Review The Double-Torsion Testing Technique For Determination of Fracture Toughness and Slow Crack Growth Behavior of Materials: A Review
Review The Double-Torsion Testing Technique For Determination of Fracture Toughness and Slow Crack Growth Behavior of Materials: A Review
Review The Double-Torsion Testing Technique For Determination of Fracture Toughness and Slow Crack Growth Behavior of Materials: A Review
Review
The double-torsion testing technique for
determination of fracture toughness and slow crack
growth behavior of materials: A review
A . S H YA M , E . L A R A - C U R Z I O ∗
Metals and Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6069, USA
E-mail: laracurzioe@ornl.gov
Published online: 15 June 2006
The double-torsion testing technique for fracture toughness and slow crack growth
determination has been critically reviewed. The analytical compliance and finite element stress
analyses of the double-torsion test specimen are summarized. The fracture toughness and
crack growth testing procedure using this test configuration is described along with the
applicable relationships. The strengths and limitations of this testing technique vis-à-vis other
standardized techniques have been critically evaluated. While the double-torsion test method
has some limiting features it has been demonstrated that its applicability is not limited as long
as these are addressed correctly. Recommendations for conducting double-torsion experiments
have been provided and potential avenues for improvement of this test method have been
identified. It is concluded based on the review that standardization of this test method is
required in order to make it more practicable. C 2006 Springer Science + Business Media, Inc.
where, L represents the total length of the torsion bar. where, is the load-point displacement and its value is
Derivation of a similar expression for bars of rectangu- small compared to the moment arm Sm (the symbols have
4094
been identified in Fig. 1). Equation 2 can be rearranged to double-torsion testing. The independence of the stress in-
give the analytical expression for the compliance (C) of a tensity factor value with crack length, however, is valid
double-torsion test specimen only for a range of crack lengths in the double-torsion test
specimen because edge effects lead to a deviation from
3S 2 a the linear crack length-compliance relationship.
C≈ ≈ 3m (3) Experimentally, it has been found for several materi-
P St G
als that the compliance varies with crack length in the
Fuller [43 ] derived a more exact version of Equation 3 following manner [25, 44, 46]
with a finite beam thickness correction factor ψ(τ ) given
by
C= = Ba + D (9)
P
3Sm2 a
C≈ (4) where, B and D are scaling constants. The experimen-
St 3 Gψ (τ ) tal form of the compliance relationship is slightly differ-
ent from the expression in Equation 3. While this does
where, τ = 2t/S is the thinness ratio and for values up to τ not alter the form of the stress intensity relationship, it
= 1, a simplified expression with an accuracy better than does have implications for slow crack growth measure-
0.1 percent is given by [43] ments. The linear compliance-crack length relationship
and other assumptions involved in the derivation of the
ψ = 1 − 0.6302τ + 1.20τ exp (−π/τ ) (5) analytical expression for compliance have been exam-
ined in Section 7. In the next section, these analytical
The validity of this thickness correction factor has been expressions are compared to predictions obtained with fi-
experimentally confirmed with the evaluation of glass ce- nite element stress analyses of the double-torsion testing
ramic test specimens [49]. This factor can be significant configuration.
for thick beams (relative to width) and arises due to con-
tact stresses between the two rectangular bars. If it is
further assumed that the shape of the crack front remains 3. Finite element stress analysis of the
unchanged as the crack propagates, then the following double-torsion loading geometry
expression is obtained for the elastic strain energy release The first comprehensive finite element stress analysis
rate (G) [50] of the double-torsion test specimen was performed by
Trantina [51]. This analysis concluded that most assump-
P2 dC 3P 2 Sm2 tions inherent in the derivation of the analytical expres-
G= = (6)
2t da 2St 4 Gψ sion are reasonable. The stress intensity factor calculation
from the analytical analysis Equation 8 was shown to be
where the Young’s modulus is E = 2G (1 + ν), nearly equal to the value obtained from the finite element
where ν is Poisson’s ratio. With the application stress analysis. Additionally, it was shown that the stress
of the linear elastic fracture mechanics (LEFM) re- intensity factor remains nearly constant (to within 5%)
lationship [50] in the range of crack lengths; a > 0.55∗ S and unbroken
ligament lengths of (L − a) > 0.65∗ S. This implies that
1 2 the range of crack lengths for which the stress intensity
K = E G / (7)
factor is independent of the crack length is a function of
the length to width (L/S) ratio of the test specimen. For
where, E = E/(1 − ν 2 ) for plane strain and E = E for L/S = 2, the middle 40% of the test specimen displays
plane stress, the expression for stress intensity factor takes crack length independent stress intensity whereas for L/S
the following form = 3, the middle 60% of the test specimen displays this
1/ 2 property.
