Unit 1 | unit 1 introduction to mechanics of solid

Fundamentals of Mechanical Engineering & Mechatronics

Unit - 1

Introduction to Mechanics of Solid

Q1) Explain the Normal and shear Stress, strain

A1) a) Normal Stress –

A force which acts normal to a surface causes stress which also acts normal to that surface. Provided that the force passes through the centroid of the surface, then the stress will be uniform over the whole surface.

This type of stress is known as a direct stress or normal stress.

b) Normal Strain –

A body subject to a direct stress will deform and be in a state of strain. The direct strain (E), which is measured in the same direction as the direct stress (cr), is given by the expression E= change in length of body (&L), original length of body (L) where both '&L and L are measured in the direction of the applied normal force (P).

a) Shear Stress –

The shear stress is defined to be the ratio of the tangential force to the cross-sectional area of the surface upon which it acts,

= F tan / A

b) Shear Strain –

The shear strain is defined to be the ratio of the horizontal displacement to the height of the block,

α = δ x / h

Q2) Explain the Hookes’ law

A2)

According to Hook’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.

Q3) Explain the Poisson’s ratio

A3) In mechanics, Poisson’s ratio is the negative of the ratio of transverse strain to lateral or axial strain. It is named after Simeon Poisson and denoted by the Greek letter ‘nu’. It is the ratio of the amount of transversal expansion to the amount of axial compression for small values of these changes.

What is Poisson’s Ratio?

Poisson’s ratio is “the ratio of transverse contraction strain to longitudinal extension strain in the direction of the stretching force.” Here,

Compressive deformation is considered negative

Tensile deformation is considered positive.

v = - d trans / d axial

v = resulting Poisson’s ratio

trans = transverse strain

axial = axial strain

Q4) What are the Elastic constants

A4) When an elastic body is subjected to stress, a proportionate amount of strain is produced. The ratio of the applied stresses to the strains generated will always be constant and is known as elastic constant. Elastic constant represents the elastic behaviour of objects.

Different elastic constants are as follows:

Young’s modulus

Bulk modulus

Rigidity modulus

Poisson’s ratio

Q5) Explain the different type of electric constant

A5) Young’s Modulus

According to Hooke’s law, when a body is subjected to tensile stress or compressive stress, the stress applied is directly proportional to the strain within the elastic limits of that body. The ratio of applied stress to the strain is constant and is known as Young’s modulus or modulus of elasticity.

Young’s modulus is denoted by letter “E”. The unit of modulus of elasticity is the same as the unit of stress which is megapascal (Mpa). 1 Mpa is equal to 1 N/mm2.

Bulk Modulus

When a body is subjected to mutually perpendicular direct stresses which are alike and equal, within its elastic limits, the ratio of direct stress to the corresponding volumetric strain is found to be constant. This ratio is called bulk modulus and is represented by letter “K”. Unit of Bulk modulus is Mpa.

Rigidity Modulus

When a body is subjected to shear stress the shape of the body gets changed, the ratio of shear stress to the corresponding shear strain is called rigidity modulus or modulus of rigidity. It is denoted by the letter’s “G” or “C” or “N”. Unit of rigidity modulus is Mpa.

Rigidity modulus (G) = Shear Stress/ Shear Strain

shear deformation

Fig: Shear Deformation of Body

4. Poisson’s Ratio

When a body is subjected to simple tensile stress within its elastic limits then there is a change in the dimensions of the body in the direction of the load as well as in the opposite direction. When these changed dimensions are divided with their original dimensions, longitudinal strain and lateral strain are obtained.

Poisson’s ration ( μ) = Lateral Strain / Longitudinal Strain

The ratio of the lateral strain to the longitudinal strain is called Poisson’s ratio. It is represented by the symbol “µ”. Poisson’s ratio is maximum for an ideal elastic incompressible material and its value is 0.5. For most of the engineering materials, Poisson’s ratio lies between 0.25 and 0.33. It has no units.

Q6) What is their relationship between electric constant?

A6) Relationship between Elastic Constants

The relationship between Young’s modulus (E), rigidity modulus (G) and Poisson’s ratio (µ) is expressed as:

E = 2G(1+μ)

The relationship between Young’s modulus (E), bulk modulus (K) and Poisson’s ratio (µ) is expressed as:

E = 3K(1-2μ)

Young’s modulus can be expressed in terms of bulk modulus (K) and rigidity modulus (G) as:

E = 9KG/ (3K + G)

Poisson’s ratio can be expressed in terms of bulk modulus (K) and rigidity modulus (G) as:

μ=(3K-2G) / (6K + 2G)

Q7) Describe the Stress-strain diagram for ductile

A7) For Ductile Metal –

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

Fig. Stress-Strain Diagram (Ductile Metal)

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.

σT = K(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

Q8) Describe the Stress-strain diagram for brittle

A8) For Brittle Metal –

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

Fig. Stress-Strain Diagram (Brittle Metal)

In figure (a), the specimen is loaded only up to point A, when 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 up to point B beyond the elastic limit E. When the specimen has gradually loaded the curve follows path BC, resulting in a residual strain OC or permanent strain.

Q9) Define terms Factor of safety

A9) Factor of safety

The ratio of ultimate to allowable load or stress is known as factor of safety i.e. The factor of safety can be denoted as the ratio of the material strength or failure stress to the allowable or working stress.

The factor of safety must be always greater than unity.

It is easier to refer to the ratio of stresses since this applies to material properties.

FOS = failure stress / working or allowable stress

Q10) Explain the Shear force and bending moment in beams

A10) Shear Force

A shear force that tends to move the left of the section upward or the right side of the section downward will be regarded as positive. Similarly, a shear force that has the tendency to move the left side of the section downward or the right side upward will be considered a negative shear force (see Figure 4.2c and Figure 4.2d).

Bending Moment

A bending moment is considered positive if it tends to cause concavity upward (sagging). If the bending moment tends to cause concavity downward (hogging), it will be considered a negative bending moment (see Figure 4.2e and Figure 4.2f).

Fig. Sign conventions for axial force, shearing force, and bending moment.

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