Mechanical properties updating of a non-uniform

natural fibre composite panel by means of a parallel

genetic algorithm

Giuseppe Petrone1, Viviana Meruane2

1Department of Industrial Engineering, Aerospace SectionUniversita degli Studi di Napoli ”Federico II”, Via Claudio 21, 80125 Naples, Italy

email: giuseppe.petrone@unina.it2Department of Mechanical Engineering

Universidad de Chile, Beauchef 851, Santiago, Chileemail: vmeruane@ing.uchile.it

Abstract

The mechanical properties of natural fibre composite panels have a largevariability, which depends on the manufacturing process, the quality of thefibres and the humidity level, among others. Thus, their properties mustbe evaluated experimentally to determine the correct values. This arti-cle presents an investigation on the mechanical properties of a compositepanel made of unidirectional flax fibres embedded in a polyethylene matrix(flax-PE). An initial set of mechanical properties was identified by classicalstatic tests. Then, an experimental modal analysis was performed in orderto get information on natural frequencies and mode shapes, which are re-lated to the mechanical properties. The experimental modal results werecompared with the numerical ones, obtained by means of a finite element(FE) model using the initial set of mechanical properties. Finally, in orderto get a good numerical-experimental correlation, the mechanical propertiesthroughout the panel were updated using an inverse modeling method basedon parallel genetic algorithms.

Keywords: Flax fibres, experimental modal analysis, genetic algorithms,model updating, non-uniform, uncertainties

Preprint submitted to Composites Part A July 27, 2016

1. Introduction

In recent years a significant amount of interest has been shown in the po-tential of natural fibres, such as cotton, jute, flax, bamboo, hemp, and sisalto replace wood fibre and glass fibre as reinforcements. These fibres offerspecific benefits such as low density, low pollutant emissions, biodegradabil-ity, high specific properties and low cost [1, 2]. In Table 1 the mechanicalproperties of some representative natural fibres are reported. It can be seenfrom these data that the density of glass fibre is higher in almost 60% thanthat of natural plant based fibres, providing that the specific stiffness of glassfibre is comparable to that of some natural fibres [3].

The properties of natural fibres are very difficult to measure with a con-siderable number of fibres (between 500 and 4000) needing to be tested toobtain statistically significant mean values. These properties are also stronglyinfluenced by many factors, particularly chemical composition and internalfibre structure, which differ between different parts of a plant as well as be-tween different plants [4]. Other factors that may affect the fibre propertiesare maturity, separating process, microscopic and molecular defects such aspits and nodes, type of soil and weather conditions under which they weregrown [5]

Table 1: Mechanical properties of some natural and synthetic fibres [2]

Fibre Diameter Density Elongation Tensile Youngat break strength modulus

[µm] [g/cm3] [%] [MPa] [GPa]

Cotton 12 - 13 1.5 7.0 - 8.0 287 - 597 5.5 - 12.6Jute 10 - 25 1.3 1.5 - 1.8 393 - 773 26.5Flax 5 - 38 1.5 2.7 - 3.2 345 - 1035 27.6Sisal 8 - 41 1.5 2.0 - 2.5 511 - 635 9.4 - 22.0Glass 10 2.5 2.8 2000 - 3500 70

Carbon 7 - 10 1.4 1.4 - 1.8 4000 230 - 240

This paper focusses its attention on bio-composite panels made of flaxfibres which have been studied by Petrone in his PhD thesis [6]. Many re-searchers have investigated flax fibre properties at the levels of elementaryand technical fibres [7, 8]. Flax fibres are often employed only in low gradecomposite applications [9] because of their large variability in the measured

2

mechanical properties [10, 11] due to uncertainties in the fibres propertiesvariability, and the influence of this variability on the final mechanical prop-erties of the composites. Table 2 represents the tensile properties of elemen-tary flax fibres measured in various studies. The reason for the scattering hasbeen attributed partly to uncertainties in the measurement of the fibre cross-sectional area along the fiber length (i.e. due to the irregularity in the crosssectional fibre shape) [8]. In addition, flax fibres are delicate materials withcross-sectional dimensions in the order of micrometers, and a few millimetersin length; this enhances measurement errors during the mechanical testingof the fibres [12]. Altogether, any uncertain evaluation of the cross-sectionalarea brings a variation into the determined mechanical properties of the fibres[13]. The tensile strength data has been mostly described using a Weibulldistribution function [14]. Bos [10] showed that tensile strength of the fibresdepends on the isolation procedure. Hand isolated fibres are stronger thanmechanically isolated fibres. Bos found also that the mechanical treatmentsinduce kink bands in the fibres and thereby reducing their tensile strengthand that, instead, the fibre strength decreases with increasing fibre gaugelength. Baley [7] determined the tensile strength of elementary flax fibres.He suggested that the fibre kink bands and micro compression defects causea loss of tensile strength of the fibres, and they act as points of fracture ini-tiation during fibre failure. The tensile strength and Young modulus of thefibres were found to decrease with increasing fibre diameter. The not-perfectknowledge of the mechanical properties of a structure is an important issuewhen the representative numerical model has to be built.

