elastic wave dispersion in microstructured...

Proc. R. Soc. A (2010) 466, 1789–1807doi:10.1098/rspa.2009.0516

Published online 13 January 2010

Elastic wave dispersion in microstructuredmembranes

BY MARIATERESA LOMBARDO* AND HARM ASKES

Department of Civil and Structural Engineering, University of Sheffield,Mappin Street, Sheffield S1 3JD, UK

The effect of microstructural properties on the wave dispersion in linear elasticmembranes is addressed in this paper. A periodic spring-mass lattice at the lower levelof observation is continualized and a gradient-enriched membrane model is obtainedto account for the characteristic microstructural length scale of the material. In thefirst part of this study, analytical investigations show that the proposed model is ableto correctly capture the physical phenomena of wave dispersion in microstructuredmembrane which is overlooked by classical continuum theories. In the second part, afinite-element discretization of microstructured membrane is formulated by introducingthe pertinent inertia and stiffness terms. Importantly, the proposed modifications do notincrease the size of the problem compared wiith classical elasticity. Numerical simulationsconfirm that the vibrational properties are affected by the microstructural characteristicsof the material, particularly in the high-frequency regime.

Keywords: continualization; higher-order continuum; lattice model; microstructure;Padé approximation; wave dispersion

1. Introduction

A tensioned membrane can be considered as the simplest example of atwo-dimensional vibrating system, although more complex structures may alsobe considered as membranes, the fundamental characteristic of which is theincapability of conveying moments or shear forces. Currently, innovative practicalapplications exist for these structural models, such as antennas and solarsails for space applications, biotechnological prostheses and lightweight roofsdesigned in architecture and structural engineering (Jenkins & Korde 2006).The use of composite materials offers excellent mechanical performance combinedwith lightweight characteristics appropriate for membrane structures. Compositemembranes tend to possess relevant microstructure that often significantlyinfluences the corresponding macroscopic response. Consequently, accuratemodelling of such structures and materials requires that the microstructure isaccounted for.

In mechanical problems involving materials with microstructure, the existenceof microscopic characteristic lengths, whose origin may be the lattice period ofa discrete system or the inclusion size of an inhomogeneous material, introduces*Author for correspondence ([email protected]).

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1790 M. Lombardo and H. Askes

a size dependence into the structural response on the macroscopic level. Theseeffects may be accurately analysed, including each individual microstructuralelement in a discrete model. However, this approach leads to a considerablecomputational effort because the number of degrees of freedom is normallyvery large. The strategy of replacing such a discrete model with an equivalentcontinuum, characterizing the average mechanical behaviour of the actual discreteor inhomogeneous material, offers an efficient alternative, as the continuummechanics approach is computationally less expensive.

Although classical continuum theories are adequate for many static loadingconditions, they may not be appropriate for problems involving dynamic analysisin the high-frequency regime, wherein microstructural effects are more significant.For instance, classical continuum theories cannot capture wave dispersion.Dispersion is the phenomenon that waves with different wavelengths propagatewith different velocities, and wave dispersion in heterogeneous or discretemedia is extensively documented by experimentalists (Erofeyev 2003; Jakata &Every 2008).

Waves that propagate through a classical medium are not dispersive, i.e.the wave speed is constant, independent of wavelength. In contrast, enhancedcontinuum models account for the intrinsic length of microstructural details inthe macrostructural material model, normally by means of additional (temporalor spatial) derivatives of the relevant state variables. It was shown that thewave dispersion can be captured by the addition of higher-order inertia termsand higher-order stiffness terms in the governing equations, as discussed in therecent overview by Papargyri-Beskou et al. (2009). To obtain a transparentlink between the microstructural material properties and the higher-order termson the macroscopic level, continualization or homogenization strategies wereproposed to account for the heterogeneous or discrete nature of materials—inthe context of wave propagation studies, see, for instance, the studies of Chang &Gao (1995), Mühlhaus & Oka (1996), Chen & Fish (2001), Suiker et al. (2001),Fish et al. (2002), Metrikine & Askes (2002) or Suiker & de Borst (2005).

Finite-element implementations are essential for widespread disseminationof material models. Unfortunately, finite-element implementations tend to becomplicated for the enhanced continuum models mentioned above. As discussed,for example, by Askes et al. (2008), many formulations of enhanced continuarequire special treatment, such as Hermitian or mixed formulations, which makestheir implementation in available finite-element packages cumbersome. Thus, it isof interest to pursue enhanced continuum formulations that allow straightforwardfinite-element implementation.

