Fracture – continuum mechanics and materials aspects

This chapter presents a review of the field of fracture mechanics, with particular concern for its application to steels. Section 1 gives a very brief history of the developments within the field, while the particular aspects of fracture mechanics are discussed in Section 2. Section 3 concerns the main features of the continuum mechanics fields in relation to cracks. Although no direct use of e.g. micromechanical criteria for cleavage fracture is made in the papers presented in this thesis, reference to such models are included Section 4,with Section 5 focusing on their use to predict cleavage fracture. The discussions on micromechanisms and local criteria for cleavage are introduced to put the results presented in the papers into perspective. The main features of ductile fracture are presented in Section 6, with special reference to the Gurson model and its use for simulation of ductile crack growth. Finally, Section 7 discusses special aspects of the ductile to brittle transition in steels.

1A brief history of fracture mechanics

The problems with fracture have puzzled man for centuries. In early structural design, using brittle materials such as rock, strategies to avoid fracture was sought by mainly introducing compressive loads. With the development of more efficient ways of manufacturing metals, mainly steels, more ductile materials opened for new design possibilities allowing for primary tensile loads to operate. However, although more ductile behaviour could be expected, the problems with fracture were becoming of increasing importance. In the 19th century a fundamental understanding of the mechanics of fracture was still lacking, and the rational basis for what was to become the field of fracture mechanics was first put forth in the beginning of the 20th century. The first important result was presented by Inglis in 1913, when he showed that the stress level ahead of an elliptical hole would approach infinity in the limit of the ellipse becoming a sharp crack. This result underlined the importance of discontinuities as stress raisers, and added to the understanding of why fracture could occur in structures even if the general load level would be well below the fracture stress of the material. However, there remained one important question. If the stress level would be infinite ahead of a crack, and this in itself would be a sufficient criterion for fracture, fracture should occur at infinitesimal loads in the presence of a crack. This was not supported by experience, and there was obviously something lacking in the theory to explain the behaviour of cracks. The solution to this question was presented by Griffith in 1920. He recognised the need to consider the energy involved in the fracture process. Fracture would create new surfaces, which would represent a higher energy state than the bulk material. Griffith argued that this energy would have to be supplied from the energy stored in the surrounding material, and fracture would only occur if the available stored energy was equal to or larger than the energy needed to create the new fracture surfaces. From this Griffith was able to derive a relationship between the critical defect size and the stress level, by including the surface energy of the newly created fracture surfaces. This was to become the fundament for the field we today know as fracture mechanics. However, some 30 more years remained before serious effort was made to turn fracture mechanics into a tool available for making fracture assessment of engineering structures. The work in transforming fracture mechanics into an engineering tool must mainly be attributed to Irwin. The introduction of the stress intensity factor, together with the demonstration of its relation to the energy release rate, opened for a more widespread use of fracture mechanics. In the 50s/60s a great interest was also shown in understanding the local micromechanisms controlling fracture, much aided by theories to explain dislocation movements. The development of better plasticity theories was to become very important for the further development of fracture mechanics. In the 60s and early 70s people like Rice, Hutchinson and McClintock made large contributions to how plasticity effects at the crack tip were to be accounted for in fracture mechanics. At the same time the exploration of oil in the North Sea and the nuclear industry led to a need for more formal ways of assessing structural integrity in presence of cracks, spawning the Failure Assessment Diagram (FAD), the CTOD design curve and the R6 procedure. The fracture mechanics research in the 80s and 90s has largely been influenced by the development of better and faster computers. This has opened for the use of more complicated, and usually highly non-linear, material models for simulation of e.g. ductile crack growth. Further, significant improvements have been made on how to quantify the effect of constraint from both geometry and material mismatch on the fracture toughness, and progress has also been made in how to characterise the fracture behaviour of steels in the ductile to brittle transition region.

2Particular aspects of fracture mechanics

The basic Griffith model states that the relation between the critical stress level and defect size can be expressed as:

(1)

where c is the critical stress level, ac is the critical defect or crack size, E is the elastic modulus,  is the surface energy of the crack, and f is a constant depending on the crack shape. Cottrell (1998) uses the Griffith equation to underline that fracture mechanics cannot be viewed as a purely continuum mechanics theory. There is no possibility to relate the critical crack size to the critical stress level by means of only regular continuum properties like the E-modulus, and there is a need to introduce a length scale into the problem. At the same time, fracture consists of crating new surfaces, which indicates that a relation between bulk properties and surface properties should be important in the mechanics of fracture. Such a relation would also require the introduction of a length scale. This is where Griffith’s surface energy comes in to provide the necessary length dimension and “saves the day”.