Table 2: Tensile properties of elementary flax fibres

Method Gauge Avg. Strength Young Modulus Failure strainLength ± std. dev. ± std. dev. ± std. dev.[mm] [MPa] [GPa] [%]

Green fibre 5 678 ± 216 - -Dew retted 5 906 ± 224 83 - 118 0.13 - 0.34

Enzyme retted 9 591 ± 250 57 ± 22 1.4 ± 0.9Scutched 5 732 ± 220 - -Hackled 3 1522 ± 400 - -Hackled 10 945 ± 190 57 ± 35 2.0 ± 0.4

The traditional approach to characterize an elastic material is through

3

static tests, where the direct measurement of stresses and strains duringtensile, compression, bending or torsion tests allows to identify the materialelastic constants. The disadvantage of these methods is that the are slow,expensive and destructive. Identifying the elastic properties of materialsby dynamic tests is an attractive alternative. Its non-destructive, economic,accurate and easy to implement. If vibrations are induced in a certain sample,then its dynamic response is a function of the geometry, density, boundaryconditions and the elastic constants. Therefore, by analyzing the dynamicresponse of a sample with any shape it is possible to determine the elasticconstants in a single non-destructive test.

Recently, a large number of mechanical characterization techniques basedon the dynamic response of both isotropic and orthotropic materials havebeen proposed [15]. This trend is particularly noticeable in the design oforthotropic panels, in which, except the parameter types like mass densityor shell thickness, the mechanical properties are most uncertain.

In an elastic, homogeneous and isotropic or orthotropic specimen witha certain shape it is possible to identify the elastic constants from the firstresonant frequencies, provided that a theoretical solution for the resonant fre-quencies is known. Unfortunately, exact analytical solutions are only avail-able for a limited number of geometries and boundary conditions. Therefore,different techniques have been used to obtain an approximate solution for theresonant frequencies. The most frequently used is the Rayleigh-Ritz method[16–18], other methods such as superposition based on series expansion [19]and Warbutton equations [20] for thin plates [21, 22] have also been used.

When analytical expressions for the resonant frequencies are not available,its possible to estimate the material constants indirectly by model updat-ing methods, which are usually refereed as “mixed numerical-experimentaltechniques” in the mechanical characterization literature. Model updatingmethods correlate a numerical model of a structure with its experimentaldata to improve the numerical model. In general, the numerical model isderived from finite element analysis (FEA) and the measurements are the vi-bration characteristics. The algorithm updates a set of parameters from thenumerical model to obtain the minimum difference between the numericaland experimental data. Building an accurate numerical model, setting upthe objective function and using a robust optimization algorithm are crucialfactors in model updating. Traditional model updating methods make useof modal information as resonant frequencies and mode shapes. Although,for the mechanical characterization of homogeneous materials only resonant

4

frequencies are required.Several approaches based on model updating to characterize composite

rectangular plates have been presented in literature. For example, Peder-sen and Frederiksen [23] identified the elastic constants of an orthotropicplate with free boundary conditions from its resonant frequencies using theRayleigh-Ritz method and an optimization algorithm. Soares et al. [24]determined the material properties of composite plates, which are modeledusing the linear shear deformation theory of Mindlin. Araujo et al. [25, 26]identified the elastic material modulus of thick composite plates, using ahigher-order finite element method and an optimization technique. Beldzkiet al. [27] determined the elastic constants of glass/apoxy unidirectionallaminates. Cunha and Piranda [28] used a sensitivity-based model updatingapproach to characterize composite materials. Hwang and Chang [29] usedthe optimization module of the ANSYS program to determine the elastic con-stants an aluminum and carbon/epoxy plates. Pagnotta and Stigliano [30, 31]investigated the feasibility of identifying the elastic constants in isotropic ma-terials with samples in any shape. Ismail et al. [32] used a model updatingapproach to study the identification properties of orthotropic plates withdifferent boundary conditions.

