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The elastic properties of rutile transition metal dioxides XO₂ (X = Ru, Rh, Os, and Ir) are investigated using first-principles calculations based on density functional theory. Elastic constants, bulk modulus, shear modulus, and Young's modulus as well as Possion ratio are given. OsO₂ and IrO₂ show strong incompressibility. The hardness estimated for these dioxides shows that they are not superhard solids. The obtained Debye temperatures are comparative to those of transition metal dinitrides or diborides.

We begin the mathematical study of Isogeometric Analysis based on NURBS (non-uniform rational B-splines). Isogeometric Analysis is a generalization of classical Finite Element Analysis (FEA) which possesses improved properties. For example, NURBS are capable of more precise geometric representation of complex objects and, in particular, can exactly represent many commonly engineered shapes, such as cylinders, spheres and tori. Isogeometric Analysis also simplifies mesh refinement because the geometry is fixed at the coarsest level of refinement and is unchanged throughout the refinement process. This eliminates geometrical errors and the necessity of linking the refinement procedure to a CAD representation of the geometry, as in classical FEA. In this work we study approximation and stability properties in the context of h-refinement. We develop approximation estimates based on a new Bramble–Hilbert lemma in so-called "bent" Sobolev spaces appropriate for NURBS approximations and establish inverse estimates similar to those for finite elements. We apply the theoretical results to several cases of interest including elasticity, isotropic incompressible elasticity and Stokes flow, and advection-diffusion, and perform numerical tests which corroborate the mathematical results. We also perform numerical calculations that involve hypotheses outside our theory and these suggest that there are many other interesting mathematical properties of Isogeometric Analysis yet to be proved.

We derive geometrically linearized theories for incompressible materials from nonlinear elasticity theory in the small displacement regime. Our nonlinear stored energy densities may vary on the same (small) length scale as the typical displacements. This allows for applications to multiwell energies as, e.g. encountered in martensitic phases of shape memory alloys and models for nematic elastomers. Under natural assumptions on the asymptotic behavior of such densities we prove Gamma-convergence of the properly rescaled nonlinear energy functionals to the relaxation of an effective model. The resulting limiting theory is geometrically linearized in the sense that it acts on infinitesimal displacements rather than finite deformations, but will in general still have a limiting stored energy density that depends in a nonlinear way on the infinitesimal strains. Our results, in particular, establish a rigorous link of existing finite and infinitesimal theories for incompressible nematic elastomers.

Using a generalized mean field theory, we have studied relations among effective meson masses, effective meson–nucleon couplings and equations of state (EOS) in asymmetric nuclear matter. If the effective ω-meson mass becomes smaller at high density, the EOS becomes stiffer. However, if we require that the ω-meson mean field is proportional to baryon density, the effective ω–nucleon coupling also becomes smaller at the same time as the effective ω-meson mass becomes smaller. Consequently, the EOS becomes softer. A similar relation is found for the effective ρ-meson mass and the effective ρ–nucleon coupling. We have also studied relations among effective meson masses, effective meson–nucleon couplings and a radius R of a neutron star. R depends somewhat on the value of the effective ω-meson mass and the effective ω–nucleon coupling.

In this paper we study how σ-finite measures on ℝ^d evolve under a class of "stochastic flows" associated to stochastic differential equations with (resp. without) jumps in ℝ^d. First we show the related measure evolution processes are càdlàg (resp. continuous), strongly Markovian and weakly Fellerian. Then we extend the existing results on incompressibility in Harris [8] and Kunita [14], and prove strong Markov property of the process describing how compact subsets evolve under incompressible "stochastic flows" under a certain condition.

