Nonlinear Compliance Research Articles

Influence of physical ageing on the non-linear viscoelasticity of polycarbonate

Abstract The molecular origin of the non-linear viscoelasticity of polycarbonate (PC) has been studied by creep measurements at room temperature for three different states of physical ageing. The non-linearity of the isochronous stress-strain diagrams was analysed according to the Boyd-Jansson model. It was found that although the stresses (up to 27 MPa) caused marked non-linearity, they did not influence the physical ageing stage, as measured with a control load. The linear compliance component of the Boyd-Jansson model, in both longitudinal and transverse directions, was barely influenced by the physical ageing state of the PC, although the non-linear compliance component, in both longitudinal and transverse directions, changed considerably. It is proposed that the origins of the non-linearity of quenched polycarbonate at room temperature are stress activated changes in the α′ and β mechanisms.

Redesign of robust controllers for rigid-link flexible-joint robotic manipulators actuated with harmonic drive gearing

The authors have redesigned a previously developed robust tracking controller for rigid-link flexible-joint (RLFJ) robot manipulators to handle flexibilities specifically induced by harmonic drive gearing. A more realistic, and consequently more complex model for the torque transmission dynamics which includes frictional losses, kinematic error and nonlinear compliance is utilised. The stability result achieved for the authors' proposed controller ensures that the link tracking error is 'globally uniformly ultimately bounded' (GUUB), in spite of additive bounded disturbances, parametric uncertainty, and the presence of complex nonnegligible actuator dynamics.

Nonlinear Compliance Testing Applied to Indentation Cracking in Ceramics

Indentation cracking under sharp‐pointed indenters is analyzed using compliance‐based, nonlinear, fracture mechanics. The stress intensity factor, K, in linear elastic fracture mechanics is well known to be proportional to the load, P; in this nonlinear analysis K is proportional to P3/4. The observed relation between indenter load‐point displacement and crack length is based on similitude of crack lengths with load‐point displacements as a strain‐controlled fracture. The equations that relate load and a function of the crack length to the crack driving force, J, have been found for Vickers indentations. Analysis of the nonlinear load vs displacement assumes an equilibrium crack in the elastic material surrounding the indent. The hardness that describes the load vs load‐point displacement during cracking is derived on a constant J line in load‐displacement space. The crack length is shown experimentally to be proportional to the load‐point displacement after crack initiation for several different indenters in ZnS. The measured loading curves are nonlinear and display crack initiation during loading. The KIc expressions found here are very similar to correlations that have been applied to indentation cracks.

Identification of the three-element windkessel model incorporating a pressure-dependent compliance.

A new one-step computational procedure is presented for estimating the parameters of the nonlinear three-element windkessel model of the arterial system incorporating a pressure-dependent compliance. The data required are pulsatile aortic pressure and flow. The basic assumptions are a steady-state periodic regime and a purely elastic compliant element. By stating two conditions, zero mean flow and zero mean power in the compliant element, peripheral and characteristic resistances are determined through simple closed form formulas as functions of mean values of the square of aortic pressure, the square of aortic flow, and the product of aortic pressure with aortic flow. The pressure across as well as the flow through the compliant element can be then obtained so allowing the calculation of volume variation and compliance as functions of pressure. The feasibility of this method is studied by applying it to both simulated and experimental data relative to different circulatory conditions and comparing the results with those obtained by an iterative parameter optimization algorithm and with the actual values when available. The conclusion is that the proposed method appears to be effective in identifying the three-element windkessel even in the case of nonlinear compliance.

Modelling storage behaviour in a fractured rock mass

A transient model was used to represent single borehole injection tests in fractured rock masses, on the basis of a three-dimensional stochastic description of the fracture network and of the mechanical behaviour of the fractures. The rock matrix remains impervious and storage occurs in the fractures, with a nonlinear compliance effect depending on both fluid pressure and normal closure stress acting on the joints. The aim of the study is to show how the shortest, weakly connected joints can be removed and replaced in the system by an equivalent parameter. These efforts can save computer time or enlarge the modelled rock volume. Moreover, this approach may be helpful in addressing the troublesome up-scaling problem in discontinuous media.

Chaotic Normal Vibrations and Friction at Mechanical Joints with Nonlinear Elastic Properties

This paper describes an investigation of the dynamics of contact systems that exhibit nonlinear stiffness characteristics. A dynamic contact model that accommodates normal motions with either analytical or empirical descriptions of nonlinear interface compliance has been formulated. Other joint nonlinearities, such as play and discontinuous changes in friction, which have been considered previously, are avoided in this study. The interface friction is related to the normal motion through the adhesion theory of friction. Numerical analyses of the model using common joint component materials and interface finishes under dynamic loads reveal that chaotic joint motion and, by implication, chaotic friction can occur over a significant range of contact parameters and operating conditions. Even though the chaotic motions are found to be on the order of microns, the instantaneous friction force is found to fluctuate considerably in the presence of such micro-scale normal motions.

