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Usually this "indestructible" material is unnaturally strong for its weight; the armor "just exists" and doesn't go into the feasibility of such a material.
Given that the material isn't "hard to obtain" (anybody is capable of obtaining the armor) and passive (no type of "energy" should be required for it to work), how might I explain that nobody simply creates bullets or other offensive items that are made out of the same material in order to overcome the armor?
Make the material flexible.
Consider Kevlar: it's an ultra-strong fiber that can be woven into sheets of highly tear-resistant fabric. Thick, interlocked layers of this fabric are used to spread the force of a bullet out over a large area. Using it to make a weapon just doesn't work: it's too light to make an effective bullet, too floppy for a long-rod penetrator, and even if you somehow make it stiff enough to look like a sword, it can't hold an edge worth mentioning. I suppose you can make a Kevlar whip, but you'd look rather silly running around a battlefield with one.
Your indestructible armor would be an "ultra-Kevlar": strong enough to resist cutting (swords) as well as tearing (bullets), but not stiff enough to make a sensible weapon.
Give it a very high strength to weight ratio.
A material that is very strong but weighs almost nothing would be lousy for any kind of weapon that relies on mass.
A bullet made of aerogel or styrofoam isn't going to do any real damage, especially if you are firing it at something as hard as steel.
At any distance the air resistance will simply rob it of all inertia and drop it out of the air.
It could be used as an unbreakable line, but as a weapon there aren't many advantages to that over normal braided steel cable.
On a side note, you could braid a space elevator with it, which would be awesome!
I second the kevlar answer, but here are soe thoughts as well.
There is such thing as non-Newtonian fluid which flows quite easily when pushed slowly but becomes almost rigid when you hit it. Google videos, they are fun - very filthy fun. Can be made at home as well.
One could theorize about a non-Newtonian cloth or plastic which is soft and elastic when moved slowly, but extremely rigid and tough when accelerated and/or deformed rapidly. Such material would be a godsend when it comes to armor: you can move, walk, run, jump all the way you want, but a bullet or club will only push, rotate, and shake you in the same pose you were before the hit. No more broken bones (but you can still suffer concussion if the projectile has enough impulse).
However, such property is useless for weapons. Make a sword that what, deforms when you swing it and stiffens when it hits the target? Would make for great supersonic whips though, but this type of armor should be immune to whipping.
The armor can still be penetrated by slowly moving knife, think Frank Herbert's Dune. Maybe physical armor (unlike force shield) could be strengthened by layers of kevlar or something.
I can also think of a sonic weapon which doesn't harm the enemy per se but binds them in place because the armor reacts to vibration as it would to a hit or fall.
But still it's a great armor overall, and no benefits for making weapons.
The simplest answer to the question Why don't they turn it into weapons is...
They don't need to.
In no logically consistent world will this material be as cheap as standard armor and munitions...more people will want it meaning cost will matter so making munitions out of it makes less and less sense.
At a certain point the limiting factor in an armored human scenario is the human. Your armor may be able to take a hit from a tank shell without being damaged but that is irrelevant because the human on the inside would end up liquified.
Blades would be pretty worthless unless you are talking about a heavy two hander (they are better for blunt damage than piercing/slashing).
Fire still works. Heat up the armor, it may still be shiny but it will have a cooked person inside.
Blunt damage still works. Hit an armored human with a heavy enough weight and the armor is irrelevant.
Electricity...still works unless your armor is an insulator.
Water. Armor is generally heavy, it's hard to swim with armor on...even kevlar.
Standard munitions (larger calibers) still work too. A .50 cal vehicle mounted machine gun can kill without actually hitting a human target, the velocity alone can snap necks and other bones...
Indestructible weaponry isn't a problem for indestructible armor (or even necessarily destructible armor). The goal of armor is not to break the weapon, but rather to protect the user. For a real world analogy, consider a bullet-proof vest. If someone shoots you, the vest functions by stopping the bullet before it penetrates your body and it protects you, despite the fact that the bullet was not "destroyed" (although it may end up squished). In fact, someone could shoot an ordinary kevlar vest with an indestructible bullet, and it might get a little further into the vest as it doesn't lose energy to being squished, but the vest would still function. So, indestructible weaponry isn't even necessarily a problem for normal armor. Why would it be a problem for indestructible armor? Anything that comes at you just bounces off your armor.
Of course, it is notable that one feels a large impact if they are shot wearing a bullet-proof vest, since all that energy and momentum has to go somewhere. Such an armor would be useful to people, but it can only do so much: If a tank shoots you, there's just too much energy and momentum there for you to survive, regardless of whether the shell actually hits you - so weapons made of lesser materials can still overcome indestructible armor. That said, if you start armoring machines and buildings in this stuff (i.e. things that are less sensitive to changes in momentum), there's not much an adversary can do.
Warning: attempt at internally-consistent handwaving. Based on answers clarifying what asker is trying to achieve, I see only one explanation: Make it susceptible to acceleration.
