Arch. Rational Mech. Anal. 138 (1997) 1±35.

Ó Springer-Verlag 1997

On The Concepts of State and Free Energy

in Linear Viscoelasticity
Dedicated to the memory of Ignace Kolodner
``... ben tetragono ai colpi di ventura'' (Par., XVII, 24)

GIANPIETRO DEL PIERO & LUCA DESERI

Communicated by D. OWEN

Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Histories and segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3. States and processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4. Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5. Free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6. Relaxation functions of exponential type . . . . . . . . . . . . . . . . . . . . . . . . . 22
7. Completely monotonic relaxation functions . . . . . . . . . . . . . . . . . . . . . . . 29
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1. Introduction

The constitutive equation of linear viscoelasticity is the Boltzmann-Vol-

terra equation
Z‡1
T t† ˆ G0 E t† ‡ G_ s†E t ÿ s† ds; 1:1†
0

in which the stress at the time t in a material element subject to the defor-
mation process E is determined by the current deformation E t† and by the past
history Et of E at t:
Et s† :ˆ E t ÿ s†; s 2 0; ‡1†: 1:2†

If we ®x a time t > 0 and if we consider the restriction E 0;t† of E to the

time interval 0; t† as a continuation of the history E0 , we can separate the
in¯uence of E 0;t† and E0 on the stress response by rewriting the equation (1.1)
in the form
2 G. DEL PIERO & L. DESERI

Zt Z‡1
T t† ˆ G0 E t† ‡ G_ t ÿ r†E r† dr ‡ G_ t ‡ r†E0 r† dr: 1:3†
0 0
0 00
This equation tells us that two histories E ; E have the same in¯uence on the
stress response if and only if their di€erence H :ˆ E0 ÿ E00 satis®es the con-
dition
Z‡1
G_ t ‡ r†H r† dr ˆ O for all t 3 0: 1:4†
0

The third axiom of NOLL'S new theory of simple materials [23] says that
``if two states are di€erent . . . then there must be some process which pro-
duces di€erent stresses with the two states as initial states''. If we accept this
axiom, and if we agree that in our case the processes are the continuations, we
may conclude that two histories whose di€erence satis®es the condition (1.4)
must correspond to the same state. If we assume that the current deformation
is independent of the past history, then we are led to de®ne a state as a pair
whose entries are an equivalence class of histories and a deformation. The
deformation is the current deformation, and two histories are equivalent if
their di€erence satis®es the condition (1.4).
With these de®nitions of process and state, we identify a system in the
sense of the theory of COLEMAN & OWEN [5], and we use the general results of
that theory to study some basic questions of linear viscoelasticity which have
long been debated by several authors. One such question is the character-
ization of the state space. Usually, states are identi®ed, at least implicitly,
with history-deformation pairs [3, 10, 13, 16]. There is, however, an impor-
tant exception, that of the viscoelastic materials of rate type, for which the
relaxation function is a linear combination of exponentials. For such mate-
rials, a state is usually identi®ed with a ®nite array of internal variables [2,
18]. It is not clear, however, whether this choice is a matter of convenience or
is dictated by some sort of general requirements. In the approach that we
present here, the possible de®nitions of a state are strictly limited by the
structure of the solution set of equation (1.4): Indeed, a state can be repre-
sented by a history-deformation pair if and only if the solution set reduces to
the null history alone, so that the equivalence classes which constitute the ®rst
entry of a state reduce to singletons.
For a relaxation function of exponential type, we show in Section 6 that
the ®nite dimensional characterization of a state is compatible with our
de®nition, while the characterization as a history-deformation pair is not.
This result can be easily extended to all viscoelastic materials of rate type. We
also produce an example of a class of completely monotonic relaxation
functions for which the equivalence classes are singletons, and therefore the
states are correctly described by history-deformation pairs.
Another question which we consider here is that of the topology of the
state space. When a state is de®ned as a history-deformation pair, it is natural
 States and Free Energies in Linear Viscoelasticity 3

to de®ne the state space as the product of the space of histories and the space
of deformations, and to endow it with the product norm of the two spaces
[13]. The norm chosen for the space of histories is usually the fading memory
norm of COLEMAN & NOLL [4], suggested by the physical consideration that
the response of a material with memory is more in¯uenced by the deforma-
tions undergone in the recent past than by those that occurred in the far past.
In e€ect, as shown by the weaker fading memory assumptions made by
VOLTERRA [24], GRAFFI [17], and DAY [11], a fading memory e€ect is implicit
in the constitutive equation (1.1), provided that the relaxation function de-
cays to its equilibrium value suciently fast. The main reason for the success
of the approach of COLEMAN & NOLL lies in the far-reaching consequences of
the principle of the fading memory, which is an assumption of continuity of
the constitutive functionals in the topology induced by the fading memory
norm. Under this assumption, many general properties of materials with
memory have been proved, such as some restrictions and interrelations for
the constitutive functionals, and the minimality of the equilibrium free energy
in the set of all states having the same current deformation. In this paper, in
the more limited context of linear viscoelasticity, we obtain the same results
in a more direct way. We endow the space of histories with a seminorm,
which is a norm for the set of the equivalence classes determined by the
solution set of the equation (1.4). The sum of this seminorm and the norm of
the space of deformations is a norm for the state space, and we use the
topology induced by that norm.
This choice plays an important role in the de®nition of the free energy,
which is the central subject of the paper. Among the de®nitions present in the
literature, we focus our attention on the de®nition given by COLEMAN &
OWEN, who de®ne the free energy as a lower potential for the work [6, Sec. 5].
The general results of their theory are then used to prove the existence of a
maximal and of a minimal free energy, characterized as the minimum work
done to approach a state starting from the natural state, and as the maximum
work which can be recovered from a given state, respectively. In the special
case of linear viscoelasticity, we ®nd two additional properties beyond those
shared by all systems and by all free energies. Namely, we prove that every
state can be approached from every other state by a sequence of processes
with the property that the sequence of the works done in these processes is
convergent, and we prove that the minimal free energy is lower semicontin-
uous with respect to the topology that we have adopted for the state space.
The last two sections are devoted to the study of two particular classes of
viscoelastic material elements, characterized by relaxation functions of ex-
ponential type and by completely monotonic relaxation functions, respec-
tively. For the ®rst class, we generalize a result of GRAFFI & FABRIZIO [19],
which asserts that there is just one free energy, whose explicit expression was
determined by BREUER & ONAT [2]. We also show that some other functions,
which are usually considered as appropriate to describe the free energy, are
indeed not acceptable because they do not de®ne a function of state for this
speci®c class of relaxation functions.
4 G. DEL PIERO & L. DESERI

For completely monotonic relaxation functions, we discuss here the

characterization of the state space, while the characterization of the free
energy is still under study. With the aid of Bernstein's representation formula
for completely monotonic functions and of MuÈntz's theorem on the com-
pleteness of powers, we determine a class of completely monotonic functions
for which states are characterized by history-deformation pairs. We also
determine a class, consisting of completely monotonic functions of expo-
nential type, for which the state space is ®nite-dimensional. These two classes
are not exhaustive, and a comprehensive classi®cation of completely mono-
tonic functions based on the structure of their state space is not yet available.
We conclude this introduction with some remarks on notation. We denote
by Sym the set of all symmetric linear transformations on the vectors,
equipped with the inner product A
B :ˆ tr ABT † and, consequently, with the
norm
jAj :ˆ A
A†1=2 ; 1:5†
and by LinSym we denote the set of all linear transformations on Sym,
equipped with the norm
jCAj
kCk :ˆ sup : 1:6†
A2SymnfOg jAj

By I and O we denote the identity and the zero mapping in Sym, and by I and
O we denote the corresponding mappings in LinSym. For each pair C, D of
elements of LinSym, the notations
C > D; C 3 D; 1:7†
mean that C ÿ D† is positive-de®nite and positive-semide®nite, respectively.
We also denote by B A; d† the open ball of radius d centered at A, and for
every function H de®ned over an interval of the real line we denote by (var H )
the variation of H in its domain of de®nition [12, Eq. 2.1].

2. Histories and segments

In the constitutive equation (1.1), E is a function from the reals into Sym,
and G_ is a function from the non-negative reals into LinSym. For these
functions, we keep the regularity assumptions made in [12. Sec. 2]; namely,
for E we assume that (i) for each t 2 R, the history Et de®ned by equation
(1.2) is a function of bounded variation, and that (ii) E is continuous from the
right. For G_ we assume that (iii) G_ is Lebesgue integrable. This assumption
implies that G_ admits a primitive G, determined to within an additive con-
stant, which we ®x by setting
G 0† :ˆ G0 ; 2:1†
where G0 is the tensor appearing in equation (1.1). It also implies that G is
absolutely continuous and bounded, and that the limit
 States and Free Energies in Linear Viscoelasticity 5

G1 :ˆ lim G s† 2:2†
s!‡1

exists. The tensor G0 is the instantaneous elastic modulus, and G1 is the

equilibrium elastic modulus. We also assume that (iv) both G0 and G1 are
symmetric, and
G0 > G1 3 O: 2:3†
The regularity assumptions made on E lead to the following de®nition of a
history.

2.1. De®nition. A history is a function from the positive reals into Sym, of
bounded variation and continuous from the left.
It can be shown that every history H is bounded and has right and left
limits H s‡ †, H sÿ † at every s > 0, with H sÿ † ˆ H s† by assumption (ii).
Moreover, a history has continuous extensions to zero and to in®nity:
H 0‡ † :ˆ lim‡ H s†; H 1† :ˆ lim H s†: 2:4†
s!0 s!‡1

A segment of duration d is a function K : 0; d Š ! Sym, of bounded

variation and continuous from the left. The truncation of a history H at r > 0
is the segment Hr of duration r de®ned by
Hr s† :ˆ H s†; 0 < s 2 r; 2:5†
and the r-section of H is the history H r given by
H r s† :ˆ H r ‡ s† for s > 0: 2:6†
The continuation K  H of a history H by a segment K of duration d is the
history 
K s† for 0 < s 2 d,
K  H † s† :ˆ 2:7†
H s ÿ d† for s > d.
Notice that
Hr  H r ˆ H 2:8†
for all histories H and for all r > 0.
In quite a similar way, one can de®ne the truncation Kr of a segment K to
be its restriction to 0; rŠ, and the r-section K r of K to be the segment of
duration d ÿr† de®ned by K r s† ˆ K r ‡ s†: The continuation of a segment K
of duration d by a segment K 0 of duration d 0 is the segment K 0  K taking the
value K 0 s† for 0 < s 2 d 0 and K s ÿ d 0 † for d 0 < s 2 d ‡ d 0 . Note that,
since histories and segments need not be continuous, any history and any
segment can be continued by any segment.
The constant history associated with the deformation A is the history Ay
with Ay s† ˆ A for all s > 0, and a ®nite history is a continuation K  Ay of a
constant history. The collection of all histories obeying De®nition 2.1 is a
vector space which is denoted by C. The set of all constant histories and the
set of all ®nite histories are subspaces of C, and the null element of C is the
constant history Oy associated with the null deformation.
6 G. DEL PIERO & L. DESERI

Another distinguished subspace of C is the set C0 of all solutions of

equation (1.4). The function
Z‡1
kH kC :ˆ sup G_ t ‡ s†H s† ds 2:9†
t30
0

is a seminorm for C and a norm for the quotient space C=C0 . Let us prove
some relevant properties of this seminorm.