Over the last several decades, the Finite Element Method (FEM) has emerged as a numerical approach method of choice for the solution of problems in solid mechanics. Part of the reason for the success of FEM is that it provides a unified framework for discretizing even complex differential equations. However, despite this overall unification, FEM still requires specific variants or corrections depending on the problem at hand. For instance, problems with skewed meshes, discontinuity, singularity, incompressible media, etc. require the analyst to modify the discretization approach in order to preserve robustness. We speculate that local-polynomial bases such as those used in FEM do not sufficiently represent local physics and more “physics-informed” approaches may be more universal. Accordingly, in this paper, we evaluate the feasibility of one such approach — the recently developed Distributed Physics Informed Neural Network (DPINN) approach — to provide a truly unified framework for addressing problems in Solid Mechanics. The DPINN approach utilizes a piecewise-neural network representation for the underlying field, rather than the piece-polynomial representation that is common in FEM. We solve a series of problems in solid mechanics using either the single or domain-distributed version of DPINN and demonstrate that the approach is able to seamlessly solve varied problems with no special treatment required for volumetric locking or capturing discontinuities. Further, we also demonstrate that the DPINN approach, due to its meshless nature, is able to avoid the curse of dimensionality. We discuss the relative merits and demerits of the DPINN approach in comparison to FEM. We expect this work to be useful to researchers looking to develop unified computational frameworks for problems in solid mechanics.

A quadratic triangular fluid element based on Lagrangian frame of reference is formulated for solving coupled fluid-structure interaction problems. The mesh-locking behavior due to simultaneous enforcement of the incompressibility and the irrotational constraints are studied in detail. In addition, their relationship to the number of active degrees of freedom and the number of integration points used to evaluate the stiffness matrix is established. It is found that the order of numerical integration used in the evaluation of stiffness matrix has pronounced effect on the element behavior. It is found that for a given mesh the formulated element does not lock in the absence of irrotationality constraints when the stiffness matrix is fully integrated. The same mesh locks when the stiffness matrix is fully integrated and both the constraints are enforced simultaneously. However, when the volumetric stiffness matrix is fully integrated and the rotational stiffness matrix is reduced integrated, the twin constraints are satisfied giving superior performance. The utility of the derived element to solve some coupled fluid-structure interaction problems is demonstrated and the solutions are compared with the available results.

The isoscalar and isovector nuclear matter properties are investigated in the Skyrme Hartree-Fock (SHF) and relativistic mean field (RMF) models. The correlations between the nuclear matter incompressibility and the isospin dependent term of the finite nucleus incompressibility is elucidated by using various different Skyrme Hamiltonians and RMF Lagrangians. Microscopic HF+random phase approximation (RPA) calculations are performed with Skyrme interactions for Sn isotopes to study the strength distributions of isoscalar giant monopole resonances (ISGMR). The symmetry term of nuclear incompressibility is extracted to be K_τ = -(500 ± 50) MeV from the recent experimental data of ISGMR in Sn isotopes.

A nuclear three-body force based on the meson-exchange approach is constructed using the same meson parameters and the exponential form factors as in the Nijmegen potential, involving four kinds of important mesons, π, ρ, ω, and σ [f₀(975) and ∊(760)]. For the 2π-exchange three-nucleon component, we adopt the new expansion strength constants a, b, c consistent with the contemporary πN-scattering data base and the corresponding dipole form factor. An effective two-body interaction is derived by averaging out the third nucleon, and is self-consistently used together with the Nijmegen potential in the Brueckner-Hartree-Fock approximation. The empirical nuclear matter saturation properties are reproduced very well. At higher density the equation of state becomes rather stiff due to the strong repulsion from the (σ, ω)-N three-body contribution.

In the framework of Skyrme energy-density functional approach together with the extended semi-classical Thomas-Fermi method, the nuclear equation of state (EOS) is investigated using the nucleus-nucleus potential. The potentials of ⁵⁸Ni+⁵⁸Ni and ³²S+⁸⁹Y are studied in the point of view that the energy at touching point is equal to the threshold energy for hindrance. It is found that MSK3-MSK5, V110 are suitable Skyrme parameters sets and the incompressibility values are in the range of K = 231-234 MeV.