Previous discussed techniques involve an optimization procedure that re-quires an initial searching point. These mechanisms are highly dependenton the initial searching point, which causes non-uniqueness in the updatedmodel. Moreover, the ill-conditioned inverse problem leads to instabilitiesand the convergence is not always assured. A suitable alternative are Ge-netic Algorithms (GAs), GA is a global optimization method; it provides arobust search in the entire solution space, thus reaching the global optimum.For these reasons, Maletta and Pagnotta [33] and Teixera et al. [34] haveinvestigated the use of GAs in the mechanical characterization of thin andthick composite plates. Nevertheless, the problem with sequential GAs isthat they are inherently slow when they work with complicated or time con-suming objective functions. To improve the searching speed, Parallel GeneticAlgorithms (PGAs) are proposed. PGAs are particularly easy to implementand provide a superior numerical performance. Meruane and Heylen [35]investigated the advantages of PGAs in a structural damage detection prob-lems. They concluded that PGAs always provide an improved and fastersearch in the solution space compared to sequential GAs.

Frequently composite fabrication leads to non-uniform property distribu-tions, which is particularly true in manufacturing of fiber-reinforced com-

5

posites where the fibers may be nonuniformly distributed throughout thematrix. As an example, Stokes [36] found that the elastic modulus of a glassfiber reinforced polymer panel varied as much as a factor of three. There-fore, it is necessary to measure in situ globally averaged properties as wellas the distribution of properties within the components. All the investiga-tions presented so far are focussed in the identification of global properties,to the authors knowledge the only work that has investigated the experimen-tal identification of non-uniform materials is the one presented by Chen andGibson [37]. They identified the fiber volume fraction distribution along thelength of nonuniform composite beams.

In the present article a methodology to identify the mechanical prop-erties of a nonuniform flax-PE panel is presented. The modal parameters,natural frequencies, mode shapes and modal damping, are identified fromexperimental tests. A finite element model is built and validated by com-paring the numerical and experimental results. Finally, a model updatingalgorithm that correlates mode shapes and natural frequencies, and uses aPGA to solve inverse problem is implemented. Special consideration is takenon the variation of the mechanical properties throughout the panel due tomanufacturing or due to the variability of flax fibres.

2. Model Updating Algorithm

This section describes the formulation of the optimization problem andthe optimization algorithm. The model updating algorithm presented byMeruane [38] has been adapted to the mechanical characterization problem.

2.1. Formulation of the optimization problem

A model updating algorithm updates a set of parameters from the nu-merical model to obtain the minimum difference between the numerical andexperimental data. Defining βi as the ith updating parameter, the modelupdating problem is a constrained nonlinear optimization problem, whereβ = β1, β2, , βn are the optimization variables. The objective function corre-lates mode shapes and natural frequencies.

The error in frequency is given by,

εω(β) =∑i

∣∣∣∣λA,i(β)

λE,i− 1

∣∣∣∣ =∑i

∣∣∣∣∣ω2A,i(β)

ω2E,i

− 1

∣∣∣∣∣ , (1)

6

where λi is the ith eigenvalue and ωi is the ith natural frequency. Thesubscripts A and E refer to analytical and experimental respectively.

The mode shape error is computed as,

εMAC(β) =∑i

|1−MAC (φA,i(β), φE,i)| , (2)

where φA,i and φE,i are the ith analytical and experimental mode shaperespectively. The modal assurance criterion (MAC) is a factor that expressesthe correlation between two modes. A value of 0 indicates no correlation,whereas a value of 1 indicates two completely correlated modes. Scaling ofthe modes is not required here. If the number of measured degrees of freedom(DOF) is less than the numerical DOF, a partial MAC is used. Hence, nomode shape expansion is needed.

The objective function J considers these two error functions plus a regu-larization term,

J(β) =ελ(β)

ελ,0+εMAC(β)

εMAC,0

+ γ∑i

(βi − β0

i

), (3)

where ελ,0 and εMAC,0 refer to the initial values of ελ and εMAC respectively,β0i are initial guess values for the parameters βi, and the regularization termγ∑

i (βi − β0i ) promotes that the updated parameters are not too far from

their initial values. The optimization problem is defined as,

min J(β)subject to lui ≤ βi ≤ ubi

, (4)

where lui and ubi are the lower and upper bounds of the ith updating pa-rameter. The aim is to minimize the objective function J , but GA alwaysmaximizes a problem. The minimization problem is transformed into a max-imization problem by defining the objective function as a suitable numbersubtracted by J .

