ASM Metals HandBook Vol. 8 - Mechanical Testing and Evaluation

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Multiaxial Testing Methods

The vast majority of material creep tests have been performed using uniaxial tests and this will probably

continue to be in the future. For modeling and validation purposes, multiaxial tests are indispensable due to

their more representative stress state. The following multiaxial test methods have been used (Ref 11):

• Tubular specimen under tension and torsion

• Pressurized tube under internal pressure and end loading

• Notched uniaxial specimen (“Bridgman notch”)

• Fracture mechanic specimen (for example, compact tension or single-edge notched bend)

• Biaxial plate

• Triaxial cruciform specimen under triaxial tension (Ref 12)

A tubular specimen under combined tension and torsion provides a small and inexpensive specimen tested in

machines that are commercially available. However, the stress state in the tension-plus-torsion test is such that

the second principal stress, σ

, is negative, which is not representative of pressurized industrial components.

A tube under internal pressure and axial end load provides also a relatively small specimen, and test services

are available from many laboratories. The loading is relatively well representative of the piping components,

with axial loading giving the possibility to vary the stress ratio of σ

/σ

. For thin-walled tubes with negligible

radial stress, the stress state is close to biaxial, but the stress state becomes triaxial when thick-walled tubes or

pipes are used with considerable (negative) radial stress. Pressurized tubes can be also tested with axial or

circumferential notches or cracks. For safety reasons, pressure tests should be carried out only in dedicated

concrete test cells.

A notched uniaxial specimen is certainly the cheapest multiaxial specimen and can be tested in a conventional

creep testing machine. Axial and diametral extensometry will provide information about the deformation during

the test. Recent work on notched bars provides detailed information about the stress distribution in the test

specimen with various notch geometries (Ref 13).

In a fracture-mechanics specimen, there is a high degree of tensile triaxiality at the notch tip when the local

deformation has been constrained by the surrounding material. This multiaxiality will probably enhance the

creep damage process.

The biaxial plate and the three-dimensional cruciform specimen provide an ideal tool for the researcher with a

possibility to vary the individual stress components, but they are extremely expensive to test in highly

specialized test rigs and only very few laboratories in the world can offer these services.

References cited in this section

11. J. Henderson and B.F. Dyson, Measurement of High Temperature Mechanical Properties of Materials,

M.S. Loveday, M.F. Day, and B.F. Dyson, Ed., National Physical Laboratory, London, 1982, p 171–

184.

12. D.J. Gooch and I.M. How, Techniques for Multiaxial Creep Testing, Elsevier, 1986, p 241

13. G.A. Webster, S.R. Holdsworth, M.A. Loveday, I.J. Perrin, and H. Purper, A Code of Practice for

Conducting Notched Bar Creep Rupture Tests and the Interpretation of the Data, High Temperature

Mechanical Testing Committee, ESIS TC11, 1999 (draft)

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)

References

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

6. R. Seshardi, Design and Life Prediction of Fired Heater Tubes in the Creep Range, J. Pressure Vessel

Technol. (Trans. ASME), Vol 110, Aug 1988, p 322–328

7. R.W. Bailey, Creep Relationships and Their Application to Pipes, Tubes and Cylindrical Parts under

Internal Pressure, Proc. Inst. Mech. Eng. A, Vol 164, 1951, p 425

8. I. Finnie and W.R. Heller, Creep of Engineering Materials, McGraw-Hill, 1959

9. P. Segle, P. Andersson, and L.Å. Samuelson, A parametric study of creep crack growth in

heterogeneous CT specimens by use of finite element simulations, Mater. High Temp., Vol 15, 1998, p

101

10. B.W. Roberts, Influence of Multiaxial Stressing on Creep and Creep Rupture, Mechanical Testing, Vol

8, ASM Handbook, ASM International, 1985, p 343–345

11. J. Henderson and B.F. Dyson, Measurement of High Temperature Mechanical Properties of Materials,

M.S. Loveday, M.F. Day, and B.F. Dyson, Ed., National Physical Laboratory, London, 1982, p 171–

184.

