Article

Elastodynamics in micropolar fractalsolids

Mathematics and Mechanics of Solids2014, Vol. 19(2) 117–134©The Author(s) 2012Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/1081286512454557mms.sagepub.com

Hady Joumaa and Martin Ostoja-StarzewskiUniversity of Illinois at Urbana-Champaign, USA

Paul DemmieSandia National Laboratories, New Mexico, USA

Received 15 June 2012; accepted 18 June 2012

AbstractThis research explores elastodynamics and wave propagation in fractal micropolar solid media. Such media incorporatea fractal geometry while being modelled constitutively by the Cosserat elasticity. The formulation of the balance lawswhich govern the mechanics of fractal micropolar solid media is presented. Four eigenvalue-type elastodynamic problemsadmitting closed-form analytical solutions are introduced and discussed. A numerical procedure to solve general initialboundary value wave propagation problems in three-dimensional micropolar bodies exhibiting geometric fractality isthen applied. Verification of the numerical procedure is discussed using the analytical solutions.

KeywordsElastodynamics, fractal solid, micropolar elasticity

1. IntroductionBodies exhibiting non-smooth or highly irregular geometric features are classified as ‘fractals’. Fractals areabundant in nature, e.g. rocks, tree leaves, and even in living entities such as the neural structure or the surfaceof the human brain [1–3]. Geometric fractals are idealized sets described as self-invariant, i.e. they are math-ematically constructed in such a way that every small part in the set is a rescaled copy of the larger ‘mother’part. The Koch curve and the Cantor set are well-known examples of geometric fractals. Fractals encounteredin real applications do not possess this powerful property of self-similarity. Nevertheless, they display a weakeror statistical version of self-similarity where randomness plays a key role in generating the body’s geometry[4, 5]. For this reason, fractal models have been adopted to characterize random and even porous materials lead-ing to mechanics based on fractal concepts [6, 7]. Fractal sets are characterized by the Hausdorff dimension D,which is the scaling exponent characterizing the fractal pattern’s power law. For regular fractals, D is a constantand it is mathematically determined. But in the case of random fractals, such as the one shown in Figure 1, Dbecomes a random variable and its evaluation is restricted to statistical methods [5].

The mechanics of fractal media is still in a developing stage. It is inadequately explored and rarely uti-lized in comparison to continuum mechanics. Nevertheless, fractal mechanics can generate elegant models forproblems where continuum mechanics fails particularly for bodies with highly irregular geometries [8–12]. Inour research, we adopt the elastodynamic model developed in [13–15] to investigate wave motion in fractal

Corresponding author:Hady Joumaa, Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206 W. Green St. # 244, Urbana, IL61801, USA.Email: hjoumaa2@illinois.edu

118 Mathematics and Mechanics of Solids 19(2)

Figure 1. A two-dimensional randomly generated Cantor set. Fractal domains of this type can be analysed in the average sense byadopting micropolar elasticity. A generalization to mechanics of random fractal media (accounting for scatter in geometry and scatterin response) is presently under development [17].

(non-continuous) micropolar (non-classical) solid materials. This model is useful for solving complex mechan-ics problems involving fractal materials composed of microstructures of inherent length scale, generalizingthe universally applied classical theory of elastodynamics for continuous bodies. An inquiry is raised about theadvantages of incorporating a fractal geometry and a micropolar elasticity in one material model. Adopting theclassical Hookean constitutive law simplifies the problem and eliminates the hassle of considering additionaldegrees of freedom (microrotations) and curvatures and, at the same time, removes the angular momentum bal-ance equation from the analysis. However, the major mechanical attribute of a fractal geometry in studying theelastodynamics of solids is the failure of the application of Hooke’s law which is manifested by the symmetryof the stress tensor [13, 14]. Indeed, angular momentum balance, as will be shown, cannot be satisfied unlessthe stress tensor’s symmetry property is ruled out. Only waves of the dilatational type can be explored in theHookean elasticity’s framework as shown in [16] where angular momentum balance is trivially satisfied withvanishing shear stress components. It thus becomes evident that the introduction of fractal effects into the prob-lem necessitates the adoption of a non-classical elastic constitutive model, the Cosserat model, for the balancelaws to be satisfied.

The wave propagation problem is governed by linear and angular momentum equations which lead to anextended version of the three-dimensional (3d) Navier field equations whose variables are the linear displace-ments and microrotations. Owing to the complexity of the resulting field equations, general modal analysisbased upon the Helmholtz decomposition cannot be performed analytically. As a result, exact solutions, crucialfor the numerical solver’s verification, seem unfeasible at first sight. However, particular problems, in whichthe dependence of the displacement and microrotation variables on the Cartesian coordinates is suppressed,generate a system of decoupled equations that is solvable analytically and can be used for verification.

The development of the fundamental balance laws in the fractal framework requires the application of ahomogenization process referred to as ‘dimensional regularization’ which transforms fractional integrals overfractal sets to equivalent continuous integrals over Euclidean sets [18]. This regularization results in having thebalance laws expressed in continuous form, thereby simplifying their mathematical manipulation both analyti-cally and numerically. A product measure ci is utilized to achieve this transformation. The mathematical analysisdemonstrating the application of ci to regularize the balance laws is thoroughly explained in [13, 14]. The dif-ferentiation process in this ‘continuous fractal model’ has to be altered to preserve mathematical consistency.

