kinematics of cm 04 material time derivatives

8/13/2019 Kinematics of CM 04 Material Time Derivatives

1/4

Section 2.4

Solid Mechanics Part III Kelly239

2.4 Material Time Derivatives

The motion is now allowed to be a function of time, t,Xx , and attention is given totime derivatives, both the material time derivativeand 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

tdt

),(),(

XXV (2.4.1)

In the motion expression t,Xx , Xand tare independent variables and Xisindependent of time, denoting the particle for which the velocity is being calculated. The

velocity can thus be written as tt /),(X or, denoting the motion by ),( tXx , as

dttd /),(Xx or tt /),(Xx .

The spatial description of the velocity field may be obtained from the material description

by simply replacing Xwith x, i.e.

ttt ),,(),( 1 xVxv (2.4.2)

As with displacements in both descriptions, there is only onevelocity, ),(),( tt xvXV

they are just given in terms of different coordinates.

The velocity is most often expressed in the spatial description, as

dt

dt

xxxv ),( velocity (2.4.3)

To be precise, the right hand side here involves xwhich is a function of the material

coordinates, but it is understood that the substitution back to spatial coordinates, as in2.4.2, is made (see example below).

Similarly, the acceleration is defined to be

2

2

2

2

2

2),(),(

),(t

t

dt

d

dt

d

dt

tdt

XVxXXA (2.4.4)

Example

Consider the motion

331

2

222

2

11 ,, XxXtXxXtXx


2/4

Section 2.4


The velocity and acceleration can be evaluated through

21122

2

2112 22),(,22),( eex

XAeex

XV XXdt

dttXtX

dt

dt

One can write the motion in the spatial description by inverting the material description:

334

1

2

224

2

2

11 ,

1,

1xX

t

xtxX

t

xtxX

Substituting in these equations then gives the spatial description of the velocity andacceleration:

2 21 2 1 1 2

1 24 4

2 21 2 1 1 2

1 24 4

( , ) ( , ), 2 21 1

( , ) ( , ), 2 21 1

x t x x t xt t t t t

t t

x t x x t xt t t

t t

v x V x e e

a x A x e e

2.4.2 The Material Derivative

One can analyse deformation by examining the current configuration only, discounting

the reference configuration. This is the viewpoint taken in Fluid Mechanics one focuses

on material as it flows at the current time, and does not consider where the fluid was.In order to do this, quantities must be cast in terms of the velocity. Suppose that the

velocity in terms of spatial coordinates, ),( txvv is known; for example, one could

have a measuring instrument which records the velocity at a specific location, but the

motion itself is unknown. In that case, to evaluate the acceleration, the chain rule of

differentiation must be applied:

dt

d

ttt

dt

d x

x

vvxvv

),(

or

vvv

a grad

t acceleration (spatial description) (2.4.5)

The acceleration can now be determined, because the derivatives can be determined

(measured) without knowing the motion.

In the above, the material derivative, or total derivative, of the particles velocity wastaken to obtain the acceleration. In general, one can take the time derivative of any

physical or kinematic property

expressed in the spatial description:


3/4

Section 2.4


v

grad

tdt

d Material Time Derivative (2.4.6)

For example, the rate of change of the density ),( tx of a particle instantaneously at

xis

v

grad

tdt

d (2.4.7)

The Local Rate of Change

The first term, t / , gives the local rate of changeof density at xwhereas the second

term gradv gives the change due to the particles motion, and is called the convective

rate of change.

Note the difference between the material derivative and the local derivative. For example,the material derivative of the velocity, 2.4.5 (or, equivalently, ( , ) /d t dt V X in 2.4.4, with

Xfixed) is not the same as the derivative ( , ) /t t v x (with xfixed). The former is the

acceleration of a material particle X. The latter is the time rate of change of the velocity

of particles at a fixed locationin space; in general, differentmaterial particles will occupy

position xat different times.

The material derivative dtd/ can be applied to any scalar, vector or tensor:

vAAA

A

vaaa

a

v

grad

grad

grad

tdt

d

tdt

d

tdt

d

(2.4.8)

Another notation often used for the material derivative is DtD / :

Df dff

Dt dt (2.4.9)

Steady and Uniform Flows

In a steady flow, quantities are independent of time, so the local rate of change is zero

and, for example, v grad . In a uniform flow, quantities are independent of

position so that, for example, t /

Example

Consider again the previous example. This time, with only the velocity ( , )tv x known,

the acceleration can be obtained through the material derivative:


4/4

Section 2.4


24

2

2

114

1

2

2

42

2

1

4

1

2

2

4

3

4

44

3

242

2

114

1

2

2

12

12

01

2

12

000

012

12

01

2

1

2

12

12

grad),(

ee

ee

vvv

xa

t

xtx

t

xtx

txtxt

t

xtxt

tt

tt

t

t

t

t

txtxt

txtxt

t

tt

as before.

The Relationship between the Displacement and Velocity

The velocity can be derived directly from the displacement 2.2.42:

dt

d

dt

d

dt

d uXuxv

)(, (2.4.10)

or

vuuuv grad tdtd (2.4.11)

When the displacement field is given in material form one has

dt

dUV (2.4.12)

2.4.3 Problems

1. The density of a material is given byxx

te

2

The velocity field is given by

213132321 2,2,2 xxvxxvxxv

Determine the time derivative of the density (a) at a certain position x in space,

and (b) of a material particle instantaneously occupying position x.

kinematics of cm 04 material time derivatives

Documents