Mechanical

Uploaded on

01 Jul 2022

Skill-Lync

When the material is loaded, it deforms and changes its shape. Figure 1.1 shows the typical engineering stress-strain behavior of the material under loading. There are two types of deformations: elastic and plastic.

Elastic deformation is a temporary deformation, which is recoverable. When the applied load is removed, the object returns to its original shape and thus the deformation disappears. The structure, which undergoes very small deformations, is studied with the elastic deformations.

Plastic deformation is a permanent deformation, which is non-recoverable. Plastic deformation gives rise to strain hardening, necking and fracture. The elastic region is governed by Hooke's law, it is a linear relation of applied stress and elastic deformation of the material within the elastic limit.

σ=Eε (1)

Here, σ is the applied stress, ε is the developed strain and E is the Young’s modulus. Elastic limit is a point at which the material first starts deforming plastically. This is the point where the material changes its behavior from elastic to plastic. When the material is loaded below the elastic limit it returns to the original shape and size. On the other hand if the material is loaded above the elastic limit it will undergo some permanent deformation. Yield strength from the figure 1.1, is a very important material property in simulation. It is defined as the amount of stress that can produce a specific amount of permanent deformation. This means by the time the yield strength is reached the material has already been permanently deformed (small deformation). This yield point lies just after the elastic limit. In some of the materials (ductile materials) this point is not well defined and stress-strain behavior shows a gradual change in elastic to plastic behavior rather than straight horizontal line.

It is difficult to detect at which point the material changes from elastic to plastic. In such cases, 0.2% offset yield strength is calculated to distinguish the elastic and plastic region. It is defined as the stress value corresponding to the 0.2% plastic strain. This is also called as proof stress and sometimes denoted asR_p0.2.

This 0.2% offset value is often mentioned in the material certificates by material suppliers. For fragile materials the offset is 0.05% - 0.1% because the plastic deformation is small.

In the following section, the R_p0.2 value will be calculated for a stress-strain curve. Figure 1.2 depicts a typical stress-strain curve for a ductile metal. The curve is in the near elastic region, so has low strain values along the X axis. The 0.2% offset yield strength is calculated by constructing a parallel line offset by 0.002 strain to the linear stress-strain line.

Young’s modulus (E) can be found through the linear elastic part of the curve. It is basically the slope of the linear part of the stress-strain curve (Elastic region). Here we will discuss how to calculate the slope in excel. The ‘SLOPE’ function in excel returns the slope of the linear regression line through the given data points.

Figure 1.3 depicts the stress-strain (marked in green) points selected from columns B and C to calculate the slope (Young’s modulus). The X data points are selected as Stress (N/m2) values starting from 3.56E+06 N/m2 to 1.29E+08 N/m2 whereas for Y data points, Strain [-] values are selected from 2.44E-05 to 8.89E-04.

This can be done by selecting any cell in excel, then type ‘=slope’ in that cell. Then select known Y values first and then select known X values separated by a comma as discussed above and hit enter to get the final result. The calculated slope value in excel is 147 GPa.

Young’s modulus can even be calculated by choosing two points and finding the slope by a two-point-slope equation. There might be some difference in the final answers by both methods.

** **

NOTE: Figure 1.3 and 1.4 shows only the first 25 points of the stress-strain curve shown above (figure 1.2)

All the strain values are offset by 0.2% strain that is 0.002 strain. This is done by adding 0.002 to every strain value in column B. Equation 2, shows the calculation for the first strain point ‘2.44E-05’. Similarly, after adding 0.002 to every strain value in column B we get column E (figure 1.4), which is 0.2% offset strain values.

0.0000244 + 0.002 = 0.0020244 (2)

For calculating 0.2% offset stress, all the strain values from column B should be multiplied with the calculated Young’s modulus. This gives the linear behavior of the stress line which is parallel to the elastic part of the curve. So, basically we are considering a complete linear behavior for all stress-strain points. Equation 3 shows calculation of the 0.2% offset stress value for the first point. Similarly, calculating the 0.2% offset stress for every value we get column F (figure 1.4).

0.2% offset yield stress = Strain * Young's modulus

0.2% offset yield stress = 0.0000244 * 147.3E+09 = 3.59E+06 Pa (3)

** **

Now, plot the Column E and F data which is 0.2% offset stress-strain data. This will be a straight line parallel to the linear elastic region of the stress-strain curve.

The intersection of this straight line with the stress-strain curve gives the 0.2% offset yield strength value. Figure 1.5 depicts the 0.2% offset yield strength marked with a green dot. The corresponding stress value on the Y-axis is the 0.2% offset yield stress.

Author

Anup KumarH S

Author

Skill-Lync

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