Stress-Strain Curve

Stress-Strain Curve is a very crucial concept in the study of material science and engineering. It describes the relationship between stress and the strain applied on an object, We know that stress is the applied force on the material, and strain, is the resulting change(deformation or elongation) in the shape of the object. For example, when force(stress) is applied to the spring, its length changes under that stress. But as stress is removed spring came to its initial position.

Stress-Strain curve provides insights into the different materials under different levels of stress. This can help engineers to design more efficient and strong structures. In this article, we will learn about, stress, strain, and the relation between them and others in detail.

Stress Definition

When forces are applied to bodies that are elastic in nature, a temporary deformation is caused in them which depends on the nature of the material. This deformation is usually not visible, but it produces a restoring force that tends to bring back the body to its natural state. The magnitude of the restoring force is equal to the force that is applied to the body. Stress has been defined as the restoring force per unit area. 

The mathematical formula for stress is given as follows:

Stress =  F/A

where,

• F is the applied force
• A is the area of the object

SI Unit of stress is given by N/m² or Pascal(Pa). The dimensional formula of stress is [ML^-1T^-2]. 

Types of stress

Stress can be classified into three categories,

• Tensile Stress
• Shearing Stress
• Hydraulic Stress

Tensile Stress

Tensile Stress is the force perpendicular to the cross-section of the material which causes the material to compress or elongate depending upon the direction of the stress applied. Some examples of Tensile Stress are the stretching of rubber bands, muscles in weight lifting experiencing tensile stress, columns in buildings and bridges being under tensile stress due to the weight on these columns, etc.

Shearing Stress

Shearing Stress is the force parallel to the cross-sectional area of the material which causes the material to deform in the shearing or sliding manner. The restoring force per unit area developed. In this case, it is called shearing stress. Cutting paper with scissors, ruptured ground during the earthquake, friction-based break mechanism, etc. are examples of shearing stress.

Hydraulic Stress

Hydraulic stress is the force applied to the complete surface of the material which causes the material to inflate or compress under the stress. Various hydraulic systems such as breaks, jacks, etc, and the force experienced by objects in very deep seas are examples of hydraulic Stress.

Different types of stress applied to an object and their effect is shown in the image below,

Strain Definition

Whenever forces are applied such that they cause stress in the material. These forces bring changes in the dimension of the object. Strain is the ratio of change in dimension to the original dimension.  For example, if a cylinder is kept under some stress and causes it to deform accordingly, then the ratio of change in the dimension of the cylinder is whether it is along the axis or parallel to the axis to its original dimension here is strain.

Stain can be classified based upon the acting up stress in three types, such as:

• Tensile Strain
• Shearing Strain
• Hydraulic Strain

Tensile Strain

In the case of compressive or tensile stress, the length of the cylinder is changed. Let ΔL be the change in length of the cylinder and L be the original length. This is called longitudinal strain. It is given by, 

Longitudinal Strain = ΔL/L

Shearing Strain

In the case of shearing stress, the object deforms in a shearing or a sliding which can be measured in the form of an angle from the original dimension. Thus Shearing Strain is given as follows:

Shearing Strain = Δθ/θ

Here, θ is the angular displacement of the cylinder from its mean position

Hydraulic Strain

When hydraulic stress is applied, the body changes its volume. In this case, volumetric strain is used and is given by:

Volumetric Strain = -ΔV/V

Hooke’s Law

Stress and Strain take different forms based on the way forces are applied to the body. In the case where the deformation is small, Hooke’s law is applicable. Hooke’s law is based on empirical evidence and is valid for almost all materials. However, this law is only applicable to small displacements. 

According to Hooke’s law, “For small deformations, the stress and the strain produced in the body are directly proportional to each other.”

Stress ∝ Strain 

Stress = k × Strain 

Here, k is the proportionality constant and is called the Modulus of Elasticity(Younge Modulas).

What is Stress-Strain Curve?

