International Journal of Engineering Science 48 (2010) 599–611

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International Journal of Engineering Science

journal homepage: www.elsevier .com/locate / i jengsci

Elastic characterization of membranes with a complex shape usingpoint indentation measurements and inverse modelling

Jef Aernouts a,*, Ivo Couckuyt b, Karel Crombecq c, Joris J.J. Dirckx a

a Laboratory of Biomedical Physics, University of Antwerp, Groenenborgerlaan 171, 2020 Antwerp, Belgiumb Department of Information Technology (INTEC), Ghent University – IBBT, Gaston Crommenlaan 8, 9050 Ghent, Belgiumc Department of Computer Science, University of Antwerp, Middelheimlaan 1, 2020 Antwerp, Belgium

a r t i c l e i n f o

Article history:Received 5 November 2009Received in revised form 21 January 2010Accepted 1 February 2010Available online 20 February 2010Communicated by K.R. Rajagopal

Keywords:Inverse modellingMembrane elasticityComplex shapePoint indentationMoiré profilometry

0020-7225/$ - see front matter � 2010 Elsevier Ltddoi:10.1016/j.ijengsci.2010.02.001

* Corresponding author.E-mail address: jef.aernouts@ua.ac.be (J. AernoutURL: http://www.ua.ac.be/bimef (J. Aernouts).

a b s t r a c t

The elasticity parameters of membranes can be obtained from tensile experiments onstrips if adequate quantities of the material are available. For biomedical specimens, how-ever, it is not always possible to harvest strips of uniform and manageable geometry suit-able for tensile tests. A typical example is the human tympanic membrane. This smallstructure has a complex conical shape. In such case, elasticity parameters need to be mea-sured in situ.

A possible way to determine elasticity parameters of complex surfaces is the use of pointindentation measurements. In this paper, this characterization procedure was applied on ascaled phantom model of the tympanic membrane. The model was built of natural latexrubber.

In the characterization procedure, a point indentation is carried out on the membranesurface while forces and three-dimensional shapes are measured. Afterwards, a finite ele-ment simulation of the experiment is performed and parameters are found using an opti-mization routine. For validation purposes, the rubber was also subjected to a uniaxialtensile test.

Several hyperelastic constitutive models are available to describe rubber-like behaviour.Among these, Mooney–Rivlin and Ogden models are the most popular. We used a loworder Mooney–Rivlin and a higher order Ogden model to describe our experiments.

Results show that there is a reasonable agreement between the tensile experiments out-put and the output of the inverse modelling of the indentation experiments.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

In order to determine the elastic properties of membranes, tensile and inflation tests are standard procedures becausethey allow a direct identification. For technical materials, samples of adequate dimensions can easily be prepared and mate-rial is available in ample quantity. For a biomedical structure, however, it is often not feasible to produce a sample of welldefined dimensions since its shape is dictated by nature and one has no control of the geometry. For small structures it iseven impossible to prepare a strip which is large enough to get accurate results from a tensile or inflation test. For such cases,the inverse finite element method may be employed to determine the material parameters.

The basic method of inverse analysis, often referred to as material identification, has been known for over 30 years[13,11,26]. It is receiving renewed attention by investigators in biomechanics because of its usefulness in characterizing

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s).

600 J. Aernouts et al. / International Journal of Engineering Science 48 (2010) 599–611

biological tissue having complicated properties, e.g. [15], or shapes, e.g. [16]. A review on membrane biomechanics is givenin [10].

In our laboratory, we are interested in the mechanical properties of the tympanic membrane. This is a very thin mem-brane with an inhomogeneous thickness, having a conical shape. In the middle ear, sound pressure variations are capturedby the tympanic membrane and transferred to the middle ear ossicles. Since the early 90s, finite element modelling has beenused to study this complex system, e.g. [7,8,14,24]. However, good data for the mechanical properties of the tympanic mem-brane are still lacking [6].

Due to the complex geometry of the tympanic membrane, it is not suitable for standard tests in which the exact solutionof the experimental situation is available. Nevertheless, the inverse finite element method could provide a solution. In doingso, one has to apply a specific loading with known boundary conditions. In the tympanic membrane case, applying a uniformpressure does not provide such a situation, since this results in a motion of the first ossicle with unknown counteractingforces. However, some preliminary experiments performed in our laboratory showed that loading the membrane locallyby an indenter yields a well-defined boundary value problem [1].

