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Research Papers

# Rapid Computation of Statistically Stable Particle/Feature Ratios for Consistent Substrate Stresses in Printed Flexible Electronics

[+] Author and Article Information
T. I. Zohdi

Department of Mechanical Engineering,
University of California,
Berkeley, CA 94720-1740

The Poisson ratio was set to ν = 0.3.

L scaled out of the results and was irrelevant.

We note that the quantity $σ':σ'$ is invariant under the rotational coordinate transformation, in other words, $σ',car:σ',car=(RT(θ)·σ',cyl·R(θ))T:(RT(θ)·σ',cyl· R(θ))=σ',cyl:σ',cyl$, thus this metric remains perfectly acceptable to use in the presence of non-normal loading.

Contributed by the Manufacturing Engineering Division of ASME for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received September 16, 2014; final manuscript received December 5, 2014; published online February 4, 2015. Assoc. Editor: Donggang Yao.

J. Manuf. Sci. Eng 137(2), 021019 (Apr 01, 2015) (5 pages) Paper No: MANU-14-1476; doi: 10.1115/1.4029327 History: Received September 16, 2014; Revised December 05, 2014; Online February 04, 2015

## Abstract

This paper develops a statistically based computational method to rapidly determine stresses in flexible substrates during particle printing processes. Specifically, substrate stresses due to multiple surface particle contact sites are statistically computed by superposing point load solutions for different random particle realizations (sets of random loading sites) within a fixed feature boundary. The approach allows an analyst to rapidly determine the number of particles in a surface feature needed to produce repeatable substrate stresses, thus minimizing the deviation from feature to feature and ensuring consistent production. Three-dimensional examples are provided to illustrate the technique. The utility of the approach is that an analyst can efficiently ascertain the number of particles needed within a feature, without resorting to computationally intensive numerical procedures, such as the finite element method.

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## Figures

Fig. 1

Placing particles into a surface and applying pressure to properly bond them to the substrate

Fig. 2

Left: a single particle; middle: a set of particles on the surface under loading; and right: the stress intensity

Fig. 3

The algorithm for computation of statistically stable particle/feature resolution

Fig. 4

LEFT: Three feature examples tested: (a) a solid circular pattern, (b) a ring pattern, and (c) a figure 8 pattern. RIGHT: two random realizations (sets of loading sites) from the 100 tested for each N-set.

Fig. 5

Example 1—solid circle (with surface area of As = π(0.15L)2): overall quality metrics (the average, δN,M and the standard deviation, δN,M) as a function of the number of particles in the feature (for an applied unit load Fz = 1). Results are shown for the deviatoric stresses (σ') and the total stresses (σ). We note that the number of particles in the area should be interpreted as number of loading sites in the feature boundary.

Fig. 6

Example 2—ring (with surface area of As = π((0.15L)2 - (0.1L)2)): overall quality metrics (the average, δN,M and the standard deviation, δN,M) as a function of the number of loading sites in the feature (for an applied unit load Fz = 1). Results are shown for the deviatoric stresses (σ') and the total stresses (σ). We note that the number of particles in the area should be interpreted as number of loading sites in the feature boundary.

Fig. 7

Example 3—figure 8 (with surface area of As = 2π((0.15L)2 - (0.1L)2)): overall quality metrics (the average, αN,M and the standard deviation, δN,M) as a function of the number of particles in the feature (for an applied unit load Fz = 1). Results are shown for the deviatoric stresses (σ') and the total stresses (σ). We note that the number of particles in the area should be interpreted as number of loading sites in the feature boundary.

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