# Fundamental Concepts of Fracture and Continuum Damage Mechanics

DOI: 10.4018/978-1-4666-6379-4.ch009

## Abstract

Some fundamental concepts of fracture mechanics, those needed for the description of concrete cracking in framed structures, are presented in a simplified way in the first section of this chapter. The second section introduces the fundamentals of continuum damage mechanics. The third section describes a physical phenomenon called localization; this is a very important effect during the process of structural collapse. In this section, it is shown that damage mechanics can lead to ill-posed mathematical problems. Finally, the relationship between localization and ill-posedness is discussed. In this chapter, fracture and damage mechanics or localization concepts are not yet applied to the analysis of framed structures; that is the subject of the following chapters.
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## 9.1 Griffith Criterion And Fracture Mechanics

In the theories of elasticity and plasticity, it is assumed that the boundaries of the solid are fixed and non-modifiable; they cannot be changed during an analysis. This simplifying hypothesis eliminates the possibility of representing crack propagation in solids; however, this phenomenon is one of the main causes of structural failure. Fracture mechanics differs from the aforementioned theories because it does not discard the possibility of modifications in the boundaries of a solid due to cracks propagation. Specifically, the primary goal of fracture mechanics is the determination of conditions for crack propagation in elastic or elasto-plastic solids.

### 9.1.1 Stress Concentration Factors in Solids

It is often necessary to drill holes in structural components. When the structural modifications occur in this way (see Figure 1), the local stresses are increased. The ratio of the local maximum stress and the average stress in the structure is called “stress concentration factors” SCF:

(9.1.1)
Figure 1.

Infinite plate with elliptical hole

An important subject of the theories of elasticity and strength of materials is the determination of these factors for changes in the cross-section, structures with holes and other discontinuities. A particularly relevant case in fracture mechanics is the one shown in Fig 1: Consider an infinite plate of thickness t with an elliptic hole of mayor diameter 2a and minor diameter 2b in its center; the plate is subjected to constant stresses σavg far away from the hole as shown in Figure 1. It can be shown that the stress concentration factor in a linear elastic case is given by:

(9.1.2)

Mathematically, a crack can be defined as an infinitesimally narrow ellipse; notice that the local maximum stress tends to infinite when the shorter semi-axis b tends to zero, regardless of the magnitude of the tensile force applied to the plate. It can also be shown that this is always the case at any crack tip for any kind of structure.

Therefore, even if the stress concentration factors are a common and useful concept in the design of structural components, they cannot be used as a condition for crack propagation. Furthermore, it must now be explained why there are cracks in brittle solids that do not propagate at all if elastic stresses at the crack tips tend to infinite; the answer to this question can be found in an energy balance as presented in the following sections.

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