MAXIMUM RECOVERABLE WORK, MINIMUM FREE ENERGY AND

Transcription

MAXIMUM RECOVERABLE WORK, MINIMUM FREE ENERGY AND
```MAXIMUM RECOVERABLE WORK, MINIMUM FREE ENERGY
AND STATE SPACE IN LINEAR VISCOELASTICITY
GIORGIO GENTILI
Abstract. The various formulations of the maximum recoverable work used
in literature are proved to be equivalent. Then an explicit formula of the
minimum free energy is derived starting from the formulation of the maximum
recoverable work given by Day. The resulting expression is equivalen to that
found by Golden and other authors. However the particular formulation allows
to prove that the domain of definition of minimum free energy is the whole
state space. Finally the maximum recoverable work is shown to be put as
the basis of the thermodynamics of viscoelastic materials under isothermal
condidtions. In this context the usual relation between the Clausius-Duhem
inequality and the dissipation of the material is restored.
1. Introduction.
All the definitions of Helmholtz free energy for a viscoelastic material (for instance the one given by Graffi [11, 18, 19, 20] and that stated by Coleman and Owen
[3, 7, 8]) do not identify a unique functional. Moreover it has been shown [23] that
in the convex set of all free energies there exist maximum and minimum element.
An explicit form of the maximum free energy, according to Graffi’s definition, has
been given by Fabrizo, Giorgi and Morro in [11]. Instead the minimum free energy,
both according Graffi’s and Coleman-Owen’s definitions, has been proved to be
obtained by maximizing the recoverable work.
For this reason, in the literature we can find many papers discussing about the
expression of the maximum recoverable work in linear viscoelasticity.
However, the procedure of finding such an expression, and hence an explicit form
of the minimum free energy, is nothing but easy.
The most important work in this sense is due to Day [4, 5]. Nevertheless, he
gave an interesting characterization of the maximum recoverable work, but not an
explicit formula. Recently an expression for the maximum recoverable work, and
the minimum free energy, was found by Golden [17] in the scalar case and then
extended to the general isothermal case [9, 13].
However, some questions seem to be still open.
For instance, although the definition of maximum recoverable work is clear, some
different formulations have been used to obtain the above quoted results. So, the
question whether such formulations are equivalent arises.
Secondly, it is worth recalling that the space of definition of any free energy in
general is a proper subset of the whole state space. In other words a free energy
1991 Mathematics Subject Classification. Primary(73B30,73F99);Secondary(45E10).
This work has been done under the auspices of G.N.F.M. of I.N.D.A.M. and partially supported
by the Italian MURST through the COFIN’98 Research Program “Mathematical Models for
Materials Science”. The author wishes to thank professors M. Fabrizio, C. Giorgi and J. M. Golden
1
2
G. GENTILI
may be unbounded in correspondence with some strain history. On the other hand,
it is reasonable to think that the domain of the minimum free energy is wider than
the one corresponding to any other free energy . As a consequence we should expect
that such a space does coincide with the whole state space if properly defined.
In this paper we try to answer the above questions.
First of all, we give an alternative definition of the state space based on the
boundedness of the work rather than the the stress (Sec. 3).
Secondly, we give a rigorous proof that all the formulations of the maximum
recoverable work present in the literature are equivalent (Sec.4). What is more,
this is proved avoiding any topology on the state space. In materias with memory
and in particular in viscoelasticity, this is important because many different and
unequivalent topologies may be given, but the objective characteristics of the material, as well as thermodynamics and its implications, must be unaffected by the
particular choice of the topology.
Then (Sec. 5), thanks to previous results given in [17, 9], we are able to prove
that the point of view of Day leads to an explicit expression of the minimuim free
energy that is perfectly equivalent to that found in [17] and [9] . Moreover (Sec.
s:sdef) such an expression allows us to prove that the domain of definition of the
minimum free energy is the whole state space as defined in Sec. 3.
Finally (Sec. 7), we show that the existence and the boundedness of the maximum recoverable work on the whole state space can be put as a basis of the
thermodynamics of viscoelastic materials under isothermal conditions, as already
draft by Fabrizio, Giorgi and Morro. Such an approach allows us to formulate the
dissipativity properties of the material without reference to any particular topology. Moreover, in describing the properties of the maximum recoverable work and
identifying it with the minimum free energy, we are lead to an interesting physical
consideration which enables us to go deep into the connection between the ClausiusDuhem inequality and the dissipation of the material. In fact some authors (se e.g.
[23]) claimed that they are unrelated when memory effects occur because of the
non uniqueness of the free energy functional. On the contrary we prove that such
a relation is restored whenever the freee energy involved in the Clausius-Duhem
inequality is the minimum one.
2. Notations and basic assumptions for a linear viscoelastic solid
Let Sym be the space of symmetric second order tensors acting on R3 endowed
1
with the inner product A · B := tr(AB> ) and norm |A| = (A · A) /2 where “> ”
denotes the transpose. Also let Lin(Sym) denote the space of the fourth order
tensors operating on Sym. For any IK ∈ Lin(Sym), the transpose IK> is such
that IK> A · B = IKB · A, ∀A, B ∈ Sym. The reason of this notation is clear
by reminding that Sym is isomorphic to R6 . Therefore, if Ci , i = 1, ..., 6 is an
orthonormal basis of Sym, so that
A=
6
X
i=1
Ai Ci
,
B=
6
X
i=1
ABi Ci
,
A·B=
6
X
i=1
Ai Bi .
MAXIMUM RECOVERABLE WORK AND STATE SPACE
3
Analogously any tensor IK ∈ Lin(Sym) will be identified with an element of
Lin(R6 ) by the representation
IK =
6
X
i,i=1
Kij Ci ⊗ Cj
and IK> is the transpose of IK as an element of Lin(R6 ). According to that, the
norm | IK| of IK ∈ Lin(Sym) may be given by


6


X
| IK|2 = tr IK IK> = 
Kij Kji  .
i,j=1
In the sequel we deal with complex valued tensors. Denoting with Ω the complex
plane and with Sym(Ω) and Lin(Sym(Ω)) respectively the tensors represented by
the above forms with Ai , Bi , Kij ∈ Ω, then the norms |A| and | IK| of A ∈ Sym(Ω)
and IK ∈ Lin(Sym(Ω)) will be given respectively by


6
X


Kij K ji 
|A|2 = A · A
, | IK|2 = tr ( IK IK∗ ) = 
i,j=1
>
where the overhead bar indicates complex conjugate and IK∗ = IK . The above
representations allows all results of [16] to be easily extended to tensors belonging
to Lin(Sym(Ω)).
For any function f ∈ L1 (R, V ) ∪ L2 (R, V ), where V is a finite-dimensional
vector space (in the present
R ∞ context Sym or Lin(Sym)), let fF denote its Fourier
transform viz. fF (ω) = −∞ f (s) e−iωs ds. Also, we define
Z 0
Z ∞
−iωs
f (s) e−iωs ds
f+ (ω) =
f (s) e
ds , f− (ω) =
−∞
0
(2.1)
Z ∞
Z ∞
fs (ω) =
f (s) sin ωs ds , fc (ω) =
f (s) cos ωs ds .
0
0
All these relations are to be understood as applying to each component of the
tensor quantities involved.
In the sequel, we need to consider the Fourier transform of functions that do not
vanish at large times and thus do not belong to L2 for the appropriate domain. The
standard procedure is adopted of introducing an exponential decay factor, calculating the Fourier transform and letting the time decay constant tend to infinity. Thus,
if f is a function defined on R+ such that lims→+∞ f (s) = f∞ , and g : R+ → V,
defined by g(s) = f (s) − f∞ , belongs to L1 (R+ , V) ∩ L2 (R+ , V), then
fF (ω) = gF (ω) −
f∞
,
iω +
ω + = lim+ (ω + iα)
α→0
Henceforth Ω+ and Ω(+) denote the following sets
Ω+ = {ζ ∈ Ω : =m ζ ≥ 0} ,
Ω(+) = {ζ ∈ Ω : =m ζ > 0}
and analogous meaning is stated for Ω− and Ω(−) . The quantities f± (z) defined in
(2.1) are analytic for z ∈ Ω(∓) .
4
G. GENTILI
A linear viscoelastic material is described by the classical Boltzmann-Volterra
constitutive equation between the stress tensor T(t) ∈ Sym and the strain history
tensor E : (−∞, t] → Sym, of the form:
Z ∞
t
T(t) = G
l 0 E(t) +
G(s)E
l̇
(s)ds
(2.2)
0
where E(t) ∈ Sym is the instantaneous value of the strain and Et : (0, +∞) → Sym
denotes the past history defined as
Et (s) := E(t − s),
s ∈ R++ := (0, +∞) .
Henceforth for simplicity we identify the strain history with the couple (E(t), Et )
and use the term “history” just to indicate the past history Et . Since G
l̇ ∈
l : (0, +∞) → Lin(Sym)
L1 (R+ , Lin(Sym)), its primitive, the relaxation function G
such that
Z
t
G(t)
l
=G
l0+
ds
G(s)
l̇
0
is well defined. The quantity G
l 0 = G(0)
l
is named as the instantaneous elastic
modulus. Moreover there exists the limit
G
l ∞ := lim G(t)
l
∈ Lin(Sym) ,
t→∞
G
l ∞ is named as the equilibrium elastic modulus.
We also assume that G
ľ : (0, +∞) → Lin(Sym), defined as
−G
G(t)
ľ
= G(t)
l
l∞,
(2.3)
is symmetric, integrable and its integral does not vanish, viz
Z

 ∞

>

G(s)
ľ
G(t)
ľ
=G
ľ (t) ,
0<
ds < ∞
(2.4)
G
l̇ s (ω)E · E < 0
(2.5)
0
The thermodynamic properties of a linear viscoelastic material imply (see[15, 14])
l>
G
l∞=G
∞
,
∀E ∈ Sym \ {0} ∀ω ∈ (0, +∞) .
Note that (2.5)2 may be written in terms of the Fourier transform of G
ľ by virtue
of the identity
G
l̇ s (ω) = −ω G
ľ c (ω) .
(2.6)
Moreover, (2.4)2 and (2.5)2 imply
G
ľ c (ω)E · E > 0
∀E ∈ Sym \ {0} ∀ω ∈ R .
(2.7)
∀E ∈ Sym \ {0} .
(2.8)
Moreover on the basis of phisycal arguments, the equilibrium elastic modulus of
a viscoelastic solid is assumed to be positive definite, namely
Further, we assume
G
l ∞ E · E > 0,
G(0)E
l̇
·E<0
,
∀E ∈ Sym \ {0}
(2.9)
whereas inequality (2.5)2 just implies that G(0)
l̇
is negative semidefinite (see [15] ).
