3.14 RATE OF CHANGE OF VOLUME

The volume change between the initial and current con¬guration was given

in Section 3.7 in terms of the Jacobian J as,

dv = J dV ; J = det F (3.124)

Di¬erentiating this expression with respect to time gives the material rate

of change of the volume element as* (see Figure 3.15),

™

d ™ dV = J dv

(dv) = J (3.125)

dt J

The relationship between time and directional derivatives discussed in

Section 3.11.2 can now be used to enable the material rate of change of the

Jacobian to be evaluated as,

™

J = DJ[v] (3.126)

Recalling Equations (3.76“77) for the linearized volume change DJ[u] gives

™

a similar expression for J where now the linear strain tensor µ has been

* Note that the spatial rate of change of the volume element is zero, that is, ‚(dv)/‚t = 0.

37

3.14 RATE OF CHANGE OF VOLUME

X 3, x 3 time = t + dt

`

dv

time = t

X1 , x1 dV

X 2 ,x2

time = 0

FIGURE 3.15 Material rate of change of volume.

replaced by the rate of deformation tensor d to give,

™

J = J trd (3.127)

Alternatively, noting that the trace of d is the divergence of v gives,

™

J =J ·v (3.128)

™

An alternative equation for J can be derived in terms of the material

™ ™

rate tensors C or E from Equations (3.127), (3.100), and (3.97) to give,

™

J = Jtrd

™

= Jtr(F ’T EF ’1 )

™

= Jtr(C ’1 E)

™

= JC ’1 : E

™

1

= 2 JC ’1 : C (3.129)

™

This alternative expression for J is used later, in Chapter 5, when we con-

sider the important topic of incompressible elasticity.

Finally, taking the material derivative of Equation (3.59) for the current

density enables the conservation of mass equation to be written in a rate

38 KINEMATICS

form as,

dρ

+ρ ·v =0 (3.130)

dt

Alternatively, expressing the material rate of ρ in terms of the spatial rate

‚ρ/‚t using Equation (3.84) gives the continuity equation in a form often

found in the ¬‚uid dynamics literature as,

‚ρ

+ ·(ρv) = 0 (3.131)

‚t

3.15 SUPERIMPOSED RIGID BODY MOTIONS AND

OBJECTIVITY

An important concept in solid mechanics is the notion of objectivity. This

concept can be explored by studying the e¬ect of a rigid body motion su-

perimposed on the deformed con¬guration as seen in Figure 3.16. From the

point of view of an observer attached to and rotating with the body many

quantities describing the behavior of the solid will remain unchanged. Such

quantities, like for example the distance between any two particles and,

among others, the state of stresses in the body, are said to be objective.

Although the intrinsic nature of these quantities remains unchanged, their

spatial description may change. To express these concepts in a mathematical

framework, consider an elemental vector dX in the initial con¬guration that

deforms to dx and is subsequently rotated to d˜ as shown in Figure 3.16.

x

The relationship between these elemental vectors is given as,

d˜ = Qdx = QF dX (3.132)

x

where Q is an orthogonal tensor describing the superimposed rigid body

rotation. Although the vector d˜ is di¬erent from dx, their magnitudes are

x

obviously equal. In this sense it can be said the dx is objective under rigid

body motions. This de¬nition is extended to any vector a that transforms

according to a = Qa. Velocity is an example of a non-objective vector

˜

˜

because di¬erentiating the rotated mapping φ = Qφ with respect to time

gives,

˜

‚φ

v=

˜

‚t

‚φ ™

=Q + Qφ

‚t

™

= Qv + Qφ (3.133)

39

3.15 SUPERIMPOSED RIGID BODY MOTIONS AND OBJECTIVITY

X3, x 3

` dx

Q

˜

dx

dX p

X1, x 1 X2, x 2

˜

p

P

˜

time = 0

`

FIGURE 3.16 Superimposed rigid body motion.

Obviously, the magnitudes of v and v are not equal as a result of the presence

˜

™

of the term Qφ, which violates the objectivity criteria.

For the purpose of extending the de¬nition of objectivity to second-order

tensors, note ¬rst from Equation (3.132) that the deformation gradients with

respect to the current and rotated con¬gurations are related as,

˜

F = QF (3.134)

Using this expression together with Equations (3.15,18) shows that material

strain tensors such as C and E remain unaltered by the rigid body motion.

In contrast, introducing Equation (3.132) into Equations (3.17) and (3.19b)

for b and e gives,

˜ = QbQT (3.135a)

b

˜ = QeQT (3.135b)

e

Note that although ˜ = e, Equation (3.19a) shows that they both express

e

the same intrinsic change in length given by,

12

(ds ’ dS 2 ) = dx · e dx = d˜ · ˜ d˜ (3.136)

xex

2

In this sense, e and any tensor, such as b, that transforms in the same

manner is said to be objective. Clearly, second-order tensors such as stress

and strain that are used to describe the material behavior must be objective.

An example of a non-objective tensor is the frequently encountered velocity

40 KINEMATICS

™

™

gradient tensor l = F F ’1 . The rotated velocity gradient ˜ = F (F )’1 can

˜˜

l

be evaluated using Equation (3.134) to give,

™

˜ = QlQT + QQT (3.137)

l

Again it is the presence of the second term in the above equation that renders

the spatial velocity gradient non-objective. Fortunately, it transpires that

the rate of deformation tensor d is objective. This is easily demonstrated

by writing the rotated rate of deformation d in terms of ˜ as,

˜ l

˜ 1 l lT ™T

™

d = 2 (˜ + ˜ ) = QdQT + 1 (QQT + QQ ) (3.138)

2

Observing that the term in brackets is the time derivative of QQT = I

and is consequently zero shows that the rate of deformation satis¬es Equa-

tion (3.135) and is therefore objective.

Exercises

1. (a) For the uniaxial strain case ¬nd the Engineering, Green™s, and Al-

mansi strain in terms of the stretch ».

(b) Using these expressions show that when the Engineering strain is

small, all three strain measures converge to the same value (see Chap-

ter 1, Equations [1.6] and [1.8]).

2. (a) If the deformation gradients at times t and t + ∆t are F t and F t+∆t

respectively, show that the deformation gradient ∆F relating the incre-

mental motion from con¬guration at t to t + ∆t is ∆F = F t+∆t F ’1 .

t

(b) Using the deformation given in Example 3.5 with X = (0, 0), t = 1,

∆t = 1, show that ∆F = F t+∆t F ’1 is correct by pushing forward the

t

T to vectors g and g

initial vector G = [1, 1] t+∆t at times t and t + ∆t

t

respectively and checking that g t+∆t = ∆F g t .

3. Consider the planar (1-2) deformation for which the deformation gradient

is,

®

F11 F12 0

F = ° F21 F22 0 »

0 0 »3

where »3 is the stretch in the thickness direction normal to the plane

(1-2). If dA and da are the elemental areas in the (1-2) plane and H and

h the thicknesses before and after deformation, show that,

da J

=j and h = H

dA j

where j = det(Fkl ), k, l = 1, 2.

41

3.15 SUPERIMPOSED RIGID BODY MOTIONS AND OBJECTIVITY

4. Using Figure 3.4 as a guide, draw a similar diagram that interprets the

polar decomposition Equation (3.34) dx = V (RdX).

5. Prove Equation (3.44b), that is, F ’T N ± = »1± n± .

6. The motion of a body, at time t, is given by,

1 t t2

®

F (t) = ° t2 1 t » ;

x = F (t)X;