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In strength training, concentric motion means the muscle or muscle group we are addressing is shortening, and eccentric means it is contracting while lengthening, under control. Of course, whenever one muscle contracts, its opposing muscle must lengthen. It's complicated, though, because during controlled effort neither relaxes.
Getting stronger entails breaking down muscle so it can regrow stronger and it seems as if the eccentric motion does a better job of that than the concentric although the latter tends to be where we feel we are working hardest.In the simplest example, when you do a barbell squat the squatting requires eccentric contraction for your quads and glutes, and the stand-up is concentric for them. Vice versa for the hamstrings. That's why your trainer may demand that, in a curl or bench press, for examples, you lower the weight to a count of 5 or 10 instead of letting gravity do more of the work. Some people call that "negative" training, or just say "control it down."
Strength training offers a fun chance to brush up on your human anatomy. While most powerlifts engage the entire body to some extent (which is why they are efficient strength-training tools, like the deadlift), generally most of the work is done by specific muscle groups. Let's consider the bench press, which is designed to not be a full-body exertion but instead to isolate upper body muscles - chest, upper back, shoulders, and arms. The concentric and eccentric contractions of the bench are explained well here.You can see why your biceps get pumped during bench even though pecs and triceps do the lifting - the control down is an eccentric move for the triceps and pecs but concentric/stabilizing for the biceps. After all, you can't drop that barbell.
The only place I can think of where you let gravity do most of the work is in the deadlift where the control-down is less important. You can almost let the bar drop.
Sorry to beat a dead horse, but you're still having a great deal of trouble with this "muscles get stronger by being broken down" theory.
Unless you have abundant levels of testosterone, most strength gains are due to NEUROLOGICAL changes, not changes in the muscle itself, i.e. for pre-puberty kids and old farts, strength gains are largely a function of more/better signals getting to the muscles.
It is significant though as different types of training make use of the different pathways that affect strength from a muscle vs nerve standpoint.
Even within the realm of how it affects the musculature, different training regimens differ in how they affect sarcoplasmic growth (body builder type growth) vs myofibrillar growth (growth of the contracting elements).
Even though neurological improvements may play a more significant role in strength gains for old farts, training tools aimed at affecting myofibrillar and sarcoplasmic growth should not necessarily be ignored. Even if old farts can't expect much growth in these areas, tools aimed at these areas may PREVENT a deterioration of these qualities thus imparting a "relative" improvement.
At the end of the day, the more we learn, the more we realize how little we know. Work hard, work harder at recovering, pray...and NEVER run out of ammo.