In this paper, the inverse modelling of a point indentation experiment is applied on a scaled phantom model of the tym-panic membrane. The phantom model was made of natural latex rubber. Indenter force, deflection and sheet deformationwere measured and the parameters in a Mooney–Rivlin and Ogden model were determined. The elasticity parameters ofthe rubber were also determined by a tensile test. In this way, the tensile experiment output can be compared with the in-verse analysis output.

2. Material and constitutive modelling

In our experiments, we used rubber from medical gloves, which mainly consists of natural latex. The thickness is0:18� 0:02 mm and the material is isotropic and homogeneous. Like other rubber-like materials, natural latex exhibits verylarge strains with strongly nonlinear stress–strain behaviour. For this reason, the rubber can be described as a hyperelasticmaterial. In our study, we neglect irreversible phenomena like the Mullins effect [19], viscoelasticity and strain rate depen-dency. These assumptions are valid when doing tests slowly and on the basis of previous studies [23,18].

A hyperelastic material is typically characterized by a strain energy density function W . A well known constitutive law forrubber-like materials is the Mooney–Rivlin law [25]:

W ¼XN

iþj¼1

CijeI1 � 3� �i eI2 � 3

� �jþ 1

2K ln Jð Þ; ð1Þ

with N the order of the model, eI1 and eI2 the invariants of the deviatoric part of the right Cauchy–Green deformation tensor,Cij the Mooney–Rivlin constants, K the bulk modulus and J the determinant of the deformation gradient which gives the vol-ume ratio. The tilde’s above the invariants indicate the removal of any effect due to volume change. The invariants are givenas:

eI1 ¼ ek21 þ ek2

2 þ ek23; ð2Þ

eI2 ¼1ek2

1

þ 1ek22

þ 1ek23

; ð3Þ

with ekiði ¼ 1;2;3Þ the principal stretches in which the volume change is removed. It is common to assume that rubber mate-rials are incompressible when the material is not subjected to large hydrostatic loadings, so that the last term in Eq. (1) canbe neglected [2].

In a first approach to describe our experiments, we will only consider the incompressible first-order Mooney–Rivlin equa-tion (N ¼ 1). In this case, Eq. (1) becomes:

W ¼ C10eI1 � 3� �

þ C01eI2 � 3� �

: ð4Þ

This low order strain energy function is described by two constants: C10 and C01. Just considering the first-order terms, theequation can only describe the first concave increase of a typical rubber engineering stress–strain curve. In general, at largerstrains, a convex increase or upturn is observed after the initial concave part.

Therefore, in a second approach a higher order Ogden constitutive equation will be used. In general, the incompressibleOgden equation is described as follows:

W ¼XN

i¼1

li

ai

ekai1 þ ekai

2 þ ekai3 � 3

� �; ð5Þ

with N the order of the model, ~kiði ¼ 1;2;3Þ the principal stretches of the deviatoric part of the right Cauchy–Green defor-mation tensor and li and ai the Ogden constants.

In Ogden’s paper [20], experimental data from simple tension, pure shear and biaxial tension on a rubber were fitted withthe first-, second- and third-order Ogden equation. The single-term theory gave good results only up to k ¼ 1:4, since it is not

J. Aernouts et al. / International Journal of Engineering Science 48 (2010) 599–611 601

able to describe the typical stress–strain curve upturn at higher stretches. In the two-term case, correspondence betweenexperiments and theory was good up to k ¼ 7 for simple tension and pure shear and up to k ¼ 2 for biaxial tension. Thethree-term theory gave the best fittings. We will use the second-order Ogden model in our second modelling approach sinceit gives significant better results than the first-order case and it is easier for optimization purposes in comparison with thethird-order case. Using the second-order theory, Eq. (5) reduces to a sum of two terms described by four constants: l1;a1;l2

and a2.It is not a priori known how much strain that will arise in the indentation experiments. Therefore, it is rather difficult to

ensure that the functional forms that are used are not over-parameterized [10]. This will be discussed in the discussion.

