Kinematics_of_CM_04_Material_Time_Derivatives - Section 2.4...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 01/20/2012 for the course ENGINEERIN 3 taught by Professor Staff during the Fall '11 term at Auckland.

Page1 / 4

Kinematics_of_CM_04_Material_Time_Derivatives - Section 2.4...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online