Joumaa et al. 119

Figure 2. Layout of a Carpinteri column, the physical domain on which analytical and numerical elastodynamic problems are solved.

In brief, the regularized fractal derivative operator and the product measures are defined as

∇Di = 1

ci

∂

∂xi(no sum on i), i = 1, 2, 3 (1a)

ci (xi) = Di (Li − xi)Di−1 , 0 < Di ≤ 1 (1b)

cv = c1c2c3 (1c)

where Li and Di are the length and the Hausdorff dimension in the ith direction, and cv is the product of allthree product measures. In all our work, we consider idealized problems characterized by an anisotropic fractalstructure embedded in a 3d topological domain. This domain, shown in Figure 2, is composed of a Sierpinskicarpet in the (x1x2) cross-section where L1 = L2 = 1, and swept longitudinally (along x3, L3 = 2) in conjunctionwith a Cantor set [19]. The corresponding Hausdorff dimensions for this column are determined to be [20]

D1 = D2 = 1

3

ln 18

3

D3 = ln 2

ln 3

(2)

The consideration of a parallepipedic (box shaped) domain is imperative for problem solving as will be assertedlater from the field equations. Indeed, the domain characteristics Li and Di are inherent in the product measure’sdefinition and consequently in the entire formulation. As a result, the field equations are meaningful only in aCartesian coordinate system.

2. Elastodynamic modelling proceduresIn this section, we apply the constitutive, kinematic and kinetic balance laws to explain the general elastody-namic model under investigation. The outcome of this work is two field equations that constitute the cornerstoneon which the entire analytic and computational work is based. The detailed formulation is presented in [14].

The development of the micropolar theory of elasticity is mainly attributed to the work of the Cosseratbrothers in 1909 [21]. With regard to kinematics, this theory grants every ‘quasi-infinitesimal’ point in thecontinuum two degrees of freedom: a displacement ui and a microrotation φi. Accordingly, two independent

120 Mathematics and Mechanics of Solids 19(2)

measures for the continuum’s deformation are generated: the strain tensor γij and the curvature tensor κij. Thesetensors are defined as (eijk being the Levi-Civita permutation tensor)

γij = ∇Di uj − ekij

φk

ci(no sum on i) (3a)

κij = ∇Di φj (3b)

The kinetic interaction between ‘quasi-infinitesimal’ elements is not limited to forces as is the case withHookean (classical) elasticity; it includes moment loading too. As a result, a couple-stress tensor μij asso-ciated with this loading, must be incorporated into the fundamental balance laws. The Cosserat force-stresstensor, or simply stress tensor τij, along with the newly introduced couple-stress tensor μij are, for the case ofsmall deformation, linearly related to the strain and curvature tensors [21],

τij = C(1)ijklγkl (4a)

μij = C(2)ijklκkl (4b)

where C(1)ijkl and C(2)

ijkl are two elastic moduli tensors that vary in complexity. For the case of isotropic continua,they are defined as (δij being the Kronecker delta)

C(1)ijkl = (β − α)δjkδil + (β + α)δjlδik + λδijδkl (5a)

C(2)ijkl = (ψ − ε)δjkδil + (ψ + ε)δjlδik + ηδijδkl (5b)

Here β and λ are the usual Lamé’s constants, measured in Pa, ψ , ε, and η are micropolar constants, measuredin Pa.m2 and α is also a micropolar constant but measured in Pa. Consequently, the direct stress versus strainand couple-stress versus curvature relations become

τij = (β + α)γij + (β − α)γji + λγkkδij (6a)

μij = (ψ + ε)κij + (ψ − ε)κji + ηκkkδij (6b)

This theory, in contrast to classical continuum theories, possesses an intrinsic length scale that can be easilydetected if we scrutinize the units of the components of the stiffness tensors. This remark explains the signifi-cance of the micropolar theory in modelling problems involved with finite microstructures (e.g. granular media,lattice structures, liquid crystals).

In classical elasticity problems, the angular momentum balance is ignored in the mathematical analysis sinceit is satisfied through the symmetry property of the Cauchy stress tensor. However, in micropolar elasticity, τij

and μij are in general non-symmetric. As a result, the angular momentum equation becomes an indispensablepart of the problem and has to be considered alongside with the linear momentum balance to fully achieve theelastodynamic setting of the problem. According to [14] they are

ρui = ∇Dj τji (7a)

Iφi = ∇Dj μji + eijk

τjk

cj(7b)

and in terms of the displacement and rotation fields

ρui = (β + α)

[∇D

j ∇Dj ui − ∇D

j

(ekjiφk

cj

)]+ (β − α + λ) ∇D

i ∇Dj uj

+ (β − α) ∇Dj

(ekijφk

ci

)(no sum on i)

(8a)