Understanding the pressure cost of ventilation: Why does high-frequency ventilation work?

To understand when the use of high-frequency ventilation would be advantageous, we formulated the problem of achieving adequate alveolar ventilation at minimal pressure cost by dividing it into two simpler problems: a) the pressure cost per unit of convective oscillatory flow; and b) the convective flow cost necessary to achieve a unit of alveolar ventilation. Simple solutions for each of these cost functions were formulated using established models of gas exchange and lung mechanics, including the effects of lung inflation tidal volume and respiratory frequency in alveolar ventilation, nonlinear lung tissue compliance, and alveolar recruitment and derecruitment. Solutions to these models were combined to assess the total pressure cost of high-frequency ventilation as a function of the ventilatory settings and the pathophysiologic variables of the patient. The model predicted that for variables applicable to an infant with respiratory distress syndrome, the selection of positive end-expiratory pressure (PEEP) becomes critical because the penalties in pressure cost are amplified for both high and low values of PEEP. The selection of frequency is not as critical for frequencies > 10 Hz, although it is more important than in the normal neonatal lung. This analysis illustrates the importance of using high-frequency ventilation in infant respiratory distress syndrome and of optimizing the amount of PEEP. It also points out the danger of barotrauma in the derecruited lung. When the lungs are in a derecruited state, the combinations of frequency, PEEP, and tidal volume that yield adequate ventilation with safe distention of recruited alveoli are severely limited.

Anisotropic creep behaviour of oriented polycarbonate

The anisotropic character of the creep behaviour of polycarbonate, oriented at a temperature 12°C above the glass transition to different draw ratios, has been studied using the model of Boyd and Jansson. This model is based on molecular assumptions and results in a second-order relation, ε = D1σ + D2σ2, where D1 is the linear and D2 the non-linear compliance component. The segmental orientation was determined by infra-red dichroism and the chain extension by shrinkage. It was found that both the linear and non-linear compliance components decreased with increasing orientation, although the non-linear component is affected much more by the orientation. The same result was found both in tension and contraction.

Arterial Compliance and Its Pressure Dependence in Hypertension and Vasodilation

Arterial compliance has been recognized to be pressure-dependent. Its variation due to changing systolic and diastolic blood pressures in hypertension and subsequent vasodilation has not been investigated. The authors examined this aspect by combining an animal experiment and a recently established nonlinear windkessel model of the arterial system that incorporates a pressure-dependent arterial compliance, C(P). Aortic pressure and flow were simultaneously measured in experimental dogs during control and during methoxamine-induced hypertension and nitroprusside-induced vasodilation. A numerical procedure was implemented to compute the nonlinear compliance and account for the pressure dependence. Results show that within the cardiac cycle, C(P) reached its maximum at end-systole and increased in diastole when diastolic aortic pressure decayed. The magnitude of C(P) and its variation within the cardiac cycle was larger at low pressures, while the reverse was found when blood pressure was high. C(P) decreased significantly in hypertension and increased during subsequent vasodilation.

Creep‐based characterization of nonlinear relaxation behavior of plastics

AbstractIt is a matter of fact that creep experiments can be conducted more easily and accurately than stress relaxation experiments, since it is easier to maintain a stress constant (for instance by a “dead weight”) than a strain constant. Nevertheless, in practice, structural parts made of plastics (which are nonlinear viscoelastic materials) are very often loaded under stress‐relaxation conditions. The present paper presents an approach to predict the behavior of a nonlinear viscoelastic material under stress‐relaxation‐type loading, based on data obtained from creep‐type experiments. The nonlinear creep compliance is described mathematically by an exponential series with a limited number of terms and a single nonlinearity function depicting the transient behavior. The nonlinear behavior of the material under constant strain (i.e., stress relaxation) is then obtained by dividing the considered time range into very short time intervals in which constant stresses are acting, while the different values of the applied stresses are chosen in a manner that guarantees the same stain at the end of each interval. In this way, one performs a numerical nonlinear superposition of the effects of the loadings in the various intervals, leading to the desired results under stress relaxation. A comparison of theoretical results with experiments conducted on some thermoplastic materials shows good agreement.

