True Strain Calculator

Consider an infinitesimal element of length $$l$$ stretched by $$dl$$. The incremental strain is $$d\varepsilon = dl/l$$. Integrating from $$L_0$$ to $$L$$:

$$\varepsilon_{\text{true}} = \int_{L_0}^{L} \frac{dl}{l} = \ln\left(\frac{L}{L_0}\right)$$

This integral definition is why true strain is also called logarithmic strain or Hencky strain. Key relationships:

• Connection to engineering strain: Since $$L/L_0 = 1 + \varepsilon_{\text{eng}}$$, we get $$\varepsilon_{\text{true}} = \ln(1 + \varepsilon_{\text{eng}})$$
• Stretch ratio: $$\lambda = L/L_0$$, so $$\varepsilon_{\text{true}} = \ln\lambda$$
• Volume conservation: For incompressible plasticity, $$A_0 L_0 = A L$$, giving $$L/L_0 = A_0/A$$, hence $$\varepsilon_{\text{true}} = \ln(A_0/A)$$

Why the difference matters: At $$\varepsilon_{\text{eng}} = 0.01$$ (1%), the error from using engineering strain instead of true strain is only 0.5%. But at $$\varepsilon_{\text{eng}} = 0.5$$ (50%), the true strain is $$\ln(1.5) = 0.405$$ — an 18.9% discrepancy. In deep drawing, forging, or rubber elasticity where strains exceed 100%, using engineering strain leads to serious errors.

Additivity: If a bar is stretched from $$L_0$$ to $$L_1$$, then from $$L_1$$ to $$L_2$$, the total true strain is $$\ln(L_1/L_0) + \ln(L_2/L_1) = \ln(L_2/L_0)$$. Engineering strain does not satisfy this: $$\varepsilon_1 + \varepsilon_2 \neq \varepsilon_{\text{total}}$$.

The True Strain is the logarithmic measure of deformation — physically meaningful for large deformations. The Engineering Strain is the simpler linear ratio. The Difference shows how much engineering strain overestimates the deformation (for tension) or underestimates it (for compression). The Relative Error quantifies this discrepancy as a percentage. The Stretch Ratio $$\lambda = L/L_0$$ is commonly used in rubber elasticity and hyperelastic material models. The True Strain from Area uses the volume-conservation approach, which is valid for metals undergoing plastic deformation.