International Journal of Mechanical Sciences 194 (2021) 106218 Contents lists available at ScienceDirect International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci Functionally Graded Materials Beams Subjected to Bilateral Constraints: Structural Instability and Material Topology Talal Salem c , Pengcheng Jiao a,b,∗ , Imen Zaabar d , Xuyang Li c , Ronghua Zhu a,b , Nizar Lajnef c a Institute of Port, Coastal and Offshore Engineering, Ocean College, Zhejiang University, Zhoushan 3216021, Zhejiang, China. b Engineering Research Center of Oceanic Sensing Technology and Equipment, Zhejiang University, Ministry of Education, China c Department of Civil and Environmental Engineering, Michigan State University, East Lansing, MI 48824, USA d Department of Computer Science and Engineering, Michigan State University, East Lansing, MI 48824, USA a r t i c l e i n f o Keywords: Functionally graded materials (FMG) beams Bilateral constraints Material function Structural instability Material topology a b s t r a c t In recent years, the study of Functionally graded materials (FGM) opened exciting new venues for the control and manipulation of engineered materials and structures. In this study, we investigate bilaterally constrained FGM beams with programmable material functions. The FGM beams are fabricated using 3D printing techniques, and tested t understand the behavior of structural instability (i.e., postbuckling) under the bilateral confinements. Theoretical and numerical models are developed to investigate the postbuckling response, and the results are compared to experimental observations with satisfactory agreements. Material topology optimization is then carried out to investigate the influences of the material functions on the release of the stored energy during the bucking mode transitions in the FGM beams. It is found that stored energy variations can be used to optimize the material functions, which allows for the guided design of bi-walled FGM beams with well-defined controllability over the structural instability. The reported bilaterally constrained FGM beams with optimized material functions can be used in a multitud of different applications. 1. Introduction Functionally graded materials (FGM), a type of functional compos- ites focused on exploring mechanical responses through material de- sign and optimization, have marked their debut decades ago and have been experiencing rapid developments ever since then, which rekindles the popularity of research on composite materials [1]. Compared to their traditional counterparts, FGM have several advantages and un- usual properties such as enhanced deformation resistance, enhanced toughness, ultra-light, well recoverability, etc. [2-4]. Surpassing the me- chanical characteristics of natural materials, FGM are reported with su- perior response behavior due to their effectively tuned and engineered material properties [5]. A research field has therefore emerged aiming to design, characterize and harness the functional patterns of FGM in order to obtain desirable performances for specific applications. This is a highly multidisciplinary research community that comprises struc- tural analysis, material science, mechanics, and engineering. As a conse- quence, research efforts have been dedicated to designing composites’ functionally graded material properties, such that to obtain advanced physical and mechanical responses that are otherwise not accessible in nature materials [6-9]. ∗ Corresponding author. E-mail address: pjiao@zju.edu.cn (P. Jiao). A significant number of studies have been conducted in recent years to characterize and design FGM under different conditions. Static, free vibration and wave propagation of bi-material beams fused with a FGM layer were investigated analytically using the first-order shear defor- mation theory [10]. Structural behavior of functionally graded tapered clamped free Euler–Bernoulli beams was studied to determine the crit- ical buckling loads and free frequencies in the longitudinal and trans- verse directions [11]. General formulas were proposed to obtain the ef- fective stiffness coefficients of the elastic beams made of FGM and piece- wise homogeneous materials [12]. Modified couple stress theory and von Kármán geometric nonlinearity were used to investigate the nonlin- ear vibration of FGM microbeams, where the material properties were assumed to be graded in the thickness direction [13,14]. The authors dis- cussed the influences of the length scale parameter and material prop- erties on the nonlinear mechanical response of the FGM microbeams. Higher-order shear deformation beam theories were developed for bend- ing and free vibration of FGM beams [15]. The authors reported that varying the power law index in the material functions significantly af- fected the stiffness of the FGM beams. Effects of material composition on the thermal buckling and vibration of FGM beams were also investigated [16-19]. Higher order shear deformation theory was adopted to develop finite element models for refined mixed beam element [53]. The mixed finite beam model was proposed to explore the vibrational behavior of FGM beams, in which material properties were described using a power law distribution [54]. The mechanical behavior of porous beams, made https://doi.org/10.1016/j.ijmecsci.2020.106218 Received 11 July 2020; Received in revised form 10 November 2020; Accepted 23 November 2020 Available online 26 November 2020 0020-7403/© 2020 Elsevier Ltd. All rights reserved.