Energy Splitting Theorems for Materials with Memory

Transcription

1 J Elast : DOI /s y Energy Splitting Theorems for Materials with Memory Antonino Favata Paolo Poio-Guiugli Giuseppe Tomassetti Receive: 29 July 2009 / Publishe online: 2 March 2010 Springer Science+Business Meia B.V Abstract We exten to materials with faing memory an materials with internal variables a result previously establishe for materials with instantaneous memory: the aitive ecomposability of the total energy into an internal an a kinetic part, an a representation of the latter an the inertial forces in terms of one an the same mass tensor. Keywors Internal energy Kinetic energy Simple materials Faing memory Internal variables Mathematics Subject Classification A20 74D99 1 Introuction The purpose of this paper is to exten to two classes of materials with memory a result establishe in [6] for materials that, as exemplifie by stanar thermoelastic materials, can only respon to the current values of their state variables. The result we aim to exten is calle in [6] the Energy Splitting Theorem: it is shown that the total energy an the inertia force have consistent representations, uner the assumptions that i the power expeniture of the inertia force be linear in the velocity; an 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 an kinetic, with the internal energy inepenent of velocity an the kinetic energy a quaratic form in the A. Favata P. Poio-Guiugli G. Tomassetti Dipartimento i Ingegneria Civile, Università i Roma Tor Vergata, Via Politecnico 1, Rome, Italy tomassetti@ing.uniroma2.it A. Favata favata@ing.uniroma2.it P. Poio-Guiugli ppg@uniroma2.it

2 60 A. Favata et al. velocity, base on a time-inepenent mass tensor, the same that etermines also the workeffective part of the inertial force. The two material classes we here consier are: the class of simple materials in the sense of Truesell an Noll [8], whose mechanical response is etermine by the history of the eformation graient; an the class of materials with internal state variables, as consiere, e.g., by Coleman an Gurtin [1] an Lubliner [5], whose evolution is governe by a generally nonlinear ifferential equation that the Energy Splitting Theorem ha to be extenable to this material class was suggeste by M.E. Gurtin in 1994, on reaing a preprint of [6]. Since these two material classes have a nonempty intersection but o not overlap, we are oblige to prove the entry part of our generalize Energy Splitting Theorem twice; we give the reasons for this at the en of next section. Luckily, as we shall see, the rest of the proof is not as sensitive to the chosen class. Our paper is organize as follows. In Sect. 1, we introuce the quantities that are subject to a constitutive prescription, we stipulate their invariance properties, an we summarize the Energy Splitting Theorem which we aim to generalize. In Sect. 2, we provie a constructive proof of the Energy Splitting Theorem uner the assumption that the constitutive functionals be smooth relative to a norm having the faing-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] givenin[7]. 2 Setting the Stage We work in a referential setting. At any given boy point in this paper, we leave all space epenencies tacit, we introuce scalar volume ensities of the total energy, enote by τ, an of the inertial power: π in = in v, where v is the velocity vector an in is the inertia force vector. Next, we efine the internal power ensity to be α = τ + π in 1 a superpose ot enotes time ifferentiation. Both τ an in are constitutively prescribe at a later stage. For now, it suffices for us to stipulate that, in principle, they both epen on one an the same list of state variables, that we split as follows:, v, where the list inclues only translationally invariant variables. Precisely, a translational change in observer is a mapping leaving the time line unchange: t, x t, x + = t, x +, such that, at some fixe time t, the current shape of the boy uner stuy is pointwise preserve, while the velocity fiel varies by a uniform amount w: x x + = x + t tw, v v + = v + w. 2 Thus, as to state-variable pairs,, v, v + =, v + w.

