On the wave propagation in isotropic fractal media

Transcription

1 On the wae propagation in isotropic fractal media Hady Joumaa & Martin Ostoja-Starzewski Zeitschrift für angewandte Mathematik und Physik Journal of Applied Mathematics and Physics / Journal de Mathématiques et de Physique appliquées ISSN Z. Angew. Math. Phys. OI / s

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3 Z. Angew. Math. Phys. c 011 Springer Basel AG OI /s Zeitschrift für angewandte Mathematik und Physik ZAMP On the wae propagation in isotropic fractal media Hady Joumaa and Martin Ostoja-Starzewski Abstract. In this paper, we explore the wae propagation phenomenon in three-dimensional (3) isotropic fractal media through analytical and computational means. We present the goerning scalar wae equation, perform its eigenalue decomposition, and discuss its corresponding modal solutions. The homogenization through which this fractal wae equation is deried makes its mathematical analysis and consequently the formulation of exact solutions possible if treated in the spherical coordinate system. From the computational perspectie, we consider the finite element method and derie the corresponding weak formulation which can be implemented in the numerical scheme. The Newmark time-marching method soles the resulting elastodynamic system and captures the transient response. Two solers capable of handling problems of arbitrary initial and boundary conditions for arbitrary domains are deeloped. They are alidated in space and time, with particular problems considered on spherical shell domains. The first soler is elementary; it handles problems of purely radial dependence, effectiely, 1. Howeer, the second one deals with general adanced 3 problems of arbitrary spatial dependence. Mathematics Subject Classification (000). 74H45 Vibration. Keywords. Fractal media Wae propagation Spherical shell. 1. Introduction Fractal media are abundant in nature (e.g., rocks, tree leaes, island distributions, clouds) and in liing bodies (e.g., brains, cardioascular systems) [1 3]. Thus, the understanding of their mechanics in the sense of a field theory capable of handling initial-boundary alue problems (IBVP) is sorely needed. Fractal bodies are characterized by highly irregular (nonsmooth) topologies; as a result, the Eucledian geometry and consequently the classical continuum mechanics theory fail to proide an expressie model able to simulate the mechanical behaior under scrutiny. The failure of the continuum theory opens the gate for new approaches, the aim of which is the exploration of the laws that goern the general dynamical behaior of fractal media. In fact, fractal mechanics is at a formatie stage [4 9] in contradistinction to the extensiely explored, adanced, and ubiquitously applied conentional continuum mechanics. A practical and well-known problem that demonstrates the necessity as well as the effectieness of fractal mechanics is the wae propagation in porous media. Indeed a porous material, cannot be truthfully regarded as a continuum because of the successie oid regions spread throughout its olume. The crucial parameter on which the fractal nature of a gien body is to be considered or disregarded in the inestigation, is the length scale. In other words, fractal features must be incorporated in the mechanical analysis only when it is inestigated within a scale range limited from aboe by the oerall macroscopic length of the body and from below by the microscopic scale of the smallest fractal feature [10]. In addition, the total mass M of a gien porous body of characteristic length L does not scale in power law with the Eucledian dimension of the space that contains that body. It neertheless scales with the Hausdorff dimension, which is noninteger and smaller than the Eucledian dimension, i.e., M kl where <<3fora fractal body that is contained in a three-dimensional Eucledian space [11].