3 More recently, Ciccotti and co-workers [32, 33, 47,
K = P Sm for plane strain (8a)
St (1 − ν) ψ
4 52] performed detailed three-dimensional finite element
stress analyses for “large” double-torsion test specimens
(L > 17 cm and S > 6 cm) and concluded that appre-
1/2 ciable deviations occurred from the classical analytical
3 (1 + ν) solution predictions of strain energy release rate (G in
K = P Sm for plane stress (8b)
St 4 ψ Equation 6). They provided correction factors to account
for experimental variables such as crack shape, groove
The stress intensity factor given by Equation 8 is a func- width and depth, notch length and test specimen geom-
tion of the applied load, the test specimen geometry and etry and found deviations (up to 40%) in the value of
Poisson’s ratio but independent of crack length. The lat- strain energy release rate (G) from the analytical solution
ter characteristic is one of the most attractive features of [47]. Of all the effects considered, the effect of test spec-
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imen geometry on the calculated stress intensity factor [3, 53]. The pre-crack originates from the tensile side of
was found to be the most significant and complex. They the double-torsion specimen and the shape of the result-
provided tables of correction factors for various combina- ing crack evolves before reaching a stable crack front. It is
tions of values of thickness (t): width (S): length (L). No therefore desirable to reach the steady state crack front be-
general conclusions could be obtained from their analy- fore performing fracture toughness or slow crack growth
sis except that the discrepancies in stress intensity values measurements. Depending on the material and specimen
predicted by analytical and numerical solutions decrease geometry, the stable crack front shape may be reached
with decreasing the thickness of the test specimen [52]. after 2–5 mm of pre-crack size ahead of the notch.
Additional finite element studies on smaller sized test The determination of fracture toughness using a pre-
specimens having commonly used length to width ratio cracked test specimen should be carried out at a fast load-
of 2:1 or 3:1 and dimensions representative of experi- ing rate so as to avoid slow crack growth. This is because
mental double-torsion test specimens are necessary. This slow crack growth prior to failure would lead to artifi-
would help in standardized test specimen designs with cially lower values of fracture toughness. For example,
well-defined operational range of constant stress intensity it has been shown that the fracture toughness value of
regions. yttria stabilized zirconia (YSZ) increases with increas-
ing crosshead displacement rates until a displacement
rate of 4 mm/min and thereafter remains constant [54].
4. Double-torsion testing for fracture toughness The experimental error in measuring the values of vari-
determination ables included in the formula for fracture toughness in
Fracture toughness can be determined in double-torsion double-torsion is comparable to that of six other geome-
testing simply by loading a pre-cracked test specimen tries (chevron notched four-point bend, double cantilever
rapidly and recording the maximum load at failure (PIC ). beam, direct crack measurement, single edge notched ten-
The fracture toughness expression is obtained by substi- sion, single edge notched specimen tested in three-point
tuting the failure load in Equation 8 bend and single edge notched tension, single edge notched
specimen tested in four-point bend) and better than inden-
1/ 2 tation strength by four-point bending [55]. In summary,
3
K IC = PIC Sm for plane strain double-torsion testing provides fracture toughness values
St (1 − ν) ψ
4
comparable to those obtained from other standardized test
(10a) methods provided the above-mentioned experimental re-
quirements are satisfied.
1/ 2
3 (1 + ν)
K IC = PIC Sm for plane stress (10b)
St 4 ψ
5. Double-torsion testing for crack growth study
There are, however, some experimental aspects that need One of the most important characteristics of the double-
to be considered both during precracking and fracture torsion testing approach is that the rate of slow crack
toughness testing. The tip of the precrack should be in growth can be derived without having to monitor the crack
the region where stress intensity factor is independent of length on a continuous basis. Additionally, the cyclic fa-
crack length. A small precrack leads to an artificially en- tigue crack growth response of a material can be deter-
hanced fracture toughness value and a small remaining mined if crack length is monitored continuously. The ini-
ligament length results in a value lower than the fracture tial analyses for load relaxation and constant displacement
toughness of the material [44]. This can be avoided by rate techniques were given by Evans [46] although several
making a starter notch with a length such that any crack corrections have since then been proposed to refine that
that initiates from it is in the constant stress intensity re- approach (Section 8). We describe in the present section,
gion. It is important to conduct the fracture toughness test four commonly employed procedures for crack propaga-
on a precracked test specimen since a blunt notch (without tion studies with the double-torsion testing configuration.
