1. 2008 ASME PVP Conference
3-D Stress Intensity Factors for Arrays of Inner
Radial Lunular or Crescentic Cracks in Thin and
Thick-Walled Spherical Pressure Vessels
by
Mordechai Perl & Vadim Bernstein
2. Motivation
*Most of Spherical Pressure Vessels are manufactured
by composing a series of double curved petals welded
along their meridional lines.
*These pressure vessels are susceptible to radial
cracking due to one or more of the following factors:
• Cyclic pressurization-depressurization
• The presence of tensile residual stresses in the
heat-effected zone near the welds
• Large temperature gradients
• The presence of corrosive agents
3. Stress Intensity Factor (SIF)
To predict :
* Fatigue life of the vessel
* Fracture endurance
The 3-D Stress Intensity Factor due to internal pressure,
KI , needs to be evaluated.
KI is numerically evaluated for each cracked
configuration as no analytical results are available.
Then, it can be compared with its
critical value-the material toughness.
4. Objective of The Present Analysis
To evaluate the 3-D KI distribution along the
crack fronts of lunular or crescentic radial cracks
emanating from the inner surface of a spherical
pressure vessel for:
– Numerous crack array configurations
– Different sphere geometries
– A wide range of crack depths
– Various crack ellipticities
5. The Mathematical Model
Boundary Conditions:
a. Three planes of symmetry
• θ=0
• θ=π/n
• φ=0
b. Pressure is applied to the
internal surface of the sphere
and it fully penetrates the
cracks cavities
6. The Mathematical Model
Boundary Conditions:
a. Three planes of symmetry
• θ=0
• θ=π/n
• φ=0
b. Pressure is applied to the
internal surface of the sphere
and it fully penetrates the
cracks cavities
7. The Numerical Solution
• Finite Element method - using
the standard ANSYS 11.0 code
• At least 25,000 degrees of freedom
• SIFs are extracted using the crack-face
displacement extrapolation procedure
built into ANSYS
8. 3-D Finite Element Model
Three types of elements are used:
• 1st layer, 3-D singular elements,
to accommodate the singular
stress field.
• Up to 4 layers of 20-node brick
elements along the crack front.
• The rest of the model is meshed
with both brick and 10-node
tetrahedron elements.
9. Summary of Solved Cases
• Due to the parameterization of the FE model
a large number of cases were solved for:
– Spheres of radii ratio of:
R0/Ri=1.01, 1.05, 1.1, 1.7 and 2.0.
– Number of cracks in the array n=1 to 20.
– Crack ellipticities a/c= 0.2 to 1.5.
– Relative crack depths a/t= 0.025 to 0.8.
• KI is evaluated every Δψ=1º÷3º degrees along
the crack front
10. Definition of a Lunular Crack
A planar, part-through crack, whose shape is enclosed
by two circular arcs of different radii, one concave
and one convex, which intersect at two points, having
an ellipticity of a/c=1
11. Definition of a Crescentic Crack
A planar, part-through crack whose shape is enclosed
by two intersecting arcs, the concave one which is
elliptical, and the convex one which is circular, having
an ellipticity of a/c≠1
a/c<1 a/c>1
Slender Crescentic Crack Transverse Crescentic Crack
14. The Normalizing factor K0
Average hoop stress
Ri2 pRi 1
in the sphere’s wall p 2
RO Ri2
2t (1 t / 2 Ri )
a
K0
Q a
1.65
Q 1 1.464 for a c 1
c
Shape factor 1.65
Q 1 1.464 c
for a c 1
a
15. Validation of the 3-D model for a Single Crack
The ratio between the 3-D SIF and the 2-D* SIF KI D / KI D
3 2
1.
RoRoRiRii .01 1
R/ // R 1 1.05
o
* KI for a through-crack in a thin spherical shell (Erdogan and Kibler, 1969)
23. The Influence of the Sphere Geometry on the SIF
1.01, 1.05 , 1.1, 1.7 and 2.0
a / c 0..5
a / c 15 0
24. Conclusions
The SIFs are considerably affected by the
three - dimensionality of the problem.
The shallower the crack the larger the difference
between the 3-D SIF and the 2-D approximation.
Therefore, crack growth rate based on the 2-D
model over estimates the realistic 3-D crack
growth rate, yielding a very conservative total
fatigue life estimate.
25. Conclusions (con’t)
KI increases :
♦ As crack depth a/t increases;
♦ As crack ellipticity a/c decreases;
♦ As the number of cracks in the
array n decreases.
The location of the maximum SIF, Kmax , along
the crack front is either at ψ=90º or at ψ=ψ0 and
depends on crack ellipticity,
a/c, and crack depth, a/t.
26. Conclusions (con’t)
The crack’s opening stress and the finite-body
effect have opposing influence on KI.
For any given crack of relative depth a/t and
ellipticity a/c, KI for a Thin spherical shell is
larger than for a thicker one.