3 Energy Splitting Theorems for Materials with Memory 61 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 ˆ 0, referre to as the work-effective inertia force mapping, such that ˆ in, v v = ˆ 0 v, 3 for every state an velocity v. Moreover, let the internal power be invariant uner translational changes in observer: ˆα +, v + = α + = α =ˆα,v. 4 Finally, let the constitutive functions ˆ in, v an ˆτ,v be, respectively, continuous an twice-continuously ifferentiable. Then, the energy τ an the inertial force in have consistent representations, parameterize by i the mass tensor, a symmetric tensor M, inepenent of, v an obeying the mass conservation law Ṁ = 0; ii the internal energy, a scalar-value mapping ˆɛ efine over the state space. These representations are: with D in, v a skew-symmetric tensor. ˆτ,v =ˆɛ + 1 v Mv, 2 5 ˆ in, v = M v + D in, vv, 6 Note that 1, 3, an 2 imply that the invariance requirement 4 takes the form: ˆτ t, ˆvt + w ˆτ t, ˆvt + ˆ 0 t w = 0, 7 t for every constitutive process t t, ˆvt an for every vector w. With this in min, we are in a position to inicate why, in the last part of the Introuction, we state that the entry part of the proof of a theorem of this sort epens on the material class for which it is meant to hol: the first an crucial step in the proof is to achieve a preliminary aitive splitting of the energy into an internal part, that oes not epen on velocity, an a kinetic part. To take that step, it is necessary to compute the erivative of ˆτ with respect to its first argument. This is easy in the case consiere in [6]. It is not so when, as we here o, the constitutive epenence of energy an work-effective inertia force on the current value of is replace either by a functional epenence on the history of up to time t or by the current value of the pair, ˆβ, with ˆβ a solution of the generally nonlinear orinary ifferential equation 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 an, more importantly, as to the possibility of giving any process an arbitrary short continuation in time. We will iscuss these technical issues at the appropriate stage of our evelopments.

4 62 A. Favata et al. 3 Materials with Faing Memory Our proof of an Energy Splitting Theorem for faing-memory materials is constructive, an is organize 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-imensional inner prouct space L, an we let V enote the velocity space. By a state process an a velocity process ˆv we mean a smooth ifferentiable curve in C an V, respectively. Given a state process, its 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 t r of t to the open half-line 0, + ; an its instantaneous value is t 0 = t; neeless to say, t an t, t r carry the same information. To escribe the energy an inertia-force response of materials with faing memory to state,velocity processes, we introuce three constitutive functionals, τ for the energy an in, 0 for the inertia force; we set: an we consistently rephrase assumption 3: τ v t := τ t, t r, ˆvt, in v t := in t, t r, ˆvt, 8 0 t := 0 t, r t ; in v t ˆvt = 0 t ˆvt. 9 Consequently, the invariance requirement 7 can now be written formally as follows: τv+w t τ v t + 0 t w = 0; 10 t for this to have a precise mathematical sense, we have to specify the regularity of the functionals τ an 0. Step 2. Faing-Memory Property an Chain-Rule Formula. For h : 0, + R + a nonnegative measurable function, chosen once an for all an such that + 0 hs 2 s<+, we enote by L r the Banach space of all measurable functions r : 0, + L, with the norm 1 2 r = hs r s s Two histories are close in the topology etermine by the norm if their values are close in the recent past no matter how far apart they are in the istant past. Thus, since the

5 Energy Splitting Theorems for Materials with Memory 63 constitutive mappings are smooth, changing the history of in the istant past oes not affect appreciably the instantaneous values of τ an in. In Coleman an Noll s terminology [3, 8], introucing the norm enows a material class with the Faing-Memory Property. We let the omain of the functional 0 be an open subset C of the Banach space L = L L r, an we let the common omain of functionals τ an in be C V. Moreover, we require that in be continuous an that τ be twice-continuously Fréchet ifferentiable. As pointe out in Remark 1 of [2], the continuous ifferentiability of τ an the smoothness of t t guarantee that the time erivatives in 10 are well efine, an that the following Chain-Rule Formula hols true: t τ vt = τ t, t r, vt t + δ r τ t, t r, vt[ t r ]+ v τ t, t r, vt vt. Here, t = t an vt = t t ˆvt; moreover, τ an v τ are the partial erivatives of τ with respect to its first an thir argument, respectively, an δ r τ is the unique boune linear functional on L r satisfying τ, t r + r, v = τ, t r, v + δ r τ, t r, v[ r]+o r, for all r in L r. By combining the invariance requirement 10 with the Chain-Rule Formula, we obtain an expression involving both the instantaneous value an the past history of the time erivative of the state process ˆ : τ t, tr, ˆvt + w τ t, tr, ˆvt t + δ r τ t, tr, ˆvt + w δ r τ t, tr, ˆvt [ t r ] + v τ t, tr, ˆvt + w v τ t, tr, ˆvt vt + 0 t, t r w = Step 3. Energy Splitting. As in [6] for instantaneous-memory materials, we achieve the esire result for materials with faing memory by showing that τ oes not epen on v. The argument use in [6] relies on the existence, given a state process an an arbitrary in L, of a state process Ɣ whose instantaneous value coincies with t, an whose rate Ɣt equals ; by replacing with Ɣ, an by invoking the arbitrariness of, one euces that the term multiplying t in the first line of 12 must vanish, an then conclues that τ oes not epen on v. The argument we use in the present proof is similar, an is base on a result ue to Coleman an Mizel [2, Remark 2]: for any given L an L, an for every positive ε, there is a state process ε Ɣ such that whose past history ε Ɣ t r satisfies t ε Ɣt <ε an ε Ɣt =, t r ε Ɣ t r <ε, t r ε Ɣ t r <ε. Replacing with ε Ɣ in 12, letting ε vanish, an using the smoothness of τ an in,we obtain that τ t, tr, ˆvt + w τ t, tr, ˆvt + =0