This contribution addresses a number of aspects of enhanced continuummodels mentioned above. The main aim is to validate the accuracy of enhancedcontinuum analysis of a membrane with microstructure subject to dynamicloading. The continualization approach of Rosenau (1987) and Andrianov &Awrejcewicz (2008) will be applied in §2 to translate microstructural propertiesinto a macroscopic model. An analysis of the dispersion properties of transversewaves propagating along different directions is presented in §3. The spatialdiscretization with standard finite elements is detailed in §4. Finally, numericalanalyses of a microstructured membrane are treated in §5 with the aim to verifythat the enhanced continuum model provides more realistic results than theclassical continuum theory.

Proc. R. Soc. A (2010)



Waves in microstructured membranes 1791

m, n + 1

m – 1, n m + 1, n

m, n – 1x

y

m, n

Figure 1. Two-dimensional discrete lattice in the (x , y) plane.

2. Transverse vibrations of a two-dimensional lattice andcontinualization methods

In this section, the equations governing the transverse vibrations of a linearelastic two-dimensional lattice have been derived. The continualization approachis employed for the construction of continuum models from associated periodicstructures. Truncated Taylor series expansions are used to approximate thedifference equations by differential equations. In this manner, the microstructuraleffects are accounted for by means of higher-order gradients naturally appearingin the continuum equations. The equation of motion obtained via the standardcontinualization approach is shown to reduce to classical elastic continuum whenonly the first terms in the series are accounted for. Higher-order terms increasethe accuracy of the approximation, but also the order of the partial differentialequations, which requires supplementary boundary conditions to define a well-posed problem. In order to avoid this inconvenience, the Padé approximantsare introduced, yielding a more effective continuum model; incidentally, Padéapproximants also help in overcoming instabilities as explained in §3.

(a) Discrete model

The two-dimensional spring-mass lattice that will be analysed consists ofa repeating arrangement of N1 × N2 identical particles of mass M , connectedby pretensioned springs. The pretensioning of the springs occurs entirely inplane, but provides stiffness against out-of-plane deflection. The interparticledistances in the x and y directions are denoted by �x and �y , respectively. Asoutlined in Rosenau (1987), owing to the spatial periodicity of the system, thedynamics of such structures can be studied by considering only a unit cell,as depicted in figure 1. If the lattice experiences small displacements in the





transverse (i.e. z) direction compared with its initial position in the x–y plane,the membrane vibrations can be studied in the linear regime (Andrianov &Awrejcewicz 2008).

The equation of motion governing the free transverse vibrations of an elementlocated in the mth row and nth column is obtained by balance of forces intransversal direction as

M∂2um,n

∂t2= [(Fm+1,n − Fm−1,n) + (Fm,n+1 − Fm,n−1)], (2.1)

m = 0, 1, . . . , N1; n = 0, 1, . . . , N2,

where um,n = um,n(t) = u(xm , yn ; t) is the transverse displacement of the mass andFi,j is the recall force of the stretched springs connecting the nearest-neighbourparticles.

For small displacements, we may assume that the contribution of Fi,j at allpoints in the lattice remains close to its equilibrium value, and each contributionhas the form

Fm+1,n − Fm−1,n = Fx

�x[(um+1,n − um,n) − (um,n − um−1,n)] (2.2a)

and

Fm,n+1 − Fm,n−1 = Fy

�y[(um,n+1 − um,n) − (um,n − um,n−1)], (2.2b)

where now Fi is the force of the spring in direction i.For the sake of simplicity, we employ a square lattice, so that Fx = Fy and

�x = �y , and, inserting equations (2.2) into equation (2.1), the equation of motionreads

M∂2um,n

∂t2= T [um+1,n − 2um,n + um−1,n] + T [um,n+1 − 2um,n + um,n−1], (2.3)

where T = (F/�) is an in-plane tensile load. The following boundary and initialconditions are applied:

u0,n = uN1,m = um,0 = um,N2 = 0 (2.4a)

and

um,n(t = 0) = ∂

∂tum,n(t = 0) = 0. (2.4b)

To continualize equation (2.3), the simplest strategy suggests to replace thediscrete degrees of freedom by a continuous variable field.

