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Abstract Liquid crystal elastomers (LCEs) marry the large deformation response of a cross-linked polymer network with the nematic order of liquid crystals pendent to the network. Of particular interest is the actuation of LCE sheets where the nematic order, modeled by a unit vector called the director, is specified heterogeneously in the plane of the sheet. Heating such a sheet leads to a large spontaneous deformation, coupled to the director design through a metric constraint that is now well-established by the literature. Here we go beyond the metric constraint and identify the full plate theory that underlies this phenomenon. Starting from a widely used bulk model for LCEs, we derive a plate theory for the pure bending deformations of patterned LCE sheets in the limit that the sheet thickness tends to zero using the framework of$$\Gamma $$ $\Gamma$-convergence. Specifically, after dividing the bulk energy by the cube of the thickness to set a bending scale, we show that all limiting midplane deformations with bounded energy at this scale satisfy the aforementioned metric constraint. We then identify the energy of our plate theory as an ansatz-free lower bound of the limit of the scaled bulk energy, and construct a recovery sequence that achieves this plate energy for all smooth enough midplane deformations. We conclude by applying our plate theory to a variety of examples. 

Abstract We present proper motion measurements of the oxygen-rich ejecta of the LMC supernova remnant N132D using two epochs of Hubble Space Telescope Advanced Camera for Surveys data spanning 16 years. The proper motions of 120 individual knots of oxygen-rich gas were measured and used to calculate a center of expansion (CoE) of α = 5 h 25 m 01.ˢ71 and δ = −69°38′41.″64 (J2000) with a 1 σ uncertainty of 2.″90. This new CoE measurement is 9.″2 and 10.″8 from two previous CoE estimates based on the geometry of the optically emitting ejecta. We also derive an explosion age of 2770 ± 500 yr, which is consistent with recent age estimates of ≈2500 yr made from 3D ejecta reconstructions. We verified our estimates of the CoE and age using a new automated procedure that detected and tracked the proper motions of 137 knots, with 73 knots that overlap with the visually identified knots. We find that the proper motions of the ejecta are still ballistic, despite the remnant’s age, and are consistent with the notion that the ejecta are expanding into an interstellar medium cavity. Evidence for explosion asymmetry from the parent supernova is also observed. Using the visually measured proper motion measurements and corresponding CoE and age, we compare N132D to other supernova remnants with proper motion ejecta studies. 

Abstract We describe a process for cross-calibrating the effective areas of X-ray telescopes that observe common targets. The targets are not assumed to be “standard candles” in the classic sense, in that we assume that the source fluxes have well-defined, but a priori unknown values. Using a technique developed by Chen et al. that involves a statistical method called shrinkage estimation , we determine effective area correction factors for each instrument that bring estimated fluxes into the best agreement, consistent with prior knowledge of their effective areas. We expand the technique to allow unique priors on systematic uncertainties in effective areas for each X-ray astronomy instrument and to allow correlations between effective areas in different energy bands. We demonstrate the method with several data sets from various X-ray telescopes.