TOWARDS THERMOELASTICITY OF FRACTAL MEDIA

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1 ownloae By: [University of Illinois] At: 21:04 17 August 2007 Journal of Thermal Stresses, 30: , 2007 Copyright Taylor & Francis Group, LLC ISSN: print/ x online OI: / Keywors: TOARS THERMOELASTICITY OF FRACTAL MEIA M. Ostoja-Starzewski epartment of Mechanical Science an Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA An extension an generalization of thermomechanics with internal variables an thermoelasticity to fractal porous meia is outline. First, a fiel form of the secon law of thermoynamics is erive. In conraistinction to the conventional Clausius uhem inequality, it involves generalize rates of eformation an internal variables. Upon introucing a issipation function an postulating the thermoynamic orthogonality on any length-scale, constitutive laws of elastic-issipative fractal meia naturally involving generalize erivatives of strain an stress can then be erive. ith a focus on thermoelasticity, a new form of uhamel s ifferential equation of heat conuction is erive. Fractals; Heat conuction; Ranom meia; Thermoelasticity; Thermomechanics Continuum mechanics is naturally suite to eal primarily with meia exhibiting spatially homogeneous properties. If the materials are heterogeneous an ranom, then it is hope that the statistics are escribable by the conventional Eucliean geometry an, therefore, by the conventional calculus. Neeless to say, many other meia with non-eucliean geometries are ubiquitous in nature, yet they fall outsie the realm of classical continuum mechanics. Another motivation of the present stuy is that the constitutive responses of many materials are best escribe by fractional calculus. Thus, it is well known that various man-mae as well as biological polymers are moele by quite simple equations with fractional erivatives, e.g., [1 3]. In physics the connection between transport phenomena in fractal geometries an fractional moels has been known for quite some time [4]. Although many avances have been mae, solution of bounary value problems of fractal meia in the vein of continuum mechanics is still an open issue. However, a first step in that irection has recently been taken by Tarasov [5, 6], who evelope continuum-type equations of conservation of mass, linear momentum an energy for fractals, an, on that basis stuie some flui mechanics an wave problems. Receive 7 April 2007; accepte 14 May This article was presente at the 70th birthay of Józef Ignaczak Symposium at the 7th International Congress on Thermal Stresses, Taipei, June 4 7, Aress corresponence to M. Ostoja-Starzewski, epartment of Mechanical Science an Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA. martinos@ uic.eu 889

2 890 M. OSTOJA-STARZESKI ownloae By: [University of Illinois] At: 21:04 17 August 2007 In this article, we combine Tarasov s approach with the thermomechanics of Ziegler [7 11] to evelop a continuum-type expression of the secon law of thermoynamics, which in turn, by invoking thermoynamic orthogonality, allows a erivation of constitutive laws of elastic-issipative fractal meia via generalize fractional integrals. A classification of complex versus compoun processes carries over to this new setting. Basically, our goal is to generalize continuum thermomechanics an thermoelasticity to the realm of ranom fractal materials, in the vein of thermomechanics of ranom non-fractal materials [12, 13]. As a final result, we present a new form of uhamel s ifferential equation an the equation of heat conuction, generalize to fractal meia. CONTINUUM MECHANICS IN THE SETTING OF FRACTAL MEIA e consier the meium B uner consieration to have a ranom fractal geometry. That is, B is taken as a set of all the realizations B parametrize by sample events of the space B = B (1) Each of the realizations B follows eterministic laws of classical mechanics in that it is a specific heterogeneous material sample; probability is introuce to eal with the set B. Now, following Tarasov [5, 6], the mass m in each B obeys a power law relation m R = kr <3 (2) where R is a box size (or a sphere raius, effectively a lengthscale of measurement), is a fractal imension of mass, an k is a proportionality constant. It follows that the power law (2) escribes the scaling of mass with R. Focusing on fractal porous meia, the power law relation (2) is rewritten as m R = m 0 ( R R p ) (3) where R p is the average raius of a pore, an m 0 is the mass at R p = R; this is a reference case. ith (3) we use to enote the fractal imension of mass in a omain, while the bounary of has imension. In general, equals neither 2 nor 1. At this point, the conventional equation giving mass in a three-imensional region of volume V m = r 3 r (4) has to be generalize to m 3 = 23 3/2 /2 r r r r (5)

3 THERMOELASTICITY AN FRACTAL MEIA 891 Assuming the fractal meium to be spatially homogeneous ownloae By: [University of Illinois] At: 21:04 17 August 2007 Eq. (4) is replace using a fractional integral r = 0 = const (6) m 3 = /2 /2 R 3 3 r (7) where R = r r 0. That is, the fractal meium with a non-integer mass imension is escribe using a fractional integral of orer. This allows an interpretation of the fractal (intrinsically iscontinuous) meium as a continuum. In particular, the next step is Tarasov s reformulation of the Green Gauss Theorem fv k n k A = where f is an arbitrary function, v is the velocity, an c 1 3 R iv c 2 R fv V (8) A = c 2 R A 2 V = c 3 R V 3 (9) On account of (9), the left-han sie in (8) is a fractional integral, equal to a conventional integral c 2 R fva 2. Similarly, the right-han sie in (8) is a fractional integral, equal to a conventional integral iv c 2 R Av V 3. BALANCE LAS IN THERMOMECHANICS OF FRACTALS The above formulation allows the erivation of fractional-type balance equations of fractal meia [5, 6]: the fractional equation of continuity: ( ) = k t v k (10) the fractional equation of balance of ensity of momentum: ( ) v t k = f k + l kl (11) the fractional equation of balance of ensity of energy: ( ) u = c R t kl v k l k q k (12) In the above kl is the Cauchy stress (symmetric accoring to the balance of angular momentum, employe just as in non-fractal meia), an the following operators

