Abstracts:This paper presents an analytical framework to construct approximate homogenization solutions for the macroscopic elastic dielectric response — under finite deformations and finite electric fields — of dielectric elastomer composites with two-phase isotropic particulate microstructures. The central idea consists in employing the homogenization solution derived in Part I of this work for ideal elastic dielectric composites within the context of a nonlinear comparison medium method — this is derived as an extension of the comparison medium method of Lopez-Pamies et al. (2013) in nonlinear elastostatics to the coupled realm of nonlinear electroelastostatics — to generate in turn a corresponding solution for composite materials with non-ideal elastic dielectric constituents. Complementary to this analytical framework, a hybrid finite-element formulation to construct homogenization solutions numerically (in three dimensions) is also presented.

Abstracts:This paper puts forth homogenization solutions for the macroscopic elastic dielectric response—under finite deformations and finite electric fields—of ideal elastic dielectric composites with two-phase isotropic particulate microstructures. Specifically, solutions are presented for three classes of microstructures: (i) an isotropic iterative microstructure wherein the particles are infinitely polydisperse in size, (ii) an isotropic distribution of polydisperse spherical particles of a finite number of different sizes, and (iii) an isotropic distribution of monodisperse spherical particles. The solution for the iterative microstructure, which corresponds to the viscosity solution of a Hamilton–Jacobi equation in five “space” variables, is constructed by means of a novel high-order WENO finite-difference scheme. On the other hand, the solutions for the microstructures with spherical particles are constructed by means of hybrid finite elements.

Abstracts:The micromorphic theory of Eringen and Mindlin, including special cases like strain gradient theory or Cosserat theory, is widely used to model size effects and localization phenomena. The heuristic construction of such theories based on thermodynamic considerations is well-established. However, the identification of corresponding constitutive laws and of the large number of respective constitutive parameters limits the practical application of such theories.

Abstracts:This paper develops an advanced, image-based crystal plasticity finite element (CPFE) model, for predicting explicit twin formation and associated heterogeneous deformation in single crystal and polycrystalline microstructures of hexagonal close-packed or hcp materials, such as magnesium. Twin formation is responsible for premature failure of many hcp materials. The physics of nucleation, propagation and growth of explicit twins are considered in the CPFE formulation. The twin nucleation model is based on dissociation of sessile dislocations into stable twin loops, while propagation is assumed by atoms shearing on twin planes and shuffling to reduce the thermal activation energy barrier. The explicit twin evolution model however has intrinsic issues of low computational efficiency. Very fine simulation time steps with enormous computation costs are required to simulate the fast propagating twin bands and associated strain localization. To improve the computational efficiency, a multi-time scale subcycling algorithm is developed. It decomposes the computational domain into sub-domains of localized twins requiring very fine time-steps and complementary domains of relatively low resolution. Each sub-domain updates the stress and the deformation-dependent variables in different rates, followed by a coupling at the end of every coarse time step to satisfy global equilibrium. A 6-fold increase in computing speed is obtained for a polycrystalline Mg microstructure simulation in this paper. CPFE simulations of high purity Mg microstructures are compared with experiments with very good agreement in stress-strain response as well as heterogeneous twin formation with strain localization.

Abstracts:Bio-inspired fibrillar surfaces with reversible adhesion to stiff substrates have been thoroughly investigated over the last decade. In this paper we propose a novel composite fibril consisting of a soft tip layer and stiffer stalk with differently shaped interfaces (flat vs. curved) between them. A tensile stress is applied remotely on the free end of the fibril whose other end adheres to a rigid substrate. The stress distributions and the resulting adhesion of such structures were numerically investigated under plane strain $(2D)$ and axisymmetric $(3D)$ conditions. The stress intensities were evaluated for different combinations of layer thickness and Young’s moduli. The adhesion strength values were found to increase for thinner layers and larger modulus ratio; these trends are also reflected in selected experimental results. The results of this paper provide a new strategy for optimizing adhesion strength of fibrillar surfaces.

