# Stress Strain Curve : Strength of Materials

## Why do we need Stress Strain Curve?

Parts or components in the real world have to withstand external forces during its lifetime. These external forces with defined factor of safety are considered during product design. Therefore new parts are designed in such a way that it can withstand these external forces. Stress Strain Curve represents the behavior of a material when external force is applied to it.

Product design engineers, FEA engineers use stress strain diagrams for manual calculation and simulation studies to understand the behavior of a material during product actual working conditions. In this article we will discuss “Engineering and True Stress-Strain-curve” for ductile and brittle materials.

To understand the Stress-Strain graph, let’s first understand what is Stress and Strain?

## What is Mechanical Stress and Strain?

Mechanical Stress is a measure of internal resistance exhibited by a body or material when an external force is applied to it. Mathematically mechanical stress is equal to the internal resisting force acting on a body per unit area. Strain in mechanics measures the deformation in a material when stress is applied to it. Mathematically Mechanical strain is equal to the ratio of change in length to the original length. ## How Stress-Strain Graph is Plotted?

Tensile testing on standard test specimen is done using Universal Testing Machine to plot Engineering and True Stress Strain Curve for a material. As shown in above image, UTM consists of two claws that are used to hold and pull the extreme ends of the test specimen at a uniform rate.

During tensile testing, change in the length of the test specimen with respect to applied load is recorded in various time stamps until test sample fractures. These values are used to determine  variation in stress acting on the test sample with respect to strain value.

Afterwards the stress strain graph is plotted by keeping mechanical stress values on the vertical axis and strain value on the horizontal axis.

## Stress-Strain Curve for Ductile Materials

The Stress Strain Curve for ductile material is plotted using standard test specimens on a universal testing machine. During this testing various observations are taken and plotted on the graph. We will discuss each of these important points on Stress-Strain Graph in detail:

###### Proportional Limit (From "O" to "A")

According to Hooke’s Law, Proportional limit (O-A) is the limit where stress is directly proportional to strain. Within the proportional limit, The Stress-strain curve is a straight line (from “O” to “A”). Young Modulus of Elasticity ( ratio of stress and strain) for a material is constant within proportional limit.

###### Elastic Limit (From "O" to "B")

Elastic Limit for a material is the limit beyond which the material will not come back to its original shape when the external force is removed. In the stress strain curve, from point A to B (Yield Point) material exhibits elastic properties.

If external load (stress) is increased beyond the elastic limit, the material will not come back to its original shape.

###### Upper Yield Point (Point B)

Beyond the elastic limit, a ductile material exhibits plastic properties. Upper yield point is the point where maximum stress is required to initiate plastic deformation inside the material. Strength of a material corresponding to Point B is known as yield strength.

###### Lower Yield Point (Point C)

After Point C, material length will increase with a very small increase in tensile load (stress). In other words Lower Yield Point is the point where minimum load is required to exhibit plastic behavior in the material.

###### Ultimate Tensile Strength (Point D)

Material Strength corresponding to Point D on the stress strain diagram indicates ultimate tensile strength of the material. Ultimate tensile strength of a material is the maximum stress a material can withstand before breaking. After this point necking starts inside the material.

###### Rapture / Fracture / Breaking Strength (Point E)

Point E is the point where material fracture or breaks. Stress associated with this point is known as breaking strength of a material.

##### Stress Strain Diagram comparison for Ductile Brittle and Plastic Materials

Most materials available in the market can be classified in three categories.

1. Ductile
2. Brittle
3. Plastic Materials

Each of these materials exhibit different behavior when external force is applied to them. We can understand behavior of these materials by analyzing their stress-strain curve.

###### Ductile Materials

As shown, ultimate stress point and fracture point are not the same in the stress strain diagram for ductile materials. Ductile material exhibits elastic as well as plastic deformation. Copper, aluminum, steel etc. are the examples of ductile materials.

For Example, During sheet metal bending, up to the elastic limit steel sheets regain their initial position. But after the elastic limit, material starts showing plastic behavior and does not come back to its original position. If we continue applying force beyond this limit. material will break at fracture point.

###### Brittle Materials

When an external force is applied to a Brittle material, it breaks with very small elastic deformation and without plastic deformation. For brittle materials the value of elastic limit, yield strength, ultimate tensile strength and breaking strength are the same.

In other words brittle material absorbs relatively little energy prior to fracture. For example, Brittle materials such as Pencil or glass will break suddenly with a snapping sound and little deformation. Ceramic, wood, glass, PMMA, graphite and cast iron are the examples of brittle materials.

##### Plastic Materials

Similar to ductile materials, plastic materials also exhibit elastic properties up to proportional limit. But plastic material requires very less stress compared to ductile materials to produce deformation. Plastic materials do not show any work hardening during plastic deformation.

For example, When external force is applied to bend a plastic spoon. After a certain limit, Plastic spoon will not retain its original position,

## Why Stress Strain Curve is Used?

Product design engineers, FEA engineers use stress strain diagrams for manual calculation and simulation studies to understand the behavior of a material during product actual working conditions.

For example if you are working on the design of a bracket to hold 100kg weight. To ensure the bracket will work in working conditions, maximum stress acting on the bracket is calculated and a factor of safety is added to it.

Afterwards total stresses acting on material in working conditions are compared with the stress strain curve. Simulation tools also utilizes the stress strain curve to understand behavior of a product in actual working conditions.

## Engineering Stress-Strain Curve vs True Stress Strain Curve

For engineering materials, true stress and true strain values are different from engineering stress and engineering strain values. When tensile force is applied to a test specimen. After necking, the area of the test specimen starts reducing rapidly. Engineering stress-strain diagrams do not consider this reduction in area.

Engineering Stress Strain Diagram considers initial area of the test specimen. Whereas in True Stress-Strain Diagram, test specimen actual area and change in length is considered.

From above engineering stress-strain and true stress-strain graph we can conclude following points:

• When Tensile force is applied, engineering stress is always less than the corresponding true stress. Because in engineering stress test specimen initial cross-section area is considered that is always greater than actual cross section area.
• When tensile force is applied, engineering strain is always greater than the corresponding true strain.
• Only after the necking starts, considerable change in the cross-section is observed. Therefore variation between engineering and true stresses becomes more prominent after necking.

Stress-Strain graph indicates the impact of applied external force per unit area on a material. Therefore it is used by product design engineers, FEA engineers for manual calculation and simulation studies to understand the behavior of a material during product actual working conditions.

Area under stress-strain diagram till yield point comes in elastic region.

To draw a true stress-strain curve the actual cross section area (at the time of measurement) of the test specimen is considered.

Whereas in Engineering Stress strain Diagram initial cross section area (at the start of test) is considered.

Area under the stress-strain graph till break point indicates the toughness of a material.

To sum up, Stress strain diagram for a material is required during product design for material selection and structure analysis. It helps designers in material selection and creating more optimized designs.