J Elast (2010) 101: 59–67 DOI 10.1007/s10659-010-9244-y
Energy Splitting Theorems for Materials with Memory Antonino Favata · Paolo Podio-Guidugli · Giuseppe Tomassetti
Received: 29 July 2009 / Published online: 2 March 2010 © Springer Science+Business Media B.V. 2010
Abstract We extend to materials with fading memory and materials with internal variables a result previously established for materials with instantaneous memory: the additive decomposability of the total energy into an internal and a kinetic part, and a representation of the latter and the inertial forces in terms of one and the same mass tensor. Keywords Internal energy · Kinetic energy · Simple materials · Fading memory · Internal variables Mathematics Subject Classification (2000) 74A20 · 74D99
1 Introduction The purpose of this paper is to extend to two classes of materials with memory a result established in [6] for materials that, as exemplified by standard thermoelastic materials, can only respond to the current values of their state variables. The result we aim to extend is called in [6] the Energy Splitting Theorem: it is shown that the total energy and the inertia force have consistent representations, under the assumptions that (i) the power expenditure of the inertia force be linear in the velocity; and that (ii) the inertial power plus the rate of change of the energy be translationally invariant. More precisely, it is shown that the energy can be split in two parts, internal and kinetic, with the internal energy independent of velocity and the kinetic energy a quadratic form in the A. Favata · P. Podio-Guidugli · G. Tomassetti () Dipartimento di Ingegneria Civile, Università di Roma Tor Vergata, Via Politecnico 1, 00133 Rome, Italy e-mail:
[email protected] A. Favata e-mail:
[email protected] P. Podio-Guidugli e-mail:
[email protected]
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velocity, based on a time-independent mass tensor, the same that determines also the workeffective part of the inertial force. The two material classes we here consider are: the class of simple materials in the sense of Truesdell and Noll [8], whose mechanical response is determined by the history of the deformation gradient; and the class of materials with internal state variables, as considered, e.g., by Coleman and Gurtin [1] and Lubliner [5], whose evolution is governed by a generally nonlinear differential equation (that the Energy Splitting Theorem had to be extendable to this material class was suggested by M.E. Gurtin in 1994, on reading a preprint of [6]). Since these two material classes have a nonempty intersection but do not overlap, we are obliged to prove the entry part of our generalized Energy Splitting Theorem twice; we give the reasons for this at the end of next section. Luckily, as we shall see, the rest of the proof is not as sensitive to the chosen class. Our paper is organized as follows. In Sect. 1, we introduce the quantities that are subject to a constitutive prescription, we stipulate their invariance properties, and we summarize the Energy Splitting Theorem which we aim to generalize. In Sect. 2, we provide a constructive proof of the Energy Splitting Theorem under the assumption that the constitutive functionals be smooth relative to a norm having the fading-memory property. In Sect. 3, we sketch a proof of the Energy Splitting Theorem for materials with internal variables. Apart for some technicalities that we try to explain as they arise, the structure of the proofs we give is the same as the variant of the proof in [6] given in [7].
2 Setting the Stage We work in a referential setting. At any given body point (in this paper, we leave all space dependencies tacit), we introduce scalar volume densities of the (total) energy, denoted by τ , and of the inertial power: π in = din · v, where v is the velocity vector and din is the inertia force vector. Next, we define the internal power density to be α = τ˙ + π in
(1)
(a superposed dot denotes time differentiation). Both τ and din are constitutively prescribed at a later stage. For now, it suffices for us to stipulate that, in principle, they both depend on one and the same list of state variables, that we split as follows: (, v), where the list includes only translationally invariant variables. Precisely, a translational change in observer is a mapping leaving the time line unchanged: (t, x) → (t, x)+ = (t, x + ), such that, at some fixed time t¯, the current shape of the body under study is pointwise preserved, while the velocity field varies by a uniform amount w: x → x + = x + (t − t¯)w, v → v+ = v + w. Thus, as to state-variable pairs, (, v) → (, v)+ = (, v + w).
