Finite element stress analysis of a push-out test. Part II: Free interface with nonlinear friction properties.

In this second part of a two-part paper, nonlinear frictional properties measured at the bone/porous-surfaced metal interface are used to perform the stress analysis of a push-out test assuming free interface. In this case, the friction at the interface is the only mechanism to resist the externally applied load. Similar to the part I, the model is axisymmetric and consists of two cylinders in contact with each other through the interface. Various relative material properties and boundary conditions are simulated in order to examine their effects on the interface stresses and overall push-out resistance. The role of the force-fit and the load direction (push-out versus pull-out) on the results is also investigated. The computed radial and shear stresses are found to markedly vary both with location along the interface and with the testing configuration. The ultimate push-out resistance is also found to significantly alter as the material arrangement and boundary conditions change. The predicted push-out load augments with an increase in the force-fit and diminishes to nil in the absence of a press-fit. For the cases studied here, there is a relative difference of as large as 13 percent between the push-out response and the pull-out response so far as the interface stresses and the maximum resistance are concerned. Therefore, any comparison between the results of push-out (or pull-out) tests performed with different design configurations appears to be invalid.     

An approach for the stress analysis of transversely isotropic biphasic cartilage under impact load.

Stress analysis of contact models for isotropic articular cartilage under impacting loads shows high shear stresses at the interface with the subchondral bone and normal compressive stresses near the surface of the cartilage. These stress distributions are not consistent, with lesions observed on the cartilage surface of rabbit patellae from blunt impact, for example, to the patello-femoral joint. The purpose of the present study was to analyze, using the elastic capabilities of a finite element code, the stress distribution in more morphologically realistic transversely isotropic biphasic contact models of cartilage. The elastic properties of an incompressible material, equivalent to those of the transversely isotropic biphasic material at time zero, were derived algebraically using stress-strain relations. Results of the stress analysis showed the highest shear stresses on the surface of the solid skeleton of the cartilage and tensile stresses in the zone of contact. These results can help explain the mechanisms responsible for surface injuries observed during blunt insult experiments.     

Numerical study of how creep and progressive stiffening affect the growth stress formation in trees

It is not fully understood how much growth stresses affect the final quality of solid timber products in terms of, e.g. shape stability. It is, for example, difficult to predict the internal growth stress field within the tree stem. Growth stresses are progressively generated during the tree growth and they are highly influenced by climate, biologic and material-related factors. To increase the knowledge of the stress formation, a finite element model was created to study how the growth stresses develop during the tree growth. The model is an axisymmetric general plane strain model where material for all new annual rings is progressively added to the tree during the analysis. The material model used is based on the theory of small strains (where strains refer to the undeformed configuration which is good approximation for strains less than 4%) where so-called biological maturation strains (growth-related strains that form in the wood fibres during their maturation) are used as a driver for the stress generation. It is formulated as an incremental material model that takes into account elastic strain, maturation strain, viscoelastic strain and progressive stiffening of the wood material. The results clearly show how the growth stresses are progressively generated during the tree growth. The inner core becomes more and more compressed, whereas the outer sapwood is subjected to slightly increased tension. The parametric study shows that the growth stresses are highly influenced by the creep behaviour and evolution of parameters such as modulus of elasticity, micro-fibril angle and maturation strain.     

Evaluation of bone plate with low-stiffness material in terms of stress distribution

The aim of this study is to evaluate a newly developed bone plate with low-stiffness material in terms of stress distribution. In this numerical study, 3D finite element models of the bone plate with low-stiffness material and traditional bone plates made of stainless steel and Ti alloy have been developed by using the ANSYS software. Stress analyses have been carried out for all three models under the same loading and boundary conditions. Compressive stresses occurring in the intact portion of the bone (tibia) and at the fractured interface at different stages of bone healing have been investigated for all three types of bone-plate systems. The results obtained have been compared and presented in graphs. It has been seen that the bone plate with low-stiffness material offers less stress-shielding to the bone, providing a higher compressive stress at the fractured interface to induce accelerated healing in comparison with Ti alloy and stainless-steel bone plate. In addition, the effects of low-stiffness materials with different Young's modulus on stress distribution at the fractured interface have been investigated in the newly developed bone-plate system. The results showed that when a certain value of Young's modulus of low-stiffness material is exceeded, increase in stiffness of the bone plate does not occur to a large extent and stress distributions and micro-motions at the fractured interface do not change considerably.     

