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The Difference Between True Stress and Engineering Stress

Think about pulling a bar in tension.  Load divided by cross-sectional area is force, or stress.  But what cross section are you considering?  Before starting  pull, the bar had a known cross-section of (lets say) 0.5" wide x metal thickness.  It's easy to measure these, since it is your starting material.  At any load, the engineering stress is the load divided by this initial cross- area.  While you are pulling, the length increases, but the width and thickness shrink.  At any load, the true stress is the load divided by the cross-area at that instant.  Unless thickness and width are being monitored continuously during the test, you cannot calculate true stress.  It is, however, a much better representation of how the material behaves as it is being deformed, which explains its use in forming simulations.  In circle grid analysis, engineering strain is the % expansion of the circle compared to the  initial diameter of the circle.  The relationships between engineering values and true values are: 

σ = s (1+e)    ε = ln (1+e)

Where "s" and "e" are the engineering stress and strain, respectively, and "" and "" are the true stress and strain, respectively.