This article investigates the theoretical and numerical analysis as well as applications of the three-dimensional theory of first strain gradient elasticity. The corresponding continuous and discrete variational formulations are established with error estimates stemming from continuity and coercivity within a Sobolev space framework. An implementation of the corresponding isogeometric Ritz–Galerkin method is provided within the open-source software package GeoPDEs. A thorough numerical convergence analysis is accomplished for confirming the theoretical error estimates and for verifying the software implementation. Lastly, a set of model comparisons is presented for revealing and demonstrating some essential model peculiarities: (1) the 1D Timoshenko beam model is essentially closer to the 3D model than the corresponding Euler–Bernoulli beam model; (2) the 3D model and the 1D beam models agree on the strong size effect typical for microstructural and microarchitectural beam structures; (3) stress singularities of reentrant corners disappear in strain gradient elasticity. The computational homogenization methodologies applied in the examples for microarchitectural beams are shown to possess disadvantages that future research should focus on.

本文研究了第一应变梯度弹性三维理论的理论和数值分析以及应用。相应的连续和离散变分公式是用源自 Sobolev 空间框架内的连续性和矫顽力的误差估计建立的。开源软件包 GeoPDEs 中提供了相应等几何 Ritz-Galerkin 方法的实现。完成彻底的数值收敛分析以确认理论误差估计和验证软件实现。最后，提出了一组模型比较，以揭示和证明一些基本模型特性：(1) 一维 Timoshenko 梁模型比相应的 Euler-Bernoulli 梁模型更接近于 3D 模型；(2) 3D 模型和 1D 梁模型在显微结构和微建筑梁结构典型的强尺寸效应上一致；(3) 凹角应力奇异性在应变梯度弹性中消失。在微结构梁的例子中应用的计算均匀化方法被证明具有未来研究应该关注的缺点。