New Double Indentation Technique for Measurement of the Elasticity Modulus of Thin Objects
Abstract
In this paper we introduce a new method to determine the Young’s modulus of thin (biological) samples. The method is especially suitable for small objects with a thickness of a few hundred micrometers. Such specimens cannot be examined with existing tests: compression and tensile tests need well-known geometry and boundary conditions while classic indentation tests need relatively thick pieces of material. In order to determine the elastic modulus we use the indentation theory as proposed by Sneddon and correct it with a finite element calculated κ factor to compensate for the small thickness. In order to avoid material deformations at the contact zone between the sample bottom and the sample stage, we replace the sample stage by a second indentation needle. In this way the sample can be clamped between two identical needles and a virtual mirror plane is introduced. The new method was used on four test-materials and results agreed well with the outcome of a standard compression method applied on large samples of the same materials. As an application example the technique was applied on thin biological samples, namely middle ear ossicles of rabbits.
Keywords
Double indentation Elasticity modulus Thin objects FE modelling Measurement method Middle ear ossicle boneIntroduction
Finite element (FE) models are widely used to investigate (bio-)mechanical problems. However, to create accurate models, the exact geometry, the boundary conditions and the material properties of the components have to be known precisely [1]. A very important parameter for (linear elastic) materials is the Young’s modulus (E). For large material samples, standard tensile and compression methods for measuring Young’s moduli are widely available (e.g. ASTM D695-02a ‘Standard Test Method for Compressive Properties of Rigid Plastics’). Those methods are very useful to determine the modulus of larger biomaterials, like the femur bone [2]. These tests require a precise description of the geometry of the specimen and usually a cylinder or a cuboid needs to be created. In order to avoid slipping or stretching, the boundary conditions at the contact zone have to be controlled very well. However, when applying this technique on small materials, these conditions are that difficult to realize that even a very fine preparation cannot avoid inaccuracies [3, 4].
Another way to obtain material parameters is through backwards engineering: the material properties in the model are changed to get a best fit between simulation and experiment. Although good results on small specimens can be obtained, this technique needs unique experiments and time-consuming simulations [5, 6, 7].
In this paper we will validate our new method on four test-materials with different thickness by comparing the results from the new method to a standard compression test (ASTM D695-02a). Finally, we will apply our method on biological samples and we will determine the elasticity modulus of middle ear (ME) ossicle bone. In current modelling of middle ear mechanics, those bones are treated either as rigid bodies or standard values for the Young’s modulus are used (12 ±3 GPa) [23, 24]. A correct value of the elasticity modulus becomes very important when bending of the auditory ossicles needs to be taken into account. Quasi-static pressure variations, such as atmospheric pressure variations, can be several orders of magnitude larger than the loudest sound pressures and ossicle bending may become important in this case. In the high frequency range ossicle bending occurs due to inertia effects and it will influence the middle ear transfer function [25, 26].
Experimental Procedure
Experimental Setup
Calibration and Loading Protocol
Should the indentation test have been performed with a perfect experimental setup and on a perfectly linear, isotropic and homogenous material, a typical relation between indentation depth and reaction force will be found. This curve, with slightly increasing stiffness for increasing indentation, can be found from FE-simulation, dealing with geometric nonlinearity, and is the same for loading and unloading [4, 10]. For real measurements, however, there will be some factors, such as nonlinear material behavior and inaccuracies of the experimental setup which will influence this ideal result. We minimized these factors by doing a calibration and using a proper loading protocol as follows.
First, the loadcell has a rather high compliance as compared to the materials under test, so the measured displacement distances were significantly larger than the actual indentation. The deformation of the loadcell is however solely dependent on the force which is applied to it, which means we could compensate this effect with a calibration. In order to perform the calibration the needles were pushed against each other without a sample in between. Seeing there was no testing material between the needles the force measured on the loadcell corresponded to the deformation of the loadcell. Actually, not only the loadcell’s compression was corrected, also all smaller deformations of the entire setup were taken into account.
