Stress Strain Curve

In engineering and materials science , a stress-strain curve for a material gives the relationship between stress and strain . This is achieved by gradually applying a load to a test coupon and measuring the deformation , from which the strain and strain can be determined ( see tensile test ). These curves reveal many properties of a material , such as Young’s modulus , yield strength , and ultimate tensile strength .

Stress Strain Curve
Stress Strain Curve


Generally speaking, curves representing the relationship between stress and strain in any type of deformation can be thought of as a stress–strain curve. Tension and stress may be normal, shear, or mixing, may also be uniaxial, biaxial, or multiaxial, even changing over time. The form of deformation may be compression, stretching, torsion, twisting etc. Unless otherwise noted, the stress–strain curve refers to the relationship between the axial normal stress and the axial normal stress of a material measured in a stress test.

engineering stress and strain

Consider a bar of original cross-sectional area to be subjected to equal and opposite forces pulling at the ends so that the bar is under tension. The material is experiencing stress defined as the ratio of the force to the cross sectional area of ​​the bar plus the axial length:

{\displaystyle \sigma ={\frac {F}{A_{0}}}}
{\displaystyle \varepsilon ={\frac {L-L_{0}}{L_{0}}}={\frac {\Delta L}{L_{0}}}}

Subscript 0 denotes the original dimensions of the sample. The SI unit for stress is the newton per square meter or pascal (1 Pascal = 1 Pa = 1 N/m 2 ), and the strain is unitless. The stress–strain curve for this material is plotted by increasing the sample and recording the strain variation with strain until the sample is dissolved. By convention, the tension is set on the horizontal axis and the tension is set on the vertical axis. Note that for engineering purposes we often assume that the cross-section area of ​​a material does not change during the entire deformation process. This is not true because the actual area will decrease when deformed due to elastic and plastic deformation. A curve based on the original cross-section and gauge length is called an engineering stress-strain curve ., whereas a curve based on instantaneous cross-section area and length is called a true stress-strain curve . Unless otherwise stated, engineering stress-strain is commonly used.

true stress and tension

Due to the undiscovered effect of the shrinkage of the segment area and the further elongation of the developed elongation, the actual stress and strain are different from the engineering stress and strain.

{\displaystyle \sigma _{\text{t}}={\frac {F}{A}}}
{\displaystyle \varepsilon _{\text{t}}=\int {\frac {\delta L}{L}}}

Here the dimensions are instantaneous values. Assuming the volume of the sample is conserved and deformation occurs uniformly,

{\displaystyle A_{0}L_{0}=AL}

The actual stress and strain can be expressed by engineering stress and strain. For true stress,

{\displaystyle \sigma _{\text{t}}={\frac {F}{A}}={\frac {F}{A_{0}}}{\frac {A_{0}}{A}}={\frac {F}{A_{0}}}{\frac {L}{L_{0}}}=\sigma (1+\varepsilon )}

for stress,

{\displaystyle \delta \varepsilon _{\text{t}}={\frac {\delta L}{L}}}

Integrate both sides and apply boundary conditions,

{\displaystyle \varepsilon _{\text{t}}=\ln \left({\tfrac {L}{L_{0}}}\right)=\ln(1+\varepsilon )}

So in a stress test, the actual stress is greater than the engineering stress and the actual stress is less than the engineering stress. Thus, a point defining the actual stress–strain curve is shifted upward and to the left to define the same engineering stress–strain curve. The difference between actual and engineering stresses and strains will increase with plastic deformation. At low strains (such as elastic deformation), the difference between the two is negligible. As for the tensile strength point, it is the maximum point in the engineering stress-strain curve, but not a special point in the actual stress-strain curve. Since the engineering stress is proportional to the force applied to the sample, the criterion for neck formation can be determined as .

{\displaystyle \delta F=0}
{\displaystyle \delta F=\sigma _{\text{t}}\,\delta A+A\,\delta \sigma _{\text{t}}=0}
{\displaystyle -{\frac {\delta A}{A}}={\frac {\delta \sigma _{\text{t}}}{\sigma _{\text{t}}}}}

This analysis suggests the nature of the UTS point. The work strengthening effect is perfectly balanced by the shrinkage of the section area at the UTS point.

After the formation of necking, the sample undergoes heterogeneous deformation, so the above equations are not valid. The tension and strain on the neck can be expressed as:

{\displaystyle \sigma _{\text{t}}={\frac {F}{A_{\text{neck}}}}}
{\displaystyle \varepsilon _{\text{t}}=\ln \left({\frac {A_{0}}{A_{\text{neck}}}}\right)}

An empirical equation is commonly used to describe the relationship between actual stress and strain.

{\displaystyle \sigma _{\text{t}}=K(\varepsilon _{\text{t}})^{n}}

Here, is the strain-hardening coefficient and is the strength coefficient. Strictness is a measure of the behavior of a material. Materials with higher necks have more resistance. Typically, metals at room temperature range from 0.02 to 0.5.