2.2. Proposition. For every pair of histories H , H 0 the following properties

hold:
(i) For every segment K,

kK  H 0 ÿ K  H kC 2 kH 0 ÿ H kC : 2:10†
(ii) For each e > 0 there is a positive real m such that, for all segments K of
duration greater than m,
kK  H 0 ÿ K  H kC < e: 2:11†
(iii) If Hd is the truncation of H at d, then
lim kHd  H 0 ÿ H kC ˆ 0: 2:12†
d!‡1

(iv) For every history H and for every segment K of duration d,

lim kK r  H ÿ H kC ˆ 0; 2:13†
r!d

where K r is the r-section of K.

Proof. If K is a segment of duration d, it follows from the de®nition of k
kC that

Z‡1
0
kK  H ÿ K  H kC ˆ sup G_ t ‡ s† H 0 s ÿ d† ÿ H s ÿ d†† ds
t30
d
Z‡1
ˆ sup G_ t 0 ‡ s 0 † H 0 s 0 † ÿ H s 0 †† ds 0 ; 2:14†
t0 3 d
0
0 0
with s ˆ s ÿ d and t ˆ t ‡ d. The inequality (2. 10) then follows by taking the
supremum over t 0 3 0 instead of t 0 3 d. To prove the second assertion, we set
M :ˆ sup jH s†j ‡ sup jH 0 s†j 2:15†
s>0 s>0

and note that M < ‡1 because both H and H 0 are bounded. Thus, from
equation (2.14),
Z‡1 Z‡1
0
kK  H ÿ K  H kC 2 M sup _
kG t ‡ s†k ds ˆ M kG_ s†k ds; 2:16†
t30
d d
 States and Free Energies in Linear Viscoelasticity 7

and, because G_ is integrable over 0; ‡1†, there is a positive real m such that
for all d > m the last integral is less than eM ÿ1 . This proves the inequality
(2.11). If we now take K ˆ Hd , from the identity (2.8) and from the inequality
(2.16) we have
Z‡1
kHd  H 0 ÿ H kC ˆ kHd  H 0 ÿ Hd  H d kC 2 M kG_ s†k ds; 2:17†
d

and (2.12) follows. The last statement asserts that the supremum of the in-
tegral
Z‡1
G_ t ‡ s† H s† ÿ K r  H † s†† ds
0
Z‡1 Zdÿr
ˆ G_ t ‡ s† ÿ G_ t ‡ s ‡ d ÿ r††H s† ds ‡ G_ t ‡ s†K r ‡ s† ds; 2:18†
0 0

taken over all non-negative t, converges to zero when r ! d. By setting

r 0 ˆ d ÿ r and integrating by parts, the same integral is transformed into
Z‡1
ÿ G t ‡ s† ÿ G t ‡ s ‡ r 0 †† dH s† ÿ G t† ÿ G t ‡ r 0 ††H 0‡ †
0
Zr 0
ÿ G t ‡ s† dK d ÿ r 0 ‡ s† ‡ G t ‡ r 0 †K d† ÿ G t†K d ÿ r 0 †; 2:19†
0

and therefore it is less than

varH † sup kG t ‡ s† ÿ G t ‡ s ‡ r 0 †k ‡ jH 0‡ †j kG t† ÿ G t ‡ r 0 †k
s30

Zr 0
‡ varK† kG t ‡ s†k ds ‡ kG t ‡ r 0 †kjK d† ÿ K d ÿ r 0 †j
0
‡ kG t ‡ r 0 † ÿ G t†kjK d ÿ r 0 †j: 2:20†

By the absolute continuity of G, for every e > 0 there is a d > 0 such that
r 0 < d implies that

Z
s‡r 0

kG s† ÿ G s ‡ r 0 †k < e; kG s 0 †k ds 0 < e 2:21†

s

for all s 3 0, and the fact that K is continuous from the left implies that there
is a d 0 > 0 such that
8 G. DEL PIERO & L. DESERI

jK d† ÿ K d ÿ r 0 †j < e 2:22†
for all r 0 < d 0 . Thus, for suciently small r 0 the expression (2.20) is less than
var H † ‡ jH 0‡ †j ‡ var K† ‡ sup kG s†k ‡ sup jK s†j†e; 2:23†
s30 d>s 3 0

and because this bound is independent of t we conclude that the supremum of

the integral (2.18) taken over all t 3 0 converges to zero when r 0 ! 0: (

The constitutive equation (1.1) de®nes the stress reached by the material
when subjected to the history Et and to the current deformation E t†; because
E is only continuous from the right, E t† need not coincide with the ®nal
value E t‡ † of Et , and therefore history and current deformation are unre-
lated. Thus, the response functional can be de®ned as the function
T~ : C  Sym ! Sym which with every history-deformation pair (H, A) as-
sociates the stress
Z‡1
T~ H ; A† :ˆ G0 A ‡ G_ s†H s† ds: 2:24†
0

The response functional is linear. Moreover, the inequality

jT~ H ; A†j 2 jG0 Aj ‡ kH kC ; 2:25†
which follows from the de®nition (2.9) of the seminorm k
kC , shows that T~ is
continuous in the topology induced by k
kC . It also follows from the
property (2.11) of k
kC that, if we continue two histories H 0 and H with the
same segment K, the di€erence between T~ K  H 0 ; A† and T~ K  H ; A† can be
made arbitrarily small by taking K of suciently long duration. This prop-
erty of linear viscoelastic materials was called the principle of dissipation of
hereditary in¯uence by VOLTERRA [24]. As shown in the proof of Proposition
2.2, this property is a consequence of the assumed integrability of G._ In fact,
_
the integrability of G was taken by GRAFFI as a weak form of the principle of
fading memory [17].

3. States and processes

On the basis of the motivations discussed in the Introduction, we identify

the states of the material with the elements of the Cartesian product
R :ˆ C=C0 †  Sym; 3:1†
i.e., with ordered pairs whose ®rst item is an equivalence class of histories
modulo C0 , and whose second item is a deformation. We use the notation
r  H ; A† 3:2†
to mean that the state r is represented by the history-deformation pair
H ; A†. Of course, r may be represented as well by any other pair H 0 ; A†
 States and Free Energies in Linear Viscoelasticity 9

such that H 0 ÿ H 2 C0 . In particular, rA  Ay ; A† is the equilibrium state

associated with the deformation A, and r0  Oy ; O† is the natural state.
The state space R is a vector space, and can be made a normed space by
setting
krkR :ˆ jG0 Aj ‡ kH kC 3:3†
whenever r  H ; A†.
It is clear from the linearity of T~ and from the inequality (2.25) that, if
H ; A† and H 0 ; A† represent the same state, then the stresses T~ H ; A† and
T H 0 ; A† are the same. This fact tells us that the response functional T~ is
~
indeed a function of state, and we may de®ne a function T^ : R ! Sym such
that
T^ r† ˆ T~ H ; A† 3:4†
when r  H ; A†: The function T^ is linear, and the inequality (2.25) tells us
that it is continuous:
jT^ r†j 2 krkR : 3:5†
We de®ne a process of duration d as a segment-deformation pair
P :ˆ K; B†; 3:6†
where B is a deformation and K is a segment of duration d with K d† ˆ O.
For a state r  H ; A† and a process P ˆ K; B†, the state reached from r
when subjected to P is, by de®nition,
P r  KA  H ; B ‡ A†; 3:7†
where KA is the segment
KA s† :ˆ K s† ‡ A: 3:8†
We denote by P the set of all processes, and for each state r we denote by Pr
the set of all states which can be reached from r:
Pr :ˆ fP r j P 2 Pg: 3:9†
We prove below that for every r 2 R the set Pr is dense in R. This is done by
constructing, for each r 0 2 R, a family r 7! Pr of processes such that the
states Pr r converge to r 0 when r ! ‡1.

3.1. Proposition. Given two states r  H ; A† and r 0  H 0 ; A 0 †; for each

r > 0 consider the process
Pr :ˆ Kr  K 0 ; A 0 ÿ A†; 3:10†
0 0
where K is a segment of arbitrary duration d with K d† ˆ O; and Kr is the
segment of duration r given by
Kr s† :ˆ H 0r s† ÿ A; 3:11†
with H 0r the truncation of H 0 at r. Then the family r 7! Pr r converges to r 0 in
the norm k
kR when r 7! ‡ 1.
10 G. DEL PIERO & L. DESERI

Proof. The de®nition (3.7) tells us that Pr r  H 0r  K 0A  H ; A 0 †. Thus, by the

de®nition (3.3) of k
kR ,
kPr r ÿ r 0 kR ˆ kH 0r  K 0A  H ÿ H 0 kC ; 3:12†
and the third item in Proposition 2.2 shows that the right-hand side converges
to zero when r ! ‡1: (
We now ®x a P 2 P, and consider the map r 7! P r given by the de®nition
(3.7). We prove that this map is Lipschitz continuous.

3.2. Proposition. There is a positive constant m, depending only upon the re-
laxation function G, such that
kP r 0 ÿ P rkR 2 mkr 0 ÿ rkR 3:13†
for every process P and for every pair of states r; r 0 .

