A minimum principle is set up for the quasi-static 
boundary-value problem (QSP) in linear viscoelasticity. 
A linear homogeneous and isotropic viscoelastic solid under unidimensional displacements is considered  along with the complete set of thermodynamic restrictions on the relaxation function. It is assumed that boundary conditions are of Dirichlet type and initial history data are not given. The variational formulation of QSP is set up through a convex functional based on a "weighted" $L^2$ inner product as the bilinear form and is strictly related to the thermodynamic restrictions on the relaxation function. As an aside, the same technique is proved to be applicable to analogous physical problems such as the quasi-static heat flux equation

C. GIORGI

MARZOCCHI A

GIORGI, Claudio

MARZOCCHI A.

Archivio istituzionale della ricerca - Università di Brescia

A MINIMUM PRINCIPLE FOR THE QUASI-STATIC PROBLEMIN LINEAR VISCOELASTICITYC. Giorgi (1) and A. Marzocchi (2)Abstract. A minimum principle is set up for the quasi-static boundary-value problem (QSP)in linear viscoelasticity. A linear homogeneous and isotropic viscoelastic solid under unidi-mensional displacements is considered along with the complete set of thermodynamic restric-tions on the relaxation function. It is assumed that boundary conditions are of Dirichlet typeand initial history data are not given. The variational formulation of QSP is set up through aconvex functional based on a ”weighted” L2 inner product as the bilinear form and is strictlyrelated to the thermodynamic restrictions on the relaxation function. As an aside, the sametechnique is proved to be applicable to analogous physical problems such as the quasi-staticheat flux equation.0. IntroductionThe connection between thermodynamic restrictions and variational formulations or,possibly, extremum principles in linear viscoelasticity was investigated in a number of recentpapers (see [1] for an up-to-date survey). Because of the convolution which occurs in thestress-strain relation, very often variational formulations of linear viscoelasticity are expressedby functionals based on bilinear forms of convolution type. Quite rarely an extremum propertyholds for the functional under consideration. If so, however, the extremum property provesto be a direct consequence of the restrictions imposed by thermodynamics on the relaxationfunction [2][3][4].Things are even more difficult when the quasi stationary approximation is concerned,i.e. when the inertia term is disregarded and solutions are defined on the whole time axis.A minimum principle for the quasi-static problem (QSP) was established by CHRISTEN-SEN [5] through a convex bilinear functional that becomes also stationary if an appropriateclass is considered for the displacement fields. Roughly speaking, it states that a factorizedquasi-static solution u0(x, t) = h(t)k(x) minimizes a suitable functional with respect to per-turbations of the spatial part k(x) only. Such a functional is built up by considering the L2inner product as the bilinear form. As noticed in [3], the convexity follows from thermody-namic restrictions, but the functional does not turn out to be stationary unless we assumethe same time dependence for all displacement fields. Hence the whole quasi-static solutioncannot be characterized as a minimum.(1) Universita` della Calabria, Dipartimento di Matematica, 87036 Arcavacata di Rende (CS),Italy.