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Explain the difference between the case 1 with bonded contact (top image) and the case 2 (bottom image) with the frictional coefficient of 0.01

For both cases, the top plate has displacement and the bottom plate is a fixed support.

case 1: both plates have bonded contact with the cylindrical workpiece

Case 2: both plates have frictional contact with a friction coefficient of 0.01 with the cylindrical workpiece

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A "bonded contact" is presumably the same as welding the parts together.

Presumably, the material also has non-zero Poisson's ratio.

A friction coefficient of 0.01 will allow the ends of the pin to expand radially as it is compressed. The small but non-zero friction coefficient will stop the compressed pin from being perfectly cylindrical as it would be with zero friction, and the second picture seems to show that small effect.

Note, the model might not run with zero friction unless you restrain one point on the pin (e.g. on its center line) to stop it moving sideways an arbitrary distance.

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