In the perspective of homogenization theory, strain-gradient elasticity is a strategy to describe the overall behaviour 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\geq$3), its establishment is generally a difficult task. In this paper a novel procedure to obtain this decomposition is introduced. This method, that we have called the \textit{Clebsch-Gordan Harmonic Algorithm}, allows to obtain \emph{explicit} harmonic decompositions satisfying good properties such as orthogonality and unicity. 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.
Three-dimensional iso-geometric solutions to general boundary value problems of Toupin's gradient elasticity theory at finite strains
我们目前，据我们所知，第一个完整的三维数值解的范围广泛的边界值问题的一般理论有限应变梯度弹性。我们选择了我们的工作，Toupin的理论[拱。机械分析：11（1），385-414，1962]-应变梯度弹性的更一般的公式之一。我们的框架有三个关键组成部分：第一个是等几何分析[Hughes et al.，比较方法应用机械工程，194（39-41），4135-4195，2005]，我们采用了它的直接和强大的表示C1-连续性。第二个是一个较弱的高阶狄利克雷边界条件的制定，控制应变梯度的发展在解决方案中的治疗。第三个要素是算法（自动）微分，它消除了需要线性化的“手”的相当复杂的几何和材料的非线性梯度弹性在有限的应变。我们提出了一些数值解决方案，以证明该框架是适用于任意边值问题的三维。我们讨论的尺寸效应，高阶边界条件的作用，也许最重要的是，相关的框架与弹性自由能量密度函数，是非凸应变空间的问题。
Displacement general solutions in strain gradient elasticity: review and analysis
Institute of Applied Mechanics of Russian Academy of Sciences（俄罗斯科学院应用力学研究所）；Moscow Aviation Institute（莫斯科航空学院）；Mansoura University（曼苏拉大学）；Belarussian State University（白俄罗斯国立大学）
Constitutive properties for isotropic energies in ideal nonlinear elasticity for solid materials: numerical evidences for invertibility and monotonicity in different stress-strain pairs
