Irwin introduced, for this purpose, the energy release rate G which is defined as . d dW P W P , and thus G , can be evaluated for different loading G 11 W P , and thus G , can be evaluated for different loading conditions. This definition is valid for both linear and nonlinear elastic deformation of the body. G is a function of the load (or displacement) and crack length. It is independent of the boundary conditions, in particular whether the loading is fixed-displacement or fixed-load. The Griffith criterion for fracture initiation in an ideally brittle solid can be re-phrased in terms of G such that .2 ' 2 S a We define the compliance C (inverse of the stiffness) of G 12 a cracked solid as C=u / F. It can be shown that . 2 2 da dC B F Thus measurements of compliance as a function of crack length allow the energy release rate to be evaluated. G Modes of Fracture The three basic modes of separation of the crack surfaces ( modes of fracture) are depicted below: 13 Combinations of modes (mixed-mode loading) are also possible. Modes of Fracture Definitions Mode I (tensile opening mode): The crack faces separate in a direction normal to the plane of the crack. The displacements are symmetric with respect to the x – z and x – y planes. 14 Mode II (in-plane sliding mode): The crack faces are mutually sheared in a direction normal to the crack front. The displacements are symmetric with respect to the x – y plane and anti-symmetric with respect to the x – z plane. Modes of Fracture Definitions Mode III (tearing or anti-plane shear mode): The crack faces are sheared parallel to the crack front. The displacements are antisymmetric with respect to the x – y and x – z planes. 15 to the x – y and x – z planes. The crack face displacements in modes II and III find an analogy to the motion of edge dislocations and screw dislocations, respectively. Plane Crack Problem The preceding analysis considered fracture from an energy standpoint. We now carry out a linear elastic stress analysis of the cracked body, which will allow us to formulate critical conditions for the 16 allow us to formulate critical conditions for the growth of flaws more precisely. An analysis of this type falls within the field of Linear Elastic Fracture Mechanics (LEFM). We consider a semi-infinite crack in an infinite plate of an isotropic and homogeneous solid as shown below: 17 Our goal is to develop expressions for the stresses, strains and displacements around the crack tip. Plane Crack Problem Equilibrium Equations The equilibrium equations (no body forces) are ,0 1 rrr rrr rr 18 ,0 2 1 rrr rr where and are the polar coordinates as shown previously. r Plane Crack Problem Strain-Displacement The strain-displacement relations for polar coordinates are: . 1 1 u u u r , 1 , u r r u r u rr rr 19 . 1 2 1 r u r u u r r r The strain compatibility equation in polar coordinates is: .0 11112 2 2 22 2 2 2 rrrrrrrrr rrrr rr Plane Crack Problem Hooke’s Law Hooke’s Law (for plane stress, ): 0 zz , rr E , rrrr E . 2 G G 20 For the case of plane strain ( ): 0 zz ,12 rrrr G . 2 rrr G G ,12 rr G .2 rr G . are: . 1 1 u u u r , 1 , u r r u r u rr rr 19 . 1 2 1 r u r u u r r r The strain compatibility equation in polar coordinates is: .0 111 12 2 2 22 2 2 2 rrrrrrrrr rrrr rr Plane. ): 0 zz , rr E , rrrr E . 2 G G 20 For the case of plane strain ( ): 0 zz , 12 rrrr G . 2 rrr G G , 12 rr G .2 rr G . in a direction normal to the crack front. The displacements are symmetric with respect to the x – y plane and anti-symmetric with respect to the x – z plane. Modes of Fracture Definitions Mode