Fracture processes under deviatoric stress

It is assumed here that, under a deviatoric state of stress, the principal directions of both local and applied states of stress coincide. For this reason, no distinction is made in the fol-lowing between local and applied stress states.

The various stages of the fracture processes – Comprehensive investigations into the behaviour of concrete under multiaxial stress (Newman 1973; Kotsovos 1974) have indi-cated that there are at least four stages in the process of crack proliferation under increasing stress. These are as follows.

Stage 1. When the load is first applied, micro-cracks additional to those pre-existing in the material may be formed at isolated points where the tensile stress concentrations due to the incompatible deformations of the aggregate and cement-paste phases are highest. During this stage, the micro-cracks do not propagate but remain stable.

Stage 2. As the load is increased, high tensile strain concentrations gradually develop near the tips of the micro-cracks as a result of the micro-crack geometry and/or

orientation. The stage is reached, therefore, when the initially stable micro-cracks begin to initiate branches in the direction of the maximum principal compressive stress. This branching process tends to relieve the strain concentrations and, once strain redistribution has occurred, the individual crack configurations remain stable during further increases of applied stress. Although this process may produce voids, the reduction of strain concentrations along the crack branches is such that it results in contraction of the material in localised zones near the crack tips which, in turn, causes the rate of increase of the tensile strain in the direction at right angles to that of branching to be reduced with respect to the rate of increase of the strain in the direction of branching. The start of such deformational behaviour has been termed local fracture initiation (LFI) and is considered to mark the start of the branching-initiation process.

Stage 3. When the load is increased to a higher level, a stage is reached at which the branched cracks start to propagate. During this crack-propagation stage, each crack of the system extends in a relatively stable manner, in that, if the applied load is held constant, the process ceases. Although the relief of strain concentrations continues during this process, void formation is such that it causes the rate of increase of the strain at right angles to the direction of branching to increase with respect to the rate of increase of the strain in the direction of branching. The start of such deformational behaviour is considered to mark the start of the stable crack-propagation process and has been termed onset of stable fracture propagation (OSFP).

Stage 4. The degree of cracking eventually reaches a more severe level, after which the crack system becomes unstable and failure occurs even if the load remains constant.

The start of this stage has been termed onset of unstable fracture propagation (OUFP).

Under compressive stress states, this level is easily defined since it coincides with the level at which the overall volume of the material becomes a minimum. Under predomi-nantly tensile stress states, it is marked by a rapid increase in the overall volume of the material and can be detected as described elsewhere (Kotsovos 1974).

The above four stages of crack extension and propagation are illustrated schematically in Figure 2.16 by reference to concrete under compressive stress. Such a representation may be thought of as possibly corresponding to a uniaxial cylinder test in which the frictional

LFI OSFP OUFP

Specimen:

Individual crack:

Figure 2.16 Stages in the process of crack extension and propagation for concrete under compressive stress.

effects between specimen and platens have been removed, or to the central portion of such a specimen if the end effects are present.

A simplified qualitative description of micro-cracking and macro-cracking processes: crit-ical levels of concrete behaviour – It is important to emphasise that the above four stages of the fracture process are not all as clearly defined as might have been implied by the subdivi-sion adopted. In particular, there is no easily detectable limit separating the local and essen-tially random process of stage 1 from the aligned cracking of stage 2. However, an analysis of triaxial stress–strain data can lead to a detection of reasonably distinct levels of change in the behaviour under increasing stress. As noted earlier, these levels are considered to represent the start of the stages within which crack branching (LFI), stable crack propagation (OSFP), and unstable crack propagation (OUFP) occur. (It is interesting to note that it has not been possible to detect the LFI level under triaxial ‘compression’ [C–C–C] of cylinder specimens when the maximum principal compressive stress is applied axially so that σa > σc However, an indication of this level may be obtained by using the LFI levels detected under triaxial

‘extension’ [C–C–C but with the maximum principal stress applied laterally, i.e., σa < σc] and triaxial ‘tension’ [C–C–T i.e., σa tensile] – see Kotsovos and Newman 1981b.)

The above considerations suggest that, in attempting to establish failure criteria for con-crete, three critical levels might be considered. The significance of these levels will depend on the damaging effect which the ensuing fracture process has on the structure of the mate-rial. Since branching-crack initiation appears to induce stabilisation of the material owing to the resulting relief of the stress concentrations (and, as intimated above, it is not always readily detectable), the OSFP and OUFP levels could be considered suitable for use as bases for lower-bound and upper-bound failure criteria in conformity with the limit-state require-ments of serviceability and US. The validity of this consideration is supported by experimen-tal evidence which has indicated a very close correlation between: (a) the OSFP level and the

‘fatigue strength’ of concrete, that is that level below which concrete does not suffer distress under repeated applications of load; and (b) the OUFP level and the ‘long-term strength’ of concrete, that is, that level below which concrete does not collapse under sustained load.