Proof. Let r  H ; A†; r 0  H 0 ; A 0 †; and P ˆ K; B† with K a segment of

duration d. Then by the de®nitions (3.3), (2.9) of k
kR and k
kC

kP r 0 ÿ P rkR ˆ kKA 0  H 0 ÿ KA  H kC ‡ jG0 A 0 ÿ A†j

Zd
ˆ sup G_ t ‡ s† KA 0 s† ÿ KA s†† ds
t30
0
Z‡1
‡ G_ t ‡ s† H 0 s ÿ d† ÿ H s ÿ d†† ds
d
‡ jG0 A 0 ÿ A†j; 3:14†
with KA 0 s† ÿ KA s† ˆ A 0 ÿ A by the de®nition (3.8), so that the ®rst integral
reduces to
Zd
G_ t ‡ s† A 0 ÿ A† ds ˆ G t ‡ d† ÿ G t†† A 0 ÿ A†: 3:15†
0

Because the second integral is less than or equal to kH 0 ÿ H kC , we have

kP r 0 ÿ P rkR 2 1 ‡ 2 sup kGÿ1 0 0
0 G r†k†jG0 A ÿ A†j ‡ kH ÿ H kC ; 3:16†
r30

and the inequality (3.13) follows, with

m ˆ 1 ‡ 2 sup kGÿ1
0 G r†k: 3:17†
r30

Note that m ˆ 1 if A 0 ˆ A: (
Given two processes P ; P 0 , the composition of the maps r 7! P r,
r 7! P 0 r is, by de®nition, the map r 7! P 0 P r which with each state r asso-
ciates the state P 0 P r reached from P r when subjected to P 0 . Thus, for
r  H ; A†; P ˆ K; B†; P 0 ˆ K 0 ; B 0 †, we have
 States and Free Energies in Linear Viscoelasticity 11

P 0 P r  K 0B‡A  KA  H ; B 0 ‡ B ‡ A†: 3:18†

Note that P 0 P r is the state reached from r when subjected to the process
K 0B  K; B 0 ‡ B†. This process is denoted by P 0 P and is called the continua-
tion of P with P 0 .
The density of the sets Pr in R and the de®nition of the composition
r 7! P 0 P r characterize the pair R; P† as a system in the sense of COLEMAN &
OWEN [5, Def. 2.1]. The further speci®cation of the response functional T^
completes the description of what is called here a linearly viscoelastic material
element. In the following, the adverb linearly is omitted for brevity. This
de®nition parallels the de®nition given in [5, Sec. 9], which includes the
thermal variables.
A di€erent de®nition of a linearly viscoelastic material element in the
framework of COLEMAN & OWEN's theory was given by FABRIZIO & MORRO
in the paper [13]. In accordance with the principle of fading memory of
COLEMAN & NOLL [4], these authors assume that the relaxation function
satis®es the condition
Z‡1
kG_ s†k2 hÿ1 s† ds < ‡1; 3:19†
0

where h is a positive, non-increasing function decaying to zero at a prescribed

rate when s ! ‡1. They also de®ne the fading memory space of histories as
the set C of all functions H : 0; ‡1† ! Sym for which the fading memory
norm
 Z‡1 1=2
kH kC :ˆ jH s†j2 h s† ds 3:20†
0

is ®nite. The states are then identi®ed with history-deformation pairs, and the
state space R is the product space C  Sym, endowed with the norm
krkR :ˆ jAj2 ‡ kH k2C †1=2 3:21†
with r  H ; A†. If we assume that h is integrable, then we see that the history
space C is included in C and, consequently, that the state space R is included
in R . Indeed, since every history H 2 C is bounded, its norm (3.20) is ®nite if
h is integrable. Moreover, if we use the de®nition (2.9) of kH kC and the
Cauchy-Schwarz inequality, we get
 Z‡1 1=2  Z‡1 1=2
kH kC 2 sup kG_ t ‡ s†k2 hÿ1 s† ds 2
jH s†j h s† ds : 3:22†
t30
0 0
ÿ1
The supremum is attained at t ˆ 0 because h is positive and non-decreasing.
Thus, if we denote by m2 the integral in inequality (3.19), we conclude that
kH kC 2 mkH kC 8 H 2 C; 3:23†
12 G. DEL PIERO & L. DESERI

and the de®nitions (3.3) and (3.21) allow us to conclude that there is a pos-
itive constant m 0 such that
krkR 2 m 0 krkR 8 r 2 R: 3:24†
This shows that the norm of R is continuous with respect to the norm of R .
Consequently, every open set in R is an open set in R , and therefore the
topology induced in R by the norm k
kR is coarser than the one induced by
the norm k
kR . The inequality (3.23) cannot be reversed in general. Indeed,
for every history H 2 C0 nfOy g we have kH kC > 0 and kH kC ˆ 0.

4. Work

The work done by a viscoelastic material element subject to a history H is

Z‡1
w H † :ˆ ÿ T~ H r ; H r††
dH r† 4:1†
0‡

with H r and T~ de®ned by equations (2.6) and (2.24), respectively, and with the
integral over 0‡ ; ‡1† de®ned as the limit of the integral over a; ‡1† when
a ! 0 from the right. Notice that the Stieltjes integral on the right side is well
de®ned only if T~ is continuous, and this is not the case for discontinuous
histories. Indeed, it follows from the constitutive equation (2.24) that the
stress has a jump exactly at those points at which H has a jump. Nevertheless,
if the instantaneous modulus G0 is symmetric as we have assumed from the
beginning, then it is possible to de®ne, by a limit procedure, the work done in
a discontinuous history. Indeed, it is shown by Proposition 3.5 in [12] that the
work done in a history H having a jump at a can be expressed again by
equation (4.1), with the convention that the integral over 0‡ ; ‡1† is now the
sum of the integrals over 0‡ ; a† and a‡ ; ‡1†, plus the product of the jump
H a‡ † ÿ H a†† of H by the arithmetic mean of the stresses T~ H a ; H a‡ †† and
T~ H a ; H a††. In particular, if we denote by w ~ H ; A† the work done in the
history H followed by the deformation A, then w ~ H ; H 0‡ †† ˆ w H † and
~ H ; A† ˆ w H † ‡ 12 T~ H ; A† ‡ T~ H ; H 0‡ †††
A ÿ H 0‡ ††:
w 4:2†
We ®nd it convenient to consider A as the extension of H to s ˆ 0, and to use
the notation
Z‡1
ÿ T~ H r ; H r††
dH r† :ˆ w
~ H ; A† 4:3†
0

to include in the integral the work done in the jump H 0‡ † ÿ A† at s ˆ 0.

If H and H 0 are histories corresponding to the same state, the works w H †
and w H 0 † need not be equal, as will be shown by an example in Section 6. In
other words, the work is not a function of state whenever C0 contains his-
tories other than Oy . On the contrary, the work done in a given process is
 States and Free Energies in Linear Viscoelasticity 13

always a function of the initial state. Indeed, take a history H and a process
P ˆ K; B† of duration d. Then the work done in the process is
Zd
~ KA  H ; B† ÿ w
w ~ H ; A† ˆ ÿ T~ KAr  H ; KA r††
dKA r†; 4:4†
0

with KAr the truncation of KA at r. Take another history H 0 such that

H 0 ÿ H 2 C0 . Then the history KAr  H 0 ÿ KAr  H † also belongs to C0 by
Proposition 2.2, and therefore the pairs KAr  H ; KA r†† and KAr  H 0 ; KA r††
represent the same state. Because T~ is a function of state, it follows that the
right-hand side of equation (4.4) does not change if H is replaced by H 0 .
Thus, we may conclude that the work done in a process is a function of the
process of the initial state only, and we may introduce the notation
^ P ; r† :ˆ w
w ~ H ; A†
~ KA  H ; B ‡ A† ÿ w 4:5†
to denote the work done in the process P ˆ K; B† starting from the state
r  H ; A†:
In this way, we have de®ned a function w ^ from the set P  R into the
^ is an action if it enjoys the following two
reals. According to [5, Def. 2.2], w
properties:
(i) Additivity with respect to continuations:
^ P 0 P ; r† ˆ w
w ^ P 0 ; P r† ‡ w
^ P ; r† 8P ; P 0 2 P; 8r 2 R: 4:6†
(ii) Continuity with respect to the states: For each P 2 P, the function
^ P ;
† is continuous in R.
w
We prove below that w ^ satis®es these conditions, and that the continuity with
respect to the states is indeed Lipschitz continuity.

4.1. Proposition. The function w^ : P  R ! R de®ned by equation (4.4) is an

action. Moreover, it satis®es the inequality
^ P ; r 0† ÿ w
jw ^ P ; r†j 2 m (var K†kr 0 ÿ rkR 4:7†
0
for all P ˆ K; B† in P and for all r ; r in R, with m the constant appearing in
equation (3.13).

Proof. Take a state r  H ; A† and two processes P ˆ K; B†; P 0 ˆ K 0 ; B 0 †:

The de®nitions (4.5) and (3.18) tell us that
^ P 0 ; P r† ˆ w
w ~ K 0B‡A  KA  H ; B 0 ‡ B ‡ A† ÿ w
~ KA  H ; B ‡ A†; 4:8†
0 0 0
w ~ K
^ P P ; r† ˆ w B‡A ~ H ; A†;
 KA  H ; B ‡ B ‡ A† ÿ w 4:9†
and the equality (4.6) is obtained by subtraction and substitution into
equation (4.5). Now take a process P ˆ K; B† of duration d and two states
r  H ; A†; r 0  H 0 ; A 0 †. From the de®nitions (4.4) and (4.5) and from the
fact that dKA 0 r† ˆ dKA r† ˆ dK r† it follows that
14 G. DEL PIERO & L. DESERI

Zd
0
^ P; r † ÿ w
jw ^ P ; r†j ˆ T~ KAr 0  H 0 ; KA 0 r†† ÿ T~ KAr  H ; KA r†††
dK r†
0

2 var K† sup jT~ KAr 0  H 0 ; KA 0 r†† ÿ T~ KAr  H ; KA r††j

r2 0;dŠ

4:10†
If we denote by P r the process K r ; K r††, then KAr 0  H 0 ; KA 0 r††  P r r 0 and
KAr  H ; KA r††  P r r, and from the de®nition (3.4), the linearity of T^, and
the inequalities (3.5), (3.13) it follows that
jT~ KAr 0  H 0 ; KA 0 r†† ÿ T~ KAr  H ; KA r††j
ˆ jT^ P r r 0 † ÿ T^ P r r†j ˆ jT^ P r r 0 ÿ P r r†j 4:11†
r 0 r 0
2 kP r ÿ P rkR 2 mkr ÿ rkR :
Substitution into inequality (4.10) then proves the inequality (4.7). (

According to [5, Def 3.3], a state r 0 is said to be w ^ - approachable from

another state r if there is a sequence n 7! Pn in P such that (i) the sequence
n 7! Pn r converges to r 0 , and (ii) the sequence n 7! w ^ Pn ; r† converges. In the
next Proposition 4.3, for every pair of states r; r 0 we construct a one-pa-
rameter family of processes with the above properties, showing thereby that
in a viscoelastic material element every state is w ^ - approachable from every
other state. The proof is based upon the following preliminary result, es-
sentially due to DAY1 .

4.2. Lemma. For every pair of histories H ; H 0 and for every deformation A,

~ H 0p  H ; A† ˆ w
lim w ~ H ; H 0 1†† ‡ w
~ H 0 ; A†: 4:12†
p!‡1

Proof. Let H 0s be the s-section of H 0 and let Hr0s be the truncation of H 0s at r.