2.2. Optimization algorithm

Figure 1 illustrates the optimization algorithm used here [38]. The algo-rithm employs a multiple population GA with four populations and a neigh-borhood migration with initial populations that are generated randomly.Each population runs a sequential GA that from time to time exchangesinformation with its neighbors (migration). The gene of each chromosome

7

is an updating parameter of the optimization problem. The GA uses a nor-malized geometric selection. To ensure an effective search with an adequatebalance between exploration and exploitation, each population works witha different crossover, being the following ones: arithmetic crossover, heuris-tic crossover, BLX-0.5 crossover and uniform crossover. In addition, eachpopulation applies both boundary and uniform mutations. Each populationhas a size of 40 individuals and the crossover and mutation probabilities arepc=0.80 and pm=0.02 respectively.

The migration interval is automatically adjusted. If a population has noimprovement after 40 generations, the GA stops and exchanges the individu-als with their neighbors. This exchange of individuals is synchronous i.e., thealgorithm waits until the five populations are ready to perform the migra-tion. At each migration, each population sends its best individual, whereasthe received individual replaces its worse individual. Before each migration,the algorithm compares the best individuals from all populations, if they areall the same the optimization is finished.

Migration

Create initial populations

Same solution for all populations?

End

No improvement in all populations?

T

F

F

T

GA optimization

Figure 1: Optimization Algorithm

8

3. Description of the panel and static tests

3.1. Materials and manufacturing

The investigated panel, of dimension 706 x 496 x 3.6 mm3 , is made ofunidirectional flax fibres mat (UD180-C003) of 180 g/m2 from Lineo, Belgium(at 0.3 volume fraction) combined with Linear Low Density PolyEthylene(LLDPE) films with a density of 0.91 g/cm3. The panels herein discussedwere manufactured at CACM (Centre for Advanced Composite Material)of the University of Auckland using a vacuum bag technique. Though themelting temperature of LLDPE used was 124◦C, it was necessary to have thetemperature of the oven at 170◦C and hold it constant for one hour to attainuniform consolidation throughout the specimen.

3.2. Testing

Tensile testing in the longitudinal and transverse directions was con-ducted according to ASTM D3039 [39] to determine the tensile modulusand strength in both directions. The test was conducted on an Instron 5567universal testing instrument (Fig. 2); the distance between the grips mea-sured 150 mm and a crosshead speed of 2 mm/min was used. Specimen wereprepared in accordance with standard requirement with a rectangular shapewith an overall length of 220 mm, a width of 30 mm and a thickness of 3 mmand were cut on a CNC milling machine. The width and thickness of eachspecimen was measured in three places along the narrow section and thenused to calculate the average cross-sectional area. The load cell measuredthe tensile load. The stress was calculated using the measured area and loaddata. A gauge length of 50 mm was established by marking the specimenwith two white dots. A video extensometer measured the displacement ofthe white dots and thus the strain of the specimen. Using this stress andstrain information, the tensile strength and modulus were determined. Fur-thermore, transverse and longitudinal strain was plotted and the gradient ofthe linear region, where linear elasticity is present, was used to determine thePoisson’s ratio. All test specimens were conditioned from the environmentalconditioning of the laboratory (Temperature=23,9◦C and humidity= 54%).In plane shear properties were determined using two-rail shear test as perstandard ASTM D 4255 [40]. Constantan alloy strain gauge rosette, EA-06-060RZ-120 of resistance 120 (±0.4%) and a gauge factor 2.06 was used torecord strain. The shear modulus G23 was determined using the following

9

equation,

G23 =E22

2(1 + ν23)(5)

All the mechanical properties are reported in Table 3.

Figure 2: Instron universal testing machine (a) and flax-PE specimen (b)

Material property Unit Symbol Value Test standard

Tensile modulus [GPa] E11 9.50 ASTM D 3039Tensile modulus [GPa] E22 1.30 ASTM D 3039

Shear modulus [GPa] G12 0.55 ASTM D 4255Shear modulus [GPa] G23 0.40 -

Poisson ratio ν12 0.40 ASTM D 3039Poisson ratio ν23 0.60 ASTM D 3039Sheet density [kg/m3] % 1025 Conventional method

Table 3: Mechanical properties of flax-PE composites

10

4. Modal analysis and model updating

4.1. Experimental modal analysis

The experimental modal analysis was performed with two techniques. Inthe first case the panel is excited by a modal hammer and the response iscaptured by lightweight miniature accelerometers, whereas in the second testthe panel is excited by an electrodynamic shaker and the response is capturedby a laser vibrometer. In both cases, we are modifying the properties of thepanel, first by adding mass due to the sensors (0.7g) and second by addingstiffness due to the shaker attachment. It was found that the shaker hasa more significant effect in the identified mode shapes and therefore it waspreferred to work with the experimental data obtained in the first test [6].The experimental setup and data acquisition for this case is described next.