As indicated above the importance of the length scale appears several places in fracture mechanics. Related to the geometry the crack size in itself represents a length dimension. Length scales can also be found in the material, and two different types may be identified her. One is related to the microstructure, of which a typical example may be the grain size. On the other hand length scales may also be attributed to the micromechanisms operating in the material. If we focus on the sharp crack tip the typical microstructural length scale will be the spacing between atoms. Two different micromechanisms can also be found here. One is the critical separation between neighbouring atoms, and another is the critical movement of a dislocation emitted from the crack tip. Both will be of the order of the interatomic spacing, and this could possibly be argued to be the fundamental length scale in fracture. However, other length scales may control the fracture behaviour. In cleavage fracture, second phase particles play a very important role, and the size of such particles may be the most important length scale. For ductile fracture the mean spacing between the larger void nucleating particles is often assumed to mainly control fracture.

Materials containing cracks may in general behave in two different ways when loaded. One possibility is that the material will be brittle, and the crack may extend easily under low imposed global deformation. Alternatively ductile behaviour may be encountered, in which the material can sustain large deformation without fracturing. Kelly et al. (1967) proposed that the behaviour of the material could be explained by the competition of separation of atoms and emission of dislocations at the crack tip. In cases where dislocations may easily be created at the crack tip, the crack tip will blunt and the critical separation to cause brittle fracture cannot be reached. In the case of less easy nucleation of dislocation, the critical separation of neighbouring atoms can be reached before the crack tip becomes blunt, and fracture will be initiated. Rice and Thompson (1974) used this concept to predict if different metals would behave in a brittle or ductile manner. An important part of their work was to identify the critical energies needed to either nucleate a dislocation from the crack tip or reach the critical displacement. In general the behaviour would be controlled by the mechanism requiring the least energy. This example serves to underline the importance of both micromechanisms and their related energies in fracture mechanics.

Due to the lack of an intrinsic length scale in standard continuum mechanics, strictly speaking, the only fracture criterion that can be used is that of a critical energy released at the crack tip for an infinitesimal crack extension. For this energy to be non-zero it is required that the stresses at the crack tip are infinite, i.e. there must be a singularity in the stress distribution. This requirement stems from the fact that the critical length scale for the fracture process must also be regarded as being infinitesimal. As seen above the atomic spacing represents some lower bound for the length scales involved in fracture mechanics. Although small, the length scale is still finite, and it is possible to prescribe a finite critical energy level to the process without requiring the stresses to be infinite. Thus, introducing the physical micromechanism, with its related characteristic length scale, makes it possible to predict that fracture will occur without the stresses being infinite at the actual crack tip. One of the first models to demonstrate this was Barenblatt’s (1962) cohesive zone model for brittle fracture, in which a critical stress-displacement relation at the crack tip was introduced as the fracture criterion. For ductile crack growth, where the relevant length scales related to the fracture process are 3-5 magnitudes higher than for cleavage fracture, the importance of introducing the physical mechanism is maybe of even greater importance.

When fracture mechanics was first introduced as a tool for fracture assessments, the limitations with regard to the physical crack tip mechanisms were known. However, the main assumption was that the zone where the fracture process took place was embedded inside a region, which could uniquely be characterised by one single continuum mechanics field. The critical state of this field could be identified from experiments, and no particular concern for the actual fracture process was necessary. Although this section has underlined that in principle a purely continuum mechanics approach is not sufficient in fracture mechanics, continuum mechanics have played, and continues to play, a very important role in the development of the field. The main aspects of the continuum mechanics field theories in fracture mechanics are reviewed in the next section.