Second, the indentation points themselves were no perfect rigid bodies, as assumed in equation (2), consequently, they also deformed slightly. This deformation is mainly determined by the indentation force, but also by the compliance of the material used under test. Which brought us to choosing a very stiff material and a cone shape for the indentation points, this meant that their deformation was much smaller than the deformation of the material under test and the bending of the needle itself was minimized.
FE Calculations for κ
The model had an adaptable indenter radius, material dimensions, Poisson’s ratio and Young’s modulus. We used a Young’s modulus E in the model and calculated E_{Sneddon} (in the model) using equation (2). The ratio of E_{Sneddon} and E gave the value for κ [see equation (4)]. Repeating this simulation for different input-values learnt us that κ depends on the Poisson’s ratio (ν) and on the aspect ratio of the indenter radius and the material thickness (a/h), which is the same as in the model from Hayes et al. [22].
Validation Materials and Compression Test
In order to validate our new test setup, we compared the material parameters obtained from experiments with our new indentation setup with those of a standardised test method. We performed indentation tests on thin samples of different materials of which large samples were also available. Four materials were selected, namely Aluminium (Al), Polyvinyl Chloride (PVC), Polymethyl methacrylate or acrylic glass (PMMA) and polycaprolactam or nylon 6 (PA6). These materials samples span a large range of Young’s moduli (±2 GPa to ±60 GPa ). All experiments on the large samples were performed according to the ASTM D695-02a ‘Standard Test Method for Compressive Properties of Rigid Plastics.’ According to this standard, the specimens were milled to cyclinders with a diameter of 12.7 mm and a height of 25.4 mm.
Since friction between the specimen and the pressure plate may have an important influence on the results, teflon spray was used between specimen and plate, so that the friction may be neglected for the lower load levels, used to calculate Young’s modulus. Several experiments were conducted on different specimens of the same material and the stiffness and standard deviation were calculated according to the ASTM norm. The results of these experiments are given in Section “ASTM Compression Test.”
Biological Samples: Middle Ear Ossicles
Results
ASTM Compression Test
Elasticity modulus (in GPa, ± SD) obtained from ASTM D695-02a test
Material | N | E_{ASTM} |
---|---|---|
Al | 3 | 60.9±1.4 |
PVC | 5 | 3.526±0.061 |
PMMA | 5 | 3.799±0.039 |
PA6 | 5 | 2.260±0.0884 |
Indentation Test-material
Elasticity moduli for four test-materials with different thickness, with h the material thickness, ν the Poisson’s ratio, E_{ASTM} the elasticity modulus obtained with the standard compression test (Table 1), E_{Sneddon} the elasticity modulus obtained with equation (2), E_{Indentation} the corrected elasticity modulus ± SD [equation (4)] and \({\rm deviation}=\displaystyle\frac{E_{\rm indentation}-E_{\rm ASTM}}{E_{\rm ASTM}}\)
Material | h (μm) | ν | E_{ASTM} (GPa) | E_{Sneddon} (GPa) | E_{indentation} (GPa) | Deviation |
---|---|---|---|---|---|---|
Al | 975 | 0.33 | 60.9 | 76.9 | 64.6 ± 5.5 | +6.1% |
Al | 444 | 0.33 | 60.9 | 86.3 | 64.0 ± 4.8 | +5.2% |
Al | 200 | 0.33 | 60.9 | 117 | 66.9 ± 6.7 | +9.9% |
PVC | 988 | 0.35 | 3.526 | 3.87 | 3.27 ± 0.14 | −7.2% |
PVC | 440 | 0.35 | 3.526 | 4.32 | 3.21 ± 0.17 | −8.9% |
PVC | 300 | 0.35 | 3.526 | 4.77 | 3.21 ± 0.26 | −9.0% |
PMMA | 1,010 | 0.40 | 3.799 | 4.97 | 4.23 ± 0.22 | +11.4% |
PMMA | 430 | 0.40 | 3.799 | 5.65 | 4.21 ± 0.20 | +10.8% |
PMMA | 275 | 0.40 | 3.799 | 6.50 | 4.27 ± 0.22 | +12.6% |
PA6 | 1,030 | 0.39 | 2.260 | 2.70 | 2.30 ± 0.14 | +1.8% |
PA6 | 450 | 0.39 | 2.260 | 2.72 | 2.05 ± 0.15 | −9.1% |
E_{Sneddon}, which is the Young’s modulus calculated without the correctionfactor proposed in equation (4), is also given in Table 2. Furthermore, the values of the sample thickness h, and the Poisson’s ratio ν are also shown in this table. Finally, the deviation between these two techniques, defined as \({\rm deviation}=\frac{E_{\rm indentation}-E_{\rm ASTM}}{E_{\rm ASTM}}\), is also shown in Table 2.