4 H. Joumaa and M. Ostoja-Starzewski ZAMP In this research, we report a study of wae propagation in 3 isotropic fractal media. This study, although not directly connected with a real application, promotes the general understanding of wae motion and elastodynamics in fractal bodies, the issue that allows scientists and engineers to incorporate this analysis into some mechanics problems where fractal effects are influential and their negligence is erroneous. The 3 fractal wae equation is deried following the approach of homogenizing fractional integrals into the Eucledian space through dimensional regularization [4,5]. The full deriation of the wae equation is proided in the appendix A. This equation, expressed in terms of conentional (nonfractional) deriaties, is gien by (Γ being the Euler Gamma function) p t = [ c 3 Γ ( ) ] [(3 Γ ( ) 3 R ) R p + R p ] (1) While linear in principle, Eq. (1) inoles additional complex terms reflecting the fractal nature of the medium, hindering the chances of obtaining analytical solutions to general problems. In addition to the wae celerity c, the fractal dimension (a medium-dependent constant) plays a significant role in wae propagation phenomena. We can readily erify that this equation reduces to the classical 3 wae equation in the classical continuum case. ue to the isotropic nature of homogenization, the position ector R, appears in the equation. For this reason, obtaining analytical solutions to this equation in the Cartesian or cylindrical coordinate system is impossible. Howeer, the spherical coordinate system allows one to decompose the equation into three independent modal equations, each of which can be soled analytically. The objectie of this work is to first introduce the modal decomposition by explaining the mathematical steps leading to the exact solutions to the reduced modal equations. As an application, we consider a spherically symmetric eigenalue problem inoled with radial wae propagation on a spherical shell, which we treat analytically and asymptotically. Then, we construct the first Finite Element Method (FEM) soler which handles problems of radial dependence only, thus numerically demonstrating this special eigenalue problem. This soler is alidated in space and time with the exact solution. Finally, the general soler which handles full 3 problems is presented, and all applied procedures leading to the final elastodynamic model are explained.. Modal decomposition The aim of the modal decomposition is to be able to formulate analytical solutions to general problems using the technique of modal superposition. For the continuum case, modal decomposition in Cartesian (x, y, z), cylindrical (r, θ, z), and spherical (r, θ, φ) coordinates generates three decoupled ordinary differential equations [1, Chap. 9]. In the fractal domain, the first two frames of references are futile; they fail to produce independent equations. Fortunately, the spherical system produces the desirable decoupled equations. We start our modal analysis by separating time and space ariables and introducing the frequency ω, thus p (R,t)=P (R) e iωt () Substituting () into(1), we obtain what we will refer to as the fractal Helmholtz equation expressed in compact and expanded forms, respectiely, P + k P = 0 (3a) [ c 3 Γ ( ) ] [(3 Γ ( ) 3 R ) R P + R P ] + k P = 0 (3b) where the waenumber k = ω/c. Expressing the dependent ariable P in () as a product of three independent modal functions, P (r, θ, φ) F (r) G (θ) H (φ), and substituting back into (3b), we obtain

5 On the wae propagation in isotropic fractal media three independent Sturm Liouille equations for a 3 eigenalue problem expressed as φ : H + m H = 0 ] θ : G + [l G tan θ + (l +1) m sin G = 0 θ [ ] r : r F +(5 ) rf + (λk) r 4 l (l +1) F = 0 (4a) (4b) (4c) Recall that, in spherical coordinates, we hae r = R (5a) R P = r P (5b) r P = 1 ( r r P ) ( 1 + r r r sin θ P ) 1 P + sin θ θ θ r sin θ φ (5c) The constant λ showing in (4c) will always be pre-multiplied by the wae number k, it is defined as follows λ = Γ ( ) 3 Γ ( ) 3 (6) and l and m are two independent integers characterizing the modes, so-called mode numbers. Let us physically interpret these equations. First, we realize that the Hausdorff dimension appears in the radial equation only. This is because Taraso s approach to homogenization relies solely on the radial distance from the origin to the point of interest, so that the azimuthal and transcendental modes are free of fractal effects [4]. Consequently, the azimuthal and transcendental equations are exactly identical to those of the continuum problem, see [1, p. 51]. In addition, upon setting = 3, the radial mode equation reduces to the spherical Bessel equation, which offers an additional erification of this analysis. Haing realized that the radial mode equation is similar to the spherical Bessel equation triggers an important hint in forming its solution as will be later explained. The analytical solutions to the modal equations are now discussed. Concerning (4a), it has the wellknown harmonic solution H (φ) =h 1 sin (mφ)+h cos (mφ) (7) The transcendental equation (4b) can be transformed into a Legendre differential equation if we apply the transformation η = cosθ. The transformed equation is ( 1 η ) d G dg η [l dη dη + (l +1) m 1 η ] G = 0 (8) One solution to this equation is the Legendre function P m l (η). Note that, if we set m = 0, we obtain the Legendre polynomial, denoted as P l (η). The other homogeneous solution to (8) is not analytic at η = ±1. This renders it physically meaningless, for this problem, therefore the Legendre function is the solely considered solution for the spherical wae problem. The radial mode equation (4c) is the most challenging to sole no closed form solution is obious at a glance. We propose a series form solution for this equation so as to obtain two linearly independent homogeneous solutions gien as ( ) F ν (1) (r, k, ) =r 4 λkr J ν (9a) ( ) F ν () (r, k, ) =r 4 λkr Y ν (9b) F (r) =f 1 F ν (1) (r)+f F ν () (r) (9c)