Frequency response measurements were performed through modal testsadopting the so-called roving hammer technique, where the excitation is pro-vided by the modal hammer in all the nodes of the mesh grid of the panel.The panel was assumed to be not constrained to overcome any kind of prob-lems arising from the boundary conditions. Thus the panel was suspendedwith soft rubber bands to simulate a free-free boundary condition. An impacthammer (ENDEVCO Modal Hammer 2302) was preferred over an electro-dynamic shaker to provide the excitation since the panel is small and light.Two accelerometers (PCB 352B10) were used to measure the response at thebottom corner and on sideward of the panel. Because an impact test is notreplicable (unless a mechanical device is used to hit the panel with the ham-mer) and in order to reduce the noise level, each measurement was obtainedas the spectrum averaging of the responses of five different impacts, ensuringa coherence as much as possible close to the unity. Vibration measurementswere taken in the frequency range 0-256 Hz with a resolution of 0.25 Hz. Aschematic diagram of the experimental set-up is shown in Figure 3.

The mesh consisted of seven points along x-axis and ten along the y-axis, equally spaced. As a rule of thumb, 2nw+1 grid points are required todescribe nw half-waves, which means that the used mesh allowed to describefour half-waves along the longest side and three half-waves along the shortestone. The frequency response data was recorded using the acquisition systemLMS SCADAS III and then analysed by means of the software LMS Test.Lab8B. Figure 4 presents the first six identified mode shapes.

11

1.5 2 2.5 3 3.5 4x 104Frequency (Hz)

Pow

er S

pect

ral D

ensi

ty

35171

Impact hammer

Data acquisition and signal processing Rubber bands

Plate

Accelerometer

Figure 3: Experimental set-up: impact hammer and accelerometer

(a) Mode 1, 8.88 Hz (b) Mode 2, 17.58 Hz (c) Mode 3, 24.57 Hz

(d) Mode 4, 28.56 Hz (e) Mode 5, 38.85 Hz (f) Mode 6, 47.37 Hz

Figure 4: Experimental Mode Shapes

12

4.2. Numerical model

The numerical model was built in Matlab R© using the Structural DynamicToolbox (SDT) [41]. The panel was modelled as a shell orthotropic materialusing 4-nodes quadrilateral elements. The numerical mesh consisted of 26x 36 nodes along the in plane-directions. The number of elements in thenumerical model was defined after a convergence analysis, where it was foundthat adding more elements did not modify the first ten modes. The panelwas not constrained along its edges, as assumed in the experimental tests.The initial elastic material properties used in the model are reported in Table3. Figure 5 presents the first nine numerical mode shapes.

(a) Mode 1, 8.07 Hz (b) Mode 2, 17.07 Hz (c) Mode 3, 22.92 Hz

(d) Mode 4, 23.73 Hz (e) Mode 5, 27.88 Hz (f) Mode 6, 43.96 Hz

(g) Mode 7, 47.72 Hz (h) Mode 8, 53.19 Hz (i) Mode 9, 62.72 Hz

Figure 5: Numerical Mode Shapes

4.3. Model updating

The numerical model was updated in two stages: First, the global me-chanical properties were updated, thus assuming an even distribution of prop-erties throughout the panel. In the second stage, the panel was divided in60 regions and the properties of each region were updated individually. Theupdated parameters were: the tensile modulus Ex and Ey, the shear modulus

13

Gxy, Gxz and Gyz, the Poisson ratio νxy and the density. The first six exper-imental modes were used during the updating process. Each experimentalmode is paired with a numerical one based on the MAC (Modal AssuranceCriterion) correlation and the frequency differences. It should be noted thatthe experimental measuring points coincide with nodes of the finite elementmodel, therefore there is no mismatch error between the experimental andnumerical nodes. Table 4 presents the initial mode shape pairs and theircorrelation.

It is possible to see that the third numerical modes was not identifiedexperimentally. This may be due to the fact that the frequencies of thethird and fourth mode modes are very close, thus hindering the identificationduring the experimental modal analysis.