(b) Classical continuum approximation

Following the hypothesis of a dense lattice, a continuum approximationinvolves introducing a continuum displacement field um,n ≡ u(x , y; t), whereasthe displacements of the neighbouring particles um±1,n and um,n±1 are replaced





by u(x ± �, y) and u(x , y ± �), respectively. Using Taylor series expansions, thedisplacements of the neighbouring particles can be approximated according to

um±1,n = u(x ± �, y; t) = u ± �∂u∂x

+ �2

2∂2u∂x2

+ · · · (2.5a)

and

um,n±1 = u(x , y ± �; t) = u ± �∂u∂y

+ �2

2∂2u∂y2

+ · · · . (2.5b)

For notational simplicity, the dependence of u on the spatial coordinates andon the time t is not explicitly shown. The accuracy of the derived continuumdepends on the number of terms included in the Taylor series. Taking into accountthe terms up to the second derivatives and introducing in equation (2.3) thecontinuum field variables, a membrane-like continuum approximation is obtained,

ρ∂2u∂t2

= T[∂2u∂x2

+ ∂2u∂y2

], (2.6)

where ρ = (M/�2) is the mass density. If the uniform tension T is sufficiently large,the transverse displacement is small (so that T does not change significantly).Equation (2.6) represents the classical equation of transverse motion for avibrating membrane uniformly stretched, bounded by a surface Γ , with initialconditions u0 and (∂u0/∂t), and boundary conditions fixed along a portion ofthe edge ΓD, i.e. u = 0 on ΓD and free along the boundary ΓN, i.e. T (∂u/∂n) = 0on ΓN. The applicability of classical continuum models in dynamics is, however,restricted to a limited range of frequencies as will be highlighted in §3.

It is also worth noting that, for the sake of simplicity, coupling betweentransversal and longitudinal vibration of the membrane is neglected in theproposed formulation. This simplification is acceptable as long as the slope ofmembrane is small. Further studies will be needed to assess the effect of thecoupled motion of the membrane.

(c) The quasi-continuum method and Padé approximation

Although the application of a model with derivative of more than second orderallows to include the effect of microstructure via the parameter � in the continuummodel, the higher order of the resulting differential equation complicates thesolution of continuous problems. An alternative continualization procedure canbe realized using the quasi-continuum approximations. This technique consists ofthe reduction of the differential-difference equation (2.3) to a partial differentialequation. The continuum model is not obtained by simply replacing the differenceoperator in equation (2.3) by the analogous differential operator. In the firststage, the backward and forward shift operators are introduced, and for thedisplacements of adjacent masses, the following symbolic notations hold:

um±1,n =[exp

(±�

∂

∂x

)]um,n (2.7a)





and

um,n±1 =[exp

(±�

∂

∂y

)]um,n . (2.7b)

By means of equations (2.7a,b) and the relation um,n = u, the discreteequation (2.3) is converted into a pseudo-differential equation as follows:

M∂2u∂t2

= T[exp

(�

∂

∂x

)− 2 + exp

(−�

∂

∂x

)]u

+ T[exp

(�

∂

∂y

)− 2 + exp

(−�

∂

∂y

)]u. (2.8)

The differential operators in square brackets can be developed into Taylor seriesin the neighbourhood of zero (McLaurin series),

exp(

�∂

∂x

)− 2 + exp

(−�

∂

∂x

)= �2 ∂2

∂x2+ �4

12∂4

∂x4+ �6

360∂6

∂x6+ · · · (2.9a)

and

exp(

�∂

∂y

)− 2 + exp

(−�

∂

∂y

)= �2 ∂2

∂y2+ �4

12∂4

∂y4+ �6

360∂6

∂y6+ · · · . (2.9b)

Taking into account only the first term in the expansions (2.9), one obtains againthe classical continuum approximation given in equation (2.6). Furthermore, theaccuracy of the approximation can be increased keeping the next terms in theseries, and the higher-order approximation reads

M∂2u∂t2

= T[�2∇2u + �4

12

(∂4u∂x4

+ ∂4u∂y4

)], (2.10)

where ∇2 = (∂2/∂x2) + (∂2/∂y2).An alternative continuum model has been developed by using the one-point

Padé approximation of the differential operator appearing on the right-hand sideof equation (2.10). For more details regarding the Padé approximants, we referthe reader to the work of Rosenau (1986), Wattis (2000), Kevrekidis et al. (2002)and Andrianov & Awrejcewicz (2005). Taking into account only two terms inthe expansion, the operator in square brackets can be developed by means of thePadé approximation

�2∇2 + �4

12

(∂4

∂x4+ ∂4

∂y4

)≈ �2∇2 − (�4/6)(∂4/(∂x2∂y2))

1 − (�2/12)∇2, (2.11)

which leads to the following continuum model:

ρ

(1 − �2

12∇2

)∂2u∂t2

= T[∇2u − �2

6∂4u

∂x2∂y2

]. (2.12)

The characteristic length � accompanies the higher-order inertia term on theleft-hand side as well as the higher-order stiffness term on the right-hand side.