6 H. Joumaa and M. Ostoja-Starzewski ZAMP (a) (b) Fig. 1. The fractal radial harmonic function of first kind, shown for different, k =1.a l =0b l =1 (a) (b) Fig.. The fractal radial harmonic function of second kind, shown for different, k =1.a l =0b l =1 4l(l+1)+( 4) where the constant ν = ( ) > 0, while J ν and Y ν are, respectiely, Bessel functions of the first and second kind of order ν. Effectiely, in (9a) and (9b) we hae introduced the fractal radial harmonic functions of the first and second kind. Figures 1 and show some plots of these two functions for seeral alues of and l. The unnormalized sinc function is reproduced for the case =3,l =0. Haing soled all three Eq. (4a 4c), we can now express the solution to wae propagation problems in terms of these functions while satisfying the necessary initial and boundary conditions as shown in the next section.

7 On the wae propagation in isotropic fractal media (a) (b) Fig. 3. The spherical shell domain in two iews along with the spatial meshing along the radial direction for the one-dimensional reduced problem. a Outer iew of the shell b Cross-section iew of the shell 3. Fractal IBVP case study We demonstrated in the modal decoupling analysis that fractal effects are obserable in the radial harmonics only. Therefore, 3 problems can be reduced to 1 (in the radial coordinate) while presering the fractal effects that we intend to explore. As an application, a Sturm Liouille (eigenalue) problem inoled in the wae propagation inside a spherical shell is considered. The analytical solution is presented in terms of the fractal radial harmonic functions described before. In addition, the WKB asymptotic method is applied, proiding an approximate but meaningful solution to the problem Exact solution Consider the spherical shell centered at the origin with inner radius R in = 1 and outer radius R out =, as shown in Fig. 3. Suppressing the dependence of θ and φ on p, the resulting wae equation for this reduced problem simplifies to p ( c ) [ t = r 4 (5 ) r p ] λ r + r p r (10) The corresponding reduced fractal Helmholtz equation is r P +(5 ) rp + λ k r 4 P = 0 (11) The boundary conditions are of irichlet type at both ends of the domain. For simplicity, we choose P ( ) R in = P (Rout ) = 0 The general homogeneous solution for P (r) is obtained in terms of harmonic functions as P n (r) =a n F ν (1) (r, k n,)+b n F ν () (r, k n,) (1) where ν = 4 ( ) The nontriial alues (eigenalues) of k n, along with the corresponding relation between constants a n and b n, are determined through the application of boundary conditions which result in the