Table 4: Correlation between numerical and experimental models

Mode pair Experimental Mode Numerical Mode ∆ω (%) MAC1 1 1 9.00 0.942 2 2 2.77 0.933 3 4 3.14 0.924 4 5 2.20 0.645 5 6 13.55 0.486 6 7 1.47 0.86

4.3.1. First stage

During the first stage a total of seven parameters were updated, allowinga variation up to 40% with respect to their initial values. The optimumsolution is achieved after 258 generations. Table 5 shows the updated valuesand their variation.

The correlation between numerical and experimental models after thefirst updating is presented in Table 6. It is possible to see an improvementin the natural frequencies differences, but this is not reflected in the modeshape correlation. This may be due to the fact that the hypothesis of auniform panel is not correct, making it necessary to update the parametersof different elements individually.

14

Table 5: Updated parameters

Parameter Updated value Variation (%)Ex [GPa] 8.98 -5.5Ey [GPa] 1.22 -6.1νxy 0.25 -37.2Gxy [GPa] 0.61 13.4Gxz [GPa] 0.55 0.0Gyz [GPa] 0.40 0.0ρ [kg/m3] 980.33 -4.4

Table 6: Correlation between numerical and experimental models after first updating

Mode pair Experimental Mode Numerical Mode ∆ω (%) MAC1 1 1 1.39 0.932 2 2 3.39 0.943 3 4 0.00 0.924 4 5 0.00 0.645 5 6 17.65 0.486 6 7 0.00 0.85

4.3.2. Second stage

During the second stage a total of 420 parameters were updated, i.e. sevenmechanical properties for each of the 60 regions, allowing a variation of 40%with respect to their initial values. It was decided to discard the fifth modeshape pair during the second updating because the MAC value of this pair islower than 0.6 and the frequency difference is much higher than others pairs.In this case the optimum solution is achieved after 298 generations. It shouldbe noted that even that there are much more parameters in this stage theconvergence is not much slower. Therefore adding more parameters duringthe updating process does not change significatively the convergence speed.

The correlation between numerical and experimental models after thesecond updating is summarized in Table 7 and Figure 6. It is possible tosee a large improvement in the correlation, which demonstrates that themechanical properties are not evenly distributed throughout the panel.

15

Table 7: Correlation between numerical and experimental models after the second updat-ing

Mode pair Experimental Mode Numerical Mode ∆ω (%) MAC1 1 1 0.79 0.942 2 2 0.53 0.963 3 4 0.68 0.964 4 5 0.09 0.895 6 7 0.38 0.91

wn=8.81, w

e=8.88, MAC=0.94

Numerical Experimental

(a) Mode pair 1

wn=17.5, w

e=17.6, MAC=0.96

Numerical Experimental

(b) Mode pair 2

wn=24.7, w

e=24.6, MAC=0.96

Numerical Experimental

(c) Mode pair 3

wn=28.6, w

e=28.6, MAC=0.89

Numerical Experimental

(d) Mode pair 4

wn=47.2, w

e=47.4, MAC=0.91

Numerical Experimental

(e) Mode pair 5

Figure 6: Mode pairs between numerical and experimental modes

Figures 7 to 9 show the distribution of the mechanical properties afterupdating. These figures illustrate the variation with respect to the globalproperties obtained in the first stage, a value of one indicates no variation,whereas a value of 1.4 or 0.6 indicate an increment or a reduction in 40%. Themost significant variations were found to be the elastic moduli Ex, Ey, the

16

shear modulus Gxy and the density. Looking at the elastic properties, in fact,it is possible to observe regions with a higher stiffness (or elastic modulus) andregions with a lower stiffness, this is probably related to the manufacturingprocess. The density, on the other hand, seems to be randomly distributed,which might be related to the density of the fibres distributed through thepanel.

(a) Ex (b) Ey

Figure 7: Variation of the elastic modulus

(a) Gxy (b) Gxz (c) Gyz

Figure 8: Variation of the shear modulus

17

(a) νxy (b) ρ

Figure 9: Variation of the Poisson ratio and density

5. Conclusions

In this paper a composite panel made of unidirectional flax fibres embed-ded in a polyethylene matrix (flax-PE) flat panel was investigated numericallyand experimentally. The mechanical properties of the panel was evaluatedexperimentally by classical static tests and used in the Finite Element modelfor the numerical modal investigation. Then, an experimental modal anal-ysis was performed in order to get information on natural frequencies andmode shapes which were compared with numerical ones. Discrepancies be-tween FEA and experimental results due maybe to uncertainty in the use ofinappropriate element material lead the authors to update the mechanicalproperties, used in the FE model, using an inverse modelling method basedon parallel genetic algorithms in order to obtain the minimum differencebetween the numerical and experimental data. The numerical model wasupdated in two stages. First, the global mechanical properties were updatedassuming an even distribution of properties throughout the panel. The cor-relation between numerical and experimental models presented an improve-ment in the natural frequencies differences, but this is not reflected in themode shape correlation probably due to the hypothesis of a uniform panel.In the second stage, the panel was divided in 60 regions and the propertiesof each region were updated individually. This correlation, instead, obtainedwith a total updating of 420 parameters (seven for each sixty region), showeda large improvement in the mode shape correlation, demonstrating that themechanical properties are not evenly distributed throughout the panel.

