If net force is 0, why does elongation occur in a rod? I was taught that stress is the restoring force per unit area (let us assume a rod). This stress is developed on order to resist the motion of the rod. My questions are as follows :

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*If the rod is subjected to equal and opposite forces why is stress developed as the rod has 0 net force acting on it

My reasoning : the rod can be assumed to be made of differentially small rod elements in a line. When the elements on the edges are pulled a tension force is generated to oppose it's motion.


*Why does elongation occur  if the net force is 0 (forces applied on the ends of the rod) ? In order for elongation to occur there must be some net force on the elements on the edges for them to start elongation but since tension is generated as soon as we apply equal and opposite external forces, there won't be any force imbalance.

My reasoning: The forces are equal and the net force is 0. The rod is expanding but it's center of mass is 0.
I think I'm confusing elongation and net acceleration. Please let me know the errors in my reasoning
 A: 
I think I'm confusing elongation and net acceleration. Please let me
know the errors in my reasoning

When the force is initially applied (at $t=0$) the rod acts very much as a spring but not necessarily a 'perfect', Hookean spring where $F=kx$, but more like:
$$F(x)=k(x)x$$

During a brief period of time the rod is being elongated from $x=0$ to $x$. During that period there was acceleration until the restoring force exerted by the rod equals the externally exerted force $F_{ext}$ and:
$$F_{ext}=k(x)x$$
There's now no net force and no more acceleration but during that initial acceleration the net force was:
$$F_{net}=F_{ext}-F(x)$$
A: "I was taught that stress is the restoring force per unit area (let us assume a rod). This stress is developed on order to resist the motion of the rod."
I'm not sure this framework is very useful if it's leading you to contradictions or paradoxes. I think of a stress state as arising through sets of balanced forces: 2 forces for normal stress, 4 forces for shear stress:


(Images from my site)
Broadly, we define stress this way (i.e., we subtract unbalanced forces that would tend to make a body accelerate) because we want to focus on elasticity and ignore dynamics. You want there to be zero net force and zero net moment on an object before you start your static stress/strain analysis. This way, you'll never conflate elongation and acceleration because they've been decoupled.
One you have your balanced sets of forces, you can determine the stresses (normal and shear, in all three axes) from the associated force/area pairs and then determine the strain.
A: When Newton wrote that $F=ma$, there was another guy at Cambridge, Mr Hooke that wrote that $F=kx$.
In a way Newton prevails and his equation is now the definition of force. But force as the net force.
If this force is done by pulling an object with a spring, it is expected that $kx = ma$. If that correspondence is not exact, we say that the spring is not perfectly linear elastic, and $F \approx kx$ for that spring.
Hooke's concept of force is not related to movement, so a rod can be static, and subject to a force $F$ at both ends. Or it can be accelerated with the force $F$ only at one of the ends. Both ways: $F=kx$.
A: If you suddenly apply a substantial force to the ends of a rod, the same thing will happen as if you apply a force to a spring having mass:  there will be oscillations (i.e., acceleration), even if Young's modulus is perfectly constant.  But the oscillations will die out with time (as a result of small damping effects that are present). If you gradually apply a force to a rod, the inertial effects will be substantially less, and the main thing that you will get is a rod under static tension.
In a rod under static tension, the internal molecular structure of the solid rod will experience deformation due to the applied loading, just the same as when you apply tension to a spring with equal forces at its ends.
A: Stress takes into account the restoring force only. And restoring force is the internal force produced by the body. It's the next external force that is zero. But since internal force is not zero, there would be a stress and thus a corresponding elongation. The body won't have acceleration since net external force is zero.