(2)
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In the next two sections, we generalize the following result in [6]. Energy Splitting Theorem Let the inertial power be linear in the velocity, in the sense that there is a mapping → dˆ 0 (), referred to as the work-effective inertia force mapping, such that dˆ in (, v) · v = dˆ 0 () · v,
(3)
for every state and velocity v. Moreover, let the internal power be invariant under translational changes in observer: ˆ v). α( ˆ + , v+ ) = α + = α = α(,
(4)
Finally, let the constitutive functions dˆ in (, v) and τˆ (, v) be, respectively, continuous and twice-continuously differentiable. Then, the energy τ and the inertial force din have consistent representations, parameterized by (i) the mass tensor, a symmetric tensor M, independent of (, v) and obeying the mass conservation law ˙ = 0; M (ii) the internal energy, a scalar-valued mapping → ˆ () defined over the state space. These representations are: 1 τˆ (, v) = ˆ () + v · Mv, 2 in dˆ (, v) = −M˙v + Din (, v)v,
(5) (6)
with Din (, v) a skew-symmetric tensor. Note that (1), (3), and (2) imply that the invariance requirement (4) takes the form: d (t), vˆ (t) + w − τˆ (t), vˆ (t) + dˆ 0 (t) · w = 0, τˆ dt
(7)
(t), vˆ (t)) and for every vector w. With this in mind, we for every constitutive process t → ( are in a position to indicate why, in the last part of the Introduction, we stated that the entry part of the proof of a theorem of this sort depends on the material class for which it is meant to hold: the first and crucial step in the proof is to achieve a preliminary additive splitting of the energy into an internal part, that does not depend on velocity, and a kinetic part. To take that step, it is necessary to compute the derivative of τˆ with respect to its first argument. This is easy in the case considered in [6]. It is not so when, as we here do, the constitutive is replaced dependence of energy and work-effective inertia force on the current value of up to time t or by the current value of either by a functional dependence on the history of ˆ with βˆ a solution of the (generally nonlinear) ordinary differential equation , β), the pair ( governing the time evolution of a chosen list of internal variables β: β˙ = f(, β). Both replacements entail a rethinking of the structure of state space, as to the accessibility of its points and, more importantly, as to the possibility of giving any process an arbitrary short continuation in time. We will discuss these technical issues at the appropriate stage of our developments.
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3 Materials with Fading Memory Our proof of an Energy Splitting Theorem for fading-memory materials is constructive, and is organized in five steps. Step 1. Translational Invariance of the Internal Power. We let the space of the translationally invariant state variables be a open set C of a finite-dimensional inner product space L, and a velocity process vˆ we and we let V denote the velocity space. By a state process , its mean a smooth differentiable curve in C and V , respectively. Given a state process history up to time t is the mapping t : [0, +∞) → C,
(t − s); t (s) :=
its past history up to time t is the restriction tr of t to the open half-line (0, +∞); and (t); needless to say, t and ( (t), tr ) carry the same its instantaneous value is t (0) = information. To describe the energy and inertia-force response of materials with fading memory to (state,velocity) processes, we introduce three constitutive functionals, τ˜ for the energy and d˜ in , d˜ 0 for the inertia force; we set: (t), tr , vˆ (t) , τ˜v (t) := τ˜ t ˜ in ˆ (t) , (8) d˜ in v (t) := d (t), r , v t (t), r ; d˜ 0 (t) := d˜ 0 and we consistently rephrase assumption (3): ˆ (t) = d˜ 0 (t) · vˆ (t). d˜ in v (t) · v
(9)
