Analysis of Stress, Strain, and Elastic Properties of Materials

This document presents the solutions to a mechanical engineering assignment. The first part of the solution, Task 3, involves calculating strains in x, y, and z directions for a component subjected to tensile and compressive stresses, considering the modulus of elasticity and Poisson's ratio. The solution provides the strain equations and calculates the change in dimensions. The second part, Task 4, focuses on a cube solid object subjected to external pressure, determining the bulk modulus, volumetric strain, and change in volume using the given modulus of elasticity and Poisson's ratio. The solutions demonstrate the relationships between elastic constants and provides detailed calculations. The document also includes references to external resources used for the solutions.

SOLUTIONS OF TASKS 3 AND 4
TASK 3) A component is subjected to a tensile stress in the x-direction of 100MPa, a
compressive stress in the y direction of 120MPa and tensile stress in the z-direction of 85MPa. If
modulus of elasticity is 205GPa and a poisson ratio of 0.34, the following can be determined:
(a) Equation of strains in the x, y and z directions when loaded in the 3 directions:
Firstly, normally in simple stresses calculations,
Modulus of elasticity is given by:
E= ϭ/έ
And poison’s ratio: v= έx/ έy
Now, in three dimensions where stress occurs in all the three directions:
Strain will be due to the stress in that particular direction plus the strain contributed by
stresses in the other two directions (coursehero.com, 2018). Hence the following are
strain equations for x, y and z directions:
έx= 1/E{ϭx –v(ϭy+ ϭz)}…(1)
έy= 1/E{ϭy –v(ϭx+ϭz)}…(2)
έz= 1/E{ϭz –v(ϭy+ ϭx)}…(3)
(b) Calculate 1-strain in x direction
Given ϭx= 100MPa, ϭy=-120MPa, ϭz=85MPa , E= 205GPa, v=0.34
Substituting in eqn 1 above: έx= 106/205x109{100 –0.34(120+ 85)}
=111.9/205 000= 0.0005458536
2-strain in the y direction
Substituting in eqn 2 above: έy= 106/205x109{-120 –0.34(85+ 100)}
=-182.9/205 000= -0.000892195

3-Strain in the z direction
Substituting in eqn 3 above: έz= 106/205x109{85–0.34(100-120)}
= 91.8/205 000= 0.0004478048
(c) Now, considering the compressive load in the y direction deforming the material by
squeezing it. If the original length was 3m, and the width is 40mm and the depth is 20mm
Calculating the change in dimension in the x, y and z directions
In ten x direction, the original length is 3m,
Change in length l’ = έx xlo= 3x0.0005458536= 0.00163775608m
In the y direction, the original dimension ho= 20mm
h’= έx xho= -0.000892195x20= -0.0178439mm
In z-direction, the original width, wo= 40mm
Hence w’ = 40 x 0.0004478048 = 0.017912192mm
TASK 4) In this question, consider a cube solid object of volume Vo= 9000mm3subjected to an
external pressure of 110MPa. The modulus of elasticity is 80GPa and Poissons ratio is 0.25
The following are determined:
(a) The four elastic constants
 Bulk modulus K
 Modulus of elasticity E
 Modulus of rigidity G
 Poisons Ratio
(b) Relationship of the four constants; K, G, E and v
Generally, Bulk modulus is given by K= ϭ/έv

Suppose now we consider a material being compressed by a pressure P hence the acting
pressure is –P
The resulting bulk modulus is K= -P/ έv
But έv = 3 έ…(4)
And suppose that ϭx= ϭy= ϭz
Combining equations 1, 2, 3 and 4 we obtain:
έ= 1/E(-p+v2p)….(5a)
έv= 3/E(-p+v2p)…(5b)
k= -p/3/E(-p+v2p) = E/3(1-2v) … (5c)
(c) (i) Bulk modulus K
Substituting in equation 5c above:
k= 80x 109/3(1-2x0.25) = 80/1.5x109= 53.33GPa
(ii) Volumetric strain
έv= 3x106/80x109(-110+2x0.25x110)
= 3/80x103(-55) = -165/80x 103= -2.0625x10-3
(iii) change in volume (comsol.com, 2018)
v’/vo= έv
Substituting in the above equation:
v’ = (-2.0625x10-3) x 9000 = -18.5625mm3

Evidence checked Summary Evidence presented
3 -Effects of 3 dimensional
loading on the given material
Equations, method and
solution under question 3;
check on comments e1, e2, e3
4 -Relationship of the four
elastic constants
Demonstrated how the four
constants are related; check on
comments e4 and e5
References
Comsol.com. (2018). Volumetric strain. [online] Available at:
https://www.comsol.com/community/forums/general/thread/15089/ [Accessed 27 Mar.
2018].
Coursehero.com. (2018). Complex Stress Tutorial 3 complex stress and strain This tutorial is not
part of the Edexcel unit mechanical Principles but covers elements of the. [online]
Available at: https://www.coursehero.com/file/12176072/t3/ [Accessed 27 Mar. 2018].