3. Mechanical testing procedures

3.1. Uniaxial tensile test

Uniaxial tensile tests were carried out on different cut-out rubber strips to obtain a first estimation of the rubber’s first-order Mooney–Rivlin and second-order Ogden parameters. Strips of 80 mm by 5 mm were cut-out and the useful part of thespecimen, that is the part with uniform deformation, was delimited to 20 mm with markers. The specimen was clamped be-tween grips, with the left one connected to a load cell (Sensotec Model 31). Elongations up till 200% were applied and thestrains in the loading direction were measured with a high resolution camera (AVT PIKE F-505B).

Let k1 ¼ k be the stretch ratio in the direction of elongation and r1 ¼ r the corresponding stress. The other two principalstresses are zero since no lateral forces are applied ðr2 ¼ r3 ¼ 0Þ. For constancy of volume the incompressibility conditionk1k2k3 ¼ 1 gives:

k2 ¼ k3 ¼ k�12: ð6Þ

The engineering stress f (force per unit unstrained area of cross-section) in the case of a uniaxial tensile test for the first-or-der Mooney–Rivlin model becomes:

f ¼ rk�1 ¼ @W@k¼ 2 k� 1

k2

� �C10 þ

C01

k

� �: ð7Þ

In the case of the second-order Ogden model, the equation for the engineering stress in the case of a uniaxial tensile test isgiven as:

f ¼X2

i¼1

li kai�1 � k�ðai=2þ1Þ� �

: ð8Þ

Eqs. (7) and (8) were fitted to the experimental data with a trust-region algorithm in Matlab’s cftool function. In the first-order Mooney–Rivlin case, limited validity was taken into account while in the second-order Ogden case the entire straindomain was considered.

It should be noted that performing a uniaxial tensile test yields homogeneous deformation states in which the variableseI1

and eI2, as defined in Eqs. (2) and (3), are related to each other. Such experiments provide too restricted a basis for the der-ivation of the true form of W . It is only by considering all types of strain, e.g. general biaxial strain, and covering as wide arange of strain as possible that the true picture can be assessed [11,25,2]. However, since maximal stretches that will arise inthe indentation experiments will be relatively small (kmax < 2), we can assume that the states of deformation encountered inthe uniaxial tensile test allow comparison with those of the indentation experiments.

3.2. Indentation measurements

3.2.1. Point indentation setupThe phantom model was created in the following way: a latex rubber sheet was flatly clamped along a circular boundary

and then deformed into a complex conical shape by pressing a triangular shaped rod along a radius. The creation process isillustrated in Fig. 1. Another view on the complex shape creation can already be seen in Fig. 7a and b, showing the finite ele-ment simulation of the experiment. The diameter of the phantom model was 50 mm and the height was 16 mm, approxi-mately eight times the size of a human tympanic membrane. The initial stress created during the circular tightening canbe neglected in comparison with the stress originating from the ‘conical shape creation’.

The phantom model was placed on a translation and rotation stage as shown in Fig. 2. Indentations in- and outwards in adirection locally perpendicular to the initial surface were carried out using a stepper motor with indentation depths up till2 mm. The needle had a cylindrical ending with a diameter of 1:7 mm and indentations were carried out slowly(_r ¼ 0:125 mm s�1), meaning that quasi-static behaviour was studied. By doing the indentation perpendicular to the initialsurface and not too close to the boundary, slippage between the needle and the rubber was avoided.

The resulting force was measured with a load cell (Sensotec Model 31). The exact indentation depth was assessed with anLVDT (HBM KWS3071). All signal processing was done via A/D conversion in Matlab (using NI DAQPad-6015).

Fig. 1. Pictures of phantom model creation. Top: step 1, a circularly tightened flat rubber membrane. Bottom: step 2, resulting complex conical shape afterpressing a triangular shaped rod along a radius.

Fig. 2. Schematic drawing of the point indentation setup: (1) translation and rotation stage, (2) phantom model, (3) needle connected to a load cell, (4)stepper motor and (5) LVDT.

602 J. Aernouts et al. / International Journal of Engineering Science 48 (2010) 599–611

3.2.2. Shape measurementThree-dimensional shapes were obtained with projection liquid crystal moiré profilometry [3,4]. Another example of the

use of (shadow) moiré measurements in material identification research can be found in [17]. In our projection moiré tech-nique, a grid is projected onto the surface of interest and a camera, placed at an angle with the projection direction, recordsthe sum of the demodulated grid and the original one. In this way, a fringe pattern is observed. After averaging out the gridnoise and recording multiple phase-shifted fringe maps, a topographic map is calculated.