8 H. Joumaa and M. Ostoja-Starzewski ZAMP Fig. 4. First three orthonormal modal functions. Note the remarkable match between the exact and asymptotic solutions following relations F ν (1) (1,λk n,) F ν () (,λk n,) F ν (1) (,λk n,) F ν () (1,λk n,) = 0 (13a) a n = F ν () (1,λk n,) b n F ν (1) (1,λk n,) (13b) Equation (13a) is of transcendental nature; it admits infinite solutions in k. A general root-finding algorithm for nonlinear algebraic equations (bisection) is applied to sole for the eigenalues. Rewriting (11) in the Sturm Liouille form [13, p. 9] as [ ] d 5 dp r +(λk) r 1 P = 0 (14) dr dr leads to the orthonormality condition expressed as The first three orthonormal modal functions are shown in Fig P n P m r 1 dr = δ mn (15) 3.. Asymptotic modal analysis The determination of the higher frequency modes can be performed effectiely through an asymptotic approach. The WKB method is a well-known asymptotic procedure that predicts higher modes in eigenalue problems particularly when the formulation of the exact solution represents a mathematical challenge. The WKB method is thoroughly explained in reference [13, Chap. 10]. We will briefly present the steps followed to reach the asymptotic solution. The WKB method expresses the solution P (r) as an infinite series of the form P (r) =e iλkσ(r) j=0 Ψ j (r) (λk) j (16)

9 On the wae propagation in isotropic fractal media Table 1. Exact and asymptotic (WKB) solution of the eigenalue problem for the first fie modes, =.5 n λk n λ k n Rel. error a n bn A n B n Rel. error Substituting this form into (11), we obtain a series of decreasing power of k. Satisfying each and eery order of k, the following relations are obtained O ( k ) : r 6 σ +1=0 σ (r) =± r (17a) O (k) : rψ 0 +Ψ 0 =0 Ψ 0 (r) = 1 (17b) r d(rψ j+1 ) = i d ( ) r 5 Ψ j (17c) dr r dr Equation (17c) proides the recursie relation to sole for Ψ j+1 knowing Ψ j. Truncating the series to one term only (Ψ 0 ), the resulting nth mode corresponding to the nth waenumber k n is approximated by: P n (r) = 1 [ ( λkn r ) ( λkn r )] A n cos + B n sin (18) r Applying the boundary conditions on the asymptotic solution P n, a transcendental equation for the waenumber k n and the (A n,b n ) relation are obtained as follows nπ ( ) k n = λ ( (19a) 1) ( ) A n nπ = tan B n (19b) 1 If the asymptotic modes were to be orthonormalized by the same condition of (15), the resulting form is obtained ( ) ( ) 1 P n (r) = 1 r sin nπ r 1 (0) 1 The exact and asymptotic solutions for the case =.5 are listed in Table 1. We performed the analysis of the first fie modes. Note the remarkable matching between both solutions at higher modes. Een though the WKB method targets higher modes, the lower modes are still well approximated. The plot of the modal functions in Fig. 4 illustrates the significance of the asymptotic approximation. Interestingly, the components of the WKB homogeneous solution shown in (18) are nothing but the asymptotic expansion of the harmonic functions F (1) and F () at large k. References [13, p. 57] and [1, App. A4] proide a detailed asymptotic expansion to the Bessel functions leading to a first order approximation for F (1) and F () gien as F (1) 1.5 F () 1.5 (1) 1 F 1.5 (r, k, =.5) = 1 r πλk cos ( λk r ) as k + () 1 F 1.5 (r, k, =.5) = 1 r πλk sin ( λk r ) as k + (1a) (1b) As such, we can perform the asymptotic analysis by applying the boundary conditions on P n (r) = A n F (1) + B n F () and reproduce the same results of the WKB work. The inconenience of this approach

