Session: 12-26-01: Mechanics and Materials of Soft Electronics

Paper Number: 147301

147301 - A Shear-Lag Model for Laminated Beams With Extreme Modulus Mismatch Between Layers 

The advent of flexible electronics has had a significant impact on a diverse range of applications, including flexible displays [1], flexible sensors [2], wearable and implantable electronics [3], and so on. A widely adopted structural design in flexible devices involves layering stiff functional layers with soft insulating or isolating layers. When there is a substantial mismatch in Young’s modulus between the functional layers and the insulating or isolating layers, these laminated beams can be much more compliant than the prediction of Euler–Bernoulli beam theory due to the shear-lag effects. Prior work in this field has primarily concentrated on specific deformation types and a fixed number of layers, with a focus on strain distribution analysis to prevent failure during bending [4,5]. However, the importance of evaluating the structural stiffness of laminated beams is manifested by the growing significance of soft implantable bioelectronics such as neuroprobes [6], where minimizing bending stiffness is key to reducing immune responses. 

In this study, we developed a comprehensive analytical framework for multilayer laminated beams with extreme modulus mismatch under arbitrary deformation. The model adopts an energy approach and calculates the normal strains and the internal moments. The equivalent flexural rigidity is defined as the ratio of the maximum internal moment to the maximum curvature. Finite element analyses were conducted to verify the proposed framework.

 We discerned a unique control parameter that reflects the interplay between modulus mismatch and the aspect ratio, which in turn determines the deviations from the Euler–Bernoulli beam theory. It was found that the equivalent flexural rigidity varies with cross-sectional properties, beam length, and loading conditions when there is a significant shear-lag effect. Moreover, when the layer number  is sufficiently large, the flexural rigidity was found to scale with  rather than , which is distinct from the Euler-Bernoulli theory.

We further applied the proposed framework to study the wrinkling of multilayer beams on a soft substrate. A simple relationship between the flexural rigidity and the wavelength, soft substrate modulus was obtained. The results were validated experimentally using a triple-layer beam made of SU-8 photoresist and PDMS wrinkling on a pre-stretched VHB tape.

This work will guide the future design of soft electronics, taking advantage of shear-lag effects to achieve devices with higher compliance and bendability. Moreover, the proposed wrinkling experiment could serve as an experimental platform to directly measure the flexural rigidity of soft electronics, which is otherwise hard due to their extreme compliance.

 

[1]. Zhao, Z., Liu, K., Liu, Y., Guo, Y., Liu, Y. 2022. Natl. Sci. Rev. 9 (6), nwac090

[2]. Luo, Y., Abidian, M.R., Ahn, J.-H., Akinwande, D., et. al. 2023. ACS Nano 17 (6), 5211-5295, PMID:36892156

[3]. Liu, S., Rao, Y., Jang, H., Tan, P., Lu, N., 2022, Matter 5 (4), 1104-1136.

[4]. Li, L., Lin, H., Qiao, S., Zou, Y., Danto, S., Richardson, K., et al. 2014. Nat. Photon. 8 (8), 543-640

[5]. Li, S., Su, Y., Li, R., 2016. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 472(2190), 20160087.

[6]. Lecomte, A., Descamps, E., Bergaud, C., 2018. J. Neural Eng. 15 (3) 031001.

Presenting Author: Zheliang Wang The University of Texas at Austin

Presenting Author Biography: Zheliang Wang is a postdoctoral researcher in the Department of Aerospace Engineering and Engineering Mechanics at the University of Texas at Austin. He earned his Ph.D. in Mechanical Engineering from Johns Hopkins University in 2022. His research focuses on elucidating the material-structure-performance relationships of soft electronics, employing a combined approach of constitutive modeling, finite element simulation, and analytical modeling.

Authors:

Zheliang Wang The University of Texas at Austin
Xinyi Lin Harvard University
Hao Sheng Harvard University
Jia Liu Harvard University
Nanshu Lu The University of Texas at Austin