For ®xed values of s and r, the identity H 0s  Hr0s ˆ H 0r‡s and the de®nition
^ tell us that
(4.5) of w
~ H 0r‡s  H ; A† ˆ w
w ~ H ; H 0 r ‡ s†† ‡ w
^ Prs ; r r ‡ s†† ‡ w
^ Ps ; rr s††; 4:13†
0
where r r ‡ s†  H ; H r ‡ s††; rr s†  Hr0s 0
 H ; H s††; and Prs ; Ps are the
processes of duration r and s de®ned by

Prs :ˆ Krs ; H 0 s† ÿ H 0 r ‡ s††; Krs r 0 † :ˆ H 0 r 0 ‡ s† ÿ H 0 r ‡ s†; 4:14†

Ps :ˆ Ks ; A ÿ H 0 s††; Ks s 0 † :ˆ H 0 s 0 † ÿ H 0 s†; 4:15†

1
[11, p. 63]. Note that, while DAY's proof relies upon the fading memory property
implicit in the assumption that G_ is Lebesgue integrable, the proof given here is based
on the assumption that every history has bounded variation.
 States and Free Energies in Linear Viscoelasticity 15

respectively. If we keep r ®xed and let s grow to ‡1, the work

~ H ; H 0 r ‡ s†† converges to w
w ~ H ; H 0 1†† by its de®nition (4.2). Moreover,
Zr
s s
^ Pr ; r r ‡ s††j ˆ
jw T r; s ‡ r 0 †
dH 0 s ‡ r† 2 sup jT r; s ‡ r 0 †j var H 0 r †;
r 0  0;rŠ
0
4:16†
with
Zr
T r; s ‡ r † :ˆ T~ H ; H 0 r ‡ s†† ‡
0
G_ r00 †H 0 s ‡ r00 † dr00 : 4:17†
r0

It is not dicult to prove that jT r; s ‡ r 0 †j has a ®nite upper bound, inde-

pendent of s, r, and r 0 . On the other hand, var H 0 sr † 2 (var H 0s † and
var H 0s † ! 0 when s ! ‡1 because H 0 has bounded variation. Thus, we
may conclude that for every ®xed r > 0 the work w ^ Prs ; r r ‡ s†† converges to
zero when s ! ‡1.
It remains to consider the last term in equation (4.13). Consider the state
r 0 s†  H 0s ; H 0 s†† and the work
^ Ps ; r 0 s†† ˆ w
w ~ H 0s  H 0s ; A† ÿ w
~ H 0s ; H 0 s††: 4:18†
The ®rst term on the right side is equal to w~ H 0 ; A† by the identity (2.8), and
the second converges to zero when s ! ‡1 because the total variation of H 0s
converges to zero. If we now compare this work with the last term in equation
(4.13), with the aid of inequality (4.7) and of the de®nition (3.3) of k
kR we
®nd that
^ Ps ; r 0 s††j 2 var K† k rr s† ÿ r 0 s† kR
^ Ps ; rr s†† ÿ w
jw
s 4:19†
ˆ var K† k H 0 r  H ÿ H 0s kC ;
and therefore, when s ! ‡1,

lim w ~ H 0 ; A† 2 (var K† k H 0r  H ÿ H 0 kC :
^ Ps ; rr s†† ÿ w 4:20†
s!‡1

Equation (4.13) then implies the inequality

~ H 0r‡s  H ; A†ÿ w
lim w ~ H ; H 0 1††ÿ w
~ H 0 ; A† 2 var K† k H 0r  H ÿ H 0 kC :
s!‡1
4:21†
By item (iii) in Proposition 2.2, for every e > 0 there is an re > 0 such that
(var K† k H 0r  H ÿ H 0 kC < e for all r > re . The arbitrariness of e then leads
to the equality (4.12). (

We are now ready to prove that every state r 0 is w-approachable

^ from
every other state r. Because we have already constructed a family of pro-
cesses starting at r and converging to r 0 , it is now sucient to prove that the
work done in the same family of processes converges.
16 G. DEL PIERO & L. DESERI

4.3. Proposition. Each state r 0  H 0 ; A 0 † is w-approachable

^ from every state
r  H ; A†: Indeed, not only does the family r 7! Pr r de®ned by equations
(3.10), (3.11) converge to r 0 as stated in Proposition 3.1, but also
lim w ~ H 0; A 0† ‡ w
^ Pr ; r† ˆ w ^ P ; r† 4:22†
r !‡1
0 0
with P ˆ K ; H 1† ÿ A†:

^
Proof. By equations (3.10), (3.11) and by the de®nition (4.5) of w,
w ~ H 0r  K 0A  H ; A 0 † ÿ w
^ Pr ; r† ˆ w ~ H ; A†; 4:23†
and, by the preceding lemma,
lim w ~ H 0r  K 0A H ; A 0 † ˆ w
~ K 0A  H ; H 0 1†† ‡ w
~ H 0 ; A 0 †: 4:24†
r !‡1

Again by the de®nition (4.5),

~ K 0A  H ; H 0 1†† ˆ w
w ~ H ; A† ‡ w
^ P ; r†; 4:25†
and the desired result (4.22) follows. (

5. Free energy

In the more general context of the theory of simple materials with

memory, COLEMAN [3] de®nes the Helmholtz free energy as a function
w~ : C   Sym ! R; continuous and continuously di€erentiable with respect
to both of its arguments. Here C denotes the fading memory space of his-
tories de®ned in Sec. 3, and continuity and di€erentiability with respect to C
are referred to the fading memory norm (3.20). It is also proved in [3] that, as
a consequence of the second law of thermodynamics, w~ has the following
properties:2
(P1) w~ satis®es the integrated dissipation inequality
w~ KA  H ; B† ÿ w~ H ; A† 2 w
~ KA  H ; B† ÿ w
~ H ; A† 5:1†
for every pair of deformations A, B, for every history H, and for every
segment K of duration d with K d† ˆ O.
(P2) For every deformation A and for every history H,
w~ Ay ; A† 2 w~ H ; A†: 5:2†
(P3) For every deformation A and for every history H, the derivative of
w~ H ;
† at A is equal to the stress T~ H ; A†:

(P4) For every deformation A,

w~ Ay ; A† ÿ w~ Oy ; O† ˆ 12 G1 A
A: 5:3†

2
The property (P4), implicit in [3, Remark 11], is stated explicitly in [8, Sec. 8].
 States and Free Energies in Linear Viscoelasticity 17

In linear viscoelasticity, several attempts have been made to de®ne the free
energy under assumptions weaker than continuity with respect to the fading
memory norm. For instance, VOLTERRA [25], DAY [8, 9], and GRAFFI [15]
proposed some explicit forms of the free energy, satisfying the properties
listed above. Later, GRAFFI [16], MORRO & VIANELLO [22], and FABRIZIO,
GIORGI & MORRO [14] took a more general viewpoint: They de®ned a free
energy to be any function which satis®es those properties.
It is common, in thermodynamics, to assume that the free energy is a
function of state. The notion of state introduced in Section 3 supplies the
following restriction that w~ has to satisfy to be a function of state:

w~ H 0 ; A† ˆ w~ H ; A† 5:4†
for all A 2 Sym and for all H 0 , H 2 C with H 0 ÿ H 2 C0 . If this condition is
satis®ed, then it is possible to de®ne a function w : R ! R such that
w r† ˆ w~ H ; A† 5:5†
whenever r  H ; A†. In terms of the function w, the integrated dissipation
inequality (5.1) takes the form
^ P ; r†
w P r† ÿ w r† 2 w 5:6†
for all P 2 P and for all r 2 R.
In their theory of thermodynamic systems, COLEMAN & OWEN [5, 6] de®ne
the free energy as a lower potential for the work. In the present context, a
^ if for every
function w : R ! R is called a lower potential for the action w
e > 0 and for every r; r 0 2 R there is a d > 0 such that
w r 0 † ÿ w r† < w
^ P ; r† ‡ e 5:7†
for every process P such that P r 2 B r 0 ; d†. If w is a lower potential for w,
^
then it satis®es the inequality (5.6), and therefore it has the property (P1).
Indeed, if r 0 ˆ P r, then P r belongs to B r 0 ; d† for all d > 0, and therefore
the inequality (5.7) is satis®ed with r 0 ˆ P r for every e > 0. We now prove

^ satis®es the properties (P2)±(P4).

5.1. Proposition. Every lower potential for w

Proof. Consider a state r  H ; A†, the equilibrium state rA  Ay ; A†, and

the process Pr :ˆ Oyr ; O†. By Proposition 3.1, Pr r ! rA when r ! ‡1. Thus,
for each d > 0 there are values of r suciently large to ensure that
Pr r 2 B rA ; d†. If w is a lower potential for w, ^ then for each e > 0 the in-
equality (5.7) can be written for r 0 ˆ rA , for the given r, and for P ˆ Pr with
suciently large r. Keeping in mind that w ^ Pr r† ˆ 0 because Pr has constant
values, we get w rA † ÿ w r† < e, and the property (5.2) follows from the
arbitrariness of e. To prove (P3), take two states r  H ; A†; r 0  H ; A 0 †
corresponding to the same history H, and the process P 0r :ˆ Oyr ; A 0 ÿ A†. The
equality
k P 0r r ÿ r 0 kR ˆ k Ayr  H ÿ H kC 5:8†
18 G. DEL PIERO & L. DESERI

and the last item in Proposition 2.2 tell us that P 0r r ! r 0 when r ! 0.

Therefore, for every d > 0 we have P 0r r 2 B r 0 ; d† for suciently small
^ the inequality (5,7) yields
values of r. If w is a lower potential for w,
w~ H ; A 0 † ÿ w~ H ; A† < w
^ P 0r ; r† ‡ e 5:9†
0
for every ®xed e > 0 and for suciently small r > 0. The work w ^ P r ; r†
is
concentrated at the jump from A to A 0 ; according to equation (4.2), it is given
by
^ P 0r ; r† ˆ 12 T~ Ayr  H ; A 0 † ‡ T~ Ayr  H ; A††
A 0 ÿ A†;
w 5:10†
and, in view of the identity
T~ H ; A 0 † ÿ T~ H ; A† ˆ G0 A 0 ÿ A†; 5:11†
it can be given the form
^ P 0r ; r† ˆ T~ Ayr  H ; A†
A 0 ÿ A† ‡ o A 0 ÿ A†:
w 5:12†
The linearity of T~, the inequality (2.25) and the convergence of r !
7 Ayr  H to
~ y ~
H imply the convergence of r 7! T Ar  H ; A† to T H ; A† when r ! 0. Thus,
for r ! 0 the inequality (5.9) yields
w~ H ; A 0 † ÿ w~ H ; A† < T~ H ; A†
A 0 ÿ A† ‡ o A 0 ÿ A† ‡ e; 5:13†
and, by the arbitrariness of e,
w~ H ; A 0 † ÿ w~ H ; A† 2 T~ H ; A†
A 0 ÿ A† ‡ o A 0 ÿ A†: 5:14†
0
If we now repeat the whole procedure with A and A interchanged, we get
w~ H ; A† ÿ w~ H ; A 0 † 2 T~ H ; A 0 †
A ÿ A 0 † ‡ o A 0 ÿ A†
5:15†
ˆ ÿT~ H ; A†
A 0 ÿ A† ‡ o A 0 ÿ A†;
with the last step following from the identity (5.11). We then conclude that
the equality sign holds in inequality (5.14). This proves that w~ H ;
† is dif-
ferentiable and that its derivative is T~ H ;
†.
To prove the property (P4), we consider a segment K of duration d with
K d† ˆ O, and for every a > 1 we de®ne the a-retardation of K as the segment
Ka of duration ad† given by
Ka as† :ˆ K s†; s 2 0; dŠ: 5:16†
It is known (see, e.g., [12, Sec. 3]) that, if rB and rA are the equilibrium states
associated with B and with A :ˆ B ‡ K 0‡ †, and if Pa ˆ Ka ; K 0‡ ††, then
lim Pa rB ˆ rA ; 5:17†
a!‡1

