PSI - Issue 42

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

421

2

are reported showing that classical fracture mechanics cannot be applied to describe dynamic behavior in terms of stress intensity factor characteristics. The instability of dynamic fracture toughness for rocks is qualitatively similar to the increasing dependence of the dynamic rock strength with strain rate. In both cases, the dynamic behavior of two materials is compared using the dynamic increase factor that normalizes the dynamic behavior of the material to the static one. Based on this parameter, the strain rate dependence is approximated in the form of analytic polynomials (Zhou and Hao (2008), Hao and Hao (2012), Chakraborty et al. (2016)). The parameters of the analytic polynomials depend on the strain rate and thereby limit each polynomial to an acceptable range of strain rates. In this work, the dynamic fracture of dry and saturated sandstone (Zhou et al. (2019)) is considered from the viewpoint of the force and energy limiting criteria, formulated based on the structural-temporal approach. 2. Force structural-temporal approach in terms of fracture toughness Using the force incubation time criterion (Petrov et al. (2012a), Petrov et al. (2012b)), the dependencies of the strength of rocks or concrete depending on the strain rate are effectively predicted, as well as to explain the influence of the structure on the dynamic strength of the rock (Selyutina and Petrov (2018), Selyutina and Petrov (2020)). In this study, the fracture condition for dynamic three-point bending experiments based on force incubation time criterion is written as in studies (Selyutina (2022), Bratov (2011)): where K Is is the static stress intensity factor in mode I fracture, τ Κ is the fracture incubation time, K I ( t ) is the time dependence of the stress intensity factor in three-point tests. Fracture time t * is defined as the earliest time at which an equality sign is reached in the condition (1). The parameter τ Κ is incubation time associated with the dynamics of the relaxation processes preceding the macro-fracture event. The independence of the incubation time τ Κ from the loading history K I ( t ) is a key feature of the application of the incubation time criterion. In this paper, linear loading of the sample is used, according to the analyzed experimental data (Zhou et al. (2019)): ( ) ( ) K t K t H t I =    (2) where ̇ is the fracture toughness rate, H ( t ) is the Heaviside function. Taking into account that the condition of equality (1) leads to mode I fracture in the specimen and Eqs. (2), we can calculate the dynamic stress intensity factor of mode I fracture K Id = K I ( t * ) as a dependence on the fracture time and fracture toughness rate:  t −  K K s ds K  ( ) t Is K  1 (1)

2 1

   

+ K K 

,

,

t







( ) I 

*

Is

I K

K

Id K K

=

(3)

K K  

2

,

t





*

Is K I

K

To construct theoretical dependences (3), it is necessary to determine the incubation time, which is estimated from experimental data by the least squares method.

3. Energy structural-temporal approach The structural-temporal approach is also used to calculate the fracture work of a material (Bragov et al. (2013); Selyutina (2022)). In this case, the criterion has the form