# Stress Strain Diagram: Find Constants E, K, n, and α

• Dell
In summary, the stress-strain graph is divided into three stages: linear, uniform plastic, and necking. In order to find the values of the constants E, K, n, and α, data was plotted for σ(ε). The constant E was found by taking the incline of the linear portion. For the uniform plastic stage, different values of K were obtained by using different data points. However, for the necking stage, there was confusion about whether the maximum point on the graph should be used. To find the relationship between δ and ε, the equation δ=ε+c was used, but this did not give consistent results. The issue lies in the fact that the graph is for σ(ε) but the
Dell
given the following points of the stress strain graph

and knowing that the stages are defined

linear σ=Eε
uniform plastic σ=Kδ2
necking σ=αδ

where ε is an elastic strain and δ plastic strain

find the values of the constants E K n α

using the given data i plotted σ(ε) and got the following

now to find the constant E, i take the linear portion and find its incline, i get
E=33.33MPa

as for the others
for the platic deformation, can i take an of the points after (3,9) ?
for each point i chose i get a different value
K=σ/δ2
K=4e4/(13e-4)2=2.366864e10
K=4.5e4/(20e-4)2=1.125e10

is the necking portion not meant to be after a maximum point in the graph? in this graph i have no such point.

Linear, curved, linear.

okay i see what you are saying, so then i can find the curve of the last part as i did for the first part, but how about the curved portion?

as i said for each point i take i get a different value
K=σ/δ^2
K=4e4/(13e-4)^2=2.366864e10
K=4.5e4/(20e-4)^2=1.125e10

i think that the problem is the relationship between δ and ε, since the graph i have and the data given is for is σ(ε), but the constants i need are for σ(δ) what is the connection between δ and ε?
i thought that δ=ε+c (c being the permanent deformation after the linear portion of the graph,) but i solved the equations using δ=ε+c

σ=αδ
σ=Kδ^2
using the given data

but i cannot solve for K, α, every time i plug in different data i get different values,

i have been using the 1st 3 sets of data for the 1st linear section
the second 3 sets for the parabolic section
the last 2 sets for the final linear section

clearly the δ cannot be raplaced directly by ε since when the stress=0 the function MUST also be 0, and the strain will not be 0 after plastic deformation, looking at the graph we can also see that the parabolic and linear sections will not reach (0,0)
i can find the functions for these curves as a function of the strain ε but not δ which i need in order to find the constants

## 1. What is a stress-strain diagram?

A stress-strain diagram is a graphical representation of the relationship between the stress (force applied per unit area) and strain (change in length per unit length) of a material. It helps to determine the mechanical properties of a material and its ability to withstand external forces.

## 2. How do you find the constants E, K, n, and α in a stress-strain diagram?

To find the constants, you need to plot the stress-strain curve for the material and determine the slope of the linear portion of the curve. E is the slope of the elastic region, K is the slope of the yield region, n is the slope of the strain hardening region, and α is the slope of the strain softening region.

## 3. What is the significance of the constants in a stress-strain diagram?

The constants E, K, n, and α provide important information about the mechanical behavior of a material. E is the Young's modulus, which measures the stiffness of a material. K is the yield strength, which indicates the amount of stress the material can withstand without permanent deformation. n is the strain hardening exponent, which shows the rate of increase in strength with strain. α is the strain softening exponent, which represents the material's ability to deform plastically under high stress.

## 4. How do the constants vary for different materials?

The values of the constants E, K, n, and α vary for different materials depending on their composition and microstructure. For example, metals tend to have higher values of E and K compared to polymers and ceramics, which have lower values. The strain hardening exponent n also varies significantly for different materials, with some materials exhibiting a higher rate of strain hardening than others.

## 5. How does temperature affect the constants in a stress-strain diagram?

Temperature has a significant impact on the constants in a stress-strain diagram. Generally, increasing temperature leads to a decrease in E, K, and n, as well as a decrease in the strength and stiffness of the material. This is because at high temperatures, the atoms in the material have more thermal energy, making it easier for them to move and deform, resulting in lower values for the constants.

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