^ Pa ; rB † ˆ 12 G1 A
A ÿ 12 G1 B
B:
lim w 5:18†
a!‡1

If we ®x d > 0, equation (5.17) tells us that Pa rB belongs to B rA ; d† for

suciently large values of a. Then we can use the inequality (5.7) and the
property (5.18) of the retardations to get
 States and Free Energies in Linear Viscoelasticity 19

^ Pa ; rB † ‡ e < 12 G1 A
A ÿ 12 G1 B
B ‡ 2e
w rA † ÿ w rB † < w 5:19†
for every pair of deformations A; B. In particular, for B ˆ O and e arbitrarily
small we have
w rA † ÿ w r0 † 2 12 G1 A
A 5:20†
where r0  Oy ; O† is the natural state, and for A ˆ O we get the opposite
inequality
w rB † ÿ w r0 † 3 12 G1 B
B: 5:21†
The equality (5.3) follows from the arbitratiness of A and B. (

In this paper we adopt COLEMAN & OWEN'S de®nition of free energy, with
the requirement that the domain of the free energy be all of R rather than
merely a dense subset of R as required in [6]. Moreover, we add for conve-
nience the normalization condition w r0 † ˆ 0.

5.2. De®nition. A free energy for the viscoelastic material element is a func-
tion w : R ! R which is a lower potential for w^ and satis®es w r0 † ˆ 0.

The set of all free energies is denoted by F. It has been proved above that
all functions in F have the properties (P1)±(P4). In particular, a consequence
of (P4) and of the positive-semide®niteness of G1 is that the free energy of
every equilibrium state is non-negative; moreover, the property (P2) tells us
that the free energy of every state is non-negative. It is also known that F is
convex, i.e., that every convex combination of free energies is a free energy [6,
Sec. 3].
We now discuss the question of the existence of free energies. More
precisely, we wish to provide necessary and sucient conditions under which
the set F is not empty. We observe that, if w is a lower potential for w,^ then
the inequality (5.7) implies that
^ P ; r† > ÿe
w 5:22†
for all P 2 P with P r 2 B r; d†. This property of w ^ is called the dissipation
property at r [6].
^ has a lower potential, it has the dissipation property at all states
Thus, if w
in R. It has been proved in [6, Theorem 3.3] that if w ^ is an action with the
dissipation property at some r 2 R, then it has a lower potential whose do-
^
main is the set of all states which are w-approachable from r. By Proposition
4.3, this set is here the whole state space R. Therefore, for a viscoelastic
element, the assertions
w^ has the dissipation property at some r 2 R,
w^ has the dissipation property at all r 2 R,
 F is not empty
are equivalent. Moreover, the ®rst of them is equivalent to
 the relaxation function G is dissipative.
20 G. DEL PIERO & L. DESERI

Indeed, we recall that G is dissipative if the work done in any process

starting from the natural state is non-negative. Thus, if G is dissipative the
inequality (5.22) with r ˆ r0 is satis®ed for all e > 0, and therefore w ^ has
^ has the dissipation property
the dissipation property at r0 . Conversely, if w
at r0 , then F is not empty, and the inequality (5.6) together with the
condition w r0 † ˆ 0 and the fact that w takes non-negative values tell us
that
^ P ; r0 † 3 w P r0 † ÿ w r0 † 3 0
w 5:23†
for all P 2 P, i.e., that G is dissipative.
Consider the function w0 de®ned over R by
w0 r† :ˆ lim‡ inf fw
^ P ; r0 † j P 2 P; P r0 2 B r; d†g: 5:24†
d!0

This is the minimum work expended to approach r by a process starting from

^ has the dissi-
the natural state. It follows from Theorem 3.3 in [6] that, if w
pation property at r0 , then w0 is a lower potential for w
^ and is lower semi-
continuous. Moreover, Theorem 3.5 of [6] implies that w0 is the maximal free
energy in the sense that, for every function w in F,
w r† 2 w0 r† 8r 2 R: 5:25†
The same theorem also shows that, under more restricted conditions, the
function
^ P ; r† j P 2 P; P r 2 B r0 ; d†g
w0 r† :ˆ ÿ lim‡ inf fw 5:26†
d!0

is the minimal element of F. In the present setting of viscoelastic elements,

this part of the theorem can be restated as

^ has the dissipation property at r0 , and that

5.3. Proposition. Assume that w
^
(i) r0 is w-approachable from every other state in R,
(ii) w0 r† > ÿ1 for all r 2 R.
Then w0 belongs to F, and every other function w in F satis®es
w0 r† 2 w r† 8r 2 R: 5:27†
For viscoelastic material elements, property (i) has been proved in
Proposition 4.3, and (ii) is a consequence of (i). Indeed, it is sucient to take
a family d 7! Pd of processes such that k Pd r ÿ r0 kR < d and d 7! w ^ Pd ; r†
converges when d ! 0‡ , to get from inequality (5.6) that
^ Pd ; r† < ‡1:
ÿw0 r† 2 lim‡ w 5:28†
d!0

Therefore, the hypotheses (i), (ii) in Proposition 5.3 are satis®ed whenever w^
has the dissipation property at r0 . As a complement to the same proposition
we prove now that, if w^ has the dissipation property at r0 , then the function
w0 is lower semicontinuous. We begin with a preliminary result, which
characterizes w0 r† as the maximum recoverable work from r.
 States and Free Energies in Linear Viscoelasticity 21

5.4. Lemma. If w ^ has the dissipation property at r0 , then for each d > 0 and for
each r 2 R,
 
inf w^ P ; r† P 2 P; P r 2 B r0 ; d† ˆ inf w ^ P ; r† P 2 P : 5:29†

Proof. The above relation is satis®ed trivially if the equality sign is replaced
by 3 ; it is then sucient to prove the same relation with 2 in place of ˆ.
For every process P and for every state r, the non-negativity of the free
energy and inequality (5.6) imply that
^ P ; r† 3 w0 P r† ÿ w0 r† 3 ÿ w0 r†;
w 5:30†
and therefore that fw^ P ; r† j P 2 Pg has a ®nite lower bound. Then, for every
e > 0 there is a process P such that
w ^ P ; r† j P 2 Pg ‡ 13 e:
^ P ; r† 2 inf fw 5:31†
Let r  H ; A† and P ˆ K; B†. For each r > 0, consider the process
Pr :ˆ Byr  K; B† obtained by continuing K with a constant segment of du-
ration r. Because no work is done in the continuation, the work w ^ Pr ; r† is
equal to w^ P ; r†. Moreover, Pr r  B ‡ A†yr  KA  H ; B ‡ A†, so that
k Pr r ÿ rB‡A kR ˆ k B ‡ A†yr  KA  H ÿ B ‡ A†y kC ; 5:32†
and the family r 7! Pr r converges to rB‡A when r ! ‡1 by item (iii) in
Proposition 2.2. Take another segment K 0 of duration d 0 , with K 0 d 0 † ˆ 0
and K 0 0‡ † ˆ ÿB ÿ A, and let P 0a be the process K 0a ; ÿB ÿ A†, with K 0a the
a- retardation of K 0 . Then the work done in the process P 0a Pr from r is
w ^ Pa 0 ; Pr r† ‡ w
^ P 0a Pr ; r† ˆ w ^ Pa 0 ; Pr r† ‡ w
^ Pr ; r† ˆ w ^ P ; r†: 5:33†
By proposition 4.1, the convergence of r 7! Pr r to rB‡A implies the conver-
^ Pa 0 ; Pr r† to w
gence of r 7! w ^ Pa 0 ; rB‡A †. Moreover, by the properties (5.17),
(5.18) of a-retardations, the states a 7! Pa 0 rB‡A converge to r0 when
a ! ‡1, and the works ^ Pa 0 ; Pr r†
a 7! w converge to
1
ÿ2 G1 B ‡ A†
B ‡ A†, which is a non-positive real by the positive de®-
niteness of G1 . It is then possible to select r and a suciently large to have
^ Pr 0 ; Pr r† 2 w
w ^ P 0a ; rB‡A † ‡ 13 e 2 23 e ; 5:34†
and therefore, by (5.31) and (5.33),
^ P 0a ; Pr r† 2 inf fw
w ^ P ; r†j P 2 Pg ‡ e: 5:35†
0
The convergence of r 7! Pr r to rB‡A and that of a 7! Pa rB‡A to r0 also imply
that, for suciently large values of r and a, the state Pa 0 Pr r belongs to
B r0 ; d† for each ®xed d > 0. For this choice of a and r,
inf fw ^ Pa 0 Pr ; r†;
^ P ; r† j P 2 P; P r 2 B r0 ; d†g 2 w 5:36†
and the combination with the preceding inequality and the arbitrariness of e
lead to the relation (5.29) with 2 in place of ˆ. (
22 G. DEL PIERO & L. DESERI

^ has the dissipation property at r0 , then the function w0 is

5.5. Proposition. If w
lower semicontinuous.

Proof. Select an e > 0, a state r 0 , and a process P such that

^ P ; r 0 † 2 inf f^
w w P ; r 0 † j P 2 Pg ‡ 12 e 5:37†
^ with respect to the states, there is a d > 0 such that
By the continuity of w
w ^ P ; r 0 † ‡ 12 e
^ P ; r† 2 w 5:38†
0
for all r in B r ; d†. By the preceding lemma, the in®mum in inequality (5.37)
is equal to ÿw0 r 0 †, and w^ P ; r† is not less than ÿw0 r†: Thus,
ÿw0 r† 2 ÿ w0 r 0 † ‡ e 5:39†
for all r in B r 0 ; d†. This proves the semicontinuity of w0 at each r 0 in R: (

Consider the function w~MV : C  Sym ! R de®ned by

w~MV H ; A† :ˆ w
~ H ; A† ‡ 12 G1 H 1†
H 1†: 5:40†
It has been proved by MORRO & VIANELLO [22] that w~MV has the properties
P1 † ÿ P4 †, and that it is the maximal element in the class of all functions
which satisfy the same properties. Thus, if the free energy is de®ned as a
function which satis®es the properties P1 †ÿ P4 †, as was done in [22], then
w~MV is the maximal free energy.
This conclusion is no longer true for free energies de®ned according to
De®nition 5.2. Indeed, w~MV need not satisfy the condition (5.4), and therefore
need not be a function of state. This certainly occurs when C0 , the set of all
solutions of equation (1.4), consists of the null history alone. There is an
explicit example for this situation, which we discuss in Section 7 in the
context of completely monotonic relaxation functions. There is another ex-
ample in which w~MV is not a function of state, the case of a relaxation
function of exponential type, which we discuss in the next section.
Even if w~MV is a function of state, it need not be a lower potential for the
^ and therefore it need not belong to the set F of all free energies
action w,
according to De®nition 5.2. The conditions under which w~MV 2 F are not
known. However, the fact that w~MV is the maximal element in the class of all
functions which satisfy the properties P1 †ÿ P4 † and that all functions in F
satisfy the same properties tells us that, whenever w~MV belongs to F, it
coincides with the maximal element w0 of F.

