Abstract

Abstract

In multilayer ferroelectric actuators, electrode edges are the main source of fracture due to the generation of non-uniform electric fields in their vicinity. The electric fields, in turn, induce incompatible strain fields and hence concentrated stresses, which may cause the ceramic to crack. In this paper, the crack initiation from the electrode edges is simulated using a phase-field model. This model is based on variational formulations of brittle crack propagation and domain evolution in ferroelectric materials. The simulation results show different crack initiation patterns depending on the bonding conditions between the ceramic and electrode layers. Three extreme conditions are considered, which are fully cofired, partially cofired, and separated layers. The crack initiation patterns can be either delimitation along the electrode–ceramic interface or oblique cracking from the electrode into the material. The calculations suggest a mechanism explaining the experimentally observed crack branches near the electrode edges. The effects of the ceramic layer thickness and length of the internal electrode on the crack initiation are also evaluated.

Abstract

This paper investigates the variational formulation and numerical implementation of the damage evolution in solids using the discrete approach of the embedded discontinuity formulation. For this purpose, the ''kinematically optimal symmetric formulation'' (KOS) which guarantees kinematics is consistently derived. This means that rigid body motion of the parts in which the element is divided is obtained. To guarantee equilibrium at the discontinuity, a new variable is considered in the numerical formulation for the definition of the discontinuity length; leading to the equilibrium of tractions at elemental level. To illustrate and validate this approach, two examples, involving mode-I failure, are presented. The numerical results obtained are compared with those reported from corresponding experimental tests. The presented discontinuity formulation, shows been a robust method to simulation damage processes in quasi-fragile materials, without modifying the mesh topology when cohesive cracks propagate.

Abstract

The paper builds a stable variational formulation for the non-ordinary state-based peridynamics (NOSB-PD). Firstly, a new force state vector is reformulated by introducing the first Piola-Kirchhoff stress in continuum mechanics. The consistency of the new governing equation of the proposed pridynamic model and classical continuum mechanics is proved. Secondly, a stable variational formulation of non-ordinary state based peridynamics is developed to unify the boundary conditions in peridynamcis and continuum mechanics. The zero mode oscillations of non-ordinary state based peridynamics is also eliminated by penalty method in numerical implementation. Numerical examples are illustrated to validate the proposed method. Numerical solutions obtained by the proposed method also indicate that the proposed method can well capture the general nonlinear behavior of solid materials.

Abstract

In this work we present a theory of the multilayer perceptron from the perspective of functional analysis and variational calculus. Within this formulation, the learning problem for the multilayer perceptron lies in terms of finding a function which is an extremal for some functional. As we will see, a variational formulation for the multilayer perceptron provides a direct method for the solution of general variational problems, in any dimension and up to any degree of accuracy. In order to validate this technique we use a multilayer perceptron to solve some classical problems in the calculus of variations.