Biological Samples: Middle Ear Ossicles
After validating our technique on the test-materials, we used our new technique on biological samples with properties within the same range as the test-materials (2 GPa < E < 60 GPa and 200 μm < h < 1,000 μm).
After harvesting the malleus and the incus, four positions were carefully chosen (Fig. 7: 1. caput malleus, 2. collum malleus, 3. corpus incudis and 4. crus longum incudis). On these precise locations the ossicles had reasonably parallel surfaces which meant they could be clamped between the two needles with a minimal non-flatness of the contact (Fig. 2). The entire process is followed with a light microscope.
The Young’s moduli (± SD) for rabbit middle ear ossicles is obtained at different positions (caput malleus, collum malleus, corpus incudis and crus longum incudis) with different thickness (h)
Position | h(μm) | Young’s modulus (GPa) | N |
---|---|---|---|
Caput malleus | 540 | 16.3 ± 2.9 | 5 |
Collum malleus | 480 | 15.6 ± 1.8 | 5 |
Corpus incudis | 770 | 16.8 ± 3.1 | 4 |
Crus longum incudis | 440 | 17.1 ± 3.8 | 3 |
Average | 16.4 ± 2.8 | 17 |
Discussion
FE Corrected Double Indentation Method
In order to obtain the elasticity modulus for a material standard tensile tests are easy and fast to perform on large objects. Sample preparation and controlling the boundary conditions in such tests are difficult for smaller objects [3, 4]. Dedicated experiments to determine the modulus give very good results but are difficult to perform and are time consuming [5, 6, 7].
Indentation tests, based on Sneddon’s solution [8], do not need such complicated preparation and are easy to use on micro- and nanoscale. They are, however, designed for relatively thick materials which is an important drawback when applying the method to biomaterials: decreasing the needle radius to measure thinner objects can change the results drastically because the bulk properties are not measured anymore. This behaviour is even more important for biological materials which are often built up with cells and fibers [21]. Therefore we made two additions to Sneddon’s solution.
First, we introduced a second needle to prevent surface deformations not caused by the indentation. By adding this second needle in such a way that a virtual mirror plane is introduced, the experimental setup becomes equivalent to a sample with half thickness which lays on a perfect sliding sample stage. Furthermore, the samples could be tested faster since they only had to be clamped between those two punches.
We should taken into account that we need to suppose that a perfect mirror plane develops in the specimen between the two indenters. This condition will only be met if the two indentation points are perfectly aligned. In order to acquire this condition, the loadcell is mounted on crossed translation tables with a translation precision of better than 1 μm. The indentation points are brought towards each other, and microscope observation from two perpendicular directions is used to align the indentation surfaces. The surfaces of the indentation points themselves are nearly perfectly perpendicular to the indentation direction. We achieved this by putting the cylindrical part of the points in a custom made holder which is placed perfectly perpendicular on a polishing disk, in order to polish the pointed end to a small plane which is perpendicular to the indentation axis. In addition the sample itself should be symmetrical on mechanical relevant places, which are the places with higher Von Mises stress in Fig. 4. Furthermore, an offset indentation is necessary to assure full contact between material and indenter surface. Otherwise, the stiffness will be underestimated, as seen in equation (3). When the full contact requirements are not reached, in case of rough materials or inclined surfaces, sample preparation and polishing is needed. Observation with the light microscope allowed us to conclude that the bone samples which we used were smooth enough.
