along the interface. As indicated, the peak of the
tion elasticity analysis, this pressure was found to
normal stress σx (y) occurs at the mid-depth of the
be approximately (5/6)E. Gent and Lindley vali-
seal, and the maximums of the tangential stress ty
dated this solution with experimental results, and
(y) occur at the upper and lower edges of the seal.
used the solution with normal stress predictions
For a given material and joint extension, the effect
from small deformation relations like eq 5 to pre-
of an increasing depth-to-width ratio on the peak
dict the nominal stress and extension values of a
normal or tangential stress is dramatic. Also, for
rubber structure at failure.
higher d/w values, the contribution of the shear
Applying Gent and Lindley's analysis tech-
deformation to the total, normal stress is much
nique to the plane strain structure of a butt joint
seal, the critical average stress σx at which an
′
greater than the contribution of the homogeneous
deformation, which is constant at the σx (y)/Ee
internal rupture occurs can be found as a function
ratio of 4/3.
of the material elastic modulus E and the depth-
to-width ratio of the seal. This relation is
0.4
4 w 2 1
d/w = 4
5
σx =
+ E .
′
2
(9)
0.2
1
3
3 d
3
0
The critical extension, e′ = σx /Ea, is
′
0.5
0.2
0.4
2
5 w
e′ =
.
(10)
3 d
0
2
4
6
8
10 4
2
0
2
4
σx(y)/Ee
ty(y)/Ee
Figure 6. Nondimensionalized normal and tangential
These relations are illustrated in Figure 7 for
stress distributions, σx (y)/E and ty(y)/E, divided by
depth-to-width ratios of a seal from 4 to 8. Below
the nominal strain e, for joint seals with depth-to-
d/w = 4, the relations should not be applied since
width ratios 0.5, 1, 2, and 4.
the resulting critical extensions are large and vio-
0.10
Elastic instability
Gent and Lindley (1958) observed in
0.08
experiments of rubber cylinders that an
internal rupture was possible at a compar-
0.06
atively small tensile load when the di-
ameter-to-thickness ratio of the cylinder
was high. The rupture was described as
0.04
consisting of the sudden appearance of
internal cracks at a repeatable, small ten-
0.02
sile load. The experiments and analysis
described by Gent and Lindley showed
1.05
the internal rupture to be governed by an
elastic instability and the failure stress to
0.95
depend upon the elastic modulus. The
cracking stress was found to be indepen-
dent of the extensibility and strength of
0.85
the rubber material.
The elastic instability was shown by
0.75
Gent and Lindley to occur when a small
cavity or imperfection within the rubber is
0.65
subjected to a tensile and primarily hydro-
4
5
6
7
8
d/w
static stress, such as the maximum normal
Figure 7. Critical rupture extension, e′, and ratio of the critical
stress of the curve for d/w = 4 in Figure 6.
rupture stress to the Young's modulus, σx /E, as functions of the
′
At a critical pressure the cavity expands,
depth-to-width ratio d/w.
forming a crack. Using a large deforma-
6
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