This paper is devoted to the theoretical modeling of the mechanical response of bonded structures and particularly focuses on the single-lap joint which corresponds to the simplest and most fundamental bonding configuration. Such a geometry brings into play all the features of any bonded assembly in terms of mechanical response (stress/strain heterogeneities, singularities, adhesion at the interfaces,...). It can especially be used to characterize the mechanical behavior of new adhesives within an assembly. In all cases, an accurate description of the stress/strain distribution (specifically in the adhesive layer) is required both for the calibration of an adhesive behavior and for dimensioning purposes. Accordingly, the present study aims at developing a specific 1D enriched finite element devoted to the numerical modeling of single-lap joints, especially of the overlap region. First, analytical solutions for a single-lap joint under tensile forces are investigated in the framework of elasticity. They are particularly based on the choice of a 2D representation of the adhesive layer with polynomial displacement fields in terms of the thickness coordinate. Such a preliminary study allows one to identify optimally the appropriate kinematics for each layer (adhesive but also substrates) and highlights the importance of non-linear terms in the polynomial expressions of both longitudinal and transverse displacements within the adhesive. A three-layer finite element model is then formulated for the overlap region, based on the retained kinematics, and involving an elastoplastic constitutive law for the adhesive material. The numerical integration through the adhesive thickness and the assembly of the three layers lead to the definition of a very low-cost 1D finite element, which provides nevertheless a complete and accurate description of the stress fields, especially within the adhesive layer. This new finite element (used simultaneously with a more classical beam finite element for the unbonded parts of the adherends) is finally validated by comparison with 2D reference results computed using Abaqus software.

本文致力于键合结构的机械响应的理论建模，尤其关注与最简单和最基本的键合配置相对应的单搭接接头。这样的几何形状在机械响应（应力/应变异质性，奇异性，界面处的附着力等）方面发挥了任何键合组件的所有功能。它尤其可用于表征组件中新粘合剂的机械性能。在所有情况下，都需要对应力/应变分布（特别是在粘合层中）进行准确描述，以用于校准粘合行为和确定尺寸。因此，本研究旨在开发一种专门用于单膝关节数值建模的特定一维富集有限元，尤其是重叠区域。首先，在弹性框架下研究了单搭接接头在拉力作用下的解析解。它们尤其基于对粘合剂层的二维表示的选择，该二维表示具有根据厚度坐标的多项式位移场。这样的一项初步研究使人们能够最佳地确定每一层（粘合剂以及基材）的合适运动学，并强调了非线性项在粘合剂内纵向和横向位移的多项式表达式中的重要性。然后，基于保留的运动学，并针对粘合剂材料的弹塑性本构定律，为重叠区域制定了一个三层有限元模型。通过粘合剂厚度和三层组装的数值积分得出了非常低成本的一维有限元的定义，尽管如此，它仍然提供了对应力场的完整而准确的描述，尤其是在粘合剂层内部。通过与使用Abaqus软件计算的2D参考结果进行比较，最终验证了这种新的有限元（与用于粘附体未粘结部分的更为经典的梁有限元同时使用）。