Fracture of Materials

Liberty Ship

Liberty Ship

Comet

Northridge earthquake

Fracture Mechanics

Fracture mechanics provides powerful criteria for the prediction of crack propagation.

Fracture Mechanics

Linear elastic fracture mechanics theory was developed in 1920.

Fracture mechanics was used successfully in design for metallic and brittle materials early on; however comparatively few applications were found for concrete.

Linear Elastic Fracture Mechanics

Griffith is often regarded as the founder of fracture mechanics.

He observed experimentally that small imperfections have a much less damaging effect on the material properties than the large imperfections.

Griffith suggested an energy balance approach based not only on the potential energy of the external loads and on the stored elastic strain energy but also on another energy term: the surface energy. 16

Crack in a plate

Crack in a plate

Griffith used a result obtained by Inglis17that the change in strain energy due to an elliptical crack in an uniformly stressed plate is and therefore the change in potential energy of the external load is twice as much .

a2 2 E

The change of energy of the plate due to the introduction of the crack

is given by:

U cracked Uuncracked = 2 a

2 2E

+ a2 2E

+ 4 a

Crack in a plate

Minimizing the energy in relation to the crack length,

a

a2 2E

+ 4 a = 0

gives the critical stress (for plane stress):

= 2 E a

Importance of the equation

This equation is significant because it relates the size of the imperfection (2a) to the tensile strength of the material.

It predicts that small imperfections are less damaging than large imperfections, as observed experimentally.

Critical energy release rate Irwin proposed that instead of using the

thermodynamic surface energy, one should measure the characteristic surface energy of a material in a fracture test.

He introduced the quantity Gc as the work required to produce a unit increase in crack area. Gc is also referred to as the critical energy release rate.

Gc is determined experimentally, normally using simple specimen configuration. 18

Critical energy release rate

The energy release per unit increase crack area, G, is computed; if the energy release rate is lower than the critical energy release rate (G < Gc ) the crack is stable.

Conversely, if G> Gc, the crack propagates. In the case when the energy release rate is equal to

the critical energy release rate (G=Gc), a metastable equilibrium is obtained.

Example

Modes

Mode I: opening or tensile mode, Mode II: sliding or in-plane shear mode Mode III: tearing or antiplane shear mode.

Modes

Stresses at the tip of the crack for mode I

y = KI

2 r cos2

1 + sin 2

sin3 2

x = KI

2 r cos2

1 sin 2

sin3 2

xy = KI

2 r sin2

cos2

cos3 2

Impressed??Hope so but noNeed to memorize

Stress-intensity factor

KI is called stress-intensity factor for Mode I.

Dimensional analysis of indicates that the stress-intensity factor must be linearly related to stress and to the square root of a characteristic length.

Assuming that this characteristic length is associated with the crack length:

K I = a f g( )

Fracture toughness

Suppose we measure the value of the stress at fracture in a given test.

Using the previous equations, we determine the critical stress intensity factor, Kc, or fracture toughness as it is usually called in the literature. .

Example

A ceramic has a strength of 300 MPa and a fracture toughness of 3.6 MPam0.5.

What is the largest-size internal crack that this material can support without fracturing?

ICK a =-8

2 4.58 x 10 mICKa = =