Finite element formulation of metal foam microbeams via modified strain gradient theory

Abstract

Size-dependent behaviours of metal foam microbeams with three different porosity distribution models are studied in this paper. Based on finite element model, a normal and shear deformation theory has been employed for the first time to investigate their structural behaviours by using modified strain gradient theory and considering the effects of variable material length scale parameter. The equations of motion and boundary conditions of system are derived from Hamilton's principle. Finite element models are presented for the computation of deflections, vibration frequencies and buckling loads of the metal foam microbeams. The verification of proposed models is carried out with comparison of the numerical results available in the literature. Calculations using the different parameters reveal the effects of the porosity parameters (distribution and coefficient), small size, boundary conditions and Poisson's ratio on the displacements, frequencies and buckling loads of metal foam microbeams. Some benchmark results of these structures for both models (modified couple stress theory and modified strain gradient theory with constant and variable material length scale parameter) and with/without Poison's effect are provided for future study.

A new higher-order beam theory for buckling and free vibration responses of laminated composite and functionally graded porous beams

“…Tables 10 and 11 present the NFF and NCBL of beams, respectively. The results obtained from the proposed theory are compared to those obtained from FBT by Chen et al, 37,39 Jamshidi and 65 Quasi-3D 8.509 15.468 -Arghavani, 13 HBT by Nguyen et al, 47 Karamanli et al, 67 and Quasi-3D by Fang et al 64 and Karamanli et al, 67 The outcomes demonstrate that the proposed results agree with those of previous studies, evidencing the efficiency of the proposed theory. Figures 6 and 7 display the NFF and NCBL of SPD and ASP beams with respect to L/h ratio.…”

“…The Legendre-Ritz functions are utilised to solve the problem. Numerical examples are conducted to verify the proposed theory and investigate the impact of porosity distribution, boundary 47 HBT 0.1007 0.0508 -Chen et al 37 FBT 0.1008 0.0508 0.0204 Karamanli et al 67 Quasi-3D 0.0953 --Fang et al 64 Quasi-3D 0.0875 --APD Present HBT 0.0917 0.0462 0.0185 Nguyen et al 47 HBT 0.0917 0.0462 -Chen et al 37 FBT 0.0917 0.0462 0.0185 Karamanli et al 67 Quasi-3D 0.0870 --Fang et al 64 Quasi-3D 0.0870 --C-C SPD Present HBT 0.5902 0.3159 0.1291 Nguyen et al 47 HBT 0.5900 0.3159 -Chen et al 37 FBT 0.5944 0.3166 0.1291 Karamanli et al 67 Quasi-3D 0.5654 --Fang et al 64 Quasi-3D 0.5653 --APD Present HBT 0.5476 0.2888 0.1174 Nguyen et al 47 HBT 0.5476 0.2888 Chen et al 37 FBT 0.5475 0.2888 0.1274 Karamanli et al 67 Quasi-3D 0.5248 -Fang et al 64 Quasi-3D 0.5248 - 67 HBT 0.04643 0.01295 0.0033349 Chen et al 39 FBT --0.003338 Jamshidi and Arghavani 13 FBT --0.003339 Karamanli et al 67 Quasi-3D 0.04686 0.01299 0.0033378 APD Present HBT 0.039743 0.010800 0.002760 Nguyen et al 47 HBT -0.010800 0.002760 Karamanli et al 67 HBT 0.03973 0.01080 0.0027601 Chen et al 39 FBT --0.002869 Karamanli et al 67 Quasi-3D 0.04058 0.01096 0.0027951 C-F SPD Present HBT 0.012958 0.003336 0.000840 Nguyen et al 47 HBT -0.003335 0.000840 Karamanli et al 67 HBT 0.01295 0.00333 0.0008400 Chen et al 39 FBT -0.00334 0.000840 Jamshidi and Arghavani 13 FBT --0.000840 Karamanli et al 67 Quasi-3D 0.01317 0.00336 0.0008440 APD Present HBT 0.010801 0.002760 0.000694 Nguyen et al 47 HBT -0.002760 0.000694 Karamanli et al 67 HBT 0.01080 0.00276 0.0006934 Chen et al 39 FBT -0.00276 0.000694 Karamanli et al 67…”

“…Therefore, to account for this effect, the Quasi-3D theory (Quasi-3D), which approximates axial and transverse displacements as high-order variations through the beam thickness, was proposed. [64][65][66][67][68] Although Quasi-3D can predict the behaviour of LC and FGP beams more accurately than CBT, FBT, and HBT, it involves more unknowns, making it more complex.…”

Mixed series solution for vibration and stability of porous bi-directional functionally graded beams

“…Numerous studies focus on the effect of porosity in one-directional FGBs with finite element methods. These studies highlight the influence of porous structures on the mechanical behavior and performance of the beams, providing valuable insights into critical parameters such as durability and load-carrying capacity [1][2][3][4][5][6][7][8][9][10][11]. On the other hand, research conducted using finite element methods on the porosity effect of bi-directional functionally graded beams is also noteworthy.…”

Bending and Buckling Analysis of Porous 2D Functionally Graded Beams with Exponential Material Property Variation

“…The size-dependent behaviours of metal foam microbeams with three different porosity distribution models were studied by Karamanli et al [30] using FEM. Turan et al [31] presented an analytical solution based on FSDT was performed for free vibration and buckling analysis of porous FG beams subjected to various boundary conditions using FE and artificial neural network methods.…”