Our liquid crystal moiré apparatus is a high-resolution projection technique in which both projection and optical demod-ulation are realized with liquid crystal light modulators, see Fig. 3. The computer generated grids, realized on thin film tran-sistor matrices, allow phase-stepping and grid averaging without the need for any mechanically moving component. Thedevice is able to perform topographic measurements with a height resolution of 25 lm on a 1392� 1040-pixel map. Themoiré shape measurement was used to locate the exact point of indentation and to determine if the membrane is locallyperpendicular with the indentation needle. The three-dimensional shape measurements were also used to compare modeland experimental outcome.

3.2.3. Finite element modellingThe numerical simulations were performed with the nonlinear finite element software package FEBio, which is specifi-

cally designed for biomechanical applications. The finite element mesh of the initial flat circular membrane was made of tri-angular shell elements that can employ the full 3D constitutive relations, meaning that the plane stress condition does notneed to be enforced. In the contact areas, mesh density was increased. The membrane showed almost no bending stiffness, sothe well-known locking effect in bending problems can be considered absent. Moreover, a convergence study was carriedout.

In the first modelling approach, the rubber membrane was modelled as a first-order Mooney–Rivlin material. In thesecond approach, a second-order Ogden material was used. The incompressibility constraint was enforced by means of

J. Aernouts et al. / International Journal of Engineering Science 48 (2010) 599–611 603

the augmented Lagrangian technique. The border of the membrane was constrained at its circumference. In the first mod-elling steps, the triangular shaped rigid body was moved vertically into the flat membrane using a combination of rigid – andsliding contact. In the following modelling steps, perpendicular needle indentation was modelled as a rigid body translationwith frictionless sliding contact.

3.2.4. Parameter optimizationUsing this finite element model construction, the elasticity parameters that describe the rubber elasticity were found by

minimizing the mean square force error (MSE) defined as:

Fig. 3.modula

Fig. 4.selectio

errorforce ¼1N

XN

j¼1

FexpðqjÞ � FmodðqjÞ� �2

; ð9Þ

in which N is the number of evaluated points, qj the indentation depth, FexpðqjÞ the experimental force and FmodðqjÞ the sim-ulated force.

Schematic drawing of the projection liquid crystal moiré profilometry setup with LCD projector, lenses (L1, L2, L3) and CCD camera. The liquid crystaltor (LCM) grids + polarizers allow phase stepping and grid averaging.

Flowchart of the hyperelastic material identification procedure: a surrogate model is built in a given input domain using an intelligent samplen. After enough iterations, optima can be found by inspecting the output.

604 J. Aernouts et al. / International Journal of Engineering Science 48 (2010) 599–611

The optimization was done with a surrogate modelling routine. We used the Matlab Surrogate Modelling toolbox (SUMO)to create a surrogate model mapping the elasticity parameters to the errorforce function in a given input domain [9]. A sim-plified flowchart is depicted in Fig. 4.

At first, an initial set of at least 20 samples arranged in an optimized Maximin Latin Hypercube design, implemented as in[21], together with the corner points are evaluated. Using these initial samples, a kriging model is constructed [22]. Thisrough approximation model is refined by intelligently selecting and evaluating extra samples according to some samplingstrategy. Subsequently, the kriging model is updated with this new information until a convergence criterion is met.

In this study, two sampling strategies were combined. Firstly, the Local Linear Adaptive sampling algorithm (LOLA) auto-matically identifies nonlinear regions in the domain and samples these more densely compared to more linear regions [5]. Inaddition, for this work the LOLA algorithm has been changed slightly to allow for non-convergence of the simulator. The sec-ond sampling strategy determines the current minima of the surrogate model using the Dividing Rectangles optimizationalgorithm [12] and samples more densely in these locations.

By combining these two sampling strategies, a final surrogate model is obtained which is a global accurate approximationof the errorforce function in the input domain. Finally, the optimization of Eq. (9) is easily achieved by analysing this cheapsurrogate model.

4. Results

4.1. Uniaxial tensile test

In Fig. 5 a plot of uniaxial tensile test data from three different samples is shown. The different dots represent experimen-tally measured engineering stress–stretch data. The full lines represent the first-order Mooney–Rivlin fittings with Eq. (7)and the dotted lines represent the second-order Ogden fittings with Eq. (8).

