Apparent Stiffness Research Articles

Inter-sonic propagation of shear zone as an effect of longitudinal deformation

Examples are presented when propagation of the boundaries of the zones of frictional sliding or elastic shear propagate with the speed of the longitudinal wave (or p-wave) and thus can be confused with the instances of intersonic fracture propagation. In these examples the mechanism of intersonic sliding/shear zone propagation is the longitudinal elastic deformation of the material surrounding the zone and hence controlled by the longitudinal wave velocity. Two simple models of intersonic sliding/shear zone propagation are considered: a frictional sliding of an elastic rod and shearing of an elastic rod on shear Winkler foundation. It is presumed that the shear Winkler layer can have either the conventional positive stiffness or apparent negative stiffness. In all these cases the sliding/shear zone propagates with the longitudinal wave velocity. In the case of negative stiffness of the Winkler layer the deformation rapidly increases with time reflecting the presence of influx of the external energy required by the system that realises negative stiffness.

Analysis of Stress Relaxation in Bulk and Porous Ultra-High Molecular Weight Polyethylene (UHMWPE).

The reported study was devoted to the investigation of viscoelastic behavior for solid and porous ultra-high molecular weight polyethylene (UHMWPE) under compression. The obtained experimental stress curves were interpreted using a two-term Prony series to represent the superposition of two coexisting activation processes corresponding to long molecular (~160 s) and short structural (~20 s) time scales, respectively, leading to good statistical correlation with the observations. In the case of porous polymer, the internal strain redistribution during relaxation was quantified using digital image correlation (DIC) analysis. The strongly inhomogeneous deformation of the porous polymer was found not to affect the relaxation times. To illustrate the possibility of generalizing the results to three dimensions, X-ray tomography was used to examine the porous structure relaxation at the macro- and micro-scale levels. DIC analysis revealed positive correlation between the applied force and relative density. The apparent stiffness variation for UHMWPE foams with mixed open and closed cells was described using a newly proposed three-term expression. Furthermore, in situ tensile loading and X-ray scattering study was applied for isotropic solid UHMWPE specimens to investigate the evolution of internal structure and orientation during drawing and stress relaxation in another loading mode.

Impact of extracellular matrix and collagen network properties on the cervical intervertebral disc response to physiological loads: A parametric study

Current intervertebral disc finite element models are hard to validate since they describe multi-physical phenomena and contain a huge number of material properties. This work aims to simplify numerical validation/identification studies by prioritizing the sensitivity of intervertebral disc behavior to mechanical properties. A 3D fiber-reinforced hyperelastic model of a C6-C7 intervertebral disc is used to carry out the parametric study. 10 parameters describing the extracellular matrix and the collagen network behaviors are included in the parametric study. The influence of varying these parameters on the disc response is estimated during physiological movements of the head, including compression, lateral bending, flexion, and axial rotation. The obtained results highlight the high sensitivity of the disc behavior to the stiffness of the annulus fibrosus extracellular matrix for all the studied loads with a relative increase in the disc apparent stiffness by 67% for compression and by 57% for axial rotation when the annulus stiffness increases from 0.4 to 2MPa. It is also shown that varying collagen network orientation, stiffness, and stiffening in the studied configuration range have a noticeable effect on rotational motions with a relative apparent stiffness difference reaching 6.8%, 10%, and 22%, respectively, in lateral bending. However, the collagen orientation does not affect disc response to axial load.