6. Relaxation functions of exponential type

Let us consider a relaxation function of the type

 s† ˆ G
G  0 eÿsB ; 6:1†
where G s† :ˆ G s† ÿ G1 . We assume that the tensor G  0 :ˆ G
 0† is sym-
metric and positive-de®nite, and that all eigenvalues of B have a positive real
 States and Free Energies in Linear Viscoelasticity 23

part. It has been proved in [12, Sec. 7] that this condition on B is necessary
and sucient to ensure that G_ is Lebesgue integrable over 0; ‡1†, as as-
sumed here in Section 2.

6.1. De®nition. A relaxation function of exponential type is a function G

 0 symmetric and positive de®nite
which admits the representation (6.1), with G
and with all eigenvalues of B having a positive real part.

For this class of relaxation functions, the constitutive equation (2.24)

takes the form
Z‡1
T r† ˆ G0 H r† ÿ G0 B eÿsB H r ‡ s† ds; 6:2†
0

where by T r† we have denoted the stress T~ H r ; H r††. An important property

of exponential relaxation functions, already studied by GRAFFI & FABRIZIO
[18], is that the states can be identi®ed with the pairs formed by the current
stress and the current deformation. In our approach, this is a consequence of
the fact that for an exponential relaxation function the number
jT 0‡ † ÿ G0 H 0‡ †j is a seminorm for the history space C, and that this
seminorm is equivalent to k
kC .

6.2. Proposition. If G is a relaxation function of exponential type, then there is

a positive constant m such that
jT 0‡ † ÿ G0 H 0‡ †j 2 kH kC 2 mjT 0‡ † ÿ G0 H 0‡ †j 6:3†

for all histories H 2 C.

Proof. By the de®nition (2.9) of k
kC and by equations (6.1) and (6.2), we

have
Z‡1
kH kC ˆ sup G0 Beÿ t‡s†B H s† ds ˆ sup jG0 eÿtB Gÿ1 ‡ ‡
0 T 0 † ÿ G0 H 0 ††j;
t30 t30
0
6:4†

where in the last step we have taken advantage of the fact that B commutes
with eÿtB . If we take t ˆ 0, we get the ®rst of the inequalities (6.3). To prove
the second inequality, we use the identity
ÿtG0 BGÿ1
G0 eÿtB Gÿ1
0 ˆe
0 6:5†

and the fact that the eigenvalues of G0 BGÿ10 are the same as those of B, and
therefore have a positive real part. Under these conditions, it has been proved
 ÿ1
in [12, Prop. 7.2] that keÿtG0 BG0 k is bounded by a positive constant m, and the
second of inequalities (6.3) follows. (
24 G. DEL PIERO & L. DESERI

This result tells us that, for a relaxation function of exponential type, the
equivalence classes modulo C0 in the history space C can be characterized by
the tensor T 0‡ † ÿ G0 H 0‡ ††: Consider now a state r  H ; A† and denote
by T the current stress T~ H ; A† given by equation (2.24). The same equation
tells us that

T ÿ G0 A ˆ T 0‡ † ÿ G0 H 0‡ †: 6:6†

Consequently, a state r  H ; A† can be identi®ed with the pair T ; A† formed

by the current stress and the current deformation, and the state space R can
be identi®ed with the product space Sym2 , which is ®nite-dimensional. Thus,
the de®nition (3.3) of the norm k
kR and the equivalence result (6.3) tell us
that k
kR is equivalent to any norm of Sym2 . In particular, we ®nd it con-
venient to identify the state r with the pair T ; A†, and to take

jT ÿ G1 Aj ‡ jAj 6:7†

as the norm in the state space.

^ takes a spe-
For relaxation functions of exponential type, the action w
cial form that we wish to determine now. Let us begin with a preliminary
result.

6.3. Proposition. Let G be of exponential type. Then for every history H 2 C

the function r ! T r† ÿ G0 H r†† is di€erentiable, and its derivative is

T r† ÿ G0 H r††_ ˆ ÿG_ 0†Gÿ1

0 T r† ÿ G1 H r††; 6:8†
with
G_ 0† ˆ ÿG0 B: 6:9†

Proof. With the change of variable r ‡ s ˆ t, equation (6.2) takes the form
Z‡1
T r† ÿ G0 H r† ˆ ÿG0 B eÿ tÿr†B H t† dt; 6:10†
r

which shows that r 7! T r† ÿ G0 H r†† is di€erentiable and that its deriva-

tive is
Z‡1
T r† ÿ G0 H r††_ ˆ G0 B H r† ÿ G0 B2 eÿ tÿr†B H t† dt: 6:11†
r

The desired equality (6.8) follows after subtracting from equation (6.11) the
equation (6.10) premultiplied by G0 B G0 :ÿ1 (

Let us now multiply both sides of equation (6.8) by Gÿ1

0 T r† ÿ G1 H r††
and integrate over 0; s†:
 States and Free Energies in Linear Viscoelasticity 25

Zs
T r† ÿ G0 H r††_
Gÿ1
0 T r† ÿ G1 H r†† dr
0
Zs
ˆÿ Gÿ1 _ ÿ1
0 G 0†G0 T r† ÿ G1 H r††
T r† ÿ G1 H r†† dr: 6:12†
0

The left-hand side is equal to

Zs
1 s
ÿ1
2 G0 T r† ÿ G0 H r††
T r† ÿ G0 H r†† 0 ‡ T r† ÿ G0 H r††_
H r† dr;
0
6:13†
and can be further transformed by integration by parts, to get
1 ÿ1

2 G0 T r† ÿ G0 H r††
T r† ÿ G0 H r††
Zs
1
s
‡T r†
H r† ÿ 2 G0 H r†
H r† 0 ÿ T r†
dH r†: 6:14†
0
If we now set
w T ; A† :ˆ 12 Gÿ1 1
0 T ÿ G1 A†
T ÿ G1 A† ‡ 2 G1 A
A; 6:15†
we see after some computation that the term in brackets is equal to
w T r†; H r††: Therefore, equation (6.12) takes the form
Zs
ÿ T r†
dH r† ˆ ÿ ‰w T r†; H r††Šs0
0
Zs
ÿ Gÿ1 _ ÿ1
0 G 0†G0 T r† ÿ G1 H r††
T r† ÿ G1 H r†† dr:
0
6:16†
By the de®nitions (4.4), (4.5), the integral on the left-hand side is the work
done in the process Hs ÿ H s††ys ; A ÿ H s†† starting from the state
T s†; H s††  H s ; H s††. More generally, if we consider a state r ˆ T ; A†
and a process P ˆ K; B† of duration d, then equation (6.16) tells us that
^ P ; r† ˆ w P r† ÿ w r†
w
Zd
ÿ Gÿ1 _ ÿ1
0 G 0†G0 T r† ÿ G1 KA r††
T r† ÿ G1 KA r†† dr:
0
6:17†
^ for a relaxation function of exponential
This is the expression of the action w
type. From it, the following result on the existence of a free energy in the
sense of De®nition 5.2 can be deduced.
26 G. DEL PIERO & L. DESERI

6.4. Proposition. Let G be a relaxation function of exponential type. Then the

function w de®ned by equation (6.15) is a free energy in the sense of De®nition
5.2 if and only if G_ 0† is negative-semide®nite.

Proof. It has already been proved in Section 5 that if there is a free energy,
then the material element is dissipative. The negative semide®niteness of G_ 0†
is a well-known property of the dissipative viscoelastic material element; see,
e.g., [11, Sec. 6.1].
Conversely, if we assume that G_ 0† is negative-semide®nite, then equation
(6.17) shows that w satis®es the integrated dissipation inequality (5.6). Be-
cause w is continuous with respect to the norm (6.7) and this norm is
equivalent to k
kR , we deduce that the inequality (5.7) is satis®ed, and
therefore w is a lower potential for w.
^ Because the normalization condition
w O; O† ˆ 0 is also satis®ed, we conclude that w is a free energy in the sense of
De®nition 5.2. (

A relaxation function with a scalar exponent

G s† :ˆ G0 eÿas 6:18†
is a particular case of equation (6.1), obtained by taking B ˆ aI. For relax-
ation functions of this type, it has been proved by GRAFFI & FABRIZIO [18]3
that there is just one function of state which satis®es the properties P1 †ÿ P3 †
listed at the beginning of Sec. 5, and that this is the function w de®ned by
equation (6.15). It follows then from Proposition 5.1 that w is the only
possible free energy according to De®nition 5.2. Here we extend this result to
all relaxation functions of exponential type with G_ 0† 2 O. We recall that by
Proposition 6.4 the assumption G_ 0† 2 O implies that w belongs to the set F
of all free energies, and that the functions w0 ; w0 de®ned by equations (5.24)
and (5.26) are the maximal and the minimal element of F, respectively.

6.5. Proposition. Let G be a relaxation function of exponential type, with

G_ 0† 2 O. Let w be the element of F de®ned by equation (6.15) and let w0 and
w0 be the maximal and the minimal elements of F. Then w0 ˆ w0 ˆ w. 

