Crack Paths 2009

θ2 is the

where u is the radial distance from the vertex to the point under consideration,

angle at the cone vertex. Here, the weight function Φis assumed to be the maximum

principal stress

1 σ . The functional Lfrom Eq. 2 leads to the following equation

() v u

u ∫ + 2

L M =

du

2 2 2 sin'

(9)

θ

2

1

,

3 sin θ π

3

u

u 1

where vis the angle between two generatrixs on the cone surface. The Euler-Lagrange

equation takes the following form

θ

v

sin'

2/1

2

v u u

(10)

=

C

.

+

() 2 2 2 sin'

(

θ

)

1

The solution of this equation leads to the crack path

(11)

()θsin2sin0v u= u ,

where

0 u is a constant.

It can be seen that Eq. 11 allows describing the crack path in the double circular cone

caused by a torque (Fig. 1).

Figure 1. Fractured surface of the double Plexiglas circular cone under torsion.

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