Joumaa et al. 121

Iφi = (ψ + ε) ∇Dj ∇D

j φi + (ψ − ε + η) ∇Di ∇D

j φj − (β + α)φi

∑j �=i

1

c2j

+ (β + α) eijk

∇Dj uk

cj+ (β − α) eijk

∇Dk uj

cj+ 2 (β − α)

φici

cv(no sum on i)

(8b)

where ρ is the mass density and I is the mass moment of inertia assuming, and for simplicity, an isotropic inertiatensor (Iij = Iδij). Clearly, no Helmholtz decomposition seems viable, the issue that renders the mathematicalanalysis cumbersome. In the next section, we will impose the kinematics for some problems to generate ana-lytical solutions. We verify the correctness of our work by examining the case where Di = 1 for all i in whichfractal effects vanish and the field equations proposed by Eringen for elastodynamics in micropolar continuaare reproduced [21]. For more details, review Chapter 5 of [21] where a variety of fundamental problems inwave motion in micropolar media are discussed. From this perspective, this research builds upon the findingsof Eringen by extending the exploration of micropolar elastodynamics into fractal structures.

3. Analytical approachIn this section, we consider several special problems for which the reduced field equations admit analyticalsolutions. These problems may not always be physically meaningful. Nevertheless, their solutions are math-ematically valuable, especially when it comes to verifying the computational solver. For some problems, thekinematic assignment is not sufficient to obtain closed-form solutions and additional constraints on the stiffnessconstants have to be enforced to fulfil this purpose. The approach in analysing the following four problems isto achieve a modal decomposition. Thus, all of these problems are of eigenvalue type. The first two problemsare similar because they both reduce to solving a 1d modal equation, while the last two are more complex byrequiring the solution of a higher-dimensional modal equation.

3.1. Dilatational wave problem

Consider the elastodynamic problem where the following ui and φi fields are imposed

ui ≡ u (xi) (9a)

φi ≡ 0 (9b)

This kinematic setting actually models the propagation of waves that are of dilatational type. This problemis thoroughly discussed in [16]. We briefly introduce it here because its modal equation is fundamental forall subsequent problems. Applying the prescribed kinematics into Equations (3) and (6), the only non-zerocomponents are the axial stress and strain tensors. Thus, the induced deformation is volumetric (dilatation orcontraction) and non-distortional; the problem designation is consistent with that introduced in Chapter 5 of [22]where waves in continuous domains are investigated. Having the curvature, the shear strain, and consequently,the shear stress and the couple-stress components identically zero, micropolar effects become abolished, theissue that makes the consideration of this particular problem in the Hookean elasticity paradigm fully consistentwith the micropolar study. The angular momentum balance stated in Equation (8b) is trivially satisfied, while thelinear momentum equation (8a) branches into three identical decoupled equations, one for each displacementvariable. Consider the governing equation for u1

ρ

λ+ 2βu1 = ∇D

1 ∇D1 u1 = 1

c1

(u1,1

c1

),1

(10)

Performing the Fourier analysis by setting u1 (x1, t) = U1 (x1) ejωt, (j = √−1), we obtain an eigenvalue problemgoverned by the 1d fractal Helmholtz equation. The modal function U1 (x1) and its corresponding wavenumberk obey the following equation (ϑd is the dilatational wave speed)

k2U1 = −∇D1 ∇D

1 U1 = − 1

c1

(U,1

c1

),1

(11a)

122 Mathematics and Mechanics of Solids 19(2)

k = ω

ϑd, ϑd =

√λ+ 2β

ρ(11b)

The general modal solution for the eigenvalue problem, derived in detail in Appendix A, is given by

U1 (x1; k) = C1f1 (x1, k) + C2f2 (x1, k) (12)

where f1 and f2 are the homogeneous solutions of the fractal Helmholtz equation, also denoted as fractalharmonic functions,

f1 (x, k) = cos[k (L − x)D]

(13a)

f2 (x, k) = sin[k (L − x)D]

(13b)

These solutions for u2 (x2, t) and u3 (x3, t) have the same functional form of u1, but with a different fre-quency which is L and D dependent. The application of the problem boundary conditions (BCs) determinesthe eigenvalue k, the constants C1 and C2. Clearly, for D = 1, the fractal harmonics become the standard har-monic functions which are indeed the solutions for the continuum case. In the case of homogeneous BCs, theeigenvalue problem presented in Equation (11a) admits the following modal orthonormality property in everydirection, ∫ Li

0UmUnci (xi) dxi = δmn (14)

where Um = Um (xi, km) is the mth mode in the ith direction, and Un is the nth mode for the same direction.

