Most
materials show a tendency to fracture when stressed beyond some critical level. However,
the fracture strength of a given material is not, in general, highly reproducible,
fluctuating in some cases by as much as an order of magnitude. Changes in test conditions,
e.g. temperature, chemical environment, load rate, etc., leads to systematic variations in
strength values. Different material types seem to fracture in radically different ways: for instance, glasses behave elastically up to a
critical point to fail suddenly under the action of a tensile stress component, while many
metallic solids deform extensively by plastic flow prior to rupture under shear.
To understand these behaviors, we must understand how the applied stresses are transmitted to the inner regions where fracture actually takes place. What is the nature of the fracture mechanism itself? The answers to these questions constitute the key to an understanding of all fracture phenomena.
The
breakthrough came in 1920 with a classic paper by A. A. Griffith. Griffith considered an
isolated crack in a solid subjected to an applied stress, and formulated a criterion for
its extension in terms of the fundamental energy theorems of classical mechanics and
thermodynamics.
The
principles laid down in this pioneering work, and the implications drawn from these
principles, effectively foreshadowed the entire field of present-day fracture mechanics.
In this objective we will briefly show the contributions of Griffith and some of his
contemporaries: this serves to introduce the reader to many of the basic concepts of
fracture theory.
Stress concentrators
An important
precursor to the Griffith study was the stress analysis by Inglis (1911) of an elliptical
hole in a uniformly stressed plate. This analysis showed that the local stresses about a
sharp notch or corner could raise to a level several times that of the applied stress. It
thus became apparent that even submicroscopic flaws are potential sources of weakness in
solids. More importantly, the Inglis equations provided the first real clue to the
mechanism of fracture: the limiting case of an infinitesimally narrow ellipse could be
considered to represent a crack.
Let us
summarize briefly the essential results of the Inglis analysis. Consider a plate (Figure
below) containing an elliptical hole with the small elliptical radies being r, which is
subjected to a uniform applied tensile stress AL along the plaate.
The object is to examine the modifying effect of the hole on the distribution of stress in
the solid. If it is assumed that Hooke's law holds everywhere in the plate, that the
boundary of the hole is stress-free (a requirement for equilibrium), and that the radius
is small in comparison with the plate dimensions, the problem reduces to a relatively
straightforward exercise in linear elasticity theory.
The results
can be expressed as sLO/ sA @ 2(l/r)1/2. This ratio between sLO/ sA is often referred to as an elastic stress concentration factor. It is immediately evident that this factor can
take on values considerably larger than unity for narrow holes. We note that the stress
concentration depends on the shape of the hole rather than the size.
The
variation of the local stresses is also of interest. sLO drops from its maximum at
the crack tip and approaches the value of sA asymptotically as it moves
away from the crack tip. The general result is that major perturbations to the applied
stress field occur only within a distance @ l from the boundary of the hole, with the
greatest stress gradients confined to a highly localized region of dimension @ r surrounding the position of maximum
concentration.
Inglis went on to consider a number of stress-raising configurations,
and concluded that the only geometrical feature, which had a marked influence on the
concentration power, was the form of the highly curved region where the stresses actually
focused. A tool was now
available for a ready appraisal of the potential weakening effect of a wide range of
structural irregularities, including, presumably, a real crack. Inglis finding can also be
applied to dentistry because it shows that sharp details in particularly brittle materials
such as ceramics and enamel are contraindicated.
Griffith
energy-balance concept
Griffith's developed later a model for a
crack system in terms of a reversible thermodynamic process. Griffith simply sought
the configuration which minimized the total free energy of the system: the crack would
then be in a state of equilibrium, and thus on the verge of extension.
Conclusion
From the above we can conclude that a sharp defect inside or on the surfaace of a body concentrate stresses and facilitate crack growth. We can also conclude that a material with high surface energy makes crack propagation more difficult because of unfavorable energy considerations. The latter is important to remember, because dental materials are stored in a moist environment. If water reaches the internal surfaces surrounding defects, the syrface energy of these defects may decrease which could facilitate crack growth. The latter phenomen is often used by individuals working with fracturing glass. By fracturing glass under water, it becomes easier to break the glass because of the lower surface energy of the glass at the glass-water interface. The surface energy reducing approach is also used during grinding and polishing of different materials. Surface active compounds are used as cutting lubricants and as surface energy reducing agents to facilitate micro crack growth.
