1.6 Objective Tensors

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Observers are not always on the same place and they do not necessarily use the same time.

Consequently, observations are made by people in totally different places characterized by

local coordinate systems for time and space. In space, these coordinate systems are related

by a translation described by a vector c(t) and a rotation defined by an orthogonal matrix

DISPLACEMENTS, STRAIN, STRESS AND ENERGY 23

x1

x2

x3

x1_

x2_

x3_

c

x

x

_

Figure 1.4 Frames of different observers

Q(t) (Figure 1.4). Notice that, since the observers generally move with a different speed,

c and Q are a function of the time t . The different wall-clock time can be expressed by a

shift of time. Hence,

x

_

(X, t

_

) = c(t) +Q(t) · x(X, t) (1.179)

t

_ = t − a. (1.180)

Since Q is an orthogonal matrix Q

−1 = QT and detQ = 1. Here, only rigid body motions

excluding reflections are considered and hence detQ = 1. The transformation in

Equation (1.179) conserves the distance and angles. Indeed,

dx

_ = Q· dx (1.181)

and consequently

(ds

_

)2 = dx

_ · dx

_ = dx ·QT ·Q· dx = dx · dx = ds2 (1.182)

and

dx

_ · dy

_ = dx ·QT ·Q· dy = dx · dy. (1.183)

It is generally accepted that material properties should be independent of the coordinate

frame of the observer. Hence, in describing these material properties, we would like to use

quantities that ensure that the frame independence is guaranteed. For a time-independent

rigid body motion, it is known that vectors a and second-order tensors b in the spatial

description transform according to

a

_ = Q· a (1.184)

24 DISPLACEMENTS, STRAIN, STRESS AND ENERGY

and

b

_ = Q· b ·QT. (1.185)

Requiring this to be true for time-dependent rigid motions guarantees the spatial frame

indifference of any material law using such quantities. Vectors and tensors obeying

Equation (1.184) and Equation (1.185) for time-dependent rigid body motions are called

objective. From Equation (1.181), it is clear that dx is objective while time-differentiation

of Equation (1.179) reveals that the velocity v and the acceleration are not:

v

_ = ˙Q · x +Q· v (1.186)

a

_ = ¨Q · x + 2 ˙Q · v +Q· a. (1.187)

Accordingly, v and a should not be used to describe material laws. That the acceleration is

not objective is well known and is the reason for the Coriolis force in mechanics. Since the

transformation in Equation (1.179) conserves the distance, one obtains (Equation (1.164)):

D

Dt

(ds

_

)2 = 2dx

_ · d

_ · dx

_

= 2dx ·QT · d

_ ·Q· dx

= D

Dt

ds2 = 2dx · d · dx (1.188)

and consequently,

d = QT · d

_ ·Q. (1.189)

This shows that the deformation rate tensor is objective. Notice that a second-order tensor

a, which maps an objective vector b into another objective vector c, is objective. Indeed,

c

_ = a

_ · b

_ (1.190)

implies

c = (QT · a

_ ·Q) · b (1.191)

yielding

a = QT · a

_ ·Q. (1.192)

The time derivative of an objective vector or tensor is generally not objective. Indeed,

time differentiation of Equation (1.184) and Equation (1.185) yields

˙

a

_ = ˙Q · a +Q· ˙a (1.193)

˙

b

_ = ˙Q · b ·QT +Q· ˙b ·QT +Q· b · ˙Q

T

. (1.194)

The terms that are underlined are the reason for the lack of objectivity.

Finally, all vectors and tensors in the material description (such as C) are objective

since they are not influenced by a change of the spatial frame of reference.

DISPLACEMENTS, STRAIN, STRESS AND ENERGY 25

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