18

In conclusion it is possible to underline that a very good agreement be-tween numerical and experimental results has been obtained after model up-date confirming the potentiality of this method which results to be essentialto reach a good level of knowledge when dealing with this kind of structuresaffected by a large variability of properties.

The results can be improved if more information is included during themodel updating process such as measurements of the thickness and shapethroughout the panel (the panel might not be perfectly flat). This informa-tion can be included in the numerical model, improving its precision.

References

[1] A. K. Mohanty, M. Misra, L. T. Drzal, Sustainable Bio-Compositesfrom Renewable Resources: Opportunities and Challanges in the GreenMaterials World, Journal of Polymers and the Environment 10 (2002)19–26.

[2] A. K. Mohanty, M. Misra, L. T. Drzal, Natural fibers, biopolymers, andbiocomposites, Taylor & Francis, 2005.

[3] S. V. Joshi, L. Drzal, A. Mohanty, S. Arora, Are natural fiber com-posites environmentally superior to glass fiber reinforced composites?,Composites Part A: Applied science and manufacturing 35 (3) (2004)371–376.

[4] P. C. B. English, D. Bajwa, Paper and Composites from Agro-BasedResources, CRC Press, 1996.

[5] R. Rowell, J. Han, S. Bisen, Changes in Fiber Properties During theGrowing Season, 1996, Ch. 2, pp. 23–36.

[6] G. Petrone, Characterisation of bio-based structures: models and ex-periments, Phd thesis, phD Thesis, ISBN 978-88-98382-07-1.

[7] C. Baley, Analysis of the flax fibres tensile behaviour and analysis ofthe tensile stiffness increase, Composites Part A: Applied Science andManufacturing 33 (7) (2002) 939–948.

[8] K. Charlet, C. Baley, C. Morvan, J. P. Jernot, M. Gomina, J. Breard,Characteristics of hermes flax fibres as a function of their location in the

19

stem and properties of the derived unidirectional composites, Compos-ites Part A - Applied Science and Manufacturing 38 (2007) 1912–1921.

[9] A. Bledzki, J. Gassan, Composites reinforced with cellulose based fibres,Progress in Polymer Science (Oxford) 24 (2) (1999) 221–274.

[10] H. Bos, M. Van Den Oever, O. Peters, Tensile and compressive prop-erties of flax fibres for natural fibre reinforced composites, Journal ofMaterials Science 37 (2002) 1683–1692.

[11] R. Joffe, J. Andersons, L. Wallstrm, Strength and adhesion characteris-tics of elementary flax fibres with different surface treatments, Compos-ites Part A: Applied Science and Manufacturing 34 (7) (2003) 603–612.

[12] M. Aslan, G. Chinga-Carrasco, B. Srensen, B. Madsen, Strength vari-ability of single flax fibres, Journal of Materials Science 46 (19) (2011)6344–6354.

[13] A. Virk, W. Hall, J. Summerscales, Failure strain as the key designcriterion for fracture of natural fibre composites, Composites Scienceand Technology 70 (6) (2010) 995–999.

[14] W. Weibull, A statistical distribution function of wide applicability,Journal of Applied Mechanics 18 (1951) 293–299.

[15] L. Pagnotta, Recent progress in identification methods for the elasticcharacterization of materials, International Journal of Mechanics 2 (4)(2008) 129–140.

[16] W. De Wilde, B. Narmon, H. Sol, M. Roovers, Determination of thematerial constants of an anisotropic lamina by free vibration analysis,in: Proceedings of the 2nd International Modal Analysis Conference,Vol. 44, 1984.

[17] L. R. Deobald, R. F. Gibson, Determination of elastic constants of or-thotropic plates by a modal analysis/rayleigh-ritz technique, Journal ofSound and Vibration 124 (2) (1988) 269–283.

