The present contribution aims to revisit higher-order homogenization schemes towards micromorphic media based on variational principles and an extension of Hill macrohomogeneity condition. Starting from the microscopic Cauchy balance equations, the local balance equations of the micromorphic continuum are formulated, highlighting the micromorphic stress measures. Relying on both energy and complementary energy expressions combined with the extended Hill macrohomogeneity condition, the complete homogeneous microscopic displacement field representative of the effective micromorphic continuum is obtained as a quartic expansion of the macroscopic micromorphic kinematic variables. This procedure leads to a higher-grade micromorphic theory, with the relative stress and hyperstress tensors including respectively second-order and third order polynomials of the relative position within the unit cell. The microscopic displacement is completed by a fluctuating part evaluated from a variational principle and characterized by three unit cell boundary value problems. Numerical applications are done for inclusion based composite materials. The obtained results highlight that the higher-order moduli converge very quickly with unit cell size, due to the consideration of correction factors based on the higher-order moments of area.

本文稿旨在基于变分原理和Hill宏观均匀性条件的扩展，重新审视针对微形态介质的高阶均质化方案。从微观柯西平衡方程出发，拟定了微形连续体的局部平衡方程，突出了微形应力的测度。依赖于能量和互补能量的表达式，结合扩展的Hill宏观均匀性条件，获得了代表有效微观形态连续体的完整的均匀微观位移场，作为宏观微观形态运动变量的四次展开。这个过程导致了更高等级的微晶理论，相对应力和超应力张量分别包括单位晶格内相对位置的二阶和三阶多项式。微观位移由根据变化原理评估的波动部分完成，并具有三个单位晶胞边界值问题的特征。对包含物的复合材料进行了数值应用。所获得的结果表明，由于考虑了基于面积的高阶矩的校正因子，因此高阶模量与单位晶胞尺寸的收敛非常快。对包含物的复合材料进行了数值应用。所获得的结果表明，由于考虑了基于面积的高阶矩的校正因子，因此高阶模量与单位晶胞尺寸的收敛非常快。对包含物的复合材料进行了数值应用。所获得的结果表明，由于考虑了基于面积的高阶矩的校正因子，因此高阶模量与单位晶胞尺寸的收敛非常快。