ASM Metals HandBook Vol. 8 - Mechanical Testing and Evaluation
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Fig. 13 Tapered, constant curvature cantilever beam strip specimen bending test
Fig. 14 Percent stress remaining versus time for 70-30 brass (C26000) for test using three methods at
three laboratories. Initial stress equals 80% of 0.2% yield strength.
Torsion Testing
In torsion testing, a cylindrical or tubular specimen is subjected to a constant torsional strain. The change in
restraining torque associated with the stress relaxation can be determined by continuous monitoring with a
torque transducer or by periodically applying a slight torque to transfer the remaining torque from the end grips
to the transducer. Alternatively, the twist springback on unloading can be used to determine how much
remanent twist angle remains, and consequently, how much torque would be required to twist the specimen
back to its original angle of twist prior to relaxation.
Because of the increased complexity of testing and analysis of torsion stress relaxation, there is interest in
predicting torsion stress relaxation based on tensile data. Using combined tension and torsion relaxation tests, it
has been shown that in thin-walled copper tubes tested at 250 °C, the results of pure tensile stress relaxation
tests can be used to predict complex stress relaxation behavior on the basis of octahedral shear stress
equivalence (Ref 26).
Test on Springs
Designers need to be able to estimate how much force remains in a spring after a given service exposure. The
experimental methods are similar to those for the other stress states and include:
• Monitoring the decrease in spring force at a constant temperature after deflection to a given
displacement
• Determining the remaining spring load by transferring the load periodically from the deflecting member
to a load measuring device
• Using the elastic springback on unloading to calculate the spring force remaining immediately prior to
unloading using the spring constant
Figures 15 and 16 (Ref 27) illustrate the latter two methods.
Fig. 15 Setup for stress relaxation testing of a clamp spring
Fig. 16 Application of load during a stress-relaxation test of a helical compression spring at room
temperature. Original free length, L
0
, compressed length, L
1
; free length upon release from load at
temperature, L
2
; permanent set, ΔL
Figure 15 shows the apparatus used to determine the force remaining as a function of time and temperature in a
complex relay pileup clamp spring. To measure the force of the spring on the pileup, the spring is clamped into
a rigid steel fixture that is on a compression load cell. Force is applied using a universal testing machine by
means of a plunger, which is depressed and then twisted to lock in place. Electrical contact is made between the
plunger and a spring-loaded contact. The force required to break the electrical contact is measured as a function
of time to give the relaxation curve.
In Fig. 16, a helical compression spring is deflected and held at the test temperature. The force remaining after
time, t, is determined by measuring the permanent set, ΔL, on unloading by the relationship:
where P
0
is the initial load, ΔP is the change in load, L
0
, is the initial length, and L is the permanent set.
References cited in this section
4. J. Miyake, T. Kimura, and T. Endo, Stress Relaxation of Cu-Ti Alloy C199, Creep and Stress
Relaxation in Miniature Structures and Components, H.D. Merchant, Ed., TMS, 1996, p 57–74
11. Standard Methods for Stress Relaxation Tests for Materials and Structures, E 328-86, Annual Book of
ASTM Standards, ASTM, 1991
23. A. Fox, Ed., Stress Relaxation in Bending of AISI 301 Type Corrosion Resistant Steel Strip, STP 676,
Stress Relaxation Testing, ASTM, 1978, p 78–88
24. E.W. Filer and C.R. Scorey, Stress Relaxation in Beryllium Copper Strip, STP 676, Stress Relaxation
Testing, A. Fox, Ed., ASTM, 1978, p 89–111
25. P. Parikh, Report on Bending Stress Relaxation Round Robin, STP 676, Stress Relaxation Testing, A.
Fox, Ed., ASTM, 1978, p 112–125
26. J. Henderson and J.D. Sneddon, Complex Stress Creep Relaxation on Commercially Pure Copper at 250
°C, Advances in Creep Design, A.I. Smith and A.M. Nicolson, Ed., John Wiley & Sons, Inc., 1971, p
163–180
27. A. Fox, Effect of Temperature on Stress Relaxation of Several Metallic Materials, Residual Stress and
Stress Relaxation, Proceedings 28th Sagamore Army Materials Research Conference, E. Kula and V.