the precrack) would not allow equivalent stress intensifi-
cation at the notch tip. Precracking is generally done at a
slow crosshead displacement rate until a load drop can be 5.1. Load relaxation technique
discerned or a load plateau is reached where the increase The load relaxation technique [46] is commonly employed
in load is balanced by relaxation of the test specimen from to indirectly obtain the sub-critical crack propagation be-
crack growth. A notch with smaller width and a tapered havior of brittle materials. According to this technique,
end, such that it goes from full thickness to a thin liga- a pre-cracked double-torsion test specimen is loaded to
ment at the tensile surface, facilitates precracking at loads below the expected fracture load (0.90–0.95∗PIC ). The
lower than PIC [45]. Other methods of precracking, such crosshead of the testing machine is then held at a fixed
as indenting the region in front of the notch to generate position and the increase in compliance of the test speci-
sharp pre-cracks contiguous to the machined notch, have men from sub-critical crack growth leads to a relaxation
also been successfully applied in double-torsion testing of the load with time. To illustrate this concept, Fig. 3
4096
load and average velocity, the corresponding stress inten-
sity value can be calculated from Equation 8. In principle,
the entire v-K curve can be obtained from a single load
relaxation experiment. In practice, however, this method-
ology works better at relatively higher crack growth rates
(>10−6 –10−7 m/s [44]) due to temperature fluctuations
affecting load measurements at very low velocities. This
method is also susceptible to spurious factors, such as
load train relaxation, which is discussed further in Sec-
tion 7. It is for this reason that it has been recommended
to generate complementary portions of the v-K curve by
combining the load relaxation technique with one of the
two techniques discussed below [56].
Figure 3 An illustration of temporal load variation obtained from a load
relaxation test in a double-torsion test specimen of 3-YSZ.
5.2. Constant load technique
The earliest application of the double-torsion loading con-
presents a load relaxation curve for a test specimen of figuration to measure slow crack growth was through the
3-YSZ. Mathematically, this can be described by differ- constant load method [57]. The average crack velocity
entiating Equation 9 with respect to time to obtain the corresponding to the stress intensity factor, which is cal-
following relationship culated from the applied constant load, can be obtained
by measuring the crack length before and after the ex-
d dP da periment and the elapsed time. The main disadvantage
= (Ba + D) + PB (11)
dt dt dt of this technique is that only one data point can be ob-
tained per experimental run since crack length measure-
where, the left hand side (LHS) of the equation equals zero ments are required. This technique, however, is suitable
if the crosshead is arrested. Additionally, if remains for the calculation of very low crack velocities where load
constant, and the tip of the crack remains in the crack relaxation measurements cannot be performed [44]. Ad-
length independent stress intensity region, then ditionally, at elevated temperatures, inelastic deformation
and machine relaxation render constant load crack growth
P (Ba + D) = Pi (Bai + D) = P f Ba f + D (12) measurements as the most reliable technique to measure
slow crack growth rates [44].
where, the subscripts ‘i’ and ‘f’ denote the initial and
final loads and crack lengths. The physical meaning of
Equation 12 is that the increase in the compliance of the 5.3. Constant displacement rate technique
test specimen due to increase in crack length is exactly Evans [46] introduced another technique for evaluating
compensated by its temporal decrease in load. By setting slow crack growth behavior that involved changing the
the LHS of Equation 11 equal to zero and rearranging displacement rate incrementally. In this technique, the
it with Equation 12, an expression for the crack growth crosshead is moved at a constant rate and the load value
velocity (v = da/dt) can be derived is allowed to reach a plateau where the increase in load
from crosshead movement is balanced by relaxation of the
−Pi D dP test specimen load from crack growth. If the plateau load
v= 2 ai + (13)
P B dt value is given by P, Equation 11 reduces to the following
form (with dP/dt ∼ 0)
which reduces to the following simplified expression for
the case ai (D/B), i.e. large crack lengths or high mod- d
= PBv (15)
ulus materials dt
−ai Pi d P The crack velocity can be calculated from the displace-
v= (14) ment rate and the value of the load plateau. The main
P 2 dt
disadvantage of this technique is also that only one data
Even though Equation 14 is popularly applied, it is noted point can be obtained per test run even though crack length
that in many cases the assumption that ai is much larger measurements are not required. By changing the displace-
than (D/B) is not true. Equation 14 should be applied ment rate over a few decades, Evans showed that the slow
when material availability is at a premium and experimen- crack growth exponent can be determined from this test-
tal compliance-crack length curves cannot be generated. ing methodology [46].
