Stress & Strain

# Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering PDF Download

 Table of contents Stress Tensile Stress Compressive Stress (σc) Strain True stress and True Strain Hook’s law Volumetric Strain Poisson’s Ratio For bi-axial stretching of sheet Elongation Elongation of composite body Factor of Safety Thermal or Temperature stress and strain Thermal stress on Brass and Mild steel combination Maximum stress and elongation due to rotation

## Stress

When a material is subjected to an external force, a resisting force is set up within the component. The internal resistance force per unit area acting on a material or intensity of the forces distributed over a given section is called the stress at a point.

• It uses original cross section area of the specimen and also known as engineering stress or conventional stress.
• Therefore, σ = P⁄A

Stress

• P is expressed in Newton (N) and A, original area, in square meters (m2), the stress σ will be expresses in N/m2. This unit is called Pascal (Pa).
• As Pascal is a small quantity, in practice, multiples of this unit is used.

1 kPa = 103 Pa = 103 N/ m2                 (kPa = Kilo Pascal)

1 MPa = 106 Pa = 106N/ m2 = 1 N/mm2            (MPa = Mega Pascal)

1 GPa = 109 Pa = 109 N/ m2                    (GPa = Giga Pascal)

Question for Stress & Strain
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What is the unit of stress?

• The resultant of the internal forces for an axially loaded member is normal to a section cut perpendicular to the member axis.
• The force intensity on the shown section is defined as the normal stress.

I

## Tensile Stress

If σ> 0 the stress is tensile. i.e. The fibres of the component tend to elongate due to the external force. The beam is subjected to an external force tensile F and tensile stress distribution due to the force is shown in the figure. Tensile Stress

## Compressive Stress (σc)

If σ< 0 the stress is compressive. i.e. The fibres of the component tend to shorten due to the external force. A member subjected to an external compressive force P and compressive stress distribution due to the force is shown in the given figure.

### Shear Stress (τ)

When forces are transmitted from one part of a body to other, the stresses developed in a plane parallel to the applied force are the shear stress. Shear stress acts parallel to plane of interest. Forces P is applied transversely to the member AB as shown. The corresponding internal forces act in the plane of section C and are called shearing forces. The corresponding average shear stress

(T) = P/Area

## Strain

The displacement per unit length (dimensionless) is known as strain.

• ### Tensile strain

The elongation per unit length as shown in the figure is known as tensile strain.

ε= ΔL/L0

It is engineering strain or conventional strain. Here we divide the elongation to original length not actual length (L0 +ΔL)

Tensile Strain

Question for Stress & Strain
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What is the definition of tensile stress?

• ### Compressive strain

If the applied force is compressive then the reduction of length per unit length is known as compressive strain. It is negative. Then εc = (-ΔL)/L0

• ### Shear strain (γ)

When a force P is applied tangentially to the element shown. Its edge displaced to dotted line. Where E is the lateral displacement of the upper face of the element relative to the lower face and L is the distance between these faces.

Shear StrainThen the shear strain is : formula

## True stress and True Strain

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

True stress

Where σ and ε is the engineering stress and engineering strain respectively.

### True strain

True strain

or engineering strain (ε) = eε-1

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

[ ∵ AoL= AL ] It is valid till the neck formation.

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

stress vs strain

• 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 engineering stress-strain curve, stress drops down after necking since it is based on the original area.
• In true stress-strain curve, the stress however 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= L(εT)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

Relations

Question for Stress & Strain
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What is compressive strain?

Question for Stress & Strain
Try yourself:
What is the unit of stress?

## Hook’s law

The co-efficient E is called the modulus of elasticity i.e. its resistance to elastic strain. The co-efficient G is called the shear modulus of elasticity or modulus of rigidity.

### Volumetric Strain

A relationship similar to that for length changes holds for three-dimensional (volume) change. For volumetric strain(εv)

the relationship is (εv) = (V-V0)/Vor (εv) = ΔV/V= P/K

• Where V is the final volume, V0 is the original volume, and ΔV is the volume change.
• Volumetric strain is a ratio of values with the same units, so it also is a dimensionless quantity.
• ΔV/V= volumetric strain = ε+ ε+ εz= ε + ε2 + ε3
•  Dilation: The hydrostatic component of the total stress contributes to deformation by changing the area (or volume, in three dimensions) of an object. Area or volume change is called dilation and is positive or negative, as the volume increases or decreases, respectively.  e = P/K ; Where P is pressure.

modulus

• For a linearly elastic, isotropic and homogeneous material, the number of elastic constants required to relate stress and strain is two. i.e. any two of the four must be known.
• If the material is non-isotropic (i.e. anisotropic), then the elastic modulii will vary with additional stresses appearing since there is a coupling between shear stresses and normal stresses for an anisotropic material.

