Blog posts

Suppose if somebody asked you the following question, and more importantly, wanted the answer to an accuracy of 100-digits:

• Problem A: A particle at the center of a 10 x 1 rectangle undergoes Brownian motion (i.e., two-dimensional random walk with infinitesimal step lengths) until it hits the boundary. What is the probability that it hits at one of the ends rather than at one of the sides?

Or, this question (again, demanding the answer to an accuracy of 100-digits):

Abstract: Flexible electronic devices often require hermetic coatings that can withstand applied strains. This paper calculates the critical strains for various configurations of channel cracks in a coating consisting of organic and inorganic layers. We show that the coating can sustain the largest strain when the organic layer is of some intermediate thicknesses.

Flexible electronics are promising for diverse applications, such as rollable displays, conformal sensors, and printable solar cells. These systems are thin, rugged, and lightweight. They can be manufactured at low costs, for example, by roll-to-roll printing. The development of flexible electronics has raised many issues concerning the mechanical behavior of materials. This paper examines a particular issue: channel cracks in hermetic coatings.

Electronic devices (e.g., organic light-emitting devices, OLEDs) often degrade when exposed to air. Developing hermetic coatings has been a significant challenge. Organic films are permeable to gases, and inorganic films inevitably contain processing flaws, so that neither by themselves are effective gas barriers. These considerations have led to the development of multilayer coatings consisting of alternating organic and inorganic films. To be used in flexible electronics, these coatings must also withstand applied strains without forming channel cracks...

This paper presents some developments related to the idea of covariance in elasticity. The geometric point of view in continuum mechanics is briefly reviewed. Building on this, regarding the reference configuration and the ambient space as Riemannian manifolds with their own metrics, a Lagrangian field theory of elastic bodies with evolving reference configurations is developed. It is shown that even in this general setting, the Euler-Lagrange equations resulting from horizontal (referential) variations are equivalent to those resulting from vertical (spatial) variations. The classical Green-Naghdi-Rivilin theorem is revisited and a material version of it is discussed. It is shown that energy balance, in general, cannot be invariant under isometries of the reference configuration, which in this case is identified with a subset of R^3. Transformation properties of balance of energy under rigid translations and rotations of the reference configuration is obtained. The spatial covariant theory of elasticity is also revisited. The transformation of balance of energy under an arbitrary diffeomorphism of the reference configuration is obtained and it is shown that some nonstandard terms appear in the transformed balance of energy. Then conditions under which energy balance is materially covariant are obtained. It is seen that material covariance of energy balance is equivalent to conservation of mass, isotropy, material Doyle-Ericksen formula and an extra condition that we call ‘configurational inviscidity’. In the last part of the paper, the connection between Noether’s theorem and covariance is investigated. It is shown that the Doyle-Ericksen formula can be obtained as a consequence of spatial covariance of Lagrangian density. Similarly, it is shown that the material Doyle-Ericksen formula can be obtained from material covariance of Lagrangian density.

The Department of Mechanical and Aerospace Engineering at North Carolina State University invites applications for a tenure-track faculty position in the general area of mechanical sciences. Candidates must have an earned doctorate in Mechanical or Aerospace Engineering or closely-related field. Successful candidates will be expected to teach at the undergraduate and graduate levels, to advise graduate students, and to establish a high quality, nationally-visible externally funded research program.

I would like to share some of our more recent findings on nano-pillar compression, namely the role of the surface treatment in plastic deformation at the nano-scale. Recent advances in the 2-beam focused ion beams technology (FIB) have enabled researchers to not only perform high-precision nanolithography and micro-machining, but also to apply these novel fabrication techniques to investigating a broad range of materials' properties at the sub-micron and nano-scales. In our work, the FIB is utilized in manufacturing of sub-micron cylinders, or nano-pillars, as well as of TEM cross-sections to directly investigate plasticity of metals at these small length scales. Single crystal nano-pillars, ranging in diameter between 300 nm and 870 nm, were fabricated in the FIB from epitaxial gold films on MgO substrates and subsequently compressed using a Nanoindenter fitted with a custom-fabricated diamond flat punch. We show convincingly that flow stresses strongly depend on the sample size, as some of our smaller specimens were found to plastically deform in uniaxial compression at stresses as high as 600 MPa, a value ~25 times higher than for bulk gold. We believe that these high strengths are hardened by dislocation starvation. In this mechanism, once the sample is small enough, the mobile dislocations have a higher probability of annihilating at a nearby free surface than of multiplying and being pinned by other dislocations. Contrary to this, if the dislocations are trapped inside the specimen by a coating, the strengthening mechanism is expected to be different. Here we present for the first time the comparison of plastic deformation of passivated and unpassivated single crystal specimens at the sub-micron scale. The role of free surfaces is investigated by comparing stress results of both as-FIB'd, annealed, and alumina-passivated pillars. Preliminary results show that ALD-coated pillars exhibit much higher flow stresses at equivalent sizes and strains compared with the uncoated samples. We also found that while FIB damage during pillar fabrication might account for a small portion of the strength increase, it is not the major contributor.