Materials Science And Engineering Properties
Materials Science And Engineering Properties
1st Edition
ISBN: 9781111988609
Author: Charles Gilmore
Publisher: Cengage Learning
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Chapter 6, Problem 6.9P
To determine

The relationship between change in resistance and strain.

Expert Solution & Answer
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Answer to Problem 6.9P

The relationship between change in resistance and strain is dR=Rε(1+2v) .

Explanation of Solution

Formula Used:

Write the expression for the resistance of wire..

  R=ρlA …… (I)

Here, R is the resistance, ρ is the resistivity, l is the length of wire and A is the area of wire.

Write the expression for the derivative of volume in terms of length and area.

  dV=Adl+ldA …… (II)

Here, dV is the change in volume, dl is the change in length and dA is the change in area.

Write the expression for change in volume.

  dV=l(1+ε)A(1vε)2lA …… (III)

Here, ε is the strain and v is the Poisson’s ratio.

Calculation:

Differentiate equation (I) with ρ as constant.

  dR=d(ρlA)=ρAdl+ρld(1A)=ρAdlρldAA2=ρ(AdlldAA2) …… (IV)

Rewrite equation (III) in expanded form.

  dV=l(1+ε)A(1vε)2lA=l(1+ε)A(12vε+v2ε2)lA

For small strain, neglect ε2 .

  dV=l(1+ε)A(12vε)lA=(lA+lAε)(12vε)lA=lAε(12v)=Adl(12v) …… (V)

Equate equation (II) and (V).

  Adl+ldA=Adl(12v)Adl+ldA=Adl2AvdlldA=2Avdl …… (VI)

Substitute 2Avdl for ldA in equation (IV).

  dR=ρ(Adl(2Avdl)A2)dRR=ρR(Adl+2AvdlA2)=ρAdlR(1+2vA) ……. (VII)

Substitute ρlA for R and ε for dll in equation (VII).

  dRR=ρAdl(ρlA)(1+2vA)=dll(1+2v)=ε(1+2v)dR=Rε(1+2v)

Conclusion:

Thus, the relationship between change in resistance and strain is dR=Rε(1+2v) .

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Materials Science And Engineering Properties

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