When performing tensile tests samples must have constant crossection alongside the axis. Then two marks are made and we can define engineering stress $\sigma=\frac{F}{A_0}$ and engineering strain $\varepsilon=\frac{l-l_0}{l_0}$, where $F$ is applied force, $A_0$ cross-section area prior the loading applied, $l_0$ distance between the marks prior the loading and $l$ is the actual distace between the marks.

But how can I estimate the strain when the cross-section is changing alongside the axis?

I'm thinking of expanding $$l-l_0=\varepsilon\ l_0$$ to $$l(t)-l_0=\int_0^{l_0}\varepsilon(x,t)\ \mathrm dx$$ but I'm stuck how to bind the measured values $l_0$, $l(t)$ and $A_0(x)$ to be able to derive $\varepsilon(x,t)$.

Edit according to Previous' comment:

  • The experiment goes beyond validity of Hooke's law $\sigma=E\ \varepsilon$;
  • I would like to find $\varepsilon(x,t)=f(A_0(x),l(t)-l_0)$;
  • After that I'd like to derive $\sigma/\varepsilon$ curve from as-measured $F/ \Delta l$ curve;
  • I can measure $\sigma/\varepsilon$ curve using another sample and different device to obtain $\varepsilon=f(\sigma)$ function, if necessary.
  • $\begingroup$ Do you want to measure Young's modulus? Or calculate total strain of the sample based on Young's modulus? E=stress/strain is constant (in elastic range), the integral of (F/E) * dx/A(x) over the total length will give the total change in length. $\endgroup$ – Previous Jul 1 '16 at 3:30
  • $\begingroup$ @Previous Thank you for your ideas. I tried the integral, you suggest but I don't need the overall change, I'm looking for the local changes and local strain. $\endgroup$ – Crowley Jul 1 '16 at 8:23
  • $\begingroup$ The formula is (derived from) Hooke's law, which is valid only for elastic deformation. when we are in elasto-plastic deformation state the strain is not proportional to stress. $\endgroup$ – Crowley Jul 1 '16 at 10:54
  • $\begingroup$ Not sure if I understand, seems that when plastic deformation is included the only options are using Ramberg–Osgood equation (includes strain hardening) or a purely numerical method, calculate for segments dx, with type of strain and formula dependent on the local stress, and vary parameters to find the best-fitting curve for the total strain. Either automatically, testing a range of values for each parameter and using least squares method to select best result, or manually change parameters to get good fit (elastic modulus, yield point stress...) $\endgroup$ – Previous Jul 3 '16 at 3:50

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