Hi

In discussing about the Vibration amplitude response against failure modes, Acceleration versus Force relationship is straightforward and could be explained using Newton's second law. Is there any way to understand the relationship between Displacement Vs Stress and Velocity Vs Fatigue failure modes.

Best regards

Shanmuga Sundaram

Original Post

Displacement and strain have a simple relation.  Strain and stress have a linear relation (for small quantities).  For simple states of stress, one may have a simple relation.
Accel, Vel, and Disp relate by time derivatives.  Strain is related by derivatives in space not time.

Is there any way to explain the relationship between Displacement Vs Stress and Velocity Vs Fatigue mathematically? Like F = M*A in which the relationship between Force and Acceleration is very clear.

Strain and displacement are related - look up definition for strain.  Strain and stress in are related; look up linear relationship between stress and strain (basically E*strain - stress , but E is not the simple constant of Young's modulus.

Velocity and fatigue - seems to simple or simplistic - you would need some assumptions.

You like force and acceleration - then you should like displacement and force.