Consequently, the invariance requirement (7) can now be written formally as follows: d τ˜v+w (t) − τ˜v (t) + d˜ 0 (t) · w = 0; dt
(10)
for this to have a precise mathematical sense, we have to specify the regularity of the functionals τ˜ and d˜ 0 . Step 2. Fading-Memory Property and Chain-Rule Formula. For h : (0, +∞) → R+ a nonnegative measurable function, chosen once and for all and such that
+∞
|h(s)|2 ds < +∞,
0
we denote by Lr the Banach space of all measurable functions r : (0, +∞) → L, with the norm ∞ 12 r = h(s)|r (s)|2 ds . (11) 0
Two histories are close in the topology determined by the norm · if their values are close in the recent past no matter how far apart they are in the distant past. Thus, since the
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in the distant past does not constitutive mappings are smooth, changing the history of affect appreciably the instantaneous values of τ and din . In Coleman and Noll’s terminology [3, 8], introducing the norm · endows a material class with the Fading-Memory Property. We let the domain of the functional d˜ 0 be an open subset C of the Banach space L = L ⊕ Lr , and we let the common domain of functionals τ˜ and d˜ in be C ⊕ V . Moreover, we require that d˜ in be continuous and that τ˜ be twice-continuously Fréchet differentiable. As pointed out in Remark 1 of [2], the continuous differentiability of τ˜ and the smooth(t) guarantee that the time derivatives in (10) are well defined, and that the ness of t → following Chain-Rule Formula holds true: d ˙ (t), tr , v(t)) · (t) τ˜v (t) = ∂ τ˜ ( dt ˙ tr ] + ∂v τ˜ ( (t), tr , v(t))[ (t), tr , v(t)) · v˙ (t). + δr τ˜ ( ˙ (t) and v˙ (t) = d vˆ (t); moreover, ∂ τ˜ and ∂v τ˜ are the partial derivatives Here, (t) = dtd dt of τ˜ with respect to its first and third argument, respectively, and δr τ˜ is the unique bounded linear functional on Lr satisfying τ˜ (, tr + r , v) = τ˜ (, tr , v) + δr τ˜ (, tr , v)[ r ] + o( r ), for all r in Lr . By combining the invariance requirement (10) with the Chain-Rule Formula, we obtain an expression involving both the instantaneous value and the past history ˆ of the time derivative of the state process : ˙ (t), tr , vˆ (t) + w) − ∂ τ˜ ( (t), tr , vˆ (t)) · (t) ∂ τ˜ ( ˙ tr ] (t), tr , vˆ (t) + w) − δr τ˜ ( (t), tr , vˆ (t)) [ + δr τ˜ ( (t), tr , vˆ (t) + w) − ∂v τ˜ ( (t), tr , vˆ (t)) · v˙ (t) + ∂v τ˜ ( (t), tr ) · w = 0. + d˜ 0 (
(12)
Step 3. Energy Splitting. As in [6] for instantaneous-memory materials, we achieve the desired result for materials with fading memory by showing that ∂ τ˜ does not depend on v. and an arbitrary The argument used in [6] relies on the existence, given a state process (t), and whose rate
in L, of a state process whose instantaneous value coincides with ˙ with (t) equals ; by replacing , and by invoking the arbitrariness of , one deduces ˙ that the term multiplying (t) in the first line of (12) must vanish, and then concludes that ∂ τ˜ does not depend on v. The argument we use in the present proof is similar, and is based ∈ L and ∈ L, and on a result due to Coleman and Mizel [2, Remark 2]: for any given for every positive ε, there is a state process ε such that (t) − ε (t)| < ε |
and
ε
˙ = , (t)
whose past history ε rt satisfies tr − ε rt < ε,
˙ tr − ε ˙ rt < ε.
with ε Replacing in (12), letting ε vanish, and using the smoothness of τ˜ and d˜ in , we obtain that (t), tr , vˆ (t) + w) − ∂ τ˜ ( (t), tr , vˆ (t)) · + · · · = 0 ∂ τ˜ (
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(the dots stand for the remaining terms of (12)), whence, by the arbitrariness of , it follows that, for all w in V , (t), tr , vˆ (t) + w) − ∂ τ˜ ( (t), tr , vˆ (t)) = 0, ∂ τ˜ ( i.e., that ∂ τ˜ does not depend on velocity. We conclude that, for every triplet (, tr , v) of a state , a past history tr , and a velocity v, the constitutive mapping that delivers the total energy splits as follows: τ˜ (, tr , v) = (, ˜ tr ) + κ( ˜ tr , v).