3.2. Torsional wave problem

Consider the problem where the displacement field is suppressed to zero and the microrotation field is permittedin such a way that every rotation solely depends on the direction about which it occurs. Mathematically, we have

ui ≡ 0 (15a)

φi ≡ φi (xi) (15b)

This problem resembles that discussed in the previous section with slight differences. The shear componentsof the curvature and couple-stress tensors are all identically zero. The axial components of the strain and stresstensors are also zero (rendering the deformation equivoluminal), but not the shear components. As a result, theelastodynamic equations for the rotation field cannot be solved exactly unless certain restrictions on the elasticmoduli are enforced. For example, if we consider the governing equation for φ1, we have

Iφ1 = (2ψ + η) ∇D1

(∇D1 φ1

) + φ1

[2β − α

c2c3− (β + α)

(1

c22

+ 1

c23

)](16)

The presence of c2 and c3 which, respectively, depend on x2 and x3, is not allowed in the φ1(x1) equation. As aresult, the second term on the right-hand side must be completely eliminated and this can be realized only bysetting α = β = 0. This restriction applies to the equations of φ2 and φ3 as well. In such a case, we reproducethe same modal equation of the dilatational wave problem. Consequently, by setting φi (xi, t) ≡ � (xi) ejωt, �satisfies the 1d fractal Helmholtz equation but with a wave speed and wavenumber given as

ϑt =√η + 2ψ

I, k = ω

ϑt(17)

As a result, the general solution for �, derived in Appendix A, is given, in terms of the fractal harmonicfunctions, by

� (xi) = C1f1 (xi, k) + C2f2 (xi, k) (18)

Joumaa et al. 123

3.3. In-plane problem

The so-called first planar problem is a generalization of the in-plane elasticity where the displacement and therotation fields are prescribed to be

u1 ≡ u1 (x1) u2 ≡ u2 (x2) u3 ≡ 0 (19a)

φ1 ≡ 0 φ2 ≡ 0 φ3 ≡ φ3 (x1, x2) (19b)

The resulting strain and curvature tensors become

γij =

⎡⎢⎢⎢⎣

∇D1 u1 −φ3

c10

φ3

c2∇D

2 u2 0

0 0 0

⎤⎥⎥⎥⎦ , κij =

⎡⎢⎢⎢⎣

0 0 ∇D1 φ3

0 0 ∇D2 φ3

0 0 0

⎤⎥⎥⎥⎦ (20)

The corresponding non-zero components of the stress tensor expressed in terms of the unknown fields are

τ11 = (2β + λ)∇D1 u1 + λ∇D

2 u2 (21a)

τ22 = λ∇D1 u1 + (2β + λ)∇D

2 u2 (21b)

τ33 = λ(∇D1 u1 + ∇D

2 u2) (21c)

τ12 = (β + α)γ12 + (β − α)γ21 (21d)

τ21 = (β − α)γ12 + (β + α)γ21 (21e)

Similarly, for the non-zero couple-stress tensor components, we obtain

μ13 = (ψ + ε)∇D1 φ3 (22a)

μ31 = (ψ − ε)∇D1 φ3 (22b)

μ23 = (ψ + ε)∇D2 φ3 (22c)

μ32 = (ψ − ε)∇D2 φ3 (22d)

The resulting linear momentum equations for u1 and u2 along with the angular momentum equation for φ3become

ρu1 = (2β + λ) ∇D1 ∇D

1 u1 + (β − α) ∇D2

(−φ3

c1

)+ (β + α) ∇D

2

(φ3

c2

)(23a)

ρu2 = (2β + λ) ∇D2 ∇D

2 u2 + (β − α) ∇D1

(φ3

c2

)+ (β + α) ∇D

1

(−φ3

c1

)(23b)

Iφ3 = (ψ + ε)[∇D

1 ∇D1 φ3 + ∇D

2 ∇D2 φ3

] − (β + α)

(1

c21

+ 1

c22

)φ3 + (β − α)

φ3

c1c2(23c)

Since the in-plane displacement u1 is set to only depend on x1 and similarly for u2 and x2, the above equationsmust be configured to preserve this mathematical consistency. In other words, the x2 terms of the u1 equationmust vanish and similarly the x1 terms of the u2 equations. This consistency is achieved once we set α = β = 0.Thus, the field equations become the 1d fractal wave equation for u1 and u2, and the 2d fractal wave equationfor φ3:

ρ

λu1 = ∇D

1 ∇D1 u1 = 1

c1

(u1,1

c1

),1

(24a)

ρ

λu2 = ∇D

2 ∇D2 u2 = 1

c2

(u2,2

c2

),2

(24b)

124 Mathematics and Mechanics of Solids 19(2)

I

ψ + εφ3 = ∇D

1 ∇D1 φ3 + ∇D

2 ∇D2 φ3 =

[1

c1

(φ3,1

c1

),1

+ 1

c2

(φ3,2

c2

),2

](24c)

Clearly, the waves involved in the dynamics of u1 and u2 are of dilatational type. As a result, the modal analysisfor these two variables is no different from that presented in Section 3.1. The celerity of the wave reduces to

ϑd =√λ

ρ, because β = 0. Expressing ui = Uiejωt, we obtain

Ui = C1f1 (xi, k) + C2f2 (xi, k) , i = 1, 2 (25a)

k = ω

ϑd(25b)

f1 and f2 being the general fractal harmonic functions presented in Equation (13). Concerning φ3, we introducethe space–time separation of variables where we have φ3 ≡ �3 (x1, x2) ejωt. The modal function, �3, satisfiesthe 2d fractal Helmholtz equation, which can be solved as explained in Appendix A if the spatial decomposition

�3 ≡ F (x1) G (x2) is performed. Setting the corresponding wave speed ϑr =√ψ + ε

Iand k = ω

ϑr, we obtain

∇D1 ∇D

1 F

F+ ∇D

2 ∇D2 G

G+ k2 = 0 (26)

By further assigning k2 = k21 + k2

2 , the above equation can be decoupled where F and G each satisfy the 1dHelmholtz equation. Thus, the decomposed eigenmodes are expressed as

F (x1, k1) = M1f1 (x1, k1) + M2f2 (x1, k1) (27a)

G (x2, k2) = N1f1 (x2, k2) + N2f2 (x2, k2) (27b)

All four constants, in addition to the two wavenumbers (k1 and k2), are BC dependent.