Normal Vibrations and Friction at a Hertzian Contact Under Random Excitation: Perturbation Solution

Closed form solutions for the non-linear stochastic contact vibrations and friction of Hertzian contact systems are derived using a perturbation technique. The vibrations are excited either externally to the contact region by a white Guassian random normal load, or within the contact region by a rough surface input. Due to the non-linear Hertzian compliance, the mean normal contact compression under dynamic excitation is smaller than the static deflection in the absence of vibrations. One also finds a reduction in the average area of contact and, by implication, in the mean friction force. Expressions for the mean values of contact separation and friction force during steady sliding are presented directly in terms of the system and excitation parameters. The results are in good agreement with those previously calculated using the Fokker-Planck equation, which agreed well with instantaneous and average measured friction data.

Feedback effects in heart-arterial system interaction.

Dynamic arterio-ventricular interaction in terms of left ventricle elastance and nonlinear arterial system compliance, pulse wave reflections and coronary blood flow was investigated, combining the use of analog models of the circulation with data from animal experiments. Results show that dynamic factors, as well as "feedback" effects are important.

Representation and learning of nonlinear compliance using neural nets

A new approach to compliant motion control using neural networks is presented. In the paper, "compliance" is treated as a nonlinear mapping from a measured force to a corrected motion. The nonlinear mapping by a multilayer neural network is outlined, this allows one to deal with complex control strategies that cannot be represented by linear compliance, such as in stiffness and damping control.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Nonlinear mechanical responses of mouse cochlear hair bundles.

The stiffness of sensory hair bundles of both inner (IHC) and outer (OHC) hair cells was measured with calibrated silica fibres in mouse cochlear cultures to test the hypothesis that the mechanical properties of the hair bundle reflect processes underlying mechanotransduction. For OHCs, the displacement of the hair bundle relaxed with time constants of 6 ms for displacements which open transducer channels and 4 ms for displacements which close the channels. The corresponding values of the time constants for IHCs were 10 ms and 8 ms, respectively. A displacement-dependent change in the stiffness of the hair bundle was not observed when the bundle was displaced orthogonally to the direction of excitation. The stiffness of the hair bundle as a function of nanometre displacements from the resting position was remarkably nonlinear. The stiffness declined to a minimum from the resting stiffness by about 12% for OHCs and 20% for IHCs when the hair bundle was displaced by about 20 nm in the excitatory direction, and it increased by a similar amount when the bundle was displaced by 20 nm in the inhibitory direction. The displacement at which the stiffness reached a minimum was within the most sensitive region of the hair-cell transducer function (receptor potential as a function of hair-bundle displacement), and the displacement at which the stiffness reached a maximum was at the point of saturation of the transducer function in the inhibitory direction. The nonlinear displacement-dependent compliance change is reversibly abolished, and the time constant of relaxation of the bundle for excitatory displacements is reversibly reduced, when mechanotransduction is blocked by the addition of either neomycin sulphate or cobalt chloride to the solution bathing the hair cells. The displacement-dependent compliance change was not apparently reduced when the receptor potential was attenuated through the substitution of sodium in the bathing solution with a less permeant cation, tetraethylammonium. These findings suggest that the nonlinear mechanical properties of the hair bundle are associated with aspects of the hair-cell mechanotransducer process. The mechanical properties of the hair bundle are discussed in relation to the 'gating-spring' hypothesis of hair-cell transduction.

Modal Analysis of Multistage Gear Systems Coupled With Gearbox Vibrations

This paper presents an analytical procedure to simulate vibrations in gear transmission systems. This procedure couples the dynamics of the rotor-bearing gear system with the vibration in the gearbox structure. The modal synthesis method is used in solving the overall dynamics of the system, and a variable time-stepping integration scheme is used in evaluating the global transient vibration of the system. Locally each gear stage is modelled as a multimass rotor-bearing system using a discrete model. The modal characteristics are calculated using the matrix-transfer technique. The gearbox structure is represented by a finite element model, and modal parameters are solved by using NASTRAN. The rotor-gear stages are coupled through nonlinear compliance in the gear mesh while the gearbox structure is coupled through the bearing supports of the rotor system. Transient and steady state vibrations of the coupled system are examined in both time and frequency domains. A typical three-geared system is used as an example for demonstration of the developed procedure.

