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Consider a metal rod being heated. As metal rod is a good conductor, it spreads heat all through it. If so I think all the linear,area and cubical expansion takes place simultaneously practically. Am I right?

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  • $\begingroup$ It's very broad question, it actually depends on type of metal and it's characteristics, while volume expansion is always accompanied by linear or aerial or both. $\endgroup$
    – Qwerty
    Jan 4 at 9:17
  • $\begingroup$ It really depends on the material and on the constraints of the system: as an example, if the medium is contained between two parallel rigid walls there could be no area expansion in the direction perpendicular to those walls (for small $\Delta T$). In this example, thermal expansion is opposed by a stress state of compression, produced by the constraints $\endgroup$
    – basics
    Jan 4 at 9:26

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Yes, you are right, provided the rod is unconstrained. The rod strain linear expansion occurs equally in all directions, and this guarantees that the area expansion and volume expansion are also occurring.

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  • $\begingroup$ false. Even with a linear non-isotropic medium, strain as a function of temperature needs a tensor relation $\varepsilon \hspace{-4pt} \varepsilon = \alpha \hspace{-5.5pt} \alpha \Delta T $. And you're not even considering constraints and stress fields $\sigma \hspace{-4.5pt} \sigma$ $\endgroup$
    – basics
    Jan 5 at 0:08
  • $\begingroup$ @basics I was trying to answer the question at the level of the OP, so I was only focusing on isotropic solids. also, what do the words "provided the rod is unconstrained" mean to you? $\endgroup$ Jan 5 at 11:40
  • $\begingroup$ it means to me that you're not considering constraints $\endgroup$
    – basics
    Jan 5 at 12:00
  • $\begingroup$ @basics That's right. I have explicitly excluded cases where constraints are present. Again, where do the words "provided the rod is unconstrained" mean to you. Do you think I should have also included laminated composites, where the rod can bend. $\endgroup$ Jan 5 at 12:09
  • $\begingroup$ even much simpler examples: isotropic material between two rigid (ideally, in real life much more stiff than the material) parallel walls: in this case, compression stress has opposite effects w.r.t. thermal effects on strains and thus deformation, and these two cancel out in the direction orthogonal w.r.t. the walls. A laminated composite could be another example of non-isotropic behaviour, that can experience contraction with temperature rise along some directions. I'd only like to stress the role of constraints and boundary conditions in many applications in real life $\endgroup$
    – basics
    Jan 5 at 13:20

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