If we define the vectorial space
Z ∞





t
t


Γ := E : (0, +∞) → Sym ; 
G(s
l̇ + τ )E (s) ds < ∞ ∀τ ≥ 0 , (2.10)
0
MAXIMUM RECOVERABLE WORK AND STATE SPACE
5
whose definition depends on the relaxation function, the Boltzman-Volterra equation (2.2) defines the linear functional T̃ : Sym × Γ → Sym such that:
Z ∞
t
t
(s)ds
T̃(E(t), E ) = G
l 0 E(t) +
G(s)E
l̇
(2.11)
0
Given the strain (E(t), Et ) continued with E(t + a) = E(t), a ∈ R+ , it is easy
to check that the yielded stress is given by
Z ∞
T(t + a) = G(a)E(t)
l
+
G(s
l̇ + a)Et (s) ds .
0
It has been shown (see [9], Proposition 2.2, (ii)), that the fading memory property
G
l̇ ∈ L1 ensures that for every ε > 0 there exists a(ε, Et ) sufficiently large such that
Z ∞



t
t


G(s
l̇
+
a)E
(s)
ds
(2.12)

 < ε , ∀a > a(ε, E )
0
3. A new definition of the space of the processes and of the state
space
The notions of the state and the process for a linear viscoelastic solid, has been
discussed by various authors (see [10, 8, 21, 24]). They can be resumed as follows.
Remark 3.1. According to the definition given in [6] and [15], a process P of duration d < +∞, is given by ĖP : [0, d) → Sym. For simplicity, given the strain
(E(t), Et ) ∈ Sym × Γ, we relate P to
Z τ
EP : (0, d] → Sym
EP (τ ) = E(t) +
,
ĖP (s0 ) ds0 τ ∈ (0, d] . (3.1)
0
Thus the strain Ef (τ 0 ) = (EP ∗ E)(τ 0 ), τ 0 ≤ t + d, yielded by Et and ĖP , is related
to the couple (EP (d), (EP ∗ E)t+d ) and is given by

EP (d − s)
for 0 ≤ s < d
Ef (t + d − s) = (EP ∗ E)(t + d − s) :=
. (3.2)
E(t + d − s) for s ≥ d
In the sequel we use the symbol“ ∗ ” to denote both the continuation of histories
and the continuation of processes (see [8]).
Definition 3.2. (see [10]) Two stain histories (E1 , Et1 ) and (E2 , Et2 ) are said to
be equivalent if for every EP : (0, τ ) → Sym and for every τ > 0, they satisfy
T̃(EP (τ ), (EP ∗ E1 )t+τ ) = T̃(EP (τ ), (EP ∗ E2 )t+τ )
(3.3)
As a consequence, it is easy to show that (0, Et ) is equivalent to the zero history
(0, 0† ) if
Z ∞
Z ∞
t+τ
∀τ > 0 .
(3.4)
G(s)E
l̇
(s) ds =
G(s
l̇ + τ )Et (s) ds = 0
τ
0
Equality (3.4) represents an equivalence relation, i.e. two past histories Et1 and
are said to be equivalent if their difference Et = Et1 − Et2 satisfies (3.4).
According to the definition of state σ given by Noll [24], two couples (E1 (t), Et1 )
and (E2 (t), E2t ) such that E1 (t) = E2 (t) and Et1 − Et2 satisfies (3.4) are represented
by the same state σ(t). In this sense σ(t) may be thought as the “minimum” set
of variables allowing a univocal relation between ĖP : (0, τ ) → Sym and the stress
T(t + τ ) = T̃(EP (τ ), (EP ∗ E)t+τ ) for every τ > 0.
Et2
6
G. GENTILI
In other words (see [8, 21]), denoting with Γ0 the set of all the past histories
of Γ satisfying (3.4), and with Γ/Γ0 the usual quotient space, the state σ of a
linear viscoelastic material may be identified by a couple (E, h) with E ∈ Sym and
h ∈ Γ/Γ0 and the state space may be identified as
Σ = Sym × (Γ/Γ0 ) .
(3.5)
In this paper we present a different point of view in defining the state space.
Roughly speaking, if Σ = Sym × (Γ/Γ0 ) is related to the boundedness of the stress
(see e.g. (2.10)), we now construct a new state space related to the boundedness
of the work. Such a procedure also induces a suitable definition of the space of the
processes.
Given the initial strain described by the pair (Ei (t), Eti ) and the process P given
f (Ei (t), Et ; ĖP ), viz.
by ĖP : (0, d] → Sym the work done on P , is denoted by W
i
Z d
f (Ei (t), Et ; ĖP ) :=
W
T̃(EP (τ ), (EP ∗ Ei )t+τ ) · ĖP (τ ) dτ
i
=
=
Z
0
t+d
T̃(E(τ ), Eτ ) · Ė(τ )dτ
t
(3.6)
1
1
G(0)E(t
l
l
+ d) · E(t + d) − G(0)E(t)
· E(t)
2
2
Z t+d Z ∞
τ
(s) · Ė(τ )dτ
+
G(s)E
l̇
(3.7)
t
0
where E = EP ∗ Ei as defined in (3.2).
Given a process ĖP : [0, d) → Sym and the null strain history (0, 0† ), where
†
0 (s) = 0, ∀s > 0, let (E0 (t), E0t ) denotes the ensuing strain given by (for simplicity
we now take the initial instant as t = 0)
 R t−s
Z t
ĖP (τ ) dτ for 0 < s ≤ t
0
. (3.8)
E0 (t) :=
ĖP (τ ) dτ , E0t (s) :=
for s > t
0
0
Definition 3.3. A process ĖP : [0, d) → Sym (d ≤ ∞) is said to be a finite work
process if
Z d
f (0, 0† ; ĖP ) =
W
T̃(E0 (t), Et0 ) · ĖP (t) dt < ∞
0
where (E0 (t), E0t ) is defined by (3.8).
f (0, 0† ; ĖP ) > 0 by virtue of the Dissipation Principle stated by
Observe that, W
Gurtin and Herrera (see[22]), however it can also be derived as a consequence of
(2.5)2 and (2.8).
When d < ∞ we can consider ĖP : R+ → Sym with ĖP (t) = 0 for t > d so that

Z τ
Z ∞
f (0, 0† ; ĖP ) =
W
G
l 0 E0 (τ ) +
G(s)E
l̇
0 (τ − s) ds · ĖP (τ ) dτ
0
Z0 ∞ Z τ
G(τ
l − s)ĖP (s) · ĖP (τ ) ds dτ
=
0
Z ∞0Z ∞
1
(3.9)
G(|τ
l
− s|)ĖP (s) · ĖP (τ ) ds dτ
=
2 0
0
MAXIMUM RECOVERABLE WORK AND STATE SPACE
7
Rt
l̇
ľ ∈ L1 (R+ , Lin(Sym)), such
ds. Using the function G
where G(t)
l
= G
l 0 + 0 G(s)
ľ
−G
l ∞ we have
that G(t)
= G(t)
l
Z Z
1 ∞ ∞
f (0, 0† ; ĖP ) = 1 G
W
− s|)ĖP (s) · ĖP (τ ) ds dτ
l ∞ E0 (d) · E0 (d) +
G(|τ
ľ
2
2 0
0
Z ∞
1
1
= G
G
ľ c (ω)ĖP + (ω) · ĖP + (ω) dω (3.10)
l ∞ E0 (d) · E0 (d) +
2
2π −∞
Therefore the set of the finite work processes can be characterised by the following
set:


Z ∞
+
+
G
ľ c (ω)ϕ+ (ω) · ϕ+ (ω) dω < ∞
ȞG (R , Sym) := ϕ : R → Sym :
(3.11)
−∞
Observe that, since G
ľ c (ω) is positive definite ∀ω ∈ R, ȞG (R+ , Sym) may be
endowed with the inner product (·, ·) defined as
Z ∞
(ϕ1 , ϕ2 )G =
G
ľ c (ω)ϕ1+ (ω) · ϕ2+ (ω) dω
−∞
and the norm || · ||G such that ||ϕ||2G = (ϕ, ϕ)G . Therefore we can define the space
of the processes as the Hilbert space HG (R+ , Sym) obtained by the completion of
ȞG (R+ , Sym) with respect to the norm || · ||G .
Definition 3.4. The set of admissible strain histories is defined as the set of all
f (E(t), Et ; ĖP ) is finite for every ĖP ∈
the couple (E(t), Et ) such that the work W
+
HG (R , Sym).
Thus, the domain of definition of the stress is the set of all those strain histories
rendering the work well defined when the process belongs to HG (R+ , Sym).
Observe that
f (E(t), Et ; ĖP )
W

Z ∞
Z τ
Z ∞
−
s)
ds
G(s)E
l̇
(τ
+
G
l 0 E(t + τ ) +
G(s)E(t
l̇
+
τ
−
s)
ds
=
· ĖP (τ ) dτ
P
τ
0
0

Z ∞
Z τ
Z ∞
G(τ
l̇ + s)Et (s) ds · ĖP (τ ) dτ
=
G(τ
l )E(t) +
G(τ
l − s)ĖP (s) ds +
0
0
0

Z ∞ Z ∞
1
G(|τ
l
− s|)ĖP (s) ds − I(τ, E(t), Et ) · ĖP (τ ) dτ
(3.12)
=
2 0
0
where
I(τ, E(t), Et ) = − G(τ
l )E(t) −
Z
∞
G(τ
l̇ + s)Et (s) ds ,
0
τ ≥0
(3.13)
Let Ĕt (s) = Et (s) − E(t) be the history relatively to the instantaneous value, it is
obvious that a strain history can be described equivalentely by the couple (E(t), Et )
or by (E(t), Ĕt ). Let Ĭ(·, Ĕt ) : R+ → Sym be defined by
Z ∞
Ĭ(τ, Ĕt ) = −
(3.14)
G(τ
l̇ + s)Ĕt (s) ds , τ ≥ 0
0
we have:
I(τ, E(t), Et ) = − G
l ∞ E(t) + Ĭ(τ, Ĕt ) , τ ≥ 0
(3.15)
8
G. GENTILI
and
Z Z
1 ∞ ∞
f (E(t), Et ; ĖP ) = 1 G
G(|τ
ľ
− s|)ĖP (s) · ĖP (τ ) ds dτ
W
l ∞ E0 (d) · E0 (d) +
2
2 0 0
Z ∞
(3.16)
Ĭ(τ, Ĕt ) · ĖP (τ ) dτ ,
+G
l ∞ E(t) · E0 (d) −
0
where E0 is given by (3.8)1 . Expression (3.16) shows how the work may be written
c (E(t), Ĕt ; ĖP ) =
in terms of the couple (E(t), Ĕt ) and the process ĖP . We denote W
t
f
W (E(t), E ; ĖP ). Moreover, by means of Plancherel’s theorem we have
Z ∞
1
1
t
c
W (E(t), Ĕ ; ĖP ) =
G
ľ c (ω)ĖP + (ω) · ĖP + (ω) dω
G
l ∞ E0 (d) · E0 (d) +
2
2π −∞
Z ∞
1
Ĭ+ (ω, Ĕt ) · ĖP + (ω) dω (3.17)
+G
l ∞ E(t) · E0 (d) −
2π −∞
R∞
where Ĭ+ (ω, Ĕt ) = 0 Ĭ(τ, Ĕt )e−iωτ dτ and
Z ∞
Z ∞
1
f+ (ω) · ϕ+ (ω) dω ,
f (t) · ϕ(t) dt =
hf , ϕi =
2π −∞
0
provided the integral is finite.