(2) Universita` Cattolica del S.Cuore, Dipartimento di Matematica, Via Trieste 17, 25121 Brescia,Italy.2 C.GIORGI-A.MARZOCCHITo avoid this objection a “maximum-minimum” principle was built up in [6] by con-structing a suitable family of convolution type functionals involving Fourier transformation.Thereby a solution to QSP can be characterized as a saddle point relative to the additivedecomposition (with respect to time) of the displacement field into its even and odd parts.What is more, this property is crucially related to the thermodynamic restrictions.Unfortunately, there is no possibility of getting a minimum principle by consideringvariational formulations of this kind. Nevertheless, on the basis of the thermodynamic re-strictions, we are able to get a minimum principle for QSP in a general class of displacementfields through a single functional.In order to keep up the convexity property we search for extremum principle via a“weighted” L2 inner product as the bilinear form. However, since a convolution inescapablyoccurs through the stress-strain relation, the stationary property should involve appropriateand shrinking assumptions. Indeed, unidimensional displacements only are allowed as well ashomogeneity and isotropy of the relaxation function are required.Since our approach rests heavily upon the special structure of the bilinear form, it cannotbe generalized to two- and three-dimensional displacements, apparently. Nevertheless it isapplicable without restrictions to other analogous physical problems such as the quasi-staticheat flux equation.1. Setting of the problemLet us consider a linear viscoelastic homogeneous isotropic body which occupies abounded regular domain Ω of IR3. Moreover, we consider unidimensional displacementsu : Ω× IR→ IR, so that the constitutive stress-strain relation takes the formT(u)(x, t) = G0∇u(x, t) +∫ ∞0G′(s)∇u(x, t− s) ds, (1.1)where G0 is the istantaneous elastic modulus and G′ is the Boltzmann memory function.For example, an equation of type (1.1) can be found in the study of vibrations of aspatially homogeneous viscoelastic membrane and in the case of a simple extension (or torsion)of a viscoelastic wire with one extremity fixed and the other subjected to the traction (ortorque) T.As usual in materials with memory, the Boltzmann function is required to comply witha fading memory principle. Following DAY [7], we may state it as follows:G′ ∈ L1(IR+). (1.2)Finally, we may account for the body being a solid by lettingG∞def= G0 +∫ ∞0G′(s) ds > 0. (1.3)A MINIMUM PRINCIPLE IN LINEAR VISCOELASTICITY 3Substituting (1.1) into the equation of motion in the quasi-static approximation andassuming homogeneous Dirichlet boundary conditions we haveG0∆u(x, t) +∞∫0G′(s)∆u(x, t− s) ds+ f(x, t) = 0 (x, t) ∈ Ω× IRu|∂Ω = 0 t ∈ IR.(1.4)We shall refer to (1.4), where G0 and G′ comply with (1.2)-(1.3) as quasi-static problem inlinear viscoelasticity (QSP).Taking into account that G′ can be defined on the whole real line by assumingG′(s) = 0 ∀s ∈ (−∞, 0),we can get ∫ ∞0G′(s)∆u(x, t− s) ds = (G′ ∗∆u)(x, t),where ∗ denotes convolution on IR. Using the Dirac δ-function , we introduceΓdef= G0δ +G′, (1.5)so that (1.4) may be rewritten in the following compact formΓ ∗∆u+ f = 0u|∂Ω = 0.(1.6)In view of the variational formulation we introduce the followingDefinition. A function u is called a strict solution to QSP (1.4) with source function fbelonging to L2(IR, L2(Ω)) if u belongs to L2(IR,H10 (Ω)) and satisfies∫IR∫Ω[(Γ ∗ ∇u)(x, t) · ∇v(x, t)− f(x, t) · v(x, t)] dx dt = 0for any v ∈ L2(IR,H10 (Ω)).For later convenience we denote by fˆ the (formal) Fourier transform of any function fdefined on Ω× IR, namelyfˆ(x, ω)def=∫IRf(x, t) exp (−iωt) dt ;similarly, letting the subscript s (c) denote the half-range Fourier sine (cosine) transform, forany function g defined on Ω× IR+ we havegˆs(x, ω) =∫ ∞0g(x, t) sinωt dt , gˆc(x, ω) =∫ ∞0g(x, t) cosωt dt.4 C.GIORGI-A.MARZOCCHI2. Thermodynamic restrictionsIn the sequel we shall assume further conditions on G0 and G′ that are derived fromThermodynamics. Mainly, we recall here the so called Graffi’s inequalityωĜ′s(ω) ≤ 0 ∀ω ∈ IR (2.1)which is a necessary and sufficient condition that the work in sinusoidal processes is non-negative. As proved by FABRIZIO & MORRO [8], (2.1) is quite equivalent to the Second Lawof Thermodynamics in the form of the Clausius property for isothermal processes