Morphological and numerical studies of the stress compatibility of patella implants]

Failure of total knee arthroplasty is relatively often caused by problems of the patellofemoral replacement. The purpose of this study was to analyze the distribution of stresses within an anatomical patella and the changes in stress distribution after patellar resurfacing with a Miller-Galante I patellar implant using two- and three dimensional finite element models (FEM). To assess validity, FEM results were compared with morphological findings from contact radiographs and densitographs. Internal orientation of bone trabeculae is in good agreement with the orientation of theoretically calculated principal stresses. Almost unchanged principal tensile stresses after implantation, together with the lack of extreme stress peaks within the cancellous bone ensure stress compatibility of the implant. In the case of a firmly seated implant with good bone ingrowth, increased von Mises stresses are found near the fixation peg/plate junction. Their relevance for improved bone ingrowth near this part of the interface is emphasized. At the same time, material failure at the peg/plate junction can be better understood. An analysis of the early postoperative period assuming nonlinear interface conditions failed to demonstrate an uniform distribution of normal and tangential interface forces.     

Three-dimensional stress and strain distribution in a two-layer model of a coronary artery

For a right coronary artery, three-dimensional stress and strain distributions at a physiological intraluminal pressure and an axial extension ratio were computed on the basis of a two-layer elastic model. To validate the model, curves of external radius versus pressure and of axial force versus pressure were computed for three axial extension ratios. To analyze mechanical properties, stress-free configurations of media and adventitia, and the constitutive law of each layer in literature, were used. The present study showed that the peak circumferential stress and the peak axial stress appear in the media at the boundary between the media and adventitia. This result is due to the opening angle of the media being larger than π (rad) and the larger value of a material constant of the strain energy function for the media than for the adventitia. The circumferential stress and strain were discontinuous at the boundary. On the other hand, the radial stress was continuous at the boundary because of the boundary condition for stress. The circumferential stress and axial stress in the adventitia were almost uniformly distributed, and smaller than in the media. The residual stress and strain were also computed. The circumferential residual stress and strain were almost linearly distributed in each layer, although discontinuity appeared at the boundary between the two layers.     

Initial stress and nonlinear material behavior in patient-specific AAA wall stress analysis

Rupture risk estimation of abdominal aortic aneurysms (AAA) is currently based on the maximum diameter of the AAA. A more critical approach is based on AAA wall stress analysis. For that, in most cases, the AAA geometry is obtained from CT-data and treated as a stress free geometry. However, during CT imaging, the AAA is subjected to a time-averaged blood pressure and is therefore not stress free. The aim of this study is to evaluate the effect of neglecting these initial stresses (IS) on the patient-specific AAA wall stress as computed by finite element analysis. Additionally, the contribution of the nonlinear material behavior of the AAA wall is evaluated.Thirty patients with maximum AAA diameters below the current surgery criterion were scanned with contrast-enhanced CT and the AAA's were segmented from the image data. The mean arterial blood pressure (MAP) was measured immediately after the CT-scan and used to compute the IS corresponding with the CT geometry and MAP. Comparisons were made between wall stress obtained with and without IS and with linear and nonlinear material properties.On average, AAA wall stresses as computed with IS were higher than without IS. This was also the case for the stresses computed with the nonlinear material model compared to the linear material model. However, omitting initial stress and material nonlinearity in AAA wall stress computations leads to different effects in the resulting wall stress for each AAA. Therefore, provided that other assumptions made are not predominant, IS cannot be discarded and a nonlinear material model should be used in future patient-specific AAA wall stress analyses.     

