How To Find Stress Strain Curve:Ductility,Yield strength,Elongation,Modulus Of Elasticity

From the Hook’s Law we get a clear concept about the topic of “How to find the stress strain curve”. In this article we will briefly summarize in below the topic of how to find stress strain curve.

If in a testing object the load is applying from the external side and deformation of the testing object is measuring then we find stress strain graph very easily. From the tensile testing we were getting how to find stress stain curve. By the help of this the material’s property can estimate such as,

How to find stress strain curve
Stress-strain curve
Image Credit – Wikimedia Commons

Yield strength:

By the help of the yield strength we can recognize whether a testing object is malleable or stubborn. In the yield strength point a testing object is cease into the elastic and after that it transforms into plastic.

Read more about Hook’s law : It’s applications and 10 important facts

With the help of yield we can decide which material is suitable for the particular testing object.

Modulus of Elasticity:

If we go through the Hook’s law get the clear concept of the Modulus of Elasticity. The other name for the Modulus of the elasticity is Young’s Modulus.

The Modulus of Elasticity states that if load is applied in a testing object within the limit of elastic then the stress and strain relation is directly proportional to each other.

Mathematically it can be written as,

σ = ∈

σ = E x ∈


E = Constant of the proportionality and it is the Modulus of the elasticity.

Read more about Shear modulus: Modulus of rigidity: It’s important facts and 10+ FAQ’s


In the field of both engineering and the manufacturing the ductility is uses to define the suitability of the material for the operations in the field of manufacturing and also to understand the capacity for the materials absorb.

One of the most important mechanical term is uses in the engineering field is ductility. With this criterion we can draw amenability of a material. Ductility can be defined as the when a testing object is sustain deformation of plastic before the failure goes under the tensile stress.

Ductility is good charectertics for the metal but all metals are not ductile some are brittle in character. Polymer is also ductile material. The metals which are consider as a good ductility property is present such as copper, gold, tungsten.


Elongation of a material can be defines as the increases the length of the gauge is measured after the testing object’s fracture within the length of the gauge which is expressed its original length of the gauge in percentage.

Mathematically it can be written as,

(final length of the material – original length of the material) / original length of the material x 100

Stress strain curve ductility:

If a ductile metal testing object is situated in the compression testing machine and the external axial load is applied then the total amount of the elongation over the length of the gauge is measured in each and every increment of the axial load and process is continued until the failure of the metal testing object is takes place.

In the ductile metal testing object the area of cross sectional is known as stress and the length is known as strain. When the graph is plotted the stress is placed along the y axis and the strain is plotted along the x axis. The diagram is known as stress strain curve ductility.

In the stress strain graph various points are appear during the process. They are,

Stress strain graph ductility
Stress strain graph
Image Credit – Wikimedia Commons

Proportional limit:

Proportional limit can be defined as the region of the stress strain curve which obeys the law of the Hook’s. In this limit the strain and stress is directly proportional to each other.In the picture AB represent proportional limit.

Elastic limit:

The elastic limit for the ductile material can be defined as if the axial load is removed from the testing object then it is the point limit from where the object cannot be back to its original form or in another word it also can be explain as if the ultimate maximum stress is developed in that way into the ductile testing object where residual deformation is no longer if axial load is permanently removed from it.In the picture BC represent elastic limit.

Yield point:

Yield point can be explained as the point of the region where the ductile metals start to deform into plastic.In the picture CD represent yield point.

The yield point in categorized in two sections,

  • Upper yield point
  • Lower yield point

Ultimate strength:

Ultimate strength can be defined as the ductile metal’s faces the maximum stress beyond the failure. The failure is appear beyond this region point.In the picture DE represent ultimate strength.

Breaking point:

At which point the failure is facing is called the breaking point. In the picture E represent breaking point.

How to find ductility from stress strain curve?

Ductile can be explain as a material is absorb the total amount of the tensile stress before taking the enduring deformation. This damage is mainly often in the decreasing the amount of area of cross sectional without any fracturing.

The ductility is measure from the stress strain curve in two methods,


The length of the gauge is increases of a particular material when tensile force acted on it is divided by the length of original. Percentage of the material’s original length is elongation.

Amount of the area of the cross sectional is decreasing:

It can be expressed in mathematically is,

Amount of the area of the cross sectional is decreasing (%) = 100 * (A_0 – A_f)/A_o


A_o = Original area of the cross section

A_f = Final area of the cross section

In this testing method temperature plays a vital role.

Both the formula is expressed in percentage and denoted that the material’s ductility is performed in correct way.

Stress strain curve yield strength:

By the help of the yield strength curve we easily can understand which application is more suitable for the testing of the material. Each and every material is facing the transition in another stage of the point elasticity to plasticity and ultimate facing the breakage.