3Continuum field theories in fracture mechanics

3.1Linear elastic fracture mechanics

The basis for linear elastic fracture mechanics (LEFM), in which a linear relation between stresses and strains is assumed to exist, is the stress intensity factor, KI:

(2)

where  is the applied stress, a is the crack depth, and f is a function depending on the relative crack depth, a/W, and the mode of loading. The subscript I refers to the global loading being applied normal to the crack plane. The stress intensity factor includes both the effect of the load level, , and the size of the crack, a. Similar expressions also exists in case of a shear stress being applied in the crack plane either normal or parallel to the crack front, and the resulting stress intensities are referred to as KII and KIII, respectively. Based on the Westergaard (1939) analysis of the singular stress field ahead of the cracks in bearings, Irwin (1957) showed that the stress intensity factor, KI, is related to the energy release rate, G, for crack growth through the following relation:

(3)

where E=E in plan stress and E=E/(1-2) in plane strain. The energy release rate represents the energy available to cause fracture in case of a small extension of the crack length. Further, it is to the sign equal to the change in potential energy in the body for an infinitesimal crack extension. In elastic fracture mechanics the critical energy release rate, GC, is regarded as a material parameter and fracture will happen at a critical value of the stress intensity factor, KIC, termed the fracture toughness of the material found from (3).

Williams (1957) solved the stress field for a crack in elastic material under tensile loading by performing an eigenvalue expansion, and showed that the solution is on the form:

(4)

where r is the radial distance from the crack tip and  is the angular position as shown in Figure 1. KI is the applied stress intensity, T is a stress that scales in proportion with the applied load, and ij is the Kronecker delta. From (4) it is noted that the amplitude of the stress in the leading term of the expansion scales linearly with the applied stress intensity factor. Further, the leading term is singular giving infinite stress at the crack tip (r=0). There is also no coupling between the angular and radial dependence of the stress field. The second term, usually referred to as the T-stress, gives a constant stress level acting parallel to the crack plane. From a classical fracture mechanics point of view the first singular term is assumed to control the behaviour at the crack tip, and is the only term necessary to consider. However, as will be seen below this is not always the case and the T-stress can also play an important role. From a mathematical point of view solutions also exist for terms being more singular than the first term in (4). However, such terms would lead to infinite energy at the crack tip, and they must be abandoned on physical grounds.


Fig.1.Definition of stress components and co-ordinate systems

3.2Elastic-plastic fracture mechanics

For materials with a non-linear relation between the stress and strain, e.g. metals above the yield stress, this will lead to different conditions at the crack tip compared to the linear elastic case. This was early recognised, and Irwin (1961) proposed corrections to include plasticity effects at the crack tip. Even though corrections to include plasticity effects were proposed, they still required that this zone be embedded inside a region where the LEFM fields would be valid. However, many structural steels would show too high resistance to initiation of fracture so that LEFM could not be applied directly. Thus, there was a need to extend the fracture mechanics concepts to be applicable also in such situations. Wells (1961) proposed that the opening of the blunted crack tip, Figure 2, could be used as an alternative fracture mechanics parameter in cases where LEFM would not be applicable. This parameter would later be known as the crack tip opening displacement (CTOD), often just referred to as . The idea behind this proposal was that fracture would occur once the opening reached a critical value, c. This value would then represent the fracture toughness of the material in cases where no valid KIC could be established.


Fig. 2.Physical definition of the crack tip opening displacement, .

Another major contribution to elastic-plastic fracture mechanics was Rice’s (1968) proposal of the J-integral as a parameter to characterise the crack tip loading in case of non-linear material behaviour. The J-integral is expressed as:

(5)

where W is the strain energy density, Ti and ui are the traction and displacement, respectively, in direction i on the path over which the integral is evaluated, as shown in Figure 3. In his derivation Rice assumed that the material would obey a non-linear elastic constitutive relation (equivalent to a deformation plasticity assumption). Further, he showed that the value of the integral would be independent of the path over which it was evaluated due to the divergence free stress/strain field. From this the J-integral could serve as a means to characterise the deformation at the crack tip. The form of the J-integral is similar to the energy momentum tensor earlier derived by Eshelby (1951,1970), which Eshelby interpreted as the force acting on a defect in an elastic medium. Under the assumption of non-linear elastic behaviour the J-integral is equal to the reduction in potential energy for a small amount of crack extension; thus, it is equal to the energy release rate G. However, Rice was careful to warn about using the J-integral as the energy release in case of more realistic material behaviour where unloading of the material will follow a different path.


Fig. 3.Arbitrary counter, , for evaluation of the J-integral.

Despite this limitation, the J-integral is still a very valuable fracture mechanics parameter for characterising the intensity of deformation at the crack tip in elstoplastic materials. This was demonstrated by Hutchinson (1968) and Rice and Rosengren (1968) when they showed how the J integral could be used to describe the asymptotic stress and deformation fields in front of the crack tip in materials following a power law hardening rule. They assumed a Ramberg-Osgood constitutive relation between the stress and strain, which in three dimensions has the form:

(6)