4 892 M. OSTOJA-STARZESKI (or generalize erivatives) are use ownloae By: [University of Illinois] At: 21:04 17 August 2007 where k f = c 3 R c x 2 R f c 3 R k c 2 R f k ( ) f = f t t + c R v f k x k c R = R /2 3/2 /2 2 2 c 2 R = R 2 /2 c 3 R = R /2 /2 c R = c 1 3 R c 2 R Henceforth, for simplicity of notation, we write c, c 2, an c 3. Note that, in a non-fractal meium ( = 3, = 2) c R = 1, whereby one recovers conventional forms of local relations of continuum mechanics. Of further use will also be the fact that the relation k fg f k g + g k f oes not hol, an shoul instea be replace by (13) (14) k fg = f k g + cg kf (15) To erive the fiel equation of the secon law of thermoynamics in a fractal meium B, we begin with the global form of that law in the volume V, having a Eucliean bounary, that is or, equivalently, Ṡ = Ṡ r + Ṡ i with Ṡ r = Q T an Ṡ i 0 (16) Ṡ Ṡ r (17) Here Ṡ an Ṡ r stan for the rate of total entropy prouction an the rate of reversible entropy prouction in V, respectively. Thus, we can write (17) as s V t = Ṡ Ṡ r q = k n k T A (18) Employing the efinitions of Section, we fin the local form of the secon law in terms of time-fractional rates of strains an internal parameters ( ) [( ) ] ( ) 0 T s i = ij u t t i j + ij t ij c R T k q k (19) T

5 THERMOELASTICITY AN FRACTAL MEIA 893 ownloae By: [University of Illinois] At: 21:04 17 August 2007 The above is a generalization of the Clausius uhem inequality to fractal meia, an it reuces to its classical form for non-fractal boies 0 T ṡ i = ij ij + ij ij T k q k T It is interesting that the temporal generalize erivative of 13 2 appears only for the time rates of external an internal strains but oes not arise in the thir term except for the coefficient c R. However, the spatial generalize erivative of 13 1 arises in processes of heat transfer in a fractal rigi conuctor an couple thermoelasticity of fractal eformable meia as is shown next. THERMOYNAMICS AN THERMOELASTICITY The inequality (19) inicates the presence of three velocity-like arguments {[( ) ( ) } v = u t i ] j t ij q k (21) analogous to the conventional non-fractal situation. Clearly, in a non-fractal meium ( = 3, = 2) one recovers v = ij ij q k. Thus, the velocity space (or V space) is mae of all triples v. One may therefore introuce a issipation functional in V ( ) v = T s i 0 (22) t (20) In view of (19), conjugate to the triple (21) we have the issipative force vector [ A = ij ij c R T ] k (23) T e next postulate the Principle of Maximal Rate of Entropy Prouction in the V-space: If the force vector A is prescribe, then the actual velocity vector v maximizes the issipation rate L = A v, subject to the sie conition v = A v 0 (24) This principle may be justifie in a way completely analogous to that given by Ziegler [7, 8]. Also, it may be state as an extremum problem with a Lagrangian multiplier, so as to yiel v k { A v j [ v A j j ]} = 0 (25) A i = + 1 Henceforth, we simply put = / + 1. (26) v k

6 894 M. OSTOJA-STARZESKI ownloae By: [University of Illinois] At: 21:04 17 August 2007 Constitutive laws of fractal meia now follow from the above, whereby one can istinguish two funamental cases: compoun or complex thermoynamical processes. The key thing to observe is that, when the issipation function is taken as a functional in erivatives /t ij an /t ij, a number of key relations of Ziegler s theory such as laws governing complex an compoun processes an the associate Onsager Casimir an Legenre transformations carry over to fractal meia. The equation of classical thermoelastoynamics, linking the thermal an mechanical fiels is generalize to cṫ = T 1 k q k (27) where is the coefficient of thermal expansion an subscript (1) inicates the first basic invariant. In the special case of 1 = 0 we have cṫ = k q k (28) which, equivalently, escribes heat conuction in a fractal rigi conuctor. Assuming a Fourier-type heat flow everywhere in the fractal meium, Eq. (27) leas to a generalization of uhamel s ifferential equation of heat conuction ( cṫ = T 1 k K T ) (29) x k while the equation (28) becomes cṫ = k ( K T ) x k (30) where K is the thermal conuctivity of the material. One possible application of the thermoelasticity of fractal meia uner evelopment is the extension of the Maxwell Betti reciprocity relation. Thus, in place of t 1 u 2 A = t 2 u 1 A (31) for an elastic meium, where is a non-fractal omain, we have t 1 u 2 A = t 2 u 1 A (32) for a fractal meium, where is a fractal omain (recall Eq. (7)). In (31) an (32) i = 1 2 enote tractions an isplacements in the first an secon loaing system, respectively. As an example, consier the classical problem of calculation of the reuction in volume V of a linear elastic isotropic boy ue to two equal, collinear, opposite forces F, separate by a istance L. Clearly, (31) gives V = FL/3, where is the bulk moulus. On the other han, for a fractal boy, (32) yiels V = c 2 FL/3 with c given by (14) 1.