Discretization error when using finite element models: Analysis and evaluation of an underestimated problem

Mesh convergence tests are often insufficiently performed in finite element analyses. There are many parameters which may have an effect on the mesh convergence behavior. The aim of this study was to identify the influence of different parameters on the mesh convergence behavior.For this purpose we used a simplified axis-symmetrical model of a single pedicle screw flank with surrounding bone to simulate a pull-out test. In parameter studies, the flank radii and the contact conditions at the bone–screw interface were varied. These parameter studies were carried out using an implicit and explicit solver. Thereby, the convergence criteria and the number of the substeps for the implicit nonlinear iteration process as well as the velocity and the material density for the explicit approach were considered.The mesh convergence behavior was influenced by varying the flank radii and the contact conditions. The implicit calculations led to a reaction force, which converged rapidly to a certain value with increasing mesh density, whereas the maximum von-Mises stress showed substantial convergence problems. The number of substeps and the convergence criteria of the iteration process strongly influenced the implicit solutions. In contrast, the maximum von-Mises stresses resulting from explicit calculations converged to a certain value after only a few refinement steps. Different pull-out velocities substantially affected the mesh convergence behavior, while the material density showed only a negligible influence.The results indicated the need to perform an appropriate mesh convergence test when using finite element methods. We were able to show that different parameters strongly influence the mesh convergence behavior and we demonstrated that convergence tests do not always lead to a satisfactory or acceptable solution.     

Finite element estimates of interface stress in the trans-tibial prosthesis using gap elements are different from those using automated contact

When compared with automated contact methods of finite element (FE) analyses, gap elements have certain inherent disadvantages in simulating large slip of compliant materials on stiff surfaces. However, automated contact has found limited use in the biomechanical literature. A non-linear, three-dimensional, geometrically accurate, FE analysis of the trans-tibial limb-socket prosthetic system was used to compare an automated contact interface model with a gap element model, and to evaluate the sensitivity of automated contact to interfacial coefficient of friction (COF). Peak normal stresses and resultant shear stresses were higher in the gap element model than in the automated contact model, while the maximum axial slip was less. Under proximally directed load, compared with automated contact, gap elements predicted larger areas of stress concentration that were located more distally. Gap elements did not predict any relative slip at the distal end, and also transmitted a larger proportion of axial load as shear stress. Both models demonstrated non -linear sensitivity to COF, with larger variation at lower magnitudes of COF. By imposing physical connections between interface surfaces, gap elements distort the interface stress distributions under large slip. Automated contact methods offer an attractive alternative in applications such as prosthetic FE modeling, where the initial position of the limb in the socket is not known, where local geometric features have high design significance, and where large slip occurs under load.     

Load transfer mechanics between trans-tibial prosthetic socket and residual limb--dynamic effects

The effects of inertial loads on the interface stresses between trans-tibial residual limb and prosthetic socket were investigated. The motion of the limb and prosthesis was monitored using a Vicon motion analysis system and the ground reaction force was measured by a force platform. Equivalent loads at the knee joint during walking were calculated in two cases with and without consideration of the material inertia. A 3D nonlinear finite element (FE) model based on the actual geometry of residual limb, internal bones and socket liner was developed to study the mechanical interaction between socket and residual limb during walking. To simulate the friction/slip boundary conditions between the skin and liner, automated surface-to-surface contact was used. The prediction results indicated that interface pressure and shear stress had the similar double-peaked waveform shape in stance phase. The average difference in interface stresses between the two cases with and without consideration of inertial forces was 8.4% in stance phase and 20.1% in swing phase. The maximum difference during stance phase is up to 19%. This suggests that it is preferable to consider the material inertia effect in a fully dynamic FE model.     

Stress-modulated growth, residual stress, and vascular heterogeneity.