Auxetics and Other Systems with Unusual Characteristics

Auxetics and Other Systems with Unusual Characteristics

Apparent Negative Stiffness Induced by Constriction

Apparent Negative Stiffness Induced by Constriction

Behavior and Damage Characterization of Impulsively Loaded Cross-Laminated Timber Panels

This research bridges the gap between the quasi-static and high-strain-rate loading regimes in cross-laminated timber (CLT) by investigating two areas that have remained unstudied or elusive, i.e., rolling shear failure of CLT under impulsive, blast-like loading and intermediate strain rates in CLT. To study the conditions that would promote shear modes of failure, a novel, highly adaptable center-point testing system and methodology were developed that permitted the application of impulsive loading to undamaged CLT panels in a highly controlled and repeatable manner. The loading condition and low span-to-depth ratio (6.40 ≤ L:h ≤ 6.55) CLT were selected to encourage the development of shear modes of failure. Changes to the rotational rigidity at the boundary conditions allowed for the empirical simulation of realistic boundary conditions. Digital Image Correlation (DIC) and load cell data were used to identify failure modes following loss in resistance in the specimens. Overall, the experiment was successful in consistently eliciting shear modes of failure and providing damage characterization in impulsively loaded CLT. Shear modes of failure resulted in the dramatic loss of resistance in all specimens tested. Strain-rate enhancement in the dynamic apparent flexural stiffness of CLT of 1.3 to 7.2 times was observed. Lower levels of damage were observed in specimens with higher levels of boundary-condition rotational rigidity.

Indentation of pore-spanning lipid membranes: Spring-stiffening or -softening responses and apparent stiffness prediction

Atomic force microscopy (AFM)-based nanoindentation has emerged as a key nanomechanical mapping method to probe mechanical properties of two-dimensional engineering and biological materials. In comparison with extensive analytical studies on the indentation of freestanding solid thin films, surprisingly little attention has been drawn on analytical predictions of mechanical responses of freestanding lipid membranes to indentation in the framework of Helfrich membrane theory. Here we perform systematic theoretical studies with formulated analysis on indentation of nanopore-spanning lipid membranes. Two types of lipid membranes are considered. One is the membrane subject to uniform area stretching and the membrane is found to exhibit a spring-stiffening behavior during indentation, while the other is the membrane of constant tension and a spring-softening behavior at intermediate indentation is exhibited. The smaller the membrane tension is, the clearer the spring-softening feature is. For both types of membranes, analytical solutions of load–depth relationship at small indentation and apparent membrane stiffnesses are obtained at small and large tension. For the stretched lipid membranes, a cubic nonlinearity arising from membrane stretching is specified. Roles of membrane bending rigidity, (pre-)tension, or area compression modulus, and indenter size in membrane indentation are elucidated. Moreover, pressure of contact between the membrane and indenter and a line load of contact acting along the contact edge are obtained based on the shell theory. Our results provide fundamental insights on nonlinear mechanical behaviors of the lipid membrane indentation.

Apparent Negative Stiffness Induced by Constriction

Lateral constriction (transverse elastic constraint) of elastic rod controls its apparent (longitudinal) stiffness and can even make it negative. (The total energy of system rod + constriction device remains positive.) The apparent stiffness depends upon moduli of the rod material and the constriction modulus relating lateral normal stress and strain components. Apparent negative stiffness can arrest wave propagation along the rod. This opens a way to design hybrid materials consisting of a stiff matrix and parallel lubricated elastic rods. The matrix is constricting the rods making them exhibit negative stiffness and thus ensuring elevated sound or vibration isolation of the hybrid. A mechanism of controlling the constriction modulus by lateral compression is proposed.

Bending and stretching behavior of graphene structures using continuum models calibrated with modal analysis

Two-dimensional nanomaterials exhibit specific mechanical characteristics essential for commercialization and industrial applications, such as nano electro-mechanical systems. When analyzing a mechanical field, it is important to determine these materials' bending stiffness and stretching properties. The apparent bending stiffness and stretching of graphene structures at the macro-scale differ from theoretical predictions at the nano-scale. This discrepancy results from thermally generated dynamic waves in these atomic-based structures. Therefore, characterization methods based on atomistic static methods were unable to capture these effects. This study uses hybrid atomistic-continuum models to apply modal analysis to determine these key parameters. The proposed approach exploits the advantages of both atomistic and continuum models. Our approach is based on optimization techniques such as the simulated annealing algorithm. The unknown parameters of the continuum models, such as the bending stiffness and stretching, were acquired from molecular dynamics simulations and continuum mechanics resonance frequencies. The effects of diameter on the mechanical properties and value of different mode shapes of graphene nanostructures are studied using this method, which incorporates the effects of nano and macro-scale models. The presented approach provides insight into the mechanical vibrational characteristics of two-dimensional graphene structures at an atomic level.

