# Stress and Strain

## Definition & Explanation

Stress and strain are two fundamental concepts in materials science and engineering used to describe the behavior of a material in response to external forces acting on it.

### What is Stress?

**Stress**, denoted by the letter (σ) “sigma,” is defined as the distribution of an applied force over the cross-sectional area of a material. There are several types of stress, including tensile stress (stretching), compressive stress (squeezing), and shear stress (sliding or twisting).

For instance, when you pull a material from both sides, internal forces will develop within the material, causing it to stretch. The distribution of those internal forces over the cross-sectional area is what we call stress.

Stress is expressed in units of force per unit area, such as pounds per square inch (psi) or newtons per square meter (Pascal). The magnitude of stress depends on the applied force and the cross-sectional area of the material.

### What is Strain?

**Strain**, denoted by the letter (ε) “epsilon,” is defined as the relative deformation or change in shape that occurs in a material in response to an external force or load. It is a measure of the degree to which a material has changed shape due to stress. There are several types of strain, including axial strain (stretching or compressing in one direction), shear strain (sliding or twisting), and volumetric strain (change in volume).

Strain is a unitless value as it’s the ratio between two similar quantities.

The relationship between stress and strain is important in materials science and engineering because it can help predict how a material will behave under certain conditions. By understanding the stress-strain relationship, engineers and scientists can design materials and structures that can withstand specific loads or forces without failing.

## Types of Stress & Strain

There are two main types of mechanical stress that a material can experience depending on the type of force acting on it: Normal stress and Shear stress.

### 1) Normal stress

**Normal stress** is the result of a force acting perpendicular to the cross-sectional area of a material, and it can cause different types of strains in the material depending on its properties and the direction of the stress. The types of normal stress and their resulting strains are:

**Tensile stress**is a type of stress that occurs when a material is subjected to a pulling or stretching force in opposite directions. It is a normal stress that acts perpendicular to a cross-sectional area of the material, and it is defined as the force per unit area that is required to produce the elongation or deformation of the material.When a material is subjected to tensile stress, it experiences a positive deformation called

**tensile strain**, which causes the material to elongate along the direction of the applied force. The tensile strain is defined as the change in length divided by the original length of the material.**Compressive Stress**is a type of stress that occurs when a material is subjected to a compressive or squeezing force in opposite directions. It is a normal stress that acts perpendicular to a cross-sectional area of the material, and it is defined as the force per unit area that is required to produce the compression or shortening of the material.When a material is subjected to compressive stress, it experiences a negative deformation called

**compressive strain**, which causes the material to shorten along the direction of the applied force. Same as the tensile strain, the compressive strain is also defined as the change in length divided by the original length of the material, however in the case of compression the change in length is negative.**Bulk stress**, also known as**volume stress**, is a type of stress that occurs when a material is subjected to a change in volume due to an external force or pressure. It is a normal stress that acts uniformly in all directions, and it is defined as the force per unit area that is required to produce the change in volume of the material.When a material is subjected to bulk stress, it experiences a volumetric deformation called

**bulk strain**, which causes the material to change in volume. The bulk strain is defined as the change in volume divided by the original volume of the material.

### 2) Shear Stress

**Shear stress** is the result of two opposite forces acting parallel to the cross-sectional area of a material. For example, suppose you hold a book tightly between the palms of your hands, then with one hand, you press and pull on the front cover away from you, while with the other hand you press and pull on the back cover toward you. In such a case, when deforming forces act tangentially to the object’s surface, we call them shear forces and the stress they cause is called shear stress.

When a material is subjected to shear stress, it experiences a deformation called **shear strain**, which is a measure of the distortion or change in shape of the material caused by the sliding of one layer of the material over another layer. The shear strain is defined as the angular deformation caused by the shear stress.

## Stress-Strain Curve

A stress-strain curve describes the relationship between stress and strain for a given material. It’s obtained by applying an incrementing load (usually a tensile load) to a material and recording the resulting deformation at each increment, from which the strain can be calculated. It contains a couple of key points that describe several properties of a material, such as the elastic limit, yield strength, and ultimate strength.

