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Explaining the deformation of metal solids

Metals are polycrystalline; inside each crystal, atoms are regularly arranged and close together. Left alone, the atoms attract their neighbours and at the same time repel each other, with forces that just cancel each other out. Each atom is, on average, in equilibrium. However, if you compress a rod or wire, pushing atoms together, you can feel how the forces of repulsion increase more than the forces of attraction. If you stretch a rod or wire, the forces of repulsion decrease and you can feel the forces of attraction holding the wire together. 
Both the repulsive forces between atoms and the attractive ones are electrical in origin. They arise from the charged particles composing one atom disturbing those of a neighbour, in ways that are specified by quantum mechanical rules. Since these forces are due to complexes of charges, they fall off more rapidly with distance than the inverse square law of force between isolated charges. 
The attractions between atoms in metals are fairly short-range forces; two pieces of metal placed close together do not attract each other. Forces of repulsion must also be there or attractive forces would collapse solids. They cannot have the same falling off with distance as the attractive forces do, otherwise the atoms in solids would never settle down to a definite spacing as they do. So the repulsions are very-short-range forces, not appearing until atoms are much closer than when they first feel attractions. The repulsion rises sharply as atoms move closer together until it balances attraction. 

Intermolecular forceSketch A

The graph of force against distance between atoms looks something like sketch A. The graph of potential energy of two atoms looks something like sketch B. If there are only two atoms in a system, their equilibrium position will be ro on sketch A (where the resultant force between atoms is zero), or on sketch B (where the potential energy is a minimum). In practice, atoms do not settle down but remain in vibration about that equilibrium position. If you think of one atom as fixed, then the other one is, so to speak, sliding up and down the sides of the potential energy bowl around ro.Potential EnergySketch B

An atom in the middle of a solid has neighbours on every side, so the principle is the same but picture is more complicated. The difference between attractions which grow with decreasing distance and repulsions which grow much more steeply with decreasing distance explains the elastic properties seen at macroscopic level. 

Hooke’s law (elastic) behaviour of a stretched wire: the stiffness (Young modulus) of a wire when stretched will depend on the way that its interatomic forces change near their equilibrium positions. 
Plastic behaviour of a stretched wire: beyond the elastic limit, planes of atoms inside metal crystals slip over one another and so the wire is permanently deformed.