Stress and Strain in Strength of Materials

In this tutorial, you will learn the fundamentals of Stress and Strain in materials, including definitions, types, and the formulas used to calculate them. You will explore different stress types like tensile, compressive, and thermal stress, as well as principal stress and its applications in design. Additionally, you will understand the stress-strain relationship, the significance of Poisson’s ratio, and the behavior of mild steel under tension.

Contents:

1. What is Stress?
2. Types of Stress
3. What is Strain?
4. Types of Strain
5. Tensile Stress and Its Role in Design
6. Thermal Stress
7. Compressive Stress and Compressive Strength
8. Principal Stress
9. Relationship Between Stress and Strain
10. Poisson’s Ratio
11. Stress-Strain Relationship in Mild Steel Under Tension

What is Stress?

Stress is defined as the internal force exerted per unit area within a material when subjected to an external force. It quantifies how the material resists deformation under applied forces.

Formula for Stress: Stress(σ) = \(\frac{F}{A}\)

Where:

• F = Applied force (N),
• A = Cross-sectional area (m²).

Types of Stress

• Tensile Stress: Occurs when forces act to stretch or elongate a material. Common in components like cables or rods under a pulling force.
• Compressive Stress: Develops when forces push a material together, compressing it. Found in columns or beams under load.
• Shear Stress: Arises from forces that cause layers of the material to slide against each other. Typical in bolts, rivets, or connections experiencing a force parallel to the surface.

What is Strain?

Strain is the measure of deformation or displacement that occurs in a material as a result of applied stress. Unlike stress, strain is a dimensionless quantity, representing the ratio of change in dimension to the original dimension.

Formula for Strain: Strain(ε)= \(\frac{ΔL}{L}\)

Where:

• ΔL = Change in length (m),
• L = Original length (m).

Types of Strain

• Tensile strain: When a tensile force is applied to a body, the resulting strain is called as a tensile strain. The tensile strain results in an increase in length. Thus, tensile strain is the ratio of increase in length to the original length.
• Compressive strain: When a compressive force is applied to a body, the resulting strain is called as compressive strain. The compressive strain always results in a decrease in length. Thus, compressive strain is the ratio of decrease in length to the original length.
• Shear strain: When a shear force is applied to a body, the resulting strain is called as shear strain. As the result of the shear strain, the body deforms from its plane.
• Volumetric strain: If a force is applied to a bulk body, the body undergoes deformation. Thus the ratio of change in volume to the original volume of the body is called as volumetric strain.

Tensile Stress and Its Role in Design

When two equal and opposite forces are applied in the direction of pull, the stress-induced in the material is called tensile stress and the corresponding strain is called tensile strain.

• Due to the tensile force, the length of the bar is increased. Thus, tensile strain is given as the ratio of increase in length to the original length.
• When the load is applied within the elastic limit, the body can return to its original shape completely or partially. When loaded beyond the elastic limit, the body undergoes permanent deformation and fractures.
• The maximum tensile stress the body can withstand without permanent deformation is called as tensile strength or yield strength of the material. Tensile strength is a property of the material. When designing, the design stress is always considered to be less than the yield strength of the material.

Thermal Stress

When the deformation of a body due to a temperature change is restricted, stress is induced in the material because it resists this deformation. This stress, induced by a change in temperature, is called thermal stress, and the related strain is known as thermal strain.

• Stress from Resistance: Normally, stress in a material is induced when an external force is applied. However, stress can also be caused by temperature changes.
• Thermal Expansion and Contraction: Metals expand with heat and contract with cooling. If allowed to freely expand or contract, no stress arises. However, if this movement is restricted, thermal stress is induced.

