In engineering and materials science, a stress–strain curve for a material gives the relationship between the applied pressure, known as stress, and amount of deformation, known as strain. It is obtained by gradually applying load to a test object and measuring the deformation, from which the stress and strain can be determined (see tensile testing). These curves reveal many of the properties of a material, such as the Young's modulus, the yield strength, and the ultimate tensile strength.
Definition Generally speaking, curves that represent the relationship between stress and strain in any form of deformation can be regarded as stress–strain curves. The stress and strain can be normal, shear, or a mixture, and can also be uniaxial, biaxial, or multiaxial, and can even change with time. The form of deformation can be compression, stretching, torsion, rotation, and so on. If not specified otherwise, the term "stress–strain curve" typically refers to the relationship between the axial normal stress and axial normal strain of materials measured in a tension test. There are two ways that stress and strain are commonly defined mathematically. These are Engineering stress-strain, and True stress-strain. The difference between the two definitions depends on whether the change in area of material cross section is being considered. Engineering Stress is defined:
σ = F A 0 {\displaystyle \sigma ={\frac {F}{A_{0}}}}
Engineering Strain is defined:
ϵ = Δ l l 0 {\displaystyle \epsilon ={\frac {\Delta l}{l_{0}}}}
Where:
F {\displaystyle F} is defined as the instantaneous load applied perpendicular to the sample cross section.
A 0 {\displaystyle A_{0}} is defined as the original cross sectional area of the sample.
l 0 {\displaystyle l_{0}} is defined as the original measured length of the sample.
Δ l {\displaystyle \Delta l} is defined as the difference between instantaneous measured length and original length of the sample. True Stress is defined:
σ T = F A i {\displaystyle \sigma _{T}={\frac {F}{A_{i}}}}
True Strain is defined:
ϵ T = l n ( l i l 0 ) {\displaystyle \epsilon _{T}=ln({\frac {l_{i}}{l_{0}}})}
Where:
F {\displaystyle F} is defined as the instantaneous load applied perpendicular to the sample cross section.
A i {\displaystyle A_{i}} is the instantaneous cross-sectional area of the sample.
l i {\displaystyle l_{i}} is the instantaneous measured length of the sample.
l 0 {\displaystyle l_{0}} is the original measured length of the sample.
Stages A schematic diagram for the stress–strain curve of low carbon steel at room temperature is shown in figure 1. There are several stages showing different behaviors, which suggests different mechanical properties. To clarify, materials can miss one or more stages shown in figure 1, or have totally different stages.
Linear elastic region The first stage is the linear elastic region. The stress is proportional to the strain, that is, obeys the general Hooke's law, and the slope is Young's modulus. In this region, the material undergoes only elastic deformation. The end of the stage is the initiation point of plastic deformation. The stress component of this point is defined as yield strength (or upper yield point, UYP for short).
Strain hardening region The second stage is the strain hardening region. This region starts as the stress goes beyond the yielding point, reaching a maximum at the ultimate strength point, which is the maximal stress that can be sustained and is called the ultimate tensile strength (UTS). In this region, the stress mainly increases as the material elongates, except that for some materials, such as steel, there is a nearly flat region at the beginning. The stress of the flat region is defined as the lower yield point (LYP) and results from the formation and propagation of Lüders bands. Explicitly, heterogeneous plastic deformation forms bands at the upper yield strength and these bands carrying with deformation spread along the sample at the lower yield strength. After the sample is again uniformly deformed, the increase of stress with the progress of extension results from work strengthening, that is, dense dislocations induced by plastic deformation hampers the further motion of dislocations. To overcome these obstacles, a higher resolved shear stress should be applied. As the strain accumulates, work strengthening gets reinforced, until the stress reaches the ultimate tensile strength.
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