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1. The Mathematics of Thin Structures

   Babadjian, J.-F.; Di Fratta, G.; Fonseca, I.; Francfort, G.; Lewicka, M.; Muratov, C. B. (January 2023, Quarterly of applied mathematics)

   This articles offers various mathematical contributions to the behavior of thin films. The common thread is to view thin film behavior as the variational limit of a three-dimensional domain with a related behavior when the thickness of that domain vanishes. After a short review in Section 1 of the various regimes that can arise when such an asymptotic process is performed in the classical elastic case, giving rise to various well-known model in plate theory (membrane, bending, Von Karmann, etc...), the other sections address various extensions of those initial results. Section 2 adds brittleness and delamination and investigates the brittle membrane regime. Sections 4 and 5 focus on micro-magnetics, rather than elasticity, this once again in the membrane regime and discuss magnetic skyrmions and domain walls, respectively. Finally, Section 3 revisits the classical setting in a non-Euclidean setting induced by the presence of a pre-strain in the model. 

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2. Building a universal orientation system for thin sections

   https://doi.org/doi.org/10.1016/j.jsg.2018.09.019  

   Tikoff, Basil; Chatzaras, Vasileios; Newman, Julie; Roberts, Nicolas (January 2019, Journal of structural geology)

   We present an orientation system for thin sections used for microanalysis, applicable to both billets and cores. The orientation system enables spatially referenced observations and consists of three parts. First, we establish a reference corner that is the uppermost corner of the sample on the thin section, in its original geographic orientation in the field or laboratory setting. This corner is tied to a right-hand coordinate system, in which all reference axes point downward. A geographic direction-based, rather than uppermost corner-based, convention for a reference corner can be substituted for projects that utilize sub-horizontally oriented thin sections. The reference corner - combined with orientation metadata - define a unique position of the thin section in geographic space. Second, we propose a system of small saw cuts (notches) that minimizes the number of notches required on the sample, to distinguish both the reference corner and the orientation of the thin section relative to fabric (e.g., foliation/lineation), if present. The utility of a notching standard is that it provides an inherent doublecheck on thin section orientation and facilitates sharing between users. Third, we develop a grid system in order to locate features of interest on the thin section, relative to the reference corner. Any of these systems – referencing, notching, and gridding – can be used independently. These systems are specifically designed to work with digital data systems, which are currently being developed, allowing researchers to share microstructural data with each other and facilitating new types of big data science in the field of structural geology. 

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3. Mechanically Programming the Cross-Sectional Shape of Soft Growing Robotic Structures for Patient Transfer

   Osele, G; Barhydt, K; Sullivan, T; Asada, H; Okamura, A (October 2025, IEEE)

   Pneumatic soft everting robotic structures have the potential to facilitate human transfer tasks due to their ability to grow underneath humans without sliding friction and their utility as a flexible sling when deflated. Tubular structures naturally yield circular cross-sections when inflated, whereas a robotic sling must be both thin enough to grow between a human and their resting surface and wide enough to cradle the human. Recent works have achieved flattened cross-sections by including rigid components into the structure, but this reduces conformability to the human. We present a method of mechanically programming the cross-section of soft everting robotic structures using flexible strips that constrain radial expansion between points along the outer membrane. Our method enables simultaneously wide and thin inflated profiles, and maintains the full multi-axis flexibility of traditional slings when deflated. We develop and validate a model relating geometric design specifications to fabrication parameters, and experimentally characterize their effects on growth rate. Finally, we prototype a soft growing robotic sling system and demonstrate its use for assisting a single caregiver in bed-to-chair patient transfer. 

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4. Surface densities prewet a near-critical membrane

   https://doi.org/10.1073/pnas.2103401118  

   Rouches, Mason; Veatch, Sarah L.; Machta, Benjamin B. (October 2021, Proceedings of the National Academy of Sciences)

   Recent work has highlighted roles for thermodynamic phase behavior in diverse cellular processes. Proteins and nucleic acids can phase separate into three-dimensional liquid droplets in the cytoplasm and nucleus and the plasma membrane of animal cells appears tuned close to a two-dimensional liquid–liquid critical point. In some examples, cytoplasmic proteins aggregate at plasma membrane domains, forming structures such as the postsynaptic density and diverse signaling clusters. Here we examine the physics of these surface densities, employing minimal simulations of polymers prone to phase separation coupled to an Ising membrane surface in conjunction with a complementary Landau theory. We argue that these surface densities are a phase reminiscent of prewetting, in which a molecularly thin three-dimensional liquid forms on a usually solid surface. However, in surface densities the solid surface is replaced by a membrane with an independent propensity to phase separate. We show that proximity to criticality in the membrane dramatically increases the parameter regime in which a prewetting-like transition occurs, leading to a broad region where coexisting surface phases can form even when a bulk phase is unstable. Our simulations naturally exhibit three-surface phase coexistence even though both the membrane and the polymer bulk only display two-phase coexistence on their own. We argue that the physics of these surface densities may be shared with diverse functional structures seen in eukaryotic cells. 

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5. Towards an integrated texture toolkit, 1: unveiling the complex relationship between crystal shape and fabric in EBSD data

   https://doi.org/10.1007/s00410-024-02128-x  

   Currier, Ryan M; Mittal, Tushar; Hidalgo, Paulo J (April 2024, Contributions to Mineralogy and Petrology)

   Rock textures observed via thin section are skewed from their true 3D nature. This is due to various cut effects—artifacts introduced due to the lower dimensional nature of the thin section relative to the rock. These cut effects can be corrected, and several methods have been developed to invert crystal shape and crystal size, but with each process performed separately and sequentially. With the ongoing adoption of electron backscatter diffraction (EBSD) by petrologists, an additional data stream has now become available: the 3D orientation of 2D grain sections. For EBSD analysis, no stereological corrections are typically applied for interpreting the data. This study tests whether this orientational information is skewed due to a fabric cut effect. We test this by numerically generating synthetic crystal datasets representative of several crystal shapes and population sizes. We find that EBSD orientational data has a fabric cut effect since crystals oriented with long axes perpendicular to the thin section are more likely to be sampled compared to those with long axes oriented parallel to it. This effect must be accounted for to interpret the true 3D fabric accurately. Towards this end, we develop two new tools for working with EBSD-derived fabric: (1) a simple first-order test for determining if a measured fabric exceeds that of the fabric cut effect, and (2) a method of inverting cut fabrics that provides robust error estimations. We demonstrate the applicability and accuracy of these methods using a range of synthetic examples and a natural sample. With these newly developed tools, there is clear potential for a new textural toolkit framework, to further our ability to correct for the various cut effects while also providing accurate uncertainty estimates. 

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