# Mechanical Metallurgy Questions and Answers – Elastic Behaviour – Hydrostatic & Deviator Component of Stress – 1

This set of Mechanical Metallurgy Multiple Choice Questions & Answers (MCQs) focuses on “Elastic Behaviour – Hydrostatic & Deviator Component of Stress – 1”.

1. Matrix (2-dimensional arrays) is a second rank tensor.
a) True
b) False

Explanation: Vector is the first rank tensor because it requires only 3 components for their specification. A scalar number is zero rank tensor because it needs only one component to define it. In the case of stress and strain, it requires nine components to define, which form a square matrix. If the number of components required to define a quantity is equal to 3n, then n is the rank of the tensor.

2. The number of components required to define an elastic constant is ______
a) 3
b) 9
c) 27
d) 81

Explanation: Elastic constant is fourth rank tensor.
According to Hooke’s law:
Stress= elastic constant* strain
In the tensor form, it will be σij=Cijkl* εkl: where σij is stress acting in i face in j direction; similarly strain in k face acting in l direction. So elastic constant is made of a combination of 2 matrices. The final matrix of order 9×9 is with 81 components.

3. For the following stress matrix, the first invariant of the stress tensor is the trace of a matrix, the sum of diagonal element: I1 = σ11 + σ22 + σ33. The second invariant of matrix is equal to _________
$$\begin{vmatrix}\sigma_{11} & \sigma_{12} & \sigma_{13}\\ \sigma_{21}& \sigma_{22} & \sigma_{23}\\ \sigma_{31}& \sigma_{32} & \sigma_{33}\end{vmatrix}$$
a) Sum of all the elements of the matrix
b) Sum of first column elements
c) Sum of first raw elements
d) Sum of all the principal minors of the matrix

Explanation: The Second invariant of the stress tensor is given by:
$$\begin{vmatrix}\sigma_{11} & \sigma_{13}\\ \sigma_{31} & \sigma_{33}\end{vmatrix} + \begin{vmatrix}\sigma_{11} & \sigma_{12}\\ \sigma_{21} & \sigma_{22}\end{vmatrix} + \begin{vmatrix}\sigma_{22} & \sigma_{23}\\ \sigma_{32} & \sigma_{33}\end{vmatrix}$$
This is the sum of all the principal minor of the matrix.

4. Dilation of the mechanical body is ____________
a) change in the length of the body
b) change in the area of the body
c) change in the volume of the body
d) change in the width of the body

Explanation: Applying the stress on a body causes the distortion in the material at the micro and macro level. This change in the shape of the body is known as dilation.
Mathematically it is defined as the = change in volume/original volume.

5. Dilation on parallelepiped with strains in x, y, z direction equals to εx, εy, εz respectively:
The total volume change is given as Δ = εx + εy + εz for the small strain. The mean stress or hydrostatic (spherical) component of stress will be equal to ____
a) Δ
b) Δ/2
c) Δ/3
d) Δ/4

Explanation: Volume strain (dilation) is equal to the first invariant of strain matrix i.e. εx + εy + εz. So, the mean strain is average of all three strains in all 3 direction, which is given as εm=(εx + εy + εz)/3 = Δ/3.
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6. The part of strain tensor which involves in the shape change rather than the volume change is called strain deviator ε’ij.
a) True
b) False

Explanation: Volume strain is responsible for volume change of material, whereas the shape change of material along any certain direction is given by deviator strain. The strain deviator is obtained by simply subtracting the εm from the diagonal terms of the strain tensor.

7. Experimentally, strain in the material with high precision is measured by _________ method.
a) calorimetry
b) hardness testing
c) bonded wire resistance gauge
d) microscopy

Explanation: Bonded wire resistance gauge or the SR-4 strain gauge is used for strain measurement. It measures the change in electrical resistivity of wire attached to the material with a change in the dimension of material.

8. Calculate the hydrostatic strain, if εx, εy, εz are 0.01, 0.2, 0.05 respectively?
a) 0.2
b) 0.866
c) 0.125
d) 1

Explanation: Hydrostatic strain is given as εm=(εx + εy + εz)/3
=> [0.01+0.2+0.05]/3
=> 0.866.

9. Calculate the dilation of body, given that the initial volume is 250 mm2 and final volume is 235.5 mm2.
a) 0.058
b) 0.061
c) 0.050
d) 0.060

Explanation: dilation is defined as change in volume/original volume
change in volume=250-235.5=14.5 mm2
=> 14.5/250 = 0.058.

10. What does σij represent in a 3×3 stress matrix?
a) Stress is applied on i face of the material in the +ve j direction
b) Stress is applied on i face of the material in the -ve j direction
c) Stress is applied on j face of the material in the +ve i direction
d) Stress is applied on j face of the material in the -ve i direction

Explanation: For a 3-dimensional material, stress is always defined by nine components. Each component represents the face on which the stress is applied along with the direction in which the stress is applied. Standard notation σij always represents stress is applied on i face on the material in -ve j direction.

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