Every once in a while I find an excuse to go back to the basics so here we go:
Firstly, with regards to the underlying mathematics, there are a few methods to solve governing partial differential equations and numerical analysis is one of those methods. Although there is a wide spectrum of methods, there are two major classes of schemes: implicit and explicit.
With the exception of the finite difference method, the governing PDEs have to be reformulated in a way suitable for a given method (e.g., finite element or finite volume or immersed boundary .. .. method). In principle, either implicit or explicit scheme should give you the exact same result (simply because you are solving the same underlying set of equations). In practice, that rarely happens and one is forced to choose one or the other class of methods to numerically solve a given problem. Software vendors working on real problems from some industries end up coming up with their tweaks on their specific implementation of versions of these methods. Abaqus/Standard (implicit) and Abaqus/Explicit (explicit) are two major commercial versions from Simulia. Abaqus/Standard lets you choose to disregard inertia (static analysis) or take it into account (implicit dynamic, as one example).
So far, I did not bring up any particular physics (elasticity, fluid dynamics, electromagnetics, etc.) in this discussion.
Now, to the physics, as part of a tall pile of approximations one makes in a numerical model, inertia may play little role in a specific phenomenon of interest perhaps because, say, the natural frequency of the structure is too high compared with the loading frequency. In such a case, you may want to perform a static analysis or a pseudo/quasi-static (using either implicit dynamic or explicit scheme) analysis. In the former case, physical time has no meaning and the "time" simply is a proportionality constant for the external load/BC being applied i.e., at "time" = 0, no load is applied whereas at "time" = 0.5, 50% of the load/BC is applied. In the quasi-static case, time carries meaning and inertia comes into play.
To reiterate, in principle, whether you choose Abaqus/Standard static analysis or Abaqus/Standard implicit dynamic (with quasi-static) analysis or Abaqus/Explicit, you will get the exact same answer. However, depending on the model, it can be anywhere from straightforward to a daunting task.
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