Therefore the set of admissible strain histories can be thought as the set of all
the couple (E(t), Ĕt ) such that, E(t) ∈ Sym and the quantity Ĭ(·, Et ), related to
Ĕt by (3.14), belongs to the dual of HG (R+ , Sym), namely to


0
HG
(R+ , Sym) := f : R+ → Sym : |hf , ϕi| < ∞ , ∀ϕ ∈ HG (R+ , Sym)
Remark 3.5. : The quantity Ĭ(τ, Ĕt ) is related to the stress associated to the static
contionuation of Et defined as:

for s ≤ τ
E(t)
(s)
, τ >0
Et+τ
=
c
Et (s − τ ) for s > τ
In fact in this case, we have:
) = G(τ
l )E(t) +
T̃(Et+τ
c
Z
0
∞
G(τ
l̇ + s)Et (s) ds = G
l ∞ E(t) − Ĭ(τ, Ĕt ) .
Therefore the requested regularity on Ĭ(τ, Ĕt ) is the regularity of
)−G
l ∞ E(t) .
T̆(Et+τ
) = T̃(Et+τ
c
c
) = 0.
In particular the fading memory property implies that limτ →∞ T̆(Et+τ
c
In the previous section we called as equivalent two strain histories (E1 (t), Et1 )
and (E2 (t), Et2 ) that yield the same stress when subjected to the same process. The
analogue equivalence relation may be done by means of the work.
Definition 3.6. Two strain histories (E1 (t), Ĕt1 ) and (E2 (t), Ĕt2 ) are said to be
w-equivalent if for every ĖP : (0, τ ) → Sym and for every τ > 0, they satisfy
c (E1 (t), Ĕ1t ; ĖP ) = W
c (E2 (t), Ĕt2 ; ĖP )
W
(3.18)
It is easy to check the following proposition
Proposition 3.7. For every viscoelastic material described by the constitutive equation (2.2) two strain histories are w-equivalent if and only if they are equivalent in
the sense of Definition 3.2.
MAXIMUM RECOVERABLE WORK AND STATE SPACE
9
Proof. It is obvious that if (E1 (t), Et1 ) and (E2 (t), Et2 ) satisfy Definition 3.2, then
they are w-equivalent, namely (E1 (t), Ĕt1 ) and (E2 (t), Ĕ2t ) satisfy (3.18) for every
EP : (0, d) → Sym and for every d > 0, since we have
Z d
Z d
T̃(EP 1 (τ ), (EP 1 ∗ E1 )t+τ ) · ĖP (τ )dτ =
T̃(EP 2 (τ ), (EP 2 ∗ E1 )t+τ ) · ĖP (τ )dτ
0
0
being the left [ right] hand side the definition of the work done by ĖP starting
from (E1 (t), Et1 ) [(E2 (t), Et2 )]. On the other hand, by virtue of (3.16) and of the
arbitrariness of ĖP and d, if (E1 (t), Ĕt1 ) and (E2 (t), Ĕt2 ) are w-equivalent, then they
satisfy:
E1 (t) = E2 (t)
,
I(τ, Ĕt1 ) = I(τ, Ĕt2 ) ∀τ > 0
(3.19)
By virtue of (3.13) and (3.15), equalities (3.19) imply
Z ∞
Z ∞
G(τ
l̇ + s)Et1 (s) ds =
G(τ
l̇ + s)Et2 (s) ds ∀τ > 0
0
0
namely the difference Et = Et1 − Et2 satisfies (3.4).
Therefore if Γw and Γw0 denote rispectively
n
o
0
Γw = Ĕt : R+ → Sym : Ĭ(·, Ĕt ) ∈ HG
(R+ , Sym)
o
n
Γw0 = Ĕt ∈ Γw : Ĭ(τ, Ĕt ) = 0 , ∀τ > 0
,
(3.20)
the new state space may be defined as
Σw = Sym × (Γw /Γw0 ) .
(3.21)
As a consequence of Definition 3.2 the work is a function of the state and the
process. Therefore henceforth we also use the notation
c (E(t), Ĕt ; ĖP ) = W
f (E(t), Et ; ĖP )
W (σ(t), P ) = W
and view a process as a function P : Σw → Σw which associates with an initial
state σ i ∈ Σw a final state P σ i = σ f ∈ Σw . The set of finite work processes P will
be denoted by Π, namely P ∈ Π if the related ĖP satisfies Definition 3.3.
4. Equivalent formulations of the maximum recoverable work
The maximum recoverable work is defined as follows
Definition 4.1. Given a state σ ∈ Σw , the maximum recoverable work WR starting
from σ is defined as
WR (σ) := sup {−W (σ, P ) : P ∈ Π } .
(4.1)
The name of “maximum recoverable work” for WR (σ) is to indicate that it is the
maximum quantity of work we can “extract” from the material at the given state
σ, in other word it is the amount of energy that is avaliable at σ.
Note that WR (σ) is a function of state and it is non negative, since the null
process belong to Π and yields a null work.
Even if (4.1) is widely recognized as the definition of the maximum recoverable
work, nevertheless in literature alternative formulations have been used in order to
obtain its expression in terms of the past histories, and consequentely the explicit
formula of the minimum free energy. Usually such alternative formulations are
motivated by reasonable arguments, but not rigorously proved. For instance Breuer
10
G. GENTILI
and Onat in [1, 2] implicitely assume that WR is obtained by means of processes
such that the final strain vanishes, namely
where
WR (σ) = sup {−W (σ, P ) : P ∈ Π0 (σ) } ,
Π0 (σ) := {P ∈ Π : P σ = (0, h) for some suitable h ∈ Γw /Γw0 } .
(4.2)
(4.3)
On the other hand, Day ([4, 5]) obtains WR by using processes such that the final
strain has the same value as the initial one, viz.


1
WR (σ) = G
l ∞ E · E + sup −W (σ(t), P ) , P ∈ Π0 ,
(4.4)
2
where
)
(
Z d
0
(4.5)
Π := P ∈ Π :
ĖP (t) dt = 0 ,
0
and d is the duration of the process 1 . Finally, Golden in [17], finds the minimum
free energy by maximazing the recoverable work in the form
WR (σ) = sup {S(∞) − W (σ(t), P ) , P ∈ Π } .
(4.6)
where S(∞) = limt→∞ S(t), and
1
G
l 0 E(t) · E(t) ,
(4.7)
2
turns out to be the opposite of the work done by a jump from E(t) to 0 (see also
[9]).
Since such formulation are quite different and since they must be related to the
minimum free energy which is supposed to be unique, one can ask whether they
are equivalent.
A first proof, showing the equivalence between (4.1) , (4.2) and (4.6), has been
given in [9]. Here we extend such results to prove the equivalence between (4.4)
and (4.1). Moreover this result leads to a generalization of (4.4) that has been used
in [13].
The proofs will be given without involving any topology or norm, so that the
results are not affected by the choice of the topology for the description of the state
space.
S(t) = T(t) · E(t) −
Theorem 4.2. For every state σ ∈ Σw , the maximum recoverable work WR (σ)
defined in (4.1) is given by
WR (σ) = sup {−W (σ, P ) , P ∈ Π0 (σ) } ,
(4.8)
where Π0 (σ) is given by (4.3).
Before giving the proof we need a preliminary lemma:
Lemma 4.3. Given a state σ(t) related to the couple (E(t), Et ), with E(t) = E,
for every ε > 0 there exist two suitable parameters a, r, such that the process
Pa,r ∈ Π0 (σ), of duration a + r related to

E
for τ ∈ (0, a)
EP (τ ) =
,
(4.9)
E − Er (τ − a) for τ ∈ (a, a + r]
1 Actually, Day’s statement is slightly different. Nevertheless, for our purpose it can be resumed
to eqs. (4.4-4.5). This will be precised before Corollary 4.6.
MAXIMUM RECOVERABLE WORK AND STATE SPACE
yields a work W (σ(t), Pa,r ) satisfying




<ε
W (σ(t), Pa,r ) + 1 G
l
E
·
E
∞


2
11
(4.10)
Proof.
−W (σ(t), Pa,r )
=
=
Z
r
E
T(t + a + τ )dτ ·
r
0


Z rZ τ
1
E
E
G
l 0E · E +
G(s)
l̇
E − (τ − s) dsdτ ·
2
r
r
0
0
Z r Z τ +a
E
G(s)dsdτ
l̇
E·
+
r
Z0 r Zτ ∞
E
t
+
(s − a − τ )dsdτ ·
G(s)E
l̇
r
0
τ +a
ľ
Integrating and making use of (2.3), i. e. G(t)
= G(t)
l
−G
l ∞ , we have
Z rZ τ
1
E E
1
−W (σ(t), Pa,r ) − G
l 0E · E + G
ľ 0 E · E =
·
G(s)dsdτ
ľ
2
2
r r
0
0
Z r
Z rZ ∞


E
E
G(s
l̇ + τ + a)Et (s)dsdτ ·
ľ ) dτ E · +
+
G(τ
ľ + a) − G(τ
r
r
0
0
0
(4.11)
Observe that
−W (σ(t), Pa,r ) −
1
1
1
ľ 0 E · E = −W (σ(t), Pa,r ) − G
G
l 0E · E + G
l ∞E · E
2
2
2
Moreover, by virtue of (2.4)
Z Z
Z
 r τ
E E

G(s)dsdτ
ľ
· +

r r
0
0
0
r



E  M
G(τ
ľ + a) − G(τ
ľ ) dτ E ·  <
r
r
(4.12)
(4.13)
for some M < ∞. Moreover, by virtue of (3.1) we have
Z ∞



t

G(s
l̇ + τ + a)E (s)ds < δ , ∀(τ + a) > a(δ, Et )

0
so that
Z



0
r
Z
0
Choosing r =
∞

E 
G(s
l̇ + τ + a)E (s)dsdτ ·  < δ|E|
r
2M
ε ,
t
δ=
ε
2|E|
,
∀a > a(δ, Et )
(4.14)
and a > a(δ, Et ), (4.11)–(4.14) yield (4.10).
