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Experimental determination of Young modulus and Poisson ratio in cortical bone tissue using high resolution scanning acoustic microscopy and nanoindentation
Abstract
Nanoindentation allows measurements of local mechanical properties of bone tissue. Scanning acoustic microscopy (SAM) provides images related to bone density and elasticity. In both techniques, the estimation of Young modulus (E) relies on the accuracy of Poisson's ratio value (sigma). In cortical bone, sigma varies between 0.15 and 0.45 but, is classically set to 0.3, resulting in an approximate value of E. This study describes a new method combining SAM and nanoindentation techniques to locally evaluate sigma in human femoral cortex. A 200 MHz SAM-based acoustic impedance (8 mum lateral resolution) was combined with synchrotron microtomography (to provide local bone mineral density) to map the distribution of near surface elastic modulus. Whereas, nanoindentation modulus was calculated on several osseous regions. Assuming the equalization rule, the intersection of both modulus curves versus sigma permits to accurately derive sigma. The method was tested on aluminium, PMMA and polycarbonate samples of known sigma and provided experimental sigma values with a precision better than 3%. In bone, sigma was 0.42+/-0.01 corresponding to E=20+/-1 GPa. Our preliminary results indicate that combination of high-resolution SAM and nanoindentation may be relevant to accurately determine both Poisson ratio and Young modulus of bone tissue.
... Therefore, the densities of the skin, subcutaneous tissue and phalanx in each finger region model were defined as 1090, 919.6, 1920 kg/m 3 , respectively; the average elastic moduli of the finger phalanx and skin were defined as 14.7 GPa and 18.0 MPa, respectively. Taking into consideration that the finger skin might have the maximum Poisson's ratio, followed by the subcutaneous tissue and the phalanx, and with the Poisson's ratios of some tissues obtained by Ren et al. (2018), Thieulin et al. (2020) and Rupin et al. (2008), the Poisson's ratios of the skin, subcutaneous tissue and phalanx in each finger region model were assumed in the ranges of 0.45-0.49, 0.30-0.45 ...
... Hence, only the thickness of the subcutaneous tissue was set as a possible independent variable to reflect the geometric size of different tissues in a finger region model. According to the studies by Oehman et al. (2011), Ren et al. (2018), Thieulin et al. (2020), Rupin et al. (2008) and Yeh et al. (2014), each potential independent variable was set as three levels. To reduce the number of simulations, a L 18 (3 7 ) orthogonal design was used. ...
Soft grasping is a great challenge for picking robots and its bionic inspiration originates from human fingers. In this study, the hand was scanned to obtain the internal structure of fingers by a computerized tomography (CT) scanner, and the soft contact mechanical index a was defined for characterizing the degree of softness of a finger region during gentle grasping. The effects of mechanics and structure of finger tissues on the soft contact mechanical index were investigated by finite element analysis and multiple linear regression. The finite element models of the 14 finger regions were split into 6 different groups by a hierarchical cluster analysis. In each group, a mathematical model was established to link the soft contact mechanical index with the mechanics as well as the structure of finger tissues. In most finger regions, their soft contact mechanical index significantly depended on the elastic moduli of the skin and subcutaneous tissue (Eskin, Etissue), the Poisson’s ratio ʋtissue and the thickness Ttissue of the subcutaneous tissue (p < 0.05). The Etissue showed the most contribution on the soft contact mechanical index of a finger region, followed by ʋtissue, Ttissue, and Eskin. This study demonstrated how the mechanics and structure of the human finger quantitatively affect its soft contact mechanical behavior during gentle grasping and further provided a bionic basis for developing robotic fingers with varying degrees of softness, particularly for fruit picking.
... The material parameters, namely the Young's modulus is set to 20 GPa and the Poisson's ratio to 0.42 (see e.g. [74]). In addition, the load is given by f = (0, 0, 0), the Dirichlet data by u D = 0 on Γ D ª Γ represented as the left dark gray region of the boundary in Figure 6.12 and g the traction data is defined as g = (0, 0, 0) on the center light gray region of the boundary and is constant on the right dark gray region of the boundary g = (−10 −7 , −10 −7 , 10 −6 ). ...
... The material parameters, namely the Young's modulus is set to 20 GPa and the Poisson's ratio to 0.42 (see e.g. [228]). In addition, the load is given by f = (0, 0, 0), the Dirichlet data by u D = 0 on Γ D Γ represented as the left dark gray region of the boundary in fig. ...
