PSI - Issue 6

Ivan Smirnov et al. / Procedia Structural Integrity 6 (2017) 34–39 Ivan Smirnov et al. / Structural Integrity Procedia 00 (2017) 000 – 000

36

3

3. The structural – temporal approach

A failure criterion on the basis of the incubation time concept was formulated by Petrov and Utkin (1989), Petrov and Morozov (1994). In recent papers of Bragov et al. (2012), Petrov et al. (2017) this criterion was successfully applied for describing of strain/stress rate dependencies of critical stresses in rocks and concrete under dynamic compression and splitting. The criterion can be written as:

t

  ,

 

, J J J 

, x t dt J

,







c  

(1)

c

c

t



or more simply:

t

 

x t dt

( , )





c  

(2)

t



where J is the local power pulse; J c is the critical pulse; σ ( x , t ) is the stress in a point of failure, σ c is the static strength of the material, and τ is the incubation time of failure. The incubation time of failure is a parameter of a material which characterizes duration of micro-fracture processes before macro-rupture of the material. Thus, failure will occur, if in any point within the incubation time, the pulse will accumulate the value not less than σ c τ . The failure time t* is defined as the moment of achievement by the pulse of the critical value. The constants τ and σ c form the system of constitutive parameters of the failure process at a given structural scale level . 4. Results and discussion

4.1. Limiting stresses at different pulse durations

Fig. 2 shows the test results with different projectile lengths. The tests were performed for samples of gabbro diabase. Both compression and splitting tests showed different the threshold limit stress for the different durations (length projectile) of a stress pulse. The threshold limit stress means that the reduction of a stress pulse amplitude (projectile speed) will not cause failure of a sample. However, these threshold values exceed the quasi-static strength of the material.

b

50

10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 100 200 300 400 500 600 700 d  /dt (GPa/s * 10 3 ) 300 mm 100 mm calculation thresholds

a

250 mm 75 mm calculation

45

40

thresholds

35

  (MPa)

  (MPa)

30

10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 25

d  /dt (GPa/s * 10 3 )

c

Fig. 2. Thresholds of limiting stresses in gabbro-diabase samples under stress pulses with different duration. a) compression tests; b) splitting tests.