3. Dispersion analysis

The dispersive character of plane waves owing to inherent material characteristiclength is consistent with experimental observations. Specifically, this effectbecomes significant when the wavelength of the physical phenomena of interestdecreases to the order of the internal length, as, for example, in problemsrelated to shock waves. The regularization procedures introduced in theprevious section are developed with the aim to define a non-local continuumthat models dispersion phenomena by capturing the characteristics of themicrostructure. The effectiveness of the enhanced continuum models to representthe wave propagation characteristics of a lattice system can be assessed byevaluating the dispersion relations.

(a) Dispersion relations for the lattice model and the classicalcontinuum approximation

In the analysis of plane-wave propagation in two-dimensional periodic discretestructures, a dispersion equation can be obtained by substituting a harmonicwave into equation (2.3) as

um,n = A exp[i(ωt − kxxm − kyyn)], (3.1)

where ω is the angular frequency, A is the amplitude, i = √−1 is the imaginaryunit and xm = m� and yn = n� are the spatial coordinates, respectively. In a two-dimensional model, transverse waves travel along an arbitrary direction that isat an angle θ , which represents the angle between the wavevector k and thex axis. In equation (3.1), kx = k cos θ and ky = k sin θ define the components ofthe wavenumber in the x and y directions, respectively. The dispersion relationis a function ω = f (kx , ky), which contains information of the wave propagationcharacteristics of the considered material and, from the equation of motion (2.3),for the discrete lattice model is obtained as follows:

ω = ±√

4 sin2(

kx�

2

)+ 4 sin2

(ky�

2

), (3.2)

where ω = ω√

(T/ρ�2) is a non-dimensional frequency parameter. In the analysisof wave dispersion, the non-dimensional wavenumbers kx� and ky� are varied toinvestigate the frequencies at which waves propagate and the direction of thebody wave propagation.

The dispersion relation expressed in equation (3.2) is evidently periodic inthe interval kx�, ky� ∈ [−π , π ], and it is also symmetric with respect to theaxes kx� = 0 and ky� = 0. These properties allow us to restrict the analysisto the range kx�, ky� ∈ [0, π ] that reflects the so-called first Brillouin zone(Brillouin 1946). A particularly convenient representation of the dispersionrelations employs iso-frequency curves whose graphical interpretation providesinformation on the dispersion properties. The curves determined by equation (3.2)are represented in normalized form (kx�−ky� plane for fixed values of ω)in figure 2a.





0.51

1.5

22.5

y

x

0.5 1.0 1.5 2.0 2.5 3.00

0.5

1.0

1.5

2.0

2.5

3.0(a) (b)

2 4 6 8 100

1

2

3

4

5

ωFigure 2. Dispersion curves for a square lattice and the classical continuum: (a) directionalcharacteristics for wave propagating through the square lattice and (b) comparison of dispersioncurves for the square lattice at different direction of propagation and the classical continuumapproximation. Black dot on a solid line, classical continuum; solid line, θ = 0; dashed line, θ = π/12;dot-dashed line, θ = π/6; dotted line, θ = π/4.

The wave group velocity cg for undamped structures is the velocity ofpropagation of vibrational energy, and may be expressed in the form

cg = dω

dk= ∂ω

∂kxi + ∂ω

∂kyj , (3.3)

where i and j are the unit vectors in the x and y directions, respectively. Itshould be noted that the group velocity is normal to the curves in the kx−kyplane for fixed ω and its magnitude is equal to the gradient of the dispersionsurface. Hence, the direction of propagation of a wave at a given frequency can beestimated by taking the normal to the iso-frequency contour lines (Langley 1997).This property, in particular, can be used to identify regions within the structurewhere waves at certain frequencies do not propagate in specified directions.

In §2, the discrete equation of motion has been transformed in a continuummodel taking advantage of the continualization approach. As previouslyillustrated, if, in the series expansion, only the terms up to the second derivativesare retained, the equation of motion yields the wave equation based upon aclassical continuum model expressed by equation (2.6). Substituting into theequation of motion the harmonic wave of the continuum form

u(x , y; t) = A exp[i(ωt − kxx − kyy)], (3.4)

we find the solutionω = k�. (3.5)

This equation shows that the classical continuum is not dispersive because thefrequency ω is directly proportional to the wavenumber k.

The dispersion relations of the square lattice and the classical continuummodel, as represented by equations (3.2) and (3.5), respectively, have been plottedin figure 2b, which displays the behaviour of the dimensionless frequency ω in





terms of the normalized wavenumber k�. The dispersion curve for the continuummodel is represented as a solid straight line through the origin, while thediscontinuous lines represent the dispersion properties of the lattice for differentangles θ in the range [0, π/4]. The comparison illustrates that the physical motionof the lattice system is well described by the classical continuum model for longerwave limits. This means that when the wavelengths are much larger than thestructural inhomogeneity of the lattice (k� → 0 or k� � 1), the waves are notaffected by the heterogeneity of the membrane.