Since the first-order Mooney–Rivlin equation can only describe the first concave part of the engineering stress–straincurve, fitting in this case was done only for stretches up to k ¼ 1:7. It is not possible to obtain good overall fitting for stretchesup to k ¼ 3. The mean output of the three fittings was C10 ¼ ð43� 5Þ kPa and C01 ¼ ð159� 6Þ kPa.

In the case of the second-order Ogden equation, fitting was done in the entire domain of applied strains. As one can see,good overall fitting is obtained. The mean output was l1 ¼ ð42� 11Þ kPa, a1 ¼ 2:2� 0:4, l2 ¼ ð�213� 4Þ kPa anda2 ¼ �3:6� 0:2.

4.2. Indentation measurements

4.2.1. Experimental outcomeIn Fig. 6a and b force-indentation curves are shown from two needle indentation experiments at different locations.

Indentations were applied, respectively, 5.5 mm and 8.4 mm away from the border at a position 90� and 135� away fromthe triangular shaped rod horizontal axis. Due to the different indentation locations, a difference of 20 mN at maximumindentation was observed. A small hysteresis is present, which will be neglected in further analysis.

1 1.5 2 2.5 30

1

2

3

4

5

6

7

8 x 105

stretch

engi

neer

ing

stre

ss [P

a]

experimental data 1experimental data 2experimental data 31st−order MR fit data 1 1st−order MR fit data 21st−order MR fit data 32nd−order Ogden fit data 12nd−order Ogden fit data 22nd−order Ogden fit data 3

Fig. 5. Uniaxial tensile test: the different symbols represent experimental data from three samples. The full lines represent the first-order Mooney–Rivlinfittings, the dotted lines the second-order Ogden fittings.

0 500 1000 1500 20000

20

40

60

80

100

120

140

160

indentation depth [μm]

forc

e [m

N]

experimental datamodel data: C10=31 kPa; C01=172 kPamodel data: μ1=59 kPa; α1=1.94;μ2=−290kPa; α2=−2.44

0 500 1000 1500 20000

20

40

60

80

100

120

140

160

indentation depth [μm]

forc

e [m

N]

experimental datamodel data: C10=33 kPa; C01=163 kPa model data: μ1=50 kPa; α1=2.36;μ2=−224 kPa; α2=−2.92

Fig. 6. Force-indentation data of the first (a) and the second (b) indentation experiment. The full line represents experimental data, the circular dotsrepresent first-order Mooney–Rivlin model data and the rectangular dots represent second-order Ogden model data.

J. Aernouts et al. / International Journal of Engineering Science 48 (2010) 599–611 605

4.2.2. Finite element modellingThe non-uniform membrane mesh built with 5853 triangular shell elements used for the finite element calculations of the

first indentation experiment is shown in Fig. 7a. In the point indentation neighbourhood and the triangular shaped rod con-tact area, mesh density was increased. The vertical line in bold represents the path from which nodal displacements wereextracted for further analysis. For the second indentation experiment another mesh was created with 6483 number of ele-ments. Since this was a very similar one, it is not depicted in the text.

Fig. 7b shows the resulting finite element model for a random pair of first-order Mooney–Rivlin parameters. The colourmap represents the effective strain, defined as:

eeff ¼1ffiffiffi2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðexx � eyyÞ2 þ ðeyy � ezzÞ2 þ ðexx � ezzÞ2 þ 6e2

xy þ 6e2yz þ 6e2

xz

� �r: ð10Þ

This definition allows a one-dimensional interpretation of three-dimensional strain fields. The corresponding effectivestretch is then given as keff ¼ 1þ eeff . The effective strain in the finite element model is dependent of the input parameters,but in general it rises up to a maximum of 120% at the top of the cone, with a corresponding maximal principal strain of100%, and up to approximately 40% in the point indentation area, with a corresponding maximal principal strain of 30%.For the second indentation experiment, the finite element model is equivalent and hence it is not depicted in the text.

