PSI - Issue 42

Nina Selyutina et al. / Procedia Structural Integrity 42 (2022) 420–424 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

422

3

1

 t

 W W t dt W  ( ' ) '

(4)

c



t

−

W

where W ( t ) is the time dependence of the fracture work per unit area of the specimen, W C is specific fracture energy under quasi-static loading, and τ W is the incubation time under the fracture work limit condition for a U-notched specimen. Based on the data of (Chen et al. 2009), the typical curves of time versus applied load F(t) and the resulting deflection u(t) in the middle of the span obtained in a three-point bending test on a notched semi-circular specimen are linear up to the point of mode I fracture. The work of fracture ( )  = u A t ku du 0 ' '

in the case of linear loading is specified as ( ) ( ) kv t H t A t 2 2 1 2 =

(5)

where k is the stiffness coefficient of the system, and v =d u /d t is the deflection growth rate. Substituting Eq. (5) into condition (4) and taking into account W ( t )= A ( t )/ S , we obtain an expression for the fracture time t * , from which we can derive expressions for the fracture work per unit area of an unnotched specimen S =( R - l n ) B :

(

)

2 3

      

2

6

2 kS k SW v c



W

,

,

t





*

W

( ) I 

d W K

=

(6)

2

  

  

(

)

2

3

+ − 3

24

kv

kv

kSW

+





W

W

c

,

t





*

W

72

kS

4. Water-saturation effect Let us apply force and energy approach for calculation of rate dependencies of fracture toughness (Eq. (3)) and fracture energy based on experimental data of sandstone (Zhou et al. 2019) tested in three-point bending tests with following parameters: l n = 5 mm, B =20 mm ( B is thickness of specimen), R =25 mm, k =0.74 MN/m. Quasi-static fracture toughness in Zhou et al. 2019 K Is = 0.51 MPa·m 1/2 for dry sandstone and K Is = 0.29 MPa·m 1/2 for wet sandstone were obtained. Fig. 1 shows theoretical stress intensity rate dependencies of fracture toughness and experimental data (Zhou et al. (2019)) for dry and saturated sandstone. Fig. 2 shows theoretical dependencies of fracture energy depending on fracture toughness and experimental data (Zhou et al. (2019)) for dry and saturated sandstone. Plotted theoretical dependencies for dry and saturated sandstone in Fig. 1 and Fig. 2 have a good correspondence with an experimental data (Zhou et al. (2019)). Incubation times for saturated sandstone is bigger than for dry sandstone. The resulting incubation time estimates obtained from strain rate dependence (6) for fracture work and strain rate dependence (3) differ significantly. This is probably due to the fact that limit condition (1) is a force condition and limit condition (4) is an energy one from a physical point of view.