Proof of Theorem 4.2. Obviously, since Π0 (σ) ⊂ Π, it holds
WR (σ) = sup {−W (σ, P ) : P ∈ Π} ≥ sup {−W (σ, P ) : P ∈ Π0 (σ)} .
(4.15)
In view of our hypothesis WR (σ) < ∞ so that, for every ε > 0 there exists a process
Pε ∈ Π, of duration d < ∞, such that
ε
WR (σ) − < −W (σ, Pε )
(4.16)
2
Suppose that Pε σ = (Eε , h) for some suitable h ∈ Γw /Γw0 and continue Pε with
a process Pa,r described by (4.9). Then, by virtue of (4.16) and Lemma 4.3, there
12
G. GENTILI
exist two values of a and r such that:
−W (σ, Pa,r ∗ Pε ) = −W (σ, Pε ) − W (Pε σ, Pa,r ) > WR (σ) +
1
G
l ∞ Eε · Eε − ε
2
(4.17)
Finally observe that (Pa,r ∗ Pε ) ∈ Π0 (σ) so that (4.17) and (2.8) imply
WR (σ) ≤ sup {−W (σ, P ) : P ∈ Π0 (σ)} .
(4.18)
Inequalities (4.15) and (4.18) ensure (4.8).
An interesting alternative formulation of the maximum recoverable work is the
following.
Theorem 4.4. For every state σ ∈ Σw , and for any fixed E ∈ Sym, the maximum
recoverable work WR (σ) defined in (4.1) is given by
1
WR (σ) = G
l ∞ E · E + sup {−W (σ, P ) , P ∈ ΠE (σ) } .
(4.19)
2
where
ΠE (σ) := {P ∈ Π : P σ = (E, h)
for some suitable h ∈ Γw /Γw0 } .
(4.20)
The proof will be given with the aid of the following lemma.
Lemma 4.5. Given a state related to the couple (0, Et ), for every ε > 0 and every
fixed strain E ∈ Sym there exist two suitable parameters b, r, such that the process
Pb,r ∈ ΠE (σ), of duration b + r related to
EP (τ ) =

0
E
r (τ
− b)
for τ ∈ (0, b)
for τ ∈ (b, b + r]
yields a work W (σ(t), Pb,r ) satisfying




W (σ(t), Pb,r ) − 1 G
l ∞ E · E < ε

2
Proof.
Z
r
,
(4.21)
(4.22)
E
T(t + b + τ )dτ ·
r
0
Z rZ ∞
1
E
G
l 0E · E +
+ b + τ − s)dsdτ ·
=
G(s)E(t
l̇
2
r
0
0
Z rZ τ
1
E
E
=
G
l 0E · E +
(τ − s) dsdτ ·
G(s)
l̇
2
r
r
0
0
Z rZ ∞
E
+
(4.23)
G(s
l̇ + b + τ )Et (s)dsdτ ·
r
0
0
Integrating (4.23) and making use of (2.3), we have
1
W (σ(t), Pb,r ) − G
l ∞E · E
2
Z rZ τ
Z rZ ∞
E
E E
G(s)dsdτ
ľ
G(s
l̇ + τ + a)Et (s)dsdτ ·
=
· +
(4.24)
r
r
r
0
0
0
0
Observe that, by virtue of (2.4)

Z r Z τ

E E  M

G(s)dsdτ
ľ
·
(4.25)
<

r r
r
0
0
W (σ(t), Pb,r )
=
MAXIMUM RECOVERABLE WORK AND STATE SPACE
13
for some M < ∞. Moreover, by virtue of (3.1) we have
Z ∞



t
t


+
+
b)E
(s)ds
G(s
l̇
τ

 < δ , ∀(τ + b) > a(δ, E )
0
so that
Z



r
0
Z
0
Choosing r =
∞

E 
G(s
l̇ + τ + b)E (s)dsdτ ·  < δ|E|
r
2M
ε ,
t
δ=
ε
2|E|
,
∀b > a(δ, Et )
(4.26)
and a > a(δ, Et ), (4.24)–(4.26) yield (4.19).
Proof of Theorem 4.4. Denoting with W0 (σ) and WE (σ) the following sets
W0 (σ)
:=
WE (σ) :=
{−W (σ, P ); P ∈ Π0 (σ)} ,


1
G
l ∞ E · E − W (σ, P ); P ∈ ΠE (σ) ,
2
(4.27)
(4.28)
Observe that, for every PE ∈ ΠE (σ) there exists a sequence of Pn ∈ Π0 (σ) such
that
1
l ∞ E · E − W (σ, PE ) .
(4.29)
− lim W (σ, Pn ) = G
n→∞
2
In fact if we continue any process PE ∈ ΠE (σ), yielding a work W (σ, PE ), with
the process Pa,r given by (4.9) we obtain a process Pa,r ∗ PE ∈ Π0 (σ). Moreover
(4.10) and the following
W (σ, Pa,r ∗ PE ) = W (σ, PE ) + W (PE σ, Pa,r )
implies that for any ε > 0 there exist a process Pa,r ∗ PE ∈ Π0 (σ) such that




W (σ, Pa,r ∗ PE ) − W (σ, PE ) + 1 G
<ε,
E
l
E
·
∞


2
from which the limit (4.29) follows. This implies that
WE (σ) ⊆ W0 (σ)
(4.30)
On the oher hand, for every P0 ∈ Π0 (σ) there exists a sequence of Pn ∈ ΠE (σ) such
that
1
G
l ∞ E · E − lim W (σ, Pn ) = −W (σ, P0 )
n→∞
2
In fact if we continue any process P0 ∈ Π0 (σ), yielding a work W (σ, P0 ), with
the process Pb,r given by (4.21) we obtain Pb,r ∗ P0 ∈ ΠE (σ). Moreover relation
(4.22) and the following
W (σ, Pb,r ∗ P0 ) = W (σ, P0 ) + W (P0 σ, Pb,r )
implies that for any ε > 0 there exist a process Pb,r ∗ P0 ∈ ΠE (σ) such that




W (σ, Pb,r ∗ P0 ) − W (σ, P0 ) − 1 G
l ∞ E · E < ε

2
This implies that
W0 (σ) ⊆ WE (σ)
(4.31)
14
G. GENTILI
Equations (4.30) and (4.31) imply W0 (σ) = WE (σ) so that sup W0 (σ) = sup WE (σ).
By virtue of Theorem 4.2 the thesis follows.
Theorem 4.4 is a generalization of the formulation of the maximum recoverable
(D)
work given by Day. For sake of precision, the maximum recoverable work WR


(D)
0
defined by Day is just the quantity WR (σ(t)) = sup −W (σ(t), P ) , P ∈ Π
.
†
However, since any free energy ψ evaluated at the constant strain history (E, E )
l ∞ E · E, he stated that the minimum free energy ψm relatively to
gives ψ = 12 G
(D)
the strain history (E(t), Et ) is given by ψm (t) = 12 G
l ∞ E(t) · E(t) + WR (σ(t)).
Therefore, for our purpose (see (5.1)), Day’s point of view can be stated can be
stated as follows:
Corollary 4.6. For every state σ ∈ Σw , and for any fixed E ∈ Sym, the maximum
recoverable work WR (σ) defined in (4.1) is also given by


1
l ∞ E(t) · E(t) + sup −W (σ, P ) , P ∈ Π0 .
WR (σ) = G
(4.32)
2
where Π0 is defined by (4.5).
Moreover Theorem 4.4 may be generalized in the following form that has been
used by Fabrizio and Golden [13]. Let φ(t) = 21 G
l ∞ E(t) · E(t) and φ(∞) =
limt→∞ φ(t). Observe that in our context the limite φ(∞) does exist, since any
process P ∈ Π has finite duration. Therefore by continuing any process P of duration d with the null process, the work done doesn’t change and the ensuing strain
is such that E(t) = E(d), ∀t ≥ d. Thus the limit E∞ = limt→∞ E(t) exists and
l ∞ E∞ · E∞ .
correspond to the final strain yielded by P , whereas φ(∞) = 12 G
Corollary 4.7. For every state σ ∈ Σw , the maximum recoverable work WR (σ)
defined in (4.1) is given by
WR (σ) = sup {φ(∞) − W (σ, P ) , P ∈ Π } .
(4.33)
Proof. First observe that P ∈ Π0 (σ), of duration d < ∞, continued with the null
process yields a strain that vanishes at infinity, i.e. E∞ = 0, so that φ(∞) = 0.
Therefore, letting W0 (σ) and W∞ (σ) be defined by (4.27) and by
we have:
W∞ (σ) := {φ(∞) − W (σ, P ); P ∈ Π} ,
(4.34)
W0 (σ) ⊂ W∞ (σ) .
(4.35)
Now fix a state σ(t) and a process P ∈ Π yielding a final strain such that
lim E(τ ) = E∞ 6= 0 .
τ →∞
Let Pd be the truncation of P up to the time d, i.e. the process of duration d
continuing σ(t) with the strain Ed such that Ed (τ ) = E(τ ), τ ∈ [t, t + d). Then
since
1
1
lim W (σ, Pd ) = W (σ, P ) , lim G
l ∞ Ed · Ed = G
l ∞ E∞ · E∞
d→∞
d→∞ 2
2
for every ε there exists a suitable dε such that
ε
1
ε
1
|W (σ, Pd ) − W (σ, P )| <
(4.36)
, | G
l ∞ Ed · Ed − G
l ∞ E∞ · E∞ | < .
3
2
2
3
holds for every d > dε .
MAXIMUM RECOVERABLE WORK AND STATE SPACE
15
Now we consider a process Pd,a,r = Pa,r ∗ Pd ∈ Π0 (σ(t)), with: d such that
(4.36) is satisfied and Pa,r is described in Lemma 4.3, related to the strain (4.9)
with E = Ed and a, r such that Pa,r satisfies (4.10) with ε replaced by 31 ε.
Then, for every ε > 0 there exists a process Pε ∈ Π0 (σ(t)) such that
|φ(∞) − W (σ, P ) + W (σ, Pε )| < ε .
