A fully electromechanically coupled, three dimensional phenomenological constitutive model for relaxor ferroelectric materials was developed for the use in a finite-element-method (FEM) solution procedure. This macroscopic model was used to simulate the macroscopic electromechanical response of lead-free ergodic $0.94\mathrm{N}{\mathrm{a}}_{1/2}\mathrm{B}{\mathrm{i}}_{1/2}\mathrm{Ti}{\mathrm{O}}_3-0.06\mathrm{BaTi}{\mathrm{O}}_3$ and non-ergodic $0.90\mathrm{N}{\mathrm{a}}_{1/2}\mathrm{B}{\mathrm{i}}_{1/2}\mathrm{Ti}{\mathrm{O}}_3-0.06\mathrm{BaTi}{\mathrm{O}}_3-0.04{\mathrm{K}}_{0.5}\mathrm{N}{\mathrm{a}}_{0.5}\mathrm{Nb}{\mathrm{O}}_3$ relaxor materials. The presented constitutive model is capable of accounting for the observed pinched hysteretic response as well as non-deviatoric polarization induced strain and internal order transitions. Time integration of the history dependent internal variables is done with a predictor-corrector integration scheme. The adaptability of the constitutive model regarding the pinching of the hystereses is shown. Simulations are compared to experimental observations.

开发了一种完全机电耦合的三维现象学本构模型，用于弛豫铁电材料，用于有限元法 (FEM) 求解过程。该宏观模型用于模拟无铅遍历的宏观机电响应。$0.94\mathrm{N}{\mathrm{一个}}_{1/2}\mathrm{乙}{\mathrm{一世}}_{1/2}\mathrm{钛}{\mathrm{哦}}_3-0.06\mathrm{巴蒂}{\mathrm{哦}}_3$ 和非遍历 $0.90\mathrm{N}{\mathrm{一个}}_{1/2}\mathrm{乙}{\mathrm{一世}}_{1/2}\mathrm{钛}{\mathrm{哦}}_3-0.06\mathrm{巴蒂}{\mathrm{哦}}_3-0.04{\mathrm{钾}}_{0.5}\mathrm{N}{\mathrm{一个}}_{0.5}\mathrm{铌}{\mathrm{哦}}_3$松弛剂材料。所提出的本构模型能够解释观察到的收缩滞后响应以及非偏极化引起的应变和内部顺序转变。历史相关内部变量的时间积分是通过预测器-校正器积分方案完成的。显示了本构模型关于滞后收缩的适应性。将模拟与实验观察进行比较。