THE INTERPRETATION OF FRACTURE TOUGHNESS DATA
University of Pretoria, Pretoria 0002, South Africa, and Laboratory for Advanced Engineering (Pty) Ltd, Pretoria, South Africa
Keywords: Fracture Toughness, Fracture Mechanics, Life Extension, Linear-elastic, Elastic-plastic, Petro-chemical
The rising importance of accurate fracture mechanics analyses for life extension purposes has placed larger responsibilities on the analyst to ensure accurate analysis inputs. Often, the most costly and confusing part of this process is to establish the fracture toughness data. This paper presents a summary of the different experimental methods that is used and puts into perspective the interpretation of such results in terms of the global objectives and restraints. It is shown that the standard methods for determining linear elastic, as well as elastic-plastic fracture toughness parameters are largely outdated and should be replaced with a single, unambiguous method.
Fracture mechanics has found its application for the most part, in the petro-chemical and power generation industries, as well as the aircraft industry, where the consequence of fracture failure makes it imperative that the existence of defects must be accepted and the effect thereof on the integrity of the structures must be controlled. These industries are on average and for the most part a few decades old. The original design lives are now often approached or exceeded and current economic and environmental concerns have put enormous pressure on extending the operational lives of these structures beyond what was originally intended.
Designs done a few decades ago have been based on analysis techniques that has since been improved and have incorporated safety factors (or ignorance factors) that may now be reduced. The life extension drive therefore implies that integrity assessments must now be redone with increased accuracy to enable design lives of old structures to be increased.
The fracture mechanics analysis process essentially entails comparing the crack driving force, which is a function of the component and crack geometry and the loading, with the material resistance to crack propagation, which is commonly called the fracture toughness. This comparison yields an estimation of the critical, or allowable crack size, or the critical, safe loading for a given crack size. The accuracy of any analysis is somewhat dependent on the inherent accuracy of the analysis method (any analysis method is based on a mathematical model of a real system or process, which always implies that some assumptions and approximations are made), but is in practice largely dependent on the accuracy of the input data.
It can, or should be argued, that the most influential factor on the accuracy of the analysis, is the accuracy of the estimation of the input loading. This, however is not seen as part of the fracture mechanics science. Be it as it may, often the most costly and confusing part of the process to establish the inputs for a fracture mechanics analysis is the determination of the fracture toughness data. The present paper summarizes the different experimental methods for determining fracture toughness and puts into perspective the interpretation of such results in terms of the global objectives and constraints.
Linear elastic fracture mechanics
Linear elastic fracture mechanics (LEFM) has been based on a stress field solution of the stresses in the vicinity of an infinitely sharp crack in an arbitrary elastic body with arbitrary loading. It was found that the stress state of any element near the crack tip is a function of the position of the element in relation to the crack, as well as a common parameter, called the stress intensity factor (K).
This factor is proportional to the applied stress and inversely proportional to the square
root of the crack length. The proportionality factor (b) is a function of the component
geometry and the crack geometry:
The stress field solution implies that the elastic stresses would reach infinity at the crack tip, which in a real material could not be the case. It is clear that a plastic zone would exist at the tip of the crack. It has been shown that the size of this zone is a function of K, implying that K still characterizes the elastic/plastic stress state at the tip of the crack and therefore could be used as a parameter to model the fracture process. This means that if K at fracture (Kc) is known for a material, failure of a component could be predicted if K, calculated for the specific loading and geometry of the component, is equal to Kc.
However, the size of the plastic zone would also depend on the constraint conditions (ie. plane strain or plane stress), leading to the conclusion that when Kc is determined for a material through a material test, it must be ensured that similar constraint conditions will exist in the test specimen, than will exist in the component for which the toughness value will be used in an analysis. Commonly, plane strain conditions is called for, since this would cause minimum toughness values to be obtained during the test. Such a toughness value is then named a plane strain fracture toughness (KIC).
Also, a further discrepancy exits in terms of the stress solution far away from the crack tip, where the stresses tend to disappear. This is obviously not true, since the stresses far away from the crack should tend to have the magnitudes of the undisturbed stress field. The complete stress field solution in fact has further terms:
The reason for only having considered the first, K dominated term, is that the fracture process takes place in the region near to the crack tip where x is small and the other terms are insignificant. Should the plastic zone become so large that it engulfs the region dominated by the K-term, K will no longer characterize the stress/strain state in the fracture process zone and can then not be used as a fracture parameter.
Elastic-plastic fracture mechanics
In materials where the toughness is large compared to the yield strength, LEFM will not apply. Elastic-Plastic Fracture Mechanics (EPFM) has evolved as a non-linear extension of LEFM in an energy formulation:
LEFM failurecriterion: (4)
Strain energy released by crack extension:
= energy required for crack extension. (5)
EPFM failurecriterion: J = Hsea = Jr (6)
The non-linear increase of strain with increasing stress during plasticity is then accounted for. This failure criterion can then be applied when the plastic zone engulfs the fracture process zone dominated by K.
The high toughness materials usually dealt with in EPFM tend to display a rising fracture toughness with crack extension (Jr does not remain constant after the initiation of cracking). This phenomena, although not well understood, is analogous to the strain hardening phenomena exhibited by some materials. This implies that the fracture criterion may be expanded to an instability criterion:
Such an analysis can result in less conservative results than an initiation analysis, since the reserve strength given by the rising fracture energy is taken into account. It however requires that toughness data as a function of stable crack growth needs to be determined. The analysis process followed is dealt with later in this paper.