However, the slow crack growth exponents calculated us- Weiderhorn [58] and later Quinn and co-workers [3,
ing Equations 13 or 14 will remain identical. For a given 59] successfully combined the constant displacement
4097
rate technique with the load relaxation technique to ob- Chevalier and co-workers [53, 61, 63–69] have exten-
tain slow crack growth information without having to sively studied the static and cyclic crack growth behavior
make any crack length measurements. In this method, of zirconia using the double-torsion configuration. Using
the crosshead is moved at a slow, constant rate un- this testing approach they found the existence of thresh-
til the load decrease due to slow crack growth exactly olds for crack growth (in both static and cyclic conditions)
offsets the load increase due to crosshead movement. and the presence of environmentally-assisted degradation
The peak load (Pi ) and the crosshead displacement rate mechanisms [61]. It has also been demonstrated that a
(d/dt) can now be used to obtain the crack veloc- cyclic effect exists in several materials in terms of faster
ity (vi ) according to Equation 15. In this version of crack growth at equivalent value of stress intensity fac-
the test method, the crosshead is arrested at the peak tor when compared to stress corrosion alone [61, 62]. A
load and a load relaxation experiment is subsequently cyclic effect may also exist in that the crack propagation
carried out. The initial crack velocity (vi ) from the threshold values are lower under cyclic fatigue loading
load relaxation experiment can be obtained by substi- [61].
tuting P = Pi in Equation 13. Assuming that the ini-
tial velocity obtained by application of Equations 13
and 15 is the same, the following relationship is 6. Advantages of the double-torsion testing
obtained: technique
Some of the advantageous characteristics of the double-
(d/dt) d Pi (Bai + D) torsion testing configuration for fracture toughness and
vi = =− (16)
Pi B dt Pi B slow crack growth characterization have already been
identified in the previous sections. The simple test speci-
This relationship allows the expression of the compliance men geometry and loading configuration involving four-
of the specimen at the beginning of the load relaxation point loading of a rectangular bar results in a low-cost
experiment as [58] setup [43]. Even the rear supports of the test specimen
(Fig. 1) are not critical since they are designed only for
(d/dt) convenience in mounting and aligning the test specimen
Bai + D = − (17)
(d Pi /dt) [25]. The most important characteristic of this testing con-
figuration, as mentioned earlier, is that the stress intensity
The initial compliance is related to the compliance at any factor resulting from it is independent of (or has a weak
other instant in the load relaxation experiment according dependence on) the crack length in the mid-section of the
to Equation 12. Substitution of Equation 17 in Equation 13 test specimen. The above factors are responsible for the
therefore, allows the calculation of crack velocity accord- fact that the application of this test method is commonly
ing to extended to elevated temperatures and controlled envi-
ronments, e.g. [15]. In addition, it has been noted that a
d/dt Pi d p/dt low compliance loading system is not required to apply
v= (18) the compressive loads required for double torsion testing
B d Pi /dt P2
[34]. Although the tapered width double cantilever beam
Since Equation 18 does not involve any crack length test specimen also has the property of the stress intensity
term, it is ideally suitable for applications such as factor being independent of crack length, these test speci-
elevated temperature and/or controlled environment mens require a lot more material for machining compared
testing. to a flat double-torsion test specimen [70]. The double-
torsion test specimen geometry is also ideally suited for
material manufactured in a planar configuration such as
5.4. Cyclic fatigue crack growth polycrystalline diamond compacts [31]. This testing con-
measurements figuration is also uniquely suitable for determining the
There have been few reports of cyclic fatigue studies us- fracture toughness of rocks [33], adhesive joints [71] and
ing the double-torsion testing methodology [45, 60–62]. diffusion bonds [72].
This is due to the fact that cyclic fatigue studies typically In double-torsion test specimens, precracking is
require continuous monitoring of crack length with load- achieved in a controlled manner and can be detected from
ing cycles, e.g. [61]. In principle, the crack length in a deviations from linearity in the load versus displacement
double-torsion test specimen can be estimated from the curve [43]. Unlike double-torsion, some other geometries
compliance of the test specimen. However, it has been for measuring fracture toughness involve separate fixtur-
reported that accurate determination of the crack length ing for precracking test specimens (e.g. the precracked
from the compliance of the test specimen can be difficult beam method).
