Do I bear sin and expiation for a traffic accident that occurred between two cars behind the car that stopped for me to cross the road, knowing that the accident did not result in human injuries but rather material damages?

There is nothing upon you regarding what you mentioned, as you were neither directly involved nor the cause of the accident. Liability falls upon the one who directly caused the collision, or upon the one who slowed down their speed, or upon both of them, depending on the assessment of experts. The general principle is that liability rests with the direct perpetrator, unless the indirect cause was negligent and the direct perpetrator was not negligent. They may both be liable if there was negligence on the part of both parties. The Islamic Fiqh Academy has clarified that traffic accidents are subject to the rulings of criminal offenses, and the driver is responsible for damages unless the accident was a result of an irresistible force, an act of the injured party, or the fault and transgression of a third party. If the driver and the injured party both contributed to causing the damage, each of them bears responsibility for what was damaged to the other. The general principle is that the direct perpetrator is liable, and the indirect cause is not liable unless they were negligent or reckless.