Hierarchical and synergistic organization of control variables during the multi-digit grasp of a free and an externally fixed object.

In the referent control theory, grip force emerges by designating the referent aperture (Ra) as a threshold position inside the object. This study quantified Ra and investigated whether the synergistic control of digit referent coordinate (RC) and apparent stiffness (k) depend on the external mechanical constraints on the hand-held object. Subjects held a motorized handle capable of adjusting the grip width and performed static multi-digit prehension tasks in which the handle was free and externally fixed in different conditions. The RC and k of individual digits were reconstructed from the changes in digit normal forces and the positions as the grip width was modulated. RCs of the thumb and virtual finger were used to calculate the width and midpoint of Ra, and synergy indices quantifying the task-specific covariation in the space of the digit normal forces and {RC, k} variables were computed. We found that the k and width of the Ra were larger when holding a free handle than the fixed handle. The higher stiffness in the free condition could be a strategy to ensure grip stability. The midpoint of Ra was skewed toward the virtual finger, reflecting different magnitudes of k for the two digits. Further, the normal forces and control variables {RC, k} displayed synergistic covariation for stabilization of the total grasping force. Finally, the synergies were weaker when the handle was externally fixed, demonstrating the dependence of synergies on external constraints. These results add to the current literature by demonstrating that grasp control involves modulation of digit apparent stiffness in addition to the referent coordinate and by identifying the synergistic organization of the control variables during static grasp.

Variability in the Natural Frequencies of a Nine-Story Concrete Building from Seconds to Decades

Abstract Since 2001, the Southern California Seismic Network has archived continuous waveform data from strong-motion station CI.MIK in Caltech Hall (formerly Millikan Library)—a nine-story reinforced concrete building. Simple spectral analysis of this 20 yr record reveals that the building’s fundamental frequencies have increased gradually by 5.1% (east–west) and 2.3% (north–south), with larger long-term variability up to 9.7% (east–west) and 4.4% (north–south). This finding is unexpected, as previous analysis of forced vibration tests and strong-motion records has shown that between 1968 and 2003 the fundamental frequencies decreased by 22% (east–west) and 12% (north–south), largely attributed to minor structural damage and soil–structure system changes from major earthquakes. Today, the building’s apparent structural stiffness is comparable to what it was in 1986, before the Whittier Narrows earthquake. Using data from earthquakes and forced vibrations, we also document the building’s nonlinear dynamic elasticity, which is characterized by a rapid softening (decrease in apparent frequencies) at the onset of strong motion, followed by a slower, log–linear recovery trend over the scale of minutes. This nonlinear behavior does not appear to have changed with time.

Beam element resonance-based prediction and parametric analysis of sound transmission of laminated plates

Laminated plates are complex structures important to practical applications, so it is essential and valuable to develop efficient and accurate models for predicting their sound transmission loss. An efficient and unified approach based on the frequency-dependent bending stiffness of a beam element taken from a laminated plate is proposed to predict the sound transmission loss of the laminated plate. First, the finite element method is used to calculate the resonant frequencies of the multilayer beam under different boundary conditions, based on which the apparent bending stiffness of the corresponding multilayer plate is estimated. Next, with the aid of the so obtained apparent bending stiffness, the dynamic behavior of a multilayer composite plate is described by a simplified fourth-order governing equation. Finally, the sound transmission loss is conveniently calculated with the modal superposition method. In this approach, the influences can be conveniently and accurately simulated of the plate size, shear modulus of the core, loss factor and boundary conditions. Furthermore, the predicted sound transmission loss is validated through comparisons with the measured values of typical laminated plates. Finally, based on calculations with the proposed approach, the influence is revealed of the shear modulus on the coincidence region and sound transmission loss, and the relationship is discussed between the weighted sound reduction index and the shear modulus. These results suggest that to achieve good acoustic performance of a designed laminated plate, the coincidence region should be narrow and located at frequencies as high as possible.

