. Thus, the zigzag theory gives the same number of kinematic variables regardless of the number of layers in the laminate.
To demonstrate the power of our method in predicting the behavior of sandwich panels with concave cores under bending loads, we compared our results with classical theories (i.e. our approach with computational models (i.e. finite elements) and experimental data (i.e. three-point bending of 3D printed sandwich panels).To this end, we first derived the displacement relationship based on the zigzag theory, and then obtained the constitutive equations using the Hamilton principle and solved them using the Galerkin method.The results obtained are a powerful tool for design corresponding geometric parameters of sandwich panels with auxetic fillers, facilitating the search for structures with improved mechanical properties.
Consider a three-layer sandwich panel (Fig. 1). Geometric design parameters: top layer \({h}_{t}\), middle layer \({h}_{c}\) and bottom layer \({h}_{ b }\) thickness. We hypothesize that the structural core consists of a pitted lattice structure. The structure consists of elementary cells arranged next to each other in an ordered manner. By changing the geometric parameters of a concave structure, it is possible to change its mechanical properties (i.e., the values of Poisson’s ratio and elastic stiffness). The geometrical parameters of the elementary cell are shown in Figs. 1 including angle (θ), length (h), height (L) and column thickness (t).
The zigzag theory provides very accurate predictions of the stress and strain behavior of layered composite structures of moderate thickness. Structural displacement in the zigzag theory consists of two parts. The first part shows the behavior of the sandwich panel as a whole, while the second part looks at the behavior between layers to ensure shear stress continuity (or the so-called zigzag function). In addition, the zigzag element disappears on the outer surface of the laminate, and not inside this layer. Thus, the zigzag function ensures that each layer contributes to the total cross-sectional deformation. This important difference provides a more realistic physical distribution of the zigzag function compared to other zigzag functions. The current modified zigzag model does not provide transverse shear stress continuity along the intermediate layer. Therefore, the displacement field based on the zigzag theory can be written as follows31.
in the equation. (1), k=b, c and t represent the bottom, middle and top layers, respectively. The displacement field of the mean plane along the Cartesian axis (x, y, z) is (u, v, w), and the bending rotation in the plane about the (x, y) axis is \({\uptheta} _{x}\) and \ ({\uptheta}_{y}\). \({\psi}_{x}\) and \({\psi}_{y}\) are spatial quantities of zigzag rotation, and \({\phi}_{x}^{k}\ left (z \right)\) and \({\phi}_{y}^{k}\left(z\right)\) are zigzag functions.
The amplitude of the zigzag is a vector function of the actual response of the plate to the applied load. They provide an appropriate scaling of the zigzag function, thereby controlling the overall contribution of the zigzag to the displacement in the plane. Shear strain across the plate thickness consists of two components. The first part is the shear angle, uniform across the thickness of the laminate, and the second part is a piecewise constant function, uniform across the thickness of each individual layer. According to these piecewise constant functions, the zigzag function of each layer can be written as:
in the equation. (2), \({c}_{11}^{k}\) and \({c}_{22}^{k}\) are the elasticity constants of each layer, and h is the total thickness of the disc. In addition, \({G}_{x}\) and \({G}_{y}\) are the weighted average shear stiffness coefficients, expressed as 31:
The two zigzag amplitude functions (Equation (3)) and the remaining five kinematic variables (Equation (2)) of the first order shear deformation theory constitute a set of seven kinematics associated with this modified zigzag plate theory variable. Assuming a linear dependence of the deformation and taking into account the zigzag theory, the deformation field in the Cartesian coordinate system can be obtained as:
where \({\varepsilon}_{yy}\) and \({\varepsilon}_{xx}\) are normal deformations, and \({\gamma}_{yz},{\gamma}_{xz} \ ) and \({\gamma}_{xy}\) are shear deformations.
Using Hooke’s law and taking into account the zigzag theory, the relationship between stress and strain of an orthotropic plate with a concave lattice structure can be obtained from equation (1). (5)32 where \({c}_{ij}\) is the elastic constant of the stress-strain matrix.
where \({G}_{ij}^{k}\), \({E}_{ij}^{k}\) and \({v}_{ij}^{k}\) are cut force is the modulus in different directions, Young’s modulus and Poisson’s ratio. These coefficients are equal in all directions for the isotopic layer. In addition, for the returning nuclei of the lattice, as shown in Fig. 1, these properties can be rewritten as 33.
Application of Hamilton’s principle to the equations of motion of a multilayer plate with a concave lattice core provides the basic equations for the design. Hamilton’s principle can be written as:
Among them, δ represents the variational operator, U represents the strain potential energy, and W represents the work done by the external force. The total potential strain energy is obtained using the equation. (9), where A is the region of the median plane.
