With the development of science and technology, precision positioning technology becomes one of the key technologies driving the prosperity and development of modern science and advanced industrial technology in the 21^st century. It is widely used in high-tech fields such as precision machining and aerospace. Among the core components of precision positioning technology, the micro-displacement actuator with a high positioning accuracy plays a critical role. However, conventional micro-displacement actuators suffer from several drawbacks, including long transmission chain, complex structure and low precision, which pose some challenges to precision positioning system. Electrostrictive effect has attracted considerable attention due to its advantageous properties, such as the absence of hysteresis and independence from the direction of the electric field. The relaxor ferroelectrics, which exhibit minimal hysteresis are commonly studied for their electrostrictive behavior. Although extensive research has been conducted on the large electrostrictive effect of materials, there is still no analytic expression to describe electrostrictive strains. In this paper, the electrostrictive strain of high-energy electron irradiated P(VDF-TrFE) 68/32 relaxor ferroelectric copolymers was measured as a function of applied electric field using a laser-assisted micro-displacement measurement setup. An analytic expression of electrostrictive strain as a function of applied electric field was derived based on the thermodynamic phenomenological theory.

An electrostrictive effect test platform was built, and the electrostrictive effect of high-energy electron irradiated P(VDF-TrFE) 68/32 relaxor ferroelectric copolymers was examined. The tested results were analyzed. The phenomenological theory was used to derive the analytic expression of electrostrictive strain as a function of applied electric field for relaxor ferroelectrics.

The results reveal that the electrostrictive strain has a quadratic relationship with the electric field at lower electric fields. At higher electric fields, the relationship transitions to a power of 2/3, and at even higher electric fields, it further shifts to a power of 2/5. The electrostrictive strain as a function of electric field is analyzed for relaxor ferroelectric ceramics and polymers using this analytical expression. The fitting results confirmed the validity of this relationship across a wide range of electric fields. The strain and crossover electric field of the material can be designed in terms of the analytical expression of strain as a function of electric field.

The high-energy electron irradiated P(VDF-TrFE) relaxor ferroelectric polymers were prepared, and their electrostrictive strain as a function of electric field was determined. In addition, an analytical expression was also derived based on the thermodynamic phenomenological theory, which was used to fit the electrostrictive strain as a function of electric field for irradiated copolymers. The electrostrictive strain had a quadratic relationship with the electric field at lower electric fields. At higher electric fields, the relationship transitioned to a power of 2/3, and at even higher electric fields, it further shifted to a power of 2/5. These relationships were consistent with the experimental results. Furthermore, the expression was used to fit the electrostrictive strains versus electric field for relaxor ferroelectric ceramics and polymers, and it was procured that the expression could be applicable. Moreover, the transition electric field of the electrostrictive strain versus electric field was proportional to α^3/2, and inversely proportional to β^1/2.