Formula for Thermal Stress:
Thermal stress (p) can be calculated as: p = E α t
Where:

• p = Thermal stress
• E = Young’s Modulus of the material (stiffness of the material)
• α = Coefficient of thermal expansion (material-specific constant)
• t = Temperature change

Compressive Stress and Compressive Strength

• Compressive stress: When a force acts on a body in a normally inward direction, the stress induced in the body is called as compressive stress. The compressive load always acts in the direction of a push to the body. Compressive stress is normal stress that resists the decrease in length. The corresponding strain is called as compressive strain.
• Compressive stress plays a major role in all structural members. Whenever an axial force in the direction of push acts in a member, compressive stress will be induced.
• In columns, compressive stress is considered as the crushing stress and is the major stress considered for design. In bending of beams, one of the outer layers of the beam undergoes compression depending on the direction of loading.
• Compressive strength: With an increase in compressive stress, the material contracts and undergoes permanent deformation. The maximum compressive stress a material can withstand without undergoing permanent deformation is called as the compressive strength of the material.
• It is a property of the material and is a constant. While designing, the stress induced is always kept below the compressive strength of the material to prevent failure.

Principal Stress

Theoretically, stresses like compressive, tensile, and shear act along the line of action of the force. However, in many engineering applications, the induced stresses may not always act along the force’s line of action and instead act on an inclined plane. This stress on the inclined plane is called principal stress.

• Stress on an Inclined Plane: When both normal stress (compressive or tensile) and shear stress are present in a body simultaneously, the resultant stress will be neither purely normal nor purely tangential. It acts on an oblique or inclined plane.
• Principal Plane: The plane with no shear stress is called the principal plane. Principal planes are, therefore, zero-shear-stress planes carrying only normal stress.
• Principal Stress Types:
  • Maximum Principal Stress: The highest normal stress on a principal plane.
  • Minimum Principal Stress: The lowest normal stress on a principal plane.
• Calculation Methods: Analytical Method and Graphical Method are used to calculate principal stresses.
• Design Stress: Principal stress is used as the design stress in engineering applications and must be kept below the material’s strength to ensure safety.

Relationship Between Stress and Strain

The relationship between stress and strain varies across one-, two-, and three-dimensional stress systems.

• One-Dimensional Stress System: Here, stress and strain relate directly according to Hooke’s Law, which states that stress is proportional to strain within the elastic limit. Mathematically, σ = E⋅ϵ, where σ is stress, ϵ is strain, and E is Young’s modulus.
• Two-Dimensional Stress System: When stress acts in two directions, strain occurs along the applied stress and perpendicular to it. The two-dimensional strain is calculated with Poisson’s ratio (m) to account for lateral contraction.
• Three-Dimensional Stress System: In a bulk material, normal stresses act in three directions (x, y, and z), with each producing strain in all directions.

Poisson’s Ratio

Poisson’s ratio (m) defines the ratio of lateral to longitudinal strain when stress is applied, remaining constant for a given material within the elastic limit. It provides insight into the material’s deformation characteristics under loading.
\(m = – \frac{Lateral \,strain}{Longitudinal \,strain}\)

Stress-Strain Relationship in Mild Steel Under Tension

The below stress-strain diagram is obtained by applying tensile force in a UTM machine to a mild steel specimen. The ratio of stress-strain will be proportional within a certain limit, i.e., point A, by following Hooke’s law. This limit is known as the limit of proportionality.

• Hooke’s law states that within the proportional limit, stress and strain are directly proportional to each other. The constant of proportionality in Hooke’s law is E, young’s modulus.
• Any deformation that occurred to the specimen within Point B will return to its original shape and size, thus point B is the elastic limit. Point B is also known as the upper yield point and point C is the lower yield point. At any force above point B, the specimen will experience plastic deformation.
• The material undergone plastic deformation cannot return to its original shape, thus plastic deformation is also known as permanent deformation.
• If the force applied is continued, the specimen reaches the maximum yield point at D, which is the Strength of the specimen, and continues to fracture at E.
• The area under OA is the modulus of Resilience, which is the energy absorbed per unit volume within the elastic limit.
• The area under the whole curve is the modulus of toughness, which is the energy absorbed per unit volume up to the breaking point.