On the other hand, the dispersive effects become evident in the short-waverange, when the phase velocity cp of the harmonic waves, that is, the speed atwhich the phase of any one component of the wave travels, is different from thegroup velocity cg. The phase velocity cp can be expressed as

cp = ω√k2x + k2

y

, (3.6)

and is graphically represented by the secant slope of the dispersion curves, whilethe group velocity cg is represented by the tangential slope. By inspection of thecurves, the local continuum model appears unable to predict the decrease of phasevelocity for increasing wavenumbers.

(b) Dispersion relations for higher-order continua

Retaining more terms in the series gives an enhanced continuum modelrepresented by equation (2.10). Explicitly, we seek solutions of the harmonicform expressed by equation (3.4) that, inserted into equation (2.10), givesthe relationship between the frequency ω and the wavenumber k, which innon-dimensional form turns out to be

ω = ±k�

√√√√1 − (k�)2

12

(1 − sin2 2θ

2

). (3.7)

From a mathematical point of view, the model governed by equation (3.7)produces wave dispersion if the function under the square root is greater thanzero. When k exceeds a critical value for a fixed direction of propagation, thewave does not possess real frequency and the solution is unstable. The dispersioncurves for the higher-order continuum, depicted in figure 3, show a downwardbranch crossing the axis ω at values of k� at which instability occurs. Because ofthe limitation owing to the stability criterion, the applicability of equation (2.10)is restricted to the dynamic phenomena that do not involve short waves.

If asymptotic analysis is resorted to, a way to deal with the instabilityproblem in the fourth-order continuum model suggests replacing the fourth-orderderivative in equation (2.10) with mixed double time double space derivatives(Pichugin et al. 2008). With the aim of restoring, in the continuum approximation,the dispersion property owing to the discreteness of the square lattice while,at the same, avoiding instabilities, the Padé approximations can be employed. Inthe following, the model introduced in equation (2.12) is referred to as a Padécontinuum.





1 2 3 4 50

1

2

3

4

5

ω

Figure 3. The ω–k� dispersion curves for the fourth-order approximation of equation (3.7). Blackdot on a solid line, classical continuum; solid line, θ = 0; dashed line, θ = π/12; dot-dashed line,θ = π/6; dotted line, θ = π/4.

Solving the dispersion equation for the expected harmonic travelling wave, thefollowing expression can be found:

ω = ±k�

√1 + ((k�)2/24) sin2 2θ

1 + ((k�)2/12). (3.8)

It is worth remarking that the model is unconditionally stable, as the frequenciesdefined by equation (3.8) are real in the complete range of wavenumbers.

In figure 4, the dispersion curves for the Padé model are plotted inaccordance with the relations obtained for the square lattice and the classicalcontinuum model for different directions of propagation. The plots show that,at the long-range limit k� → 0, the tangential slopes of the dispersion curves,corresponding to the group velocity vg = (∂ω/∂k), are equal to those of thelattice model and the local continuum. For increasing wavenumbers, curvesstart to deviate from those of the discrete lattice model. However, they arelocated below the bold line that represents the local continuum model, andthe group velocity is smaller than the one obtained for the local model, for alldirections.

We note that the dispersion curve for the Padé continuum with θ = 0 tendsto a horizontal asymptote. In this case, the slope (group velocity) for transversewaves is zero and frequencies above the limit value cannot propagate. Thus, themodel acts as a filter, where only relatively low frequencies are transmitted alongthe θ = 0 direction.

By considering its definition, the group velocity of waves for the Padécontinuum can be found to depend on the wavenumber and the direction of





0

2

4

6

8

10(a) (b)

(c) (d)

5 10 150

2

4

6

8

10

0 5 10 15

ωω

Figure 4. Comparison of dispersion curves for classical continuum (solid line), square lattice(dot-dashed line) and Padé model (dotted line) for different directions of propagation: (a) θ = 0;(b) θ = π/12; (c) θ = π/6; (d) θ = π/4.

propagation as

cg = ±c288 + (k�)2(24 + (k�)2) sin2(2θ)

√2(12 + (k�)2)

√((24 + (k�)2 sin2(2θ))/(12 + (k�)2))

, (3.9)

where c = √(T/ρ) is the classical wave speed. This expression allows us to verify

that propagating waves cannot transfer energy infinitely fast. In the long-wavelimit, the group velocity takes the following form:

limk→0

cg = ±c, (3.10)

and for short waves

limk→∞

cg = ± csin2(2θ)√

2< c, ∀θ . (3.11)

Thus, the model governed by equation (2.12) is stable and provides lowerand upper bounds for the speed of energy transfer by propagating waves(Metrikine 2006).