For structural purposes, however, the number of repeated loadings at the OSFP level which are required to cause fatigue failure of concrete is considered to be impractically high. For this reason, it seems that it is mainly the OUFP level which is relevant in struc-tural applications where the governing criterion is that of static strength. In this respect, it is found convenient to simplify the cracking processes conceptually by considering these to consist of two major stages, which occur at the microscopic and macroscopic levels of obser-vation, respectively. Accordingly, the first of these can be denoted generically by the term micro-cracking, and encompasses stages 1–3 (inclusive) described above. Micro-cracking can be said to be the underlying cause of the non-linear behaviour at the material level and thus determines the constitutive relations of concrete. One way of describing it would be as a static process in the sense that crack extension stops when the load is maintained constant. The second major stage in the fracture process is that of macro-cracking, which coincides with stage 4 discussed earlier. In contrast to micro-cracking, macro-cracking indi-cates material failure in localised regions within a structure and is a dynamic phenomenon in that crack extension continues even if the load is maintained constant. It stops when equilibrium, which is disturbed by material failure in the region of the macro-crack, is re-established through stress redistribution (elsewhere) in the structure. Due to material break-down, macro-cracking causes local discontinuities in the original geometry of a structure.

Clearly, unlike micro-cracking, macro-cracking affects concrete at the structure level and hence defines the failure criterion (or the failure ‘envelope’) of concrete.

Failure envelopes – Figure 2.17 shows typical variations in stress and strain space of LFI, OSFP, OUFP and US. (The actual values correspond to a concrete with f_c = 46.9 MPa subject

to axisymmetric triaxial conditions [Kotsovos and Newman 1977, 1981a,b; Kotsovos 1979a] in which the stress path 3 [Kotsovos 1979a,b] depicted in Figure 2.18 was followed.) The figure indicates that the OSFP forms a closed envelope in both stress and strain space. It may also be noted that, in both strain space and the wholly compressive portion of the stress space, the OSFP envelope is nearly symmetrical about the hydrostatic axis.

Such behaviour implies that, for stress and strain states enclosed by the envelope, the mate-rial may be considered isotropic. Since the orientation of crack branching under a triaxial

‘compression’ state of stress is perpendicular to that under triaxial ‘extension’ or C-C-T (see the schematic sketches in Figure 2.17a), such isotropic behaviour indicates that the fracture process up to the OSFP level causes insignificant disruption in the structure of the material.

In view of this isotropy, it is possible that the LFI envelope within the triaxial ‘compression’

zone may be represented by the reflection of the LFI envelope in the triaxial ‘extension’ and C-C-T zones with respect to the hydrostatic axis, as shown in the figure.

In contrast to the LFI and OSFP envelopes, the OUFP and US envelopes are open-ended for the range of stresses used in the tests, and are non-symmetrical with respect to the hydrostatic axis. This latter observation suggests that, for stress and strain states outside the OSFP envelope, the material becomes anisotropic, because crack extension occurs along a particular direction dictated by the maximum principal compressive stress. However, allow-ing for the scatter of results that is encountered in processallow-ing the deformational stress–strain data within the OUFP envelopes for a wide variety of concrete mixes (this scatter will be discussed in the next chapter), it seems reasonable to postulate that the description of the stress–strain relations may be approximated by means of an isotropic model up to the

Axial strain × 103 TensileCompressive LFIOSFP

OUFPUS

√(2) × Confining pressure

√(2) × Lateral strain × 10³ Uniaxial cylinder compressive strength

Hydrostatic axis

Hydrostatic axis

0 1 2 3 4

0 1 2 3 4

5 12

10

8

6

4

2

0

–4 –2 0

Tensile Compressive2 4 6

(a) (b)

Axial stress Uniaxial cylinder compressive strength

Figure 2.17 Typical LFI, OSFP, OUFP and US envelopes for concrete (with f_c = 46.9 MPa) subjected to axisymmetric triaxial stress states using the stress path 3 defined in Figure 2.18: (a) stress space;

(b) strain space.

OUFP level corresponding to macro-cracking. Moreover, it is found that the OUFP stress level forms a surface in stress space similar in shape and size with that of the ultimate-strength surface. As the ultimate-strength of the concrete increases, the two surfaces become practi-cally identical, but even for lower-strength concrete these surfaces are quite close to each other (Kotsovos 1974). Therefore, it seems reasonable to consider the OUFP and US levels as being essentially the same, at least for practical purposes. This may be represented schemati-cally, as in Figure 2.19, by reference to both restrained and unrestrained concrete specimens under uniaxial compression.

Restrained specimen

Unloaded specimen

Unrestrained specimen

State of stress: < OUFP > OUFP > ultimate

< ultimate

Figure 2.19 Schematic representations of the fracture processes for restrained and unrestrained concrete specimens under increasing compressive stress.

σ₁ > σ₂ = σ₃

σ₁ = σ₂ > σ₃

σ₁ = σ₂ > 0 > σ₃

√2 × confining pressure Hydrostatic pressure Axial stress

Compressive

Tensile

Path 3

Path 3 Paths 1 and 2

Paths 1 and 2

Figure 2.18 Schematic representations of various stress paths used in the triaxial testing of concrete cylinders.

2.3.3.2 Fracture processes under hydrostatic stress