This work concerns the study of the effective balance equations governing linear elastic electrostrictive composites, where mechanical strains can be observed due to the application of a given electric field in the so-called small strain and moderate electric field regime. The formulation is developed in the framework of the active elastic composites. The latter are defined as composite materials constitutively described by an additive decomposition of the stress tensor into a purely linear elastic contribution and another component, which is assumed to be given and quadratic in the applied electric field when further specialised to electrostrictive composites. We derive the new mathematical model by describing the effective mechanical behaviour of the whole material by means of the asymptotic (periodic) homogenisation technique. We assume that there exists a sharp separation between the micro-scale, where the distance among different sub-phases (i.e. inclusions and/or fibres and/or strata) is resolved, and the macro-scale, which is related to the average size of the whole system at hand. This way, we formally decompose spatial variations by assuming that every physical field and material property are depending on both the macro-scale and the micro-scale. The effective governing equations encode the role of the micro-structure, and the effective contributions to the global stress tensor are to be computed by solving appropriate linear-elastic-type cell problems on the periodic cell. We also provide analytic formulae for the electrostrictive tensor when the applied electric field is either microscopically uniform or given by a suitable multiplicative decomposition between purely microscopically and macroscopically varying components. The obtained results are consistently compared with previous works in the field, and can pave the way towards improvement of smart active materials currently utilised for engineering (possibly bio-inspired) purposes.

这项工作涉及控制线性弹性电致伸缩复合材料的有效平衡方程的研究，在该方程中，由于在所谓的小应变和中等电场状态下施加了给定的电场，因此可以观察到机械应变。该配方是在活性弹性复合材料的框架内开发的。后者定义为通过应力张量的加法分解为纯线性弹性贡献和另一种成分而构成性描述的复合材料，当进一步专门用于电致伸缩复合材料时，假定该成分在施加的电场中是给定的并且是平方的。我们通过渐进（周期）均化技术描述整个材料的有效机械行为，从而得出了新的数学模型。我们假设微观尺度之间存在着明显的分离，在微观尺度上，不同子相（即夹杂物和/或纤维和/或地层）之间的距离得以解决，宏观尺度，这与手头整个系统的平均大小有关。这样，我们通过假设每个物理场和材料属性都取决于宏观和微观尺度来正式分解空间变化。有效的控制方程对微结构的作用进行了编码，并且通过求解周期单元上的适当线性弹性型单元问题，可以计算出对整体应力张量的有效贡献。当施加的电场在微观上是均匀的或通过在微观上和宏观上变化的分量之间进行适当的乘法分解给出时，我们还提供了电致伸缩张量的解析公式。将获得的结果与该领域以前的工作进行了比较，