Through enrichment of the elastic potential by the second‐order gradient of deformation, gradient elasticity formulations are capable of taking nonlocal effects into account. Moreover, geometry‐induced singularities, which may appear when using classical elasticity formulations, disappear due to the higher regularity of the solution. In this contribution, a mixed finite element discretization for finite strain gradient elasticity is investigated, in which instead of the displacements, the first‐order gradient of the displacements is the solution variable. Thus, the C¹ continuity condition of displacement‐based finite elements for gradient elasticity is relaxed to C⁰. Contrary to existing mixed approaches, the proposed approach incorporates a rot‐free constraint, through which the displacements are decoupled from the problem. This has the advantage of a reduction of the number of solution variables. Furthermore, the fulfillment of mathematical stability conditions is shown for the corresponding small strain setting. Numerical examples verify convergence in two and three dimensions and reveal a reduced computing cost compared to competitive formulations. Additionally, the gradient elasticity features of avoiding singularities and modeling size effects are demonstrated.

中文翻译：

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无旋转混合有限元，在有限应变下具有梯度弹性

通过利用二次变形梯度来丰富弹性势，梯度弹性公式能够考虑非局部效应。此外，由于解决方案的规则性更高，在使用经典弹性公式时可能会出现几何形状引起的奇异性。在此贡献中，研究了有限应变梯度弹性的混合有限元离散化，其中位移的一阶梯度代替了位移，而是求解变量。因此，基于位移的有限元的梯度弹性的C ¹连续性条件松弛为C ⁰。与现有的混合方法相反，所提出的方法包含无腐约束，通过该约束将位移与问题解耦。这具有减少求解变量数量的优点。此外，示出了对于相应的小应变设置的数学稳定性条件的满足。数值示例验证了二维和三维的收敛性，并显示了与竞争公式相比降低的计算成本。此外，还演示了避免奇异性和建模尺寸效应的梯度弹性特征。