Harwood
Engineering Company, Inc.
Monitoring the Progress of Partial Autofrettage
by Donald H. Newhall
A method is shown to monitor the depth of plastic yielding, the permanent
expansion of the cylinder and the amount of residual strain during hydraulic
autofrettage. The significance of the residual stress is shown graphically.
Introduction
When autofrettage is used there exists the practical problem of estimating
the before-autofrettage bore diameter to minimize the amount of material
for machining to finish size1,2,3. To that purpose, it is necessary
to estimate the penetration of plastic flow and the permanent deformation
of the bore. Two methods are offered, either of which may be used to monitor
the progress of the autofrettage.
General
The ellipse in Figure 1 represents the well known
Von Mises-Henky description of the limit of elastic behavior when applied
to a cylinder subjected to internal pressure. The coordinates are the principal
stresses, tangential (St) and radial (Sr). The longitudinal
stress (Sz) developed by friction of the packings in an open
ended cylinder is small and is, therefore, neglected. The lines with arrows
indicate the increasing and decreasing course of pressure during autofrettage.
Elastic behavior exists with ascending pressure until it reaches the ellipse,
pressure increasing further producing plastic flow until the pressure is
released. When the pressure is released, the cylinder behaves elastically
again. Descending pressure follows a path parallel to the initial ascending
path. The increase in strength of the cylinder against pressure and the
magnitude of the residual tangential stress are indicated. This excursion
out of the elastic domain also increases the yield strength of the material.
Estimating the Elastic – Plastic Boundary
Figure 2 illustrates a cylinder at sufficient pressure
to produce partial autofrettage. The boundary stresses at the bore, the
elastic – plastic boundary and the outside surface are labeled and evaluated.
The derivations of the latter are readily made and described in standard
texts of elasticity4 and plasticity5,6. The condition
of equality of tangential stress at the elastic – plastic boundary permits
the following relation — written from inspection of Figure
2:
(1)
Remembering that W=W1 x W0, a convenient rearrangement
of terms is:
(2)
A graphical solution for W0 is shown in Figure
3.
A second method uses bonded strain gages on the outside of the cylinder.
As long as the cylinder expands reasonably uniformly (without appreciable
barreling which would produce longitudinal stress) there is a uniaxial
stress on the outside, stb, with sz=0. Therefore
ez=-met.
At the elastic – plastic boundary, the elastic portion is stressed by
Pb to the point of yielding. Using the Von Mises criterion:
(3)
where - sr=Pb, 
, 
etc.
appropriate substitution yields:
(4)
(5)
Residual Stress at the Bore
The distribution of residual stress after autofrettage is found by subtracting
the elastic distribution released when Pa goes to zero from the plastic
distribution. Thus the residual tangential stress at the bore of the cylinder
is: 
from
which a solution for Pa is found in terms of desired residual stress:
(6)
Should it be desired to limit St residual in large wall ratios
(over 2/1) to (-
),
(to avoid reverse yielding) substitution in the above shows that:
(7)
To Estimate the Expansion of the Bore
During the expansion of a vessel in autofrettage, planes remain plane so
that the elastic – plastic boundary ez, elastic zone=ez,
plastic zone. In the plastic zone strains are gross compared to the elastic,
which permits one to assume that the condition of constant volume exists
in the plastic portion. This assumption then allows the derivation of the
plastic strains and bore deformation. If the plastic zone of the cylinder
in Figure 2 is of length L, its volume is:
(8)
by differentiating both sides of the equation and placing dv=0, the following
relation may be found:
(9)
By taking advantage of the aforementioned equality of stress and strain
at the elastic – plastic boundary and rearranging terms noting:
(10a)
and
(10b)
using m=0.3, subtracting 
,
the elastic recovery strain, solving for ea and remembering
that DI.D. =ea x diameter "a", it
is found that:
(11)
Equation (11) is in reasonable agreement with experience.
Comments
Throughout, the Von Mises relation is expressed as st-sr=k.
The value "k" actually varies with wall ratio and is the ratio of the Von
Mises and Tresca theories or 
.
For full autofrettage of large wall ratios, k varies from 1.15 at the bore
to 1.00 on the outside. Using an in-between value, k=1.08, reasonable agreement
with experiment is found.
The assumed character of the stress strain relationship of material
used in the classical theory of plasticity is unrealistic when applied
to steel cylinders with bore expansions of as much as 6% in either partial
or full autofrettage. The assumption leads to the value of Poisson's ratio
of 0.5, yet Poisson's ratio occurs with a mean experimentally found value
between 0.30 and 0.33%. However, planes do remain plane in autofrettage.
Apropos of Figure 1, stresses during plastic flow
follow the ellipse if the theory were fully applicable, indicating less
resistance to expansion than shown and experienced for a given autofrettage
pressure.
Due to the limitation of space, the author has not discussed optimum
amounts of autofrettage, the effect of reverse yielding, hysteresis, nor
any other Bauschinger effect.
Stress distributions have been discussed mostly in a qualitative way.
They are of academic interest here since only the boundary conditions are
essential to engineering design.
An analysis of autofrettage written in 1909 by L.B. Turner5
is classical and of interest.
References
1. D.H. Newhall, Selected Design Data Pertaining to Gun Tubes and High-Pressure
Vessels, Watertown Arsenal Report No. WGD-4, (1943).
2. D.H. Newhall, Plastic Strains in Thick Hollow Cylinders Overstrained
by Internal Pressure, Watertown Arsenal Report No. AGD-7, (1944).
3. T.E. Davidson, C.S. Barton, A.N. Reiner, D.P. Kendall, Overstrain
of High-Strength Open-End Cylinders of Intermediate Diameter Ratio, Watervliet
Arsenal, Watervliet, New York, (1963).
4. S. Timoshenko, Strength of Materials, Part II. Van Nostrand, (1942).
5. L.B. Turner, The Stresses in a Thick Hollow-Cylinder Subjected to
Internal Pressure, Trans. Camb. Phil. Soc. XXI, No. XIV, pp. 377-396, (1909).
6. A. Nadai, Theory of Plasticity, McGraw Hill, (1931).
Harwood Engineering Company, Incorporated – © July, 1999
455 South Street, Walpole, Massachusetts 02081
phone: 508-668-3600
fax: 508-660-2276
http://www.ultranet.com/~harwood/
email: harwood@ma.ultranet.com
