Showing papers by "Zhigang Suo published in 1996"

Fracture mechanics for the design of ceramic multilayer actuators

Abstract: In a multilayer actuator, each internal electrode terminates an edge inside the active ceramic. Around the edge, the nonuniform electric field generates an incompatible strain field, which, in its turn, generates stresses and may cause the ceramic to crack. The industry has been exploring alternative electrode configurations to alleviate the stress concentration. The effort has been empirical and benefited little from numerical simulations. An inherent difficulty is that the actuator ceramics have nonlinear electro-mechanical interactions, of which no unified mathematical description is now available. In this paper, we develop a crack nucleation model that includes essential features of this nonlinearity. The model applies to both paraelectrics and ferroelectrics. Attention is focused on situations where the small-scale saturation conditions prevail. That is, the driving voltage is low enough so that the bulk of the ceramics is linearly dielectric, except for a cylinder of a small radius around the electrode edge. Inside the cylinder, large strains result from electrostriction or polar rotation. We identify a parameter group that determines the cracking condition; details in the material description only affect a dimensionless coefficient. Everything else being fixed, a critical layer thickness exists, below which a multilayer actuator will not crack around its internal electrode edges. Merits and limitations of the small-scale saturation model are discussed. We analyze this model analytically for a paraelectric with perfect polarization saturation, and estimate the value of the dimensionless coefficient in the model.

Cracking of Laminates Subjected to Biaxial Tensile Stresses

Abstract: During the processing of laminar ceramic, biaxial residual stresses can arise due to differential thermal contraction between unlike layers. A tensile stress can cause preexisting flaws to extend across the layer and into the adjacent layers and then tunnel until they meet either another crack or a free surface. A previous analysis has shown that for a given residual stress there is a critical layer thickness, below which no tunnel cracks will exist, regardless of initial flaw size. Here, the previous analysis was modified to take into account the crack extension into adjacent layers. To determine the validity of the analysis, laminates composed of alternating layers of zirconia and alumina/zirconia were fabricated by a sequential centrifugation technique. The composition of the alumina/zirconia layer was varied to change the biaxial, tensile stresses in the zirconia layer. Observations were then made to determine the critical layer thickness for tunnel cracks and their extension into the adjacent layers. These observations were compared to the theoretical predictions.

A simulation of electromigration-induced transgranular slits

Abstract: An on‐chip aluminum interconnect carries an intense electric current at an elevated temperature, motivating atoms to diffuse in the solid state, and inducing voids that may cause an open failure. Recent observations have shown that a void sometimes collapses to a slit running nearly perpendicular to the electric current direction. Such a slit often lies inside a grain rather than along a grain boundary. An earlier calculation showed that diffusion on the void surface, driven by the electric current, can cause a circular void to translate in an infinite, isotropic interconnect. It was suggested recently that this solution may be unstable, and that two forces compete in determining the void stability: surface tension favors a rounded void, and the electric current favors a slit. A linear perturbation analysis, surprisingly, revealed that the translating circular void is stable against infinitesimal shape perturbation. Consequently, the slit instability must have resulted from finite imperfections. This article reviews the experimental and theoretical findings, and describes a numerical simulation of finite void shape change. We determine the electric field by a conformal mapping of complex variables, and update the void shape for a time step by a variational method. The simulation shows that a finite void shape imperfection or surface tension anisotropy can cause a void to collapse to a slit.

Reliability of ceramic multilayer actuators : A nonlinear finite element simulation

Abstract: In a ceramic multilayer actuator, the abrupt end of an internal electrode concentrates the electric field, inducing stresses in the ceramic. Crack nucleation and growth have been observed experimentally, but have not been well modeled due to the complex material behaviors. We write a finite element program to solve this coupled electromechanical problem. The material is taken to be nonlinearly dielectric with electrostrictive strain quadratic in electric displacement. The program solves field distributions in a multilayer actuator, which are combined with fracture mechanics to obtain cracking condition. The calculations are compared with the existing analytical solution under the small-scale saturation conditions, and then extended to the large-scale saturation conditions. We show that the cracking condition established under the small-scale saturation conditions gives useful estimate even when the saturation zone is comparable to the actuator layer thickness.

A global analysis of structural evolution in a row of grains

Abstract: When a polycrystalline fiber is heated for some time, grains change shape and may evolve to one of two equilibrium configurations: isolated spheres, or truncated spheres that remain connected. To which configuration do the grains evolve? How long does this evolution take? These are global questions, and call for a global way to look at the phenomenon. The fiber is a nonequilibrium structure. The free energy consists of the surface and grain-boundary energies; it is the reduction of this energy that drives the diffusive flux of atoms on the surfaces and the grain boundaries. We describe the grain shape using two generalized coordinates, the grain length and the dihedral angle. The free energy is expressed as a function of these coordinates. In the space of the free energy and the coordinates, the energy function is represented by a surface, or a landscape. A point on the landscape represents a nonequilibrium state in general, and the bottom of a valley represents an equilibrium state. We use a variational principle to assign a viscosity matrix to every point on the landscape. The approach leads to a set of ordinary differential equations that govern the evolution of the generalized coordinates.

Competitive Motions of Grain-Boundary and Free Surface in Selecting Thin Film Morphology

Abstract: During annealing of a polycrystalline thin film, grain-boundaries and film surfaces move. If the grain-boundaries move faster, the grains having the lowest free energy grow at the expense of others, resulting in a continuous film with large grains. If the film surfaces move faster, they groove along their junctions with the grain-boundaries, breaking the film to islands. This paper describes analytic solutions for steady surface motions, and discusses the morphology selection.