A mesh convergence study was carried out to find a good balance between accuracy and computing time. Convergencewas found at approximately 17,000 elements, with a corresponding computing time of approximately 30 min on a personal

Fig. 7. (a) Finite element mesh used for the finite element calculations of the first indentation experiment. The mesh contains 5853 elements. In furtheranalysis, nodal displacements were extracted from the analysis path. (b) Resulting finite element model with applied triangular shaped rigid bodytranslation and subsequent needle indentation. The colour map represents the effective strain, which rises up to approximately 40% in the point indentationarea.

606 J. Aernouts et al. / International Journal of Engineering Science 48 (2010) 599–611

computer (Intel quad core 2.66 GHz and 4 GB of RAM). Since we need to go through an optimization routine, we used a coar-ser mesh with 5853 number of elements. Using this mesh, there is a 3.3% relative error on the force, but computing time isonly 5 min.

4.2.3. First-order Mooney–Rivlin optimizationIn the first modelling approach, the rubber was modelled as a first-order Mooney–Rivlin material. In order to find the

corresponding parameters C10 and C01 that result in a force-indentation curve which is in good agreement with the exper-imental one, an optimization was carried out. We used the surrogate modelling routine with the input domain set to

J. Aernouts et al. / International Journal of Engineering Science 48 (2010) 599–611 607

C10 ¼ ½10;140� kPa and C01 ¼ ½60;210� kPa. The final surrogate model of the first indentation experiment, mapping the elas-ticity parameters to the errorforce function, is shown in the contour plot in Fig. 8. In total, 119 samples were evaluated to guar-antee an accurate kriging model. The evaluated samples are plotted as white points and the colour map represents thesurrogate model fitted through the errorforce values. One can see that as a consequence of the chosen sampling strategy, moresamples were picked in the minimum areas in comparison with elsewhere.

The surrogate model shown in Fig. 8 has the shape of a valley, implying that there is a range of minima. The minima arelocated on a linear curve, highlighted with a dashed line, with the following linear equation:

Fig. 8.Mooneythese m

C01 ¼ �0:964 � C10 þ 201890: ð11Þ

For the second indentation experiment, a similar surrogate model was obtained. Since its equivalence, it is not depicted inthe text. However, the range of minima were also located on a linear curve with the following equation:

C01 ¼ �0:978 � C10 þ 195180: ð12Þ

It is not possible to extract a unique set of first-order Mooney–Rivlin parameters only on the basis of a load curve optimi-zation. Therefore, also a shape analysis is required since various models that exhibit identical load curves will have differentstress distributions and hence different deformations.

In Fig. 9a, the difference between model – and experimental deformation for the first indentation experiment along theanalysis path shown in Fig. 7a after 2 mm needle indentation is shown. The different dots represent model data for threepairs of first-order Mooney–Rivlin parameters that fulfil Eq. (11).

In order to extract a unique pair of first-order Mooney–Rivlin parameters out of these who fulfil Eq. (11), a shape analysiswas carried out. This was achieved by optimizing the mean square shape error defined as:

errorshape ¼1N

XN

j¼1

zexpðrjÞ � zmodðrjÞ� �2

; ð13Þ

with N is the number of evaluated nodal points, rj the position on the analysis path as in Fig. 7a, zexpðrjÞ the experimentallymeasured height after a 2 mm needle indentation and zmodðrjÞ the model height after 2 mm indentation. The errorshape wascalculated for 35 equally spread C10 values in the input domain with corresponding C01 values conform Eq. (11). The result isshown in Fig. 9b. A minimum can be seen for C10 ¼ 31 kPa and matching C01 ¼ 172 kPa. The corresponding force-indentationdata of this optimum value is plotted in Fig. 6a with circular dots.

For the second indentation experiment, an identical shape analysis was performed which is not described in detail. How-ever, the final output here was C10 ¼ 43 kPa and C01 ¼ 153 kPa and the corresponding force-indentation data is plotted inFig. 6b with circular dots.

4.2.4. Second-order Ogden optimizationIn the second modelling approach, the rubber was modelled as a second-order Ogden material. Again, a surrogate mod-

elling routine was used to find elasticity parameters that result in a force-indentation curve which is in good agreement withthe experimental one. The input interval was set to l1 ¼ ½20;80� kPa, a1 ¼ ½1:5;2:9�, l2 ¼ ½�375;�125� kPa anda2 ¼ ½�4:8;�2:4�. When optimizing four parameters, it is rather difficult to interpret the resulting surrogate model graphi-

C10 [Pa]

C01

[Pa]

2 4 6 8 10 12 14x 104

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x 105

erro

r forc

e [mN

2 ]

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

2400

Contour plot representing the final surrogate model of the first indentation experiment built up after 119 sample evaluations. The first-order–Rivlin input parameters are plotted as white dots and the resulting range of local minima is highlighted with a dashed line. The errorforce values ofinima are �8 mN2.