A simple phenomenological model is used to study interrelations between material properties, growth-induced residual stresses, and opening angles in arteries. The artery is assumed to be a thick-walled tube composed of an orthotropic pseudoelastic material. In addition, the normal mature vessel is assumed to have uniform circumferential wall stress, which is achieved here via a mechanical growth law. Residual stresses are computed for three configurations: the unloaded intact artery, the artery after a single transmural cut, and the inner and outer rings of the artery created by combined radial and circumferential cuts. The results show that the magnitudes of the opening angles depend strongly on the heterogeneity of the material properties of the vessel wall and that multiple radial and circumferential cuts may be needed to relieve all residual stress. In addition, comparing computed opening angles with published experimental data for the bovine carotid artery suggests that the material properties change continuously across the vessel wall and that stress, not strain, correlates well with growth in arteries.     

A stress compatible finite element for implant/cement interface analyses

A new finite element has been developed to enforce normal and shear stress continuity at bimaterial interface points in order to alleviate the problem of high stress discontinuity predictions by the conventional displacement finite element method. The proposed element is based on a five node isoparametric quadrilateral element where the fifth node is located at the interface boundary of the element. A series of validation tests have been carried out to assess the correctness of the stress distribution obtained by the new element at interfaces of highly dissimilar materials. The results of the tests are compared to analytical solutions and to results from convergence studies performed by the conventional finite element method (SAP-IV). Overall, the proposed element has been demonstrated to have a very satisfactory degree of reliability, especially in view of the observed inability of the conventional method to yield interpretable interface stress values for most cases analyzed. Finally, the new interface element has been applied to the analysis of an axisymmetric model of the knee tibial implant. The superiority of the proposed element over the conventional one has been demonstrated in this case by a convergence study.     

Extended push-out test to characterize the failure of bone-implant interface]

To study the mechanical behaviour of the implant-bone interface the push- or pull-out test was overtaken from material science. Most authors equate the maximum load (break point) with the failure of the implant integration. Extending the test procedure by acoustic emission analysis reveals the possibility to detect the failure of the interface more in detail and from its earliest beginning. The development of disconnection between host and implant was found to start long before the ultimate load is reached and can be monitored and quantified during this period. The active interface mechanisms are characterized by the distribution function of acoustic emissions and the number of hits per time defines the kinetics of the failure. From clinical studies a gradual subsidence of loaded implants is known starting long time before the definite implant failure. The presented extension of the push-out test with acoustic emission analysis allows the detection of a critical shear stress tc which demarks the onset of the gradual interface failure. We believe this value to represent the real critical load which should not be exceeded in the clinical application of intraosseous implants.     

Variability of a three-dimensional finite element model constructed using magnetic resonance images of a knee for joint contact stress analysis

Magnetic resonance (MR) imaging has been widely used to evaluate the thickness and volume of articular cartilage both in vivo and in vitro. While morphological information on the cartilage can be obtained using MR images, image processing for extracting geometric boundaries of the cartilage may introduce variations in the thickness of the cartilage. To evaluate the variability of using MR images to construct finite element (FE) knee cartilage models, five investigators independently digitized the same set of MR images of a human knee. The topology of cartilage thickness was determined using a minimal distance algorithm. Less than 8 percent variation in cartilage thickness was observed from the digitized data. The effect of changes in cartilage thickness on contact stress analysis was then investigated using five FE models of the knee. One FE model (average FE model) was constructed using the mean values of the digitized contours of the cartilage, and the other four were constructed by varying the thickness of the average FE model by +/- 5 percent and +/- 10 percent, respectively. The results demonstrated that under axial tibial compressive loading (up to 1,400 N), variations of cartilage thickness caused by digitization of MR images may result in a difference of approximately 10 percent in peak contact stresses (surface pressure, von Mises stress, and hydrostatic pressure) in the cartilage. A reduction of cartilage thickness caused increases of contact stresses, while an increase of cartilage thickness reduced contact stresses. Furthermore, the effect of variation of material properties of the cartilage on contact stress analysis was investigated. The peak contact stress increased almost linearly with the Young's modulus of the cartilage. The peak von Mises stress was dramatically reduced when the Poisson,s ratio was increased from 0.05 to 0.49 under an axial compressive load of 1,400 N, while peak hydrostatic pressure was dramatically increased. Peak surface pressure was also increased with the Poisson's ratio, but with a lower magnitude compared to von Mises stress and hydrostatic pressure. In conclusion, the imaging process may cause 10 percent variations in peak contact stress, and the predicted stress distribution is sensitive to the accuracy of the material properties of the cartilage model, especially to the variation of Poisson's ratio.     

