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We present a general relation between the electromechanical couplings ofinfinitesimal strain and electric field to arbitrary order, measured at fixedvoltage and at fixed electric field. We show that the improper response atfixed field can be written as the strain derivative of the $n^{\text{th}}$order susceptibility tensor, and the proper response at fixed voltage drop canbe written as the response at fixed field plus corrections for dilations and90$^{\circ}$ rotations induced by strain. Our theory correctly reproduces theproper piezoelectric response and we go beyond with the electrostrictiveresponse. We present first-principles calculations of the improperelectrostrictive response at fixed field, and illustrate how the correction isused to obtain the proper response at fixed voltage. This distinction is ofhigh importance given the recent interest in giant electrostrictors exhibitingelectromechanical responses as large as the piezoelectric ones.

Generalized relation between electromechanical responses at fixed voltage and fixed electric field