Assuming a uniform application of the load (p) in the z direction, the work of the external force can be obtained from the following formula:
Replacing the equation Equations (4) and (5) (9) and replace the equation. (9) and (10) (8) and integrating over the plate thickness, the equation: (8) can be rewritten as:
The index \(\phi\) represents the zigzag function, \({N}_{ij}\) and \({Q}_{iz}\) are forces in and out of the plane, \({M}_{ij }\) represents a bending moment, and the calculation formula is as follows:
Applying integration by parts to the equation. Substituting into formula (12) and calculating the coefficient of variation, the defining equation of the sandwich panel can be obtained in the form of formula (12). (13).
The differential control equations for freely supported three-layer plates are solved by the Galerkin method. Under the assumption of quasi-static conditions, the unknown function is considered as an equation: (14).
\({u}_{m,n}\), \({v}_{m,n}\), \({w}_{m,n}\),\({{\uptheta}_ {\mathrm {x}}}_{\mathrm {m} \text{,n}}\),\({{\uptheta }_{\mathrm {y}}}_{\mathrm {m} \text {,n}}\), \({{\uppsi}_{\mathrm{x}}}_{\mathrm{m}\text{,n}}\) and \({{\uppsi}_{ \mathrm{y}}}_{\mathrm{m}\text{,n}}\) are unknown constants that can be obtained by minimizing the error. \(\overline{\overline{u}} \left({x{\text{,y}}} \right)\), \(\overline{\overline{v}} \left({x{\text {,y}}} \right)\), \(\overline{\overline{w}} \left( {x{\text{,y}}} \right)\), \(\overline{\overline {{{\uptheta}_{x}}}} \left( {x{\text{,y}}} \right)\), \(\overline{\overline{{{\uptheta}_{y} }}} \left( {x{\text{,y}}} \right)\), \(\overline{\overline{{\psi_{x}}}} \left( {x{\text{, y}}} \right)\) and \(\overline{\overline{{ \psi_{y} }}} \left( {x{\text{,y}}} \right)\) are test functions, which must satisfy the minimum necessary boundary conditions. For just supported boundary conditions, the test function can be recalculated as:
Substitution of equations gives algebraic equations. (14) to the governing equations, which can lead to obtaining unknown coefficients in equation (14). (14).
We use finite element modeling (FEM) to computer-simulate the bending of a freely supported sandwich panel with a concave lattice structure as the core. The analysis was performed in a commercial finite element code (for example, Abaqus version 6.12.1). 3D hexahedral solid elements (C3D8R) with simplified integration were used to model the top and bottom layers, and linear tetrahedral elements (C3D4) were used to model the intermediate (concave) lattice structure. We performed a mesh sensitivity analysis to test the convergence of the mesh and concluded that the displacement results converged at the smallest feature size among the three layers. The sandwich plate is loaded using the sinusoidal load function, taking into account the freely supported boundary conditions at the four edges. The linear elastic mechanical behavior is considered as a material model assigned to all layers. There is no specific contact between the layers, they are interconnected.
We used 3D printing techniques to create our prototype (i.e. triple printed auxetic core sandwich panel) and corresponding custom experimental setup to apply similar bending conditions (uniform load p along the z-direction) and boundary conditions (i.e. . just supported). assumed in our analytical approach (Fig. 1).
The sandwich panel printed on a 3D printer consists of two skins (upper and lower) and a concave lattice core, the dimensions of which are shown in Table 1, and was manufactured on an Ultimaker 3 3D printer (Italy) using the deposition method ( FDM). technology is used in its process. We 3D printed the base plate and main auxetic lattice structure together, and printed the top layer separately. This helps to avoid any complications during the support removal process if the entire design has to be printed at once. After 3D printing, two separate parts are glued together using superglue. We printed these components using polylactic acid (PLA) at the highest infill density (i.e. 100%) to prevent any localized printing defects.
The custom clamping system mimics the same simple support boundary conditions adopted in our analytical model. This means that the gripping system prevents the board from moving along its edges in the x and y directions, allowing these edges to rotate freely around the x and y axes. This is done by considering fillets with radius r = h/2 at the four edges of the gripping system (Fig. 2). This clamping system also ensures that the applied load is fully transferred from the testing machine to the panel and aligned with the center line of the panel (fig. 2). We used multi-jet 3D printing technology (ObjetJ735 Connex3, Stratasys® Ltd., USA) and rigid commercial resins (such as the Vero series) to print the grip system.
Schematic diagram of a 3D printed custom gripping system and its assembly with a 3D printed sandwich panel with an auxetic core.
We perform motion-controlled quasi-static compression tests using a mechanical test bench (Lloyd LR, load cell = 100 N) and collect machine forces and displacements at a sampling rate of 20 Hz.
