1.5 Velocity

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In most problems time is involved. The total time rate of change of a field is denoted by

the total derivative D/Dt. It physically means that a material particle is followed while

monitoring the change of some field at the momentaneous location of the moving particle.

The partial derivative ∂/∂t is used when looking at the change in time of a field at a fixed

spatial position. Both are related by (chain rule)

D

Dt

φ(x, t) = ∂φ

∂t

+ ∂φ

∂x

· ∂x

∂t

(1.151)

20 DISPLACEMENTS, STRAIN, STRESS AND ENERGY

where φ(x, t) is some field variable. The second term in Equation (1.151) is also called

the convective time rate of change and is solely due to the nonzero velocity of the particle.

The vector field

v := ∂x(X, t)

∂t

(1.152)

is the classical velocity of a particle originally at location X. Applying Equation (1.151)

to the particle acceleration, a defined by

a := Dv

Dt

(X, t) (1.153)

one finds

a = ∂v

∂t

+ ∂v

∂x

· ∂x

∂t

(1.154)

= ∂v

∂t

+ (v ⊗∇) · v (1.155)

where

∇ := ∂

∂x

(1.156)

is a one-form. Writing

v = vkgk (1.157)

and

∇ = gl ∂

∂xl

(1.158)

leads to

∇ ⊗v = (v ⊗∇)T = gl ⊗ ∂

∂xl

(vkgk)

= vk

;lgl ⊗ gk (1.159)

and, consequently,

akgk

= ∂vk

∂t

gk

+ vk

;lvm(gk

⊗ gl) · gm

= _∂vk

∂t

+ vk

;lvl_gk (1.160)

or, in rectangular coordinates

ak = ∂vk

∂t

+ vk

,lvl . (1.161)

DISPLACEMENTS, STRAIN, STRESS AND ENERGY 21

The change of length in time is given by

D

Dt

ds2 = D

Dt

dx · dx

= _ D

Dt

dx_ · dx + dx · _ D

Dt

dx_

. (1.162)

Since

D

Dt

dx = dv = ∂v

∂x

· dx = (v ⊗∇) · dx (1.163)

one finds

D

Dt

ds2 = dx · (∇ ⊗v) · dx + dx · (v ⊗∇) · dx

= dx · (∇ ⊗v + v ⊗∇) · dx

= 2dx · d · dx (1.164)

where

d := 12

(∇ ⊗v + v ⊗∇)_ (1.165)

is called the deformation rate tensor. One also defines the velocity gradient l and the spin

tensor w:

l := (v ⊗∇)_ (1.166)

w := 12

(l − lT). (1.167)

Consequently, one obtains

d = 12

(l + lT). (1.168)

Equation (1.164) shows that 2d plays a similar role for D(ds2)/Dt as g for ds2.

Since

dx = F · dX (1.169)

one finds by taking the total derivative of both sides

(v ⊗∇) · dx = ˙F · dX

= ˙F · F

−1 · dx (1.170)

or

l = (˙F · F

−1)_ (1.171)

22 DISPLACEMENTS, STRAIN, STRESS AND ENERGY

where

˙ ( ):= ˙

( ):= D

Dt

( ). (1.172)

In component notation, Equation (1.164) reads

D

Dt

ds2 = 2dxk dxl dkl . (1.173)

Since

ds2 = (2EKL + GKL) dXK dXL (1.174)

one also finds

D

Dt

ds2 = 2 ˙EKL dXK dXL. (1.175)

Comparison of Equation (1.173) and Equation (1.175) leads to

˙E

KL = dklxk

,Kxl

,L (1.176)

or

˙E

= FT · d · F. (1.177)

Accordingly, the tensor ˙E is the pullback of d and equivalently d is the push-forward of

˙E

.

The time derivative of the Jacobian J can be derived as follows:

DJ

Dt

= DJ

Dxk

,K

Dxk

,K

Dt

= JXK

,k

_Dxk

Dt

_

,K

= JXK

,kvk

,K

= JXK

,kvk

;lxl

,K

= Jvk

;k. (1.178)

In this derivation, Equation (1.61) was used. The expression vk

;k corresponds to the divergence

of the velocity, also written as ∇ · v.

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