PSI - Issue 60

S.K. Pandey et al. / Procedia Structural Integrity 60 (2024) 665–677 S. K. Pandey/ Structural Integrity Procedia 00 (2023) 000 – 000

667 3

( , ) = − ( , )

(5)

The impedance (z), define as ratio of force (F) and particle velocity (v), is provided in Eq. 6. Where ρ and A are the density and cross section of bar respectively. The level of reflection and transmission of wave is decided by the mismatch of two bars at the interface. Higher the mismatch lower the transmission. ( , ) = = = = ( 2 ) = (6) The velocity and force at the either side of interface is must be equal. Since at the interface, faces are at the intimate contact at all time during impact hence velocity at just left and right of the material must be equal. Similar for satisfy the equilibrium, the forces at right and left of interface must be equal. Velocity of particle at interface in the incident bar is the sum of velocity of particle due to incident wave (v i ) and reflected wave (v r ). Incident and reflected wave are moving in positive x and negative x direction respectively. The velocity of left end of specimen (v s1 ), provided in Eq. 7, is equal to the velocity of incident bar due to continuity of velocity across the interface. Similarly velocity of the right end of specimen (v s2 ), provided in Eq. 8, is equal to the velocity of transmission bar. The wave in the transmission bar is moving in positive x direction. 1 = + =− + =− ( − ) (7) 2 = =− (8) The strain rate in specimen is ̇ = ( 1 − 2 ) = − ( − − ) ) (9) The force at the left end of specimen which is equal to the force in incident bar is provided in Eq. 10. Where, A s , A, σ s1 , σ i , and σ r are the cross section of specimen, cross section of incident bar, stress in specimen at left end, stress due to incident pulse in incident bar and stress due to reflected pulse in incident bar respectively. Force in the incident bar is equal to the sum of forces due the incident and reflected pulse. Similarly force at the right end of specimen is equal to the force in transmission bar. 1 = ( + ) (10) 2 = (11) Forces in the specimen will be in equilibrium after a few wave propagation. Hence from equation 10 &11, the relations in incident, reflected and transmitted stresses pulse is provided in Eq. 12 similar for strains relations is provided in Eq. 13. = + (12) = + (13) Stress in specimen (force in transmission bar/area of specimen) can be found from Eq. 14. The strain rate from Eq. 9 and Eq. 13 can be written as in Eq. 15. The strain at any time, t, in the specimen can be determine by integrating the strain rate from 0 to t as in Eq. 16. = (14) ̇ = 2 (15)