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This is a problem from Beer and Johnston's Mechanics of Materials (Amazon link). There's a problem in which there are two rods of different material, area, and length. The rods are fixed to walls, horizontally facing each other, with a gap between them. The gap is spaced such that, if the rods expand for a given rise of temperature, they can't expand to full extent and instead press against each other. The solution says that at interface, the forces are equal and opposite. But another case could be a net force at interface which gets balanced by reactions at walls.

My question is why do we not to consider the latter case, and only the former case is taken?

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As far as I can tell, the question is "Why is assuming a zero net force at the interface less appropriate than assuming a net force that is balanced by forces from the walls".

The reason is because if we assume a zero net force, then that means that we're assuming each rod is providing the same amount of force. With them being different sizes and materials, this is unlikely (though not impossible). Far more likely is that one is going to expand more quickly than the other as temperature increases, and so once they're touching, it will be pushing harder on the other rod. This means that there is a non-zero force at the interface, and if we're assuming that the system is stationary, then that force will have to be reacted to by a force in the wall.

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If there were a net force at the interface, the interface would move (i.e. one rod would expand and the other contract while they remain touching). If the interface remains in a fixed position then the net force across it must be zero.

Actually it seems quite plausible to me that the interface could move slightly from the point at which the rods first touch. If the rods are made of stiff materials that resist compression (such as metals) then the interface will move to a point of equilibrium. That is, if one rod is pushing harder, the other rod will compress slightly, and as it compresses, the force with which it resists compression will increase (Hooke's law) until the forces between the two rods are equal and opposite.

The precise position at which this equilibrium is achieved will vary depending on the temperature, but at any fixed temperature the rods will very quickly settle to this equilibrium state.

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