J Elast (2012) 107:125–130DOI 10.1007/s10659-011-9347-0

A Comment on the Formation of Objective ResponseFunctions

James Casey · Michael M. Carroll

Received: 27 March 2011 / Published online: 3 June 2011© Springer Science+Business Media B.V. 2011

Abstract The well-known method for obtaining objective response functions in continuummechanics, due to W. Noll, is reviewed and an objection to its logic, raised by R.S. Rivlinand G.F. Smith, is evaluated.

Keywords Invariance requirements · Objectivity conditions · Finite elasticity · Continuummechanics

Mathematics Subject Classification (2000) 74A20

1 Introduction

Objectivity requirements provide a powerful means for restricting the forms of physicallyacceptable constitutive functions. Notable improvements on earlier, vaguer notions of ma-terial objectivity were made during the 1950s, especially by Oldroyd [1], Richter [2], Noll[3], Rivlin and Ericksen [4], Green and Rivlin [5], and Noll [6], using somewhat differentapproaches.

The reduction procedure utilized by Noll [6], in the context of materials with memory, issimple, direct, and very general. Making clever use of invariance requirements together withthe polar decomposition theorem, Noll obtained both necessary and sufficient conditions forresponse functions to be objective. Truesdell, in his excellent lectures [7], presented Noll’smethod in a less formal setting than in [6]. The method was used extensively in Truesdell

J. Casey (�)Department of Mechanical Engineering, University of California, 6125 Etcheverry Hall, Berkeley,CA 94720-1740, USAe-mail: jcasey@me.berkeley.edu

M.M. CarrollDepartment of Mechanical Engineering & Materials Science, Rice University, MS 321, PO Box 1892,Houston, TX 77251-1892, USAe-mail: carroll@rice.edu

126 J. Casey, M.M. Carroll

and Noll’s The Non-Linear Field Theories of Mechanics [8] and has since been adopted byvirtually all authors in continuum mechanics (see e.g., [9–17]).

While not in disagreement with the outcome of Noll’s procedure, Rivlin and Smith [18]raised an objection to the logic of the supporting argument. The purpose of this Note is toevaluate this objection, which has been a source of some puzzlement.

Let us denote the set of (second-order) tensors with positive determinants by Lin+. Also,let Psym be the set of symmetric positive definite tensors and let Orth+ be the set of properorthogonal tensors. Recall that the deformation gradient F ∈ Lin+ possesses a unique rightpolar decomposition

F = RU , (1)

where R, the rotation tensor, belongs to Orth+, and U , the right stretch tensor, belongs toPsym.

In brief, Rivlin and Smith [18] hold the opinion that Noll’s method supposes that everyscalar-valued function of the right stretch tensor U would retain the same value when theargument U is replaced by the deformation gradient F .1 Thus, for example, while it isalways true that

tr(U T U

) = tr(F T F

), (2)

it is plainly not the case that for every value of F , tr(U 2) is equal to tr(F 2). But, as weshall explain, Noll’s method does not actually entail this. Rather, the idea involved may bestated as: For any given scalar-valued function φ, defined on Psym, it is always possible toconstruct a function ψ on the larger domain Lin+, such that ψ satisfies the conditions

ψ(F ) = ψ(U) = φ(U). (3)

In essence, the issue lying at the center of [18] involves the relationship between a functionand its extensions to a larger set of tensors. It will be shown that once Noll’s procedureis carefully analyzed and an understanding is reached about the freedom inherent in theconcept of the extension of a function, the objection of Rivlin and Smith [18] can be setaside.

By way of background, recall that the set dom(f ) on which a function f is defined, isan essential part of the definition of that function. Let Lin denote the set of all second-ordertensors, and let B ⊂ A ⊂ Lin. Let f be any given scalar or tensor-valued function withdomain A. The restriction of f to B denoted by f |B , is defined uniquely by evaluating f atthe tensors belonging to the subset B . Also, let g be any given function with dom(g) = B .Any function h, with dom(h) = A, that satisfies the condition h|B(B) = g(B), for all B ∈ B ,is called an extension of g to A. The original function f is obviously always an extensionof f |B to A, but there are infinitely many other extensions of f |B to A, and as we shall see,there may be a good reason for preferring a different extension than this obvious one. Also,if a function g, with dom(g) = B is given by an algebraic formula and the same formulamakes sense for all elements of A, we may call such an extension the natural extension of gto A. But, once again, there are always other extensions of the function.

