1. Introduction
Lattice structures are bio-inspired 3D configurations of repeated and open unit cells defined by interconnected struts and nodes [
1,
2]. Relative to conventional bulk materials, topologically ordered lattice structures can exhibit impressive mechanical strength, stiffness, thermal, and electrical properties while using significantly less material. In other words, they possess higher strength- and stiffness-to-weight ratios. These advantages have led to their broad applications in advanced lightweight naval, automobile, aerospace, and other engineered structures [
3,
4,
5,
6].
Increasing performance demands for these ultra-lightweight engineering applications means that cellular lattice structures need to be fabricated with greater complexity and with smaller feature sizes. Conventional manufacturing processes, such as wire weaving [
7], high-temperature forming and diffusion bonding [
8], and the interlocking method [
9], are unsuitable and too time-consuming for fabricating lattice structures with complex nodal connections. Recent advances in additive manufacturing (AM) have enabled methods to realize cellular lattice structures with intricate geometries [
10,
11]. Some of the widely used AM methods include fused deposition modeling [
12] and stereolithography [
13] for polymer-based structures, as well as extrusion [
10], powder bed fusion [
14], and direct ink write [
15] for metallic cellular lattice systems.
Despite the ability to use AM to fabricate complex cellular structures, their functional performance strongly depends on manufacturing quality. The presence of minor defects could compromise the structural integrity of the entire part [
16]. For instance, nozzle clogs, micro-voids, and pores that occur during extrusion or uncontrolled thermo-mechanical behavior in powder bed fusion may induce cracks, shrinkage, uneven surfaces, and nodal disconnections in the struts [
17]. During storage, transit, or use, these weakened struts are prone to stress concentrations, which can lead to defect propagation, broken struts, and partial or complete lattice structure failure [
18]. Therefore, quality assurance and control of AM parts require that the type of defects and damage locations be identified whether they are incurred during manufacturing or when in service.
Traditional nondestructive evaluation (NDE) methods, such as x-ray computed tomography (CT) and ultrasonic measurements, have been used to detect defects such as voids and inclusions; however, they can be inefficient for inspecting complex cellular lattice structures. For example, CT reconstruction of defects in lattice structure struts requires multiple projection slices and can be computationally intensive, slow, and expensive [
19]. Similarly, ultrasonic testing requires a dense array of transducers and complicated wave generation and propagation patterns to evaluate the different scales and locations of defects [
20]. Another approach is to physically integrate sensors as part of lattice structures for continuous monitoring even during post-production. An example is the printing of conductive paths for capacitive sensing or radio-frequency identification (RFID) [
21]. However, the sensing area for these approaches is restricted by the limited number of paths and may not be suitable for NDE of the entire lattice structure.
This study aims to overcome these limitations by leveraging 3D electrical resistance tomography (ERT) to directly detect and accurately identify the locations of damaged struts in cellular lattice structures. ERT aims to reconstruct the conductivity distribution of a conductive target that is directly correlated to damage or strain states by using only boundary electric potential measurements [
22,
23]. The utilization of a target’s electromechanical properties exempts inspection from complex operations (i.e., multiple projections of CT) [
22,
24].
A major challenge when using electromagnetic tomographic methods is that the accurate reconstruction of electrical properties (e.g., conductivity) distribution around the interior of the target is challenging due to the lower sensitivity of measurements in its interior versus near the boundaries [
25,
26]. This limitation may cause inaccurate localization and quantification of defects and may be even more severe for open cell lattice structures. To solve this problem, Baltopoulos et al. [
27] proposed reserving smaller singular value decomposition (SVD) components of the sensitivity map for efficient conductivity reconstruction of the center region by choosing a smaller hyperparameter, but this solution may introduce additional artifacts in the region of interest, hence deteriorating reconstruction quality. Li et al. [
26] used a normalized sensitivity map to compensate for the low central sensitivity, but the proposed normalized methods are element-based and are computationally intensive. The element-based normalization is effective for detecting perturbations with large-area conductivity change. However, the method is not as effective for small-area defects, such as small defects in open cell lattice structures with small cross-section struts, where reconstructions would still suffer from image artifacts. These image artifacts may result in inaccurate defect detection or incorrect decisions. Thus, to improve reconstruction performance with respect to small perturbations, the normalized sensitivity map should be adjusted to be capable of compensating for low central sensitivity and restraining image artifacts.
In this work, we present a high-throughput, 3D, iterative, absolute conductivity distribution ERT system for identifying single and multiple damaged struts in conductive cellular lattice structures. The significance of this work is that the ERT algorithm employs a strut-based normalized map that preconditions the sensitivity map for enhancing conductivity reconstruction sensitivity while mitigating artifacts due to the ill-conditioning of the ERT inverse problem. The efficacy of this method was assessed by quantifying the relationship between damage severity and the corresponding reconstructed conductivity changes. Both numerical simulations and corresponding experiments of cellular lattice structures with different damage features were performed. To demonstrate that ERT could examine conductive cellular lattice structures, experiments were performed using 3D-printed polymer cellular lattice structures, which were then coated with a soluble, sacrificial, and electrically conductive nanocomposite thin film. Damage scenarios with single and multiple damaged struts were considered.
5. Discussion
The simulation and experimental results showed that the 3D ERT method with the strut-based normalized sensitivity map was able to characterize damage accurately and quantitatively in cellular lattice structures. The strut-based normalized sensitivity map compensated for the low central sensitivity and drastically reduced image artifacts, so the reconstructions had much smaller reconstruction errors as defined by eC, eA, and eσ.
However, it is worth mentioning that the implementation of ERT in practice may face certain challenges, since electrodes need to be physically attached to the structure for propagating electrical current. Improperly attached electrodes can introduce unwanted contact impedance (especially if electrodes are not permanently mounted) and subsequently affect the reconstructed conductivity distributions. A potential solution is to use spring-loaded press-contact electrodes that can apply a consistent force at each electrode during ERT interrogation and measurements. On the other hand, extreme or varying ambient temperatures and environmental conditions can also potentially affect the conductivity of the structure and thus the recorded boundary voltages. Besides leveraging reference sensors that quantify these ambient effects, another approach can be optimizing the electrode configuration so that the minimum number of electrodes and measurements are needed to achieve the desired damage quantification resolution. Fewer electrodes and measurements mean that ERT interrogation can be performed faster, and varying ambient effects become less significant.
Overall, the 3D ERT method is an efficient method for detecting damage in lattice structures. With only a few electrodes attached to the boundary and their corresponding voltage measurements, the resistivity distribution that correlated to the damaged state could be captured. Currently, vibrational-based methods could only offer classification of different damage scenarios but could not effectively pinpoint specific damaged struts unlike the 3D ERT method [
41]. Moreover, ERT utilizes the intrinsic electromechanical properties of lattice structures and renders effective inspection by propagating current throughout the entire structure. Furthermore, X-ray CT-based measurements or other image processing methods require the structure to be placed between a source and detector while being rotated to obtain multiple projection slices, which requires extensive operational times and computational resources [
42].