A schematic diagram for the stress–strain curve of low carbon steel at room temperature is shown in Figure 1. There are several phases showing different behaviour, suggesting different mechanical properties. To clarify, the material may be missing one or more of the steps shown in Figure 1 or may be an entirely different phase. The first stage is the linear elastic field. The strain is proportional to the strain, i.e. 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 phase is the initiation point of plastic deformation. The stress component of this point is defined as the yield strength (or upper yield point, UYP for short ) .

The second stage is the stress hardening zone. This region begins when the stress goes beyond the yield point, reaching a maximum at the final strength point, which is the maximum stress that can be maintained and is called the ultimate tensile strength (UTS). In this region, the stress mainly increases as the material moves, except that for some materials, such as steel, there is initially a nearly flat region. The plane field strain is defined as the low yield point (LYP) and is in the Luder band .as a result of its formation and spread. Clearly, heterogeneous plastic deformation forms bands at upper yield strength and with deformation these bands spread along the sample at lower yield strength. After the sample is again uniformly deformed, the increase in strain with the progress of expansion results in strengthening work, i.e. the dense dislocations induced by plastic deformation impedes the further movement of dislocations. To overcome these constraints, a high resolved shear stress must be applied. As the stress accumulates, the work force becomes stronger, until the stress reaches the ultimate tensile strength.

The third stage is the neck area. Beyond the tensile strength, a neck is formed where the local cross-sectional area becomes significantly smaller than the average. The neck deformity is heterogeneous and will reinforce itself as the stress is more concentrated on the smaller section. Such a positive reaction leads to rapid growth and fracture of the neck. Note that although the pulling force is decreasing, the work reinforcement is still progressing, i.e. the actual stress continues to increase but the engineering stress decreases because the shrinking section area is not considered. This area ends with a fracture. After fracture, the percentage elongation and decrease in the segment area can be calculated.


It is possible to distinguish some common characteristics between the stress–strain curves of different groups of materials and, on this basis, divide materials into two broad categories; Namely, ductile materials and brittle materials. 

Ductile material

Ductile materials, which include structural steel and many alloys of other metals, are characterized by their ability to yield at normal temperatures.

Low carbon steel typically exhibits a very linear stress–strain relationship up to a well-defined yield point. The linear part of the curve is the elastic area and the slope is the modulus of elasticity or Young’s modulus., Many ductile materials, including some metals, polymers and ceramics, exhibit a yield point. Plastic flow begins at the upper yield point and continues at the lower level. At the low yield point, the permanent deformation is asymmetrically distributed along the sample. The deformation band formed at the upper yield point will spread along the length of the gauge at the lower yield point. The band occupies the Luder’s strain over the entire gauge. Beyond this point, hardening begins. The presence of the yield point is associated with the pinning of dislocations in the system. For example, solid solution interacts with dislocations and acts as a pin and prevents dislocations from moving. Therefore, the tension required to initiate the movement will be large. As long as the dislocation escapes the pinning, the tension required to keep it going is low.

After the yield point, the curve usually decreases slightly due to dislocations emanating from the Cottrell atmosphere . As the deformation continues, the strain increases due to stress hardening until it reaches the final tensile stress . Up to this point, the cross-sectional area decreases uniformly due to Poisson contraction . Then it begins to fracture the neck and finally.

The presence of necking in ductile materials is associated with geometric instability in the system. Due to the natural unevenness of the material, it is common to find some areas with small inclusions or porosity within it or the surface, where the stress will be concentrated, creating a locally smaller area than other areas. For stresses less than the ultimate tensile stress, the increase of the work-hardening rate in this region will be greater than the reduction rate in the region, making this region more difficult to deform than the others, leading to instability, i.e. material has the ability to weaken inhumanity before it reaches peak tension. However, as the stress becomes larger, the rate of work hardening decreases, so that for now the region with the smaller area is weaker than the other region, The reduction in area will therefore be concentrated in this area and the neck will become more and more pronounced until the fracture. After the neck is formed in the material, further plastic deformation is concentrated in the neck while the rest of the material undergoes elastic contraction due to a decrease in the tensile force.

The stress–strain curve for a ductile material can be estimated using the Ramberg–Osgood equation . [3] This equation is straightforward to implement, and only requires the yield strength, ultimate strength, elastic modulus and percentage elongation of the material.

Brittle material

Brittle materials, which include cast iron, glass, and stone, are characterized by the fact that rupture occurs without any noticeable prior change in the rate of elongation, sometimes they fracture before yielding. become.

Brittle materials such as concrete or carbon fiber do not have a well-defined yield point, and are not stress-hardened. Therefore, ultimate power and breaking power are the same. Typical brittle materials such as glass do not show any plastic deformation , but fail when the deformation is elastic . One of the characteristics of brittle failure is that the two broken parts can be reconnected to produce the same shape as the original component as the neck would not be formed in the case of ductile materials. A typical stress-strain curve for brittle materials will be linear. For some materials, such as concrete, the tensile strength is negligible compared to the compressive strength and is assumed to be zero for many engineering applications. Glass fiber has higher tensile strength than steel, but bulk glass generally does not. This is due to the stress intensity factor associated with defects in the material. As the sample size increases, the expected size of the largest defect also increases.

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