Morphological instabilities and surface wrinkling of soft materials such as gels and biological tissues are of growing interest to a number of academic disciplines including soft lithography, metrology, flexible electronics, and biomedical engineering. In this paper, we review some of the recent progresses in experimental and theoretical investigations of instabilities that lead to the emergence and evolution of surface wrinkling, folding and creasing under various geometrical constraints (e.g., thin films, sheets, fibers, particles, tubes, cavities, vesicles and capsules) and loading stimuli (e.g., mechanical forces, growth, atrophy, swelling, shrinkage, van der Waals interactions). Some representative theoretical and numerical approaches aimed at modelling the onset of instabilities as well as the postbuckling evolution involving multiple bifurcations and symmetry-breakings are discussed along with the main characteristics and some possible applications of this rich phenomenon.

Mechanics of morphological instabilities and surface wrinkling in soft materials: a review

Piezoelectric nano-structures have been regarded as the next-generation piezoelectric material due to their inherent nano-sized piezoelectricity. This review summarizes the recent theoretical and experimental findings in piezoelectric nano-structures, including piezoelectric nanowires, nanoplates, nanobeams, nanofilms, nanoparticles, and piezoelectric heterogeneous materials containing piezoelectric nano-inhomogeneities. To begin, the types of piezoelectric nano-structured materials and the wide application of piezoelectric nano-structures in recent years are delineated. Next, the theoretical foundations including the definition of surface stress and electric displacement, the surface constitutive relations, the surface equilibrium equations, and nonlocal piezoelectricity, and their applications, are illustrated. Then, the effective mechanical and piezoelectric properties are depicted. Furthermore, the experimental investigations are classified, and some important observations are discussed. Finally, the perspectives and challenges for the future development of piezoelectric nano-structures are pointed out.

Fundamental formulations and recent achievements in piezoelectric nano-structures: a review

The effects of surface and flexoelectricity have been found in the presence of strong size dependence and should be technically taken into account for nano-scaled dielectric structures. This paper proposes a Bernoulli–Euler beam model to investigate the electromechanical coupling response of piezoelectric nanostructures, in which the effects of surface elasticity, dielectricity and piezoelectricity as well as bulk flexoelectricity are all taken into consideration. The governing equations with non-classical boundary conditions are naturally derived from a variational principle. Then the present beam model is directly applied to solve the static bending problems of cantilever beams. Without considering the residual surface stresses, the bending rigidity can be defined the same as that in classical piezoelectricity theory. The bending rigidity is found to increase for silicon nanowires and decrease for silver nanowires. Also the flexoelectric effect in piezoelectric nanowires has a momentous influence on the bending rigidity. The residual surface stresses which are usually neglected are found to be more important than the surface elasticity for the bending of nanowires. However, this has no influence on the effective electromechanical coupling coefficient. The deflections reveal the significance of the residual surface stresses and the bulk flexoelectric effects. The effective electromechanical coupling coefficient for piezoelectric nanowires is dramatically enhanced, which demonstrates the significant effects of the bulk flexoelectricity and surface piezoelectricity. The effects of surface and flexoelectricity decrease with the increase of the beam thickness, and therefore these effects can be ignored for large-scale structures. This work is very helpful in designing cantilever-beam-based nano-electro-devices.

https://iopscience.iop.org/article/10.1088/0964-1726/23/3/035020/pdf

Effects of surface and flexoelectricity on a piezoelectric nanobeam

Size-dependent piezoelectricity

Two-dimensional (2D) general equations of piezoelectric shells with nano-thickness are presented in an orthogonal curvilinear coordinate system, in which the surface effect is considered by treat the shell as a bulk core plus two surface layers. The general 2D equations can be directly degenerated into those of particular shells such as flat plates, cylindrical shells and so on by setting the Lame coefficients and the principal radii of curvature to certain values. Using the derived 2D equations of cylindrical shells, the free vibration of a radially polarized piezoelectric cylindrical shell of finite length is analyzed. Numerical results show that the surface effect has a remarkable influence on the natural frequency of the shell at the nano-scale.

Two-dimensional theory of piezoelectric shells considering surface effect

In this paper, a phenomenological continuum theory of surface piezoelectricity accounting for the linear superficial interplay between electricity and elasticity is formulated primarily for elastic dielectric materials. This theory is inspired by the physical idea that once completely relaxed, an insulating free dielectric surface will sustain a nontrivial spontaneous surface polarization in the normal direction together with a tangential self-equilibrated residual surface stress field. Under external loadings, the surface Helmholtz free energy density is identified as the characteristic function of such surfaces, with the in-plane strain tensor of surface and the surface free charge density as the independent state variables. New boundary conditions governing the surface piezoelectricity are derived through the variational method. The resulting concepts of charge-dependent surface stress and deformation-dependent surface electric field reflect the linear electromechanical coupling behavior of nanodielectric surfaces. As an illustrative example, an infinite radially polarizable piezoelectric nanotube with both inner and outer surfaces grounded is investigated. The novel phenomenon of possible surface-induced polarity inversion is predicted for thin enough nanotubes.

A continuum theory of surface piezoelectricity for nanodielectrics

This paper studies the surface instability of an elastic thin solid film lying on a rigid substrate and subjected to van der Waals-like surface interactions. The effect of film?substrate interfacial slippage is accounted for by using a simplified linear cohesive interface model. It is found that the interfacial slippage generally plays a destabilizing role in the surface instability of the thin film. For highly compressible films with Poisson's ratio smaller than 0.25, the surface wrinkling behaviour previously inconceivable in the case of a perfectly bonded interface is now feasible if film?substrate interface slipping is permitted. In addition, our linear perturbation analysis shows that the critical conditions for the onset of surface instability can be modulated by adjusting the slippery stiffness of the interface. The result might be helpful for developing novel techniques to create micro-/nanosized surface patterns.

http://iopscience.iop.org/article/10.1088/0022-3727/42/5/055302/pdf

Interfacial slippage effect on the surface instability of a thin elastic film under van der Waals force

In the present paper, the issue of the thermomechanical definitions of intrinsic surface elastic properties is addressed in the context of infinitesimal elasticity, utilizing two fundamental ideas of Gibbs surface thermodynamics, i.e., the Gibbs dividing surface and the associated surface excesses. Firstly, we introduce a novel scalar yet implicitly oriented thermomechanical function, called the Reissner free energy, as a partial Legendre transformation of the conventional Helmholtz free energy function. It naturally facilitates the application of fundamental principles in classical Gibbs surface thermodynamics of fluids to purely elastic solids. In view of the diffusionless characteristic of elastic surfaces, surface elastic parameters that are invariant with altering the position of the Gibbs dividing surface are defined. Resorting to the construction of a fictitious elastic field in the comparison system, we obtain a fundamental thermodynamic identity, i.e., the isothermal surface Gibbs-Duhem type relation. Thereby the modified Shuttleworth-Herring equation as well as the generalized Nozieres-Wolf equation, which could be regarded respectively as the defining equations for the intrinsic surface stress and surface strain, are derived. The oriented thermomechanics treatment of elastic surfaces presented here is expected to serve as an essential basis for further developments of an intrinsic surface elasticity theory.

Oriented Thermomechanics for Isothermal Planar Elastic Surfaces Under Small Deformation

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Papers

A continuum theory of surface piezoelectricity for nanodielectrics

Interfacial slippage effect on the surface instability of a thin elastic film under van der Waals force

Oriented Thermomechanics for Isothermal Planar Elastic Surfaces Under Small Deformation