LEFM (K) fracture toughness testing
An ASTM and other similar standards have been established for plane strain fracture toughness testing in terms of the K parameter . The use of either of two standard specimens is proposed, namely, a compact tension specimen (CT), or a three-point bend specimen (TPB), as shown in Figure 1. The test method involves fatigue pre-cracking from the machined notch (to ensure a sharp crack front) and thereafter loading to failure whilst recording the load and the load-line displacement. The crack length is measured after failure has occurred.
From this data, a candidate fracture toughness is calculated, using the load at what is said to be a nominal 0.2% crack extension (to ensure unambiguous results), as well as the known geometry factors for these specimens, which is then evaluated by testing its compliance to certain validation requirements stipulated by the standard:
a) This requirement attempts to ensure that plane strain conditions exist.
b) This avoids that a test be done on a specimen with an insignificant crack where the failure would be by plastic collapse and not fracture.
c) This requirement ensures that the plastic zone is sufficiently small compared to the remaining ligament so as to ensure that the fracture process zone is still dominated by the K-term.
d) This requirement limits the non-linearity of the load-displacement record, caused by stable growth or plastic deformation.
EPFM (J) fracture toughness testing
Standard methods exist for the determination of the initiation toughness (JIC) , as well as the rising fracture toughness in terms of J (the Jr - curve) . In essence, these material properties could be found from any test on a specimen for which the geometry factors (b and H) are known. The standard uses the same specimens described in the previous section but unfortunately does not employ the now known elastic-plastic geometry factors for these specimens, but rather an approximate method, since at the time of establishing the standard these were not known.
Both methods involve fracturing the specimen whilst measuring the crack extension, as well as the load and loadline displacement. The measurement of crack extension can be done using various methods. One such method, named the unloading compliance method, involves partially unloading the specimen at regular intervals during the ramping application of the fracture load, making it possible to calculate the crack extension at each unloading event as a function of the linear and decreasing unloading compliance. A typical test record is shown in Figure 2.
The test results in a record of J as a function of increments of crack extension (Da). For the determination of the initiation toughness (JIC) this R-curve is then extrapolated back to zero crack growth (or rather to a blunting line to take account of crack blunting without crack extension). The curve can also be used as a Jr - curve for an instability analysis.
The size requirements stipulated in the standards for ensuring plane strain conditions is much less stringent than for the KIC test, albeit that the JIC test standard claims that this initiation value can be converted to a conservative KIC value using the LEFM relationship. This has been severely criticised by some authors , but the following argument applies:
It is argued that the thickness dependence of KIC is due, not to the thickness dependence of the initiation toughness, but rather to the thickness dependence of the resistance curve, which comes into play because of the nominal 0.2% crack extension allowance inherent to the KIC test procedure. It is further argued that the true initiation toughness is almost independent of thickness because even in relatively thin specimens, the initial, very localized, crack extension takes place in plane strain. An explanation for this is shown in Figure 3. The points marked (x) are points where instability occur. For different thicknesses there are different R (or Jr ) curves, therefore causing the observed trend of increasing toughness at 0.2% crack extension for decreasing thicknesses. The stringent KIC test size requirements therefore are there to ensure that very little stable growth would occur, to obtain a valid initiation toughness.
The JIC test procedure circumpasses these difficulties by extrapolating back to the true initiation toughness.
INTERPRETATION OF TOUGHNESS DATA
Figure 4 depicts a load displacement record from a TPB specimen of a typical pressure vessel material. This record shows the contribution towards displacement made by the elastic deflection, plastic deflection and stable crack growth after initiation.
An EPFM initiation analysis on the specimen is shown in Figure 5. This graph depicts the material Jr-curve as a function of stable crack growth, as well as the crack driving force curves in terms of J as a function of the crack length (a+Da). The Jtot-curve shows that initiation has taken place at 38 kN. Also shown is the Jel-curve (being equivalent to K) at this same load. Kc = 8.5 MP(m) 0.5 would have been determined from this test, which would not have been a valid KIC value.
Figure 6 depicts the instability event at 48 kN.
If the toughness values are now to be used to determine the failure load for a center cracked panel (CCP), the following results would be obtained:
A LEFM initiation analysis using KC would result in a failure load of 310 kN. Figure 7 shows the true initiation event determined from an EPFM analysis using JIC, which results in a load of 260 kN. Figure 8 shows the instability event at a load of 290 kN.
From this example it is clear that non-valid KC results, or JIC values converted to KIC, used in a LEFM analysis, on a material that would exhibit significant plasticity before fracture, would give un-conservative results.
Based on the above arguments, the following guidelines can be given for the interpretation of fracture toughness data for use in fracture analysis (rp = plastic zone size):
IF LEFM is applicable for the structure (rp << W-a,a):
IF plane strain in the structure (rp << B):
IF valid KIC is available
Use for initiation analysis (will be thickness and geometry independent)
Use JIC to estimate KIC (rp << W-a,a for test)
ELSE (rp » B in structure)
IF Kr or Jr curve is available
Use in instability analysis (will thickness dependent)
Use KC (will be geometry and thickness dependent)
ELSE (rp » W-a in structure)
IF plane strain in structure (rp < B)
Use JIC for initiation
IF Jr-curve is available
Use for stability analysis (will be thickness dependent)
Use Jc (will be geometry and thickness dependent)