Electrostatic-Elastic MEMS with Fringing Field: A Problem of Global Existence

In this paper, we prove the existence and uniqueness of solutions for a nonlocal, fourth-order integro-differential equation that models electrostatic MEMS with parallel metallic plates by exploiting a well-known implicit function theorem on the topological space framework. As the diameter of the domain is fairly small (similar to the length of the device wafer, which is comparable to the distance between the plates), the fringing field phenomenon can arise. Therefore, based on the Pelesko–Driscoll theory, a term for the fringing field has been considered. The nonlocal model obtained admits solutions, making these devices attractive for industrial applications whose intended uses require reduced external voltages.

On a nonlocal Cahn–Hilliard model permitting sharp interfaces

A nonlocal Cahn–Hilliard model with a non-smooth potential of double-well obstacle type that promotes sharp interfaces in the solution is presented. To capture long-range interactions between particles, a nonlocal Ginzburg–Landau energy functional is defined which recovers the classical (local) model as the extent of nonlocal interactions vanish. In contrast to the local Cahn–Hilliard problem that always leads to diffuse interfaces, the proposed nonlocal model can lead to a strict separation into pure phases of the substance. Here, the lack of smoothness of the potential is essential to guarantee the aforementioned sharp-interface property. Mathematically, this introduces additional inequality constraints that, in a weak formulation, lead to a coupled system of variational inequalities which at each time instance can be restated as a constrained optimization problem. We prove the well-posedness and regularity of the semi-discrete and continuous in time weak solutions, and derive the conditions under which pure phases are admitted. Moreover, we develop discretizations of the problem based on finite element methods and implicit–explicit time-stepping methods that can be realized efficiently. Finally, we illustrate our theoretical findings through several numerical experiments in one and two spatial dimensions that highlight the differences in features of local and nonlocal solutions and also the sharp interface properties of the nonlocal model.

Nonlocal regularization of an anisotropic critical state model for sand

Many advanced constitutive models which can capture the strain-softening and state-dependent dilatancy response of sand have been developed. These models can give good prediction of the single soil element behaviour under various loading conditions. But the solution will be highly mesh-dependent when they are used in real boundary value problems due to the strain-softening. They can give mesh-dependent strain localization pattern and bearing capacity of foundations on sand. Nonlocal regularization of an anisotropic critical state sand model is presented. The evolution of void ratio which has a significant influence on strain-softening is assumed to depend on the volumetric strain increment of both the local and neighbouring integration points. The regularization method has been implemented using the explicit stress integration method. The nonlocal model has been used in simulating both drained plane strain compression and the response of a strip footing on dry sand. In plane strain compression, mesh-independent results for the force–displacement relationship and shear band thickness can be obtained when the mesh size is smaller than the internal length. The force–displacement relationship of strip footings predicted by the nonlocal model is much less mesh-sensitive than the local model prediction. The strain localization under the strip footing predicted by the nonlocal model is mesh independent. The regularization method is thus proper for application in practical geotechnical engineering problems.

Nonlocal elastodynamics and fracture

A nonlocal field theory of peridynamic type is applied to model the brittle fracture problem. The elastic fields obtained from the nonlocal model are shown to converge in the limit of vanishing non-locality to solutions of classic plane elastodynamics associated with a running crack. We carry out our analysis for a plate subject to mode one loading. The length of the crack is prescribed a priori and is an increasing function of time.