This work studies the solid-shell finite element approach to approximate thin structures using a stabilized mixed displacement–stress formulation based on the Variational Multiscale framework. The work is divided in two parts. In Part I, the numerical locking effects inherent to the solid-shell approach are characterized using a variety of benchmark problems in the infinitesimal strain approximation. In Part II, the results are extended to formulate the mixed approach in finite strain hyperelastic problems. In the present work, the stabilized mixed displacement–stress formulation is proven to be adequate to deal with all kinds of numerical locking. Additionally, a more comprehensive analysis of each individual type of numerical locking, how it is triggered and how it is overcome is also provided. The numerical locking usually occurs when parasitic strains overtake the system of equations through specific components of the stress tensor. To properly analyze them, the direction of each component of the stress tensor has been defined with respect to the shell directors. Therefore, it becomes necessary to formulate the solid-shell problem in curvilinear coordinates, allowing to give mechanical meaning to the stress components (shear, twisting, membrane and thickness stresses) independently of the global frame of reference. The conditions in which numerical locking is triggered as well as the stress tensor component responsible of correcting the locking behavior have been identified individually by characterizing the numerical response of a set of different benchmark problems.

中文翻译：

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使用固壳单元对薄结构进行应力-位移稳定有限元分析，第一部分：关于内插应力的需要


这项工作研究了固体壳有限元方法，使用基于变分多尺度框架的稳定混合位移-应力公式来近似薄结构。这项工作分为两部分。在第一部分中，使用无穷小应变近似中的各种基准问题来表征固壳方法固有的数值锁定效应。在第二部分中，结果被扩展以制定有限应变超弹性问题的混合方法。在目前的工作中，稳定的混合位移-应力公式被证明足以处理各种数值锁定。此外，还提供了对每种类型的数字锁定、其触发方式以及克服方式的更全面分析。当寄生应变通过应力张量的特定分量超越方程组时，通常会发生数值锁定。为了正确分析它们，应力张量的每个分量的方向已根据壳导向器定义。因此，有必要在曲线坐标系中制定固体壳问题，从而独立于全局参考系为应力分量（剪切应力、扭转应力、膜应力和厚度应力）赋予机械意义。通过表征一组不同基准问题的数值响应，可以单独识别触发数值锁定的条件以及负责校正锁定行为的应力张量分量。