This manuscript is concerned with a posteriori error estimation for the finite element discretization of standard and fractional partial differential equations as well as an application of fractional calculus to the modeling of the human meniscus by poro-elasticity equations. In the introduction, we give an overview of the literature of a posteriori error estimation in finite element methods and of adaptive refinement methods. We emphasize the state–of–the–art of the Bank–Weiser a posteriori error estimation method and of the adaptive refinement methods convergence results. Then, we move to fractional partial differential equations. We give some of the most common discretization methods of fractional Laplacian operator based equations. We review some results of a priori error estimation for the finite element discretization of these equations and give the state–of– the–art of a posteriori error estimation. Finally, we review the literature on the use of the Caputo’s fractional derivative in applications, focusing on anomalous diffusion and poro-elasticity applications. The rest of the manuscript is organized as follow. Chapter 1 is concerned with a proof of the reliability of the Bank–Weiser estimator for three–dimensional problems, extending a result from the literature. In Chapter 2 we present a numerical study of the Bank–Weiser estimator, provide a novel implementation of the estimator in the FEniCS finite element software and apply it to a variety of elliptic equations as well as goal-oriented error estimation. In Chapter 3 we derive a novel a posteriori estimator for the L2 error induced by the finite element discretization of fractional Laplacian operator based equations. In Chapter 4 we present new theoretical results on the convergence of a rational approximation method with consequences on the approximation of fractional norms as well as a priori error estimation results for the finite element discretization of fractional equations. Finally, in Chapter 5 we provide an application of fractional calculus to the study of the human meniscus via poro-elasticity equations.
... The remaining connective ligaments were also modeled using this average stress strain data with the same mechanical property parameters, sans damage. To model the bones, a Young's modulus of E = 20 GPa with an assumed nearly incompressible Poisson's ratio of v = 0.42 from Rupin et al., was used [27]. ...
The anterior cruciate ligament (ACL) plays a pivotal role in support of the knee under loading. When damaged, it is known that substantial changes in the mechanics of the neighboring ligaments can be observed. However, a localized damage approach to investigating how ACL deficiency influences the neighboring ligaments has not been carried out. To do this, a finite element model, incorporating a continuum damage material model of the ACL, was implemented. Localized ACL damage was induced using high quadriceps force loading. Once damaged, anterior shear forces or tibial torque loadings were applied to the knee joint. The relative changes in stress contour and average mid-substance stress were examined for each of the neighboring ligaments following localized ACL damage. It was observed that localized ACL damage could produce notable changes in the mechanics of the neighboring knee ligaments, with non-homogenous stress contour shape changes and average stress magnitude being observed to increase in most cases, with a notable exception occurring in the MCL for both loading modes. In addition, the ligament bearing the most loading also changed with ACL deficiency. These changes carry implications as to morphological effects that may be induced following localized ACL damage, indicating that early diagnosis of ACL injury may be helpful in mitigating other complications post injury.
... Elastic modulus of cortical bone has been reported roughly in the range 10 GPa to 32 GPa, when measured with different test methods such as destructive, ultrasound and non-indentation method (Choi et al., 1990;Hunt et al., 1998). The Poisson's ratio varies in the range 0.15 to 0.45, however it is classically set to 0.3 (Rupin et al., 2008). Since the elastic modulus of bone is considerably higher than the surrounding soft tissues, the bones are considered rigid more often (Beidokhti et al., 2017;Halloran, Petrella, and Rullkoetter, 2005;Pena et al., 2005). ...
Musculoskeletal disorder of the lower limb is one of the most common health burdens that may lead to functional impairment in an individual. Although various operative management options are available, there seems no unanimity on a particular procedure that serves the best. To objectively assess disorders and effectively plan surgeries, it is essential to understand lower limb biomechanics under physiological loading conditions. With that motivation, this PhD aims to develop a comprehensive finite element based musculoskeletal modeling framework of the lower limb. The first phase of the PhD focuses on the development and evaluation of subject-specific finite element models under passive flexion. Novel approaches are proposed and evaluated for fast model development focusing on geometry and ligament properties. In the second phase, a novel finite element based approach for soft tissue artifact compensation is proposed and evaluated. This contribution allowed to effectively compensate for soft tissue artifact in motion analysis by taking subject specificity into account. The third phase of the PhD is dedicated to clinical application, where the utility of the biplanar X-ray system in evaluating Total Knee Arthroplasty implant alignment is briefly explored. Overall, this PhD may help to accurately estimate and understand lower limb biomechanics under clinically relevant loading conditions, and bring the model a step closer to clinical routine.
