Part 12 - Fracture and Failure

Both hardened cement paste and concrete are considered brittle materials. The stress-strain behavior and failure modes are governed by cracking at the paste-aggregate interface. Therefore, it is very important to understand the fracture mechanism.

Fracture Mechanics

Fracture mechanics is the study of stress-strain relationships and displacement fields in a region near a crack tip. The following is a brief introduction to the concepts of classical fracture mechanics and the extension to a composite material.

• Theoretical Cohesive Stress -- The strength of a solid depends on the strength of its atomic bonds. Therefore, we will consider the interaction between two atoms. There is a minimum energy associated with the equilibrium spacing of two atoms. The total energy required to separate these two atoms is U0. As the solid fractures and two new surfaces are formed, the energy, gs, is equally shared by both surfaces. The force is zero at the equilibrium spacing. The initial slope of the curve is the modulus of elasticity. The stress-strain curve can be approximated by half a sine wave. The area under this curve is the amount of work necessary to fracture the atomic bond, 2 gammas, the initial slope is E. The relationship between stress, sigma, and displacement , x, is

The work to fracture is the area under the stress-strain curve. By using Hooke's law for small displacement, the following relationship can be found

where sigmamax is the theoretic cohesive; for concrete this strength is about 300,00 lb/in^2.

• Griffith Theory -- Based on thermodynamic considerations the total energy is the system is

where - WL is the work due to the applied loads, UE is the strain energy stored in the system, and US is the free surface in creating the new crack surface. When dU/dc < 0 the crack will propagate and the fracture strength is

where C is half the crack length; this is similar to the cohesive stress theory strength. The values of the theoretical strength differ enormously from the measured values because the material contains flaws and microcracks that concentrate the stress.

Consider the elliptical hole in an infinite plate in tension. As C >> rho, a very sharp crack is modeled.

None of these simple fracture models are not accurate when applied to concrete. The reasons vary form multiple cracking to high varied cracking paths.

• Coulomb-Mohr Theory -- This theory attempts to describe the failure mechanism on the macroscopic level. The method is used to develop a Mohr's envelope of safe stress conditions. This theory also has difficulty in predicting ultimate strength. To overcome this the Coulomb-Mohr theory has been modified and is the best representation of a failure criterion for concrete.

Mechanisms of Failure

Since concrete is a composite material we are interested in the interactions of the components. At low levels of stress the materials are linear in nature. However, a higher levels concrete is highly nonlinear. This nonlinearity is due to the interactions of the materials and the nature of the cement-aggregate bond. Stronger concrete exhibits a more linear stress-strain curve behaviors, also linearity is increased when the stress-strain relationship of the aggregate and the cement matrix are more evenly matched.

Concrete like most brittle materials pass through three stages: (1) crack initiations, (2) slow crack growth, and (3) rapid crack growth. As the stress reaches the ultimate stress the value of the Poisson's ratio also changes.

Effect of Aggregate -- Concrete is heterogeneous and the aggregate particles are not only irregularly shaped but also imperfectly bonded to the cement. In general, the order of failure is (1) tensile bond failure, (2) shear bonds, (3) shear and tensile matrix failures, and (4) aggregate failure. Even in compression, regions of tensile stress will develop around aggregate particles.

Stabilizing Crack Growth

Since aggregate is generally stronger than the cement matrix, cracks will tend to go around aggregate and not through it. The energy required for crack extension is increased by the aggregate. Also, unhydrated cement gains will act in a similar manner. Air voids tend to blunt the crack tip. The affect of microcracking or branch cracking will increase the surface area of new cracking and tend to distribute the stress and prevent a large concentration to build-up at the crack tip.

Static Fatigue

If concrete is loaded to about 75% or more of its short-term static strength, and if the load is sustained, the concrete will eventually fail. This is referred to a static fatigue. The failure is not difficult to explain; slow crack growth continues until they reach a critical size, then fracture will occur. Crack growth is sustained in the presence of water.

Fatigue

This a phenomenon when failure occurs by repeated applications of loads which are not large enough to cause failure in a single application. Fatigue data is represented by an S-N diagram; Repeated stress S, plotted as a stress ratio, and the number of cycles of the loading to cause failure, N. Generally, S-N diagrams are presented in terms of the probability of failure. Properties of the concrete like w/c ratio, aggregates, air entrainment, etc. have no significant effects. The frequency of the loading has no effect on fatigue strength as long as the maximum stress is less than about 75% of the static strength. Even though fatigue strengths are lower than the static strength, the strains at failure are substantially larger in fatigue loadings than in simple static loadings. The shape of the stress-strain curve changes with load.

This website was originally developed by Charles Camp for his CIVL 1101 class.
This site is maintained by the Department of Civil Engineering at the University of Memphis.
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