Mechanics of the tapered interference fit in dental implants

In evaluation of the long-term success of a dental implant, the reliability and the stability of the implant-abutment interface plays a great role. Tapered interference fits provide a reliable connection method between the abutment and the implant. In this work, the mechanics of the tapered interference fits were analyzed using a closed-form formula and the finite element (FE) method. An analytical solution, which is used to predict the contact pressure in a straight interference, was modified to predict the contact pressure in the tapered implant-abutment interface. Elastic-plastic FE analysis was used to simulate the implant and abutment material behavior. The validity and the applicability of the analytical solution were investigated by comparisons with the FE model for a range of problem parameters. It was shown that the analytical solution could be used to determine the pull-out force and loosening-torque with 5-10% error. Detailed analysis of the stress distribution due to tapered interference fit, in a commercially available, abutment-implant system was carried out. This analysis shows that plastic deformation in the implant limits the increase in the pull-out force that would have been otherwise predicted by higher interference values.     

Residual stress generation and necrosis formation in multi-cell tumour spheroids

We consider how cell proliferation and death generate residual stresses within a multi-cell tumour spheroid (MCTS). Previous work by Jones and co-workers [8] has shown that isotropic growth in a purely elastic MCTS produces growth induced stresses which eventually become unbounded, and hence are physically unrealistic. Since viscoelastic materials show stress relaxation under a fixed deformation we consider the effect of the addition of a small amount of viscosity to the elastic system by examining formation of equilibrium stress profiles within a Maxwell type viscoelastic MCTS. A model of necrosis formation based upon that proposed by Please and co-workers (see [16] [17] [18]) is then presented in which necrosis forms under conditions of adverse mechanical stress rather than in regions of extreme chemical stress as is usually assumed. The influence of rheology on necrosis formation is then investigated, and it is shown that the excessive stress generated in the purely elastic tumour can be relieved either by the addition of some viscosity to the system or by accounting for an inner necrotic interface with an appropriate stress boundary condition.     

A piece-wise non-linear elastic stress expression of human and pig coronary arteries tested in vitro.

Eight human and nineteen pig unembalmed proximal left anterior descending and circumflex coronary arteries were subjected to linear volume changes (2 s ramp time) at three fixed axial extensions while immersed in a physiological saline bath at body temperature. Measured parameters included: lumen pressure, outside diameter, axial force, and axial extension. The deformations were measured using a video dimensional analyzer. The arteries were inflated to pressures well above the physiological range at each axial extension. A latex inner tube was placed inside of each specimen to prevent leakage, and its effects upon the measured stresses were corrected analytically. With this method, the average circumferential and axial stresses could be computed directly from the experimental data. In both directions the average stresses measured displayed two distinct regions: stresses occurring for small diameter changes (physiological pressures) and stresses occurring for large diameter changes (high pressures). The resulting average small strain and large strain stress components were curve-fit separately and, when reassembled, provided a piece-wise model of the stress response of coronary arteries over a wide range of inflation pressures and axial extensions.     