[Intext Question]

## Poisson’s Ratio

Poisson's ratio

(Under unidirectional stress in x-direction)

• The theory of isotropic elasticity allows Poisson's ratios in the range from -1 to 1/2.
• Poisson's ratio in various materials

Ratios of various material

• We use cork in a bottle as the cork easily inserted and removed, yet it also withstand the pressure from within the bottle. Cork with a Poisson's ratio of nearly zero, is ideal in this application.

Ratio

## Elongation

• ### A prismatic bar loaded in tension

Elongation for a bar

[Intext Question]

• ## Elongation of composite body

Elongation of a composite body

[Intext question]

• ### Elongation of a tapered body

Elongation of a tapered body

• ### Elongation of a body due to its self weight

Elongation due to self weight

Structural members or machines must be designed such that the working stresses are less than the ultimate strength of the material.

Working stress

## Factor of Safety

(n) = σor σor σult σw

## Thermal or Temperature stress and strain

• When a material undergoes a change in temperature, it either elongates or contracts depending upon whether temperature is increased or decreased of the material.
• If the elongation or contraction is not restricted, i. e. free then the material does not experience any stress despite the fact that it undergoes a strain.
• The strain due to temperature change is called thermal strain and is expressed as,

ε = α(ΔT)

• Where α is co-efficient of thermal expansion, a material property, and ΔT is the change in temperature.
• The free expansion or contraction of materials, when restrained induces stress in the material and it is referred to as thermal stress.

σt = αE(ΔT)

where, E = Modulus of elasticity

• Thermal stress produces the same effect in the material similar to that of mechanical stress. A compressive stress will produce in the material with increase in temperature and the stress developed is tensile stress with decrease in temperature.

[Intext Question]

## Thermal stress on Brass and Mild steel combination

A brass rod placed within a steel tube of exactly same length. The assembly is making in such a way that elongation of the combination will be same. To calculate the stress induced in the brass rod, steel tube when the combination is raised by tοC then the following analogy have to do.

Where, δ = Expansion of the compound bar = AD in the above figure.

δst= Free expansion of the steel tube due to temperature rise tοC = αsLt = AB in the above figure.

δsf = Expansion of the steel tube due to internal force developed by the unequal expansion = BD in the above figure.

δbt = Free expansion of the brass rod due to temperature rise tοC = αbLt = AC in the above figure.

δbt= Compression of the brass rod due to internal force developed by the unequal expansion. = BD in the above figure.

And in the equilibrium equation

Tensile force in the steel tube = Compressive force in the brass rod

Where, σs= Tensile stress developed in the steel tube.

σb= Compressive stress developed in the brass rod.

As= Cross section area of the steel tube.

Ab= Cross section area of the brass rod.

## Maximum stress and elongation due to rotation

The document Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering is a part of the Mechanical Engineering Course Strength of Materials (SOM).
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## FAQs on Stress & Strain - Strength of Materials (SOM) - Mechanical Engineering

 1. What is stress and how does it relate to mechanical engineering?
Ans. Stress is a measure of the internal forces within a material that develop in response to external loads or forces applied to it. In mechanical engineering, stress plays a crucial role as it helps determine the material's ability to withstand and deform under different loading conditions.
 2. What is the difference between tensile stress and compressive stress?
Ans. Tensile stress occurs when a material is being pulled apart, causing it to elongate or deform. On the other hand, compressive stress occurs when a material is being pushed together, causing it to shorten or deform in the opposite direction. The main difference lies in the direction of the internal forces within the material.
 3. How does Hook's law relate to stress and strain?
Ans. Hook's law states that the stress in a material is directly proportional to the strain it experiences, as long as the material remains within its elastic limit. This means that the relationship between stress and strain is linear, and the material will return to its original shape once the stress is removed.
 4. What is Poisson's ratio and what does it tell us about materials?
Ans. Poisson's ratio is a measure of the ratio of lateral strain to longitudinal strain when a material is subjected to an axial stress. It provides information about the material's ability to deform in different directions. A material with a high Poisson's ratio tends to become thinner when stretched, while a material with a low Poisson's ratio tends to bulge out.
 5. How does true stress and true strain differ from conventional stress and strain measurements?
Ans. Conventional stress and strain measurements are based on the original dimensions of the material, while true stress and true strain take into account the changes in the material's dimensions as it deforms. True stress is calculated based on the actual cross-sectional area of the material at any given point, and true strain accounts for the change in length or volume of the material. This provides a more accurate representation of the material's behavior under load.

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