Proof. With every state r ˆ T ; A†, let us associate the deformation

B :ˆ Gÿ1
0 G0 A ÿ T †: 6:19†
‡
Consider a segment K of duration d, with K d†=0 and K 0 † ˆ B, and the
process Pa ˆ Ka ; B†, with Ka the a-retardation of K. If r0 is the natural state,
it follows from the properties (5.17), (5.18) of the a-retardations that
lim Pa r0 ˆ G1 B; B†; 6:20†
a!‡1

^ Pa ; r0 † ˆ 12 G1 B
B:
lim w 6:21†
a!‡1

3
See also BREUER & ONAT [2].
 States and Free Energies in Linear Viscoelasticity 27

Moreover, if we consider the process P 0a :ˆ Ka ; A†, from the formula (5,11)

we get
lim P 0a r0 ˆ G1 B ‡ G0 A ÿ B†; A† ˆ T ; A† ˆ r; 6:22†
a!‡1

and, from (4.2) and the expression (6.19) for B,

^ P 0a ; r0 † ˆ 12 G1 B
B ‡ G1 B
A ÿ B† ‡ 12 G0 A ÿ B†
A ÿ B†
lim w
a!‡1
ˆ 12 G0 B
B ÿ G0 A
B ‡ 12 G0 A
A
ˆ 12 Gÿ1 1
0 T ÿ G0 A†
T ÿ G0 A† ‡ A
T ÿ G0 A† ‡ 2 G0 A
A:
6:23†

Now we observe that the right-hand side in the last equation is equal to
w T ; A†, as can be checked by direct computation. Thus,
^ P 0a ; r0 † ˆ w T ; A†:
lim w 6:24†
a!‡1

By equation (6.22), for every ®xed d > 0 the state P 0a r0 belongs to B r; d†

for suciently large values of a. Therefore, for each d > 0 there is an a such
that
^ P 0a ; r0 † 3 inf wfr
w ^ 0 ! B r; d†g; 6:25†
and the fact that, when d ! 0, the left-hand side converges to w T ; A† and the
right-hand side converges to w0 T ; A† allows us to conclude that
w T ; A† 3 w0 T ; A† for all states T ; A† in R, and, therefore, that w 3 w0 .
Because w0 is the maximal free energy, we have proved that w ˆ w0 .
To prove that w ˆ w0 , consider a state T ; A† and suppose that the de-
formation undergoes a jump from A to B. Then equation (5.11) tells us that
the stress jumps from T to T 0 ˆ T ‡ G0 B ÿ A†; and if we take B as in
equation (6.19), we see that T 0 ˆ G1 B. This characterizes T 0 ; B† as the
equilibrium state associated with B; indeed, the constitutive equation (2.24)
shows that G1 B is the stress associated with the history-deformation pair
By ; B†:
By equation (4.2), the work done in the jump from A to B is
T
B ÿ A† ‡ 12 G0 B ÿ A†
B ÿ A†. If we now subject the equilibrium state
T 0 ; B† to the retarded processes Pa ˆ Ka ; O†, the work done in those pro-
cesses converges to ÿ 12 G1 B
B when a ! ‡1 by equation (5.18). Thus, the
total work done is

T
B ÿ A† ‡ 12 G0 B ÿ A†
B ÿ A† ÿ 12 G1 B
B; 6:26†

and substitution of the expression (6.19) of B shows that this work is equal to
ÿw T ; A†. In this way, we have proved that it is possible to recover the work
w T ; A† from the state T ; A†. Recalling the characterization (5.29) of w0 as
the maximum recoverable work, we conclude that w0 T ; A† 3 w T ; A†, for all
states T ; A†, and therefore w0 3 w. The equality w ˆ w then follows from
0
the fact that w0 is the minimal element of F: (
28 G. DEL PIERO & L. DESERI

We now prove that the function wMV de®ned by equation (5.40) is not a
function of state when the relaxation function is of exponential type. Con-
sider the history

H s† :ˆ A f s†; 6:27†

where A4O is a ®xed deformation and f is the piecewise constant function

de®ned over 0; 1† by 8
< 1 for a < s 2 b;
f s† :ˆ ÿ1 for b < s 2 c; 6:28†
:
0 everywhere else,
with 0 < a < b < c. It is convenient to express the stress in the form
Z1
T r† ˆ G1 Af r† ÿ G0 eÿ sÿr†B A df s†; 6:29†
r
obtained from equation (6.2) by integration by parts. In particular,
T c‡ † ˆ O;
T b‡ † ˆ G1 A ‡ G0 eÿ cÿb†B A;
6:30†
T a‡ † ˆ ÿG1 A ‡ G0 eÿ cÿa†B A ÿ 2G0 eÿ bÿa†B A;
T 0‡ † ˆ G0 eÿaB A ÿ 2G0 eÿbB A ‡ G0 eÿcB A:
The last equation shows that, if a, b and c satisfy

eÿ cÿa†B ÿ 2eÿ bÿa†B ‡ 1 ˆ 0; 6:31†

then T 0‡ † ˆ O: Because H 0‡ † ˆ O, the history-deformation pair H ; O†

with H de®ned by equations (6.27), (6.28) and (6.31) represents the natural
state, just as done by the pair Oy ; O†: We wish to show that wMV H ; O† is
di€erent from wMV Oy ; O†. Because the ®rst is equal to w  H ; O† and the
 H ; O† is not identically zero. The
second is zero, it is sucient to show that w
history H being piecewise constant, the formulae (4.3) and (5.11) yield
 H ; O† ˆ 12 T c‡ † ‡ T c††
A ‡ 12 T b‡ † ‡ T b††
ÿ2A† ‡ 12 T a‡ † ‡ T a††
A
w
ˆ T c‡ †
A ÿ 2T b‡ †
A ‡ T a‡ †
A ‡ 3G0 A
A; 6:32†

and substitution of the expressions (6.30) for the stresses yields

 H ; O† ˆ G0 ÿ2eÿ cÿb†B ‡ eÿ cÿa†B ÿ 2eÿ bÿa†B ‡ 3 I†A
A:
w 6:33†
Finally, using the condition (6.31) we get
 H ; O† ˆ 2G0 I ÿ eÿ cÿb†B †A
A:
w 6:34†
That w H ; O† is not identically zero follows from the fact that the right-hand
side is a continuous function of c ÿ b†, which takes the value zero when
c ÿ b† ˆ 0 and approaches the positive value 2G0 A
A when c ÿ b† !
‡1. (
 States and Free Energies in Linear Viscoelasticity 29

6.6. Remark. A similar argument shows that the free energy of single-integral
type proposed by GURTIN & HRUSA [20]:
Z‡1 Z‡1
1 1
wG ÿ H ; A† :ˆ G0 A
A ‡ A
G_ s†H s† ds ÿ G_ s†H s†
H s† ds 6:35†
2 2
0 0

is not a free energy according to De®nition 5.2 if the relaxation function is of

exponential type. Indeed, for A ˆ O and for the history H de®ned by equa-
tion (6.27) with A replaced by B we get
Z‡1 Zc

wGH H ; O† ˆ ÿ2 G s†B
B f s† ds ˆ ÿ2 G_ s†B
B ds
1 _ 2 1

0 a

ˆ 12 G a† ÿ G c††B
B; 6:36†
and the last term is not identically zero.

7. Completely monotonic relaxation functions

We recall that a relaxation function G is completely monotonic if it has

derivatives G p† of every order p and if
ÿ1†p G p† s† 3 O 7:1†
for all s 3 0 and for all p 2 N. It has been proved in [12, Sec. 6] that G is
completely monotonic if and only if its symmetric part G s admits the rep-
resentation
Z‡1
s
G s† ˆ eÿxs dK x†; 7:2†
0

with K : ‰0; ‡1† ! LinSym; symmetric, bounded and non-decreasing with

respect to the order relation (1.7).
The problem that we consider here is the characterization of the state
space for completely monotonic relaxation functions. As we shall see, there
are di€erent characterizations for di€erent subclasses of such functions. In
particular, we are interested in determining a subclass for which states can be
identi®ed with history-deformation pairs. We know that this occurs whenever
the solution set C0 of equation (1.4) reduces to the null history. Let us take,
then, a history H 2 C0 . The equation (1.4), multiplied by H t† and integrated
over 0; 1†, yields
Z‡1 Z‡1
G_ t ‡ s†H s†
H t† ds dt ˆ 0: 7:3†
0 0

In this equation, G can be replaced by its symmetric part, and we can use the
representation (7.2) to obtain
30 G. DEL PIERO & L. DESERI

Z‡1
dK x† x1=2 H^ x††
x1=2 H^ x†† ˆ 0; 7:4†
0

where H^ x† is the Laplace integral

Z‡1
H^ x† :ˆ eÿxs H s† ds; 7:5†
0

with x real and non-negative. That K is non-decreasing implies that the

integrand in equation (7.4) vanishes at almost every x. In particular,
H^ x† ˆ O at all those x at which K is continuous and has a right derivative
which is positive-de®nite, and at all those x at which K has a jump and
K x‡ † > K xÿ †. When either of these conditions is satis®ed, we say that K is
strictly increasing at x. Thus, we may state

7.1. Proposition. Let H 2 C0 . Then H^ x† ˆ O at all those x at which K is

strictly increasing.

It is known that if H^ x† ˆ O for all x > 0, then H is the null history Oy ,

and, therefore, that C0 ˆ fOy g: But there are weaker conditions on H^ leading
to the same conclusion. We begin with the following result [26, Sec 2.6].

7.2 Proposition. If H 2 C0 and if H^ n† ˆ O for n ˆ 1; 2; . . . ; then H ˆ Oy .

Proof. The proof is based on the change of variable r ˆ eÿs , which trans-
forms equation (7.5) into
Z1
H^ x† ˆ rxÿ1 K r† dr; 7:6†
0
ÿ1
with K r† ˆ H ln r ††. Assume that there is a sequence n 7! xn of points at
which H^ xn † ˆ O: If we consider the linear combination
X
N
PN r† :ˆ Pn rxn ÿ1 7:7†
nˆ1

with coecients Pn in Sym, we get

Z1
PN r†
K r† dr ˆ 0; 7:8†
0

and therefore, by the Cauchy-Schwarz inequality,

 States and Free Energies in Linear Viscoelasticity 31

Z1 Z1
kKk2L2 :ˆ 2
jK r†j dr ˆ K r†
K r† ÿ PN r†† dr 2 kKkL2 kK ÿ PN kL2 :
0 0
7:9†
Note that K is bounded, and therefore square integrable, because K and H
take the same values and H is bounded. We may then divide by the L2 -norm
of K, to obtain
kKkL2 2 kK ÿ PN kL2 7:10†
for every PN of the type (7.7).
If xn ˆ n as stated in the proposition, then the functions PN are the
polynomials. By the Weierstrass approximation theorem, the polynomials are
dense in C 0 endowed with the supremum norm, and therefore they are dense
in L2 . Consequently, for every e > 0 there is a polynomial PN such that the
right-hand side of inequality (7.10) is less than e. This implies that kKkL2 ˆ 0,
i.e., that K r† ˆ O almost everywhere.
We now observe that K r† ˆ H f r††, with f continuous and decreasing
and H continuous from the left. Then K is continuous from the right, and a
function which vanishes almost everywhere and is continuous from the right
is identically zero. Because H takes the same values as K, we conclude that H
is identically zero as well. (

It is clear from the preceding proof that the conclusion H ˆ Oy remains

valid if the sequence n ˆ 1; 2; . . . is replaced by any other sequence xn with
the property that the linear combinations PN are dense in C 0 . The theorem of
MUÈNTZ [7, p. 100] provides a class of sequences with this property. They are
the sequences n 7! xn which satisfy the two conditions
X
1
lim xn ˆ ‡1; xÿ1
n ˆ ‡1: 7:11†
n!‡1
nˆ1

It was recognized later (see [1]) that the unboundedness of the sequence is not
essential. Thus, we can state the following fairly general conclusion.

7.3. Proposition. For a completely monotonic relaxation function, the states

can be identi®ed with the history-deformation pairs if the function K de®ned by
equation (7.2) is strictly increasing at a countable set n 7! xn of points, satis-
fying the condition (7.11).

