Micromechanics of Fracture: Connecting Physics to Engineering
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Micromechanics
of Fracture: Connecting Physics to Engineering
A. Needleman and E.Van der Giessen Introduction Humpty Dumpty sat on a wall Humpty Dumpty had a great fall All the king’s horses and all the king’s men Couldn’t put Humpty together again.* This nursery rhyme is one of the oldest, most familiar, and shortest treatises on fracture mechanics in the literature. Nevertheless, it makes several important points: (1) fracture occurs in response to some driving force (for Humpty, a “great fall”), (2) fracture involves material separation, and (3) fracture is irreversible. Of these, the driving force for fracture is probably the best-understood aspect. Fracture generally occurs because of the propagation of crack-like defects under the influence of a surrounding stress field. If the surrounding material behaves like a linear elastic solid (i.e., it follows Hooke’s law), the stress state near the crack tip is described by
ij
K 2r
fij ,
(1)
where r is the distance from the crack tip, fij is a function of the angular position relative to the crack tip (which depends on the material’s elastic constants), and K is the stress-intensity factor that depends on the
* The real Humpty Dumpty is said to have been a powerful cannon used during the English Civil War (1642–1649). It was mounted on top of St. Mary’s at the Wall Church in Colchester, defending the city against siege in the summer of 1648. The church tower was hit by the enemy and the top of the tower was blown off, sending Humpty tumbling to the ground.
MRS BULLETIN/MARCH 2001
mode of crack loading. Three modes are possible: Mode I involves opening of the crack; Mode II involves sliding of the crack faces perpendicular to the crack front; Mode III is also a sliding mode, but with the direction of sliding parallel to the crack front. The opening mode, Mode I, is of the greatest significance in practical applications and has received the most attention in the engineering and materials science fields. However, the sliding modes are what cause earthquakes, and so they are of great interest to geophysicists. In addition, in heterogeneous solids, such as composites and multilayers, the mismatch in mechanical properties leads to mixedmode behavior (i.e., opening and sliding) in the vicinity of the crack tip, even if the remote loading is purely Mode I. The function fij depends on the mode of loading, but is otherwise independent of the forces that act to cause crack growth. All of the details about the geometry and the forces only affect the value of K. The physical significance of Equation 1 is that the energy available to drive crack growth, the energy-release rate , is proportional to K2. Whether or not the driving force (for Humpty, the height of the fall) is sufficient to cause fracture depends upon the presumed presence of unavoidable flaws or small cracks (in fracture, as in many other subjects, growth is much better understood than nucleation). Given a solid body with an initial crack subject to some external forces, will the crack grow, or will it
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