12. D.J. Gooch and I.M. How, Techniques for Multiaxial Creep Testing, Elsevier, 1986, p 241

13. G.A. Webster, S.R. Holdsworth, M.A. Loveday, I.J. Perrin, and H. Purper, A Code of Practice for

Conducting Notched Bar Creep Rupture Tests and the Interpretation of the Data, High Temperature

Mechanical Testing Committee, ESIS TC11, 1999 (draft)

Superplastic Deformation at Elevated Temperatures

Farghalli A. Mohamed, University of California, Irvine

Mechanical Behavior

Superplasticity is regarded as a creep phenomenon because it has been observed at temperatures at or above

50% of the melting point. Accordingly, in studying the mechanical behavior of superplastic alloys, it is

essential to establish the following four relationships that define the basic deformation characteristics associated

with a creep process: (a) the relationship between stress and strain rate, (b) the relationship between strain rate

or stress and temperature, (c) the relationship between strain rate or stress and grain size, and (d) the

relationship between strain contributed by boundary sliding and total strain.

The Relationship between Stress and Strain Rate

In general, the relationship between stress and strain rate is determined from two types of presentation. In the

first type of presentation, the data are plotted as stress against strain rate on a logarithmic scale. The slope of

this plot yields the value of the strain-rate sensitivity, m. In the second type of presentation, the data are plotted

logarithmically as strain rate against stress. The slope of such a plot represents the stress exponent, n. Under

steady-state conditions, n is equal to 1/m.

In early work on the mechanical behavior of superplastic alloys, the plots of stress versus strain rate and the

strain-rate sensitivity, m, were determined using several procedures in which specimens were tested in tension

using a machine operated at a constant crosshead speed (a screw-driven type or a servohydraulic system) (Ref

7, 8, 9, 10, 11, 12). A summary of these procedures is given in Table 1. While the application of these

procedures has provided useful information on the variation of m with strain rate during superplastic

deformation, there are several problems that need attention when some of the procedures are used. Primary

among these problems is the fact that true steady-state conditions cannot be achieved with a tensile machine

operated at constant crosshead speed. As a result, the plot of stress against strain rate, using logarithmic scales,

can produce erroneous results when used to determine the steady-state characteristics of superplastic flow, such

as the activation energy for deformation or grain-size sensitivity. In addition, the stress-relaxation method can

lead to inaccurate data if the procedure is not corrected for the effect of the stiffness of the testing machine.

Furthermore, it is difficult to obtain very low strain-rate data on machines with constant crosshead speed or a

constant strain rate. Due to this difficulty, it is not possible to unambiguously establish the characteristics of

superplastic flow in this range of strain rates.

Table 1 Early procedures for calculating strain-rate sensitivity, m

Method

m Representation Procedure

Reference

The curve for V

extrapolated to point a′

such that the total strain at

a′ is equal to the strain at

that of point b′ (max load)

on the curve for V

Ref 7

The true stress and true

strain rate at points a and

b which represent the

maximum loads on curves

and V

Ref 8

The curve for V

extrapolated to point c

which is above point d at

which the velocity change

is made

Ref 9

The load values

immediately before (point

d) and after (point e) the

velocity change are used

Ref 10

The values of true stress

and true strain rate are

calculated at various

points along the load-time

curve and log σ is then

plotted against dε/dt

Ref 11

The cross-head is

stopped at S and the

values of the true

stresses are then

estimated at different

points along the

relaxation curve. m is

determined from the

slope of the plot of log σ

vs log dσ/dt

Ref 12

Recently, the stress dependence of strain rate, or, equivalently, the strain-rate dependence of stress in

superplastic materials has been established under double-shear conditions (Ref 13, 14, 15). As indicated by

experimental evidence, double shear specimens, shown in Fig. 2(a), have the following advantages over tensile

specimens (Ref 16, 17, 18). First, a constant stress condition is maintained under constant load, and, as a result,

it is not necessary to use a creep machine with a compact cam lever to maintain constant stress as is required in

the case of a tensile specimen. Figure 2(b) schematically represents a simple creep machine that can be used in

testing double shear specimens. Second, the problem of plastic instability can be avoided. Finally, shear

deformation remains completely uniform over the gauge length of the specimen up to 120% strain. In

conducting double-shear experiments, two different, but complimentary, procedures are used.