3.4. Out-of-plane problem

We now consider our second planar problem which is a generalization of the out-of-plane elasticity. No in-planedisplacement or out-of-plane rotation is allowed. We consider the case in which the in-plane rotations dependon the planar coordinates (x1, x2) while the out-of-plane displacement, u3, depends on x3 for a reason that willbe identified later in the problem’s analysis. The kinematics thus far is

u1 ≡ 0 u2 ≡ 0, u3 ≡ u3 (x1, x2, x3) (28a)

φ1 ≡ φ1 (x1, x2) , φ2 ≡ φ2 (x1, x2) φ3 ≡ 0 (28b)

The resulting strain and curvature tensors for this prescribed kinematics are

γij =

⎡⎢⎢⎢⎢⎢⎣

0 0 ∇D1 u3 + φ2

c1

0 0 ∇D2 u3 − φ1

c2

−φ2

c3

φ1

c3∇D

3 u3

⎤⎥⎥⎥⎥⎥⎦ , κij =

⎡⎢⎢⎢⎣

∇D1 φ1 ∇D

1 φ2 0

∇D2 φ1 ∇D

2 φ2 0

0 0 0

⎤⎥⎥⎥⎦ (29)

The non-zero components of the stress tensor expressed in terms of the unknown fields become

τ11 = τ22 = λ∇D3 u3 (30a)

τ33 = (2β + λ)∇D3 u3 (30b)

τ13 = (β + α)γ13 + (β − α)γ31 (30c)

τ31 = (β − α)γ13 + (β + α)γ31 (30d)

τ23 = (β + α)γ23 + (β − α)γ32 (30e)

τ32 = (β − α)γ23 + (β + α)γ32 (30f)

Joumaa et al. 125

and similarly for the couple-stress tensor, we have

μ11 = (2ψ + η)∇D1 φ1 + η∇D

2 φ2 (31a)

μ22 = (2ψ + η)∇D2 φ2 + η∇D

1 φ1 (31b)

μ33 = η(∇D1 φ1 + ∇D

2 φ2) (31c)

μ12 = (ψ + ε)∇D1 φ2 + (ψ − ε)∇D

2 φ1 (31d)

μ21 = (ψ + ε)∇D2 φ1 + (ψ − ε)∇D

1 φ2 (31e)

The resulting angular momentum equation for φ1 becomes

Iφ1 =∇D1 μ11 + ∇D

2 μ21 + ∇D3 μ31 + τ23

c2− τ32

c3

= (2ψ + η) ∇D1 ∇D

1 φ1 + (ψ − ε + η) ∇D1 ∇D

2 φ2 + (ψ + ε) ∇D2 ∇D

2 φ1

+ (β + α)

(γ23

c2− γ32

c3

)+ (β − α)

(γ32

c2− γ23

c3

) (32)

The above equation contains the term c3 which depends on x3. Prescribing φ1 to solely depend on the in-plane coordinates requires the c3 term to be eliminated which can be realized only by setting α = β = 0.In addition, the full decoupling of the mathematical model (where φ2 vanishes from the φ1 equation and viceversa) necessitates the condition ψ − ε + η = 0. Under such conditions, we have for φ1 and φ2

Iφ1 = (η + 2ψ)[∇D

1 ∇D1 φ1 + ∇D

2 ∇D2 φ1

]= (η + 2ψ)

[1

c1

(φ1,1

c1

),1

+ 1

c2

(φ1,2

c2

),2

](33a)

Iφ2 = (η + 2ψ)

[1

c2

(φ2,2

c2

),2

+ 1

c1

(φ2,1

c1

),1

](33b)

Concerning u3, after applying all of the necessary conditions, the corresponding linear momentum equationbecomes

ρu3 = ∇D3 τ33 = λ∇D

3 ∇D3 u3 (34)

At this point, it becomes clear why u3 is set to depend on x3. If this were not the case, u3 would not possess anymeaningful dynamics in this problem.