(4.37)
In fact, considering Pε = Pd,a,r we have
W (σ, Pε ) = W (σ, Pa,r ∗ Pd ) = W (σ, Pd ) + W (Pd σ, Pa,r )
and
|φ(∞) − W (σ, P ) + W (σ, P ε)| = |φ(∞) − W (σ, P ) + W (σ, Pd ) + W (Pd σ, Pa,r )|
1
l ∞ Ed · Ed | + |W (σ, P ) − W (σ, Pd )|
< |φ(∞) − G
2
1
+|W (Pd σ, Pa,r ) + G
l ∞ Ed · Ed | < ε
2
Estimate (4.37) means that for each fixed σ ∈ Σw and P ∈ Π yielding a work
W (σ, P ) there exists a sequence of processes Pdn ,an ,rn ∈ Π0 (σ) yielding works
W (σ, Pdn ,an ,rn ) ∈ W0 (σ) such that
c (σ, Pd ,a ,r ) = W (σ, P ) − φ(∞)
lim W
n n n
n→∞
and hence
W∞ (σ) ⊆ W0 (σ) .
(4.38)
Inclusions (4.35) and (4.38) imply that
sup W∞ (σ) = sup W0 (σ)
by virtue of definitions (4.34) and Lemma 4.3 the thesis is proved.
We conclude this section by recalling the equivalent formulation for WR used by
Golden in [17]. Such a formulation may be resumed as follows
Theorem 4.8. The maximum recoverable work WR (σ(t)) is given by
WR (σ(t)) = sup {S(∞) − W (σ(t), P ) , P ∈ Π }
(4.39)
where S(∞) = limt→∞ S(t) and S(t) defined by (4.7).
Proof. Any process P ∈ Π0 (σ), yields a final strain E∞ = 0, so that S(∞) = 0.
0
Therefore, letting W0 (σ) and W∞
(σ) be defined by (4.27) and by
0
W∞
(σ) := {S(∞) − W (σ, P ); P ∈ Π} ,
we have:
0
(σ) .
W0 (σ) ⊂ W∞
(4.40)
Now consider the strain E(τ ) τ ∈ R, related to σ(t) continuated with a process
P ∈ Π and such that limτ →∞ E(τ ) = E∞ =
6 0. Let Ed denote the truncation of E
at the time d > t, continued with the “jump”-process yielding
lim Ed (τ ) = E(d)
τ →d−
,
Ed (τ ) = 0 , τ ≥ d ,
(4.41)
16
G. GENTILI
and call Pd the related process. The work done on Pd is:
Z d
1
T(τ ) · Ė(τ ) dτ − [T(d+ ) + T(d− )] · E(d)
W (σ(t), Pd ) =
2
t
Z d
=
T(τ ) · Ė(τ ) dτ − S(d)
(4.42)
t
Since
lim W (σ(t), Pd ) = W (σ(t), P ) − S(∞) ,
d→∞
(4.43)
for every P ∈ Π we can find a family of processes Pd ∈ Π0 (σ(t)) such that
lim W (σ(t), Pd ) = W (σ(t), P ) − S(∞) .
d→∞
Therefore
0
(σ(t)) ⊂ W0 (σ(t))
W∞
(4.44)
(4.40) and (4.44) implies (4.39).
5. An expression for the minimum free energy
The minimum free energy has been shown to be given by the maximum recoverable work, viz.
ψm (σ) = WR (σ) .
(5.1)
The problem of finding an expression of the maximum recoverable work, and hence
of the minimum free energy, in the general case for a voscoelastic materal has been
considered first by Day in [4]. However such works do not solve completely the
problem because the exhibited final expression is not in terms of the given strain,
but in terms of the “reversible extension”, that is the “optimal future continuatiion”
of the strain yielding the maximum recoverable work. Now, such a reversible extension can be written in terms of the given strain by solving a Wiener-Hopf equation
of the first kind. Unfortunately this type of integral equation is not solvable unless
particular properties of the integral kernel are specified. Therefore Day’s formula
remains an interesting characterization, but not an explicit formula of the minimum
free energy. Recently Golden [17] has provided an explicit expression of the minimum free energy in the scalar case by using a variational tecnique. Such a method
has been extended to the tensorial case in [9]. However the results on factorization
of the memory kernel, which turn out to be crucial both in [17] and [9], may also be
applied to Day formulation to yield an explicit formula of the maximum recoverable
work in terms of the given strain, and hence of the minimum free energy. This is
just the aim of this section, namely to give an explicit characterization of WR (σ),
and hence of ψm (σ) starting from relation (4.19) of Corollary 4.6.
Consider a fixed time instant t and a strain history (E(t), Ĕt ) represented by
a state σ(t) ∈ Σw The work done on a process P ∈ Π0 , related to the strain
EP : (0, ∞) → Sym is given by (3.16) with E0 (d) = 0, viz.

Z ∞ Z ∞
1
t
t
c
W (E(t), Ĕ ; ĖP ) =
G(|τ
ľ
− s|)ĖP (s) ds − Ĭ(τ, Ĕ ) · ĖP (τ ) dτ , (5.2)
2 0
0
MAXIMUM RECOVERABLE WORK AND STATE SPACE
17
The “optimal process” P (m) will be the one related to the future strain E(m) :
(0, ∞) → Sym providing the maximum recoverable work, namely such that
WR (σ(t)) =
1
c (E(t), Ĕt ; Ė(m) ) .
G
l ∞ E(t) · E(t) − W
2
(5.3)
For each ĖP we put ĖP (s) = Ė(m) (s) + h(s) for a suitable h : R− → Sym.
c (E(t), Ĕt ; ĖP ) is minimized in correspondence of E(m) satisfying
Therefore W
Z ∞
G(|τ
ľ
− s|)Ė(m) (s) ds = Ĭ(τ, Ĕt ) τ ∈ R+
(5.4)
0
Such an equation turns out to be a Wiener-Hopf equation of the first kind which is
not solvable in the general case. Nevertheless the thermodynamic properties of the
integral kernel and some theorems on factorization will allow us to determine the
solution E(m) of (??).
Let us introduce a new function r : R− → Sym, defined as
Z ∞
− s|)Ė(m) (s) ds τ ∈ R−
r(τ ) =
G(|τ
ľ
−∞
which , added to (5.4), yields
Z ∞
G(|τ
ľ
− s|)Ė(m) (s) ds = Ĭ(τ, Ĕt ) + r(τ ) τ ∈ R
(5.5)
−∞
where supp(Ė(m) ) ⊆ R+ , supp(Ĭ(·, Ĕt )) ⊆ R+ and supp(r) ⊆ R− . The Fourier
transform of (5.5) gives
(m)
t
(ω) + r− (ω)
2G
ľ c (ω)Ė+ (ω) = Ĭ+
(5.6)
R
∞
where Ĭt+ (ω) = Ĭ+ (ω, Ĕt ) = 0 Ĭ(τ, Ĕt )e−iωτ dτ .
It is worth recalling that we can apply results of [16] to fourth-order tensor valued
functions (see Appendix). In particular the tensor IK(ω) = (1 + ω 2 ) G
ľ c (ω) can be
factorized as
IK(ω) = IK(+) (ω) IK(−) (ω) ,
where henceforth the subscript (±) indicates that the function f(±) (z) has zeroes
and singularities only for z ∈ Ω± . In particular IK(ω) has no zero for every real ω,
ľ c (ω) can be factorized too as follows
−∞ ≤ ω ≤ ∞. Moreover, G
ľ (−) (ω) ,
ľ (+) (ω) G
G
ľ c (ω) = G
G
ľ (±) (ω) =
Factorization (5.7) allows to rewrite (5.6) as follows:
(m)
G
ľ (+) (ω)Ė+ (ω) =
1
IK± (ω) .
1 ± iω
(5.7)
1 −1
1 −1
G
ľ
(ω)Ĭt+ (ω) + G
ľ
(ω)r− (ω)
2 (−)
2 (−)
−1
t
Applying the plemelj formulae, the quantity G
ľ (−) (ω)Ĭ+
(ω) may be rewritten as
1 −1
G
ľ
(ω)Ĭt+ (ω) = pt(−) (ω) − pt(+) (ω)
(5.8)
2 (−)
t
where p(±)
(z) has zeroes and singularities for z ∈ Ω± and they are defined by:
pt (z) :=
1
4πi
Z
∞
−∞
−1
G
ľ (−) (ω)Ĭt+ (ω)
dω
z−ω
,
pt(±) (ω) := lim∓ pt (ω + iα)
α→0
(5.9)
18
G. GENTILI
Therefore we obtain:
(m)
t
G
ľ (+) (ω)Ė+ (ω) = −p(+)
(ω) + pt(−) (ω) +
1 −1
G
ľ
(ω)r− (ω)
2 (−)
(5.10)
(m)
Observe that the quantities G
ľ (+) (z)Ė+ (z) and pt(+) (z) are analytic for z ∈ Ω− ,
−1
whereas pt(−) (z) and G
ľ (−) (ω)r− (ω) are analytic for z ∈ Ω+ . Therefore the quantity
1 −1
G
ľ
(ω)r− (ω)
2 (−)
has analytic extension on the whole complex plane. Since J vanishes at infinity it
must be zero so that
(m)
J(ω) = G
ľ (+) (ω)Ė+ (ω) + pt(+) (ω) = pt(−) (ω) +
(m)
−G
ľ (+) (ω)−1 pt(+) (ω)
(5.11)
1 −1
pt(−) (ω) = − G
(ω)r− (ω)
ľ
2 (−)
It is worth noting that the process P (m) related to E(m) does not belong to Π0 but
may be constructed as the limit of a sequence of processes Pn of Π0 .
 
Remark 5.1. : Since pt(+) (ω) = O ω1 as |ω| → ∞ and limω→∞ G
ľ (+) (ω) =
h
i1/2
1
l̇
, it follows that
iω − G(0)
Ė+ (ω)
=
(m)
lim Ė+ (ω) 6= 0 ,
ω→∞
therefore Ė(m) (τ ) has an initial singularity as τ → 0+ . This agrees with the property of the optimal continuation that must have an initial discontinuity as τ → 0+ .