Relationships between stress and strain can be plotted on a graph for most of the materials. In this experiment, the force is gradually increased, and it produces the strain. The values of the stress and the strain are plotted on a graph. This graph is called the stress-strain curve. These curves vary from material to material and are very helpful in giving a fair idea of how the material performs in different load conditions. 

In the graph, it can be seen that from O to A the graph is almost a straight line. It is the only region in this curve where Hooke’s Law is obeyed. 

Proportionality Limit

As after the point A or OA region, the graph doesn’t obey the Proportionality law or Hooke’s law, thus Point A is called the proportionality limit. 

Elastic Region

The initial region of the graph which is represented by graph OA is the Elastic Region. In this region, the material undergoes deformation under the applied street but returns to its initial state as stress is removed. In this region, Hooke’s Law is obeyed.

Elastic modulus

The slope of the Stress-Strain Curve in the elastic region is called the Elastic Modulus and this modulus represents the stiffness of the material. It is also called Younge’s Modules.

Yield point

The point in the Stress-Strain curve, from where the material started to deform plastically and can’t fully regain its initial state after stress is removed. In other words, the yield point is defined as the stress at which material starts to exhibit plastic deformation by some certain amount.

Yield strength

The required amount of stress to deform the given material 0.2-0.5% plastically, is called the yield strength of the material.

Ultimate Tensile Strength (UTS)

Ultimate Tensile Strength is the maximum amount of stress a material can handle before it breaks or fractures. It is a measure of the toughness of the material and generally measure in Pounds Per Square Inches (PSI).

Plastic Region

Plastic Region is part of the Stress-Strain curve, where the material undergoes plastic deformation permanently and can’t able to attain its initial state after the stress is removed.

Strain hardening modulus

The slope of the Stress-Strain Curve in the plastic region is called Strain hardening modulus and this modulus represents the ability of the material to resist further deformation.

Fracture point

The point in the Stress-Strain Curve, where material breaks in the experiment, is called the Fracture Point, and the stress or the force at this point is called Fracture strength/

Classification of Materials

Materials can be classified into two categories based on the Stress-Strain curve,

• Brittle Materials
• Ductile Materials

Brittle Materials

Brittle Materials are those materials, which can fracture without any warning or plastic deformation. Glass, Ceramic, Cast iron, Concrete, and some types of Plastics are examples of brittle materials.

Ductile Materials

Ductile materials are those materials, which can undergo a large amount of plastic deformation such as stretching, bending, or compressing, without breaking apart. Ductile materials can be formed into any shape without losing their structural integrity. Metals, Polymers, Rubber, and Composite materials are examples of Ductile Materials.

Read More,

• Stress, Strain and Elastic Potential Energy
• Difference Between Stress and Strain
• Difference Between Stress and Pressure

Sample Problems on Stress-Strain Curve

Problem 1: A steel rod of 1 m increases by a length of 10cm when tensile stress is applied. Find the longitudinal strain.

Answer: 

Longitudinal strain is given by the ratio of change in length with the total original length. 

Let the original length be L, and the change in length be ΔL

Longitudinal Strain = ΔL/L

Given:

• ΔL = 0.1 m
• L = 1 m 

Plugging the values into the equation, 

Longitudinal Strain =  ΔL/L

⇒ Longitudinal Strain = 0.1/1

⇒ Longitudinal Strain = 0.1 

Problem 2: A steel ball of radius 1.5m shrinks in size to a length of 1.4m when hydraulic stress is applied. Find the volumetric strain.

Answer: 

Volumetric Strain is given by, 

Volumetric Strain = -ΔV/V

Volume of a Sphere is given by, 

V = 4/3πr³

• Initial Radius: r_i = 1.5m 
• Final Radius: r_f = 1.4m 

Therefore, Change in Volume = 4/3π(r_i³ – r_f³)

⇒ Change in Volume = 4/3π[(1.5)³ – (1.4)³]

⇒ Original Volume = 4/3πr³

⇒ Original Volume = 4/3π(1.5)³

Thus, 

Volumetric Strain = Change in Volume/Original Volume

⇒ Volumetric Strain = 4/3π(r_i³ – r_f³)/4/3πr_i³

⇒ Volumetric Strain = 4/3π[(1.5)³ – (1.4)³]/4/3π(1.5)³

⇒ Volumetric Strain = [(1.5)³ – (1.4)³]/(1.5)³

⇒ Volumetric Strain = 0.631/3.375

⇒ Volumetric Strain = 0.18

Problem 3: A cube of side 1 m shrinks in size to a length of 0.5 m when hydraulic stress is applied. Find the volumetric strain.