(13)
Step 4. Representation of the Kinetic Energy. In view of (13), relation (12) becomes: ˙ tr ] ˜ tr , vˆ (t) + w) − δr κ( ˜ tr , vˆ (t)) [ δr κ( (t), tr ) · w = 0, ˜ tr , vˆ (t) + w) − ∂v κ( ˜ tr , vˆ (t)) ·˙v(t) + d˜ 0 ( (14) + ∂v κ( for all w in V . Moreover, since both mappings δr τ˜ and ∂v τ˜ are continuously-differentiable by assumption, we have that ∂v (δr τ˜ (, tr , v)[ r ]) = δr (∂v τ˜ (, tr , v))[ r ],
(15)
for every triplet (, tr , v) of a state , a past history tr , and a velocity v, and for every r in Lr .1 On differentiating (14) with respect to w and using (15), we obtain: ˙ tr ] + ∂vv κ( (t), tr ) = 0, δr ∂v κ( ˜ tr , vˆ (t) + w)[ ˜ tr , vˆ (t) + w)˙v + d˜ 0 (
(16)
for all w in V ; upon choosing w = −ˆv(t) in (16), we have that ˙ tr ] + ∂vv κ( (t), tr ) = 0; δr ∂v κ( ˜ tr , 0)[ ˜ tr , 0)˙v(t) + d˜ 0 (
(17)
and, on subtracting (17) from (16) and selecting w = 0, we obtain: ˙ tr ] + ∂vv κ( δr ∂v κ( ˜ tr , vˆ (t)) − δr ∂v κ( ˜ tr , 0) [ ˜ tr , vˆ (t)) − ∂vv κ( ˜ tr , 0) v˙ (t) = 0. (18) Finally, since the choice of vˆ (t) and v˙ (t) in (18) is arbitrary, we have that (18) itself is equivalent to ˙ tr ] = 0 ˜ tr , v) − δr ∂v κ( ˜ tr , 0) [ (19) δr ∂v κ( and
tr ), ˜ tr , v) = M( ∂vv κ(
(20)
tr ) is a second-order symand for every velocity v; note that M( for every past history metric tensor. An integration of (20) yields: tr
1 t ˜ tr ) · v + μ( ˜ tr ). κ( ˜ tr , v) = v · M( r )v + m( 2
(21)
1 This identity generalizes the theorem on the inversion of the order of partial differentiation for functions of
real variables [4].
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We incorporate the constant part μ( ˜ tr ) of κ( ˜ tr , ·) in the internal energy (·, ˜ tr ), without affecting the splitting (13); and we dispose of the linear part by requiring that the total energy ˜ tr ) ≡ 0. be an even function of the velocity, an assumption that, with (13), entails that m( With these measures, we obtain the following representation for the kinetic energy: 1 t κ( ˜ tr , v) = v · M( r )v 2
(22)
and, on combining (13) and (22), we arrive at 1 t τ˜ (, tr , v) = ˜ (, tr ) + v · M( r )v, 2
(23)
an additive energy splitting that generalizes the one in (5). Step 5. Mass Conservation Law and Representation of the Inertial Force. (22) into (19), we obtain that, for all v ∈ V ,
By substituting
t
tr )v [ ˙ r ] = 0, δr M(
that is, in view also of the Chain Rule Formula, that mass is conserved in the following general sense: d t M(r ) = 0, dt
d t
tr )[ ˙ tr ]. M(r ) := δr M( dt
(24)
It remains for us to substitute (22) in (17). This yields:
tr )˙v. d˜ 0 (, tr ) = −M(
(25)
With (25) and assumption (9) of linearity in the velocity of the power expenditure of the inertia force, we obtain a representation of the inertia force that generalizes (6):
tr )˙v +
d˜ in (, tr , v) = −M( Din (, tr , v)v,
(26)
where
Din (, tr , v) is a skew-symmetric tensor. We are now in position to summarize our findings under form of our Energy Splitting Theorem for Materials with Fading Memory Let the constitutive dependence of energy and inertia force be specified by (8)1,2 , with the mappings τ˜ and d˜ in presumed to be, respectively, twice-continuously differentiable and continuous and with the set of past histories endowed with the fading memory norm defined in (11). Assume that: (i) the inertial power satisfies (9); (ii) the internal power satisfies the invariance requirement (10); (iii) the energy mapping is an even function of its third argument. Then, the mappings τ˜ and d˜ in admit the representations (23) and (26), parameterized by
which a scalar internal-energy mapping ˜ and a symmetric-valued mass-tensor mapping M, obeys the conservation law (24)1 .
