Introduction

1 In Eq. ?? we showed that around a circular hole in an infinite plate under uniform traction, we do have a stress concentration factor of 3.

2 Following a similar approach (though with curvilinear coordinates), it can be shown that if we have an elliptical hole, Fig. ??, we would have

We observe that for a = b, we recover the stress concentration factor of 3 of a circular hole, and that for a degenerated ellipse, i.e a crack there is an infinite stress. Alternatively, the stress can be expressed in s

Elliptical Hole Stress Concentration

Figure 11.1: Elliptical Hole in an Infinite Plate s s

Figure 11.1: Elliptical Hole in an Infinite Plate

Theoretical Strength

Theoretical Strength

Diameter

Figure 11.2: Griffith's Experiments terms of p, the radius of curvature of the ellipse,

From this equation, we note that the stress concentration factor is inversely proportional to the radius of curvature of an opening.

3 This equation, derived by Inglis, shows that if a = b we recover the factor of 3, and the stress concentration factor increase as the ratio a/b increases. In the limit, as b = 0 we would have a crack resulting in an infinite stress concentration factor, or a stress singularity.

4 Around 1920, Griffith was exploring the theoretical strength of solids by performing a series of experiments on glass rods of various diameters.

5 He observed that the tensile strength (at) of glass decreased with an increase in diameter, and that for a diameter ^ « 10 O00 in., at = 500, 000 psi; furthermore, by extrapolation to "zero" diameter he obtained a theoretical maximum strength of approximately 1,600,000 psi, and on the other hand for very large diameters the asymptotic values was around 25,000 psi.

Area

Ai

<

A2

<

A3

<

A4

Failure Load

Pi

<

P2

<

P3

>

P4

Failure Strength (P/A)

"1

>

"2

>

"3

>

Furthermore, as the diameter was further reduced, the failure strength asymptotically approached a limit which will be shown later to be the theoretical strength of glass, Fig. 11.2.

6 Clearly, one would have expected the failure strength to be constant, yet it was not. So Griffith was confronted with two questions:

1. What is this apparent theoretical strength, can it be derived?

2. Why is there a size effect for the actual strength?

The answers to those two questions are essential to establish a link between Mechanics and Materials.

7 In the next sections we will show that the theoretical strength is related to the force needed to break a bond linking adjacent atoms, and that the size effect is caused by the size of imperfections inside a solid.

Figure 11.3: Uniformly Stressed Layer of Atoms Separated by a0

0 0

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