Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

introduce the concept of objectivity for tensor fields. We study further how velocity and acceleration fields behave under .changes of observers.

Observer and Euclidean transformation. The description of a physical process

.is related directly to the choice of an observer, which we denote subsequently by 0. An arbitrarily chosen observer in the three-dimensional Euclidean space and in time is equipped to measure

(i) .relative positions of points in space (with a ruler), and

(ii) instants oftime (with a clock).

An event is notked by an observer in tenns of position (place x) and time t.

Consider two arbitrary events in the Euclidean space characterized by the pairs (x0 , t0 ) and (x, t) (compare with OGDEN [1997]). We .assume that (x0 , t0 ) is 'frozen' as long as the event (x, t) occurs. An observer records that the pair of points .in space is separated by the distance fx - x0 l, and that the time interval (lapse) between the events

Figure 5.1 Map of two points preserving distance and time interval.

A spatial mapping which satis:fi.es the requirements above may be represented by the time-dependent transformation

The po.int differences x+ - x·t and x- x0 can be interpreted as vectors which are related through the orthogonal tensor Q(t), with the well-known property Q-1.(t) = Q(t)T

(compare with Section 1..2, p. 16). In order to maintain orientations we admit only

where in regard to x+ and x we think of position vectors characterizing two _points. For eq. (5.2) we have introduced a vector c(i), and a real number o· denoting the time-sh(ft, which are defined to be

c(t) = xt - Q(t)xn , (t = t(j - to . {5.3)

Note that both c and Q are continuous .functions of time which, for convenience, are assumed to be continuously differentiable.. The one-to-one mapping of the form (5.2) connecting the pair (x, t) with its corresponding pair (x+, t+) .is fre-quently referred to as a Euclidean t.ranstormation.

Change .of observer. We assert, for -example, that macroscop.ic properties of mate-

rials are not affected by the choice of an observer, a fundamental principle of physics. However, the general aim is to ensure that the stress state in a body and any physical -quantity with an .intrinsic feature must be invariant relative to a particular change of observer. Therefore, we ·expect from a change of observer that distances between arbi- trary pairs of points in space and time inten1als between events are preserved. In other words_, we require that a different observer o+ monitors the same relative distances of points and the same time intervals between events under observation.

We can show that the spatial transformation (5.1) or (5.2) 1, combined with the time- shift (5.2)2, denote the most general time-dependent change of observer from 0 too+. The event at place x and at time t recorded by observer 0 is the same event as that recorded by a different observer o+ at x+ and t+. Note that we are now considering one event recorded by two (different) observers 0 and o+ who are moving .relative to each other.

In orde:r to describe a physical process in the (three-dimensional) Euclidean space and on the real time axis we .assign to each of the observers a rectangular Cartesian coordinate system, which we characterize by a set of fixed basis vectors, i.e. {Ca} and {e~·} relative to 0 and o+, respectively. We call them reference frames of the observers. Hence, any points x and x+ may be represented by the position vectors x = ~r0e,1 and x+ = :l:te.~, with :1:a and :c:% denoting rectangular Cartesian coordinates, as usual. The shift in the time scale between the observer 0 and o+ is t+ = t +a.

In order to fonnulate the change of observer in index notation_, we identify Q(t) as the relative .rotation of the reference frames of the observers. Hence, e;t = Q(-t)ea

The mathematical express.ion (5.4)i states that the observers 0 and o+ assign, with the exception of the shift ct, the same coordinates to the corresponding points ·x and x+.

The reader should be aware that the introduced mapping, i.e. a change of observer, affects the points .in space-ti1ne and not the coordinates of points. However, a .change of reference frame changes the coordinates of points (and not the points themselves) and is simply governed by a coordinate transformation, as introduced in Section 1.5.

EXAMPLE 5.l Consider two .arbitrary points of a continuum body identified by

their position vectors x and y at time t. The events and are recorded by. an observer 0 with the reference frame ea·· A second observer o+ with the reference

counterpart u+ = y+ - x+ = u;-et and determine the components ua (of u) and u.;

(of u+), recorded by the two (different) observers 0 and o+T respectively.