In particular, the identi®cation of states with history-deformation pairs is

possible whenever K is strictly increasing on a subset of 0; ‡1† with positive
measure. However, one cannot expect that this identi®cation can be extended
to all completely monotonic relaxation functions. Indeed, there are com-
pletely monotonic relaxation functions which are of exponential type, and it
has been shown in the preceding section that for such functions the states
cannot be identi®ed with history-deformation pairs.
32 G. DEL PIERO & L. DESERI

The question arises of which functions of exponential type are completely

monotonic. In one dimension, the answer is easy: An exponential function is
completely monotonic if and only if its exponent is negative. In higher di-
mensions, the de®nition (7.1) of complete monotonicity involves only the
symmetric part of G, and it is not so immediate to relate it with the de®nition
(6.1), which involves the whole function. The result which we present here is
obtained with the aid of the S ‡ N † decomposition of linear operators on a
real ®nite-dimensional vector space [21, Sec. 6.2]. According to this decom-
position, any B 2 LinSym admits the representation
Xl
Bˆ kh Eh ‡ N: 7:12†
hˆ1

where kh ˆ ah ‡ ibh are the eigenvalues of B; E h are projections of LinSym

such that
Xl
E h E k ˆ dhk E h ; E h ˆ I; 7:13†
hˆ1

and N is a nilpotent operator, i.e., an element of LinSym such that N q ˆ O

for some q 2 N. Moreover, N commutes with every E h :
NE h ˆ E h N 8 h 2 f1; . . . :lg: 7:14†
7.4. Proposition. Let G be a relaxation function of exponential type with ex-
ponent B. Then G is completely monotonic if and only if all eigenvalues of B are
real, all tensors E h appearing in the decomposition (7.12) of B satisfy
G0 E h 3 O 8h 2 f1; . . . ; lg: 7:15†
and the tensor N appearing in the same decomposition (7.12) is zero.

Proof. The properties (7.13) of the projections E h and the de®nition of an

exponential tensor function4 imply that
X l  X l
exp kh E h ˆ ekh E h : 7:16†
hˆ1 hˆ1

Thus, by using the decomposition (7.12) of B, equation (6.1) can be rewritten

in the form
X
l
G s† ˆ eÿskh G0 eÿsN E h : 7:17†
hˆ1

Assume that G is completely monotonic. Then G is positive-semide®nite, and

this implies that
X
l
eÿskh G0 eÿsN E h E p A
E p A 3 O 7:18†
hˆ1

4
See, e.g., [12, Eq. 7.2].
 States and Free Energies in Linear Viscoelasticity 33

for all A 2 Sym and for all p 2 f1; . . . lg; and from the identity E h E p ˆ dhp E p
we conclude that
eÿskp E p †T G0 eÿsN E p 3 O: 7:19†
Because N q ˆ O; eÿsN is a polynomial of order q ÿ 1†
X
qÿ1 r
s
eÿsN ˆ ÿN†r ; 7:20†
rˆ0
r!
and therefore the inequality (7.19) can be veri®ed for every s > 0 only if kp is
real. Thus, we have proved that G completely monotonic implies that all
eigenvalues of B are real.
Now the coecient eÿskp can be dropped form inequality (7.19); what
remains is a polynomial of order q ÿ 1†, whose term of highest order must be
positive-semide®nite. Then, if we set M :ˆ ÿN †qÿ1 and use the com-
mutativity property (7.14), we get
E p †T G0 E p M 3 O 8 p 2 f1; . . . lg: 7:21†
We claim that M ˆ O. Indeed,
P if M4O, there is a C 2 Sym such that
MC4O, and the fact that lpˆ1 Ep MC ˆ MC4O tells us that there must be
some p for which E p MC4O. For one such p, the inequality (7.21) yields
G0 E p MA ‡ E p MB†
E p A ‡ E p B† 3 0 7:22†
for all A; B 2 Sym. In particular, if we take B ˆ MC, we have MB ˆ M 2 C ˆ O
because M 2 ˆ N 2qÿ2 ˆ O except for the trivial case q ˆ 1. The inequality
(7.22) then reduces to
G0 E p MA
E p A ‡ E p MC† 3 0 7:23†
for all A 2 Sym, and this implies
G0 E p MA
E p MC ˆ 0 7:24†
for all A 2 Sym. It is now sucient to take A ˆ C to get a contradiction,
because E p MC4O and G0 is positive-de®nite. Thus, we have proved that, if
G is positive-semide®nite, then N q ˆ O implies N qÿ1 ˆ O. By induction, we
may conclude that N ˆ O. The equation (7.17) then takes the form
X
l
G s† ˆ eÿsah G0 E h ; 7:25†
hˆ1

with ah the real eigenvalues of B. Now assume that G is completely mono-

tonic. In view of the de®nition (7.1), this is equivalent to assuming that
X
l
aph eÿsah G0 E h 3 O 7:26†
hˆ1
for all s 3 0 and for all p 2 N. In particular, if we set s ˆ 0 and recall that the
vectors ap1 ; ap2 ; . . . ; apl †; p 2 f0; 1; . . . l ÿ 1g are linearly independent, we ob-
tain the inequalities (7.15). This completes the proof that a function of ex-
ponential type that is completely monotonic satis®es the conditions stated in
34 G. DEL PIERO & L. DESERI

the proposition. That the same conditions are sucient for complete
monotonicity is trivially veri®ed. (

The formula (7.25) tells us that a completely monotonic relaxation

function of exponential type is a linear combination of scalar exponentials,
whose coecients G0 E h are positive semide®nite. Moreover, if we take the
eigenvalues of B with the ordering 0 < a1 < a2 <


< al and if we consider
the piecewise constant function K de®ned by
X
j
K x† :ˆ G0 Eh for aj < x < aj‡1 ; j 2 f1; . . . ; lg; 7:27†
hˆ1

we see that the formula (7.25) can be written in the form

Z‡1
G s† ˆ eÿxs dK x†; 7:28†
0
s
and the representation (7.2) of G is obtained after replacing K by its sym-
metric part K s . Because it is constant except at a ®nite number of jump
points, K s does not satisfy the condition, required by Proposition 7.3, of
being strictly increasing at a countable set of points. This agrees with the
conclusion obtained in Section 6 that for relaxation functions of exponential
type the states cannot be identi®ed with history-deformation pairs.

Acknowledgements. We thank D: R: OWEN for stimulating discussions and helpful

suggestions, and V: MIZEL for drawing our attention to reference [1] and for other
valuable comments. We also acknowledge support by the Italian Ministry for Uni-
versity and Scienti®c Research.

References

[1] BORWEIN, P. B. & T.ERDEÂLYI, MuÈntz's spaces and Remez inequalities, Bull.
Amer. Math. Soc. 32, 38±42, 1995.
[2] BREUER, S. & E.T. ONAT, On the determination of free energy in linear visco-
elastic solids, Z. Angew, Math. Phys. 15, 184±191, 1964.
[3] COLEMAN, B. D., Thermodynamics of materials with memory, Arch. Rational
Mech. Anal. 17, 1±45, 1964.
[4] COLEMAN, B. D. & W. NOLL, Foundations of Linear Viscoelasticity, Rev. Mod.
Physics 33, 239±249, 1961.
[5] COLEMAN, B. D. & D. R. OWEN, A mathematical foundation for thermody-
namics, Arch. Rational Mech. Anal. 54, 1±104, 1974.
[6] COLEMAN, B. D. & D. R. OWEN, On thermodynamics and elastic-plastic mate-
rials, Arch. Rational Mech. Anal. 59, 25±51, 1975.
[7] COURANT, R. & D. HILBERT, Methods of Mathematical Physics, Vol. I, Inter-
science, 1953.
[8] DA Y. W. A., Thermodynamics based on a work axiom, Arch. Rational Mech.
Anal. 31, 1±34, 1968.
 States and Free Energies in Linear Viscoelasticity 35

[9] DA Y. W. A., Reversibility, recoverable work and free energy in linear visco-
elasticity, Quart. J. Mech. Appl. Math. 23, 1±15, 1970.
[10] DA Y. W. A., Restrictions on relaxation functions in linear viscoelasticity, Quart.
J. Mech. Appl. Math. 24, 487±497, 1971.
[11] DA Y. W. A., The Thermodynamics of Simple Materials with Fading Memory,
Springer, 1972.
[12] DEL PIERO, G. & L. DESERI, Monotonic, completely monotonic, and exponen-
tial relaxation functions in linear viscoelasticity, Quart. Appl. Math. 53, 273±300,
1995.
[13] FABRIZIO, M. & A. MORRO, Fading memory spaces and approximate cycles in
Linear Viscoelasticity, Rend. Sem. Mat. Univ. Padova 82, 239±255, 1989.
[14] FABRIZIO, M., C. GIORGI & A. MORRO, Free energies and dissipation properties
for systems with memory, Arch. Rational Mech. Anal. 125, 341±373, 1994.
[15] GRAFFI, D., Sull'espressione dell'energia libera nei materiali viscoelastici lineari,
Annali Mat. Pura e Applicata 98, 273±279, 1974.
[16] GRAFFI, D., Sull'espressione analitica di alcune grandezze termodinamiche nei
materiali con memoria, Rend. Sem. Mat. Univ. Padova 68, 17±29, 1982.
[17] GRAFFI, D., On the fading memory, Applicable Analysis 15, 295±311, 1983.
[18] GRAFFI, D. & M. FABRIZIO, Sulla nozione di stato per materiali viscoelastici di
tipo ``rate'', Atti Acc. Lincei, Rend. Fis. 83, 201±208, 1989.
[19] GRAFFI, D. & M. FABRIZIO, Non unicitaÁ dell'energia libera per materiali vis-
coelastici, Atti Acc. Lincei, Rend. Fis. 83, 209±214, 1989.
[20] GURTIN, M. E. & W. J. HRUSA, On energies for nonlinear viscoelastic materials
of single-integral type, Quart. Appl. Math. 46, 381±392, 1988.
[21] HIRSCH, M. W. & S. SMALE, Di€erential Equations, Dynamical Systems, and
Linear Algebra, Academic Press, 1974.
[22] MORRO, A. & M. VIANELLO, Minimal and maximal free energy for materials
with memory, Boll, Un. Mat. Ital. A4, 45±55, 1990.
[23] NOLL, W., A new mathematical theory of simple materials, Arch. Rational Mech.
Anal. 48, 1±50, 1972.
[24] VOLTERRA, V., LecËons sur les fonctions de lignes, Gauthier-Villars, 1913, or: Sui
fenomeni ereditari, Rend. Acc. Lincei 22, 529±539, 1913.
[25] VOLTERRA, V., Energia nei fenomeni elastici ereditari, Acta Pont. Acad. Scient. 4,
115±128, 1940.
[26] WIDDER, D. V., The Laplace Transform, Princeton Univ. Press, 1941.

Istituto di Ingegneria
Universita di Ferrara
44100 Ferrara, Italy

(Accepted December 7, 1995)