Schematic of the behavior of a beam of original length L 0 and buckled length .
(Top panel) Normalized force and (bottom panel) normalized internal (strain) energy versus normalized displacement for a buckled (solid lines) and unbuckled beams (dashed lines), adapted from Saif. 23 Results indicate response for both displacement- (black lines) and force-controlled loading (gray lines), with arrows indicating direction of force loading.
Schematic of the MMM developed in this work.
Depiction of scales and scale transitions in the MMM: (a) microscale, (thermally induced buckling of top beam element depicted in inset); (b) mesoscale, ; and (c) macroscale, .
Benchmark results of energy derivative method for a cube of isotropic steel: (a) strain energy vs. engineering strain; (b) stress function vs. engineering strain; and (c) stiffness vs. engineering strain.
Temperature-dependent stiffness moduli vs. engineering strain of candidate microstructure for values ranging from 0–1200 K in increments of 200 K: (a) effective C 11; (b) effective C 12; and (c) effective C 66 (note: change in x-axis made due to symmetry of response). Arrows indicate increasing values of .
Effective stiffness values verses volume fraction: (a) black–C 11, gray–C 33; (b) black–C 12, gray–C 13; and (c) black–C 44, gray–C 66. Solid line—differential effective medium; dashed line—self-consistent model; dashed line with dots—Mori-Tananka; FEA with denoted by an “o”; and FEA with structured inclusion denoted by an “x.”
Effective stiffness (top panel) and loss (bottom panel) properties predicted by the Self-Consistent model for a composite containing identically oriented 2% by volume of the mesoscale inclusions. The matrix is assumed to have ν = 0.30 and η = 0.05. Stiffness and loss values ( denotes complex-valued) for —dashed black line; —solid black line; —dashed gray line; and —solid gray line.
Summary of the applied strains and expressions to determine the independent constants of an tetragonally symmetric medium using the energy derivative approach.
Strain energy and stiffness in a candidate structured inclusion using the direct FEA micro → meso scale transition model.
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