With referen~e to eqs. ( 1.23h and {1.2.l ), the components of the spatial vector fields u+ and u relative to one and to the other reference frames of the observers read

(5~7)

By means of (5.4) 1 we deduce .from (5.6)3 that

(5~8)

With (5. 7h this shows that u; = ua, signifying lhat both observers measure the same. distance. •

is said to be objective or equivalently frame-indifferent. According to Example 5.1.,

184 5 Some Aspects of Objectivity

The overbar covers the quantity to which the time differentiation is applied. Further, we define the skew tensor

with the property

The tens.or n :represents the spin of the .reference frame of observer 0 relative to the referenee frame of observer o+.

Hence, material time differentiation of the spatial part of (5.10.) gives, using and the .product rule,

v+.(x+.1 t+) = c(t) + Q(t)x + Q(t)v(x, t;) .

From (5.17) we find, with the aid of (5.13) and (5.15)1, the transformation spatial velocity field, namely

v+ = Qv + c+ n{x+ - c)

(sup.pressing the arguments of functions).

We deduce from relation (.5.18) that the spatial velocity field v is not u~.de.r changes of observers following the arbitrary transformation x+ = c(t)

Since the extra terms ·C and O(x+ - c) are present, the .requirement for obj-ectiv.ity, i.e. eq. (5.9), is not satisfied. Hence, the velocity field v is only objective if

implying a change of observer according to the fo11owing time-independent trans.formation

with Co= 0 , (5.20)

referred to as a time-.independent rigid t-ranst'ormation. Therein, the vector c0 and the orthogonal tensor·Q0 .are assumed to be time-independent (constant) quantities. For a time-independent rigid transformation the magnitudes of v and v+ are equal, so that v+ = Qv, as would be require.d for an objective vector field~

The material time differentiation of (5.1. 8) gives, by means of the product rule and

(suppressing the arguments of functions).

Finally, by means of (5.14) and eqs. (5.15)1, (5.16h, we obtain from (5.21) the

for the spatial acceleration field a. Like for the spatial velocity field, the acceleration

field is not objective for a general change of observer. The terms O(x+ - c) and

-!12 (x+ - c) are called the .Euler .acceleration and centrifugal acceleration, while the last term in eq. (5.22), i.e. 2S'l(v+- c), represents the Coriolis acceleration.

An acceleration fi.eld is objective under all changes of observer if and 011/y (f the lengths of a+ and a are equal, i.e. a+ (x+, t+) = Q(t)a{x, t), requiring that

Consequently, this implies that cis constant and that the orthogonal tensor Q is also constant. A change of observer from 0 to o+ of this .type, for which the spatial acceleration is objective, is ca~led a Galilean transformation, and is governed by

with c(t) = o , (5.24)

for all times t. Here, Q0 denotes the time-independent orthogonal tensor, and c(t) = v0t + c0 , with the initial (constant) quantities for c0 and the velocity v0 •

Objective highe-r..order tensor fields. A spatial tensor field of order n, n = 1, 2, ....,

.i.e. u1 ® ···®Un, is called objective or equivalently frame-·indifferent, if, during any change of observer, u1 0 · · · 0 Un transforms according to

For any spatial vector field u (n = .1), eq. (5.25) reduces to u+(x+, t+) = Q(t)u(x, t), which we found through eq. (5.9).

In particulart for n = 0 we have a scalar field. It is obvious that any spatial scalar

.field cJ>(x, t), recorded by 0, is unaffected by a change of observer. Hence, a spatial scalar field <I> is obj-e·ctive if, under all Euclidean transformations (5.2), <I> transforms

according to

(5.27)

186 5 Some Aspects of Objectivity

where <I>+ is the corresponding scalar field recorded by observer o+.

In summary: the requirement .of objectivity means that tensor, vector and scalar fields transform under changes of observers according to the laws

Q(t)A(x, t)Q(t)T

(.5.28)

<I>{xJ) ,

where x+ and x are related by the Euclidean transformation (5.2) .

