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Consider an ideal massless wire of length $L$, uniform cross-sectional area $A$, Young's modulus $Y$ suspended from the ceiling, with a load of weight $W$ suspended at the end. There is no variation in the value of acceleration due to gravity $g$ along the length of the wire. Any other factors that vary in real life are also constant here.

When the weight is just enough to produce the breaking stress in the wire, the wire breaks (fractures) at a point.

Which point will the wire break at? Is it entirely random (anywhere along the length) or is there a definite point (like the midpoint)?

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    $\begingroup$ From my knowledge of tensile testing, the location most likely to fracture would be at the jaw that holds the wire. For that reason, dog bone shaped test specimens are typically used where the wide ends are secured by the jaws making fracture less likely there, and with the narrow section in between being the subject of the test. $\endgroup$
    – Bob D
    Apr 14 at 15:34

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Since nothing in the question identifies any point on the wire as particularly remarkable—the stress is perfectly uniform—we can't select any particular point as a location of predicted failure. We might as well assume the failure location to be random.

In practice, the top connection point would draw attention, for at least several reasons:

  1. No material is truly massless, so the stress (arising from the suspended weight plus the weight of the wire) would be highest there.

  2. Connection points can involve additional stresses because of applied constraints; if the connection is rigid, for example, then the connection point must bear an axial load plus a bending moment.

  3. Connection points can involve discontinuities that can create stress concentrations.

  4. Connection points can feature additional surface area that may increases susceptibility to corrosion or crack propagation.

As supported by the relevant physics models, it's therefore good engineering practice to build up the connection point cross-section area, and reduce attachment constraints (use a lubricated hinge, for example), and constrain the motion of the hanging object, and protect the top end from environmental damage, and so on, to address the possible failure modes listed above.

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