P. Pallav, PhD

Linear Elastic Fracture Mechanics (LEFM) deals with the process of brittle fracture.

Brittle means that no or only very little plastic deformation occurs; the fractured surfaces fit back together like pieces of a puzzle.

According to this theory all materials contain flaws or micro cracks, which are normally very small. These cracks work as stress raisers, which effectively reduce the strength of the specimens.

Experiment: Strength of Adhesive Tape

1. If you take about 300 millimeters (~1ft) of ordinary adhesive tape, get a really firm grip on both ends and (try to) pull it in two, you will agree, that it is pretty strong.
2. Now you repeat the experiment, but before you get hold of the tape, you take a sharp knife or scissors and somewhere in the middle of the tape you make a small cut in one of the sides.

You will find that the tape breaks very easy now. This is because the small cut works as a stress raiser; all stresses concentrate at the (progressing) crack tip.

Stress Raisers

The cracks are generally drawn as ellipses or half ellipses. These two pictures may represent holes in the middle or in the sides of flat plates, but just as well may the one to the left be a cross-section through a circular, disc-shaped hole inside the bulk of a material, and the one above a cross-section through a semi-circular hole at a surface.
Generally, the ellipses are defined by a and b, but for our purpose it is more appropriate to use a and r (the radius at the tip of the crack).

Obviously, with real cracks in brittle materials, b and r are much smaller than a, i.e. the ellipses are much flatter than we could draw in the sketches on this page.

When a so-called remote tensile stress (s) is applied, the stress in the area around the tip(s) of the crack is increased. The greatest stress is found at the tip (A).

This formula is this simple only when the length and the width of the body are much greater than the length of the crack (a). With a (semi-)circular crack in a solid body, also the thickness should be much greater.

We use an "about equal to" sign, because it makes a little difference if it is an edge, surface, or center crack and if it's in a plate or in a solid body.

Cohesive Strength (s_c)

A crack will propagate when the raised stress at the crack tip (s_A) exceeds the (theoretical) cohesive strength (s_c).

The cohesive strength of ceramic materials is in fact a purely theoretical property, which cannot be measured directly. It may be inferred by multiplying the force required to pull out one atom, with the number of atoms in a unit area.

In this way, the cohesive strength of ceramics is about equal to one third of their E-module. This is in the order of a hundred times the strength of ordinary dental gold alloys!

Crack Tip Radius (r)

Ionic lattices in ceramic materials are built on the strong attraction between positive and negative electrical charges. Sliding planes, which are required for plastic deformation cannot occur, because any movement of any atom would require passage over an atom with the same charge and the local integrity will be lost, i.e. any "sliding" plane would become a fracture plane at the verge of sliding.

In the absence of any plastic yielding, the radius at the crack tip is of the same order of magnitude as the size of the atoms, typically a few tenth of a nanometer (10^-9m). This is extremely small and leads to a very great stress concentration, because r appears in the denominator in equation 1.

Example

At the tip of a ten-micrometer crack (a=10µm=10^-5m) in a ceramic material where the atoms are spaced two tenth of a nanometer apart (r=0.1nm=10^-10m) the stress at the crack tip will be several hundred times the remotely applied stress! (equation 1)

Crack Length (a)

This a measure of the cross-sectional area that is not bearing the remote stress. In this way it represents the amount of stress available to be concentrated at the crack tip.

When a glass rod is loaded increasingly in tension, ultimately one of the larger flaws or micro cracks will grow (a will increase). In a tensile test on glass like this, the crack growth is unstable. This means that when the crack length starts to increase, it grows instantly to complete fracture.

Depending on the setup and the material, cracks may grow without becoming unstable. If this occurs, the load versus extension plot will bend in a way that looks like plastic deformation. See Testing Fracture Toughness, the R-curve.

When the flaw or crack at which fracture initiates is inside the bulk of a specimen the initial value of the crack length or flaw size may be related to the grain size, the particle size, or the size of inclusions.

When fracture initiates from surface defects the initial crack length (a) may be related to the finishing procedure and the surface roughness (Ra).

Remote Stress (s)

The equations on this page assume that the crack length (a) is much smaller than the relevant dimensions of the body.

The remote stress then is the stress as applied on the structure, without the crack. In a tensile test, the remote stress is simply the force (P) divided by the area of the cross-section (A).

Fracture Toughness (K_Ic) and Stress Intensity (K_I)

The stress intensity at the crack tip (K_I) is given as

K_I=s·a^½=s_A·r^½, eq. 1a

The index I in K_I refers to the type of loading (tensile).

The crack length will increase when the stress intensity (K_I) exceeds the critical stress intensity or fracture toughness (K_Ic) of the material.

When the stress intensity, K_I is known, the entire distribution of the stresses around the crack tip can be described. This means that all the stresses (s_x, s_y, s_z, t_xy) at any point near the crack tip can be calculated.

Fracture Toughness (G_Ic) and Energy Release Rate (G_I)

Griffith, one of the founders of fracture mechanics originally proposed the energy approach, which considers the energy required to increase the crack.

When a remote stress (s) is applied the body is stretched according to its stiffness. This implies that strain (=deformation) energy is stored within the body.

When the crack grows a small increment, the material near the newly created surfaces is exposed to much less stress and strain than when the crack tip passed through; it has lost, or released most strain energy.

The amount of strain energy, which is released with each increment in crack size is called the (strain) energy release rate (G_I)

When fracture occurs G_I = G_Ic, the critical energy release rate or the fracture toughness, which is the energy required to create more surface. Although both K_Ic and G_Ic are called fracture toughness these should not be confused.

G = K² / E, eq. 3

To make it even more interesting, this G is often called J and both are interchanged with R, standing for resistance to fracture.

With absolutely brittle materials the fracture toughness (G_Ic), equals twice the surface tension (g_s) or surface free energy, which is a material property. Twice because two surfaces are extended with crack growth. Griffith (1920) originally found good agreement with his strength experiments on glass, ground to various degrees of roughness (Ra value ~ a) and equation 2 with as said G_Ic=2·g_s.

Strength (s)

This would be described as the remote stress (s) at failure. With very brittle materials strength does not appear to be a material property.

References

Anderson, TL (1991); Fracture Mechanics : fundamentals and applications; CRC Press, Inc. ; ISBN 0-8493-4277-5.

Griffith, AA (1920); The Phenomena of Rupture and Flow in Solids; Philosophical Transactions of the Royal Society of London, Series A, Vol. 221, 163-198.