Weiss, Ed., Plenum Press, 1982, p 181–203
Stress Relaxation Testing
D.A. Woodford, MPa, Inc.
Acknowledgments
In preparing this chapter, extensive use was made of the article by A. Fox, under the same title, which appeared
in the previous edition of Volume 8 of this Handbook. ASTM E 328-86 (1991) was also extremely useful.
Stress Relaxation Testing
D.A. Woodford, MPa, Inc.
References
1. J. Bolton, Design Considerations for High Temperature Bolting, Performance of Bolting Materials in
High Temperature Plant Applications, Institute of Materials, 1995
2. R.J. Glodowski and G.E. Hoff, Stress Relaxation of Steel Tendons Used in Prestressed Concrete under
Conditions of Changing Applied Stress, STP 676, Stress Relaxation Testing, A. Fox, Ed., ASTM, 1978,
p 42–58
3. S.U.V. Idermark and E.R. Johansson, Room Temperature Stress Relaxation of High Strength Strip and
Wire Spring Steels-Procedure and Data, STP 676, Stress Relaxation Testing, A. Fox, Ed., ASTM, 1978,
p 61–77
4. J. Miyake, T. Kimura, and T. Endo, Stress Relaxation of Cu-Ti Alloy C199, Creep and Stress
Relaxation in Miniature Structures and Components, H.D. Merchant, Ed., TMS, 1996, p 57–74
5. H. Mavoori, J. Chin, S. Aaynman, B. Moran, L. Keer, and M. Fine, Creep, Stress Relaxation and Plastic
Deformation in Sn-Ag and Sn-Zn Eutectic Solders, H.D. Merchant, Ed., TMS, 1996, p 173–190
6. J.M. Beeston and T.K. Burr, In-Reactor Stress Relaxation of Type 348 Stainless Steel In-Pile Tube, STP
676, Stress Relaxation Testing, A. Fox, Ed., ASTM, 1978, p 155–170
7. F.T. Geyling and P.L. Key, Stress Relaxation of Residual Metalworking Stresses, STP 676, Stress
Relaxation Testing, A. Fox, Ed., ASTM, 1978, p 143–154
8. R.M. Chenko, Weld Residual Stress Measurements on Austenite Stainless Steel Pipes, Weldments:
Physical Metallurgy and Failure Phenomena, Fifth Bolton Landing Conference, 1978, p 195–206
9. D.A. Woodford and D.F. Mowbray, Effect of Material Characteristics and Test Variables on Thermal
Fatigue of Cast Superalloys, Mater. Sci. Eng., Vol 16, 1974, p 5–43
10. G.A. Webster and R.A. Ainsworth, High Temperature Component Life Assessment,, Chapman and Hall,
1994
11. Standard Methods for Stress Relaxation Tests for Materials and Structures, E 328-86, Annual Book of
ASTM Standards, ASTM, 1991
12. Practice for Testing Stress Relaxation of Plastics, D 2991, Annual Book of ASTM Standards, ASTM
13. D.A. Woodford, Stress Relaxation, Creep Recovery, and Newtonian Viscous Flow in Silicon Nitride, J.
Am. Ceram. Soc., Vol 81 (No. 9), 1998, p 2327–2332
14. M.S. Loveday and B. King, Uniaxial Testing Apparatus and Testpieces, High Temperature Mechanical
Properties of Materials, M.S. Loveday, M.F. Day, and B.F. Dyson, Ed., NPL, 1982, p 128–157
15. A. Strang, Review of Current International Stress Relaxation Testing Standards, Performance of Bolting
Materials in High Temperature Plant Applications, Institute of Materials, 1995, p 97–114
16. J. Hacon and A Krause, Relaxation Properties of a 1CrMoV Bolting Steel under Uniaxial Tensile Load,
Paper 51, Proceedings Conference Properties of Creep Resistant Steels, VDEL, Dusseldorf, 1972
17. E. Hart, Load Relaxation Testing and Material Constitutive Relations, Stress Relaxation Testing, STP
676, A. Fox, Ed., ASTM, 1978, p 5–20
18. D.A. Woodford, Measurement of the Mechanical State of a Low Alloy Steel at Elevated Temperature,
Metall. Trans. A, Vol 6, 1975, p 1693–1697
19. D.A. Woodford, Test Methods for Accelerated Development, Design, and Life Assessment of High
Temperature Materials, Mater. Des., Vol 14 (No. 4), 1993, p 231–242
20. D.A. Woodford and K. Iijima, Creep Strength Evaluation, Design, and Life Management of Cr-Mo-V
Rotor Steel Using Stress Relaxation Testing, Advances in Turbine Materials, Design, and
Manufacturing, Fourth International Charles Parsons Turbine Conference, A. Strang, W.M. Banks, R.D.