... The parameters of the linear elasticity problem for this test cases are the following: the Young's modulus is set to 20 GPa and the Poisson's ratio to 0.42 (see e.g. [64]). In addition, the load is given by f = (0, 0, 0), the Dirichlet data by u D = 0 on Γ D Γ represented as the left dark gray region of the boundary in fig. ...
In the seminal paper of Bank and Weiser [Math. Comp., 44 (1985), pp.283-301] a new a posteriori estimator was introduced. This estimator requires the solution of a local Neumann problem on every cell of the finite element mesh. Despite the promise of Bank-Weiser type estimators, namely locality, computational efficiency, and asymptotic sharpness, they have seen little use in practical computational problems. The focus of this contribution is to describe a novel implementation of hierarchical estimators of the Bank-Weiser type in a modern high-level finite element software with automatic code generation capabilities. We show how to use the estimator to drive (goal-oriented) adaptive mesh refinement and to mixed approximations of the nearly-incompressible elasticity problems. We provide comparisons with various other used estimators. An open-source implementation based on the FEniCS Project finite element software is provided as supplementary material.
... Including this deformability improves the model because strong deformations of the softer layers would violate a general elastic model. Here, the shear modulus of the PC is taken as that of a slack muscle and its bulk modulus (K) is calculated using the shear modulus (G) and poisson ratio (v) according to the following equation: [35] 520.4 0.49 Bone [36] 20,000 0.42 Base (steel) [37] 210,000 0.3 Urethane [33] 0.025 0.0 After running simulations with a steel base to simulate a sturdy support, we also ran simulations using the properties of urethane foam, a material commonly used in seating and mattresses. When using Neo-Hookean hyperelastic properties, the simulations took 14e24 h to complete on a PC with Intel I7 4 GHz processor and 16 Gb of RAM. ...
... An experimental approach is to assess the Poisson ratio at this length scale is a site-matched analysis of the acoustic impedance and the indentation modulus E IT by SAM and nanoindentation, respectively [33,37,38]. Interestingly, in all studies consistent, rather moderate correlations between Z and the indentation modulus E IT (0.61 ≤ R 2 ≤ 0.67) have been observed. ...
Scanning acoustic microscopy (SAM) with frequencies between 50 MHz and 2 GHz is adapted to the investigation of structure and elastic properties of the bone tissue matrix, that is, the linear elastic interaction of acoustic waves with the material under investigation can be used to visualize the microarchitecture, and also to measure sound velocities and acoustic impedances at various length scales. When combined with local density estimates (e.g., derived from quantitative X-ray micro-computed tomography data), acoustic impedance estimates can be used to derive tissue stiffness. By combining multiscale experimental data with numerical and continuum mechanical homogenization approaches, structural properties can be decoupled from material properties and their respective impacts on the elastic functional behavior of the compound can be studied. The underlying principles and application are described in this chapter.
The anterior cruciate ligament plays a major role in maintaining the stability of the knee joint, is susceptible to injury under strenuous activity. ACL injuries can lead to joint instability and complications such as osteoarthritis. Despite this, there is a lack of material models capable of predicting damage at a localized fiber level, hindering our ability to understand how damage develops in real-time. The present work develops a continuum-damage material model of the ACL and applies the model to a finite element simulation of the knee undergoing high quadriceps tendon loading. Using quadriceps tendon loadings of 1000, 1500, and 2000 N, the development of microstructural damage within the ACL tissue was examined, and the effects of localized damage on the joint kinematics were investigated. Damage tended to develop in the mid-substance of the ACL in the present model in the anterior medial bundle region and could induce significant changes in the joint kinematics. Using this model, new insights into the development of ACL injury mechanisms can be investigated.
The Mississippi Alligator gar (Atractosteus spatula) possesses a flexible exoskeleton armor consisting of overlapping ganoid scales used for predatory protection. Each scale is a two-phase biomineralized composite containing bio-modified hydroxyapatite (hard) minerals and collagen (soft) fibers. The protective layer consists of a stiff outer ganoine layer, a characteristic “sawtooth" pattern at the interface with the compliant bone inner layer. The garfish scale exhibits a decreasing elastic modulus from the external to the internal layers. Scanning electron microscopy (SEM) images of the cross-section revealed a two-layered structure. Elastic moduli, measured from nanoindentation experiments, were correlated to structural changes across each layer. The “material” symmetry of this materially and geometrically nonlinear biomineralized composite is unknown. Therefore, to be able to determine the stiffness tensor requires the use of finite element analysis (FEA). The gar fish scale was computationally modeled using the representative volume element (RVE) based approach. As a result, the unknown symmetry induced by the architecture and material layering require the use of complex FEA boundary conditions. The simulation was conducted in the pure uniaxial strain regimes of tension and shear, which necessitated the mathematical determination so appropriate surface loading conditions could be applied. This paper provides the results from a highly-resolved mesoscale RVE model based on iso-strain boundary conditions (ISBC) to determine the elastic stiffness tensor for the composite system. By assuming isotropic behavior in individual elements, the results for the RVE reveal the fish scale has an “orthotropic symmetry” with slight local strain variations occurring at the sawtooth interface.