A second addition to Sneddon’s solution was a new FE calculated correction factor κ for those thin, freestanding materials. We take geometric nonlinearity, in contrast with the infinitesimal results of Hayes et al. [22] into account, but the finite deformation effect which causes stiffening during larger indentations was left out [4, 10]. We obtained κ values for a frictionless sliding contact and a fixed sliding contact, shown in Fig. 5. The real values will be in between those two situations. In accordance with the conclusions by Zhang et al. [4], we could state that the value of κ is fairly constant for Poisson’s ratios between 0 and 0.4, if the aspect ratio indenter radius and material thickness was smaller than 1. The new κ-values are higher when compared with the values of Hayes et al. [22] and Zhang et al. [4], which were obtained for an elastic layer bounded to a rigid half space. The correction factor depends on the aspect ratio of the indenter radius and the material thickness (a/h), and on the Poisson’s ratio (ν). Considering a fixed or a sliding contact at the needle-surface zone gives approximately the same solution for κ when the Poisson’s ratio is smaller than 0.4 and aspect ratio is smaller than 1, which corresponds well with the conclusions by Zhang et al. [4].
By using a more sensitive loadcell, our technique can possibly also be extended towards much softer materials such as biological tissue. Because the indentation surfaces are very small, there will be little effect from local bending on the indentation surface itself. A very soft material should be supported in a holder to prevent the material from bending around the needle. In addition, an active vibration isolation will be necessary when smaller forces need to be measured. However, from our calibration measurements we did see that no significant noise was picked up due to acceleration effects caused by external vibrations. Therefore a regular rigid table was sufficient for our setup. Providing such technical problems can be solved we think our method can have promising applications in soft tissue testing as well, but at the moment we are not equipped to demonstrate this.
Thinner objects could be measured by using a different indenter radius, but this will also change the physical contact-properties as remarked earlier. In that case κ should be tested for thinner materials. The bulk properties of thicker materials can also be determined with this method. Furthermore, larger indentation punches are produced more easily. When the aspect ratio of the indenter radius and the material thickness (a/h) is smaller than 0.025, the correction factor κ will be lower than 1.1. The double indentation will still be necessary to ensure no unwanted surface deformations. When the sample dimensions are larger than a few millimeters, tensile tests can easily be applied to determine material bulk properties. The materials which we examined with this indentation test, were considered to be homogenous, isotropic and linear (assumption in Sneddon’s solution and in the FE calculations), just like in the ASTM test.
Loading Protocol
Visco-elastic materials showed creep and relaxation and there were some plastic deformations and phase transformations due to local high stresses [12, 17]. Also, hysteresis caused the load and unload curves to be different. In order to minimize these effects, pre-loadings were used and the indentation-force curve was measured during unloading [20]. The force-indentation curves for most materials showed increasing stiffness for increasing indentation depth. This effect could be attributed to nonlinear phenomena, but it is also the result from geometric nonlinearity [4, 10]. The indentation depths should not be too deep to minimize this effect. Therefore, several offset indentations had to be tested empirically and the loading protocol as presented in Fig. 3 was introduced.
Validation
In order to test the proposed method, four test-materials with different material-properties (Al, PVC, PMMA and PA6) were selected and tested according to the ASTM D695-02a standard. The standard deviation calculated from different samples was between 1 and 4%. A second part of the four test-materials were prepared with different thicknesses between 200 and 1,000 μm and were used for the new double indentation method. As such, the FE calculated correction factor κ and the thickness independency of the measurement could be tested. A Young’s modulus E_{indentation} and a standard deviation, which ranged between 5 and 10%, is obtained and presented in Table 2. The Poisson’s ratio, used in the calculations, were also presented. It should be possible to measure the Poisson’s ratio by using two different sizes of indentation punches [28].
