Soft adhesive structures are widely employed in cutting-edge applications due to their unique mechanical properties. However, the phenomenon of delayed failure resulting from static loading is often overlooked in these scenarios. In particular, the counterintuitive delayed failure of the soft adhesive structure under shear displacement holding, where no additional energy is input into the soft adhesive structure during the macroscopic relaxation process, poses a challenge for comprehension. To address this, this work employs a timedependent cohesive zone model (CZM) to characterize the viscoelastic behavior of the soft adhesive within the soft adhesive structure during displacement holding. Two adhesive systems, namely the hard adherend bonded to the hard substrate with soft adhesive (HSH) and the soft adherend bonded to the hard substrate with soft adhesive (SSH), were investigated both experimentally and numerically. The adopted CZM was validated through single-lap shear tests on HSH, encompassing single loading, constant force, and displacement holding experiments. Subsequently, the delayed failure of the local soft adhesive in SSH was investigated, and the deformation field of the soft adherend during the holding process was observed experimentally using the particle tracking method. The results reveal that the delayed failure of SSH is attributed to the occurrence of local creep in the soft adhesive at the crack front, despite the evident macroscopic relaxation phenomenon observed during displacement holding. Aiming to eliminate the delayed failure of the soft adhesive structure, the importance of the adherend to adhesive modulus ratio, viscoelastic properties, and loading conditions were discussed to address the delayed failure of the soft adhesive structure. By shedding light on the overlooked aspect of delayed failure in soft adhesive structures, this study presents a novel perspective that extends beyond conventional static loading analyses. The understanding of delayed failure mechanisms provides valuable guidance for enhancing the performance of soft adhesive structures.