Engineering stress and
engineering strain are approximations to the internal state that may be determined from the external forces and deformations of an object, provided that there is no significant change in size. When there is a significant change in size, the
true stress and
true strain can be derived from the instantaneous size of the object.
Engineering stress and strain Consider a bar of original
cross sectional area being subjected to equal and opposite forces pulling at the ends so the bar is under tension. The material is experiencing a stress defined to be the ratio of the force to the cross sectional area of the bar, as well as an axial elongation: Subscript 0 denotes the original dimensions of the sample. The
SI derived unit for stress is
newtons per square metre, or
pascals (1 pascal = 1 Pa = 1 N/m2), and strain is
unitless. The stress–strain curve for this material is plotted by elongating the sample and recording the stress variation with strain until the sample
fractures. By convention, the strain is set to the horizontal axis and stress is set to vertical axis. Note that for engineering purposes we often assume the cross-section area of the material does not change during the whole deformation process. This is not true since the actual area will decrease while deforming due to elastic and plastic deformation. The curve based on the original cross-section and gauge length is called the
engineering stress–strain curve, while the curve based on the instantaneous cross-section area and length is called the
true stress–strain curve. Unless stated otherwise, engineering stress–strain is generally used.
True stress and strain In the above definitions of engineering stress and strain, two behaviors of materials in tensile tests are ignored: • the shrinking of section area • compounding development of elongation
True stress and
true strain are defined differently than engineering stress and strain to account for these behaviors. They are given as : Here the dimensions are instantaneous values. Assuming volume of the sample conserves and deformation happens uniformly, :A_0 L_0 = A L The true stress and strain can be expressed by engineering stress and strain. For true stress, :\sigma_\mathrm{t} = \frac{F}{A}=\frac{F}{A_0} \frac{A_0}{A} = \frac{F}{A_0} \frac{L}{L_0} = \sigma (1 + \varepsilon) For the strain, :\delta \varepsilon_\mathrm{t} = \frac{\delta L}{L} Integrate both sides and apply the boundary condition, :\varepsilon_\mathrm{t} = \ln\left(\frac{L}{L_0}\right)=\ln(1+\varepsilon) 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 \delta F = 0. :\begin{align} & \delta F =\sigma_\text{t} \, \delta A + A \, \delta\sigma_\text{t} = 0 \\ & -\frac{\delta A}{A} = \frac{\delta \sigma_\mathrm{t} }{\sigma_\mathrm{t}} \end{align} This analysis suggests nature of the
ultimate tensile strength (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: :\begin{align} \sigma_\mathrm{t} &= \frac{F}{A_\mathrm{neck}} \\ \varepsilon_\mathrm{t} &= \ln\left(\frac{A_0}{A_\mathrm{neck}}\right) \end{align} An
empirical equation is commonly used to describe the relationship between true stress and true strain. :\sigma_\mathrm{t} = K (\varepsilon_\mathrm{t})^n Here, is the strain-hardening exponent 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.
Discussion Since we disregard the change of area during deformation above, the true stress and strain curve should be re-derived. For deriving the stress strain curve, we can assume that the volume change is 0 even if we deformed the materials. We can assume that: :A_i\times \varepsilon_i=A_f\times \varepsilon_f Then, the true stress can be expressed as below: :\begin{align} \sigma_T = \frac{F}{A_f} &= \frac{F}{A_i} \times \frac{A_i}{A_f} \\ &= \sigma_e \times \frac{l_f}{l_i} \\[2pt] &= \sigma_E \times \frac{l_i+\delta l}{l_i} \\[2pt] &= \sigma_E(1+\varepsilon_E) \end{align} Additionally, the true strain can be expressed as below: :\varepsilon_T= \frac{dl}{l_0} + \frac{dl}{l_1}+ \frac{dl}{l_2}+\cdots=\sum_i \frac{dl}{l_i} Then, we can express the value as :\int_{l_0}^{l_i} \frac{dl}{l} \, dx=\ln \left (\frac{l_i}{l_0} \right )=\ln(1+\varepsilon_E) Thus, we can induce the plot in terms of \sigma_T and \varepsilon_E as right figure. Additionally, based on the true stress-strain curve, we can estimate the region where necking starts to happen. Since necking starts to appear after ultimate tensile stress where the maximum force applied, we can express this situation as below: :dF=0=\sigma_TdA_i+A_id\sigma_T so this form can be expressed as below: :\frac{d\sigma_T}{\sigma_T}=-\frac{dA_i}{A_i} It indicates that the necking starts to appear where reduction of area becomes much significant compared to the stress change. Then the stress will be localized to specific area where the necking appears. Additionally, we can induce various relation based on true stress-strain curve. 1) True strain and stress curve can be expressed by the approximate linear relationship by taking a log on true stress and strain. The relation can be expressed as below: :\sigma_T=K\times(\varepsilon_T)^n Where K is stress coefficient and n is strain-hardening coefficient. Usually, the value of n has range around 0.02 to 0.5 at room temperature. If n is 1, we can express this material as perfect elastic material. 2) In reality, stress is also highly dependent on the rate of strain variation. Thus, we can induce the empirical equation based on the strain rate variation. :\sigma_T=K'\times(\dot{\varepsilon_T})^m On the figure (a), there is only concave upward Considere plot. It indicates that there is no yield drop so the material will be suffered from fracture before it yields. On the figure (b), there is specific point where the tangent matches with secant line at point where \lambda=\lambda_Y. After this value, the slope becomes smaller than the secant line where necking starts to appear. On the figure (c), there is point where yielding starts to appear but when \lambda=\lambda_d, the drawing happens. After drawing, all the material will stretch and eventually show fracture. Between \lambda_Y and \lambda_d, the material itself does not stretch but rather, only the neck starts to stretch out. ==Misconceptions==