When comparing the results from the standard compression method (E_{ASTM}) and from our double-needle indentation method (E_{indentation}), the difference between the results obtained from the two techniques was always smaller than 13% (Table 2). We also demonstrated the importance of the correction factor κ for thin materials by calculating E_{Sneddon}. The correction factor goes up to 1.75 for thin materials and the difference between the results of our method and Sneddon’s solution goes up to 92%. From the validation measurements we learn that our method allows to determine Young’s moduli to an accuracy of better than 13% on small specimens in a large range of Young’s moduli and material thickness (2GPa < E < 60 GPa and 200 μm < h < 1,000 μm).
Biological Materials
In current mechanobiological research linear FE models are often used. Such models need information about the geometry, the boundary conditions and the material properties of all parts. The most important parameter for linear elastic models is the Young’s modulus. Often general parameters are used: for instance for bone, the values of Evans [2] which were measured on the unembalmed wet cortical bone of the human femur are used even in papers dealing with other species or other types of bone [23]. In our method we only measure the bulk elasticity parameter of the material. Biological materials often consist of fine structures, which we do not take into account. However, the results from our method are intended to be used in FE modelling of biomechanical structures, where materials are often approximated as being homogeneous. Nevertheless, great care should always be taken, as in some cases the outer surface of a biological object can have significantly different properties from its inner structure. With our method one can harvest thin samples out of such a structure, and thus obtain more detailed information on the distribution of the elasticity parameters, this is not possible in a classical compression test. It remains impossible to take into account inhomogenities within the thin sample itself.
Rabbit middle ear ossicle bone is a good example of the usefulness of our method in the mechanobiology, since we know that the Young’s modulus of bone should be in the range of the test-materials [2] and since middle ear ossicle bone is too small for standard test methods [3]. In hearing science [23, 24, 29, 30], proper models with correct material-parameters are important to investigate the functioning of normal and pathologic ears [31, 32] by describing the correct ossicle bending and so their results for the middle ear transfer function [25, 26].
On incus and malleus, four locations are found with approximately parallel surfaces so the indentation test could be performed easily. The thickness cannot be chosen freely, but was measured and it was in the same range as the thickness of the samples used in the validation experiments. Young’s moduli for all bones were found between 11 and 22 GPa, which is in the range of the test-materials. Standard deviations, which are obtained from indentation on different depths (between 5 and 10 μm) and on different specimens, ranged from 10% to 22%. When comparing the elasticity modulus between different locations no significant difference was found and the Young’s modulus from rabbit middle ear ossicle bone could be determined as 16 ±3 GPa which is higher than the value given by Evans [2] (12 ±3 GPa). When we compared this result to the nanoindentation results obtained by Rho et al. [33], we found that our value was between the highest and lowest Young’s modulus of the microstructural components of bone (13.4 GPa for trabecular and 25.8 GPa for cortical bone). These microstructural properties are very useful for a better understanding of the building up and functional behaviour of bone, but for modelling of complex biomechanical systems, such as the middle ear, a general bulk parameter will mostly suffice.
Conclusion
We have developed a portable double indentation device which allows to measure the Young’s modulus of thin samples. The material is clamped between two needles and symmetrically indented. As such, the problem of creating a perfect smooth contact zone between sample and sample stage is avoided, as a virtual plane is created due to symmetry. A correction factor κ, which compensates for the small thickness, is calculated with FE modelling and added to Sneddon’s solution. As such, the Young’s modulus of the validation materials could easily be measured, independent from thickness, with an accuracy better than 13%. As a demonstration , we applied our technique on the small ME ossicle bone of rabbits and found a Young’s modulus of 16 ± 3 GPa.
