13th International Conference on Fracture June 16–21, 2013, Beijing, China -1- Threshold Fracture Energy for Differently Shaped Particles Impacting Halfspace (Erosion-Type Fracture) Grigory Volkov1,2,*, Yuri V Petrov1,2, Nikolay Gorbushin1 1 St. Petersburg State University, Research Center of Dynamics, St. Petersburg, Russia, 198504, 2 Institute of Problems of Mechanical Engineering, St. Petersburg, Russia, 199178, * Corresponding author: volkovgrig@mail.ru Abstract Energetic aspects of erosion fracture are studied. Threshold (minimal) energy needed for initiation of fracture caused by a particle impact is estimated using the incubation time criterion. The dependences of the fracture threshold energy on impact duration are calculated for two different shapes of the impacting particle: sphere and cylinder. The difference in the threshold energy behavior between these two cases is demonstrated and discussed. Keywords Incubation time criterion, erosion, threshold energy Introduction In the present work the problem of dynamic impact of spherical and cylindrical solid particle on elastic half-space is studied. This problem is considered within the framework of fracture mechanics by neglecting of the heat transfer process and wave-process origination. Also there is an assumption that the impact is quasistatic. It permits to use the solution of the problem of quasistatic of the indenter pressing in [1]. This problem plays an important role in practical application, since the short impacts are typical for such industry process as ultrasonic assisted machining [2,3]. Threshold Energy of a Sphere Particle The normal impact of the spherical rigid particle on the elastic half-space is considered. Hertz solution for the contact problem gives the following temporal dependence of the contact force Pon the distance h between the bodies approaching each other: ( ) ( ) [ ]3/ 2 P t k h t = , (1) where ( ) [ ] 2 4 / 31 ν− = k RE , R - particle radius, E and ν - elastic constants of the half-space. If m is the mass of the particle and V is its initial velocity then the time dependence of the approach ( ) h t can be determined by solving the equation of motion: ( ) ( ) P t dt d h t m = − 2 2 . (2) The solution of the differential equation (2) can be approximated with high precision by the following expression: ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 0 0 sin t t h t h π , (3) where 2/ 5 2 0 4 5 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = mV k h is maximum approach and 1/5 2 2 0 0 3.2 2.94 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ≈ = Vk m V h t is the contact time. Also Hertz solution gives the following expression for the tensile stresses originated on the half-space surface: ( ) ( ) ( ) ( ) a t P t t t r 2 2 1 2 π ν σ σ θ − = − = , (4) ( ) ( )( ) ( ) 1/3 3/2 1/3 2 . 4 3 1 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ = ⎠ ⎞ ⎜ ⎝ ⎛ − = k R P t E R a t P t ν . (5)
RkJQdWJsaXNoZXIy MjM0NDE=