10 H. Joumaa and M. Ostoja-Starzewski ZAMP lies in its reliance on the exact solution, neertheless, it remains significant through its strong approximation of the higher modes while aoiding the differential equation soling dictated by the WKB and the implementation of the root finding algorithm required by the exact analysis. In conclusion, the two asymptotic approaches are consistent, both proiding a meaningful prediction to higher modes. 4. FEM for reduced problem In this section, we introduce the FEM as a numerical alternatie to sole the reduced problem of wae propagation. We discuss the FEM scheme construction through the weak formulation. We next sole transient problems based on modal excitation. The numerical results are erified and conergence is confirmed through error analysis FEM construction The reduced wae equation is presented in (10). We seek an FEM transient solution to the reduced problem whose domain and irichlet boundary conditions are already presented in the modal analysis. On the temporal side, the system is excited with a particular mode, i.e., p 0 = P n, a modal function. As such, the expected response is a single frequency harmonic, corresponding to the excited mode. The weak formulation is obtained following standard FEM procedures explained in [14, 15]. We introduce an admissible function ˆp, continuous throughout the domain of interest and satisfying the boundary conditions. Consider the strong (differential) form of (10), then multiplying both sides by ˆp and integrating oer the domain, the weak (integral) form is generated. ( ) λ p 5 p ˆp dr = (5 ) r ˆp dr + r 6 p ˆp dr () c t r r Integrating by parts and performing additional calculus, we obtain ( ) λ p 5 p p ˆp dr = ( 1) r ˆpdr + c t r r r6 ˆp ] Rout R in p r ˆp r r6 dr (3) Upon applying the boundary conditions, the middle term on the right-hand side anishes. The integration is then performed on an elemental basis e, where e e =. The weak form is thus rewritten as ( ) λ Ne c e Ne p t ˆp dr +(1 ) e Ne r 5 p r ˆp dr + e p ˆp r r r6 dr = 0 (4) For simplicity, the domain is meshed with a -node linear element. The integration can be performed either analytically or by implementing the Gaussian quadrature rule [14, Chap. 3]. The former method is chosen because the latter one fails to produce the exact alue due to the noninteger powers present in the kernels. The elemental integration of the first term of (4) generates the elemental mass matrix M e, the second term generates the fractal elastic matrix L e, while the third term generates the regular elastic matrix K e. The nomenclature of these matrices will be better appreciated when discussing the 3 formulation in the upcoming section. Summing all the resulting elemental matrices, the discrete formulation is produced N e M e p + ( Ne ) N e L e + K e p =0 M p +(L + K) p = 0 (5)

11 On the wae propagation in isotropic fractal media The obtained model is constitutiely similar to its continuum elastodynamic counterpart which balances inertial forces with elastic ones. As such, the numerical treatments of the continuous system remain applicable in the fractal field. In this problem, the mass matrix M and the regular elastic matrix K are symmetric positie definite (SP), while the fractal elastic matrix L is not. Thus, the total stiffness tensor L+K is in general not SP. These remarks can be proed by obsering the elemental forms of these matrices. Haing formulated the discrete elastodyanmic model, eigenalue (frequency) analysis in addition to transient analysis can be conducted. The latter topic is discussed next. 4.. Time-marching solution We explain the time-marching scheme we applied to sole (5) and obtain the transient response. The numerical analysis literature ([14, Chap. 9],[15, Chap. 9]) describes a ariety of methods to sole this linear hyperbolic system, whereby we note the Newmark method. It can be implemented either implicitly or explicitly depending on its parameters adjustments. For our analysis, we hae chosen the trapezoidal (implicit) case of Newmark s method; it is unconditionally stable, does not generate any dissipatie errors, and has a good accuracy. In conclusion, the numerical algorithm of the trapezoidal method can be implemented for fractal problems; the matching between numerical and analytical solutions is a clear proof of the method s alidity. The first three modal excitations are reported in Fig. 5. The error analysis in time and space is conducted; conergence plots are shown in Fig. 6 where the L norm of the error is plotted with respect to the element size and time step. 5. FEM for 3 problems In this section, we extend the application of the FEM to general 3 problems. The formulation is conceptually similar to that of the reduced problem but mathematically more complex. We adhere to the same problem soled on the spherical shell to ease the perception of the obtained solutions. Needless to say, the designed soler can handle general problems of arbitrary domain and boundary conditions FEM construction The weak form for this problem is obtained by considering an admissible function ˆp, multiplying both sides of the strong form shown in (1) by this function, and then integrating both sides of the equation oer the domain of interest,. The corresponding weak form is gien as ( ) λ c p t ˆpd = (3 ) ˆp R 4 R pd + ˆp R 6 pd (6) Calculus rearrangements are made for the kernel of the third integral to engage the admissible function into the gradient operator. Applying the Green Gauss theorem to this integral, we obtain ˆpr 6 p = (ˆpr 6 p ) r 6 ˆp p (6 ) ˆpr 4 ˆR p ˆpr 6 p d = ˆpr 6 p ds r 6 ˆp p d +( 6) ˆpr 4 R p d S (7a) (7b)

