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Kinematics_of_CM_04_Material_Time_Derivatives

# Kinematics_of_CM_04_Material_Time_Derivatives - Section 2.4...

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Section 2.4 Solid Mechanics Part III Kelly 239 2.4 Material Time Derivatives The motion is now allowed to be a function of time, ( ) t , X χ x = , and attention is given to time derivatives, both the material time derivative and the local time derivative . 2.4.1 Velocity & Acceleration The velocity of a moving particle is the time rate of change of the position of the particle. From 2.1.3, by definition, dt t d t ) , ( ) , ( X χ X V (2.4.1) In the motion expression ( ) t , X χ x = , X and t are independent variables and so X is independent of time, denoting the particle for which the velocity is being calculated. The velocity can thus be written as t t / ) , ( X χ or, denoting the motion by ) , ( t X x , as dt t d / ) , ( X x or t t / ) , ( X x . The spatial description of the velocity field may be obtained from the material description by simply replacing X with x , i.e. ( ) t t t ), , ( ) , ( 1 x χ V x v = (2.4.2) As with displacements in both descriptions, there is only one velocity, ) , ( ) , ( t t x v X V = they are just given in terms of different coordinates. The velocity is most often expressed in the spatial description, as dt d t x x x v = = & ) , ( velocity (2.4.3) To be precise, the right hand side here involves x which is a function of the material coordinates, but it is understood that the substitution back to spatial coordinates, as in 2.4.2, is made.

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