In (3), dom(φ) = Psym, and the function ψ is a particular extension of the function φ

to Lin+.

1Although Noll did not make any such statement, Rivlin and Smith [18] interpret his procedure to entail it.

A Comment on the Formation of Objective Response Functions 127

2 Noll’s Procedure

Consider a Green-elastic (or hyperelastic) continuum, with strain-energy W per unit mass,and let T (= T T ) be the Cauchy stress tensor. Thus,

W = W̄ (F ), T = T̄ (F ). (4)

Note that dom(W̄ ) = dom(T̄ ) = Lin+. We will discuss reduction procedures for both of theresponse functions in (4)1,2.2

For clarity, we state Noll’s procedure [6, 7] in two parts: (N1), and its converse (N2).(N1) If the response functions W̄ and T̄ , defined on Lin+, are assumed to be objective,

then

W̄ (QF ) = W̄ (F ), T̄ (QF ) = QT̄ (F )QT , (5)

for all rotations Q ∈ Orth+.3

We may choose Q = RT , where R is the rotation in the polar decomposition (1) of F . Itthen follows from (5)1,2 that

W̄ (U) = W̄ (F ), T̄ (U) = RT T̄ (F )R. (6)

Thus, the function W̄ must have the same value at the two points U and F in its domain,Lin+. Likewise, the values of the function T̄ at the two points U and F must be related toone another by (6)2. It follows immediately from (4)1,2 and (6)1,2 that

W = W̄ (F ) = W̄ (U), T = T̄ (F ) = RT̄ (U)RT . (7)

The domain of each of the functions W̄ and T̄ is still Lin+.The converse of (N1) may be stated as:(N2) If each of the functions W̄ and T̄ has the domain Lin+ and if these functions

satisfy the conditions (7)1,2, where F , R, and U are connected to each other through thepolar decomposition (1), then W̄ and T̄ satisfy the objectivity conditions (5)1,2. The proofis straightforward: Since the right stretch tensor of QF is also U and the rotation tensor ofQF is QR it follows from (7)1,2 that

W̄ (QF ) = W̄ (U), T̄ (QF ) = (QR)T̄ (U)(QR)T . (8)

Equations (8)1,2 and (7)1,2 together imply (5)1,2.An equivalent statement of the converse which is more constructive in nature may be

made:(N2*) If W̄ and T̄ are any given functions with

dom(W̄ ) = dom(T̄ ) = Psym, (9)

2Thus, our discussion will actually cover the full class of elastic (or Cauchy-elastic) materials. For hyperelas-tic materials, the response functions T̄ and W̄ are ultimately related to one another, but we may discuss theirobjectivity properties before taking this into account.3We are using invariance requirements under superposed rigid motions of the continuum. Readers who preferthe principle of material frame-indifference would allow Q to be any orthogonal tensor. This will not affectthe present arguments in any way.

128 J. Casey, M.M. Carroll

and if extensions=W and

=T of these functions to the domain Lin+ are defined by the special

recipes=W(F ) = W̄ (U),

=T (F ) = RT̄ (U)RT , (10)

where F , R, and U are related to each other through the polar decomposition (1), then the

extended functions=W and

=T satisfy the objectivity conditions (5)1,2.

That (N2*) is equivalent to (N2) can be readily seen: The extended functions=W and

=T

in (N2*) can be used as the given functions in (N2); likewise, the functions in (N2) can beregarded as extensions of their own restrictions to Psym that satisfy the conditions (10)1,2.

Remark 1 If arbitrary recipes are used to extend the functions W̄ and T̄ from Psym to Lin+then, in general, the extensions will not satisfy the objectivity conditions (5)1,2. For, if theextended functions were to satisfy (5)1,2, the proof in (N1) would imply conditions of thetype (6)1,2 which are restrictive.

