Page 25 - Marks Calculation for Machine Design

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FUNDAMENTAL LOADINGS
s
A, B, C, D, F 7
E

e
FIGURE 1.8 Stress-strain diagram (brittle material).

The stress-strain diagram for a brittle material is shown in Fig. 1.8, where points A, B,
C, D, and F are all at the same point. This is because failure of a brittle material is virtually
instantaneous, giving very little if any warning.

Poisson’s Ratio. The law of conservation of mass requires that when an axially loaded
bar lengthens as a result of a tensile load, the cross-sectional area of the bar must reduce
accordingly. Conversely, if the bar shortens as a result of a compressive load, then the cross-
sectional area of the bar must increase accordingly. The amount by which the cross-sectional
area reduces or increases is given by a material property called Poisson’s ratio (ν), and is
defined by Eq. (1.4).
lateral strain
ν =− (1.4)
axial strain
where the lateral strain is the change in any lateral dimension divided by that lateral dimen-
sion. For example, if the lateral dimension chosen is the diameter (D) of a circular rod, then
the lateral strain could be calculated using Eq. (1.5).

D
lateral strain = (1.5)
D
The minus sign in the definition of Poisson’s ratio in Eq. (1.4) is needed because the
lateral and axial strains will always have opposite signs, meaning that a positive axial strain
produces a negative lateral strain, and a negative axial strain produces a positive lateral
strain. Strangely enough, Poisson’s ratio is bounded between a value of zero and a half.
1
0 ≤ ν ≤ (1.6)
2
Again, this is a consequence of the law of conservation of mass that must not be violated
during deformation, meaning a change in shape. Values of both the modulus of elasticity (E)
and Poisson’s ratio (ν) are determined by experiment and can be found in Marks’ Standard
Handbook for Mechanical Engineers.