Conroy, and M.J. Goulette, Ed., Institute of Materials, 1997, p 613–624
21. D.A. Woodford, Accelerated High Temperature Performance Evaluation of Superalloys for Process
Optimization and Remaining Life Assessment, ASME IGTI Turbo Expo Conference (Indianapolis, IN),
ASME, 1999
22. G.G. Gryzywinski and D.A. Woodford, Creep Analysis of Thermoplastics Using Stress Relaxation
Data, J. Polym. Eng. Sci., Vol 35 (No. 24), 1995, p 1931–1937
23. A. Fox, Ed., Stress Relaxation in Bending of AISI 301 Type Corrosion Resistant Steel Strip, STP 676,
Stress Relaxation Testing, ASTM, 1978, p 78–88
24. E.W. Filer and C.R. Scorey, Stress Relaxation in Beryllium Copper Strip, STP 676, Stress Relaxation
Testing, A. Fox, Ed., ASTM, 1978, p 89–111
25. P. Parikh, Report on Bending Stress Relaxation Round Robin, STP 676, Stress Relaxation Testing, A.
Fox, Ed., ASTM, 1978, p 112–125
26. J. Henderson and J.D. Sneddon, Complex Stress Creep Relaxation on Commercially Pure Copper at 250
°C, Advances in Creep Design, A.I. Smith and A.M. Nicolson, Ed., John Wiley & Sons, Inc., 1971, p
163–180
27. A. Fox, Effect of Temperature on Stress Relaxation of Several Metallic Materials, Residual Stress and
Stress Relaxation, Proceedings 28th Sagamore Army Materials Research Conference, E. Kula and V.
Weiss, Ed., Plenum Press, 1982, p 181–203
Influence of Multiaxial Stresses on Creep and Creep
Rupture of Tubular Components
R.C. Hurst and J.H. Rantala, European Commission, Joint Research Centre, Institute for Advanced Materials
(The Netherlands)
Introduction
DESIGN OF PRESSURIZED COMPONENTS is normally based on uniaxial data because all the material data
is generated using uniaxial tests. However, typical industrial piping components operate under a multiaxial state
of stress as a result of the internal pressure, temperature gradients, and system stresses. Very general effective
stress concepts have been used in component design codes to calculate the creep life of the material under
triaxial state of stress. These concepts are based on the principle that the life of a component with a multiaxial
effective stress corresponds to the rupture time at the same uniaxial stress according to the uniaxial design data.
Influence of Multiaxial Stresses on Creep and Creep Rupture of Tubular Components
R.C. Hurst and J.H. Rantala, European Commission, Joint Research Centre, Institute for Advanced Materials (The Netherlands)
Multiaxiality and Principal Stresses
Any complex stress combination with stresses in three directions and six different shear stresses can be reduced
to just three stresses, the principal stresses σ
1
> σ
2
> σ
3
. The highest of these is called the maximum principal
stress (MPS). In the principal stress coordinate system there are no shear stresses.
The stress state is biaxial if σ
3
= 0, as is often assumed for thin-walled tubes. The stress system is reduced to
uniaxial when σ
1
= σ and σ
2
= σ
3
= 0. In thick-walled pipes and components the stress state is triaxial.
In the stress distribution figures in this article, a tube under internal pressure that has a ratio of inner radius to
outer radius of 0.6 has been used as an example.
Influence of Multiaxial Stresses on Creep and Creep Rupture of Tubular Components
R.C. Hurst and J.H. Rantala, European Commission, Joint Research Centre, Institute for Advanced Materials (The Netherlands)
Effective Stress Equations
The basic idea of effective stresses is to calculate a stress for complex stress situations with a value that gives
the correct failure time when applied to uniaxial data, which is often the basis for engineering design—and
often the only data available in absence of representative multiaxial test data.
