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finding contact time from maximum displacement for hertzian contact

While going through the derivation of contact time for a hertzian contact as given in problem 3 at the following link http://s17.postimg.org/t1kq6mlxr/Capture.png , I am not able to understand how the integral form for contact time has come into picture. Can anyone explain the in-between steps to get the same? I understand that this is a very trivial problem but it will be a great help to understand the steps. Thanks.

Dependence of fracture energy of ceramic on loading condition?

Literature suggests that the fracture strength of the ceramic tends to be higher in a dynamic loading condition than in static condition. This relates to the increase in the fracture energy in dynamic processes. Literature refers to an inelastic response prior to failure (Hugoniot elastic limit) the reason behind increased strength. Can anyone explain the phenomenon in a more lucid way, or guide me to an appropriate reference to understand this?

Which is the best model to capture brittle fracture and failure of ceramics at moderate velocities?

Several models have been developed over the past decades to capture the fracture and failure of ceramic materials. JH2, JHB models are widely used for simulating the behavior of armor plates upon ballistic impact. I have a doubt regarding these models. Are these models only valid when the impact velocity is in the order of 1000m/s, as under such circumstances material transitions from elastic to elastic-plastic regime defined by the HEL Pressure? But what about when the impact velocity of the ceramic is around 300-400 m/s (a fraction of ballistic impact)?

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