4. Discretization of the equation of motion

In this section, a spatial discretization based on the finite-element method iscarried out and the conditions for the convergence are reported.

(a) Finite-element equations

A weak form of the initial-boundary value problem is taken by premultiplyingequation (2.12) by a weight function v and integrating over the domain Ω, that is

∫Ω

v

[ρ

(1 − �2

12∇2

)∂2u∂t2

]dΩ =

∫Ω

v

[T

(∇2u − �2

6∂4u

∂x2∂y2

)+ p

]dΩ. (4.1)

The term p represents the influence of a transverse applied force per unit area, andit has been introduced in the last integral of equation (4.1) in order to generalizethe initial boundary-value problem. Next, the divergence theorem is used todistribute the differentiation among v and u, so the weak form of equation (2.12)is written as∫Ω

vρu dΩ +∫Ω

ρ�2

12

(∂v

∂x∂u∂x

+ ∂v

∂y∂u∂y

)dΩ

+∫Ω

T(

∂v

∂x∂u∂x

+ ∂v

∂y∂u∂y

)dΩ +

∫Ω

T�2

6∂2v

∂x∂y∂2u∂x∂y

dΩ

=∫Ω

vpdΩ +∮Γ

vT[nx

(1 − �2

12∂2

∂y2+ ρ

�2

12∂2

∂t2

)∂u∂x

+ny

(1 − �2

12∂2

∂x2+ ρ

�2

12∂2

∂t2

)∂u∂y

]dΓ +

∮Γ

T�2

12

(∂v

∂xny + ∂v

∂ynx

)∂2u∂x∂y

dΓ ,

(4.2)

where nx and ny are the components of the unit normal vector on the boundaryΓ , and a superimposed dot denotes the full derivative with respect to time.On the right-hand side of equation (4.2), the boundary integrals are collectedwhose contribution will be discussed afterwards in relation to the precise form ofthe boundary conditions of the problem. At this stage, we assume homogeneousboundary conditions, so the boundary integrals cancel.

In deriving the finite-element equations, the computational domain Ω ispartitioned into bilinear isoparametric quadrilateral finite elements. This finiteelement is designated in the sequel as Q4 finite element (i.e. a quadrilateralfinite element with four nodes). The classical nodal interpolation procedure ofthe displacement field is assumed, and the time dependence is separated fromthe spatial variation. The unknown displacement field u(x , y, t) is approximatedover a typical finite element Ωe by using standard polynomial shape functions Nj(j = 1, . . . , 4), as follows:

u(x , y, t) =∑

j

dej (t)N

ej (x , y), (4.3)





where dj (j = 1, . . . , 4) are the nodal displacements of the jth node of theelement. The weight function v is also discretized with the same shape functions.Because the weak form involves first-order and second-order mixed partialderivatives, C 0 continuous shape functions can be used, avoiding the cumbersomeC 1 continuity in the numerical implementation.

Substituting the finite-element approximation in the weak form, thecorresponding matrix form of equation (4.2) is given by

[M 0 + M 2] d + [K0 + K2] d = P, (4.4)

where

M (0)ij =

∫Ωe

ρN ei N e

j dΩ and K (0)ij =

∫Ωe

T(

∂N ei

∂x

∂N ej

∂x+ ∂N e

i

∂y

∂N ej

∂y

)dΩ (4.5)

are the classical elasticity contributions to the element mass and stiffnessmatrices, respectively, while

M (2)ij =

∫Ωe

ρ�2

12

(∂N e

i

∂x

∂N ej

∂x+ ∂N e

i

∂y

∂N ej

∂y

)dΩ (4.6a)

and

K (2)ij =

∫Ωe

T�2

6∂2N e

i

∂x∂y

∂2N ej

∂x∂ydΩ (4.6b)

are the higher-order contributions to the element mass and stiffness matrices, and

Pi =∫Ωe

N ei p dΩ (4.7)

is the nodal force vector given by the externally applied pressure.The equation of motion can be easily discretized and solved in the time domain

by applying Newmark’s time-integration method. For an unconditionally stablesolution, the constant average acceleration scheme is used.

(b) Convergence and accuracy of the finite-element solution

To prove the convergence of the numerical solution to the exact solution of thegoverning equations as the number of elements increases, a measurement for itsquality is required. If an exact solution is not available, the convergence can bemeasured only by the fact that some conditions contained in the mathematicalmodel must be satisfied at convergence (Reddy 1993).