.EXAM.PLE 5.2 Show that the spatial gradient of an objective vector field u = u(x, t) transforms according to

(5.29)

where (gradu)+ = au+/ox+ denotes the spatial gradient of the vector u+ recorded by an observer o+. Note that in view of (5.28) 1 the second-order tensor field gTadu

(the arguments of the functions have been omitted). With the aid of transformation

(5. l 0) 1 we obtain the desired result. Note that, in general, the gradient of an objective tensor .field of order n is also objective. •

EXERCISES

L Consider objective scalar, vector and tensor fields <I>(x, t), u(x~ ·t) and A(x, t_),

.respectively.

(a)Show that grad~!>, divu and divA are objective fields during any change of -observer.

(b)Show that the Lie time derivative of a -contravar.iant spatial vector field u, as determined in e.q. (2.192), is objective.

2..Using eqs. (5.13)-(5...15) obtain the a1ternative relation for the spatial acceleration field (5.2.1) in terms of x, v and a in the form

3.Assume an objective transformation of the body· force (per unit volume), i.e. b+(x+, t+) = Q(t)b(x, t). Show that the local form o:f Cauchts first equation of

motion in the spatial description, i.e. (4.53), is only objective under a Galilean transformation.

5.2Superimposed Rigid~body Motions

In the following section we show that a change of observer may equivalently be viewed as certain rigid-body motions superimposed on the current configuration. We apply this concept to various kinematical quantities and to some stress tensors of importance.

Rigid-body motion. As noted, the fundamental relationship (5.2) desc~ibes a change of observer, preserving both the distances between arbitrary pairs of points in space;, and ti.me intervals between -events under observation.

It .is essential to introduce an important equivalent mechanical statement of the specification (5.2): for this purpose we consider a motion x+ = x+ (X, t+) of a con- tinuum body whic.h differs from another motion x = x(X, t) of the same body by a

superimposed (possibly time-dependent) rigid-body motion and by a time-shift, as depicted in Figure 5.2. We emphasize that, in contrast to the considerations of the .Jast section, x+ = x+(x, t+) .and x = are motions of two events recorded by .a s·ingl·e observer 0. The rigid-body motion moves the region n in space occupied by the body at time t, defined by the ,motion x = x(X, t), to a new region n+ occupied by the same body at t+, which is given by x+ = x+ (X, t+). Here and elsewhere we will employ the symbol -( • )+ to designate quantities associated with the new region n+.

According to the principle of relativity, the description .nf a single motion moni- tored by two (different) obse~vers, as described in is equivalent to the description of two (different) .motions monitored by a single observer. Hence, the pairs (x, t) and (x+, t+), which are defined on regions n and n+, are precise]y related by the Euclidean transformation (5.2), i.e. x+ + = t

c_,Q

time t+

time ·t

,......

time ·t = 0

Figure 5.2 Two motions x+ and x of a body (monitored by .a single observer) which differ by

.a superimposed rigid..body motion and by a time..shift. A spatial vector field u transforms into u+ = Qu, with length ju+j = fuj.

Within this context the vector c describes a superimposed (tim-e-dependent, pure) rigid-body translation for which any material po.int moves an identical distance, with the same magnitude and direction at time t. Since Q is .a proper orthogonal tensor (detQ = +1), the orientation .is preserved and Q describes a superimposed (time~

dependent, pure) rigid-body rotation. For a pure rigid-body rotation, transformation (5.2) reduces to x+ = Q(t)x.

Hence, at each instant of time a rigid-body motion is the composition of .a rigidbody translation c and a rigid-body rotation Q about an ax.is of rotation, combined with a time-shift a = t+ - t. The material points occupy the same relative position in each motion (the angle between two arbitrary ·vectors and their lengths remain constant).

We now recall Example 5.1 and apply the described concept to the spatial vector fie:Id u = y - x located at region Q (see Figure 5.2). Hence, a rigid-body .motion maps the points x, y to the associated points x+., y+ located inn+ and the spatial vector u = y - x to u+ = y+ - x+.

With (5.2) we may conclude that the -distance between the two points y+ and x+ remain unchanged. Namely, y+ -·x+ = Q(t)(y - x) (compare with eq.. (5."I)), which immediately implies, on use of definition ( 1.15), identity (l .81) and the orthogonality condition Q(t)TQ(t) = I, that ly+ - x+f = Jy - xf. Consequently the lengths of the vectors u+ and u are equal, i.e. ju+ I = juj. We say that the spatial vector field u is objective during the rigid-body motion.