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Remark 1 Assumption (iii)—which is not stated in [6] and is not needed to arrive at (20)—is instrumental in ruling out the linear term in (21). The same can be achieved by the strengthened invariance requirement that the internal power be invariant under time-dependent translational changes of observer: if (2)2 is accordingly replaced by v → v+ = v + w(t), then, as a bonus, one can also show that the internal energy cannot depend on acceleration, a dependence that would be incompatible with the Second Law of Thermodynamics [6].
4 Materials with Internal Variables For materials with internal variables, the total energy, the inertia force, and the workeffective part of the inertia force are given by ˆ (t), β(t), vˆ (t) , τˇv (t) := τˇ ˆ ˇ in ˆ (t) , dˇ in v (t) := d (t), β(t), v ˆ (t), β(t) dˇ 0 (t) := dˇ 0 ,
(27)
ˆ where β(t) is a translationally-invariant vector of internal variables, whose evolution is governed by the ordinary differential equation: d ˆ ˆ (t), β(t)). β(t) = f( dt
(28)
By (27) and (28), the invariance statement (7) leads to ˆ ˆ ˙ (t), β(t), (t), β(t), ∂ τˇ ( vˆ (t) + w) − ∂ τˇ ( vˆ (t)) · (t) ˆ ˆ ˆ (t), β(t), (t), β(t), (t), β(t)) vˆ (t) + w) − ∂β τˇ ( vˆ (t)) · f( + ∂β τˇ ( ˆ ˆ (t), β(t), (t), β(t), vˆ (t) + w) − ∂v τˇ ( vˆ (t)) · v˙ (t) + ∂v τˇ ( ˆ (t), β(t)) + dˇ 0 ( · w = 0.
(29)
˙ By a free-continuation argument borrowed from [6], the time derivative (t) appearing in the first line of (29) can be replaced by an arbitrary rate ; the resulting relation allows one to conclude that ˆ ˆ (t), β(t), (t), β(t), vˆ (t) + w) − ∂ τˇ ( vˆ (t)) = 0, ∂ τˇ ( for all w in V , whence the energy splitting: ˆ ˆ ˆ (t), β(t), (t), β(t)) τˇ ( vˆ (t)) = ( ˇ + κ( ˇ β(t), vˆ (t)). From this point on, the proof proceeds along the steps listed in the previous section, with ˆ β(t) in the place of tr . The conclusions are, mutatis mutandis, the same as those stated in
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the Energy Splitting Theorem for materials with fading memory: the total energy and the inertial force admit the representations: 1 ˇ τˇ (, β, v) = (, ˇ β) + v · M(β)v, 2 ˇ ˇ in (, β, v)v, dˇ in (, β, v) = −M(β)˙ v+D ˇ satisfies the ˇ ˇ in (, β, v) skew; and the mass-tensor mapping M with M(β) symmetric and D conservation law d ˇ ˆ M β(t) = 0. dt Acknowledgements We thank an anonymous referee for pointing out to us that some justification was needed to rule out the linear term in (21). G.T. gratefully acknowledges the financial support of INdAMGNFM (Research Project: “Modellazione fisico-matematica di materiali e strutture intelligenti”).
References 1. Coleman, B.D., Gurtin, M.: Thermodynamics with internal state variables. J. Phys. Chem. 47, 597–613 (1967) 2. Coleman, B.D., Mizel, V.J.: A general theory of dissipation in materials with memory. Arch. Ration. Mech. Anal. 27, 255–274 (1967) 3. Coleman, B.D., Noll, W.: An approximation theorem for functionals, with applications in continuum mechanics. Arch. Ration. Mech. Anal. 6, 355–370 (1960) 4. Graves, L.M.: Riemann integration and Taylor’s theorem in general analysis. Trans. Am. Math. Soc. 29, 163–177 (1927) 5. Lubliner, J.: On fading memory in materials with evolutionary type. Acta Mech. 8, 75–81 (1969) 6. Podio-Guidugli, P.: Inertia and invariance. Ann. Mat. Pure Appl. CLXXII, 103–124 (1997) 7. Podio-Guidugli, P.: Sparse notes in thermodynamics (2009, forthcoming) 8. Truesdell, C., Noll, W.: The non-linear field theories of mechanics. In: Flugge, S. (ed.) Encyclopedia of Physics, vol. III/3. Springer, Berlin (1965)
