Why Is True Strain More Accurate?

I saw that true strain is a sum of strains over many increments. Why would this be more accurate than using engineering strain? Engineering Strain assumes the initial length to be constant which makes sense to me because strain is just a measure of deformation. What is wrong with the idea of assuming initial length to be constant?

If I strain a material twice sequentially, the individual engineering strains $$\varepsilon$$ don't even add up to the final strain:

$$\frac{L_2-L_0}{L_0}\neq\varepsilon_1+\varepsilon_2=\frac{L_1-L_0}{L_0}+\frac{L_2-L_1}{L_1}.$$

That doesn't sound very good.

However, the true strains $$e$$ do add up to the final strain:

$$\ln\left(\frac{L_2}{L_0}\right)=e_1+e_2=\ln\left(\frac{L_1}{L_0}\right)+\ln\left(\frac{L_2}{L_1}\right).$$

Nice.

I discuss more differences (including geometric and displacement symmetry) here, prepared for an MIT 3.032 (Mechanical Behavior of Materials) lecture.

• Wait is true strain just the definition of strain and engineering strain is just an approximation of true strain? Commented Sep 7, 2023 at 4:24
• The essential definition of strain is arguably $\frac{dL}{L}$ for an infinitesimal change in length. True strain is what you get when you integrate this, and engineering strain is what you get when you approximately integrate it. Commented Sep 7, 2023 at 4:52

True strain represents what is "actually happening." Materials (at least in this context) don't have a memory of the initial state they were in. At a molecular level they deform and behave based on the applied stress and the local state at the current time.

Engineering strain is essentially an average strain that is much easier to measure, but you can convert it to true strain and vise versa.

This is somewhat like other "average" quantities in physics such as average velocity, which may not ever correspond to the actual speed of the object at any time. But it may still tell you useful information.

• Thanks for the answer! I am confused by what you mean on the "what is actually happening" part. Commented Sep 7, 2023 at 4:22