Having crafted the mathematical setting for this problem, we now present its analytical solution. The proce-dure to solve for the fields is realized by considering modal analysis. As noted, φ1 and φ2 obey the same equationand thus they must admit the same general solution. The Fourier decomposition sets φ1 = ejωt�1(x1, x2) =ejωtF(x1)G(x2), where �1 satisfies the 2d fractal Helmholtz equation

∇D1 ∇D

1 �1 + ∇D2 ∇D

2 �1 + k2�1 = 0 (35)

More explicitly, F and G each satisfies the 1d fractal Helmholtz equation

∇D1 ∇D

1 F + k21F = 0 (36a)

∇D2 ∇D

2 G + k22G = 0 (36b)

k2 = k21 + k2

2 =(ω

ϑt

)2

(36c)

ϑt =√η + 2ψ

I(36d)

126 Mathematics and Mechanics of Solids 19(2)

As a result, the well-known solution for F and G is constructed using the fractal harmonic functions f1 and f2

F (x1) = C1f1 (x1, k1) + C2f2 (x1, k1) (37a)

G (x2) = C3f1 (x2, k2) + C4f2 (x2, k2) (37b)

The eigenvalue solutions to k1 and k2, in addition to that of constants C1, C2, C3, and C4 are determined throughthe application of the BCs. Note that the torsional wave speed ϑt which was introduced in the second problem,characterizes the out-of-plane waves corresponding to microrotations.

The out-of-plane displacement obeys the ‘dilatational’ 1d fractal wave equation, thus its correspondingmodal problem admits the fractal harmonic functions as the general solution. Mathematically, setting u3 (xi, t) =U3 (xi) ejωt, U3 (xi) satisfies the 1d fractal Helmholtz equation in x3, where

U3 (xi) = C1 (x1, x2) f1 (x3, k) + C2 (x1, x2) f2 (x3, k) (38a)

k = ω

ϑdϑd =

√λ

ρ(38b)

Note that C1 and C2 are allowed to depend on x1 and x2, because u3, as prescribed for this problem, depends inthe general case on all of the coordinates. Here C1 and C2 are determined through the application of the BCs.

4. Numerical solutionsHaving developed the full field equations for the generalized problem and considered particular cases of knownclosed-form solutions, our goal is now to construct a valid and accurate computational tool capable of handlingarbitrary initial boundary value (IBV) problems unsolvable by analytic methods. Owing to formulation restric-tion, the numerical tool is limited to treating box-shaped domains in Cartesian systems. Given this restraint andthe considerable size of such a 3d problem, the most efficient method to design the numerical solver would beto implement a direct discretization method for the field equations and adopt the explicit scheme for the timemarch. As such, the finite difference method is applied with discrete variables ui (xk , tn) and φi (xk , tn) whichare assigned on uniform meshes in space and time. The second-order accurate central difference discretiza-tion approximates all of the spatial derivatives while the first-order accurate (Euler’s forward method) treats theacceleration terms [23]. The resulting numerical scheme is conditionally stable and its parallel implementationfor the case of expensive problems is straightforward and highly efficient.

The validation of the numerical solver is achieved by simulating the four previously discussed problemswhere the transient response for modal excitation is evaluated. The first modal excitation is employed for allproblems and a strong match between the numerical and exact solution is observed. We now discuss the numer-ical simulations and show the computed time histories for the variables of interest and the 3d contour plots ofthe variable fields in the entire numerical domain.

For the first problem, we set λ = β = 1, and the micropolar constants whose values are irrelevant tothe solution are all set to one. The density and the inertia terms are set to one as well. These values resultin having the frequencies of first mode oscillation ω1 = ω2 = 5.441 and ω3 = 3.514. We used 1000 timesteps to span two periods of u1 oscillation, which implies that �t ≈ 2.31 × 10−3. The grid is uniform with�x1 = �x2 = �x3 = 10−2. The transient solutions for the displacement field are shown in Figure 3 and thecorresponding contour plots for the same problem are shown in Figure 4.

For the second problem, we set ψ = η = 1 in addition to the condition α = β = 0. All other constants areirrelevant to the solution. The density and the inertia terms are set to one for simplicity. The resulting oscillationfrequencies of the first mode are ω1 = ω2 = 5.441 and ω3 = 3.514. We used 1000 time steps to span twoperiods of φ1 oscillation, which implies that �t ≈ 2.31 × 10−3. The grid is identical to that of the first problem.The transient solution for the microrotation field is shown in Figure 5 and the corresponding contour plots forthe same problem are shown in Figure 6.

For the in-plane problem, we set the stiffness constants ψ = ε = λ = 1 and set α = β = 0. The remainingterms, irrelevant to the problem, assumed the value of one. The density and the inertia terms are set to one forsimplicity. The resulting frequencies for the first modal excitation of u1 and u2 are ω1 = ω2 = 3.141, while thatof φ3 for the first mode is ω3 = 6.283. We used 900 time steps to span two periods of φ3 oscillation, whichimplies that �t ≈ 2.222 × 10−3. The grid is identical to that of all previous problems. The time history of thetransient solution is shown in Figure 7 and the corresponding contour plots are shown in Figure 8.

Joumaa et al. 127

Figure 3. Transient solution for the dilatational wave problem. The exact solution is shown in solid black and the numerical solutionis shown in coloured dots.

Figure 4. First problem contour plots for ui at different times.

In the simulation of the out-of-plane problem, we set ψ = η = 1 and ε = 2 to satisfy ψ + η − ε = 0.In addition, we set α = β = 0. The Lamé’s constant λ can assume any value and was set to one. The densityand the inertia terms are set to one for simplicity. The resulting frequencies for the first modal excitation of φ1

128 Mathematics and Mechanics of Solids 19(2)

Figure 5. Transient solution for the torsional wave problem. The exact solution is shown in solid black and the numerical solutionis shown in coloured dots.