6 64 A. Favata et al. the ots stan for the remaining terms of 12, whence, by the arbitrariness of, it follows that, for all w in V, τ t, t r, ˆvt + w τ t, t r, ˆvt = 0, i.e., that τ oes not epen on velocity. We conclue that, for every triplet, t r, v of a state, a past history t r,anavelocityv, the constitutive mapping that elivers the total energy splits as follows: τ, t r, v = ɛ, t r + κ t r, v. 13 Step 4. Representation of the Kinetic Energy. In view of 13, relation 12 becomes: δ r κ tr, ˆvt + w δ r κ tr, ˆvt [ t r ] + v κ tr, ˆvt + w v κ tr, ˆvt vt + 0 t, t r w = 0, 14 for all w in V. Moreover, since both mappings δ r τ an v τ are continuously-ifferentiable by assumption, we have that v δ r τ, t r, v[ r] = δ r v τ, t r, v[ r], 15 for every triplet, t r, v of a state, a past history t r,anavelocityv, an for every r in L r. 1 On ifferentiating 14 with respect to w an using 15, we obtain: δ r v κ t r, ˆvt + w[ t r ]+ vv κ t r, ˆvt + w v + 0 t, t r = 0, 16 for all w in V ; upon choosing w = ˆvt in 16, we have that δ r v κ t r, 0[ t r ]+ vv κ t r, 0 vt + 0 t, t r = 0; 17 an, on subtracting 17 from 16 an selecting w = 0, we obtain: δ r v κ tr, ˆvt δ r v κ tr, 0 [ tr ]+ vv κ tr, ˆvt vv κ tr, 0 vt = Finally, since the choice of ˆvt an vt in 18 is arbitrary, we have that 18 itself is equivalent to δ r v κ tr, v δ r v κ tr, 0 [ t r ]=0 19 an vv κ t r, v = M t r, 20 for every past history t r an for every velocity v; note that M t r is a secon-orer symmetric tensor. An integration of 20 yiels: κ t r, v = 1 2 v M t r v + m t r v + μ t r This ientity generalizes the theorem on the inversion of the orer of partial ifferentiation for functions of real variables [4].

7 Energy Splitting Theorems for Materials with Memory 65 We incorporate the constant part μ t r of κ t r, in the internal energy ɛ, t r, without affecting the splitting 13; an we ispose of the linear part by requiring that the total energy be an even function of the velocity, an assumption that, with 13, entails that m t r 0. With these measures, we obtain the following representation for the kinetic energy: an, on combining 13an22, we arrive at κ t r, v = 1 2 v M t r v 22 τ, t r, v = ɛ, t r v M t r v, 23 an aitive energy splitting that generalizes the one in 5. Step 5. Mass Conservation Law an Representation of the Inertial Force. 22 into19, we obtain that, for all v V, By substituting δr M t r v [ t r ]=0, that is, in view also of the Chain Rule Formula, that mass is conserve in the following general sense: t M t r = 0, It remains for us to substitute 22 in17. This yiels: t M t r := δ M r t r [ t r ]. 24 0, t r = M t r v. 25 With 25 an assumption 9 of linearity in the velocity of the power expeniture of the inertia force, we obtain a representation of the inertia force that generalizes 6: in, t r, v = M t r v + D in, t r, vv, 26 where D in, t r, v is a skew-symmetric tensor. We are now in position to summarize our finings uner form of our Energy Splitting Theorem for Materials with Faing Memory Let the constitutive epenence of energy an inertia force be specifie by 8 1,2, with the mappings τ an in presume to be, respectively, twice-continuously ifferentiable an continuous an with the set of past histories enowe with the faing memory norm efine 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 thir argument. Then, the mappings τ an in amit the representations 23 an 26, parameterize by a scalar internal-energy mapping ɛ an a symmetric-value mass-tensor mapping M, which obeys the conservation law 24 1.