The accuracy of the finite-element method for the membrane problem whenthe enhanced continuum model is used may be assessed, for instance, from theconvergence of the strain energy. The strain energy actually contained in thefinite-element model can be evaluated via

U = 12dT[K0 + K2] d. (4.8)

In order to illustrate the performance of the Q4 membrane elements, theconvergence in strain energy is shown in figure 5a,b for two sequences of uniformand skewed meshes obtained by halving the element size for the static problem of a





3.9

4.0

4.1

4.2

4.3

4.4

4.5st

rain

ene

rgy

(×10

−4

J)

stra

in e

nerg

y (×

10−

4 )

100 101 102 10310−3

10−2

10−1

100

increased refinement

stra

in e

nerg

y no

rm

100 101 102 103

increased refinement

stra

in e

nerg

y er

ror

(a) (b)

(c) (d)

Figure 5. Convergence of the numerical energy and convergence rate for a sequence of meshes with(a,c) regular and (b,d) quasi-regular patterns.

membrane simply supported on the boundary and loaded by a uniform pressure.The plots give an error estimate if the convergence of strain energy has beenachieved. As both meshes are refined, the strain energy approaches the exactsolution from below.

Another, more frequently used, approach of quantifying the discretization erroris the relative energy norm defined as follows:

e =√

U − Uref

Uref, (4.9)

where the reference energy Uref corresponds to the finest mesh. This measureof error is closely related to the error on the derivative of the computeddisplacement field.

Error estimates of the type (equation (4.9)) are very useful because they givequalitative information on the accuracy of the approximate solution, whether ornot the true solution is known. The energy norm as a function of mesh refinementis reported in figure 5c,d in logarithmic scaling. The error in the energy norm isproportional to h, which represents a typical length of an element side, and isexpressed as

e = Ch−(k+1−m), (4.10)





L

Lx

y

u = 0,

∂u= 0

∂y

u = 0,∂u

= 0∂x

∂u ∂2u= 0, = 0

∂y ∂x∂y

∂u= 0,

∂x

∂2u= 0

∂x∂y

x

y

2 L

2 L

u = 0

u = 0

u = 0

u = 0

(a) (b)

Figure 6. (a) Geometry and boundary conditions of the membrane problem and (b) top-rightquarter of the membrane and boundary conditions.

where k is the degree at which the interpolant polynomial is complete and mis the order of the highest derivative of the displacement field in the weak form(Zienkiewicz & Taylor 2000; Akin 2005). The element investigated here uses apolynomial displacement complete to the degree k = 1, and in the weak form, thehighest derivative order is of first order in the spatial dimensions (m = 1). Thus,a rate of convergence p = k + 1 − m = 1 is expected. As can be observed fromfigure 5c,d, the convergence obtained in the numerical simulation compares verywell with the theoretical value. Moreover, having compared the numerical resultsobtained for the two test meshes, it is evident that the convergence rate is notaffected by the element distortion.

5. Numerical examples

In §2, the equation of motion of a microstructured membrane was derived bymeans of non-local continuum mechanics models, and in §4, the discretizationof the equation in the spatial and temporal domain was presented. Based onthe formulations obtained above, to examine the microstructure effects on thedynamic response of a membrane simply supported at the edges, the resultsobtained from classical and higher-order elasticity (Padé continuum) are discussedin this section.

The geometry and the boundary conditions of the squared microstructuredmembrane are given in figure 6a. The length and the width of the panel is 2L = 2m, the material density is ρ = 1 kg m−2. The parameter � is the microstructuralcharacteristic length. The restoring force in the vibrating membrane arises fromthe tension T at which it is stretched and the in-plane tension T = 16 N m−1 ischosen for the current analysis.

Because the dynamical response of a structure is mainly dependent on thedynamic load applied and on the dynamic characteristic of the structure itself,the analysis of natural frequencies is an important topic. The modal properties





500 1000 1500 2000 2500 30000

100

200

300

400

500

vibrational mode numbers

ω (r

ad s–1

)

Figure 7. Comparison of natural frequencies computed using classical and higher-ordercontinuum models. Solid line, l = 0; dot-dashed line, l = 0.01; dashed line, l = 0.1.

of a membrane can be obtained by solving the standard matrix eigenproblem[−ω2M + K]d = 0, (5.1)

where ω is the finite-element predicted natural frequency of the system and d isthe corresponding natural mode of the system.