Figure 6. Second problem contour plots for φi at different times.

and φ2 are ω1 = ω2 = 7.695, while that of u3 for the first mode is ω3 = 2.029. We used 1000 time steps tospan two periods of φ1 (and φ2) oscillation, which implies that �t ≈ 1.633 × 10−3. The grid is identical to

Joumaa et al. 129

Figure 7. Transient solution for the in-plane problem. The exact solution is shown in solid black and the numerical solution is shownin coloured dots.

Figure 8. In-plane problem contour plots for u1, u2, and φ3 at different times.

that of all previous problems. The transient solution for the variables of interest is shown in Figure 9 and thecorresponding contour plots are shown in Figure 10.

130 Mathematics and Mechanics of Solids 19(2)

Figure 9. Transient solution for the out-of-plane problem. The exact solution is shown in solid black and the numerical solution isshown in coloured dots.

Figure 10. Out-of-plane problem contour plots for φ1, φ2, and u3 at different times.

5. ConclusionIn this paper we explored the elastodynamics of 3d micropolar fractal solids from both analytical and com-putational perspectives. The generalized field equations in terms of displacements and microrotations weredeveloped. The analytical approach is feasible only for those problems whose kinematics is prescribed in sucha way as to decouple the field equations. The resulting simplified problems, of eigenvalue type, were analysedusing the well-known approach of modal decomposition. The pure dilatational wave propagation problem andthe pure torsional wave problem resulted in the same modal equation, the 1d fractal Helmholtz equation. Thein-plane and out-of-plane problems required the solution of the 2d fractal Helmholtz equation. Realizing the

Joumaa et al. 131

impossibility to solve a general problem of arbitrary kinematics and BCs by analytic means, the developmentof a computational tool is necessary to accomplish this purpose. Therefore, we constructed a numerical solverbased on the finite difference method and all four problems discussed earlier were solved numerically, an impor-tant step to verify the computational model. This work constitutes a step forward in the current exploration ofmechanics of fractal materials. Its significance lies in (i) the rigorous mathematical approach in formulatingthe balance laws and deriving field equations for general elastodynamic problems involving the propagation ofwaves in 3d micropolar fractal solids and (ii) verification of a numerical procedure to solve such problems.

Funding

This work was supported by Sandia-DTRA (grant number HDTRA1-08-10-BRCWMD) and the NSF (grant number CMMI-1030940).Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiaryof Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

Conflict of interest

None declared.

References

[1] Mandelbrot, BB. The Fractal Geometry of Nature. W.H. Freeman and Co, 1982.[2] Barnsely, M. Fractals Everywhere. San Mateo, CA: Morgan Kaufmann, 1993.[3] Le Mehaute, A. Fractal Geometry: Theory and Applications. Boca raton, FL: CRC Press, 1991.[4] Hastings, HM, and Sugihara, G. Fractals: A User’s Guide for the Natural Sciences. Oxford: Oxford Science Publications, 1993.[5] Falconer, K. Fractal Geometry: Mathematical Foundations and Applications. Chichester: John Wiley & Sons, 2003.[6] Tarasov, VE. Fractional hydrodynamic equations for fractal media. Ann Phys 2005; 318: 286–307.[7] Tarasov, VE. Wave equation for fractal solid string. Mod Phys Lett B 2005; 19: 721–728.[8] Ostoja-Starzewski, M. Towards thermomechanics of fractal media. ZAMP 2007; 58: 1085–1096.[9] Ostoja-Starzewski, M. Extremum and variational principles for elastic and inelastic media with fractal geometries. Acta Mech

2009; 205: 161–170.[10] Ostoja-Starzewski, M. On turbulence in fractal porous media. ZAMP 2008; 59: 1111–1117.[11] Joumaa, H, and Ostoja-Starzewski, M. On the wave propagation in isotropic fractal media. ZAMP 2011; 62: 1117–1129.[12] Li, J, and Ostoja-Starzewski, M. Fractal materials, beams and fracture mechanics. ZAMP 2009; 60: 1–12.[13] Li, J, and Ostoja-Starzewski, M. Fractal solids, product measures and fractional wave equations/ Proc R Soc Lond A 2009; 465:

2521–2536.[14] Li, J, and Ostoja-Starzewski, M. Micropolar continuum mechanics of fractal media. Int J Eng Sci 2011; DOI:10.1016/j.ijengsci.

2011.03.01.[15] Demmie, PN, and Ostoja-Starzewski, M. Waves in fractal media. J Elast 2011; DOI: 10.1007/s10659-011-9333-6.[16] Joumaa, H, and Ostoja-Starzewski, M. On the dilatational wave motion in anisotropic fractal solids. Math Comput Sim, submitted.[17] Guilleminot, J, Ostoja-Starzewski, M, and Soize, C. Mechanics of random fractal media, in preparation.[18] Jumarie, G. Table of some basic fractional calculus formulae derived from a modified Riemann–Liouville derivative for non-

differentiable functions. Appl Math Lett 2008; 22: 378–385.[19] Carpinteri, A, Chiaia, B, and Cornetti, P. A disordered micro structure material model based on fractal geometry and fractional

calculus. ZAMP 2004; 84: 128–135.[20] Manning, A, and Simon, K. Dimension of slices through the Sierpinski carpet. Trans Amer Math Soc, to appear.[21] Eringen, AC. Microcontinuum Field Theories. New York: Springer, 1998.[22] Graff, KF. Wave Motion in Elastic Solids. New York: Dover Publications, 1975.[23] Durran, D. Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. New York: Springer, 1998.