Remark 2 Clearly, if functions W̄ and T̄ are specified arbitrarily on Lin+, then the objectiv-ity conditions (5)1,2 cannot be expected to be satisfied. However, the following procedure canalways be followed for constructing objective response functions from any given functionsW̄ and T̄ :

Step 1. Take the given functions and get their restrictions W̄ |Psym and T̄ |Psym to Psym bysetting F = U in them, where U is the right stretch tensor associated with F through (1):

W̄ |Psym(U) = W̄ (U), T̄ |Psym(U) = T̄ (U). (11)

Step 2. Define extensions=W and

=T of W̄ |Psym and T̄ |Psym, respectively, to Lin+ by using the

recipes=W(F ) = W̄ |Psym(U),

=T (F ) = RT̄ |Psym(U)RT , (12)

where F , R, and U are related to each other through (1). The functions=W and

=T , thus

defined on Lin+, satisfy the objectivity conditions (5)1,2.

Some simple examples will illustrate this procedure:

Example 2.1 Suppose that

W̄ (F ) = trF ,(dom(W̄ ) = Lin+)

. (13)

Then,

W̄ |Psym(U) = trU , (14)

and, by virtue of (12)1,=W(F ) = trU . (15)

Note that the natural extension of trU to Lin+, which in this case is the original functiontrF , does not satisfy the condition (6)1, as can be seen by choosing U = I and R = F to bea rotation of 180 degrees about an axis.

A Comment on the Formation of Objective Response Functions 129

Example 2.2 Next, let

W̄ (F ) = tr(F T F

),

(dom(W̄ ) = Lin+)

. (16)

Then,

W̄ |Psym(U) = tr(U T U

) = tr(U 2

), (17)

and=W(F ) = tr

(U 2

) = tr(F T F

) = W̄ (F ). (18)

In this case, the original function works. But, note that if the argument in (16) were in-

stead F 2, then (17) would remain the same, and=W would again be tr(U 2) = tr(F T F ) and

not the natural extension tr(F 2).

Remark 3 The recipes (10)1,2 are not the only ones that produce objective extensions ofgiven functions on Psym to Lin+. For example, the following also do:

W #(F ) = β(U)W̄ (U), T #(F ) = FT̄ (U)F T , (19)

where β is any scalar-valued function with dom(β) = Psym. And, there are others (see e.g.,[8, 11]). However, they are all closely related to the basic recipes (10)1,2.

3 The Objection

Rivlin and Smith [18] observe that it is not true that every given scalar-valued functionW̄ (U) would satisfy the condition

W̄ (F ) = W̄ (U). (20)

Rivlin and Smith [18] appear to be thinking only of natural extensions. The followingslightly weaker statement may be made, which allows for arbitrary extensions:

(Z) Given a function W̄ , with dom(W̄ ) = Psym, extensions of W̄ to Lin+ exist whichdo not satisfy the condition (10)1.

As we have seen in Examples 2.1 and 2.2, an extension, even a natural one, need not sat-isfy (10)1. Thus, the observation of Rivlin and Smith [18] is correct, and so is statement (Z).Indeed, since extensions can be specified freely outside of the smaller domain, there areinfinitely many extensions of W̄ to Lin+ that do not satisfy the condition (10)1. Moreover,these would not, in general, satisfy the objectivity condition (5)1, as explained in Remark 1.

However, the observation of Rivlin and Smith [18] does not contradict the special exten-sion procedures described in Sect. 2. In particular, if a given function W̄ , with dom(W̄ ) =Psym, is extended to Lin+ according to the recipe (10)1, then that extension will satisfy theobjectivity requirement (5)1.

Similarly, it is true that not every extension of a given function T̄ , with dom(T̄ ) = Psym,will satisfy the condition (10)2. However, if T̄ is extended to Lin+ according to therecipe (10)2, then the objectivity requirement (5)2 will be satisfied.4

4Corresponding results hold for the Piola-Kirchhoff stress tensors [8].

130 J. Casey, M.M. Carroll

A secondary remark by Rivlin and Smith [18] concerns different choices for the compo-nent forms of U , R, and Q. All of these tensors belong to the same vector space Lin+, and,as seen above, invariance requirements can be completely phrased in component-free form.

4 Conclusion

Rivlin and Smith [18] observe that not every given function W̄ would automatically satisfythe condition (20). This observation is correct, and so is statement (Z). However, what isactually being asserted in Noll’s procedure is that the particular extension which is givenby the recipe (10)1 does satisfy the invariance requirement (5)1. That assertion is true. OnceNoll’s arguments are stated precisely, the objection of Rivlin and Smith [18] is seen to haveno bearing on their logical validity.

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