relative toisotropic and homogeneous viscoelastic materials. In the same work it is proved that from(2.1) it followsG0 ≥ G∞. (2.2)It is worth remarking that, according to [9], a stronger version of the Second Law can begiven so that the Clausius inequality reduces to an equality if and only if “reversible” cyclesare considered. If such is the case, then (2.1) is replaced byωĜ′s(ω) < 0 ∀ω ∈ IR \ {0}. (2.3)and (2.2) holds with the strict inequality sign. Therefore we have the followingProposition 2.1. Under assumptions (1.2)-(1.3) and (2.3), there exists δ such that|Γ̂(ω)| = |G0 + Ĝ′(ω)| ≥ δ > 0 ∀ω ∈ IR. (2.4)As to the link between the solvability of QSP and thermodynamic restrictions, we recallthe following result proved by FABRIZIO in [8].Proposition 2.2. If the viscoelastic material is a solid, i.e. (1.3) holds, then QSP has oneand only one strict solution provided that (2.3) is satisfied.Remark . In the previous Proposition 2.2, the inequality (2.3) cannot be weakened into (2.1)(see [10]). Nevertheless, it is a sufficient but not necessary condition (see [11]).3. Preliminaries on the variational formulationA systematic method for the derivation of variational formulations is that pertaining tothe theory of inverse problems of the Calculus of Variations. Here we state the main resultsof this theory (see [13]) which are relevant to our purpose.LetX be a Banach space over IR andX∗ denote the conjugate (dual) space ofX. Lettingv ∈ X∗ and z ∈ X, 〈v, z〉 represents a non-degenerate bilinear form whereby if 〈v, z〉 = 0 forA MINIMUM PRINCIPLE IN LINEAR VISCOELASTICITY 5every v ∈ X∗ (resp. z ∈ X) then z (resp. v) is the null element of X (resp. X∗). Supposethat U ⊂ X is an open set and let N be an operator U → X∗. Let u+ νh ∈ U, ν ∈ [O, ν], forsome ν > 0. If N is Gaˆteaux differentiable we denote bydN(u|h) = dN(u+ νh)dν∣∣∣ν=0the (Gaˆteaux) differential of N at u ∈ U in the direction h. If there exists N ′(u) such thatdN(u|h) = 〈N ′(u), h〉we regard N ′(u) as the derivative of N with respect to the bilinear form. Similarly, in thecase of functionals f : X → IR we write df(u|h) = 〈f ′(u), h〉. If there exists a functional fsuch that N = f ′ then we say that N is a potential operator.The existence of a potential for the given operator is crucially related to the choice of thebilinear form, as it is expressed by the following theorem whose proof is given by VAINBERG[12]:Theorem 3.1. Let 〈·, ·〉 be a non-degenerate bilinear form on X. Suppose that(i) N is an operator from X into X∗,(ii) N has a linear Gaˆteaux differential dN(u|h) at every point of the open convex setD ⊂ X,(iii) the bilinear form 〈dN(u|h), k〉 on h, k ∈ X is continuous at every point of D.Then a necessary and sufficient condition for N to be a potential operator in D is that〈dN(u|h), k〉 = 〈dN(u|k), h〉 (3.1)for every h, k ∈ X and every u ∈ D. Moreover, if (3.1) holds then N = f ′ withf(u) = f(u0) +∫ 10〈N(u0 + ν(u− u0)), u− u0〉 dν (3.2)u0 being any point of D.For any linear operator L which is symmetric with respect to the chosen bilinear formTheorem 3.1 yields a functional f , given by (3.2), which is a potential for L ; that is, in sucha case, the potentialness condition (3.1) reads〈Lu,w〉 = 〈Lw, u〉. (3.3)The application of this theorem to a given differential problem requires, as a preliminarystep, the choice of a bilinear form. Most often variational formulations are established byletting X be the Hilbert space L2(IR, L2(Ω)), and 〈·, ·〉 be the scalar product in that space.In connection with viscoelasticity, and more generally in the case of constitutive functionalsexpressed by convolutions, this choice is unsuccessful.Since the bilinear form is required to be non-degenerate, an alternative choice for 〈·, ·〉in the case of space-time dependent functions could be the L2 scalar product with respect6 C.GIORGI-A.MARZOCCHIto space and the convolution with respect to time, as made in [12]. This approach leads