In the perspective of homogenization theory, strain-gradient elasticity is a strategy to describe the overall behavior of materials with coarse mesostructure. In this approach, the effect of the mesostructure is described by the use of three elasticity tensors whose orders vary from 4 to 6. Higher-order constitutive tensors make it possible to describe rich physical phenomena. However, these objects have intricate algebraic structures that prevent us from having a clear picture of their modeling capabilities. The harmonic decomposition is a fundamental tool to investigate the anisotropic properties of constitutive tensor spaces. For higher-order tensors (i.e. tensors of order $n\ge$3), their determination is generally a difficult task. In this paper, a novel procedure to obtain this decomposition is introduced. This method, which we have called the Clebsch–Gordan Harmonic Algorithm, allows one to obtain explicit harmonic decompositions satisfying good properties such as orthogonality and uniqueness. The elements of the decomposition also have a precise geometrical meaning simplifying their physical interpretation. This new algorithm is here developed in the specific case of 2D space and applied to Mindlin’s Strain-Gradient Elasticity. We provide, for the first time, the harmonic decompositions of the fifth- and sixth-order elasticity tensors involved in this constitutive law. The Clebsch–Gordan Harmonic Algorithm is not restricted to strain-gradient elasticity and may find interesting applications in different fields of mechanics which involve higher-order tensors.

从均质化理论的角度来看，应变-梯度弹性是一种描述具有粗细观结构的材料的整体性能的策略。在这种方法中，通过使用三个阶数从4到6的弹性张量来描述介观结构的效果。高阶本构张量可以描述丰富的物理现象。但是，这些对象具有复杂的代数结构，使我们无法清楚地了解其建模能力。谐波分解是研究本构张量空间各向异性特性的基本工具。对于高阶张量（即阶张量）$ñ\ge$3），他们的决心通常是困难的任务。在本文中，介绍了一种获得这种分解的新方法。这种方法，我们称为Clebsch–Gordan谐波算法，可以使人们获得满足良好特性（如正交性和唯一性）的显式谐波分解。分解的元素还具有精确的几何含义，从而简化了其物理解释。这种新算法是在2D空间的特定情况下开发的，并应用于Mindlin的应变梯度弹性。我们首次提供了涉及本构定律的五阶和六阶弹性张量的谐波分解。该Clebsch -戈登谐波算法 不限于应变梯度弹性，并且可能在涉及高阶张量的力学的不同领域中找到有趣的应用。