In the ear of the desert locust frequency analysis arises from the mechanical properties of the tympanal membrane. Incident sound is spatially decomposed into discrete frequency components through a tympanal travelling wave that funnels mechanical energy to specific tympanal locations, where distinct groups of mechanoreceptor neurones project. Initial analysis of the travelling waves employs conventional, steady state FFT, allowing a detailed analysis of the spatial composition of different frequencies onto the membrane. To further understand the exact mechanics of the tympanal travelling wave, its motion was also measured in the time domain to characterise its response to single impulse and single frequency stimuli, with a resolution of 390 ns. This allows the measurement of instantaneous wave velocity and the direct observation of wave compression across the tympanum. The locust tympanal membrane locust exploits tonotopic frequency analysis, in a similar sense to that of the travelling waves of von Bekesy on the mammalian basilar membrane. However, von Bekesy's wave is born from interactions between the anisotropic basilar membrane and surrounding incompressible fluids, whereas the locust's wave rides on an anisotropic membrane suspended in air. The locust's tympanum thus combines the functions of both sound reception and frequency analysis.
We explored the statistical properties of the spontaneous and evoked firing activity in the brainstem auditory units of the grass frog (Rana t temporaria). We estimated the properties of spontaneous firing using the following functions: interspike interval distribution, hazard function, autocorrelation functions for original and shifted sequence of interspike intervals, function of interdependence of neighboring intervals. The dependences of Fano and Allans factors upon the values of counting time (window size) were also obtained. We observed a considerable deviation of the spontaneous activity from the renewal process for the majority of units located both in the dorsal medullar nucleus (homolog of cochlear nucleus in the mammals) and in torus semicircularis (homolog of inferior colliculus). The negative correlation between the neighboring short interspike intervals was typical for some (mainly toral) units. However, generally week but quite reliable positive correlation between neighboring intervals was typical for the great majority of neurons. The values of Fano and Allan factors were close to 1 at window size less than 100-500 ms and increased in proportion to some power for larger window sizes. The exponents of these power dependences were different for different units.
Machinery such as compressors, boilers and generators require adequate ventilation and extract. To avoid excessive noise breakout, acoustic louvres are frequently used. These are usually made of tilted hollow metallic blades filled with absorptive materials and perforated at the inside face. This study investigates the important characteristics of common shaped acoustic louvres with regard to sound insulation. First, a market survey on 109 existing acoustic louvres of 24 producers worldwide is made. Second, laboratory measurements of sound insulation on 38 prototypes of 1 m(2) acoustic louvres are compared. The influence of several parameters are studied such as louvre depth, blade angle, blade shape, blade thickness, blade spacing, kind of absorptive material, perforation degree, and edge filling.
Although small-size windows are used in a ship, shipboard windows are a multilayered structure with glasses and air-gaps for high sound insulation. This paper discusses how to improve the sound insulation performance of shipboard windows. First, the sound transmission loss (STL) obtained from various experiments with shipboard windows are introduced and studied. The results show the layer arrangement as well as the material property of each layer makes an effect on the window's STL. It is also found that the higher the sound insulation performance of the window is, the more important the effect of the window frame is. Next, theoretical investigations are carried out and discussed in comparison with the experimental results. The comparison provides possible clues to increase the STL of the window.
Using sweep spread pulses a variety of multipath components of a receive signal can be often resolved in time-frequency domain. In result of synchronisation of the receiver with the most powerful component, other multipaths can be effectively suppressed by means of a matched filter. Owing to exact synchronisation, the difference frequency will be equal to zero and outputs of matched filters will be good for estimation of (information) parameters. However, the synchronous component (as well as others) can occasionally turn out to be composite and to contain energy of several non-resolved multipaths. In this case, the synchronisation time of the receiver will be not exact and the difference frequency of demodulated signal will differ from zero. This effect will cause a systematic error in estimation of the information parameter. This paper represents the results of mathematic modeling of the systematic error (for receivers, utilising the sweep-spread carrier technology), as well as, experiments on synchronisation of the receiver by means of wideband sweep spread pulses.