Normal materials are concerned with force, not acceleration, and break apart if applied force is greater than for example it's tensile strength. This material is different: no matter what force is applied, it becomes relatively fragile when it accelerates (this is "the big lie", rest is derived from this quality). Even accelerating, it can easily outperform ordinary materials, but it's the strongest when it's still or moving at constant speed, easily defeating bullets made of same material.
In practice, this means that it's great for construction, or perfect for armours and non-moving parts of weapons (like barrels, but not locks, at least not without extreme redesign) but terrible for blades, bullets or shells: 9 mm bullet weights about 100 grams and is moving at 400 m/s giving it momentum of 40 kg*m/s, same as 80 kg human moving at 0.5 m/s. During impact, bullet is subject to extreme deceleration while it transfers almost all of it's momentum to wearer, wearer on the other hand is only slightly affected - bullet becomes much more fragile than armour and shatters.
This means, that humans wearing such armour are completely immune to small arms fire, and any "indestructible" projectile lighter than target (assuming target isn't anchored in any way) will shatter and has to rely on other properties than penetration to make a kill (120mm Abrams cannon: ~8 kg projectile, 1700 m/s, after impact 80 kg wearer would be sent flying at ~150 m/s, which without liquid breathing should kill him anyway, but shell is subjected to almost 10 times higher acceleration than armour), which means that "indestructible" projectile has no advantage over normal projectile.
In practice, this remarkable property would have very interesting impact on modern battlefield: armoured infantry becomes immune to shrapnel and small arms fire, airburst bombs and artillery become completely ineffective because ground itself protects from acceleration/deceleration. The main way to kill infantry is to use projectiles and explosions intended to send them flying, killing them within armour. Infantry on the other hand is most likely equipped with some means to quickly anchor themselves to the ground/surrounding for increased protection when located in exposed or otherwise dangerous spot. As technology develops, increased weight of powered armour becomes a great boon, in addition, anchors can easily become part of armour, deploying automatically when radar detects incoming fire (same as anti-missile systems used on modern tanks). The only sure way to kill, is to use shell heavier than target, while target is in air or space.
Edit to explore effects a bit more:
Tanks become effectively immune to any regular weapons, but are not cost effective compared to infantry, infantry on the other hand is completely immune to regular weapons wielded by other infantry, this heavily changes infantry tactics:
Anti-tank tactics:
In practice that removes tanks from battlefield, because increase in protection over infantry is not worth costs and loss in versatility and mobility, unless tanks can carry weapons that CAN damage this material while same weapons are impractical for infantry. For example, tanks could use laser/plasma weapons which are too big even for powered armours (something about fusion chamber sizes and energy requirement necessitating use of reactors?) but can penetrate this astounding material.
The way I see it, end result is a mess where infantry fights each other in melee and literally dismantles tanks if let close, while tanks rule supreme in open terrain.
Hmm, it's not actually THAT different.
I think your analogy is flawed. You are assuming that armor made from material X is unnaturally strong; so a weapon made from material X should also be unnaturally powerful(kind of like a diamond-tipped saw is required to cut diamonds).
Flaw in this logic is that the weapon and the armor are made using different goals in mind: Weapon needs to damage the enemy, not his armor. If I were to use a flamethrower or EMP on a person wearing said armor, the armor won't need to be penetrated to kill the person.
Also, as Milo Brandt said in his answer, such material will only yield indestructible weapons, not unbeatable ones.
Here's a joke question to clarify the idea:
If the black box in an airplane is never destroyed in a crash, why isn't the entire plane made of that stuff?
Make it impact sensitive
The material, like corn starch water mix becomes very hard only briefly on impact. This works well for armour as a layer of this material could be contained in the protective armour in pockets of another material or through some other containment system.
However if making bullets or blades this becomes completely useless as the moment they are not being hit against something they turn to fluid and drain away.
Simplest answer. The real reason to not use an indestructible material for making weapons.
It is indestructible
That means that you cannot not damage it. Which means you cannot shape it, sharpen it, or generally work it. Ergo, if the raw material wasn't weapon shaped to begin with, it can never be weapon shaped.
The up shot of this is, how did it become armor shaped? Well the material could come in a variety of plate shapes, which would be woven together into some sort of plate armor.
It might be that the material would be prohibitively expensive to make bullets out of it. Body armor that's made from a basically indestructible material would last basically forever. So, it's really expensive, but it's a one time purchase. If you could make a bullet out of that material that might not in and of itself be enough to defeat the armor. You might need to hit the same exact spot in the armor 10, 15 or 100 times to break it because of the way it's layered or hardened or whatever other way you explain it. So in that example, while technically, it could be destroyed by a projectile of the same material, it's effectively indestructible because nobody would buy bullets at $10,000 a round when there isn't even a guarantee that they will take the target out. It would be better to train people to make very precise shots with cheap ammo. It gives you more story hooks and is more interesting for things to have some weakness. Maybe there is one multi billionaire that can afford one all out attack with super bullets on one person and it bankrupts them. Weaknesses give you cool things to write about. I think that's why Superman has a weaknesses. Otherwise, where's the story? He just wins.
Suppose the process of manufacturing the armor produces a material that is lightweight and indestructible. It cannot even be dented. Suppose further that the process can be modified so that the finished product is of any size and shape we want, within reason.
