2. Simple Stresses and Strains, CE 3003 notes in English, Mechanics of Materials notes in English

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2. Simple Stresses and Strains

2.1 Definition of Rigid, Elastic, and Plastic Bodies

• Rigid Body: A body that does not deform under any applied load, meaning the distance between particles does not change.
• Elastic Body: A body that deforms under load but returns to its original shape once the load is removed. Elastic bodies obey Hooke's Law in the elastic limit.
• Plastic Body: A body that permanently deforms when the applied load exceeds a certain limit (the elastic limit). Once the load is removed, the body does not return to its original shape.

2.2 Deformation of Elastic Body Under Various Forces

When an elastic body is subjected to external forces, it experiences deformation. This deformation can be in the form of:

• Tensile Deformation: The body stretches under pulling forces.
• Compressive Deformation: The body compresses under pushing forces.
• Shear Deformation: The body experiences a change in shape due to forces applied parallel to its surface.

The degree of deformation depends on the magnitude of the applied force and the material's properties.

2.3 Definitions of Key Terms

2.3.1 Stress

Stress is the internal resistance offered by a material to deformation due to external forces. It is defined as the force per unit area and is given by:

$$\text{Stress} (\sigma) = \frac{F}{A}$$

where:

• $F$ is the applied force,
• $A$ is the cross-sectional area.

Stress has units of N/m² (Pascal, Pa).

2.3.2 Strain

Strain is the measure of the deformation of a material due to an applied force. It is the ratio of change in length to the original length:

$$\text{Strain} (\epsilon) = \frac{\Delta L}{L_0}$$

where:

• $\Delta L$ is the change in length,
• $L_0$ is the original length.

Strain is a dimensionless quantity (no units).

2.3.3 Elasticity

Elasticity refers to the property of a material to return to its original shape and size after the removal of the applied force, as long as the material stays within its elastic limit.

2.3.4 Hooke's Law

Hooke's Law states that the stress is directly proportional to the strain for materials that behave elastically, within their elastic limit. Mathematically:

$$\sigma = E \epsilon$$

where:

• $\sigma$ is the stress,
• $E$ is the modulus of elasticity (Young's Modulus),
• $\epsilon$ is the strain.

This law is valid only for elastic deformation.

2.3.5 Elastic Limit

The elastic limit is the maximum stress a material can withstand without permanent deformation. Beyond this point, the material enters the plastic region where it will not return to its original shape.

2.3.6 Modulus of Elasticity

The modulus of elasticity (also known as Young's Modulus, $E$) is a material constant that measures the stiffness of a material. It is the ratio of stress to strain in the elastic region:

$$E = \frac{\sigma}{\epsilon}$$

A higher modulus indicates a stiffer material.

2.4 Types of Stresses

• Normal Stress: Stress that acts perpendicular to the surface (either tensile or compressive).
• Direct Stress: Stress that acts directly on the material in a direction parallel to the applied force (same as normal stress).
• Bending Stress: Stress that occurs due to bending of a beam or structure.
• Shear Stress: Stress that acts parallel to the surface and causes deformation in the form of shearing.

2.5 Nature of Stresses (Tensile and Compressive Stresses)

• Tensile Stress: When a force stretches or pulls a material, it experiences tensile stress.
• Compressive Stress: When a force compresses or pushes a material, it experiences compressive stress.

Tensile stress causes elongation, while compressive stress causes shortening.

2.6 Standard Stress-Strain Curve for a Steel Bar under Tension

The stress-strain curve for steel (or most ductile materials) under tension shows how the material deforms as the stress increases.

• Proportional Limit: The point up to which stress and strain are proportional (Hooke's Law region).
• Elastic Limit: The maximum stress where the material can still return to its original shape.
• Yield Stress (Yield Point): The point at which the material starts to undergo permanent deformation.
• Proof Stress: A stress value taken at a small amount of permanent strain (e.g., 0.2%).
• Ultimate Stress (Tensile Strength): The maximum stress the material can withstand before necking (failure begins).
• Fracture/Breaking Stress: The stress at the point of material failure.
• Strain at Critical Points: The amount of strain associated with the above points.
• Percentage Elongation: The increase in length (after failure) expressed as a percentage of the original length.
• Factor of Safety (FoS): The ratio of the material's ultimate strength to the allowable working stress.

2.7 Deformation of Body Due to Axial Force

When an axial force is applied to a body, it causes a uniform elongation or compression (depending on the force type). The deformation depends on the material’s Young's Modulus and the applied force.

Maximum and Minimum Stress:

• The maximum stress occurs at the point of application of the axial force.
• The minimum stress occurs at points away from the force.

2.8 Composite Section Under Axial Loading

A composite section consists of two or more materials with different moduli of elasticity. When subjected to axial loading, the deformation of the composite section is the result of the deformation of individual components, calculated using the strain compatibility condition.

2.9 Concept of Temperature Stresses and Strains

Temperature-induced stress occurs when a material is subjected to a temperature change, causing it to expand or contract. If the material is constrained (unable to freely expand or contract), internal stresses develop.

The temperature strain is given by:

$$\epsilon_T = \alpha \Delta T$$

where:

• $\alpha$ is the coefficient of thermal expansion,
• $\Delta T$ is the temperature change.

The stress due to temperature change is:

$$\sigma_T = E \cdot \epsilon_T$$

where:

• $E$ is the modulus of elasticity.

2.10 Longitudinal and Lateral Strain

• Longitudinal Strain: The strain in the direction of the applied force (e.g., in tension or compression).
• Lateral Strain: The strain perpendicular to the direction of the applied force. This is influenced by Poisson's Ratio.

2.11 Modulus of Rigidity, Poisson’s Ratio, Biaxial and Triaxial Stresses, Volumetric Strain, Bulk Modulus (Introduction Only)

• Modulus of Rigidity (Shear Modulus) ($G$): The ratio of shear stress to shear strain.

$$G = \frac{\text{Shear Stress}}{\text{Shear Strain}}$$

• Poisson’s Ratio ($\nu$): The ratio of lateral strain to longitudinal strain in a material under axial loading. It is usually positive, meaning when a material is stretched, it gets thinner in the perpendicular direction.

$$\nu = -\frac{\text{Lateral Strain}}{\text{Longitudinal Strain}}$$

• Biaxial Stress: Stresses acting in two perpendicular directions, such as in thin-walled cylinders under internal pressure.
• Triaxial Stress: Stresses acting in three perpendicular directions.
• Volumetric Strain: The change in volume per unit volume, usually resulting from changes in pressure.
• Bulk Modulus (Introduction Only): Describes the material’s resistance to uniform compression. It relates to the relationship between pressure and volume change.

2.12 Relation Between Modulus of Elasticity, Modulus of Rigidity, and Bulk Modulus

The relationship between these moduli is given by:

$$\frac{1}{E} = \frac{1}{3K} + \frac{2}{3G}$$

where:

• $E$ is the modulus of elasticity,
• $K$ is the bulk modulus,
• $G$ is the modulus of rigidity.

This equation is useful when you need to calculate one modulus if the others are known.