7 THERMOELASTICITY AN FRACTAL MEIA 895 ownloae By: [University of Illinois] At: 21:04 17 August 2007 CONCLUSIONS The recent formulation of continuum mechanics of fractal meia ue to Tarasov [5, 6] makes it possible to exten such a theory to issipative fiel phenomena. The path taken is that of Ziegler s thermomechanics. In this article we erive the fiel formulation of secon law of thermoynamics of meia whose mass ensity has a fractal structure. Upon taking the special case of integer imensions of both the spatial omain an its bounary, the resulting relation reuces back to the conventional Clausius uhem inequality. The erivation hinges on the concept of internal (kinematic) variables an internal stresses, as well as the split of the total stress into its quasi-conservative an issipative parts. The issipation function is recognize to be a functional in fractional-type rates of strain an internal variables. Postulating the thermoynamic orthogonality on any lengthscale, allows a erivation of constitutive laws of elastic-issipative fractal meia involving generalize erivatives of strain an stress. e en with an observation that Ziegler s thermomechanics is very suitable for generalization to ranom meia precisely because it allows scale-epenent homogenization of elastic/issipative meia in the vein of Hill conition (e.g., [15]), where (i) either the applie strain or the applie stress are on an equal footing, an (ii) the energy or power of issipation is the key criterion for equivalence between a heterogeneous structure an a smoothing continuum. By contrast, the rational thermomechanics of Truesell s school is not consistent with the Hill conition because the stress is taken as a primary quantity while the energy as a seconary one, e.g., [16]. REFERENCES 1. R. L. Bagley an P. J. Torvik, Fractional Calculus A ifferent Approach to the Analysis of Viscoelastically ampe Structures, AIAA J. vol. 21, pp , Y. Z. Povstenko, Fractional Heat Conuction Equation an Associate Thermal Stress, J. Therm. Stresses, vol. 28, pp , Y. Z. Povstenko, Stresses Exerte by a Source of iffusion in a Case of a Non- Parabolic iffusion Equation, Int. J. Eng. Sci., vol. 43, pp , B. J. est, M. Bologna, an P. Grigolini, Physics of Fractal Operators, Springer, New York, V. E. Tarasov, Continuous Meium Moel for Fractal Meia, Phys. Lett. A, vol. 336, pp , V. E. Tarasov, Fractional Hyroynamic Equations for Fractal Meia, Ann. Phys., vol. 318, no. 2, pp , H. Ziegler, Thermomechanics, Q. Appl. Math., vol. 28, pp , H. Ziegler an Ch. ehrli, The erivation of Constitutive Relations from the Free Energy an the issipation Functions, Av. Appl. Mech., vol. 25, (e. T. Y. u an J.. Hutchinson), pp , Acaemic Press, New York, P. Germain, Q. S. Nguyen, an P. Suquet, Continuum Thermoynamics, ASME J. Appl. Mech., vol. 50, pp , G. A. Maugin, The Thermomechanics of Nonlinear Irreversible Behaviors An Introuction, orl Scientific, Singapore, I. F. Collins an G. T. Houlsby, Application of Thermomechanical Principles to the Moeling of Geotechnical Materials, Proc. Roy. Soc. Lonon A, vol. 453, pp , 1997.

8 896 M. OSTOJA-STARZESKI ownloae By: [University of Illinois] At: 21:04 17 August M. Ostoja-Starzewski, Microstructural Ranomness Versus Representative Volume Element in Thermomechanics, ASME J. Appl. Mech., vol. 69, pp , M. Ostoja-Starzewski, Towars Stochastic Continuum Thermoynamics, J. Non- Equilib. Thermoyn., vol. 27, no. 4, pp , M. Ostoja-Starzewski, Towars Thermomechanics of Fractal Meia, ZAMP, in press, M. Ostoja-Starzewski an Z. F. Khisaeva, Scale Effects in Infinitesimal an Finite Thermoelasticity of Ranom Composites, J. Thermal Stresses, vol. 30, pp , 2007; Keynote Lecture at 7th Intl. Congress on Thermal Stresses, Taipei, Taiwan, June 4 7, J. M. Ball an R.. James, The Scientific Life an Influence of Cliffor Ambrose Truesell III, Arch. Rational Mech. Anal., vol. 161, pp. 1 26, 2002.