The actual stress–strain curve for a particular material is given by σ = Cε ^n where C is a constant. Assuming that the material suffers no change in volume during plastic deformation, derive an expression for the nominal stress–strain curve and show that this has a maximum value when ε = n/(1 - n)

Question

The actual stress–strain curve for a particular material is given by [latex] \sigma=C \varepsilon^{n} [/latex] where C is a constant. Assuming that the material suffers no change in volume during plastic deformation, derive an expression for the nominal stress–strain curve and show that this has a maximum value when ε = n/(1-n).

Accepted Answer

Suppose that the actual area of cross section of the material is A and that the original area of cross section is [latex] A_{ o } [/latex]. Then, since the volume of the material does not change during plastic deformation

[latex] A L=A_{ o } L_{ o } [/latex]

where [latex] L ext { and } L_{ o } [/latex] are the actual and original lengths of the material respectively. The strain in the material is given by

[latex] \varepsilon=\frac{L-L_{ o }}{L_{ o }}=\frac{A_{ o }}{A}-1 [/latex] (i)

from the above. Suppose that the material is subjected to an applied load P. The actual stress is then given by σ = P/A while the nominal stress is given by [latex] \sigma_{ ext {nom }}=P / A_{ o } [/latex]. Therefore, substituting in Eq. (i) for [latex] A / A_{ o } [/latex]

[latex] \varepsilon=\frac{\sigma}{\sigma_{ ext {nom }}}-1 [/latex]

Then

[latex] \sigma_{ ext {nom }}(1+\varepsilon)=\sigma=C \varepsilon^{n} [/latex]

or

[latex] \sigma_{ nom }=\frac{C \varepsilon^{n}}{1+\varepsilon} [/latex] (ii)

Differentiating Eq. (ii) with respect to ε and equating to zero gives

[latex] \frac{ d \sigma_{ nom }}{ d \varepsilon}=\frac{n C(1+\varepsilon) \varepsilon^{n-1}-C \varepsilon^{n}}{(1+\varepsilon)^{2}}=0 [/latex]

i.e.

[latex] n(1+\varepsilon) \varepsilon^{n-1}-\varepsilon^{n}=0 [/latex]

Rearranging gives

[latex] \varepsilon=\frac{n}{(1-n)} [/latex]

Question 8.3: The actual stress–strain curve for a particular material is ...

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