The increasing prevalence of central nervous system (CNS) disorders has imposed a significant social and economic burden on healthcare systems. The blood–brain barrier (BBB) presents a major challenge for effective drug delivery to the brain, hindering disease treatment advancements. The BBB consists of various cell types, including microvascular endothelial cells and astrocytes, with tight junctions playing a key role in regulating molecular exchange and maintaining brain homeostasis. However, current research on the mechanical properties of the BBB mainly focuses on individual cellular components or the vasculature as a whole, with limited attention to the mechanical behavior of the tight junctions between these cells. This study develops a three-dimensional (3D) mechanical model of the BBB, incorporating tight junctions as membrane structures within the vessel wall. To simulate stress distribution within the BBB, tight junctions are described by a modified standard linear solid model, while the vessel wall is depicted by the Yeoh model. Parameters of the Yeoh model are optimized using machine learning algorithms based on experimental data. Then finite element simulations are conducted to analyze the stretching process of the BBB, yielding stress distributions and stress–strain relationships that elucidate the mechanical properties of the BBB under tensile conditions. The influences of the cell membrane elastic modulus, elastic modulus of the cytoskeleton and cytoplasmic viscosity in the modified standard linear solid model on the maximum stress in the tight junction at equilibrium are intensively discussed. These findings provide theoretical insights into the understanding of CNS disorders and have potential applications in drug delivery strategies.