Question

Two wires of same length and radius, are joined end to end and loaded. The Young's modulii of the materials of the two wires are Y₁ and Y₂. The combination behaves as a single wire whose Young's modulus is :

Answer 1

The effective Young's modulus of two different wires joined end to end is given by the rule of mixtures, where it depends on the individual Young's modulii and the lengths of the wires, assuming equal length and cross-section.

Step-by-step explanation:

Two wires made of different materials and possessing respective Young's modulii Y₁ and Y₂ are joined together and loaded. When considering the composite wire as a single entity, the effective Young's modulus of this combination cannot be derived from a simple algebraic mean because the stress experienced by both wires will be the same (due to the force being the same over both), but the strain will differ according to their respective stiffness. Therefore, the effective Young's modulus Y of the combined wire system is given by the rule of mixtures for series:

(1/Y) = (L₁/(L₁+L₂))(1/Y₁) + (L₂/(L₁+L₂))(1/Y₂)

where L₁ and L₂ are the lengths of the wires, which in this case are equal, simplifying to L. Hence, the effective Young's modulus of the wire combination is given by:

(1/Y) = (1/2)(1/Y₁) + (1/2)(1/Y₂)

This equation holds as long as the wires have the same cross-sectional area and length prior to being joined and loaded.