12 H. Joumaa and M. Ostoja-Starzewski ZAMP (a) (b) (c) Fig. 5. The system numerical (1 FEM) and exact solutions for the first three modal excitations. a First mode excitation b Second mode excitation c Third mode excitation Applying the irichlet boundary conditions on the inner and outer spherical boundaries, the surface integral anishes. Incorporating (7b) into(6), the weak form becomes ( ) λ p ˆp d = ( 3) ˆpr 4 R p d r 6 ˆp p d (8) c t The integration is conducted on the elemental leel. We thus hae ( ) λ Ne p c t ˆp d N e N e e = ( 3) ˆpr 4 R p d e r 6 ˆp p d e (9) e e e The aboe equation is similar to (4). The three goerning matrices (M, L, and K) introduced in the reduced problem are easily identified. For the continuum case ( = 3), the middle term, which produces the fractal elastic matrix L anishes. It is clear by now why L is named as such: it disappears

13 On the wae propagation in isotropic fractal media (a) (b) Fig. 6. Conergence plots in time and space. The order of accuracy is shown for each mode. a Time conergence b Space conergence (a) (b) Fig. 7. The system numerical (3 FEM) and exact solutions for the first two modal excitations. a First mode excitation. b Second mode excitation from the formulation when fractal effects are absent. On the other hand, the symmetric regular elastic matrix K regains its conentional form and reflects regular elastic effects in the case of continuous medium. The mass matrix M is indifferent to fractal effects; this is a direct consequence of the homogeneity (constant density) property of the fractal medium. The procedure to derie the elemental matrices requires integrating oer 3 elements, 4-node tetrahedral being the simplest choice. The integrals cannot be ealuated exactly; the use of a quadrature method becomes a must. A useful mathematical algorithm for the deriation of quadrature points and corresponding weights oer tetrahedral elements is proided in [16]. Assembling the goerning matrices, the discrete elastodynamic form presented in (5) is obtained but with a much larger number of degrees of freedom. The Newmark method is applied to sole for the transient solution. The first two modal excitations are simulated and the response is shown in Fig. 7.

14 H. Joumaa and M. Ostoja-Starzewski ZAMP 6. Conclusions The applicability of analytical and numerical methods in understanding the wae propagation phenomena in isotopic fractal media is demonstrated in this research work. The remarkable consistency obsered in all the results alidates all these dierse approaches to soling the problem. In pursuing an analytical solution of the reduced 3 problem, we introduce the fractal radial harmonic functions of the first and second kind, a generalization of Bessel functions of the first and second kind. The elastodynamic analysis embedding modal decomposition, soling eigenalue system, and predicting transient response is achieed for this noncontinuum type of material. The deelopment of a general 3 soler, implementing the robust FEM, and applying the stable and conergent trapezoidal time-stepping scheme, allows solution of IBVPs in media with fractal geometries. Adanced fractal materials of greater complexity, encountered in real life applications, can now be modeled and inestigated by further building upon the work presented here. Acknowledgments This research was made possible by the support of Sandia-TRA (grant HTRA BRCWM) and the NSF (grant CMMI ). Appendix A: deriation of the wae equation The wae equation in isotropic fractal media is deried by considering small perturbations to the goerning balance laws describing the fractal medium dynamics. We strongly encourage the reader to refer to reference [4], where all the dynamic laws for the fractional continuous medium model are deeloped. Two releant mathematical operators are first defined. The fractional spatial deriatie (gradient) operator k is gien as k [A] = 3 Γ ( ) Γ ( ) 3 A 3 R (30) x k and the fractional material deriatie is defined as ( ) d [A] = A dt t + u k k A (31) For the three-dimensional isotropic fractal media, the mass balance law is presented as follows ( ) d ρ = ρ k u k (3) dt while the momentum balance law is gien as ( ) d u k = f k 1 dt ρ k p (33) Assuming isentropic (reersible and adiabatic) processes for small amplitude oscillation (small disturbance ariables: u k,p,andρ ) about an equilibrium point ( p, ρ), we get, for a fluid of compressibility κ p p = κ ρ ρ p = κ ρ ρ (34) ρ Introducing the gradient operator into (33), we hae [ ( ) ] d k ρ u k = k k p (35) dt