As analytical solutions are only available for the classical continuum modeland a mesh refinement allows to obtain more accurate results, a convergenceanalysis has been carried out in order to assess how many Q4 elements shouldbe used in the numerical calculations. The frequencies based on classic elasticityand higher-order models are given in figure 7. The results associated with � = 0correspond to those of the classical elasticity theory where the microstructuraleffect is ignored. It can be seen that the difference of the frequencies for differentvalues of � becomes evident at higher vibrational modes, although this differenceis negligible at lower vibrational mode numbers. Further, the frequency decreaseswhen the characteristic internal length � increases. Hence, the frequencies arealways overestimated by the classical continuum model.

The second numerical application deals with the transient response of themicrostructured membrane subjected at its centre to an impact excitation. Thedynamic load has been modelled as an initial non-zero velocity v(t0) = 1 m s−1 atthe impact point. Owing to the biaxial symmetry of the problem, we use a uniformmesh of Q4 elements to discretize only the top-right quadrant of the domain. Theaxes of symmetry x = 0 and y = 0 become a portion of the boundary Γ of thecomputational domain and the kinematic considerations reported in figure 6 allowus to justify the assumption of homogeneous boundary conditions introduced in§4. The Newmark scheme is used for the time integration of the equations usingthe recommendations on time-step size proposed by Bennett & Askes (2009).





(a)

(b)

Figure 8. Fronts of propagating elastic waves in microstructured membrane: (a) classical continuummodel and (b) enhanced Padé continuum model.

0 0.2 0.4 0.6 0.8 1.0−3

−2

−1

0

1

2

3

x (m)

u (×

10–5

m)

0 0.2 0.4 0.6 0.8 1x (m)

(a) (b)

Figure 9. Propagating waves for enhanced continuum model (Padé model) and two microstructurallengths �: (a) direction θ = 0; (b) direction θ = (π/4). Solid line, l = 0.005; dashed line, l = 0.01.

Figure 8 shows the wavefronts at times t = 0.10, 0.20 and 0.25 s for classicalelasticity, obtained taking � = 0, and the Padé model for microstructural length� = 0.01, respectively. As previously predicted in §3 by means of the dispersionanalysis, the elastic waves obtained using the local elasticity theory propagatefaster than the case of the enhanced continua. The wavefronts for both modelsare governed by the components with higher wavelengths (small k) because theytravel faster than the shorter ones. In fact, the velocity of the long waves is notinfluenced by the direction of propagation, and the fronts have quite a circularshape. The directional properties of the microstructured membrane expected forthe enhanced continuum model can be observed from figure 8b, focussing the





attention on the short waves that propagate slower and whose velocities vary withthe direction of propagation. The directional property of the Padé model can bebetter observed in figure 9, where elastic waves propagating from the excitationpoint to the boundary for different characteristic lengths and along two specificdirections are plotted at time t = 0.20 s.

6. Conclusions

This contribution addresses the simulation of wave dispersion in compositemembranes with microstructure. As an alternative to the direct analysis of thediscrete material that, at microscopic level, has been assumed here as a periodicelastic lattice, a continuum model approximating the actual material behaviourhas been employed. A higher-order theory has been successfully resorted to,overcoming the limits of classical elasticity in the high-frequency regime.

In the first part of this study, continuum models linked with the underlyingmaterial have been introduced, taking into account the discreteness of themicrostructure. From the application of different continualization approaches, ithas been shown that the higher-order theories improve the continuum descriptionby including the microstructural effects. Specifically, enhanced continuumtheories based on the use of Padé approximants introduce higher-order inertiaand stiffness terms in classical models and improve the continuum descriptionby including the length parameter characterizing the heterogeneity. The analysisof closed-form dispersion relations demonstrated that the enhanced continuummodels deliver physically consistent results also in the limit of short waves anddo not generate numerical instability.

Further, the field equations have been discretized in space by means of simplefour-noded finite elements. In the implementation, C 1 continuity is completelyavoided. In fact, the weak form of the governing equations involves first-orderand second-order mixed derivatives of the unknown field, thus C 0 continuousshape functions are sufficient. Importantly, the higher-order terms do not lead toan increase in system size. The evaluation of the convergence rate confirms theappropriateness of the adopted finite-element discretization.

Finally, the analysis of a microstructured membrane has been conducted.The vibration frequencies have been estimated by solving the standard matrixeigenproblem for the classical and enhanced continuum models. It has been foundthat the frequencies are always overestimated by the classical continuum model.In order to illustrate the dispersion of the elastic waves, the response of themembrane to impulsive point load has been simulated. Numerical results are inagreement with the analytical dispersion analysis presented in this paper.

We gratefully acknowledge financial support of the Engineering and Physical Sciences ResearchCouncil, contract number EP/D041368/1.

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