The 'local approach' to cleavage fracture

TWI Bulletin, March/April 1994

Christoph Wiesner studied materials science and engineering at the University of Erlangen Nuremberg, Germany, gaining his degree in 1987. Following this, he was employed as a Scientific Engineer at the Swiss Institute of Technology in Lausanne, Switzerland, where he obtained his PhD in 1991. He joined TWI in the same year and has since gained experience in structural integrity evaluation. He currently leads the Defect Assessment Section of the Structural Integrity Department.

A new method to assess the fracture behaviour of metals, known as the 'local approach' to fracture, has been developed in the past ten years. Christoph Wiesner reviews the methodology for cleavage fracture and recent applications.

For a more precise analysis, complex linear and non-linear finite element calculations are necessary to obtain global fracture parameters such as the linear-elastic stress intensity factor, K, or the elastic-plastic J-integral and crack tip opening displacement, describing the crack driving force. If such a computation has established the stress-strain conditions in a cracked structure, an alternative approach is the application of a local fracture criterion.

Local criteria are based on the micromechanisms of the fracture event, and they assume that fracture takes place in a given region of material when a critical stress or strain state is reached in the vicinity of the flaw under consideration. Since the fracture event is described locally, it is possible to deal with non-symmetrical and non-isothermal loading conditions without knowing the conventional fracture toughness under such conditions. In addition, temperature, strain rate and size effects (the last including the influence of plastic constraint) can be readily accommodated, if their influence on the local fracture mechanism is known.

This method, known as the 'local approach' to fracture, enables precise predictions of the fracture behaviour to be made without having to conduct conventional fracture mechanics tests. Obviously, such tests can be used to assess whether the model accurately describes the fracture event in a given metal at a given temperature. The critical load criterion in conventional fracture mechanics specimens, whose stress and strain fields can be described by a single global parameter, will be reached when the value of the parameter reaches its critical value, e.g. K Ic for cleavage fracture.

This article assesses the current state of knowledge regarding the methodology and application of the 'local approach' to cleavage fracture for ferritic steels and their weldments. The model for the cleavage fracture mode is limited to the lower shelf of the toughness temperature transition curve where no stable tearing occurs prior to cleavage failure. The technically important transition region requires a combination of 'local approach' models for ductile rupture (which is not dealt with here) and cleavage fracture. This article is based on a TWI Members' Report. [2]

Micromechanisms

Cleavage fracture is a sequential process involving crack nucleation and propagation. Crack nucleation occurs in most steels at brittle grain boundary particles (carbides, etc) due to the stress concentration caused by dislocation pile-ups at these particles. Inclusions and/or second phase particles can also provide crack nuclei in steel. This implies that some localised plastic deformation always precedes cleavage fracture. The critical stage of cleavage fracture is the propagation of the slip initiated crack nuclei into the surrounding steel matrix.

Cleavage fracture initiation at grain boundary carbides has also been suggested for some C-Mn weld metals exhibiting large grain boundary carbides. However, the majority of cleavage fracture events in weld metals are triggered at non-metalllic inclusions or other second-phase particles.

Cleavage fracture in ferritic steels and their weldments thus occurs when the normal stress acting on stable microcracks exceeds a critical value. The stable microcracks are initiated in brittle carbides, large intermetallic inclusions or other microstructural features due to dislocation slip.

Methodology of the 'local approach'

The 'local approach' to cleavage fracture is based on the statistical weakest link concept which postulates that failure of a body of material containing a large number of statistically independent elements is triggered by the failure of one of the elements. [6] Two assumptions are made regarding the failure mechanism in each element: [7,8]

• Microcracks are formed at the onset of plastic deformation in each element;
• The microcrack becomes unstable when the normal stress acting on it has reached a critical value, or when the crack has a critical length.

The probability of failure at a given applied stress can be calculated by summing the probabilities of finding a crack longer than the critical crack length in each element. The cumulative survival probability of the body under consideration is then given by the product of the survival probabilities of all elements and is given by:

1 - P f exp(- σ w/ σ u) m     [1]

where
1 - P f is the survival probability, σ w is the so-called Weibull stress (in recognition of the early statistical work of Weibull [9] , σ u is a particular value of the cleavage fracture stress and m is the Weibull distribution shape parameter. au and m will be referred to as 'local approach' parameters.

Details of the deduction of the above relation are given in Ref.2. To determine the 'local approach' parameters, a statistical sample of P f and σ w data is analysed by the maximum likelihood method or regression analysis. [10]

It is assumed in Eq.[1] that each element of the stressed body in which plastic flow occurred contributes to the failure probability. However, it is well established that there is a minimum stress threshold value below which cleavage fracture is not initiated. Such a threshold stress value can be accommodated in the failure probability distribution, as shown by Bakker and Koers. [11]

The applicability of the 'local approach' model is based on Eq.[1], which correlates the failure probability of a cracked body with the stress distribution in the process zone (expressed by the Weibull stress). The probability that a flaw, which triggers the cleavage failure, lies in a certain region of the process zone is determined by the contribution of this region to the Weibull stress.