**The Key Points of Stress-Strain Curve:**

**Proportional Limit**indicates the maximum stress at which stress and strain are directly proportional, and Hook’s law is applicable.**Elastic Limit**indicates the maximum stress that a material can withstand before plastic (permanent) deformation occurs**Yield Point**is the point at which there is a large increase in strain with little or no increase in stress (typically taken as 0.2% strain).**Yield Strength**is the stress corresponding to the yield point on the stress-strain curve.**Ultimate Strength**indicates the maximum stress a material can withstand.**Fracture Point**indicates the stress at which a material fails or fractures.**Elastic Region**is the region where a material behaves elastically, meaning it will regain its original shape and size once the applied load is removed.**Plastic Region**is the region where a material behaves plastically, in other words, the deformation is permanent even after the applied load is removed.**Strain Hardening**occurs after the yield point in which a material’s strength increases and it becomes harder to deform. Strain hardening is the reason that the stress-strain curve increases past the yield stress.**Necking**occurs when the material’s cross-sectional area starts to decrease. At first, as the material elongates, the cross-sectional area will uniformly reduce. At some point, one section of the area becomes just a bit thinner than the rest. This concentrates the stress, essentially creating a new tensile test with a smaller gauge diameter.

### True vs Engineering Stress-Strain Curve

The engineering stress-strain curve (the one we saw earlier) is actually an approximation of the true stress and strain obtained during a stress-strain test. The true stress and strain of a material under loading are plotted on what we call the true stress-strain curve. You can easily differentiate between the two by looking at the curve during necking, the engineering curve starts to decrease when necking occurs, whereas the true curve keeps increasing.

For instance, during a tensile test, the engineering stress and strain are taken under the assumption that the length and cross-sectional area of the sample remain constant throughout the entire test. In this case, the engineering stress is taken as the applied load divided by the original cross-sectional area of the sample.

$\text{EngineeringStress}(\mathit{\sigma})=\frac{F}{{A}_{o}}$Where:

- σ is the engineering stress
- F is the applied force
- A
_{o}is the original area

And the engineering strain is calculated as the relative change in length to the original length of the sample.

$\text{EngineeringStrain}(\mathit{\epsilon})=\frac{L-{L}_{o}}{{L}_{o}}=\frac{\Delta L}{{L}_{o}}$Where:

- ε is the engineering strain
- L
_{o}is the initial length - L is the final length
- ΔL is the change in length

On the other hand, the true stress and strain correctly account for the change in cross-sectional area and the change in length during the test.

True stress is taken as the applied load divided by the cross-sectional area at that instant of the sample.

$\text{TrueStress}({\mathit{\sigma}}_{true})=\frac{F}{{A}_{i}}$Where:

- σ
_{true}is the true stress - F is the applied force
- A
_{i}is the instantaneous area

True strain is taking as the natural logarithm of the ratio of the instantaneous length to the original length of the sample.

$\text{TrueStrain}({\mathit{\epsilon}}_{true})=\mathrm{ln}\left(\frac{{L}_{i}}{{L}_{o}}\right)$Where:

- ε
_{true}is the true strain - L
_{i}is the instantaneous length - L
_{o}is the original length

### Frequently Asked Questions

- What is the relation between stress and strain?
- Up to The elastic limit, the relation between stress and strain is proportional, and it’s described by Hooke’s law which states that the strain in a solid is directly proportional to the applied stress within the elastic limit.

- When to use true vs engineering stress-strain curve?
- The engineering stress-strain curve is used for performance applications, whereas the true stress-strain curve is used for material property analysis.

- What is stress and strain curve?
- The stress-strain curve describes the relationship between stress and strain for a given material, and it’s obtained by gradually increasing the applied load to a material and measuring the resulting strain.

- What is elastic limit?
- The elastic limit is the point on the stress-strain curve that indicates the maximum stress a material can withstand without deforming permanently.

- What are elastic moduli?
- An elastic modulus is the proportionality constant of the linear region on the stress-strain curve when a material is under a specific type of stress. For example, when a material is under longitudinal stress (tension or compression), the proportionality constant in this case is Young’s modulus.

**References:**

William Moebs. “University Physics Volume 1 - Stress, Strain, and Elastic Modulus.” OpenStax, 2016. September 19. https://openstax.org/books/university-physics-volume-1/pages/12-3-stress-strain-and-elastic-modulus.