Remark 5.2. : Observe that G
ľ (+) (0) 6= 0, so that
Z ∞
(m)
ľ (+) (0)−1 pt(+) (0)
Ė(m) (τ )dτ = Ė+ (0) = − G
E(m) (∞) − E(m) (0) =
0
Therefore, the solution Ė
(m)
lim E
τ →∞
(m)
of (5.4) tends to the finite limit
(τ ) = E(m) (∞) = E(t) − G
ľ (+) (0)−1 pt(+) (0)
The substitution of (5.11) into (3), taking account of (4.36), yields
Z Z
1
1 ∞ ∞
WR (σ) =
G
l ∞ E(t) · E(t) +
G(|τ
ľ
− s|)Ė(m) (s) · Ė(m) (τ ) ds dτ
2
2 0
0
Z ∞
1
1
(m)
(m)
G
l ∞ E(t) · E(t) +
=
G
ľ c (ω)Ė+ (ω) · Ė+ (ω) dω
2
2π −∞
Z ∞
1
1
G
l ∞ E(t) · E(t) +
|pt (ω)|2 dω
=
(5.12)
2
2π −∞ (+)
Therefore , in view of (5.1), the minimum free energy takes the form
Z ∞
1
1
t
l ∞ E(t) · E(t) +
|pt (ω)|2 dω
ψm (σ(t)) = ψ̂m (E(t), p(+) ) = G
2
2π −∞ (+)
(5.13)
Actually expression (5.13) suggests that the couple (E(t), pt(+) ) provides an explicit
representation of the statet σ(t) ∈ Σw ; this question will be considered in the next
section.
Now we conclude the section by comparing expression (5.13) for the minimum
free energy with the ones already found in [17] and [9]. Let us exploit the relation
MAXIMUM RECOVERABLE WORK AND STATE SPACE
19
between pt(+) (ω) and Ĕt . We identify Ĕt with its causal extension (viz. Ĕt (τ ) = 0
(d)
l̇
of G,
l̇ viz.
for τ < 0) and consider the odd extension G

(d)
(d)
for rξ ≥ 0
G(ξ)
l̇
,
G
l̇ F (ω) = −2iω G
l̇ s (ω)
G
l̇ (ξ) =
so that
l̇
for ξ < 0
− G(−ξ)
Then (3.14) can be rewritten as
Z ∞
(d)
Ĭ(τ, Ĕt ) = −
G
l̇ (τ + s)Ĕt (s) ds ,
−∞
Moreover let Ĭ(n) (·, Ĕt ) : R− → Sym, be defined as
Z ∞
(d)
Ĭ(n) (τ, Ĕt ) = −
G
l̇ (τ + s)Ĕt (s) ds ,
τ ≥0
(5.14)
τ <0,
(5.15)
−∞
and consider the extension Ĭ(R) (·, Ĕt ) : R → Sym of Ĭ(τ, Ĕt ) to the real line as
follows

Z ∞
(d)
for τ ≥ 0
Ĭ(τ, Ĕt )
G
l̇ (τ + s)Ĕt (s) ds =
. (5.16)
Ĭ(R) (τ, Ĕt ) = −
Ĭ(n) (τ, Ĕt ) for τ < 0
−∞
Introducing the variable Ĕtn (s) = Ĕt (−s), s ≤ 0 and the ensuing extension
= 0 for s > 0, we have:
Z ∞
(d)
(R)
t
Ĭ (τ, Ĕ ) = −
G
l̇ (τ − s)Ĕtn (s) ds ,
Ĕnt (s)
−∞
so that
(R)
t (ω)
ĬF (ω, Ĕt ) = 2i G
l̇ s (ω)Ĕ+
l̇ s (ω)Ĕtn− (ω) = 2i G
(5.17)
By virtue of (5.16), we have the property
(R)
(n)
ĬF (ω, Ĕt ) = Ĭ+ (ω, Ĕt ) + Ĭ− (ω, Ĕt ) .
So that
1 −1
1 −1
1 −1
(R)
(n)
G
ľ (−) (ω)ĬF (ω, Ĕt ) = G
ľ (−) (ω)Ĭ+ (ω, Ĕt ) + G
ľ
(ω)Ĭ− (ω, Ĕt )
2
2
2 (−)
By virtue of (5.8) we have
1 −1
1 −1
(R)
(n)
t
(ω)ĬF (ω, Ĕt ) = pt(−) (ω) − p(+)
(ω)Ĭ− (ω, Ĕt )
G
ľ
ľ
(ω) + G
2 (−)
2 (−)
−1
(R)
ľ (−) (ω)ĬF (ω, Ĕt ) may be written by using
On the other hand, the quantity 12 G
the Plemelj formulae as follows
1 −1
t
t
(R)
G
ľ (−) (ω)ĬF (ω, Ĕt ) = p0 (−) (ω) − p0 (+) (ω)
2
t
where p0 (±) (z) has zeroes and singularities for z ∈ Ω± and they are defined analogously to (5.9). Thus, we have
t
t
t
(ω) − pt(+) (ω) +
p0 (−) (ω) − p0 (+) (ω) = p(−)
1 −1
(n)
(ω)Ĭ− (ω, Ĕt ) .
G
ľ
2 (−)
Observe that the quantity
t
t
J0 (ω) = pt(+) (ω) − p0 (+) (ω) = pt(−) (ω) − p0 (−) (ω) +
1 −1
(n)
(ω)Ĭ− (ω, Ĕt ) (5.18)
G
ľ
2 (−)
20
G. GENTILI
is analytic on Ω− by virtue of the first relation of (5.18), and it is analytic on Ω+
by virtue of the rightmost relation of (5.18). So J0 (ω) = 0 and
t
pt(+) (ω) = p0 (+) (ω) .
In particular, when it is possible to use (5.17)2 it holds
1 −1
−1
(R)
t (ω)
ľ (−) (ω) G
l̇ s (ω)Ĕt+ (ω) = −iω G
G
ľ (−) (ω)ĬF (ω, Ĕt ) = i G
ľ (+) (ω)Ĕ+
2
so that
Z ∞
ľ (+) (ω 0 )Ĕt+ (ω 0 ) 0
iω 0 G
1
t
dω
pt(+) (ω) = p0 (+) (ω) = lim
z − ω0
z→ω − 2πi −∞
(5.19)
(5.20)
t
The quantity Ĕ+
can be written in terms of the Fourier transform of Et as
Ĕt+ (ω) = Et+ (ω) −
so that
pt(+) (ω)
1
= lim+
z→ω 2πi
Z
∞
−∞


t
G
ľ (−) (ω 0 ) E(t) + iω 0 E+
(ω 0 )
dω 0
ω0 − z
Since
G
ľ (−) (ω) = lim+
α→0
1
E(t)
iω +
1
2πi
we have
Z
∞
−∞
(5.21)
G
ľ (−) (ω 0 )
dω 0
ω 0 − (ω + iα)
pt(+) (ω) = G
ľ (−) (ω)E(t) + iqt(−) (ω)
(5.22)
where
qt(±) (ω) = lim∓
α→0
1
2πi
Z
∞
−∞
G
ľ (−) (ω 0 )ω 0 Et+ (ω 0 ) 0
dω
ω 0 − (ω + iα)
(5.23)
It is worth noting that qt(±) (ω) are the same quantity defined in [9] as qt± (ω),
ľ c (ω) and hence ω G
ľ (±) (ω) correspond to IH± (ω) of
since IH(ω) = −ω G
l̇ s (ω) = ω 2 G
the factorization of IH (see (5.11) of [9]). Substitution of (5.22) into (5.13) yields
Z ∞
1
1
t
ľ (−) (ω)E(t) + iq(−)
(ω)|2 dω
G
l ∞ E(t) · E(t) +
|G
ψm (σ(t)) =
2
2π −∞
Z ∞
1
1
iG
=
G
l 0 E(t) · E(t) + <
ľ (+) (ω)qt(−) (ω) dω · E(t)
2
π
−∞
Z ∞
1
t
2
+
(5.24)
|q (ω)| dω .
2π −∞ (−)
On the other hand observe that the convolutive part of the stress may be written
in terms of such quantities. In fact
Z ∞
Z
Z
1 ∞
1 ∞
t
t
(s) ds =
iG
l̇ s (ω)E+
G(s)E
l̇
(ω) dω = −
G
ľ c (ω)(iωEt+ (ω)) dω
π
π
0
−∞
−∞
Z
1 ∞
=−
G
ľ (+) (ω)(qt(+) (ω) − qt(−) (ω)) dω
π −∞
MAXIMUM RECOVERABLE WORK AND STATE SPACE
Moreover, since both G
ľ (+) (z) and qt(+) (z) are analytic on Ω− we have
Z ∞
G
ľ (+) (ω)qt(+) (ω) dω = 0
−∞
Z
Z ∞
i ∞
t
G
ľ (+) (ω)qt(−) (ω) dω
G(s)E
l̇
(s) ds =
π
0
−∞
Since
<
Z
∞
−∞
iG
ľ (+) (ω)qt(−) (ω) dω
=
Z
∞
−∞
21
(5.25)
t
iG
ľ (+) (ω)q(−)
(ω) dω
substitution of (5.25) into (5.24) and the position
Z ∞
1
1
t
l 0 E(t) · E(t) +
l 0 E(t) · E(t)
S(t) = G
G(s)E
l̇
(s) ds · E(t) = T(t) · E(t) − G
2
2
0
Z ∞
1
(5.26)
|qt (ω)|2 dω
ψm (t) = S(t) +
2π −∞ (−)
that is the expression for the minimum free energy found in [9]. As a consequence,
all the porperties showed in [9] for (5.26) are satisfied by (5.13) too. In particular
it satisfies the definition of the free energy stated by Graffi and the one stated by
Coleman and Owen.
However the method used here to find expression (5.13) will be useful in the next
section to show that that the domain of definition of the minimum free energy is
the whole space of the admissible states defined in Sec. 3.
6. The space of definition of the minimum free energy
The minimum free energy ψm is a function of the state σ ∈ Σw . Moreover,
since pt(+) may be thought as an element of (Γw /Γw0 ), (E(t), pt(+) ) is a direct
representation of the state σ(t) ∈ Σw = Sym × (Γw /Γw0 ).