The effective stresses can be calculated for different cases, each of which will be described in detail in this
article:
• Elastic stresses (see the section “Elastic Stress Distribution in a Tube”)
• Steady-state creep stresses (see the section “Steady-State Creep Stress Distribution in a Tube”)
• Stresses in a fully plastic case (see the section “Plastic Stresses in a Tube”)
• Thermal stresses (see the section “Thermal Stresses in a Tube”)
The thermal stresses are often superimposed with one of the former stress systems.
Von Mises Criterion. The most commonly used effective stress concept is the one presented in 1913 by von
Mises, based on a concept of maximum energy of distortion:
(Eq 1)
The von Mises effective stress is most widely used in engineering design, both in the low and high temperature
ranges. It has been found to govern the deformation of materials under complex loading situations. It is also the
controlling parameter for creep rupture at high stresses where the rupture is associated with large deformations
and a ductile failure mechanism. However, in long-term service, the rupture is controlled either by MPS or a
mixed MPS-von Mises criterion. Also in brittle materials, the rupture tends to be more MPS controlled.
Much work has been focused on whether or not creep rupture would occur under a complete tensile triaxiality
when σ
1
= σ
2
= σ
3
= σ. Under these circumstances, von Mises would go to zero, which means that there is no
deformation or rupture in the short term, but it is believed that creep rupture would occur under MPS control
after a long exposure time with a constrained grain-boundary cavity nucleation and growth mechanism.
Experimentally this is extremely difficult to verify because of the difficulty of triaxial loading using a cruciform
specimen, for example.
For practical reasons, the von Mises equation is often used in engineering design in a different form where,
instead of principal stresses, the engineering stresses are used, including torsion stress:
(Eq 2)
where σ
h
is hoop stress (circumferential), σ
a
is axial stress, σ
r
is radial stress, and τ is torsion stress.
Often an effective stress concept has been used in the background when design rules have been determined, but
it is not always apparent to the user from the design code itself. Similarly, plenty of detailed analyses have often
been hidden behind the design safety factors.
Tresca Criterion. The yield criterion generally known as the Tresca criterion is based on the concept of
maximum shear stress energy and is expressed simply as:
σ
TR
= σ
1
- σ
3
(Eq 3)
Huddleston Criterion. This multiaxial effective stress concept is a modified version of the von Mises stress, and
it is claimed to better take the multiaxiality into account (Ref 1, 2). Here the equation is shown in the simplified
form.
(Eq 4)
J
1
= σ
1
+ σ
2
+ σ
3
(Eq 5)
(Eq 6)
While under the steady-state creep conditions, the von Mises predicts highest stresses on the inner surface; the
Huddleston distribution, on the other hand, is quite contrary and would probably provide a more reliable basis
for damage and life predictions than von Mises, at least if only stress is considered in life prediction and not
ductility. (See Fig. 1 in the section “Steady-State Creep Stress Distribution in a Tube” in this article.) This
criterion was originally developed for stainless steels but has proved to be useful for ferritic high-temperature
engineering materials as well. The Huddleston criterion was incorporated into ASME Code Case N-47-29.
Fig. 1 Distribution of normalized steady-state creep stresses in a pressurized tube with a ratio of inner
radius, R
i
, to outer radius, R
o
, of 0.6. The principal facet stress values have been normalized by dividing
by a factor of 2.24
Mixed Criteria. Often mixed criteria have been proposed because, in practical cases, none of the classical
parameters describe the material behavior sufficiently in various stress cases. One of the most common criterion
is the one developed at the former Central Electricity Generating Board in the United Kingdom, presented by
Cane and Hayhurst (Ref 3, 4):
σ
eff
= ασ
1
+ (1-α)
(Eq 7)
where α is the coefficient of thermal expansion. A similar empirical effective stress concept for high-
temperature materials has been proposed by the Russian research institute CKTI:
σ
CKTI
= ( + 0.47 )
1/n
(Eq 8)
where n is the Norton creep exponent.