Особенности применения теории силовых смещений к расчету контактной жесткости опоры шпиндельного радиально-упорного шарикоподшипника

Contact stiffness is an important parameter for describing the interaction of plenty precise engineering solutions, for example, for machine-tool manufacture, the most important element of the carrier system is the spindle assembly supports, largely determining the accuracy of machining parts on the machine. The article considers an example of a system approach to describing the variable stiffness of a high-precision spindle bearing, combining an analytical approach to calculating the variable quasi-static stiffness of an angular contact ball bearing and a methodological approach that takes into account the contact normal and tangential force displacements of the upper bearing ring while considering the friction force for tangential interactions. The proposed system approach to calculating the stiffness of a ball bearing support is based on fairly simple models, in particular, the apparatus proposed by Professor P.M. Chernyanskiy, describing the phenomena of changing the spindle supports stiffness. These changes were experimentally obtained and weren’t described in earlier works. It is shown that the experimental stiffness consists of elastic and contact components, depending on the friction forces, therefore, the concept of apparent stiffness is introduced for data that are not taken into account in the processes under study. The effect of the axial displacement of the spindle is explained on the basis of contact interactions in the support of the spindle unit. The resulting analytical system model was tested on well-known bearing supports and the obtained stiffness solutions were compared with validated experimental data from open sources

P20. Evaluation of interbody design parameters on graft subsidence in the 3D-printed era: separating fact from fiction

<h3>BACKGROUND CONTEXT</h3> Cage subsidence is a known complication of spinal interbody fusion and is associated with poor clinical outcomes. Three-dimensional-printed titanium interbodies are becoming increasingly popular and their unique manufacturing allows for alteration of several features including stiffness and porosity. Though there are many claims made regarding the influence of these features on interbody subsidence, there is a lack of rigorous scientific evaluation. <h3>PURPOSE</h3> The purpose of this study is to determine how changes in implant parameters biomechanically affect subsidence using an in-vitro sawbone model. <h3>STUDY DESIGN/SETTING</h3> In vtiro biomechanical saw bone model. <h3>OUTCOME MEASURES</h3> Subsidence measured as relative displacement of the interbody and foam block using digital imaging correlation. Stiffness measured using thethe linear portion of the load displacement curve. <h3>METHODS</h3> Eight groups were tested (n = 8/group) including seven 3D-printed titanium (Seaspine, Inc.) varying in surface area (aperture: standard, small and none), porosity (high vs low) and surface topography (nonporous vs porous) as well as one standard PEEK and one solid titanium. Subsidence testing was performed in a standardized foam block model compressed to 150 N, cycled between 50-250N for 5000 cycles at a rate of 1 Hz. Digital imaging correlation was used to determine relative displacement of the interbody and foam block. Stiffness testing was performed by compressing implants between parallel steel blocks by 500 N at a rate of 50N/sec for three cycles and measuring the linear portion of the load displacement curve. <h3>RESULTS</h3> Subsidence was decreased as the surface area of contact with the sawbone was increased (all p 0.353) or based on material property (all p>.192). Apparent stiffness of the implant was affected by porosity (p 0.146). The PEEK cage had similar stiffness to porous Ti (p= 0.959). A nonporous surface led to greater stiffness compared to the standard porous interbody (p= 0.010). Surface area was negatively correlated with subsidence (r= -0.808, p=0.0284), but was not correlated with stiffness (r= 0.568, p=0.183). <h3>CONCLUSIONS</h3> Implant surface area and surface topography have the greatest influence on interbody subsidence. Apparent stiffness, implant porosity, as well as material property do not affect subsidence in an in vitro model. Biologic response in the in vivo setting likely also influences clinical subsidence, which is a topic of future study. <h3>FDA DEVICE/DRUG STATUS</h3> This abstract does not discuss or include any applicable devices or drugs.