Appendix AThe 1d fractal Helmholtz equation is an ordinary differential equation (ODE) expressed in compact form as

∇D∇DF + k2F = 0 (39)

132 Mathematics and Mechanics of Solids 19(2)

Expanding the fractal gradient, we obtain

d2F

dx2− 1

c

dF

dx

dc

dx+ k2c2F = 0, c = D (L − x)D−1 (40)

The procedure to solve this ODE involves the application of several transformations as will be explained in thefollowing steps. First, an obvious transformation to suppress the L term is to introduce y = L − x. Then thederivatives are transformed according to

dF

dx−→ −dF

dy

d2F

dx2−→ d2F

dy2

(41)

The resulting ODE in terms of y becomes

y2 d2F

dy2− (D − 1) y

dF

dy+ k2D2y2DF (y) = 0 (42)

The presence of the non-integer power in the last term deters a quick insight for a solution form. Nevertheless,this power can be eliminated if the transformation z = ys is applied. The derivatives are transformed as follows

dF

dy−→ s

dF

dzz(1− 1

s )

d2F

dy2−→ s2z(2− 2

s ) d2F

dz2+ s (s − 1)

dF

dzz(1− 2

s )(43)

and the resulting equation in z becomes

s2z2 d2F

dz2+ sz (s − D)

dF

dz+ k2D2z

2Ds F = 0 (44)

When setting s = D, the equation reduces to one of harmonic type in z, which admits a general solution givenby

F(z) = C1 cos (kz) + C2 sin (kz) (45)

Rewriting the solution in terms of the original variable x, it becomes

F(x) = C1 cos[k (L − x)D] + C2 sin

[k (L − x)D]

(46)

We denote the two components of the general solution as the fractal harmonic functions f1 (x, k) and f2 (x, k),respectively.

For an eigenvalue problem, where homogeneous Dirichlet BCs are applied at x = 0 and x = L, the solutionis

Fn(x) = sin[kn (L − x)D]

kn = nπ

LD

(47)

The first three modes are shown in Figure 11. The numerical simulation of the first two problems involves thefirst modal excitation. In Figure 11, we also show the mode shapes for the case of continuous media (D = 1). Forthat case, odd modes exhibit an axial symmetry about the mid-span while even modes have a point symmetryabout this location. This symmetry vanishes once D �= 1 and the modes squeeze toward x = L as D is decreased.In addition, all modes become infinitely steep at x = L and the order of the first derivative singularity is(L − x)1−D.

Joumaa et al. 133

Figure 11. Mode shapes corresponding to solutions for the 1d fractal Helmholtz equation where the chosen Hausdorff dimensionsare those of the Carpinteri column and for the case of continuum.

Table 1. Modal frequencies along x3 for continuous and fractal case.

First problem Second problem

Frequency (rad/s) Continuous Fractal Continuous Fractal

ω13 5.441 3.514 5.441 3.514ω2

3 10.883 7.028 10.883 7.028ω3

3 16.324 10.541 16.324 10.541

We present in Table 1 a set of results showing the discrepancy in modal frequency between the continuousand the fractal case for the first two problems handled in our discussions. These results are indicative to theinfluence of fractal effects on elastodynamic excitation. Consider ωi

3 to be the frequency of excitation for the ithmode along the x3 direction. In calculating the frequencies, we used the same values for the stiffness constantsand densities of the numerical simulations.

The 2d fractal Helmholtz equation is expressed as

∇D1 ∇D

1 F + ∇D2 ∇D

2 F + k2F = 0 (48)

By setting F (x1, x2) = G (x1) · H (x2) and k2 = k21 + k2

2 , a separation of variables is achieved. The outcome istwo independent 1d fractal Helmholtz equations for G and H with eigenvalues k1 and k2, respectively. Figure 12shows the contour plots for some modes on the (x1x2) section (Sierpinski carpet), where homogeneous DirichletBCs are applied on the entire boundary. The first mode was excited for φ3 in the in-plane problem, and forφ1, φ2 in the out-of-plane problem. In addition, the mode shapes are considered along the (x1x3) surface. Theircorresponding contour plots are shown in Figure 13.

134 Mathematics and Mechanics of Solids 19(2)

Figure 12. Mode shapes corresponding to solutions of the 2d fractal Helmholtz equation where L1 = L2 = 1 and D1 = D2 =13

ln 18ln 3 .

Figure 13. Mode shapes corresponding to solutions of the 2d fractal Helmholtz equation where L1 = 1, L3 = 2, D1 = 13

ln 18ln 3 and

D3 = ln 2ln 3 .