in practice, especially for brittle materials [45]. This is at- Some researchers have claimed that values of the frac-
tributed to the small load point displacements associated ture surface energy calculated from double-torsion (DT)
with the deformation of stiff materials and the inherent testing are more accurate when compared with other con-
noise associated with thermal fluctuations, for example. figurations such as double cantilever beam (DCB) or sin-
4098
gle edge notch bend (SENB) testing [73]. Others have re-
ported comparable values of fracture energy release rates
obtained by DT and DCB test specimens [74] whereas
others have reported that the DCB test specimen geom-
etry has the highest tendency for slow crack growth and
yields higher values of fracture toughness compared to
SENB or DT test specimen geometries [75]. For hot-
pressed SiC, for example, comparable KIC values were
reported from Hertzian indentation and double-torsion
techniques [76]. However, uncertainties in the fracture
toughness determination from indentation techniques are
well documented and therefore fracture toughness values
from double-torsion are deemed more reliable than those
attempted from indentation methods [54]. Figure 4 The experimental variation of compliance with crack length in a
As mentioned in Section 5, slow crack growth can be double-torsion test specimen leads to the mid-region in the test specimen
investigated in three complementary modes using dou- with stress intensity values independent of crack length. The theoretically
ble torsion without the use of crack opening displace- predicted compliance variation with crack length has also been illustrated
in this figure.
ment gages. Evans and Williams [25] have demonstrated
that slow crack growth characteristics are extremely com-
parable for several materials using the DT and DCB
test specimen configurations. Bhaduri [18] reported that e.g. [40]. The justification for the former is that plane
the slow crack growth exponent calculated from inden- strain fracture toughness is suitable for brittle materials.
tation techniques is comparable to that calculated from Fracture modality selection (plane stress/strain) can in-
double-torsion. While studying stress corrosion cracking duce errors in the calculation of the stress intensity factor
in steels, Briggs et al. [77] found that an optical method according to Equation 8. The other assumption is that the
for measuring crack growth rates and the load relaxation loading at the tip of the crack is purely mode I with a
version of the double-torsion test method gave similar negligible mode III component [43]. This was shown to
results. Quinn and Quinn [59] carried out a compari- be a reasonable assumption by Evans and co-workers [25,
son of the published slow crack growth exponent val- 46], who demonstrated the experimental critical strain en-
ues in hot pressed silicon nitride between room tem- ergy release rate (Gc ) compared well with mode I values
perature and 1400◦ C. They demonstrated that in some calculated from Equation 7.
instances, the slow crack exponent values were similar The most important characteristic of the double-torsion
between double-torsion and other test methods such as test specimen is the lack of dependence of the stress inten-
static and dynamic fatigue. Quinn [3] also demonstrated sity factor on crack length in approximately the mid-range
the coincidence of the slow crack growth exponent be- of crack lengths in the test specimen. Outside this region,
tween double torsion and static fatigue testing for alumina the compliance-crack length relationship becomes non-
at 1000◦ C. Slow crack growth characteristics calculated linear due to end effects and this has been schematically
from static/dynamic fatigue experiments are indirect cal- illustrated in Fig. 4 [43]. One assumption in the analyti-
culations whereas in the double-torsion test the slow crack cal derivation of this relationship is that the torsion bars
growth behavior of long cracks can be deduced directly deform independently with no contact stresses and negli-
[9]. It has been shown by Sudreau et al. [78] that indi- gible deflection beyond the crack tip. The reason for the
rect methods can induce large errors in interpretation of slopes of compliance versus crack length curve (Fig. 4)
slow crack growth characteristics. Double torsion has also becoming lower than that of the constant slope region
been successfully applied to investigate the time depen- at smaller crack lengths is due to interaction between
dant fracture of composite materials [19, 20] and tough the two torsion bars in the test specimen contributing
polymers [21]. significantly to the total deformation of the torsion bars
[25]. At small remaining ligament lengths, the elastic de-
flection of the uncracked portion of the test specimen
7. Limitations of the double-torsion testing does not remain negligible thus increasing the slope of
technique the compliance-crack length curve in Fig. 4. Even for
It is important to be aware of the major assumptions in- the so called region of constant driving force, a number
volved in double-torsion testing that could limit its valid- of researchers have found that the stress intensity fac-
ity under certain circumstances. One assumption involves tor could be a function of the crack length [4, 5, 14,
the choice of state of plane strain or stress to describe the 44, 56, 63, 64, 79, 80]. Very often, a hysteresis in the
stress intensity at the crack tip according to Equation 8a v-K curve calculated from load relaxation is shown as
or 8b. While earlier researchers favored the plane strain indication of the dependence of stress intensity factor on
expression for fracture toughness calculation [43,45] re- crack length [44, 56, 63, 64, 79, 80]. The dependency of
cent calculations are based on the plane stress expression, stress intensity factor on crack length has been attributed
4099
to several factors which can be broadly classified as
follows:
4103
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4104