Synergies Stabilizing Vertical Posture in Spaces of Control Variables

In this study, we address the question: Can the central nervous system stabilize vertical posture in the abundant space of neural commands? We assume that the control of vertical posture is associated with setting spatial referent coordinates (RC) for the involved muscle groups, which translates into two basic commands, reciprocal and co-activation. We explored whether the two commands co-varied across trials to stabilize the initial postural state. Young, healthy participants stood quietly against an external horizontal load and were exposed to smooth unloading episodes. Linear regression between horizontal force and center of mass coordinate during the unloading phase was computed to define the intercept (RC) and slope (apparent stiffness, k). Hyperbolic regression between the intercept and slope across unloading episodes and randomization analysis both demonstrated high indexes of co-variation stabilizing horizontal force in the initial state. Higher co-variation indexes were associated with lower average k values across the participants suggesting destabilizing effects of muscle coactivation. Analysis of deviations in the {RC; k} space keeping the posture unchanged (motor equivalent) between two states separated by a voluntary quick body sway showed significantly larger motor equivalent deviations compared to non-motor equivalent ones. This is the first study demonstrating posture-stabilizing synergies in the space of neural control variables using various computational methods. It promises direct applications to studies of postural disorders and rehabilitation.

Framework of sampling the subject-specific static loads from the gait cycle of interindividual variation

Background and ObjectiveNumerous techniques for bone remodeling simulation have been developed based on Wolff's law. However, most studies have been conducted with empirically determined static loads, which cannot reflect subject-specific characteristics. We recently proposed a new concept of representative static loads (RSLs) to efficiently consider the effect of cyclically repeated dynamic loads on bone remodeling simulation. Based on this concept, the goal of this study is to sample the subject-specific static loads (SSL) from a general gait cycle of interindividual variation. MethodsA total of 15 gait cycles (ten normal and five abnormal cycles) obtained from the public database were used in this study. Each gait cycle was applied to a femur FE model constructed from the clinical CT scan data to evaluate the strain energy distribution as a reference. Then, a natural coordinate was introduced to maintain the predefined locations of extreme points (i.e., two peaks and one valley) for both normal and abnormal gait cycles. To determine the RSLs in the natural coordinate, five out of ten normal gait cycles were used. Through an inverse transformation for each gait cycle, the RSLs in the natural coordinate were converted to the SSLs in the original coordinate. Topology optimization results with the proposed SSLs were compared with those with a single full gait cycle (reference). For comparison, topology optimization was also conducted with empirically determined loads (EDLs) which have been widely used in the literature. ResultsFor normal gait cycles, the proposed SSLs reduced the average computing cost by 95.86% while suppressing the errors of bone mass distribution and apparent stiffness below maximum 4.24% and 1.72%, respectively. Even for abnormal gait cycles, the errors of bone mass distribution and apparent stiffness were suppressed below maximum 9.49% and 2.12%, respectively. Conversely, the conventional EDLs (peak loads selected in this study) showed significantly larger errors of maximum 47.28% and 30.31% in bone mass distribution and apparent stiffness for normal gait cycles. ConclusionBy virtue of using the coordinate transformation for each gait cycle, the proposed SSLs achieved a higher accuracy in the bone mass distribution and apparent stiffness than the previous RSLs and EDLs. Furthermore, this approach can be used for abnormal gait cycles which have higher interindividual variation.

Elastic shell theory for plant cell wall stiffness reveals contributions of cell wall elasticity and turgor pressure in AFM measurement

The stiffness of a plant cell in response to an applied force is determined not only by the elasticity of the cell wall but also by turgor pressure and cell geometry, which affect the tension of the cell wall. Although stiffness has been investigated using atomic force microscopy (AFM) and Young’s modulus of the cell wall has occasionally been estimated using the contact-stress theory (Hertz theory), the existence of tension has made the study of stiffness more complex. Elastic shell theory has been proposed as an alternative method; however, the estimation of elasticity remains ambiguous. Here, we used finite element method simulations to verify the formula of the elastic shell theory for onion (Allium cepa) cells. We applied the formula and simulations to successfully quantify the turgor pressure and elasticity of a cell in the plane direction using the cell curvature and apparent stiffness measured by AFM. We conclude that tension resulting from turgor pressure regulates cell stiffness, which can be modified by a slight adjustment of turgor pressure in the order of 0.1 MPa. This theoretical analysis reveals a path for understanding forces inherent in plant cells.

