Principle of least variance for dual scale reliability of structural systems

A R-integral is defined to account for the evolution of the root functions from Ideomechanics. They can be identified with, though not limited to, the fatigue crack length or velocity. The choice was dictated by the available validated data for relating accelerated testing to real time life expectancy. The key issue is to show that there exists a time range of high reliability for the crack length and velocity that correspond to the least variance of the time dependent R-integrals. Excluded from the high reliability time range are the initial time span where the lower scale defects are predominant and the time when the macrocrack approaches instability at relatively high velocity. What remains is the time span for micro-macro cracking. The linear sum (ls) and root mean square (rms) average are used to delineate two different types of variance. The former yields a higher reliability in comparison with that for the latter. The results support the scale range established empirically by in-service health monitoring for the crack length and velocity. The principle of least variance can be extended to multiscale reliability analysis and assessment for multi-component and multi-function systems.     

Pseudo global energy released locally by crack extension involving multiscale reliability

The energy release rate criterion, being mono scale by definition, is incompatible with the failure behavior of solids that are inherently dual, if not, multiscale. Time span of reliability is scale sensitive and can be addressed with consistency only by use of transitional functions that are designed to transform a function from one scale to another. A pseudo transitional energy release rate G^∗ is defined to address the cross-scaling properties of energy release rate. The reliability of such a function is found to fall quickly when the scale range deviates from that of micro-macro. In general, the time span of reliability based on G^* shortens considerably within the nano-micro and pico-nano scale ranges, resulting in fast turnover of system usability. Prediction accuracy tends to be scale range specific. Stress or strain based criteria are also mono scale. They may be adequate for some situations at the macroscopic scale, but can be ambiguous for multiscale problems. These situations are analyzed by application of the principle of least variance in conjunction with the R-integrals.Accelerated test data for the equivalent of 20 years’ fatigue crack growth in 2024-T3 aluminum panels were analyzed using the mutliscale reliability model. A time span plateau within the micro-macro range is from 8 to 17 years. This corresponds to the reliable portion of prediction, while the terminal 3 years are regarded as unreliable. A similar time span plateau were also found from 4 to 6 years within the nano-micro scale range. And an even smaller plateau hovering around 1.2 years were found for the pico-nano scale range. Time span of reliable prediction narrows with down sized scale range. The overlapping ends of the scale ranges are rendered unreliable as anticipated. These regions can be suppressed by the addition of meso scale ranges. Reference can be made to past discussions related to multiscaling and mesomechanics.     

Scale shifting laws from pico to macro in consecutive segments by use of transitional functions

Systems with parts that vary in size from pico to macro inclusive are vulnerable of being incapacitated when a single part fails owing to deterioration of material properties. The majority of system failure can be attributed to incompatibility of integrated parts that were designed individually for general purpose. Total reliability calls for all parts, small and large, to be compatible in life spans. Mass, when regarded as energized matter, can vary as a function of time. This, in retrospect, explains why non-equilibrium and non-homogeneity cannot be avoided for multiscale shifting laws. A consistent and scale invariant definition of energy dissipation gives rise to mass pulsation, a common mechanism that seems to be applicable to living and non-living organisms.Scale shifting laws are developed from the use of transitional functions that stand for the mass ratios related to absorption energies and dissipation energies . The notations j and j + 1 stand for two successive scales: pi-na, na-mi, and mi-ma. Hence, the mass ratios , and can be referred to as the transitional inhomogeneity coefficients. They make up the scale shifting laws . Connection of the accelerated test data at the different scales, say from pico to nano to micro to macro, can be made by application of the definition of a scale invariant energy density dissipation function.On physical grounds, the segmented non-equilibrium and non-homogeneous test data can be connected through a velocity dependent mass and energy relation. Energy and power efficiency are defined to explore the macroscopic experiences to those at the lower scales. The time evolution properties of the material can also be derived as a package to include the accelerated test data, a procedure normally referred to as validation. The separation of derive-first and test-later, can never be abridged without ambiguities. Hence, total reliability of a system with many parts is advocated by judiciously matching the nine primary variables consisting of the initial disorder sizes, the time rates, and increments of the absorbed and dissipated energy density. The nine controllable variables consisting of life span distribution, energy, and power efficiencies for the three scale ranges are of secondary consideration.