Strength of Materials - Metal's Property, Stress and Strain Complete Study Notes

Metal's Property, Stress, and Strain

Stress and Strain is the first topic in Strength of Materials, which consists of various types of stresses, strains, and different properties of materials that are important while working on them.

Stress: When a material is subjected to an external force, a resisting force is set up in the component. The internal resistance force per unit area acting on the material is called the stress at a point. It is a tensor quantity having a unit of N/m² or Pascal.

Force(F) is expressed in Newton (N) and A, original area, in square meters (m²), the stress σ will be expressed in N/m². This unit is called Pascal (Pa).

• As Pascal is a small quantity, multiples of this unit are used in practice.

Types of Stresses

• Normal stress

• Shear Stress

• Bulk Stress

Strain is the deformation produced in the material due to simple stress. It usually represents the displacement between particles in the body relative to a reference length.

Types of Strains

• Normal Strain: The normal strain of a body is generally expressed as the ratio of total displacement to the original length.

Since the strain is m/m, it is dimensionless.

It is of two types: Longitudinal strain and Lateral Strain

Longitudinal strain is defined as the ratio of the change in length of the body due to the deformation to its original length in the direction of the force.

Lateral Strain is defined as the ratio of the change in length (breadth of a rectangular bar or diameter of a circular bar) of the body due to the deformation to its original length (breadth of a rectangular bar or diameter of a circular bar) in the direction perpendicular to the force.

• Shear strain

Note 1: The angle is in radians, not degrees. The volume of the solid is not changed by shear strain.

• Bulk Strain or Volumetric Strain

Stress and Strain are tensor quantities, i.e., they have both magnitude and direction changes.

True Stress and True Strain

• The true stress is defined as the ratio of the load to the cross-section area at any instant.

   Where σ and ε are the engineering stress and engineering strain, respectively.

• The true strain is defined as

Lo- original length, L-successive values of the length as it changes

• The specimen volume is assumed to be constant during plastic deformation.

Stress-Strain Relationship

• The stress-strain diagram is shown in the figure. There is no appreciable change in the rate of strain in brittle materials. There is no yield point, and no necking takes place.

• In figure (a), the specimen is loaded only upto point A; when the load has gradually removed, the curve follows the same path AO and strain completely disappears. Such behavior is known as elastic behavior.
• In figure (b), the specimen is loaded upto point B beyond E's elastic limit. When the specimen is gradually loaded, the curve follows path BC, resulting in a residual OC or permanent strain.

Comparison of engineering stress and the true stress-strain curves shown below:

The true stress-strain curve is also known as the flow curve.

• True stress-strain curve gives a true indication of deformation characteristics because it is based on the instantaneous dimension of the specimen.
• In the engineering stress-strain curve, stress drops down after necking since it is based on the original area.
• In a true stress-strain curve, the stress increases after necking since the cross-sectional area of the specimen decreases rapidly after necking.
• The flow curve of many metals in the region of uniform plastic deformation can be expressed by the simple power law.

σ_T = K(ε_T)ⁿ

  Where K is the strength coefficient

•    n is the strain hardening exponent
•    n = 0 perfectly plastic solid
•    n = 1 elastic solid For most metals, 0.1< n < 0.5

Hooke's Law:

According to Hooke’s law, the stress is directly proportional to strain, i.e., normal stress (σ) ∝ , normal strain (ε)

and shearing stress ( ζ ) ∝ , shearing strain ( γ ).
σ = Eε and ζ = γG
The co-efficient E is called the modulus of elasticity, i.e., its resistance to elastic strain. The coefficient
G is called the shear modulus of elasticity or modulus of rigidity. 

Properties of Materials

Some properties of materials that judge the strength of materials are given below:

• Elasticity: Elasticity is the property by which a material is deformed under the load and is enabled to return to its original dimension when the load is removed.
• Plasticity: Plasticity is the converse of elasticity. The material in the plastic state is permanently deformed by applying load, and it does not tend to recover. The characteristic of the material by which it undergoes inelastic strains beyond those at the elastic limit is known as plasticity.
• Ductility: Ductility is the characteristic that permits a material to be drawn out longitudinally to a reduced section under the action of a tensile force (large deformation).
• Brittleness: Brittleness implies a lack of flexibility. A material is brittle when it cannot be drawn out by tension to the smaller section.

• Malleability: Malleability is a material property that permits the material to be extended in all directions without rapture. A malleable material possesses a high degree of plasticity but not great strength. Malleability is a physical property of metals that defines their ability to be hammered, pressed, or rolled into thin sheets without breaking
• Toughness: Toughness is the property of a material that enables it to absorb energy without fracture
• Hardness: Hardness is the ability of a material to resist indentation or surface abrasion. Brinell hardness test is used to check the hardness.
• Strength: The strength of the material enables it to resist fracture under load.

Engineering Stress-Strain Curve

• The stress-strain diagram is shown in the figure. The curve starts from an origin. Showing thereby that there is no initial stress or strain in the specimen.
• The stress-strain curve diagram for a ductile material like mild steel is shown in the figure below.
• Up to point A, Hooke's Law is obeyed, and stress is proportional to strain. Point A is called the limit of proportionality.

• Point B is called the elastic limit point.
• At point B, the cross-sectional area of the material starts decreasing, and the stress decreases to a lower value to point D, called the lower yield point.
• The apparent stress decreases, but the actual stress goes on increasing until the specimen breaks at point C, called the upper yield point
• From point E onwards, the strain hardening phenomena become predominant, and the strength of the material increases, thereby requiring more stress for deformation until point F is reached. Point F is called the ultimate point.

 Elongation

A colorful bar loaded in tension by an axial force P       
For a colorful bar loaded in tension by
an axial force P. The elongation of the bar
can be determined as
δ=PL/AE

Elongation of the composite body

Elongation of a bar of varying cross-section A₁, A₂,----------, A_n of lengths l₁, l₂,--------l_n respectively.

Elongation of a tapered body

Elongation of a body due to self-weight

(i) Elongation of a uniform rod of length ‘L’ due to its own weight ‘W.’

The deformation of a bar under its weight compared to that when subjected to
a direct axial load equal to its weight will help behalf.

1. ii) Total extension produced in the rod of length ‘L’ due to its own weight ‘ ω ’ per with

(iii) Elongation of a conical bar due to its self-weight

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