I'll call this material "unobtainium."
To counter the question, why wouldn't people make weapons of unobtainium in order to overcome the armor, the response is: how would that work?
Imagine we fired a twenty-first-century anti-tank round at a soldier armored in unobtainium, and suppose the soldier is well enough braced (e.g. against a wall) so that the impact of the round does not just knock him over. What happens?
The round cannot pierce or deform the armor. It strikes the armor with great force, but the armor pushes back (Newton's laws). The reaction force may deform the anti-tank round (in fact I think it almost surely will do so) or cause it to bounce back, or both.
Now put a full jacket of unobtainium around a similar anti-tank round and fire it at the same armored soldier. What happens?
I claim that the round still exerts force on the armor and the armor still pushes back with an equal and opposite force. The difference is that this time, the unobtainium jacket prevents the anti-tank round from deforming. But it does not prevent it from bouncing off, which I believe is exactly what it would do.
An unobtainium-jacketed anti-tank round would indeed be useful as a weapon against twenty-first-century tanks: their armor is not nearly as tough as unobtainium, so a round fired with sufficient force is sure to penetrate. But if nobody is using twenty-first-century tanks in your world, the ability to pierce that armor is not worth anything.
In fact, the main thing stopping you from shooting a projectile through unobtainium armor is not the strength of your projectile, it's the inability to put enough energy and momentum into any projectile in order to overcome the armor's incredible strength. (Non-projectile weapons are even worse; the amount of energy and momentum you can put into a sword by swinging it is very limited.)
So to fight unobtainium-armored soldiers, you need to get around the armor, or find something (a force-field "blade" perhaps) that can disrupt whatever material properties give unobtainium its strength; you could shoot something at a walking soldier with enough momentum to knock him over or at least slow him down, or simply hit the armor with something that imparts so much momentum to the soldier that his bones and/or organs are injured by the acceleration, killing or disabling him. None of this implies that these weapons would be made from unobtainium.
Warning: extreme physics-geek talk below this line.
Delving into the physics a little deeper, the force on the anti-tank round has to be exerted through some distance in order to reverse the direction of the round. Since the force occurs only when the unobtainium jacket touches the unobtainium armor, and neither of these can deform, how can the force act through a distance? I suppose that when we say unobtainium cannot be deformed, we mean it cannot be deformed on a macroscopic scale (anything you could measure with a ruler or even a micrometer); but it still consists of atomic nuclei and some kind of cloud of electrons, and as these components of the jacket approach these components of the armor, there is an electric repulsive force that becomes significant only when the gap between the objects is not much more than the distance between the nuclei, and that it rapidly increases as the objects approach nearer. This rapidly-increasing force provides the "springiness" that allows two colliding pieces of unobtainium to bounce off each other.
If its indestructible and cheap, then turning it into a weapon would be pointless because everyone else is indestructible.
If this unobtanium is super cheap, then everyone will have it. If everyone will have it, everyone will be "indestructible". If everyone is indestructible, then no one can be killed by weapons, and if no one can be killed by weapons then weapons become pretty useless.
An exotic option that, as far as I can tell, evades many of the mentioned creative workarounds the other answers have:
If you're willing to go the way of vibranium (Captain America's shield, absorbs kinetic energy), have a material with unusual inertial properties, like a solid-state equivalent of a spinning flywheel.
Perhaps possessing huge inertia regardless of mass, possibly manifesting at greater intensity the greater the attempted speed of change.
Superb protection against high-velocity impacts.
Mediocre protection against melee-speed weapons.
Movement impediment, possibly "tunable" to a small degree, letting you find your preferred balance of low-velocity protection and freedom of movement.
The "indestructability" part could come in a related flavor, letting the material be workable at low speeds, but otherwise resisting with a force roughly proportional to the mass of the piece (thus affecting design, and maybe still leaving steel superior for small pieces, justifying vulnerabilities and similarities to our world).
It would, at best, make a functional mace, though the property trade-off may still make it less practical than many alternatives. Not impossible, just an inferior idea.
As projectiles, I suppose they could be used in a long rifle if you held it steady long enough, though maybe not much better than a sling.
What if you wanted to use them as tiny bullets packing a huge punch?
I suppose small mass, air drag, and a need to put together a larger piece of the material to make use of the properties, could take care of that. Or low temperature tolerance. This could make for a separate SE question.
Rods from God, though? Goodbye, continents.
I can think of two explanations:
Another answer could be that the material becomes exponentially stronger in relation to the size of a given piece of it. In the same way a large mass has more gravity, larger pieces of this material could be much stronger. Pieces small enough to be wielded, would be weak and break apart easily while a contiguous piece of it the size of a suit might be able to withstand an atomic blast. Once you get past a certain size (size of a suit or maybe smaller), It might get tougher, but it doesn't matter because at the size of a suit, it can withstand being tossed into the sun.
There is a real life example of such a thing: Big safes or bank vaults. Yes, they could be destroyed, but it is impractical enough to do so that they are effectively considered indestructible. And the materials they are made of - concrete, hard steel - are not difficult to come by at all...