15 On the wae propagation in isotropic fractal media Neglecting the higher order terms (conectie ones and ρ u k product), we simplify the aboe equation to [ ] ρ uk k = k k p (36) t oing the same for the mass balance in (3) we obtain, ρ t = ρ k u k (37) ifferentiating the aboe equation with respect to time, ρ t = ρ ( ) ( ) t k u k = ρ uk k (38) t Combining (35) and (38), we finally obtain ρ t If we assign the isotropic wae celerity c = κ ρ, and expand the k the fractal wae equation presented in (1) = k k p p t = κ ρ k k p (39) operator twice on p, we reproduce References 1. Mandelbrot, B.B.: The Fractal Geometry of Nature. W.H. Freeman & Co, San Francisco (198). Barnsely, M.: Fractals Eerywhere. Morgan Kaufmann, Los Altos (1993) 3. Le Mehaute, A.: Fractal Geometries Theory and Applications. CRC Press, Boca Raton (1991) 4. Taraso, V.E.: Fractional hydrodynamic equations for fractal media. Ann. Phys. 318/, (005) 5. Taraso, V.E.: Wae equation for fractal solid string. Mod. Phys. Lett. B 19(15), (005) 6. Ostoja-Starzewski, M.: Towards thermomechanics of fractal media. ZAMP 58(6), (007) 7. Ostoja-Starzewski, M.: Extremum and ariational principles for elastic and inelastic media with fractal geometries. Acta Mech. 05, (009) 8. Ostoja-Starzewski, M.: On turbulence in fractal porous media. ZAMP 59(6), (008) 9. Ostoja-Starzewski, M., Li, J.: Fractal materials, beams and fracture mechanics. ZAMP 60, 1 1 (009) 10. Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, England (003) 11. Hastings, H.M., Sugihara, G.: Fractals: A User s Guide for the Natural Sciences. Oxford Science Publications, Oxford (1993) 1. Kinsler, L.E., Frey, A.R.: Fundamentals of Acoustics. Wiley, New York (000) 13. Bender, C.M., Orszag, Steen A.: Adanced Mathematical Methods for Scientists and Engineers. Springer, Berlin (1999) 14. Hughes, T.J.R.: The Finite Element Method. oer Publications, New York (000) 15. Bathe, K.J.: Finite Element Procedures. Prentice Hall, NJ (198) 16. Rathold, H.T., Venkatesudu, B., Nagaraja, K.V., Shafiqul Islam, Md.: Gauss Legendre-Gauss Jacobi quadrature rules oer a tetrahedral region. Appl. Math. Comp. 190, (007) Hady Joumaa and Martin Ostoja-Starzewski epartment of Mechanical Science and Engineering Uniersity of Illinois at Urbana-Champaign 106 W. Green St. Urbana IL, USA hjoumaa@illinois.edu Martin Ostoja-Starzewski martinos@illinois.edu (Receied: ecember 30, 010)