How to handle expired date in ivytask

The expiry inscription can be used to define what happens when the user does not pick up a task over a period of time. Please consult the Designer Guide for more information about expiry inscriptions in the task tab: ExpiryInscription

So, yes to me that sounds like you could use it for your requirement. You could define a responsible role/user that has to pick up the task when the initially assigned user did not pick up the task after a period of time. Furthermore you could throw an ExpiryException and implement a detailed expiry handling process with an exception start element.

Another remark: It often makes sense to calculate the expiry time based on business calendar settings instead of hard coded time periods. E.g. if you want to ensure that the task is picked during the next 3 working days (to keep an SLA). So the time-out time could often be easily calculated with the business calendar API that is aware of non-working times (weekends, night, national holidays, etc.). If you care about that feature, consult the server guide for business time administration reference. Or the public API of IDefaultBusinessCalendar for the implementation reference.