3.7 Elements of fracture mechanics of concrete

Foundations of mechanics of brittle fracture were laid by A. Griffith. Griffiths showed that crack growth must be energetically favorable process (in which the quantity of stored energy in the body decreases).

Consider the energy balance of the body in which the crack extends. For simplification we assume that the area of the loaded material has the form of a circle (figure 30).

1 - crack, 2 – area for relaxation

Figure 30 – Scheme to define the energy balance of the body with a crack

Compared to fracture mechanics of homogeneous (quasi-homogeneous) materials fracture mechanics of heterogeneous bodies is studied much less.

For the analysis of crack development in the concrete use a number of simplifying hypotheses. Based on these assumptions and using the currently available experimental and theoretical data, formulated some of the basic features of the behavior of cracks in concrete:

1. In the concrete, as in any heterogeneous body, even in the absence of cracks, the stress distribution is significantly different from the distribution in a homogeneous body. Account the differences in elastic properties of the components can be produced by utilizing existing solutions of elasticity theory of stress concentration around inclusions in a homogeneous matrix.

2. In concrete, cracks can propagate not only in the matrix or in the aggregate, but also in the area of their contact. If the fracture surface area is large enough (higher than the strength of the matrix and filler, existing on the contact surface of the crack will not go on the surface and propagate into the matrix or filler according to the laws of development of cracks in homogeneous bodies. If the fracture surface area is insufficient, the crack will travel along the surface of contact.

3.8 Over view of the phenomenological theories of concrete strength

Strength – the property of materials to resist to destruction under action of external influences.

Measure of strength is the ultimate strength – maximum stress (or some combination of voltages), which is a destruction of samples or elements of designs. Tensile strength, usually associated with the strength of concrete: tensile strength in compression, tension, shear or a combination of both.

The strength of concrete in addition to various technological and non-violent operational factors depend on the type of stress state, the components of the stress tensor and rate of load application.

Criteria or terms of strength are those criteria which allow to determine the occurrence of the limit state at the point of a material at any stress tensor.

Allocate a phenomenological, statistical and structural theory of strength of concrete.

Phenomenological (mechanical) theory of strength considering the destruction as an instantaneous process, while the time of the destruction usually associated with the attainment of some value (selected as the criterion of strength limit values. The criterion is convenient to design and analyze the axes of principal stresses σ1,σ2,σ3. In these axes the strength criteria are in the form of a surface, which contains within itself a safe stress σ1,σ2,σ3. On the contrary, the yield on this surface indicates the destruction. In the concrete, σ1,σ2,σ3 – smoothed voltage. The construction of the phenomenological strength criterion, as a rule, is reduced to finding functions of the form F(σ1,σ2,σ3)=0, which must accurately fit the experimental data to determine the strength of concrete samples for the entire range of possible σ1,σ2,σ3.

In the development of phenomenological strength criteria, the researchers go one of two ways: either trying to record continuous failure criterion for all cases the stress state (thus it is not possible to achieve equally good convergence of theoretical and experimental data for all areas of the stress-strain state), or record a variety of criteria, one for the region with the presence of only compressive stresses, the other for the zone of tensile stresses and a combination of compressive-tensile" stresses. The latter method allows to achieve more accurate coincidence of the experimental and theoretical data, but complicates the use of this calculation in the areas of structures, where the stress-strain state only with the presence of compressive stress, transferred to the zone with the presence of tensile stresses.

Statistical theory of strength compared to the phenomenological theories in more detail analyze the essence of destruction. In statistical theories assumes the existence of some isotropic medium with a separate structural defects. These defects are distributed in accordance with the statistical laws and determines the effective strength of the body.

Statistical theory usually proceeds from the assumption that the strength of the body is entirely determined by the strength of the weakest of the primary element, that is not taken into account the interaction of the defects. These theories often do not consider the real structure of the materials and related features of the stress state.(For example, the stress concentration near the pores and grains of the filler, the possible occurrence of the first cracking between the aggregate and the cement stone and others).

The applicability of the existing statistical theories of strength to the concrete is rather limited, although some areas, such as the approach to fracture of concrete as a random process of damage accumulation, seems to be quite promising.

Structural (physical) theory of the strength of a much deeper structure than the phenomenological and statistical theories of strength, and are therefore the most promising direction of the theory of strength. However, structural theories have significant drawbacks. Key among them – the complexity of mathematical constructs, and the fact that the structural theory is limited to addressing certain tensions that lead to cracking and are not able to describe the whole process of fracture of concrete, consisting in the gradual accumulation of damage and crack growth.

It should be noted that the theory of concrete strength is in its formative stage. The use of different analytical expressions of the strength criteria requires special justification, and often experimental verification for the specific concrete used in the project, and range hazardous voltages that may occur in the designed construction.