The U.S. Army needs detailed information about the blast pressure field created by large weapons to avoid negative impacts on testing and training mission capabilities due to noise. The noise software tools BNOISE2trade mark and RMTK Noise Tool, which the Department of Defense (DoD) uses to predict and assess impacts of large weapon noise, require accurate acoustic source emission models. A free-field representation of the sound source acoustical emission, containing no waveform signal perturbations due to the ground, other reflecting surfaces, ballistic waves, or propagation anomalies, is required. The measurement protocols have been developed to collect the free-field source emission data. These protocols are described along with acoustical data and emission source results for the 120 mm M-1 tank gun, the 105 mm Stryker gun, and various weights of C-4 explosives. In addition, the measured acoustic data are compared with an idealized Friedlander curve. The overall results demonstrate that the protocols do provide sufficient free-field blast waveforms.
This work presents the results of a project for the design of novel noise barriers with improved performance. The aim of the project was twofold. First, to optimize the performance of barriers by appropriately modifying both the shape and the acoustical conditions at the edge to suppress the sound pressure. Second, to quantify the acoustic performance of the novel barriers both mathematically and experimentally in a standardized, well controlled process. The insertion loss of different types of barriers with modified edge shapes and acoustical conditions were investigated systematically in comparison with conventional ones using a method based on BEM. The comparison highlighted the influence of shape and edge configuration on barrier efficiency and allowed the design of novel barriers with optimal performance. The novel barriers were tested and their efficiency was quantified experimentally using the Maximum Length Sequence (MLS) technique, which offers the advantage of in-situ measurements in the presence of extraneous noise. The experimental results are shown to be in good agreement with the experiments. Based on the aforementioned procedure, a unified framework was established, witch allows the design of potentially improved noise barriers.
The transposition of the European Noise Directive by Member States gives a new way to action plans of urban areas, underlying (article 6 and Annex III) that dose-effect relations should be used to assess the effect of noise on populations. Particular interest, in this sense, has the relationship between annoyance and perceived soundscape: it is now almost accepted that a higher annoyance is not always proportional to higher noise levels. This paper will demonstrate that a way to move from mapping noise exposure to a map of annoyance passes through the characterization of the soundscapes present on the territory. A numerical indicator ("slope"), derived from the time history of the measured soundnoise, will in fact be used to characterize the evolution along the day of the soundscapes present in the city of Pisa (I). In particular, "slope" (correlated to annoyance in previous studies) will be used to prepare a map of the annoyance in selected areas of Pisa: predicted annoyance will be to be compared with the one extrapolated by noise levels. Differences, similarities and their effects on action plans will be discussed.
Common measures for the speech intelligibility predict increasing values with increasing absorption in empty classrooms. Measurements confirm this relation, so it seems as if an anechoic room is the best place to teach. However, there are two problems: the sound pressure level at the back row is too low when compared to the levels on the front row or when compared to the noise level caused by pupils plus ventilation systems. A simple method is proposed based on U50, which gives the relation between the sound energy from direct sound (from a teacher or a pupil) plus early reflections on one side and late reflections plus noise on the other. Barron's formula for the sound level decrease in a concert hall is included to predict the sound level drop through the classroom. Model results indicate that the absorption coefficient should be in the order of 20 to 30% if the signal to noise ratio is in the range between +5 to -5 dB. These values seem low, but measurements in existing classrooms show low signal to noise levels as well.
The use of the non-negative matrix factorization (NMF) as a decomposition technique has dramatically grown in various signal processing applications over the last years. Its computation, based on the iterative minimization of a cost function, relies on several choices, among which the distance involved in the cost function itself but also the initialization of the algorithm. These choices are all the more crucial as the usual algorithms, iterative, only ensure convergence to a local minimum. In this work, we compare three typical distances in the NMF problem (namely Euclidian, Kullback-Leibler divergence and Itakura-Saito distance) and their combination with different initializations, in an audio context: decomposition of the time-frequency representation for the transcription of polyphonic music. Both the existence of global and local minima, and the efficiency of transcription are examined. Moreover, NMF update rules can be formulated in a unified framework for the three aforementioned cost functions. This formulation allows figuring out new NMF algorithms which could address the local minima question.