Notes
Acknowledgements
Financial support to this project is given by the Research Foundation–Flanders (FWO). We thank F. Wiese, J. Van Daele and W. Deblauwe for their technical assistance, V. Vandervelden and B. Soons for proofreading the English writing and J. Buytaert for his 3D model of the middle ear ossicles (www.ua.ac.be/bimef).
References
- 1.Strait DS, Wang Q, Dechow PC, Ross CF, Richmond BG, Spencer MA, Patel BA (2005) Modeling elastic properties in finite-element analysis: how much precision is needed to produce an accurate model? Anat Rec A Discov Mol Cell Evol Biol 283(2):275–287Google Scholar
- 2.Evans FG (1973) Mechanical properties of bone. Thomas Springfield, ILGoogle Scholar
- 3.Speirs AD, Hotz MA, Oxland TR, Husler R, Nolte L-P (1999) Biomechanical properties of sterilized human auditory ossicles. J Biomech 32:485–491CrossRefGoogle Scholar
- 4.Zhang M, Zheng YP, Mak AFT (1997) Estimating the effective youngs modulus of soft tissues from indentation tests nonlinear finite element analysis of effects of friction and large deformation. Med Eng Phys 19:512–517CrossRefGoogle Scholar
- 5.Kvistedal YA, Nielsen PMF (2009) Estimating material parameters of human skin in vivo. Biomech Model Mechanobiol 8:1–8CrossRefGoogle Scholar
- 6.Meunier L, Chagnon G, Favier D, Orgas L, Vacher P (2008) Mechanical experimental characterisation and numerical modelling of an unfilled silicone rubber. Polym Test 27:765–777CrossRefGoogle Scholar
- 7.Samani A, Plewers D (2004) A method to measure the gyperelastic parameters of ex vivo breast tissue samples. Phys Med Biol 49:4395–4405CrossRefGoogle Scholar
- 8.Sneddon IN (1965) The relaxation between load and penetration in the axisymmetric boussinesq problem for a punch of arbitrary profile. Int J Eng Sci 3:47–57MATHCrossRefMathSciNetGoogle Scholar
- 9.Bolduc J-E, Lewis LJ, Aubin C-E, Geitmann A (2006) Finite-element analysis of geometrical factors in micro-indentation of pollen tubes. Biomech Model Mechanobiol 5(4):227–236CrossRefGoogle Scholar
- 10.Choi APC, Zheng YP (2005) Estimation of youngs modulus and poissons ratio of soft tissue from indentation using two different sized indentors: finite element analysis of the finite deformation effect. Med Biol Eng Comput 43:258–264CrossRefGoogle Scholar
- 11.Feng C, Tannenbaum JM, Kang BS, Alvin MA (2009) A load-based multiple-partial unloading micro-indentation technique for mechanical property evaluation. Exp Mech (in press). doi:10.1007/s11340-009-9271-4.Google Scholar
- 12.Galanov BA, Domnich V, Gogotsi Y (2002) Elastic-plastic contact mechanics of indentations accounting for phase transformations. Exp Mech 43:303–308Google Scholar
- 13.Ju BF, Ju Y (2006) Video enhanced depth-sensing indentation technique for charaterizing mechanical behaviour of biomaterials. Meas Sci Technol 17:1776–1784CrossRefGoogle Scholar
- 14.Lin D, Shreiber D, Dimitriadis E, Horkay F (2008) Spherical indentation of soft matter beyond the hertzian regime: numerical and experimental validation of hyperelastic models. Biomech Model MechanobiolGoogle Scholar
- 15.Riccardi B, Montanari R (2004) Indentation of metals by a flat-ended cylindrical punch. Mater Sci Eng A 381:281–291CrossRefGoogle Scholar