Abstracts:We present a unified variational treatment of evolving configurations in crystalline solids with microstructure. The crux of our treatment lies in the introduction of a vector configurational field. This field lies in the material, or configurational, manifold, in contrast with the traditional displacement field, which we regard as lying in the spatial manifold. We identify two distinct cases which describe (a) problems in which the configurational field's evolution is localized to a mathematically sharp interface, and (b) those in which the configurational field's evolution can extend throughout the volume. The first case is suitable for describing incoherent phase interfaces in polycrystalline solids, and the latter is useful for describing smooth changes in crystal structure and naturally incorporates coherent (diffuse) phase interfaces. These descriptions also lead to parameterizations of the free energies for the two cases, from which variational treatments can be developed and equilibrium conditions obtained. For sharp interfaces that are out-of-equilibrium, the second law of thermodynamics furnishes restrictions on the kinetic law for the interface velocity. The class of problems in which the material undergoes configurational changes between distinct, stable crystal structures are characterized by free energy density functions that are non-convex with respect to configurational strain. For physically meaningful solutions and mathematical well-posedness, it becomes necessary to incorporate interfacial energy. This we have done by introducing a configurational strain gradient dependence in the free energy density function following ideas laid out by Toupin (1962, Elastic materials with couple-stresses. Arch. Ration. Mech. Anal., 11, 385–414). The variational treatment leads to a system of partial differential equations governing the configuration that is coupled with the traditional equations of nonlinear elasticity. The coupled system of equations governs the configurational change in crystal structure, and elastic deformation driven by elastic, Eshelbian, and configurational stresses. Numerical examples are presented to demonstrate interface motion as well as evolving microstructures of crystal structures.

Abstracts:To characterize the coherent interface effect conveniently and feasibly in nanomaterials, a continuum theory is proposed that is based on the concept of the interface free energy density, which is a dominant factor affecting the mechanical properties of the coherent interface in materials of all scales. The effect of the residual strain caused by self-relaxation and the lattice misfit of nanomaterials, as well as that due to the interface deformation induced by an external load on the interface free energy density is considered. In contrast to the existing theories, the stress discontinuity at the interface is characterized by the interface free energy density through an interface-induced traction. As a result, the interface elastic constant introduced in previous theories, which is not easy to determine precisely, is avoided in the present theory. Only the surface energy density of the bulk materials forming the interface, the relaxation parameter induced by surface relaxation, and the mismatch parameter for forming a coherent interface between the two surfaces are involved. All the related parameters are far easier to determine than the interface elastic constants. The effective bulk and shear moduli of a nanoparticle-reinforced nanocomposite are predicted using the proposed theory. Closed-form solutions are achieved, demonstrating the feasibility and convenience of the proposed model for predicting the interface effect in nanomaterials.

Abstracts:We consider the canonical problem of an array of rods, which act as resonators, placed on an elastic substrate; the substrate being either a thin elastic plate or an elastic half-space. In both cases the flexural plate, or Rayleigh surface, waves in the substrate interact with the resonators to create interesting effects such as effective band-gaps for surface waves or filters that transform surface waves into bulk waves; these effects have parallels in the field of optics where such sub-wavelength resonators create metamaterials in the bulk and metasurfaces at the free surfaces.

Abstracts:A novel mechanism for slithering locomotion of a snake-like soft robot is presented. A rectangular beam with an isotropic coefficient of friction of its contact surface with the flat ground can move forward or backward when actuated by a periodic traveling sinusoidal wave. The Poisson's ratio of the beam plays an important role in the slithering locomotion speed and direction, particularly when it is negative. A theoretical model is proposed to elucidate the slithering locomotion mechanism, which is analogous to the rolling of a wheel on ground. There are two key factors of slithering locomotion: a rotational velocity field and a corresponding local contact region between the beam and ground. During wriggling motion of the rectangular beam, a rotational velocity field is observed near the maximum curvature point of the beam. If the beam has a negative Poisson's ratio, the axial tension will cause a lateral expansion so that the contact region between the beam and ground is located at the outer edge of the maximum curvature (the largest lateral expansion point). The direction of the beam's velocity at this outer edge is usually opposite to the traveling wave direction, so the friction force propels the beam in the direction of the traveling wave. A similar scenario is found for the relatively large amplitude of wriggling motion when the beam's Poisson's ratio is positive. Finite element method (FEM) simulation was conducted to verify the slithering locomotion mechanism, and good agreement was found between the FEM simulation results and theoretical predictions. The insights obtained here present a simple, novel and straightforward mechanism for slithering locomotion and are helpful for future designs of snake-like soft robots.

Abstracts:Elastomeric seals are essential to two great technological advances in oilfields: horizontal drilling and hydraulic fracturing. This paper describes a method to study elastomeric seals by using the pressure-extrusion curve (i.e., the relation between the drop of pressure across a seal and the volume of extrusion of the elastomer). Emphasis is placed on a common mode of failure found in oilfields: leak caused by a crack across the length of a long seal. We obtain an analytical solution of large elastic deformation, which is analogous to the Poiseuille flow of viscous liquids. We further obtain analytical expressions for the energy release rate of a crack and the critical pressure for the onset of its propagation. The theory predicts the pressure-extrusion curve using material parameters (elastic modulus, sliding stress, and fracture energy) and geometric parameters (thickness, length, and precompression). We fabricate seals of various parameters in transparent chambers on a desktop, and watch the seals extrude, slide, rupture and leak. The experimentally measured pressure-extrusion curves agree with theoretical predictions remarkably well.