IFMASS10

309

DISCRETE AND FRACTAL ASPECTS OF FRACTURE

Michael P. Wnuk

College of Engineering and Applied Science, University of Wisconsin–Milwaukee,

USA

1. INTRODUCTION

Cohesive models of cracks have been remarkably successful in explaining certain

essential features of fracture process such as a finite stress and non-zero crack opening

displacement at the crack tip, and a certain equilibrium length of cohesive zone. This

length increases with the applied load up to the point of incipient fracture and it serves as

a measure of the material resistance to crack propagation. As a new approach, fractal and

discrete nature of fracture occurring in real materials should be incorporated in cohesive

model. Recent research on nano-cracks, Isupov and Mikhailov /1/, Ippolito et al /2/, indi-

cate that the classic failure criteria break down for very small cracks. Such examples

show the need for novel non-local failure criteria for design of structures, in which

multiscale fracture mechanisms are present. It is proposed here, in order to refine

available mathematical tools and to extend their validity for fractal (rough) cracks and for

nano-scale to merge the basic concepts of the cohesive crack model with the fractal view

of the decohesion process, employing the non-local criteria, which remain valid at

atomistic or nano-scale levels. The present model is applied to define the cohesion

modulus, which is used as a measure of material resistance to initiation and to

propagation of fracture.

Fracture mechanics is developed based on two fundamental failure theories, proposed by

Griffith /3/ and Barenblatt /4/. Griffith realized that brittle fracture happens as a result of

competition between strain energy release and surface energy required to create new fracture

surfaces. The interesting thing here is that there is no stress singularity. In the Griffith’s

theory it is predicted that for a given crack length there is a unique critical stress above

which crack grows and below which crack remains in equilibrium. Barenblatt proposed the

cohesive theory of fracture in which one assumes that there is a nonlinear region in the

vicinity of the crack tip. Increasing applied loads increase separation of crack faces in the

cohesive region and when the displacement is large enough the crack propagates.

The physically unacceptable singularities predicted by the classical fracture mechanics

had been considered by Orowan /5/ and Irwin /6/. They proposed the “plasticity correc-

tion” term that was added in the equations describing the stress intensity factors for fracture

of various modes, so that for the crack length approaching zero a finite stress resulted, corres-

ponding to the local yield stress. Novozhilov /7/ suggested discrete nature of the crack propa-

gation, introducing the concept of the minimum admissible growth step

. This entity,

known as “fracture quantum”, must be included in the energy criterion of Griffith - or an

equivalent local stress criterion for fracture. Initially, the fracture quantum was identified

with the interatomic distance

in a cubic lattice. Similar concepts proposed Eshelby /8/.

The concept of discrete nature of fracture has been investigated by many researchers,

beginning with Neuber /9/, Novozhilov /7/, Wnuk /10/, Seweryn /11/, Pugno and Ruoff

/12/. Comparison of various fracture criteria used to describe discrete fracture process