Fig. 2 Double shear testing. (a) Specimen configuration. a = ⅜in., b = ⅛in., c = in., d

= ⅜in., d

= ¾in.

(b) Creep machine for testing double shear specimens

In the first procedure, specimens are deformed under creep conditions of constant load (equivalent to true

constant stress due to the use of double-shear specimen geometry). In this case, the steady-state strain rate is

recorded for the imposed stress and the data are plotted logarithmically as strain rate against stress. Steady

strain rates are obtained by performing uninterrupted tests and/or stress-change tests. In uninterrupted tests,

several identical specimens are tested at different stresses (constant temperature and constant grain size), and

steady-state strain rates are calculated from creep curves that represent plots of strain against time. Typical

examples of creep curves reported for commercial grade Zn-22Al at intermediate and low stresses are given in

Fig. 3(a) and (b) respectively (Ref 14). Examination of these creep curves shows that the alloy exhibits a

normal primary stage followed by a well-defined steady-state stage, and that the duration of the primary stage

increases with decreasing applied stress. In stress-change tests, the steady-state strain rates are measured at

several different stresses on a single specimen. In this procedure, a single specimen is crept at a constant stress

to within steady-state creep, and then the initial stress is changed in successive steps to a final value. For each

stress change, whether an increase or a decrease, sufficient strain is allowed to ascertain whether the new

steady-state stage has been reached. Figures 4(a) and (b), in which the creep rate is plotted as a function of

creep strain, provide typical examples of the application of this procedure to high purity Zn-22Al (Ref 14). The

data presented in these figures reveal the following features: (a) After a stress increase, the creep rate decreases

and reaches the new steady-state value (Fig. 4a), and by contrast, after a stress reduction, the creep rate

increases and reaches the new steady-state value (Fig. 4a); (b) After a stress decrease or a stress increase, the

amount of transient strain preceding steady-state creep is small, approximately 0.01 to 0.02.

Fig. 3 Shear strain versus time (creep curve) for commercial purity Zn-22Al having a grain size of 3.5

μm. Temperature, 463 K. (a) τ = 4.14 MPa. (b) τ = 0.825 MPa. Source: Ref 14

Fig. 4 The results of stress-change experiments for high-purity Zn-22Al having a grain size of 3.5 μm.

Temperature, 463 K. (a) Stress is increased from τ = 0.69 MPa in steps to τ = 6.0 MPa. (b) Stress is

decreased from τ = 6.9 MPa in steps to τ = 0.69 MPa. Source: Ref 14

In the second procedure, specimens are deformed using a testing machine that imposes a constant rate of

crosshead displacement (equivalent to a true constant strain rate due to the double-shear configuration of the

specimen). In this case, the steady-state flow stress is recorded for an imposed strain rate and the data are

plotted as stress against strain rate on logarithmic scales. The steady-state stresses are obtained from tests

conducted on several specimens at true strain rates (uninterrupted tests) or from cycling tests conducted on the

same specimen (strain rate-change tests). Figure 5 illustrates both types of tests using double-shear specimens.

In Fig. 5, the results on Zn-22Al are plotted as shear stress against shear strain up to strains of 125% (Ref 19).

The solid lines show the individual stress-strain curves for the various specimens tested at different constant

strain rates; the strain rates are indicated for each curve. The broken line shows the composite stress-strain

curve obtained by cycling a single specimen through a number of constant strain rates.

Fig. 5 Shear stress plotted against shear strain for specimens of Zn-22Al at a temperature of 484 K,

obtained either by testing each specimen at a single constant strain rate (solid line) or by cycling a single

specimen through a number of constant strain rates (broken line). Source: Ref 19

Typical experimental results on the stress dependence of strain rate in commercial purity Zn-22Al (Ref 14) are

shown in Fig. 6, in which shear strain rate, , is plotted as a function of shear stress, τ, on a logarithmic scale.