608 J. Aernouts et al. / International Journal of Engineering Science 48 (2010) 599–611

cally. However, multiple minima can be found by running the toolbox sufficiently long and choosing the aforementionedsampling strategy.

For the first indentation experiment, after 316 surrogate modelling sample evaluations, nine input samples were foundthat resulted in a load curve with an errorforce value smaller than 10 mN2. The samples are listed in Table 1. It is clear that

25 30 35 40 45 50

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

r [mm]

erro

r z [mm

]

baselinemodel data: C10=31 kPa; C01=172 kPamodel data: C10=11 kPa; C01=191 kPamodel data: C10=51 kPa; C01=153 kPa

0 1 2 3 4 5 6 7 8x 104

0.005

0.006

0.007

0.008

0.009

0.01

0.011

0.012

0.013

0.014

0.015

C10 [Pa]

erro

r shap

e [mm

2 ]

Fig. 9. (a) Difference between first-order Mooney–Rivlin model - and experimental deformation along the analysis path shown in Fig. 7a after 2 mmindentation for the first indentation experiment. (b) Output of errorshape optimization. The corresponding C01 values can be found through Eq. (11). Theminimum is C10 ¼ 31 kPa and matching C01 ¼ 172 kPa.

Table 1Second-order Ogden load curve optima for the first indentation experiment found with the surrogate modelling routine.

No. l1 (kPa) a1 l2 (kPa) a2

1 21.27 2.82 �204.50 �3.592 64.04 1.87 �244.23 �2.823 57.40 1.99 �172.78 �3.864 66.42 1.74 �204.83 �3.365 41.93 2.08 �211.28 �3.376 51.18 1.85 �165.37 �4.107 33.02 2.54 �237.46 �2.998 59.17 1.94 �289.64 �2.449 45.31 2.16 �230.20 �3.00

0 1 2 3 4 5 6 7 8 9 100.008

0.009

0.01

0.011

0.012

0.013

0.014

0.015

0.016

0.017

Optimum number

erro

r shap

e [mm

2 ]

Fig. 10. Output of errorshape calculation for the first indentation experiment with second-order Ogden modelling. The values on the x-axis represent theoptimum number as listed in Table 1.

J. Aernouts et al. / International Journal of Engineering Science 48 (2010) 599–611 609

these samples are distributed over the entire input domain, so a subsequent deformation analysis is required here too. Toachieve this, the errorshape defined in Eq. (13) was calculated for all the optimum values. The result is shown in Fig. 10. Aminimum was found for optimum number 8. The corresponding force-indentation data of this optimum value is plottedin Fig. 6a with rectangular dots. Due to some modelling difficulties, load curve data in this case could only be determinedstarting from 500 lm.

For the second indentation experiment, after 147 surrogate modelling sample evaluations, 14 input samples were foundthat resulted in an optimum load curve. In a similar way to the first indentation experiment, a shape analysis was carried out.Since its equivalence, this is not shown in the text. However, a minimum was found for l1 ¼ 49:58 kPa, a1 ¼ 2:36,l2 ¼ �224:32 kPa and a2 ¼ �2:92. The corresponding force-indentation data is plotted in Fig. 6b with rectangular dots.

5. Discussion

In this paper, an elastic characterization procedure for membranes with a complex shape is proposed. The approach usespoint indentation measurements together with inverse finite element modelling. The test specimen was made of natural la-tex rubber and had an upscaled tympanic membrane shape. In this way, it represents a realistic situation as encountered inbiomedical research.

In order to validate the approach, a uniaxial tensile test on three different samples was performed. Both a first-order Moo-ney–Rivlin material and a second-order Ogden material were fitted to the data. The average output was C10 ¼ ð43� 5Þ kPaand C01 ¼ ð159� 6Þ kPa in the Mooney–Rivlin case and l1 ¼ ð42� 11Þ kPa, a1 ¼ 2:2� 0:4, l2 ¼ ð�213� 4Þ kPa anda2 ¼ �3:6� 0:2 in the Ogden case.

