Abstract
A concept of composite materials reinforced by branching micro or nanotubes optimized for both heat transfer and strength of the material is presented. Numerous examples of reinforcement by branched fibers in cells, tissues and organs of plants and animals are studied. It is shown orientation of the fibers according to principals of the stress tensor at given external load is the main principle of optimal reinforcement in nature. The measurement data obtained on venations of the plant leaves revealed clear dependencies between the diameters, lengths and branching angles that correspond to delivery of the plant sap to live cells of the leaf with minimal energy expenses. The mathematical problem on geometry of asymmetrical loaded branched fibers experienced minimal maximal stress is solved. Heat propagation in the fibers is described by generalized Guyer-Krumhansl equation. It is shown the optimality for the heat propagation, fluid delivery and structural reinforcement are based on the same relations between the diameters, lengths and branching angles. The principle of optimal reinforcement is proposed for technical constructions, advanced composite materials and MEMS devices.
Highlights
Biological tissues are mostly presented by composite materials reinforced by fibers or tubes conveying biological fluids to and from the live cells (Fung, 1981)
As it was shown in numerous experimental studies and observations, geometry of the pipelines is determined by certain relationships between the diameters of the pipes in the bifurcations (Murray, 1926a; 1926b), and between the diameters and branching angles (Rosen, 1967; Weibel, 1963; La Barbera, 1990; Kizilova and Popova, 1999; McCulloh, Sperry and Adler, 2003)
Like for the fluid flow case, the conditions of local optimality of the tube/wire will coincide with necessary conditions of global optimality of the network in the meaning of the minimal total energy expenses for the fluid/heat flux and structural support of the network
Summary
Biological tissues are mostly presented by composite materials reinforced by fibers or tubes conveying biological fluids to and from the live cells (Fung, 1981). Summarizing the above presented data on fiber reinforcement in nature, one can conclude both plants and animals use the same main principles of mechanical construction of their tissues and organs, namely − Reinforcement by relatively rigid fibers/tubes located according the principles of the stress tensor at given external mechanical load;. Like for the fluid flow case, the conditions of local optimality of the tube/wire will coincide with necessary conditions of global optimality of the network in the meaning of the minimal total energy expenses for the fluid/heat flux and structural support of the network (material and other expenses) It gives example of the functionally perfect nature inspired design with optimal branching angles α1,α2 (Fig.[6] b) instead of geometrically perfect engineered design with α1,2 = 90o (Fig.6a). Geometry of the bifurcation can be described by relative parameters (b1 + b2 ) / a ∈]0,2[, b1 / b2 ∈]0,1[
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