Fracture mechanics for the design of ceramic multilayer actuators

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.