tovariational formulations with minimum properties in dynamical viscoelasticity [2][3][4] butwith saddle-point property in the quasi-static approximation [6].In order to reach a minimum principle for QSP, in the next section we shall introduce a”weighted” L2 scalar product with respect to space and time . It is worth noting that althoughthis choice leads to a non-degenerate bilinear form which is non symmetric, nevertheless thepotentialness condition (3.3) is satisfied when L is the QSP operator.4. Minimum principleFor any pair (p,q) of vector- or scalar-valued functions on Ω × IR we introduce thefollowing bilinear form〈p,q〉Γ def=∫IR∫Ωp(x, t) · (Γ ∗ q)(x, t) dx dt=∫IR∫ ∞0∫Ωp(x, t) · [G0q(x, t) +G′(t− s)q(x, s)] dx ds dt. (4.1)where Γ is defined by (1.5).Thanks to (1.2), this bilinear form is well-defined on L2(IR, L2(Ω)) but it is easily seento be non symmetric. Nevertheless, by virtue of the thermodynamic restriction (2.3), it isnon-degenerate: i.e. if 〈p,q〉Γ = 0 for every q ∈ L2(IR, L2(Ω)), then p = 0 a.e.. In order toprove this property, we apply Plancherel’s formula to (4.1) and we obtain〈p,q〉Γ = 12pi∫IR∫ΩΓ̂(ω)p̂(x, ω) · q̂∗(x, ω) dx dω (4.2)where z∗ denotes the complex conjugate of z. Hence, if〈p,q〉Γ = 0 ∀q ∈ L2(IR, L2(Ω)) (4.3)then we have0 = 2∫ ∞0∫Ω{Re[Γ̂(ω)pˆ(x, ω)] · qˆ+(x, ω)− Im[Γ̂(ω)pˆ(x, ω)] · iqˆ−(x, ω)} dx dωwhere q+ and q− are respectively the even and odd parts of q, i.e.q+(t) =12(q(t) + q(−t)) , q−(t) = 12(q(t)− q(−t)),so that the following relations holdRe qˆ∗(ω) = qˆ+(ω) , Im qˆ∗(ω) = iqˆ−(ω).A MINIMUM PRINCIPLE IN LINEAR VISCOELASTICITY 7Because of the bijectivity of the Fourier transform from L2 into L2, qˆ+ and iqˆ− are arbi-trary and independent functions of L2(IR+, L2(Ω)) when q runs over L2(IR, L2(Ω)). Therefore(4.3) leads toRe[Γ̂(ω)pˆ(x, ω)]= 0Im[Γ̂(ω)pˆ(x, ω)]= 0almost everywhere on Ω× IR in the sense of L2(IR, L2(Ω)), that is∫IR∫Ω∣∣Γ̂(ω)∣∣2|pˆ(x, ω)|2 dx dω = 0. (4.4)Now, since Γ̂(ω) = G0 + Ĝ′(ω) is a continuous and bounded complex-valued scalar functionthat does not vanish for any ω ∈ IR by virtue of (2.4), it must be pˆ = 0 and then p = 0 inL2(IR, L2(Ω)).Remark. The inequality (2.4) implies that|||f |||Γ =∫IR∫Ω∣∣Γ̂(ω)∣∣2|fˆ(x, ω)|2 dx dωis a norm in L2(IR, L2(Ω)) equivalent to the natural one.Finally, to apply Theorem 3.1 we have to prove the potentialness condition (3.3) for theproblem (1.6) with respect to 〈·, ·〉Γ, which reads〈Γ ∗∆u,w〉Γ = 〈Γ ∗∆w, u〉Γ for any u,w in L2(IR,H10 (Ω)).Denoting by (·, ·) the usual inner product on L2(IR, L2(Ω)), we have〈Γ ∗∆u,w〉Γ = (Γ ∗∆u,Γ ∗ w). (4.5)Then, using the fact that Γ is independent of x and integrating by parts with respect to thespace variables, we find(Γ ∗∆u,Γ ∗ w) = (Γ ∗ u,Γ ∗∆w) = (Γ ∗∆w,Γ ∗ u) = 〈Γ ∗∆w, u〉Γwhich leads to the required result.The straightforward application of Theorem 3.1 givesTheorem 4.1. If the relaxation function G satisfies conditions (1.2),(1.3) and (2.3) thenu is a strict solution to QSP if and only if it is a strict minimum on L2(IR,H10 (Ω)) of thefunctionalΦ(u) =12〈Γ ∗ ∇u,∇u〉Γ − 〈f, u〉Γ. (4.6)The minimum property follows easily from the convexity of Φ with respect to the usualL2 norm, in fact8 C.GIORGI-A.MARZOCCHIΦ(u) =12(Γ ∗ ∇u,Γ ∗ ∇u)− (u,Γ ∗ f). (4.6)It is worth remarking that, in this case, thermodynamic restrictions are intimatelyrelated to the non degeneracy of the chosen bilinear form, instead of convexity of the functionalΦ.5. Other applicationsIt is easily seen that the potentialness condition (3.3) for the problem (1.6) is not satisfiedif the material is not homogeneous or not isotropic. What is more, condition (3.3) fails forthree-dimensional displacements even in the homogeneous and isotropic case.The previous technique, however, applies successfully to other quasi-static problems inmathematical physics when constitutive equations like (1.1) are concerned. For example,we take under consideration a homogeneous and isotropic rigid heat conductor