8 66 A. Favata et al. Remark 1 Assumption iii which is not state in [6] an is not neee to arrive at 20 is instrumental in ruling out the linear term in 21. The same can be achieve by the strengthene invariance requirement that the internal power be invariant uner time-epenent translational changes of observer: if 2 2 is accoringly replace by v v + = v + wt, then, as a bonus, one can also show that the internal energy cannot epen on acceleration, a epenence that woul be incompatible with the Secon Law of Thermoynamics [6]. 4 Materials with Internal Variables For materials with internal variables, the total energy, the inertia force, an the workeffective part of the inertia force are given by ˇτ v t := ˇτ t, ˆβt, ˆvt, ď in v t := ďin t, ˆβt, ˆvt, 27 ď 0 t := ď 0 t, ˆβt, where ˆβt is a translationally-invariant vector of internal variables, whose evolution is governe by the orinary ifferential equation: t ˆβt = f t, ˆβt. 28 By 27 an28, the invariance statement 7 leas to ˇτ t, ˆβt, ˆvt + w ˇτ t, ˆβt, ˆvt t + β ˇτ t, ˆβt, ˆvt + w β ˇτ t, ˆβt, ˆvt f t, ˆβt + v ˇτ t, ˆβt, ˆvt + w v ˇτ t, ˆβt, ˆvt vt + ď 0 t, ˆβt w = By a free-continuation argument borrowe from [6], the time erivative t appearing in the first line of 29 can be replace by an arbitrary rate ; the resulting relation allows one to conclue that ˇτ t, ˆβt, ˆvt + w ˇτ t, ˆβt, ˆvt = 0, for all w in V, whence the energy splitting: ˇτ t, ˆβt, ˆvt =ˇɛ t, ˆβt +ˇκˆβt, ˆvt. From this point on, the proof procees along the steps liste in the previous section, with ˆβt in the place of t r. The conclusions are, mutatis mutanis, the same as those state in

9 Energy Splitting Theorems for Materials with Memory 67 the Energy Splitting Theorem for materials with faing memory: the total energy an the inertial force amit the representations: ˇτ,β, v =ˇɛ,β v ˇMβv, ď in, β, v = ˇMβ v + Ď in, β, vv, with ˇMβ symmetric an Ď in, β, v skew; an the mass-tensor mapping ˇM satisfies the conservation law t ˇM ˆβt = 0. Acknowlegements We thank an anonymous referee for pointing out to us that some justification was neee to rule out the linear term in 21. G.T. gratefully acknowleges the financial support of INAM- GNFM Research Project: Moellazione fisico-matematica i materiali e strutture intelligenti. References 1. Coleman, B.D., Gurtin, M.: Thermoynamics with internal state variables. J. Phys. Chem. 47, Coleman, B.D., Mizel, V.J.: A general theory of issipation in materials with memory. Arch. Ration. Mech. Anal. 27, Coleman, B.D., Noll, W.: An approximation theorem for functionals, with applications in continuum mechanics. Arch. Ration. Mech. Anal. 6, Graves, L.M.: Riemann integration an Taylor s theorem in general analysis. Trans. Am. Math. Soc. 29, Lubliner, J.: On faing memory in materials with evolutionary type. Acta Mech. 8, Poio-Guiugli, P.: Inertia an invariance. Ann. Mat. Pure Appl. CLXXII, Poio-Guiugli, P.: Sparse notes in thermoynamics 2009, forthcoming 8. Truesell, C., Noll, W.: The non-linear fiel theories of mechanics. In: Flugge, S. e. Encyclopeia of Physics, vol. III/3. Springer, Berlin 1965