Answer: 

Volumetric Strain is given by,

Volumetric Strain = -ΔV/V

Volume of a sphere is given by, A

V = a³

• Initial radius: a_i = 1 m 
• Final radius: a_f = 0.5m 

Change in Volume = a_f³ – a_i³

⇒ Change in Volume = 1³ – (0.5)³

⇒ Change in Volume = 0.875

Original Volume = a³

⇒ Original Volume = 1

Thus, 

Volumetric Strain =  Change in Volume/Original Volume

⇒ Volumetric Strain = a_f³ – a_i³

⇒ Volumetric Strain = 0.875/1

⇒ Volumetric Strain =0.875

Problem 4: A cube of side 2 m shrinks in size to a length of 0.5 m when the compressive force of 500N is applied. Find the compressive stress.

Answer: 

Stress is given by, 

Stress =  F/A

In this case, 

• F = 500 N
• A = side^2. 

Side is given as 2 m 

A = side² 

⇒ A = 2²

⇒ A = 4

Stress = F/A

⇒ Stress = 500/4

⇒ Stress = 125 N/m²

Problem 5: The axis of a cylindrical rod moves by 30° when a force is applied horizontally. The length of the cylinder is 0.5m.  Find the shearing strain and the displacement of the cylinder from its mean position. 

Answer: 

Shearing strain is given by, 

Shearing Strain = Δθ/θ

Here,

• θ = 30^o
• L = 0.5 m 

⇒ Shearing Strain = tan θ

⇒ Shearing Strain = tan(30°)

⇒ Shearing Strain = 1/√3

Let Displacement be x,

Shearing Strain = x/L

x/lL = (1/√3)

x = L/√3

x = 0.5/√3

x = 0.289 m

FAQs on Stress-Strain Curve

Q1: What is a Stress-Strain Graph?

Answer:

A stress-strain curve is a graph between stress and strain for any material, which give us insight into its strength, elasticity, and ductility.

Q2: What is Stress?

Answer:

Stress is defined as force per unit area that a material experiences when it is subjected to any external force. It’s unit is newtons per square meter (N/M²) in SI system and pounds per square inch (psi) in the CGS system.

Q3: What is a Stress-Strain Curve?

Answer:

A stress-strain curve is a graph that illustrates the relationship between stress and strain in a material. It is used to characterize the mechanical properties of a material, such as its strength, elasticity, and ductility.

Q4: What is Strain?

Answer:

Strain is the deformation that a material undergoes when stress is applied. It is expressed as the ratio of change in dimension to the original dimension.

Q5: What is Elastic Deformation?

Answer:

Elastic deformation is a temporary deformation that a material undergoes when it is subjected to stress, but material returns to its original shape and size when the stress is released. In Stress-Strain curve this deformation is shown by the linear region.

Q6: What is Plastic Deformation?

Answer:

Plastic deformation is a permanent deformation that a material undergoes when it is subjected to a load or force beyond its elastic limit. This behavior is characterized by a nonlinear region on a stress-strain curve.

Q7: What is Yield Strength?

Answer:

Yield strength is the point on a stress-strain curve where a material begins to undergo plastic deformation. It is often used as a measure of a material’s strength and is typically reported in units of force per unit area.

Q8: What is Ultimate Tensile Strength?

Answer:

Ultimate tensile strength is the maximum stress that a material can withstand before it fails or breaks. It is typically reported in units of force per unit area and is often used as a measure of a material’s strength.