At which point the metal is start itself changing it elastic to plastic is known as Yield point.

Yield strength is at which region the stress transforms the metal elastic to plastic.

In ductile material the value of yield strength is more than the plastic.

How to find yield strength from stress strain curve?

If we go through the stress strain graph then we can observe there are lots point which are used to indicate from where the material start to transform itself  from elastic to plastic. From them yield strength is one of them. When the graph is plotted the yield strength is denoted in the stress axis means to the y axis of the graph.

  • In the graph along the y axis the elongation is plotted and in the stress axis means in the x axis the stress is plotted. A line which should be straight is drawn slope of the starting point to the stress strain graph.
  • In this time the new line is intersected by the stress strain curve which is plotted along the y axis. The value of the stress expressed in pounds per square inch. The plotting method is done by the for the purpose of subtracting the amount of elastic strain from the total amount of the strain where permanent offset is present to indicate the remainder.
Yield strength
Yield strength
Image Credit – Wikimedia Commons

Stress strain curve elongation:

Stress strain curve elongation can be defined as in a tensile testing machine a testing object is placed and axial load is applied gradually at that time where the load is maximum.

How to find elongation from stress strain curve?

  • In the tensile stress testing method the testing object is facing the elongation that time the width and thickness is decreases in the area of the cross section.
  • When we observe the elongation on the stress strain graph applied axial load is drunken in the peak point as a result the balancing is became difficult for work hardening and deformation is appear in the testing object.
  • When the axial load is in peak point the cross sectional area is reduced and curve of the stress strain graph is compressed. Diffuse neck is form in the middle part of the testing object.
Elongation from stress strain graph
Image Credit – Wikipedia Commons

Stress strain curve modulus of elasticity:

In the mechanical field the modulus of elasticity is very important factor to understand the property of material is suitable for the testing application. This not depends of the size, weight of the testing object.

In tensile testing machine the axial load is applied into the specimen the deformation is happened due to heavy load. The initial stage of the stress strain curve deformation is known as the modulus of elasticity.

Read more about Shear Strain and All important facts

How to find modulus of elasticity from stress strain curve?

In low strain the deformation of elastic is takes place. When we see the graph of stress strain the behaviour is very clear visible that strain is about less than 1 percent in a region of straight line. Its mean the elastic limit for the graph is 1 percent.

We know the formula of the Modulus of elasticity is,

E = \frac{\sigma}{\varepsilon}

So, at the beginning we need to identify the region in the strain stress curve where the deformation of elastic is happened. We already know that strain is about less than 1 means in other we can write the value of strain is 0.01. The stress for the stress strain curve is 250 Newton per square mm. Now putting the values in the formula easily can determine the value of modulus of elasticity.

Stress strain curve yield point:

In the stress strain curve the yield point particularly indicate the point where elasticity ended and plasticity is begins.

When the applied axial load the deformation is took place into the testing object but if load is removed from the yield point of testing object then the testing object can go back to its original shape.

How to find tangent modulus from stress strain curve?

  • At the starting of the process a straight line is drawn from the strain stress graph’s origin and need to find the slope present in the origin.
  • From the portion of the liner select two points and find the difference between their stress and strain point in the graph. 

How to find yield point in stress strain curve?

At first we need to find the point 0.2 % in the horizontally means in the strain axis when we mark the point then a line is draw parallels to the region of the elastic in the stress strain graph and finally 0.2 % need to point which denoted yield stress and draw a line where it is intersect in the stress strain curve.

Yield point in stress strain graph
Yield point in stress strain graph
Image Credit – Wikimedia Commons

How to find ultimate tensile strength from stress strain curve?

  • Where the axial load is failure that should by divided by the initial cross sectional area.
  • Below of the yield point the testing objects tends to change elasticity to plasticity for the deformation.
  • Broken testing object measured by the percentage elongation.
  • Calculate the reduction of the area in percentage.

Stress strain curve area under stress:

Another term for the area under stress in the stress strain graph is toughness.

In the strain stress curve the amount of energy absorb in per unit volume ability of the material before failure is known as stress strain curve area under stress.

The area under stress can be calculated integrating the strain stress curve.

Mathematically it can be express as,

\frac{Energy absorb by the material}{Volume} = \int {o}^{varepsilon_f}\sigma d\varepsilon


\varepsilon = Strain

\varepsilon_f = Strain when failure is appear

\sigma = Stress

How to find area under stress strain curve?

In the tensile testing machine the testing object is situated on it and axial load is gradually applied to the object that time stress strain graph is produce and underneath the curve stress – strain the area is easily can be measure.

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