Fixed Electrical Charges and Mobile Ions Affect the Measurable Mechano-Electrochemical Properties of Charged-Hydrated Biological Tissues: The Articular Cartilage Paradigm

The triphasic constitutive law [Lai, Hou and Mow (1991)] has been shown in some special 1D cases to successfully model the deformational and transport behaviors of charged-hydrated, porous-permeable, soft biological tissues, as typified by articular cartilage. Due to nonlinearities and other mathematical complexities of these equations, few problems for the deformation of such materials have ever been solved analytically. Using a perturbation procedure, we have linearized the triphasic equations with respect to a small imposed axial compressive strain, and obtained an equilibrium solution, as well as a short-time boundary layer solution for the mechano- electrochemical (MEC) fields for such a material under a 2D unconfined compression test. The present results show that the key physical parameter determining the deformational behaviors is the ratio of the perturbation of osmotic pressure to elastic stress, which leads to changes of the measurable elastic coefficients. From the short-time boundary layer solution, both the lateral expansion and the applied load are found to decrease with the square root of time. The predicted deformations, flow fields and stresses are consistent with the analysis of the short time and equilibrium biphasic (i.e., the solid matrix has no attached electric charges) [Armstrong, Lai and Mow (1984)]. These results provide a better understanding of the manner in which fixed electric charges and mobile ions within the tissue contribute to the observed material responses.     

Wall shear stress during pulsatile flow distal to a normal porcine aortic valve

The shear stress at the wall has been of interest as one of the possible fluid dynamic factors that may be damaging in the region of prosthetic valves. The purpose of this study was to measure the axial wall shear stresses in the region of a 29 mm tissue annulus diameter porcine stent mounted prosthetic aortic valve (Hancock, Model 242). Studies were performed in an in vitro pulse duplicating system. The axial wall shear stress was calculated from velocities obtained near the wall with a laser Doppler anemometer. The largest axial wall shear stress was 29 dyn cm-2 and it occurred at the highest stroke volume used (80 ml). At a stroke volume of 50 ml, the largest axial wall shear stress was 17 dyn cm-2 and at a stroke volume of 35 ml, it was 15 dyn cm-2. Stresses of these magnitudes are far below those reported to be damaging to the endothelial surface. These stresses may be high enough, however, to affect platelet function.     

Dynamics of mechanical signal transmission through prestressed stress fibers

Transmission of mechanical stimuli through the actin cytoskeleton has been proposed as a mechanism for rapid long-distance mechanotransduction in cells; however, a quantitative understanding of the dynamics of this transmission and the physical factors governing it remains lacking. Two key features of the actin cytoskeleton are its viscoelastic nature and the presence of prestress due to actomyosin motor activity. We develop a model of mechanical signal transmission through prestressed viscoelastic actin stress fibers that directly connect the cell surface to the nucleus. The analysis considers both temporally stationary and oscillatory mechanical signals and accounts for cytosolic drag on the stress fibers. To elucidate the physical parameters that govern mechanical signal transmission, we initially focus on the highly simplified case of a single stress fiber. The results demonstrate that the dynamics of mechanical signal transmission depend on whether the applied force leads to transverse or axial motion of the stress fiber. For transverse motion, mechanical signal transmission is dominated by prestress while fiber elasticity has a negligible effect. Conversely, signal transmission for axial motion is mediated uniquely by elasticity due to the absence of a prestress restoring force. Mechanical signal transmission is significantly delayed by stress fiber material viscosity, while cytosolic damping becomes important only for longer stress fibers. Only transverse motion yields the rapid and long-distance mechanical signal transmission dynamics observed experimentally. For simple networks of stress fibers, mechanical signals are transmitted rapidly to the nucleus when the fibers are oriented largely orthogonal to the applied force, whereas the presence of fibers parallel to the applied force slows down mechanical signal transmission significantly. The present results suggest that cytoskeletal prestress mediates rapid mechanical signal transmission and allows temporally oscillatory signals in the physiological frequency range to travel a long distance without significant decay due to material viscosity and/or cytosolic drag.