[18] K. Muthurajan, K. Sanakaranarayanasamy, B. N. Rao, Evaluation ofelastic constants of specially orthotropic plates through vibration test-ing, Journal of Sound and Vibration 272 (1) (2004) 413–424.

20

[19] F. Moussu, M. Nivoit, Determination of elastic constants of orthotropicplates by a modal analysis/method of superposition, Journal of Soundand Vibration 165 (1) (1993) 149–163.

[20] G. Warburton, The vibration of rectangular plates, Proceedings of theInstitution of Mechanical Engineers 168 (1) (1954) 371–384.

[21] M. Alfano, L. Pagnotta, Determining the elastic constants of isotropicmaterials by modal vibration testing of rectangular thin plates, Journalof Sound and Vibration 293 (1) (2006) 426–439.

[22] M. Alfano, L. Pagnotta, A non-destructive technique for the elasticcharacterization of thin isotropic plates, NDT & E International 40 (2)(2007) 112–120.

[23] P. Pedersen, P. Frederiksen, Identification of orthotropic material moduliby a combined experimental/numerical method, Measurement 10 (3)(1992) 113–118.

[24] C. M. Soares, M. M. De Freitas, A. Araujo, P. Pedersen, Identification ofmaterial properties of composite plate specimens, Composite Structures25 (1) (1993) 277–285.

[25] A. Araujo, C. M. Soares, M. M. de Freitas, Characterization of materialparameters of composite plate specimens using optimization and exper-imental vibrational data, Composites Part B: Engineering 27 (2) (1996)185–191.

[26] A. Araujo, C. M. Soares, M. M. De Freitas, P. Pedersen, J. Herskovits,Combined numerical–experimental model for the identification of me-chanical properties of laminated structures, Composite Structures 50 (4)(2000) 363–372.

[27] A. Bledzki, A. Kessler, R. Rikards, A. Chate, Determination of elas-tic constants of glass/epoxy unidirectional laminates by the vibrationtesting of plates, Composites Science and Technology 59 (13) (1999)2015–2024.

[28] J. Cunha, J. Piranda, Application of model updating techniques in dy-namics for the identification of elastic constants of composite materials,Composites Part B: Engineering 30 (1) (1999) 79–85.

21

[29] S.-F. Hwang, C.-S. Chang, Determination of elastic constants of mate-rials by vibration testing, Composite Structures 49 (2) (2000) 183–190.

[30] L. Pagnotta, G. Stigliano, Elastic characterization of isotropic plates ofany shape via dynamic tests: theoretical aspects and numerical simula-tions, Mechanics Research Communications 35 (6) (2008) 351–360.

[31] L. Pagnotta, G. Stigliano, Elastic characterization of isotropic platesof any shape via dynamic tests: Practical aspects and experimentalapplications, Mechanics Research Communications 36 (2) (2009) 154–161.

[32] Z. Ismail, H. Khov, W. L. Li, Determination of material properties of or-thotropic plates with general boundary conditions using inverse methodand fourier series, Measurement 46 (3) (2013) 1169–1177.

[33] C. Maletta, L. Pagnotta, On the determination of mechanical propertiesof composite laminates using genetic algorithms, International Journalof Mechanics and Materials in Design 1 (2) (2004) 199–211.

[34] M. F. Teixeira Silva, L. M. S. Alves Borges, F. A. Rochinha, L. A. V.De Carvalho, A genetic algorithm applied to composite elastic parame-ters identification, Inverse Problems in Science and Engineering 12 (1)(2004) 17–28.

[35] V. Meruane, W. Heylen, Damage detection with parallel genetic al-gorithms and operational modes, Structural Health Monitoring 9 (6)(2010) 481–496.

[36] V. K. Stokes, Random glass mat reinforced thermoplastic composites.part i: Phenomenology of tensile modulus variations, Polymer compos-ites 11 (1) (1990) 32–44.

[37] W.-H. Chen, R. Gibson, Property distribution determination for nonuni-form composite beams from vibration response measurements andgalerkins method, Journal of applied mechanics 65 (1) (1998) 127–133.

[38] V. Meruane, Model updating using antiresonant frequencies identifiedfrom transmissibility functions, Journal of Sound and Vibration 332 (4)(2013) 807–820.

22

[39] A. D3039, Standard test method for tensile properties of polymeric ma-trix composite materials, ”” (2008).

[40] A. D4255, Standard test method for in-plane shear properties of poly-meric matrix composite materials by the rail shear method, ”” (2007).

[41] E. Balmes, J. Bianchi, J. Leclere, Structural dynamics toolbox, UsersGuide, Version 6.

23