page_title: Stress–strain curve
section_title: True stress and strain
hierarchical_section_title: Stress–strain curve / Definition / True stress and strain
context_page_description: A stress–strain curve for a material gives the relationship between stress and strain. It is obtained by gradually applying load to a test coupon and measuring the deformation, from which the stress and strain can be determined. These curves reveal many of the properties of a material such as the Young's modulus, the yield strength and the ultimate tensile strength.
context_section_description: Due to the shrinking of section area and the ignored effect of developed elongation to further elongation, true stress and strain are different from engineering stress and strain.
  Here the dimensions are instantaneous values. Assuming volume of the sample conserves and deformation happens uniformly,
  The true stress and strain can be expressed by engineering stress and strain. For true stress, 
  For the strain, 
  Integrate both sides and apply the boundary condition,
  So in a tension test, true stress is larger than engineering stress and true strain is less than engineering strain. Thus, a point defining true stress-strain curve is displaced upwards and to the left to define the equivalent engineering stress-strain curve. The difference between the true and engineering stresses and strains will increase with plastic deformation. At low strains (such as elastic deformation), the differences between the two is negligible. As for the tensile strength point, it is the maximal point in engineering stress-strain curve but is not a special point in true stress-strain curve. Because engineering stress is proportional to the force applied along the sample, the criterion for necking formation can be set as .
  This analysis suggests nature of the UTS point. The work strengthening effect is exactly balanced by the shrinking of section area at UTS point.
  After the formation of necking, the sample undergoes heterogeneous deformation, so equations above are not valid. The stress and strain at the necking can be expressed as:
  An empirical equation is commonly used to describe the relationship between true stress and strain. 
  Here,  is the strain-hardening coefficient and  is the strength coefficient.  is a measure of a material's work hardening behavior. Materials with a higher  have a greater resistance to necking. Typically, metals at room temperature have  ranging from 0.02 to 0.5.
caption: The difference between true stress–strain curve and engineering stress–strain curve