- 16.Gong J, Miao H, Peng Z (2003) Simple method for determining the initial unloading slope for ceramics nanoindentation tests. J Mater Sci Lett 22:267–267CrossRefGoogle Scholar
- 17.Huang G, Lu H (2006) Measurements of two independent viscoelastic functions by nanoindentation. Exp Mech 47:87–98CrossRefGoogle Scholar
- 18.Scholz I, Baumgartner W, Federle W (2008) Micromechanics of smooth adhesive organs in stick insects: pads are mechanically anisotropic and softer towards the adhesive surface. J Comp Physiol A 194:373–384CrossRefGoogle Scholar
- 19.Shuman DJ, Costa ALM, Andrade MS (2007) Calculating the elastic modulus from nanoindentation and microindentation reload curves. Mater Charact 58:380–389CrossRefGoogle Scholar
- 20.Oliver WC, Pharr GM (1992) An improved technique for determining hardness and elastic modulus using load and displacement sensing indentaion experiments. Mater Res 7:1564–1583CrossRefGoogle Scholar
- 21.Choi K, Kuhn JL, Ciarelli MJ, Goldstein SA (1990) The elastic moduli of human subchondral, trabecular, and cortical bone tissue and the size-dependency of cortical bone modulus. J Biomech 23(11):1103–1113CrossRefGoogle Scholar
- 22.Hayes WC, Keer LM, Herrmann G, Mockros LF (1972) A mathematical analysis for indentaion tests of articular cartilage. J Biomech 5:541–551CrossRefGoogle Scholar
- 23.Koike T, Kobayashi T, Wada H (2002) Modeling of the human middle ear using finite-element method. J Acoust Soc Am 111(3):1306–1317CrossRefGoogle Scholar
- 24.Sun Q, Gan RZ, Chang K-H, Dormer KJ (2002) Computer-integrated finite element modeling of human middle ear. Biomech Model Mechanobiol 1:109–122CrossRefGoogle Scholar
- 25.Decraemer WF, Shyman MK, Funnell WRJ (1991) Malleus vibration mode changes with frequency. Hear Res 54:305–318CrossRefGoogle Scholar
- 26.Dirckx JJJ, Buytaert JAN, Decraemer WF (2006) Quasi-static transfer function of the rabbit middle ear, measured with a heterodyne interferomenter with high resolution position decoder. JARO 7(4):339–351CrossRefGoogle Scholar
- 27.Maas S, Weiss JA (2008) Febio: finite elements for biomechanics. user’s manual, version 1.0. Online publication: http://mrl.sci.utah.edu/component/docman/doc_download/1-febio-users-manual
- 28.Jin H, Lewis JL (2003) determination of poisson s ratio of articular cartilage in indentation test using different sized indenters. In: Summer bioengineering conference, FloridaGoogle Scholar
- 29.Decraemer WF, Dirckx JJJ, Funnell WRJ (2003) Three-dimensional modelling of the middle-ear ossicular chain using a commercial high-resolution x-ray ct scanner. J Assoc Res Otolaryngol 4(2):250–263CrossRefGoogle Scholar
- 30.Dirckx JJJ, Decraemer WF, von Unge M, Larsson C (1997) Measurement and modeling of boundary shape and surface deformation of the mongolian gerbil pars flaccida. Hear Res 111(1–2):153–164CrossRefGoogle Scholar
- 31.Dirckx JJJ, Decraemer WF (2001) Effect of middle ear components on eardrum quasi-static deformation. Hear Res 157(1–2):124–137CrossRefGoogle Scholar
- 32.von Unge M, Decraemer WF, Dirckx JJJ, Bagger-Sjbck D (1999) Tympanic membrane displacement patterns in experimental cholesteatoma. Hear Res 128(1–2):1–15Google Scholar
- 33.Rho JY, Tsui TY, Pharr GM (1997) Elastic properties of human cortical and trabecular lamellar bone measured by nanoindentation. Biomaterials 18(20):1325–1330CrossRefGoogle Scholar