Analytical and numerical analyses of neo-Hookean thin-wall composite spheres

Thin-wall composite spheres (TWCSs) are very common in both natural and artificial structures. Their response to mechanical loading was investigated in the past almost solely in the limit of infinitesimal deformations. We examine, within the framework of finite deformation elasticity, the mechanics of incompressible TWCSs with neo-Hookean core and shell phases subjected to general homogeneous displacement and traction boundary conditions. We derive explicit general forms for the displacement and the pressure fields in both phases in terms of a power series about the shear and the tension magnitudes and the shell volume fraction. The predictions of the analytical solutions are analyzed and compared with corresponding results of finite element simulations for TWCSs with different ratios between the phases shear moduli. In addition to an extension of the work of Weil and deBotton [22] from simple shear to general homogeneous boundary conditions, we modify the power series solution and provide a reliable solution for any combination of phases shear modulus. We demonstrate that a relatively small number of terms in the series is required for a good agreement with the numerical simulations up to a stretch ratio of 1.5 when considering the local fields, and up to a stretch ratio of 2 when considering the average fields. The analysis emphasizes the interaction between the shell and the core and reveals the different roles of the coating under different boundary conditions. We highlight interesting similarities and dissimilarities between the spatial distributions of the local stresses and the variations of the average stresses developing in TWCSs with stiff and soft shells, under displacement and traction boundary conditions.

Nonlocal Analytical Solutions for Multilayered One-Dimensional Quasicrystal Nanoplates

An exact closed-form solution for the three-dimensional static deformation and free vibrational response of a simply supported and multilayered quasicrystal (QC) nanoplate with the nonlocal effect is derived. Numerical examples are presented for a homogeneous crystal nanoplate, homogenous QC nanoplate, and sandwich nanoplates with various stacking sequences. Induced by traction boundary conditions, extended displacements and stresses reveal the important role that the nonlocal parameter plays in the structural analysis of nanoquasicrystals (nano-QCs). The natural frequencies and the corresponding mode shapes of the nanoplates further show the influence of stacking sequence and phonon–phason coupling effect. This exact solution is useful for it provides benchmark results to assess the accuracy of finite element nano-QC models and can assist engineers in tuning their quasicrystal nanoplate design.

A constitutive model for an internally balanced compressible elastic material

Many large deformation constitutive models for the mechanical behavior of solid materials make use of the multiplicative decomposition [Formula: see text] as, for example, used by Kröner in the context of finite-strain plasticity. Then [Formula: see text] describes the elastic effect by letting the potential energy of the deformation depend upon [Formula: see text]. In this paper we allow the potential energy to depend upon both portions of the multiplicative decomposition. As in hyperelasticity, energy minimization with respect to displacement gives equilibrium field equations and traction boundary conditions. The new feature, minimization with respect to the decomposition itself, generates an additional mathematical requirement that is interpreted here in terms of a principle of internal mechanical balance. We specifically consider a Blatz–Ko-type solid suitably generalized to incorporate the notion of internal balance. Conventional results of hyperelasticity are retrieved for certain limiting forms of the energy density, whereas the general form of the energy density gives rise to an overall softening response, as is demonstrated in the context of pure pressure and uniaxial loading.