Applying the point-indentation method, firstly a first-order Mooney–Rivlin modelling was used. Doing a load curve opti-mization, it seemed that there were multiple first-order Mooney–Rivlin pairs that describe experimental load curve data.Hence, a subsequent shape analysis was carried out for which a very precise shape measurement was necessary. We useddeflection information from the radial line through the indentation point because the deflection differences for different in-put parameters have maximum values along this line. In previous studies, deformation information over the entire mem-brane was used and it was argued that this leads to more accurate results, e.g. [16]. This was however not studied in thispaper, since we assumed it would not contribute to more accurate optima. It is furthermore mentioned that one could alsodefine an error that combines both the force and shape errors, leading directly to the final optimum, but it is argued thatdoing the two optimizations subsequently gives more insight.

The final output was C10 ¼ 31 kPa and C01 ¼ 172 kPa for the first indentation experiment and C10 ¼ 43 kPa andC01 ¼ 153 kPa for the second. There is a good agreement with the tensile test output, especially for the second indentationexperiment. However, there is a relative difference between the two experiments of approximately 30% for the C10 value andapproximately 10% for the C01 value. It is known that the C10 parameter has a small influence on the stress–strain curve forstretches smaller than 1.5. Since effective stretches in the point indentation area only go up to 1.4, with a correspondingmaximal principal stretch of 1.3, this apparently high relative difference can be argued. Moreover, the deformation differencebetween multiple load curve optimum models is rather small, meaning that the error on the final output after shape analysisis considerable.

In a second approach, the rubber was modelled as a second-order Ogden material. Again, multiple load curve optima werefound and a subsequent shape analysis was carried out. The final output was l1 ¼ 59:17 kPa, a1 ¼ 1:94, l2 ¼ �289:64 kPa

610 J. Aernouts et al. / International Journal of Engineering Science 48 (2010) 599–611

and a2 ¼ �2:44 for the first indentation experiment and l1 ¼ 49:58 kPa, a1 ¼ 2:36, l2 ¼ �224:32 kPa and a2 ¼ �2:92 for thesecond. The agreement with the tensile experiments is reasonable. Deviations can again originate from the very delicateshape analysis. For example, in the case of the first indentation experiment optimum number 5 is in very good agreementwith the tensile test data, but its errorshape value is not an optimum. Moreover, it is known that the parameter a2 has a rathersmall influence on the stress–strain curve for stretches smaller than 1.5.

We assumed that using a higher order modelling would result in a better agreement between numerical and experimen-tal results. Inspecting the force-indentation curves in Fig. 6a and b, one can see that there is almost no difference between thetwo modelling approaches. This can be argued when one considers the uniaxial tensile test output which shows that thefirst-order Mooney–Rivlin theory can describe the rubber behaviour very well up to stretches of 1.7 and the second-orderOgden theory up to stretches of at least 3. In the point indentation models as shown in Fig. 7b, the only region where aneffective stretch or maximal principal stretch higher than 1.7 occurs is at the top op the cone. This part of the specimenis sufficiently distant from the indentation area and hence will have a small influence on the force output on the needle. Tak-ing a look at the model deformation, the optimum errorshape values are even better in the Mooney–Rivlin case. One wouldhowever expect a smaller deformation error for the second-order Ogden case. This discrepancy can originate from the verydelicate shape analysis.

In summary, it can be concluded that the first-order Mooney–Rivlin strain energy function was an adequate descriptor ofthe observed behaviour while the second-order Ogden strain energy function was over-parameterized.

6. Conclusion

We have shown that the point indentation technique allows a determination of the elasticity parameters of a complexsurface. Due to the considerable deviations of the elasticity parameters as compared to the results of the tensile test, theapproach allows an estimation rather than an exact identification. However, on small membranes with a complex shape likethe tympanic membrane, it is inevitable to perform in situ measurements.

In hearing science, correct tympanic membrane elasticity parameters are needed for realistic modelling of middle earmechanics. Such modelling is for instance used to predict the behaviour of tympanic membrane and middle ear ossiclesprostheses.

Acknowledgements

This work was supported by the Institute for the Promotion of Innovation through Science and Technology in Flanders(IWT-Vlaanderen).

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