Nonlinear elastic constitutive relations of residually stressed composites with stiff curved fibres

Applied Mathematics and Mechanics - Mechanical models of residually stressed fibre-reinforced solids, which do not resist bending, have been developed in the literature. However, in some residually...     

13C chemical shift tensors in MOF α-Mg3(HCOO)6: Which component is more sensitive to host-guest interaction?

Metal–organic frameworks (MOFs) are a class of important porous materials with many current and potential applications. Their applications almost always involve the interaction between host framework and guest species. Therefore, understanding of host–guest interaction in MOF systems is fundamentally important. Solid-state NMR spectroscopy is an excellent technique for investigating host–guest interaction as it provides information complementary to that obtained from X-ray diffraction. In this work, using MOF α-Mg₃(HCOO)₆ as an example, we demonstrated that ¹³C chemical shift tensor of organic linker can be utilized to probe the host–guest interaction in MOFs. Obtaining ¹³C chemical shift tensor components (δ₁₁, δ₂₂, and δ₃₃, where δ₁₁ ≥ δ₂₂ ≥ δ₃₃) in this MOF is particularly challenging as there are six coordinatively equivalent but crystallographically non-equivalent carbons in the unit cell with very similar local coordination environment. Two-dimensional magic-angle-turning experiments were employed to measure the ¹³C chemical shift tensors of each individual crystallographically non-equivalent carbon in three microporous α-Mg₃(HCOO)₆ samples with different guest species. The results indicate that the δ₂₂ component (with its direction approximately being co-planar with the formate anion and perpendicular to the C−H bond) is more sensitive to the adsorbate molecules inside the MOF channel due to the weak C−H···O hydrogen bonding or the ring current effect of benzene. The ¹³C isotropic chemical shift, on the other hand, seems much less sensitive to the subtle changes in the local environment around formate linker induced by adsorption. The approach described in this study may be used in future studies on host–guest interaction within MOFs.     

Radial breathing-mode frequency of elastically confined spherical nanoparticles subjected to circumferential magnetic field

Knowledge of the vibrational properties of nanoparticles is of fundamental interest since it is a signature of their morphology, and it can be utilized to characterize their physical properties. In addition, the vibration characteristics of the nanoparticles coupled with surrounding media and subjected to magnetic field are of recent interest. This paper develops an analytical approach to study the radial breathing-mode frequency of elastically confined spherical nanoparticles subjected to magnetic field. Based on Maxwell's equations, the nonlocal differential equation of radial motion is derived in terms of radial displacement and Lorentz's force. Bessel functions are used to obtain a frequency equation. The model is justified by a good agreement between the results given by the present model and available experimental and atomic simulation data. Furthermore, the model is used to elucidate the effect of nanoparticle size, the magnetic field and the stiffness of the elastic medium on the radial breathing-mode frequencies of several nanoparticles. Our results reveal that the effects of the magnetic field and the elastic medium are significant for nanoparticle with small size.     

Bi-Sobolev solutions to the prescribed Jacobian inequality in the plane with Lp data and applications to nonlinear elasticity

We construct planar bi-Sobolev mappings whose local volume distortion is bounded from below by a given function $f\in L^p$ with $p>1$. More precisely, for any $1<q<(p+1)/2$ we construct $W^{1,q}$-bi-Sobolev maps with identity boundary conditions; for $f\in L^{\infty }$, we provide bi-Lipschitz maps. The basic building block of our construction are bi-Lipschitz maps which stretch a given compact subset of the unit square by a given factor while preserving the boundary. The construction of these stretching maps relies on a slight strengthening of the celebrated covering result of Alberti, Csörnyei, and Preiss for measurable planar sets in the case of compact sets. We apply our result to a model functional in nonlinear elasticity, the integrand of which features fast blowup as the Jacobian determinant of the deformation becomes small. For such functionals, the derivation of the equilibrium equations for minimizers requires an additional regularization of test functions, which our maps provide.     

Mechanical deformations of a liquid crystal elastomer at director angles between 0° and 90°: Deducing an empirical model encompassing anisotropic nonlinearity

Despite the wealth of studies reporting mechanical properties of liquid crystal elastomers (LCEs), no theory can currently describe their complete mechanical anisotropy and nonlinearity. Here, we present the first comprehensive study of mechanical anisotropy in an all‐acrylate LCE via tensile tests that simultaneously track liquid crystal (LC) director rotation. We then use an empirical approach to gain a deeper insight into the LCE's mechanical responses at values of strain, up to 1.5, for initial director orientations between 0° and 90°. Using a method analogous to time–temperature superposition, we create master curves for the LCE's mechanical response and use these to deduce a model that accurately predicts the load curve of the LCE for stresses applied at angles between 15° and 70° relative to the initial LC director. This LCE has been shown to exhibit auxetic behavior for deformations perpendicular to the director. Interestingly, our empirical model predicts that the LCE will further demonstrate auxetic behavior when stressed at angles between 54° and 90° to the director. Our approach could be extended to any LCE; so it represents a significant step forward toward models that would aid the further development of LCE theory and the design and modeling of LCE‐based technologies. © 2019 The Authors. Journal of Polymer Science Part B: Polymer Physics published by Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019, 57, 1367–1377