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Kirchhoff-Love shells are commonly used in many branches of engineering, including in computer graphics, but have so far been simulated only under limited nonlinear material options. We derive the Kirchhoff-Love thin-shell mechanical energy for an arbitrary 3D volumetric hyperelastic material, including isotropic materials, anisotropic materials, and materials whereby the energy includes both even and odd powers of the principal stretches. We do this by starting with any 3D hyperelastic material, and then analytically computing the corresponding thin-shell energy limit. This explicitly identifies and separates in-plane stretching and bending terms, and avoids numerical quadrature. Thus, in-plane stretching and bending are shown to originate from one and the same process (volumetric elasticity of thin objects), as opposed to from two separate processes as done traditionally in cloth simulation. Because we can simulate materials that include both even and odd powers of stretches, we can accommodate standard mesh distortion energies previously employed for 3D solid simulations, such as Symmetric ARAP and Co-rotational materials. We relate the terms of our energy to those of prior work on Kirchhoff-Love thin-shells in computer graphics that assumed small in-plane stretches, and demonstrate the visual difference due to the presence of our exact stretching and bending terms. Furthermore, our formulation allows us to categorize all distinct hyperelastic Kirchhoff-Love thin-shell energies. Specifically, we prove that for Kirchhoff-Love thin-shells, the space of all hyperelastic materials collapses to two-dimensional hyperelastic materials. This observation enables us to create an interface for the design of thin-shell Kirchhoff-Love mechanical energies, which in turn enables us to create thin-shell materials that exhibit arbitrary stiffness profiles under large deformations. 

Abstract Drought and human land use have increased dust emissions in the western United States. However, the ecological sensitivity of remote lakes to dust deposition is not well understood and to date has largely been assessed through spatial and temporal correlations. Using in situ bioassays, we investigated the effects of dust enrichment on the production, chlorophylla(Chla) concentration, and taxonomic composition of phytoplankton and microbial communities in three western US mountain lakes. We found that dust‐derived nutrients increased Chlaconcentration in all three lakes, but the magnitude of the effect varied from 32% to 226%. This variation was related to pre‐existing lake conditions, such as trophic status, pH, and nutrient limitation. In Castle Lake, co‐limited by N and P, dust bioassays showed an increase in Chlacontent per cell but suppressed primary production and increased dark¹⁴C uptake. In contrast, both Flathead Lake and The Loch were primarily P‐limited and exhibited increases in Chlaconcentration. The contrasting Chlaand primary production results from Castle Lake are consistent with the alleviation of nitrogen limitation where energy Adenosine triphosphate (ATP) is used for nutrient assimilation instead of carbon fixation. Dust additions also altered the algal and microbial communities. The latter included the addition of new phyla (e.g.,Deinococcota), indicating that dust‐delivered microbes have the potential to thrive in receiving lakes. Our study provides the first short‐term experimental in situ evidence of rapid ecosystem effects in mountain lakes following dust exposure. The results emphasize the need for continued research in this area to understand interactions of both the short‐ and long‐term consequences of dust‐induced perturbations in remote lakes in the context of global changes.