Failure maps and optimal design of metallic sandwich panels with truss cores subjected to thermal loading
Introduction
Sandwich panels with truss cores (SPTCs) are a class of novel structures that can be applied as both load bearing components and other functionality, such as thermal management, energy absorption and blast resistance [1], [2], [3], [4], [5]. A prominent characteristic of SPTCs is that their macroscopic mechanical behavior can be designed or tailored through the configuration, arrangement and material of mesostructure, of the lattice truss. There have been a variety of configurations of lattice truss materials, such as pyramidal [6], [7], tetrahedral [8], [9], [10], [11], Kagome [12], [13] and recently proposed X-type [14], [15], [16], [17]. Compared with closed or open foams, which are bending-dominated configuration, the stretching-dominated lattice truss material that have high degree of nodal connectivity is much stiffer and stronger [18]. When being used as thermal protection systems of high speed vehicles, sandwich panels typically experience a large temperature change. The high-temperature degraded properties together with intense thermal loading may lead to a failure of various mechanisms. Therefore, the failure behavior of sandwich panels to thermal loading becomes a driving design parameter before they can be applied into practice.
There have been some studies on the structural response of SPTCs at room temperature. In some cases, SPTCs have been tested in various shear and bending modes [8], [9], [19]. To study the in-plane compressive behaviors, Cote et al. [7] carried out experimental and theoretical analysis on the response of metallic sandwich columns with pyramidal truss cores made from AISI 304 stainless steel. Failure maps of the sandwich column are constructed based on three failure mechanisms: Euler buckling, shear buckling and face sheet wrinkling. For all-composite sandwich columns, face sheet crushing will appear besides the three failure modes considered in metallic sandwich columns [20]. Wicks and Hutchinson [21] carried out theoretical analysis on the optimal design of sandwich panels with either planar trusses or solid face sheets with a single material subjected to prescribed combinations of bending and transverse shear loading. Four failure modes are considered in their analysis: face yielding, face buckling, core member yielding and core member buckling. Based on these failure modes, Zok et al. [22] obtained the failure mechanism map of sandwich beams with pyramidal truss cores and compared with the three point bending test. Then, Rathbun et al. [23] conducted an systematic optimal analysis on sandwich beams with several core topologies, including pyramidal and tetrahedral truss cores, square honeycombs, and corrugated sheets. In their works, the optimal design is obtained at the confluence of three failure mechanisms. However, it should be noted that for a nonlinear system, the global optimal result is not necessarily in the intersection of constraint equations, which have obvious nonlinear characteristics in nature. In this case, the numerical programming of the optimization model is imperative.
When structures are subjected to high temperature environments, one of the undesirable effect is the development of thermal stresses, which is often happened at temperatures below those that impair the material properties considerably [24]. Compressive thermal stresses arise either from non-uniform temperature distributions or from supports which constrain the thermal expansion even when heating is uniform. The behavior of global buckling induced by the compressive thermal stresses is an important failure mode for the slender or thin-walled structure, and has been studied extensively for shells and general sandwich plates in the theoretical analysis [25], [26], [27], [28], [29], [30]. Rakow and Waas [31] also carried out experimental analysis on the thermal buckling behavior of sandwich panels with foam cores. For SPTCs, Yuan et al. [32] obtained the eigenvalue buckling and post buckling behavior of fully-clamped (CCCC) and simply-supported (SSSS) SPTCs subjected to uniform thermal loading. Later on, Yuan et al. [33] also performed experimental study on the thermal buckling behavior of SPTCs, and the full field buckling history of the panel under uniform high temperature environments was obtained. It is found that, due to defects during fabrication, the sandwich panel deformed in asymmetric mode in high temperature environments. However, it should be noted that due to the complexity of the structure, both the face sheet and the core member of the SPTC may fail in various modes, besides global buckling.
Within the authors' knowledge, there has been little theoretical analysis reported on the failure behaviour of SPTCs subjected to uniform thermal loading. In the present paper, five failure mechanisms are considered to obtain the high temperature failure maps of SPTCs, they are global buckling (GB), face sheet buckling (FB), face sheet yielding (FY), core member buckling (CB) and core member yielding (CY). The objective of this paper is to construct failure mechanism maps as well as to estimate the minimum weight design of SPTCs at a given thermal loading with the competing failure modes. The outline of the paper is as follows. Firstly, analytical expressions for critical loads of five failure modes are derived for the CCCC and SSSS SPTCs made from a single metallic material. Based on these expressions, failure mechanism maps are constructed with dimensionless geometrical parameters of SPTCs. Finally, minimum weight designs are obtained for sandwich panels with different truss core topologies by using the numerical optimal program model based on Lingo. It is verified that for this nonlinear problem, the confluence of constraint equations for various failure modes is not the optimal design.
In the present paper, failure maps of SPTCs subjected to uniform thermal loading are developed by comparing the load capacity in these mechanisms. In addition, optimal designs of the SPTC are obtained by using the failure modes of SPTCs as constraint conditions, and the dimensionless weight as objective function. Therefore, analytical expressions of SPTCs under the five failure modes should be deduced.
Section snippets
Failure modes of metallic SPTCs
Before proceeding, performance evaluation criteria for SPTCs are needed. In an optimization process, one needs to ascertain the minimum weight of SPTCs that can maintain structural integrity at a given thermal loading. Therefore, two dimensionless parameters, one is based on weight and the other based on load, are considered. The pertinent load index for strength-based designs can be expressed aswhere and ∆T are the coefficient of thermal expansion of the material of the SPTC and the
SSSS boundary condition
The preceding analysis on the failure behavior of SPTCs can be employed to generate failure mechanism maps. It is assumed that, in constructing such a map, the operative failure mode is one associated with the lowest critical temperature rise. When normalized by the edge length of the sandwich panel, dimensionless parameters can be expressed as
The corresponding weight index from Eq. (2) is
And the constraint based on the failure mode of SSSS SPTCs
Optimization methods
The analysis is extended to obtain the entire family of optimal designs for SPTCs subjected to uniform thermal loading. The goal is to find the minimum weight of SPTCs that can maintain structural integrity at a given thermal loading. However, it can be found from Eq. (18c) that the failure mode of FY is irrelevant to the geometrical configuration of SPTCs. It means when the temperature rise is lower than the critical temperature of FY, this constraint equation can be removed from the
Conclusions
The response of SPTCs subjected to uniform thermal loading has been studied analytically and numerically. Analytical formulae are developed for the failure strength of the SPTC and five possible failure modes for the four different configurations of truss cores are identified. Failure mechanism maps for the SPTC made from a single metallic material are developed when the dimensionless weight index are fixed. Using these failure modes as constraint conditions, sandwich panels with different
Acknowledgments
Financial supports from National Natural Science Foundation of China (Grant nos. 91016025, 11472276, and 11332011) are gratefully acknowledged.
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