Computational modelling of ankle-foot orthosis to evaluate spatially asymmetric structural stiffness: Importance of geometric nonlinearity.

An ankle-foot orthosis (AFO) constructed as a single piece of isotropic elastic material is a commonly used assistive device that provides stability to the ankle joint of patients with spastic diplegic cerebral palsy. The AFO has asymmetric stiffness that restricts plantarflexion during the swing phase while it is flexible to allow dorsiflexion during the stance phase with a large deflection, including buckling originating from geometric nonlinearity. However, its mechanical implications have not been sufficiently investigated. This study aims to develop a computational model of an AFO considering geometric nonlinearity and examine AFO stiffness asymmetry during plantarflexion and dorsiflexion using physical experiments. Three-dimensional AFO mechanics with geometric nonlinearities were expressed using corotational triangle-element formulations that obeyed Kirchhoff-Love plate theory. Computational load tests for plantarflexion and dorsiflexion, using idealised AFOs with two different ankle-region designs (covering or not covering the apexes of the malleoli), showed that plantarflexion moment-ankle angle relationships were linear and dorsiflexion moment-ankle angle relationships were nonlinear; increases in dorsiflexion led to negative apparent stiffness of the AFO. Both ankle-region designs resisted both plantarflexion and dorsiflexion, and out-of-plane elastic energy was locally concentrated on the lateral side, resulting in large deflections during dorsiflexion. These findings give insight into appropriate AFO design from a mechanical viewpoint by characterising three-dimensional structural asymmetry and geometric nonlinearity.

Individual diametral compression behavior of a ceramic proppant

In this work we report the systematic individual diametral compression of a commercial ceramic proppants, describe the main features of their mechanical behavior and explain them in terms of morphology, microstructure and complementary characterizations.A fragile behavior was observed and described, and mechanical parameters were easily evaluated from the stress-strain measured curves (maximum stress and displacement), an apparent stiffness parameter (slope) was proposed from the linear part of the evaluated curves as well. Weibull analysis was performed for the measured stresses, with the resulting Weibull parameters were also illustrative and allowed a better understanding of the behavior of an individual proppant submitted to diametral compression.The possibility of employing this approach or strategy for both proppants development and quality control is proposed and discussed. This was found to be descriptive and complementary to the typical collective bed crush test.

A bending stiffness criterion for sandwich panels with high sound insulation and its realization through low specific modulus layers

The sound transmission loss of a sandwich panel is usually much lower than that given by the mass law for a homogeneous panel with the same weight in the mass-controlled frequency range. How to make the sound transmission loss of the sandwich panel close to the mass law and maintain a high static stiffness is a key acoustic issue in its design. In this paper, the critical apparent bending stiffness is defined as the apparent bending stiffness at which the sound transmission loss of a sandwich panel approaches that of the mass law. Then, a method is given to determine the critical apparent bending stiffness of sandwich panels, based on which a range of structural parameters can be obtained to guide the design of sandwich panels with high acoustic performance. Compared with the apparent bending stiffness of a typical sandwich panel, the critical apparent bending stiffness decays more rapidly with the increase of frequency, exhibiting a relatively high static stiffness and a relatively low dynamic stiffness. In order to make the apparent bending stiffness of a typical sandwich panel close to the critical apparent bending stiffness, the concept of introducing an elastic layer with low specific modulus in the middle of the core layer is proposed. Both theoretical and experimental results show that the introduced elastic layer with low specific modulus reduces the dynamic bending stiffness of the sandwich panel while a high static bending stiffness is maintained, thus significantly improving the broadband sound-insulation performance of the panel.