ABSTRACT
Let us consider a sheet of thickness t with a plane crack of length L extending through the thickness. The sheet is loaded by tractions pi = P fi(x) of amplitude P along the edges. Assuming linear elastic behavior, the displacements produced are ui = P gi(x). The stored energy is
, (6.1.1)
where
. (6.1.2)
C is the flexibility coefficient or the coefficient of compliance of the structure. It must be found by solving the elasticity equations. C depends on the material properties, the geometry of the body, the distribution of load, and the support conditions. If all other parameters are fixed, C is a function of the crack length L. Let us define a generalized displacement by
. (6.1.3)
Then the stored energy is
. (6.1.4)
Thus, U is a function of P and L, and
. (6.1.5)
U = = =∫ ∫12 12 12 2τ εij ij i idV p u dA C P V S
C f g dAi i= ∫ S
D C P=
U = =1 2
1 2
2C P P D
∂ ∂
∂ ∂
∂ ∂
U U L
P C L P
C P= = 2
2 ,
If the crack elongates by a small amount dL, the change in stored energy is
(6.1.6)
The work of the external force during the crack extension is
. (6.1.7)
Additional energy dE is expended to fracture the material and create a new free surface . If G is the energy of crack growth per unit area,
. (6.1.8)
The fundamental assumption is that G is a material constant which has to be determined by materials testing. This is known as the Griffith hypothesis.