By identifying the individual datum points from the creep machine and universal testing machine (UTM) (a

screw-driven machine operated at a constant cross-head speed), it is clear that there is excellent agreement

between the two experimental procedures over intermediate stresses (and intermediate strain rates). In obtaining

the data for Fig. 6, the creep machine (constant-stress and stress-change experiments) was primarily used to

establish the deformation at low and intermediate stresses whereas the UTM (constant-strain-rate and strain-

rate-cycling experiments) was used to establish behavior at intermediate and high stresses.

Fig. 6 Shear strain rate versus shear stress (logarithmic scale) for commercial purity Zn-22Al having a

grain size of 3.5 μm at various temperatures from 433 to 493 K. Source: Ref 14

The Relationship between Strain Rate and Temperature

When the plastic deformation of a material is dictated by a thermally activated process, the strain rate for the ith

deformation mechanism may be expressed in terms of rate reaction theory as:

(Eq 1)

where A

is the frequency factor, τ is the applied stress, T is the absolute temperature, d is the grain size, ΔH

the activation enthalpy, and R is the gas constant. The activation enthalpy (ΔH

) is an important component in

the deformation equation, and its determination plays a key role in the process of identifying the rate-

controlling mechanism. According to Eq 2, ΔH

is uniquely defined by the expression:

(Eq 2)

Equation 2 suggests a simple and direct method of estimating ΔH

, which is referred to as an activation enthalpy

at constant stress and which, under creep conditions, is equivalent to the apparent activation energy, Q

Superplasticity is a thermally activated process, and Eq 2 provides the basis of various methods adopted to

determine the activation energy associated with the flow process. The following section outlines three methods

for measuring the activation energy for super-plastic deformation.

Rapid Change in Temperature at Constant Stress. If the temperature is changed very rapidly during a test and

the stress level remains constant, the apparent activation energy may be obtained from Eq 2 through the

expression:

(Eq 3)

where

and

are the instantaneous creep rates immediately preceding and following a small increase in

temperature from T

to T

(Ref 20, 21). Figure 7 provides an example of the application of this temperature

change procedure to commercial purity Zn-22Al (Ref 14).

Fig. 7 Shear strain rate versus strain for commercial purity Zn-22Al having a grain size of 3.5 μm,

showing the determination of the activation energy (Q) from the temperature change procedure. τ = 4.14

MPa; Q

ave

= 84.2 kJ/mole. Source: Ref 14

Variation of Strain Rate at Constant Stress. Under conditions of constant stress and grain size, Q

is given by Eq

2. A plot of logarithmic versus 1/T at a constant stress has a slope of -Q

/2.3R. The data of such a plot are

obtained from a logarithmic plot of strain rate against stress at a constant grain size and with a range of

temperatures (Fig. 6).

Variation of Stress at Constant Strain Rate. As mentioned earlier, tests on superplastic behavior are also

conducted at constant strain rate. According to an analysis of the activation energy for superplastic flow, under

conditions of constant strain rate, Q

is given by:

(Eq 4)

Therefore, a plot of logarithmic τ versus 1/T has a slope of Q

/2.3nR (Ref 21).

It is worth noting that due to the small values of the stress exponent for superplastic flow, the difference

between the apparent activation energy and the true activation energy is not significant and falls in the range of

the experimental error involved in measurement of Q

, that is, the correction for the temperature dependence of

the shear modulus may not be necessary (Ref 22).

The Relationship between Strain Rate or Stress and Grain Size

The relationship between strain rate and grain size can be established by conducting a series of tests at a

constant temperature and with a range of grain sizes. The data obtained from these tests are then plotted as

strain rate against stress on a logarithmic scale, as shown in Fig. 8. Finally, the grain-size exponent, s, is

obtained from the logarithmic plot of strain rate against stress by logarithmically plotting strain rate against

grain size at constant stress (Ref 14, 23).