with linearmemory occupying a fixed bounded domain Ω ⊂ IR3.If we consider only small variations of the temperature θ(x, t) from a reference tem-perature θ0 and small temperature gradients g(x, t), we may suppose that internal energyε(x, t) and heat flux q(x, t) are described by the following linearized constitutive equations(see [14]):ε(x, t) = ε0 + α0u(x, t) +∫ ∞0α′(s)u(x, t− s) ds, (5.1)q(x, t) = −k0∇u(x, t) +∫ ∞0k′(s)∇u(x, t− s) ds (5.2)where u(x, t) = θ(x, t)− θ0 and α′, k′ ∈ L1(IR+).The ensuing evolution problem, with homogeneous Dirichlet boundary conditions, isgiven byα0ut(x, t) +∫∞0α′(s)ut(x, t− s) ds− k0∆u(x, t)−∫∞0k′(s)∆u(x, t− s) ds = r(x, t)u|∂Ω = 0so that in the quasi-static approximation we have k0∆u(x, t) +∫∞0k′(s)∆u(x, t− s) ds+ r(x, t) = 0u|∂Ω = 0(5.3)or, in a more compact form,A MINIMUM PRINCIPLE IN LINEAR VISCOELASTICITY 9Γ ∗∆u+ r = 0u|∂Ω = 0(5.4)whereΓdef= K0δ + k′. (5.5)The only considerable difference between (5.4) and (1.6) is that concerning thermody-namic restrictions. In fact, as proved by G.GENTILI (see [14]), in a rigid heat conductorsatisfying (5.2) the Second Law of Thermodynamics holds if and only ifk0 + kˆ′c(ω) > 0 ∀ω ∈ IR. (5.6)Nevertheless, it is easily seen that the bilinear form〈p,q〉Γ def=∫IR∫Ωp(x, t) · (Γ ∗ q)(x, t) dx dt (5.7)is non-degenerate since (5.6) implies∣∣Γ̂(ω)∣∣ ≥ Re Γ̂(ω) = k0 + kˆ′c(ω) > 0 ∀ω ∈ IR.Moreover, if k0 6= 0 (and then k0 > 0 by virtue of (5.6)), the norm|||f |||2Γ =∫IR∫Ω∣∣Γ̂(ω)∣∣2∣∣fˆ(x, t)∣∣2 dx dωis equivalent to the natural one in L2(IR, L2(Ω)).Hence, by using the same procedure as in the previous section, a minimum principle forthe quasi-static problem (5.4) in heat conduction with memory can be established throughthe functionalΨ(u) =12〈Γ ∗ ∇u,∇u〉Γ − 〈r, u〉Γ.Acknowledgement. This work was partially supported by Italian M.P.I. 40% project ”Prob-lemi di evoluzione sui fluidi e sui solidi” and G.N.F.M. of C.N.R..References[1] M.FABRIZIO, A.MORRO, Mathematical Problems in Linear Viscoelasticity, Springer, Berlin,(to appear).[2] M.FABRIZIO, C.GIORGI and A.MORRO, Minimum Principles, Convexity, and Thermody-namics in Linear Viscoelasticity, Continuum Mech. Thermodyn., 1, 197 (1989).10 C.GIORGI-A.MARZOCCHI[3] R.REISS, E.S.HAUG, Extremum Principles for Linear Initial-Value Problems of Mathemat-ical Physics, Int. J. Engng. Sci., 16, 231 (1978).[4] C.GIORGI, A.MORRO, Extremum Principles for Viscoelastic Fluids, Int. J. Engng. Sci.(toappear).[5] R.M.CHRISTENSEN, Theory of Viscoelasticity, 2nd edn., Academic Press, New York, 1982.[6] C.GIORGI, A.MARZOCCHI, New Variational Principles in Quasi-Static Viscoelasticity, Qua-derni del Seminario Matematico di Brescia n.9/90.[7] W.A. DAY, The thermodynamics of materials with fading memory , Springer, Berlin, 1972.[8] M.FABRIZIO, A.MORRO, Viscoelastic Relaxation Functions Compatible with Thermody-namics, J. Elasticity, 19, 63 (1988).[9] M.FABRIZIO, An Existence and Uniqueness Theorem in Quasi-Static Viscoelasticity, Quart.Appl. Math., 47, 1 (1989).[10] M.FABRIZIO, A.MORRO, On Uniqueness in Linear Viscoelasticity. A Family of Counterex-amples, Quart. Appl. Math., 45, 321 (1987).[11] C.GIORGI, B.LAZZARI, Uniqueness and Stability in Linear Viscoelasticity: Some Counterex-amples, to appear in Proceedings of the V International Conference on Waves and Stabilityin Continuous Media, Sorrento (Italy), October 9-14, 1989.[12] F.BAMPI, A.MORRO, Topics in the inverse problem of the calculus of variations, FisicaMatematica, Suppl. Boll. U.M.I., 5 (1986), 93.[13] M.M.VAINBERG, Variational Methods for the Study of Nonlinear Operators, Holden-Day,San Francisco, 1964.[14] G.GENTILI, C.GIORGI, Thermodynamic properties and stability for the heat flux equationwith linear memory (to appear)

A minimum principle for the quasi-static problem in linear viscoelasticity

Abstract

Similar works

Full text

Available Versions

Archivio istituzionale della ricerca - Università di Brescia