It can be seen from Eq.[1] that the failure probability at a given applied local stress is described by a two parameter Weibull distribution. The failure probability or, equivalently, the statistical distribution of the Weibull stress at cleavage failure, is unique for a material and independent of specimen geometry and size and crack size and shape. This means that geometry and constraint effects, asymmetric loading and thermal stress conditions are taken into account via the calculation of the Weibull stress which considers the local tensile stresses (which trigger cleavage fracture) over the process zone for the situation considered.

Applications

The 'local approach' to cleavage fracture has been used to study a variety of engineering fracture problems in ferritic steels as summarised below:

Temperature and strain rate dependence and estimation of scatter of fracture toughness

For crack opening mode loading and small scale yielding, the elastic-plastic stress distribution ahead of the crack tip can be calculated analytically. Inserting the analytical field into the equation for the Weibull stress and combining the result with Eq.[1], gives a relation between fracture toughness and yield strength, thickness and fracture probability. A typical fracture toughness transition curve calculated for fracture probabilities of 10, 50 and 90%, is shown in Fig 1, together with experimental results. [7] The 10-90% failure probability range contains most of the experimental results.

Similarly, Pineau ^[12] has applied the 'local approach' methodology to the effect of strain rate in different steels and found that the effect could be predicted by measuring the yield strength at the strain rates considered. The influence of irradiation on the fracture toughness may also be predicted using the local approach methodology for cleavage fracture by quantifying the effects of irradiation on the yield strength. ^[13]

The above predictions are very much facilitated by the assumption that the 'local approach' parameters are temperature independent. This assumption is presently somewhat disputed, especially if the steep fracture toughness transition region is reached.

The effect of warm pre-stressing on fracture toughness and thermal shock predictions

If a cracked structure is loaded above the operating load level at temperatures exceeding the brittle/ductile transition temperature, the fracture toughness at the operating temperature is markedly increased. This so-called warm pre-stress effect is caused by the following factors (with increasing importance):

• Cold working of the microstructure resulting in increased cleavage resistance;
• Decrease of crack tip acuity and related modification of the stress/strain field ahead of the crack;
• Local compressive residual stress due to inhomogeneous plastic deformation.

French work ^[14] has quantified these different aspects using numerical calculations performed on axisymmetrically cracked round tensile bars and local approach methodologies. A good correlation was found between measured and predicted fracture toughness.

In Ref.15 the 'local approach' methodology was applied to a thermal shock experiment in a pressure vessel. 5000 litres of liquid nitrogen was poured in the test cylinder while monitoring the crack opening displacement and the temperature. Cleavage occurred at a temperature of -38°C. The thermal shock experiment was modelled by finite elements and the Weibull stress, the failure probability, and the probability density, F, were calculated as a function of temperature in the thermal shock experiment, see Fig.2. It can be seen that the maximum probability density of cleavage failure occurs at a temperature of approximately -41°C, close to the experimental result.

Size effects on cleavage fracture behaviour

There are two reasons for the observed effect of section size on fracture toughness:

• The effect of plastic constraint, which increases with increasing thickness, crack depth to specimen width ratio, etc;
• The statistical weakest link effect, i.e. the probability of finding a weakest spot which will initiate cleavage fracture increases with increased stressed volume sampled by the crack tip ( i.e. with increasing thickness for through-thickness cracks).

The thickness effect of bending fracture mechanics specimens was also predicted by Bakker and Koers. ^[11] Good correlation with the observed thickness effect was obtained as long as no stable tearing preceded cleavage fracture. It is worth noting that the 'local approach' methodology and the weakest link concept are equivalent formulations of the statistical size effect and are identical for pure cleavage fracture.

More recently, the 'local approach' to cleavage fracture has been applied to fracture mechanics specimens with crack depth to specimen width (a/W) ratios of less than 0.4. ^[16,17] Deep (a/W = 0.5) and shallow (a/W = 0.05) cracked single edge notch bend specimens were analysed to calculate failure probability as a function of temperature. Most of the experimental values lay within the predicted 10-90% failure probability range for both deep and shallow cracks.

Finally, the cleavage fracture behaviour of three- and four-point bend fracture mechanics and structurally representative centre cracked wide plate specimens was predicted using the 'local approach'. ^[11] The measured fracture toughness values lay in the predicted 90% confidence band as long as tests in which stable tearing preceded cleavage were not considered.

Cleavage fracture initiation in the transition region

Cleavage fracture initiation in the ductile to brittle transition region is affected by two factors:

• Large scale yielding occurs prior to cleavage fracture;
• The crack elongates by ductile tearing before cleavage fracture intervenes.

The influence of large scale yielding has been studied by Mudry et al. ^[8,18] They calculated the Weibull stress of both compact tension and centre-cracked panel fracture mechanics specimens as a function of the applied load. For high applied loads, the calculated Weibull stresses were markedly smaller than those obtained using the small-scale yielding limit. Since the cleavage fracture probability is related to the magnitude of the Weibull stress, a decrease in its value is reflected by a decrease in fracture probability or, equivalently, an increase in fracture toughness, as observed experimentally in low constraint specimen geometries.