This section presents a numerical study of the proposed sandwich structure. We assume that the top and bottom layers are made of carbon epoxy resin, and the lattice structure of the concave core is made of polymer. The mechanical properties of the materials used in this study are shown in Table 2. In addition, the dimensionless ratios of displacement results and stress fields are shown in Table 3.
The maximum vertical dimensionless displacement of a uniformly loaded freely supported plate was compared with the results obtained by different methods (Table 4). There is good agreement between the proposed theory, the finite element method and experimental verifications.
We compared the vertical displacement of the modified zigzag theory (RZT) with 3D elasticity theory (Pagano), first order shear deformation theory (FSDT), and FEM results (see Fig. 3). The first-order shear theory, based on the displacement diagrams of thick multilayer plates, differs most from the elastic solution. However, the modified zigzag theory predicts very accurate results. In addition, we also compared the out-of-plane shear stress and in-plane normal stress of various theories, among which the zigzag theory obtained more accurate results than FSDT (Fig. 4).
Comparison of normalized vertical strain calculated using different theories at y = b/2.
Change in shear stress (a) and normal stress (b) across the thickness of a sandwich panel, calculated using various theories.
Next, we analyzed the influence of the geometric parameters of the unit cell with a concave core on the overall mechanical properties of the sandwich panel. The unit cell angle is the most important geometric parameter in the design of reentrant lattice structures34,35,36. Therefore, we calculated the influence of the unit cell angle, as well as the thickness outside the core, on the total deflection of the plate (Fig. 5). As the thickness of the intermediate layer increases, the maximum dimensionless deflection decreases. The relative bending strength increases for thicker core layers and when \(\frac{{h}_{c}}{h}=1\) (ie, when there is one concave layer). Sandwich panels with an auxetic unit cell (i.e. \(\theta =70^\circ\)) have the smallest displacements (Fig. 5). This shows that the bending strength of the auxetic core is higher than that of the conventional auxetic core, but is less efficient and has a positive Poisson’s ratio.
Normalized maximum deflection of a concave lattice rod with different unit cell angles and out-of-plane thickness.
The thickness of the core of the auxetic grating and the aspect ratio (i.e. \(\theta=70^\circ\)) affect the maximum displacement of the sandwich plate (Figure 6). It can be seen that the maximum deflection of the plate increases with increasing h/l. In addition, increasing the thickness of the auxetic core reduces the porosity of the concave structure, thereby increasing the bending strength of the structure.
The maximum deflection of sandwich panels caused by lattice structures with an auxetic core of various thicknesses and lengths.
The study of stress fields is an interesting area that can be explored by changing the geometric parameters of the unit cell to study the failure modes (eg, delamination) of multilayer structures. Poisson’s ratio has a greater effect on the field of out-of-plane shear stresses than normal stress (see Fig. 7). In addition, this effect is inhomogeneous in different directions due to the orthotropic properties of the material of these gratings. Other geometric parameters, such as the thickness, height, and length of the concave structures, had little effect on the stress field, so they were not analyzed in this study.
Change in shear stress components in different layers of a sandwich panel with a lattice filler with different concavity angles.
Here, the bending strength of a freely supported multilayer plate with a concave lattice core is investigated using the zigzag theory. The proposed formulation is compared with other classical theories, including three-dimensional elasticity theory, first-order shear deformation theory, and FEM. We also validate our method by comparing our results with experimental results on 3D printed sandwich structures. Our results show that the zigzag theory is able to predict the deformation of sandwich structures of moderate thickness under bending loads. In addition, the influence of the geometric parameters of the concave lattice structure on the bending behavior of sandwich panels was analyzed. The results show that as the level of auxetic increases (ie, θ <90), the bending strength increases. In addition, increasing the aspect ratio and decreasing the thickness of the core will reduce the bending strength of the sandwich panel. Finally, the effect of Poisson’s ratio on out-of-plane shear stress is studied, and it is confirmed that Poisson’s ratio has the greatest influence on the shear stress generated by the thickness of the laminated plate. The proposed formulas and conclusions can open the way to the design and optimization of multilayer structures with concave lattice fillers under more complex loading conditions necessary for the design of load-bearing structures in aerospace and biomedical technology.
The datasets used and/or analyzed in the current study are available from the respective authors upon reasonable request.
Aktai L., Johnson A. F. and Kreplin B. Kh. Numerical simulation of the destruction characteristics of honeycomb cores. engineer. fractal. fur. 75(9), 2616–2630 (2008).
Gibson L.J. and Ashby M.F. Porous Solids: Structure and Properties (Cambridge University Press, 1999).
Post time: Aug-12-2023