Temporal relationship between left ventricular and arterial system elastances

Arterial compliance is an important component of ventricular afterload. Although its pressure dependence has been recognized, its temporal relationship to ventricular elastance (Elv(t)) has not been established. We investigated this in five open chest anesthetized dogs where simultaneous aortic pressure and flow and left ventricular pressure were measured. Elv(t) was derived using an elastance-resistance model of the left ventricle assuming an ejection fraction of 0.50 and a dead volume (Vd) of 3.0 mL. The nonlinear pressure-dependent compliance (C(P)) of the arterial system was incorporated in a three-element Windkessel model and determined by accurate prediction of aortic pressure from aortic flow. The resulting arterial elastance (Eas(t)) was computed as Eas(t) = 1/C(P). Results show that Eas(t) reaches a minimum value at or near the start of ventricular ejection and attains its peak value at or near the same time maximum LV elastance (Emax) is reached, at end-systole. Finally, numerical simulation of the model demonstrates its ability to adequately reproduce measured pressure and flow. Thus, the arterial system, in terms of elastance, is dynamically and temporally coupled to the left-ventricle during ejection.

Normal vibrations and friction at a Hertzian contact under random excitation: Theory and experiments

Non-linear stochastic contact vibrations at a Hertzian contact are studied using the Fokker-Planck equation. The vibrations are excited either externally to the contact region by a white Gaussian random normal load, or within the contact region by a rough surface input. The statistics of the stationary response are obtained for each case. The parametric dependence of the normal motions and contact area on various contact parameters is investigated. Due to the non-linear Hertzian compliance, the mean normal contact compression under dynamic excitation is smaller than the static deflection in the absence of vibrations. One also finds a reduction in the average area of contact and, by implication, in the mean friction force. It is found that for a 5% probability of contact loss, a reduction in the mean friction force of approximately 9% is expected. Average friction measurements taken during continuous sliding are in agreement with the analysis. Changes in the probability density function of the friction force with sliding, as predicted by the analysis, are also shown to be in good agreement with measured friction data. The results are consistent with the adhesion theory of friction in both an instantaneous and an average sense; even in the presence of substantial fluctuations in the normal contact force.

Diastolic mechanics and the origin of the third heart sound

The third heart sound (S3) is observed for various hemodynamic conditions in both the normal and diseased heart. A theory is proposed in which myocardial viscoelasticity is primarily responsible for S3. A mathematical model is developed based on the mechanical aspects of diastolic function: nonlinear elasticity, viscoelasticity, and pressure generation. The model is provided as an electrical analogy of the left ventricle and circulatory system. S3 is predicted for the normal heart and the heart with dilated cardiomyopathy. An elevation of S3 intensity is indicated for cardiomyopathy, as is often observed in the clinic. S3 is produced experimentally by volume loading of the open-chest canine preparation and mathematically by imposing the conditions of volume loading on the model. Consistency of theory and experiment imply that it is valid to attribute S3 to myocardial viscoelasticity. The animal whose heart possessed the largest constant of viscoelasticity produced the greatest level of S3, in both cases. Nonlinear ventricular compliance is not found to be an essential requirement for sound generation, although increased compliance led to an increase in sound. S3 is predicted to change in response to venous return, ventricular stiffness, contractility, heart rate, and duration of contraction, as observed by others. In general, the coupling of these quantities to S3 is explained in terms of an excitation of viscous properties of the ventricle.

Influence of forced vibrations on the static coefficient of friction — numerical modeling

Various experimental observations reveal that forced vibrations of a sliding system result in a reduction of the static coefficient of friction. In this paper this phenomenon is modeled numerically. A recently developed model of frictional interface, assuming the existence of non-linear normal compliance of the interface, is applied to the transient analysis of vibrations of the system. This analysis reveals that in the presence of vibrations the sliding occurs at a value of tangential force smaller than in the purely static case. However, the tangential motion of the slider consists of microscopic sticks and slips, which in the macroscopic scale are perceived as a “creep”-type motion. It was also observed that the presence of interface damping weakens the effect of reduction of the coefficient of friction, especially in the vicinity of a resonance zone. The formulation and numerical algorithm developed in this paper are applied to the analysis of a representative experimental apparatus. A good qualitative and quantitative agreement of results is obtained.

A nonlinear model of the arterial system incorporating a pressure-dependent compliance

The three-element modified Windkessel model has been widely used to study the characteristics of the systemic arterial system. This model provides most of the features of the systemic input impedance, but does not describe the nonlinear effect of the pressure dependence of arterial compliance. The current investigation examines the hemodynamic consequences of such an inclusion. Simultaneous aortic pressure and flow during control and brief descending aortic occlusion were measured in open chest anesthetized experimental dogs. A numerical procedure was implemented to compute constant compliance linear and nonlinear compliance model-predicted pressure waveforms with flow as the input. Results show that the nonlinear compliance model in general can more accurately predict the measured pressure waveforms during control and during acute pressure loading. The difference between the predicted waveforms is more pronounced when blood pressure is high and when the pulse pressure is large.