The effect of stable tearing preceding cleavage fracture has been satisfactorily predicted using the 'local approach' methodology. ^[19] However, the authors assumed that the two failure mechanisms, i.e. ductile tearing and cleavage fracture, were independent. As pointed out by Knott ^[20] this assumption is unrealistic on a microstructural level. The stable tearing mechanism, which involves nucleation, growth and coalescence of voids at large inclusions and/or other particles, 'deactivates' these microstructural features for cleavage initiation and propagation. Therefore, the population of inclusions and/or grain boundary particles for cleavage initiation after stable tearing exhibits an average size which is smaller than without stable tearing.

More recently, a method which couples both the ductile tearing and the cleavage fracture failure modes was presented. ^[21] A micromechanical damage model for ductile rupture is used and, after each loading step, the corresponding cleavage fracture probabilities are calculated using the cleavage model. A simultaneous computation of the J-value allows computation of stable crack growth and the cleavage fracture probability as a function of the applied J-value, see Fig.4.

It should be noted that the 'local approach' parameters needed for application of the models were extrapolated to the test temperature of -20°C, assuming temperature independence. An increase in cleavage fracture toughness due to preceding stable tearing is predicted by the model as experimentally observed.

Weldments

The 'local approach' methodology has also been applied to welded joints. However, because of the additional difficulties associated with the fracture behaviour of welds, the number of applications is still limited. So far, the temperature dependence and the scatter of weld metal fracture toughness, the scatter of heat affected zone (HAZ) toughness and the influence of weld mismatch have been studied. ^[22-26]

Relatively successful predictions were made given the extra difficulties associated with fracture predictions of welds, especially welding residual stresses and microstructural gradients in HAZs.

Discussion

The 'local approach' to cleavage fracture correlates the cleavage failure probability with the so-called Weibull stress. The Weibull stress is calculated via knowledge of the distribution of the maximum principal tensile stress in the plastic zone ahead of the crack tip, a critical value of which is known to trigger cleavage failure. The higher the local maximum principal stress in a volume element ahead of the crack tip, the more this element contributes to the Weibull stress and increases the cleavage fracture probability.

Since the cleavage event is predicted by the local stresses ahead of the crack tip, the mechanical factors affecting cleavage fracture are 'included' in calculation of the Weibull stress. Examples of such parameters are: size (constraint and sampled volume effect), loading mode, non-isothermal stresses, etc. An additional advantage is that there is a relation between the Weibull stress and the fracture probability. This enables scatter in the fracture toughness to be predicted.

A drawback of the methodology is the need for numerical analysis to calculate accurately the stress/strain state ahead of the crack tip. This will cause difficulties when applying the approach to weldments where complex welding residual stress distributions in and adjacent to the weld influence the stress/strain state. Furthermore, it can be assumed that the 'local approach' parameters depend on the microstructure which is sampled by the crack tip. Since the microstructures in a weldment vary considerably depending on whether the crack is located in weld metal, different heat affected zones or parent plate, the 'local approach' parameters will differ accordingly.

Some areas of the method warrant further work:

• The assumed temperature independence of the 'local approach' parameters should be checked;
• Theory predicts that identical 'local approach' parameters should be obtained whatever type of specimen is used. It remains to be demonstrated that different types of specimen, say, notched bend and notched tensile specimens, lead to identical values;
• Further test programmes investigating the predictive capability of the model with regard to structurally representative tests are needed. Different steels and welds should be used to check the applicability of the methodology to a variety of microstructures;
• Application of coupled cleavage and ductile 'local approach' methodologies in the steep ductile-to-brittle transition region requires further development and experimental validation, and will also be dependent on the temperature dependence or independence of the material's parameters.

TWI is currently active in all the above areas. A Core Research Programme project covers the first two issues, whilst a European-funded joint project with British Steel Technical is concerned with application of the 'local approach' to fracture behaviour of HAZs and its predictive capability regarding structural behaviour. Finally, a project for the UK Health and Safety Executive is studying the application of coupled micromechanical failure models for cleavage and ductile fracture.

Conclusions

Combination of micromechanical failure models with the ever-growing power of finite element stress analyses has resulted in a powerful method of predicting fracture behaviour of structures using simple laboratory tests. This so-called 'local approach' to cleavage fracture correlates the fracture probability with the stress distribution ahead of the crack tip. Since the cleavage event is predicted using the local crack tip stresses, many parameters which influence cleavage fracture are included in calculation of the local criterion. Examples are size (constraint and sampled volume effect), loading mode, non-isothermal stresses, etc.

The methodology for cleavage fracture is therefore a valuable tool to predict lower shelf fracture behaviour of structures using simple laboratory tests. However, the method requires accurate numerical analysis of the stress/strain state in the flawed structure which, for weldments, has to take into account welding residual stresses. Moreover, the 'local approach' parameters are microstructure dependent, which makes application to weldments, which exhibit steep microstructural gradients, more difficult. Further work is required to enhance the reliability of the method and to uncover possible limitations.

References