This can be proved by recalling (3.19)2 and by showing that Ĭ(τ, Ĕt ) = 0, ∀τ > 0
t
if and only if p(+)
(ω) = 0, ∀ω ∈ R. To this aim, observe that (5.5) can be rewritten
as
Z ∞
G(s)
l̇
Ĕt (s − τ ) ds , τ ≥ 0
Ĭ(τ, Ĕt ) = −
τ
A causal extension of Ĕt and an odd extension of G
l̇ provide the following representation of Ĭ in the frequency domain
Z
Z ∞
1 ∞
t (ω)eiωτ dω = − 1
iG
l̇ s (ω)Ĕ+
iω G
ľ c (ω)Ĕt+ (ω)eiωτ dω , τ ≥ 0
Ĭ(τ, Ĕt ) =
π −∞
π −∞
Moreover representation (5.20) ensures that Plemelj formula for iω G
ľ (+) (ω)Ĕt+ (ω)
may be given by
t
where
t
p00 (−) (ω)
iω G
ľ (+) (ω)Ĕt+ (ω) = pt(+) (ω) − p00 (−) (ω)
is a suitable function analytic on Ω+ . Then
Z


1 ∞
t
Ĭ(τ, Ĕt ) = −
G
ľ (−) (ω) pt(+) (ω) − p00 (−) (ω) eiωτ dω
π −∞
Z
1 ∞
= −
G
ľ (−) (ω)pt(+) (ω) eiωτ dω; , τ ≥ 0 ,
π −∞
(6.1)
22
G. GENTILI
t
the second equality holding since G
ľ (−) (ω), p00 (−) (ω), and eiωτ are analytic in the
half plane Ω+ . Consider now the following function
Z
1 ∞
t
t
G
ľ (−) (ω)p(+)
(ω) eiωτ dω; , τ ∈ R
(6.2)
Ĭ (τ ) = −
π −∞
This characterization of Ĭt allows to prove the following theorem
Theorem 6.1. For every viscoelastic material with a symmetric relaxation function, a given strain history Ĕt is equivalent to the zero history 0† in the sense of
(3.19)2 if and only if the quantity pt(+) , related to Ĕt by (6.1), is such that
pt(+) (ω) = 0 ,
∀ω ∈ R
(6.3)
Proof. Using representation (6.2), the theorem in effect states that
Ĭt (τ ) = 0
∀τ ≥ 0
⇐⇒
pt(+) (ω) = 0
∀ω ∈ R
(6.4)
The statement relating to the left arrow of (6.4) follows trivially from (6.2). In
order to prove the statement relating to the right arrow, we can consider (6.2) as
the inverted Fourier transform of
Z ∞
t
t
(6.5)
ľ (−) (ω)p(+) (ω) = −
Ĭt (τ )e−iwτ dτ .
f (ω) = 2 G
−∞
If Ĭt (τ ) = 0 ∀τ ≥ 0, it follows from (6.5) that f t is analytic in Ω+ . The zeros of
G
ľ (−) cannot cancel singularities of pt(+) since all such zeros are by construction in
Ω− . Thus pt(+) must be analytic in Ω+ and therefore in Ω. Since it goes to zero at
infinity, by Liouville’s theorem, it must vanish and (6.3) is proved.
Theorem 6.1 ensures that (E(t), pt(+) ) is a direct representation of the state
σ(t) ∈ Σw and hence ψ̂m (E(t), pt(+) ) = ψm (σ(t)) is a function of the state. More
precisely, we should have to say that (E(t), pt(+) ) describes a state σ(t) ∈ Sm ⊆ Σw
where
Sm := {σ ∈ Σw : ψm (σ) < ∞} .
(6.6)
Actually, any free energy is defined in a proper subset of Σw , namely
S := {σ ∈ Σw : ψ(σ) < ∞} ⊆ Σw ,
(6.7)
where ψ is any free energy such that ψ = 0 when evaluated at the zero history
(0, 0† ). Moreover, since 0 ≤ ψm (σ) ≤ ψ(σ), it follows that S ⊆ Sm ⊆ Σw .
Moreover we can prove the following important result that ensures that every
state of Σw has a bounded minimum free energy.
Theorem 6.2. The space of definition of the minimum free energy is the whole
state space, viz. Sm = Σw .
Proof. Observe that crucial equation in determining the expression of the mnimum
free energy is (5.4), viz.
Z ∞
− s|)Ė(m) (s) ds = Ĭ(τ, Ĕt ) τ ∈ R+ .
(6.8)
G(|τ
ľ
0
MAXIMUM RECOVERABLE WORK AND STATE SPACE
23
Substitution of (6.8) in (5.2) and (5.3) yields
Z Z
1 ∞ ∞
1
− s|)Ė(m) (s) · Ė(m) (τ ) ds dτ
WR (σ) =
G
l ∞ E(t) · E(t) +
G(|τ
ľ
2
2 0
Z ∞0
1
1
(m)
(m)
G
ľ c (ω)Ė+ (ω) · Ė+ (ω) dω
(6.9)
=
G
l ∞ E(t) · E(t) +
2
2π −∞
Therefore WR (σ) induces a natural norm and a natural funcional spaces for the
of the processes P ∈ Π0 , namely


Z ∞
0
ȞG
(R+ , Sym) := ϕ ∈ ȞG (R+ , Sym) :
ϕ(τ ) dτ = 0
(6.10)
0
Moreover, it is worth noting that (6.9) and Theorems 4.2 and 4.4 ensure that
0
ȞG
(R+ , Sym) is dense in HG (R+ , Sym) with respect to the norm k · kG . In fact,
considering the fixed value of Theorem 4.4, equations (4.30) and (4.31) and Theorem
4.2 imply


1
G
l ∞ E(t) · E(t) − W (σ, P ), P ∈ Π0
(6.11)
sup {−W (σ, P ), P ∈ Π} = sup
2
Equations (6.9) and (6.11), jointly with definitions (4.1) and (3.12) ensure that
0
ȞG
(R+ , Sym) and ȞG (R+ , Sym) admit the same completion with respect to the
norm k · kG .
As a consequence, every state σ(t), whose representative strain history (E(t)Ĕt )
0
is such that E(t) ∈ Sym and Ĭ(·, Ĕt ) ∈ HG
(R+ , Sym), belongs to Sm , so that
Σw ⊆ Sm . Since definition (6.6) implies Sm ⊆ Σw , the thesis is proved.
We conclude the section by characterising the states of Σw , and in particular
analysing the type of past strain histories Ĕt belonging to Γw /Γw0 . Remember
that an element of Γw /Γw0 can be directly represented by pt(+) as a consequence of
Theorem 6.1.
0
Proposition 6.3. Given a past strain history Ĕt , then Ĭ(t, Ĕt ) ∈ HG
(R+ , Sym) if
and only if pt(+) ∈ L2 (R, Sym(Ω)).
Proof. For every ϕ ∈ HG (R+ , Sym) we have
Z
Z ∞
G
ľ c (ω)ϕ+ (ω) · ϕ+ (ω) dω =
−∞
+
∞
−∞
2

G
ľ (+) (ω)ϕ+ (ω) dω
Consider now ϕ ∈ HG (R , Sym) and the duality product
Z ∞
Z ∞
1
Ĭ(t, Et ) · ϕ(t) dt =
Ĭt (ω) · ϕ+ (ω) dω
2π −∞ +
0
Z ∞
 >

1
−1
t
=
(ω) · G
G
ľ (−) (ω)Ĭ+
ľ (−) (ω)ϕ+ (ω) dω
2π −∞
Z ∞


1
−1
t
(ω) · G
ľ (+) (ω)ϕ+ (ω) dω
G
ľ (−) (ω)Ĭ+
=
2π −∞
The Plemelj formulae (5.8) and the fact that
Z ∞


ľ (+) (ω)ϕ+ (ω) dω = 0
pt(−) (ω) · G
−∞
24
G. GENTILI
Z ∞
Ĭ(t, Et ) · ϕ(t) dt =
0
≤
Z ∞


1
pt(+) (ω) · G
ľ (+) (ω)ϕ+ (ω) dω
2π −∞
Z ∞
Z ∞
1
1
t
2
ľ (+) (ω)ϕ+ (ω)|2 dω
|p (ω)| dω +
|G
4π −∞ (+)
4π −∞
0
Therefore if pt(+) ∈ L2 (R, Sym(ω)), then Ĭ(·, Et ) ∈ HG
(R+ , Sym).
0
On the other hand if Ĭ(·, Et ) ∈ HG
(R+ , Sym), Theorem 6.2ensures that the
maximum recoverable work is finite and the optimal process Ė(m) ∈ HG (R+ , Sym).
Thus we have
Z ∞
1
(m)
(m)
0≤
G
ľ c (ω)Ė+ (ω) · Ė+ (ω) dω = G
l ∞ E(t) · E(t) − WR (σ) < ∞ (6.12)
2
−∞
∗
ľ (±) = G
ľ (∓) ,
Substitution of (5.11) in (6.12), jointly with (5.7) and the property G
yield
Z ∞
2
 t

p(+) (ω) dω < ∞
t
Therefore Ĭ(·, E ) ∈
−∞
0
HG
(R+ , Sym)
implies pt(+) ∈ L2 (R, Sym(ω)).
7. Recoverable work and Thermodynamics in viscoelasticity
The maximum recoverable work and its properties may be taken as the basis for
the thermodynamics of a viscoelastic material under isothermal condition. From
this point of view, it is of interest to recall the Dissipation Principle of the Mechanical Energy in the form stated by Fabrizio, Giorgi and Morro in [11] that reads:
Dissipation Principle For any state state σ ∈ Σw the set of works done starting
from σ is bounded from below viz.
N (σ) := inf {W (σ, P ) : P ∈ Π } > −∞
(7.1)
In our context,we do not need to assume such a Principle. In fact we can prove
it by virtue of Theorem 6.2 and of a “Weak Dissipation Principle” that is actually a
definition of a new state space , called “space of admissible states” and that reads:
Weak Dissipation Principle There exists a suitable state space ΣN ⊆ Σw , named
as the set of admissible state, whose elements are all and only those states σ ∈ Σw
satisfying (7.1).
In other words
ΣN := {σ ∈ Σw : N (σ) > −∞} .
(7.2)
Observe that, comparing (7.1) with (4.1), it is clear that
WR (σ) = −N (σ) < ∞ .
(7.3)
Thus Definition 4.1 and comparison (7.3) clarify the physical meaning of the
name “admissible state” for a state σ satisfying (7.1). In fact, if (7.1) des not hold,
we might have a state which could provide infinite energy, and this is not physically
MAXIMUM RECOVERABLE WORK AND STATE SPACE
25
Moreover, by virtue of (5.1), it is easy to note that the spaces Sm and ΣN ,
defined rispectively by (6.6) and (7.2), coincide. As a consequence, Theorem 6.2
that ensures Sm = Σw , proves the Dissipation Principle too.
Such a thermodynamic principle, in the form introduced in [11] or in the weaker
form showed above, agrees with the general theory of dissipative dynamical systems
proposed by Willems in [25]. In fact, when a viscoelasic material under isothermal
condition is considered, the general “supply rate” of [25] corresponds to the mechanical power, the general “storage function” S of [25] is the free energy and the
“avaliable storage” S of [25] is the maximum recoverable work (i.e. the minimum
free energy). From this point of view it is worth quoting Theorem 1 of [25] that
Theorem 7.1 (Willems). The avaliable storage (i.e. the maximum recoverable
work) WR (σ) is finite for every σ ∈ Σ if and only if the material is dissipative.