The concept of principal facet stress σ
F
has been developed (Ref 5) based on the observations that creep
damage will first appear on grain boundaries transverse to the axis of MPS, and that on inclined boundaries
there will be shear deformation, which is mainly controlled by von Mises effective stress:
σ
F
= 2.24σ
1
- 0.62(σ
2
+ σ
3
)
(Eq 9)
The principal facet stress has been found to bring uniaxial and multiaxial data together especially for ferritic
and austenitic steels. However, this stress does not apply directly to engineering calculations because the stress
values calculated by Eq 9 are high. Therefore, in Fig. 1, σ
F
has been normalized by a factor of 2.24. The
principal facet stress is then seen to coincide with the von Mises stress on the outer surface of the pressurized
tube. Normalization by a factor of 2 would make σ
F
coincide with the von Mises stress in the skeletal point and
with the distribution of the Huddleston stress.
The Effect of Stress System on the Damage Processes. At least in the short term, the creep damage process of
cavity nucleation seems to be von Mises controlled because cavity nucleation requires accumulation of shear
strain. On the contrary, for brittle materials the number of cavities formed is a function of MPS and
intergranular fracture is promoted by the increasing ratio of σ
1
/σ
VM
(Ref 3). Similarly, the effect of MPS
becomes more pronounced under long service times, under which the uniaxial ductility is known to reduce
remarkably and fracture becomes more brittle.
According to Cane (Ref 3), the growth of grain-boundary cavities is controlled by MPS at low effective stress
levels and by von Mises at high stress levels. If component-integrity and life-assessment methods do not give
proper consideration to the change of the damage mechanism, they are likely to be conservative at high stresses
and short times when σ
1
/σ
VM
< 1 and not conservative at low stress levels in the long term when σ
1
/σ
VM
> 1.
References cited in this section
1. R.L. Huddleston, An Improved Multiaxial Creep-Rupture Strength Criterion, J. Pressure Vessel
Technol. (Trans. ASME), Vol 107, Nov 1985, p 421–429
2. R.L. Huddleston, Assessment of an Improved Multiaxial Strength Theory Based on Creep-Rupture Data
for Type 316 Stainless Steel, J. Pressure Vessel Technol. (Trans. ASME), Vol 115, May 1993, p 177–
184
3. B.J. Cane, Creep Damage Accumulation and Fracture under Multiaxial Stresses, Proc. Advances in
Fracture Research (Fracture 81), 1981 (Cannes, France), Vol 3, Pergamon Press, 1982
4. B.J. Cane, Representative Stresses for Creep Deformation and Failure of Pressurised Tubes and Pipes,
Int. J. Pressure Vessel Piping, Vol 10, 1982, p 119–128
5. H.K. Kim, F.A. Mohamed, and J.C. Earthman, “High Temperature Rupture of Microstructurally
Unstable 304 Stainless Steel Under Uniaxial and Triaxial Stress States,” Metall. Trans. A, Vol 22, 1991,
2629–2635
Influence of Multiaxial Stresses on Creep and Creep Rupture of Tubular Components
R.C. Hurst and J.H. Rantala, European Commission, Joint Research Centre, Institute for Advanced Materials (The Netherlands)
Effective Strain
In a multiaxial stress system as well, strains occur in three different directions, which are assumed to coincide
with the principal stress axes. It is assumed that failure will occur when the effective strain reaches the uniaxial
rupture strain value. The effective strain equation is almost identical with the respective stress equation (Eq 2):
(Eq 10)
Similarly, the effective strain rate can be expressed as:
(Eq 11)
The relationship between stress and strain rate can then be expressed:
(Eq 12)
Further, a simple criterion comes from the fact that most engineering materials are considered to be
incompressible. This rule applies for both strains and strain rates:
1
+
2
+
3
= 0
(Eq 13)
Influence of Multiaxial Stresses on Creep and Creep Rupture of Tubular Components
R.C. Hurst and J.H. Rantala, European Commission, Joint Research Centre, Institute for Advanced Materials (The Netherlands)
Elastic Stress Distribution in a Tube
The equations for the elastic stress distribution for hoop, axial, and radial stress (Ref 6) were first published by
Lame in 1852 and can be expressed as:
(Eq 14)
(Eq 15)
(Eq 16)
where p is pressure, Y is the outer radius divided by the inner radius (R
o
/R
i
), and r is the radius. The elastic
stress distribution is shown in Fig. 2 for an example tube (R
i
= 0.6R
o
) without any stress relaxation due to
plasticity or creep.