PSI - Issue 6

Y.V. Petrov et al. / Procedia Structural Integrity 6 (2017) 134–139 Author name / Structural Integrity Procedia 00 (2017) 000 – 000

136

3

the form of an impact of a trapezoidal time profile   f t . In this case, the duration of the growth and decay of the pulse will be denoted by 1 T and 3 T respectively, and 2 T the duration of the constant section. Thus, the total exposure time is 1 2 3 T T T T    . Let A is its amplitude (stress value at a constant section). In this way

t

t T

0

0,  

  

t

t T 

,

1

T

A   

 

1

f t

(2)

1 T t T T t T T T T t T T       1 2 1 2 1 2 1, ,

1     

 

3

Assuming that the time t is counted from the start of the moment of reflection, and the coordinate x corresponds to the distance to the free boundary, and solving the corresponding dynamic problem of the theory of elasticity, it can be obtained that the stress in the cross section x at time t is given by the sum of the stresses in the incident and reflected waves

x c                x c f t 

(3)

( , ) t x f t 

Here, c is the propagation velocity of the longitudinal wave. Let there be a specific temporal profile of the impact (in general, it can take the form of an isosceles or right - angled triangle, a trapezoid, a sinus, etc.) and its duration T . For a given duration T and form of action, there is the smallest amplitude * A , that leads to destruction. Such an impulse can be called a minimal destructive or threshold. For the trapezoidal pulse under consideration, the form of the effect will be given by intervals 1 T , 2 T , 3 T . Similarly to how it was done for the impact of triangular pulses in Petrov et al. 2010 can be found for the load under consideration the smallest destructive (threshold) amplitude * A , as well as the moment of failure * t and the coordinate of the spall section * x . It can be shown that the threshold amplitude will be determined by substituting (2) into criterion (1). The minimum value * A , at which (1) is violated will be the one sought by virtue of its definition. Accounting for the interaction of waves is not required for this. In order to determine the moment of failure and the coordinate of the spall section, the total stress, formed jointly by both the forward and reflected waves (3) is substituted to criterion (1). After carrying out the necessary calculations, we obtain:

2 ,  

 

T





c

T T 

   

2

2

c  

(4)

A

T

T

,

  

*

2



  2



T T T

T T 



  

   

2

2

T

c 





2



      

T

T

,







2





T T  

T

1

t

T

T

,

     

  

(5)





*

2

T T 

2 2

2

T T 

T









1

2

2