Trust is essential for automated vehicles (AVs) to promote and sustain technology acceptance in human-dominated traffic scenarios. However, computational trust dynamic models describing the interactive relationship between the AVs and surrounding human drivers in traffic rarely exist. This paper aims to fill this gap by developing a quantitative trust dynamic model of the human driver in the car-following interaction with the AV and incorporating the proposed trust dynamic model into the AV’s control design. The human driver’s trust level is modeled as a plan evaluation metric that measures the explicability of the AV’s plan from the human driver’s perspective, and the explicability score of the AV’s plan is integrated into the AV’s decision-making process. With the proposed approach, trust-aware AVs generate explicable plans by optimizing both predefined plans and explicability of the plans in the car-following interactions with the following human driver. The results collectively demonstrate that the trust-aware AV can generate more explicable plans and achieve a higher trust level for the human driver compared to trust-unaware AV in human-AV interactions.