Moreover, 0 ≤ WR (σ) ≤ ψ(σ) (∀σ ∈ Σ) and WR (σ) is itself a possible storage
function (free energy).
This theorem points out how Theorem 4.2 is crucial. In othe words, Theorem
7.1 asserts that assuming the dissipative property of the material is equivalent to
state that the domain of definition of the minimum free energy coincides with the
whole state space. Let us recall how (7.1) leads to the Clausius-Duhem inequality.
Remember that WR satisfies the following property
Proposition 7.2. The maximum recoverable work WR (σ) is non negative for every
σ ∈ Σm .
The proof is an immediate consequence of (4.1), reminding that at every state σ
it is possible to apply the null process yielding a null work.
We recall the following assumption that is quite natural at least for simple materials and in particular for viscoelastic materials (see[11]).
Assumption 7.3. There exists a “zero state” σ0 , such that WR (σ0 ) = 0.
Actually, as pointed out in [11], for viscoelastic material σ0 is the state related
to the identically vanishing strain (0, 0† ).
The fundamental principles of thermodynamics follows from the Dissipation Principle and Assumption 7.3. In fact, from (7.1) and (4.1) follows that for each fixed
pair σ ∈ Σw , P1 ∈ Π we have Π1 := {P = P 0 ∗ P1 , P 0 ∈ Π} ⊂ Π so that
WR (σ)
≥ sup {−W (σ, P 0 ∗ P1 ) , P 0 ∈ Π }
= −W (σ, P1 ) + sup {−W (P1 σ, P 0 ) , P 0 ∈ Π } = −W (σ, P1 ) + WR (P σ)
Therefore, for any σ1 , σ2 ∈ Σw and P ∈ Π such that P σ1 = σ2 , we have
WR (σ2 ) − WR (σ1 ) ≤ W (σ1 , P )
(7.4)
W (σ, P ) ≥ 0
(7.5)
As a consequence, for every cycle, that is a pair σ ∈ Σw and P ∈ Π such that
P σ = σ, (7.4) implies
that is the Second Principle of Thermodynamics in the formulation of Clausius.
Instead Assumption 7.3 implies the Dissipation Principle of Gurtin and Herrera
[22]. In fact from Assumption 7.3 and (4.1) follows that
W (σ0 , P ) , ≥ 0
∀P ∈ Π ,
26
G. GENTILI
as stated by the above mentioned Principle.
Inequality (7.5), implies that there exists a function of state ψ(σ), called free
energy that is a lowerpotential such that
ψ(σ2 ) − ψ(σ1 ) ≤ W (σ1 , P )
(7.6)
D(σ1 , P ) := W (σ1 , P ) − ψ(σ2 ) + ψ(σ1 ) ≥ 0 .
(7.7)
ψm (σ) = WR (σ) ,
(7.8)
That is the integrated Clausius -Duhem inequality for isothermal processes2 . Comparing (7.4) with (7.6), it is clear that WR is a free energy
We conclude the section discussing on the physical meaning of the quantity
In fact D is usually called as the dissipation of the material, so that a process P ,
acting on σ, should be reversible (i.e. non dissipative) if and only if D(σ, P ) = 0
and a material should be non dissipative if and only if D(σ, P ) = 0 for every σ ∈ Σ
and P ∈ Π.
However this way of relate the dissipation to the Clausius-Duhem inequality (7.6)
fails when memory effects occur, as pointed out in [12]. This is substantially due
to the non uniqueness of the free energy as a functional satisfying (7.6), whereas
the dissipation should be an osservable uniquely determined. Moreover we can find
a counterexample of a suitable free energy satisfying (7.6) and (7.7) as an equality
even if the material with memory is dissipative. This is the case of the maximum
free energy defined in [11].
Even in this situation the maximum recoverable work show an interesting property. In fact inequality (7.4), unlike (7.6), is uniquely fixed and it represents the
dissipation of the material by virtue of the definition of the maximum recoverable
work as the amount of avaliable energy. In fact (7.4) can be interpreted as follows:
the sum given by the energy provided to the material by mean of the work W (σ1 , P )
added to the avaliable energy at the state σ1 is not less the than avaliable energy at
the state P σ1 = σ2 . The difference is the part of W (σ1 , P ) that cannot be utilized
anymore, that is the dissipated energy. Therefore, denoting with Di the function
defined in (7.7) when ψi is the free energy involved, and calling ψm the free energy
given by the maximum recoverable work, viz.
in view of (7.4), (7.7) and (7.8) the quantity
Dm (σ1 , P ) := W (σ1 , P ) − ψm (σ2 ) + ψm (σ1 ) ≥ 0 .
represents the dissipation of the material in the sense outlined above.
Thus, when memory effects occur, the Clausius-Duhem inequality is still related
to the dissipation of the material provided the involved free energy is the one given
by the maximum recoverable work.
Appendix: Factorization of the integral kernel
The solution of the Wiener-Hopf equation (5.4) relies crucially on the factorization of the integral kernel. The aim of this appendix is recalling the results of
Gohberg and Kreı̆n [16] and Deseri, Gentili and Golden [9], which ensure such a
factorization. The results in [16] apply to hermitean matrices of arbitrary finite
dimension but we state them for the case of 6 × 6 matrices, moreover, as recalled
2 However inequality (7.6) allows many functionals to be a free energy, especially for a material
with memory, and they do not differ just for an additive constant
MAXIMUM RECOVERABLE WORK AND STATE SPACE
27
in Sec. 2, they can be easily extended to tensors belonging to Lin(Sym(Ω)). The
result in [9], shows sufficient conditionsfor the factorization of the integral kernel
G.
ľ
Definition A.1. A non-singular continuous function IK : R → Lin(Sym(Ω)),
has a left [right] factorization if, for every ω ∈ R, it can be represented in the form


IK(ω) = IK(+) (ω) IK(−) (ω)
IK(ω) = IK(−) (ω) IK(+) (ω)
(A.1)
where the matrix functions IK(±) admit analytic continuations, holomorphic in the
interior and continuous up to the boundary of the corresponding complex half planes
Ω± , and are such that
det IK(±) (ζ) 6= 0
,
ζ ∈ Ω±
(A.2)
Definition A.2. A non-singular continuous function IK : R → Lin(Sym(Ω)),
−
belongs to <6×6 , [<+
l 0 and a matrix
6×6 ], [[<6×6 ]] if there exists a constant matrix C
function IF(t) such that
Z ∞
∀ω ∈ R
(A.3)
IK(ω) = C
l0+
IF(t)eiωt dt ,

l0+
IK(ω) = C
Z
0
∞

iωt
IF(t)e dt ,
−∞

l0+
IK(ω) = C
Z
0
IF(t)e
−∞
iωt
dt

(A.4)
Note that if IK ∈ <±
6×6 , it has the analytic properties ascribed to IK(±) above.
The main result we use is Theorem 8.2 of [16], that can be stated in our context as
follows:
Theorem A.3 (Gohberg – Kreı̆n). In order that the non-singular (hermitian) matrix function IK ∈ <6×6 possesses a representation of the form
IK(ω) = IK(+) (ω) IK∗(+) (ω)
(A.5)
+
in which the matrix function IK(+) ∈ <6×6
and satisfies det IK(+) (ζ) 6= 0 for
+
ζ ∈ Ω , it is necessary and sufficient that IK(ω) be positive definite for every
−∞ ≤ ω ≤ ∞.
Comparison of (A.5) with the first relation of (A.1) yields
IK(−) (ω) = IK∗(+) (ω)
(A.6)
Recalling that any fourth order symmetric tensor maps into a 6 × 6 matrix under
the isomorphism discussed at the beginning of Section ??, let us consider for each
given ω ∈ R the fourth order tensor G
ľ c (ω) ∈ Lin(Sym) defined by(2.6). By virtue
ľ
of (2.7) and the assumption that G(t)
is symmetric, it follows that, G
ľ c (ω) is a real
valued, symmetric, positive definite tensor. Moreover, since
l̇ s (ω) = − G(0),
ľ c (ω) = − lim ω G
l̇
lim ω 2 G
ω→∞
ω→∞
(A.7)
it follows from (2.9) that the tensor
IK(ω) := (1 + ω 2 ) G
ľ c (ω) ,
(A.8)
is symmetric and positive definite for every −∞ ≤ ω ≤ ∞.
The result in [9] ensure that IK ∈ <6×6 , i.e. that the representation (A.3)
applies to IK.
28
G. GENTILI
ľ and G
Proposition A.4 (Deseri-Gentili-Golden). If G
l̈ are integrable tensor valued functions, and the tensor G(0)
l̇
is finite and non-singular, the tensor valued
function IK, related to G
l though (A.8), belongs to <6×6 .
Proof. Observe that integration by parts yields
l̇
ω2 G
ľ c (ω) = −ω G
l̇ s (ω) = − G(0)
−G
l̈ c (ω) ,
(A.9)
h
i
IK(ω) = − G(0)
l̇
−G
l̈ c (ω) + G
ľ c (ω) .
(A.10)
so that (A.8) becomes:
Consider now the tensors
i
1h
− G(t)
+ G(t)
ľ
,t ∈ R
(A.11)
l̈
2
where G
ľ and G
l̈ are extended on the real line as even functions, so that G
ľ F = 2 G
ľ c
l̈ F = 2 G
l̈ c . Then, in view of (A.11), (A.10) is equivalent to (A.3) and the
and G
assertion is proved.
2
Theorem A.3, Proposition A.4 and suitable hypotheses on G
l̈ ensure that IK(ω)
has a representation of the form (left factorization)
Cl 0 = − G(0),
l̇
IF(t) =
IK(ω) = IK(+) (ω) IK(−) (ω)
with IK(−) (ω) = IK∗(+) (ω), IK(+) (ω) ∈ <+
6×6 and
det IK(+) (ζ) 6= 0
for
ζ ∈ Ω+
(A.12)
(A.13)
Moreover such a factorization is unique up to a post-multiplication on the right by
a constant unitary tensor (see Remark on page 253 of [16]).
As a consequence G
ľ c (ω) can be factorized too. In fact we have
ľ (+) (ω) G
ľ (−) (ω)
G
ľ c (ω) = G
(A.14)
where
G
ľ (±) (ω) =
1
IK(±) (ω) .
1 ± iω
(A.15)
ľ (±)
Observe that, following the convention used in [17], both for IK(±) and for G
the sign indicates the half plane where any singularities of the tensors IK(±) (ζ) and
G
ľ (±) (ζ), ζ ∈ Ω and any zeros in the determinant of the corresponding matrices
may occur.
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Dipartimento di Matematica, Università di Bologna, P.za di P.ta S. Donato 5, 40126
Bologna, Italy