Hookean Springs Research Articles

Constitutive modelling and wave propagation through a class of anisotropic elastic metamaterials with local rotation

Elastic metamaterials are typically periodic materials possessing unit cells endowed with engineered architecture much smaller than the typical phenomenological length scale. The development of continuum models capable of accurately representing the effects of this aforementioned architecture is extremely challenging, and hence a sparsely developed area. This paper develops a novel 2-D continuum model capable of capturing the dynamic behaviour of a class of anisotropic elastic metamaterials with local rotational elements in the long wavelength limit. A constitutive relation incorporating these local rotational effects is proposed, and ratified using a representative discrete model using linear Hookean springs and identical rigid disks. The new continuum model is used to generate a dispersion relation for harmonic plane waves propagating in an arbitrary direction, which is subsequently compared to the dispersion behaviour of the original discrete model. The general behaviour of this continuum when subjected to 2-D planar harmonic wave propagation in the anisotropic medium is then analysed, with specific attention given to the effect of material anisotropy and wave propagation direction. This work is the first of its kind to create a new continuum model of a class of anisotropic elastic metamaterials with local rotational effects.

Effects of Contact Surface Shape on Dynamic Lifetime and Strength of Molecular Bond Clusters under Displacement- and Force-Controlled Loading Conditions.

Many experimental and theoretical studies have shown that the mechanical properties of cells and the extracellular matrix can significantly affect the lifetime and strength of the adhesion clusters of molecular bonds. However, there are few studies on how the shape of the contact surface affects the lifetime and strength of the adhesion clusters of molecular bonds, especially theoretical studies in this area. An idealized model of focal adhesion is adopted, in which two rigid media are bonded together by an array of receptor-ligand bonds modeled as Hookean springs on a complex surface topography, which is described by three parameters: the surface shape factor β, the length of a single identical surface shape L, and the amplitude of surface shapes w. In this study, systematic Monte Carlo simulations of this model are conducted to study the lifetime of the molecular bond cluster under linear incremental force loading and the strength of the molecular bond cluster under linear incremental displacement loading. We find that both small surface shape amplitudes and large surface shape factors will increase the lifetime and strength of the adhesion cluster, whereas the length of a single surface shape causes oscillations in the lifetime and strength of the cluster, and this oscillation amplitude is affected by the surface shape amplitude and the factor. At the same time, we also find that the pretension in the cluster will play a dominant role in the adhesion strength under large amplitudes and small factors of surface shapes. The physical mechanisms behind these phenomena are that the changes of the length of a single surface shape, the amplitude of surface shapes, and the surface shape factor cause the changes of stress concentration in the adhesion region, bond affinity, and the number of similar affinity bonds.

Measurement of Forces Acting on Single T-Cell Receptors.

Molecular forces are increasingly recognized as an important parameter to understand cellular signaling processes. In the recent years, evidence accumulated that also T-cells exert tensile forces via their T-cell receptor during the antigen recognition process. To measure such intercellular pulling forces, one can make use of the elastic properties of spider silk peptides, which act similar to Hookean springs: increased strain corresponds to increased stress applied to the peptide. Combined with Förster resonance energy transfer (FRET) to read out the strain, such peptides represent powerful and versatile nanoscopic force sensing tools. In this paper, we provide a detailed protocol how to synthesize a molecular force sensor for application in T-cell antigen recognition and hands-on guidelines on experiments and analysis of obtained single molecule FRET data.

A mechanically-derived contact model for adhesive elastic-perfectly plastic particles, Part I: Utilizing the method of dimensionality reduction

In this two part series (Zunker and Kamrin, 2024), we present a contact model able to capture the response of interacting adhesive elastic-perfectly plastic particles under a variety of loadings. In Part I, we focus on elastic through fully-plastic contact with and without adhesion. For these contact regimes the model is built upon the method of dimensionality reduction which allows the problem of a 3D axisymmetric contact to be mapped to a corresponding problem of a 1D rigid indenter penetrating a bed of independent Hookean springs. Plasticity is accounted for by continuously varying the 1D indenter profile subject to a constraint on the contact pressure. Unloading falls out naturally, and simply requires lifting the 1D indenter out of the springs and tracking the force. By accounting for the incompressible nature of this plastic deformation, the contact model is able to capture multi-neighbor dependent effects such as increased force and formation of new contacts. JKR type adhesion is recovered seamlessly within the method of dimensionality reduction by simply allowing the springs to ‘stick’ to the 1D indenter’s surface. Because of the mechanics-focused formulation of the contact model, only a few physical inputs describing the interacting particles are needed: particle radius, Young’s modulus, Poisson ratio, yield stress, and effective surface energy. The contact model is validated against finite element simulations and analytic theory—including Hertz’s contact law and the JKR theory of adhesion. These comparisons show that the proposed contact model is able to accurately capture plastic displacement, average contact pressure, contact area, and force as a function of displacement for contacts as well as particle volume within the elastic to fully-plastic regimes.

Scott Blair Fractional-Type Viscoelastic Behavior of Thermoplastic Polyurethane.

In this paper, the experimental characterization of the viscoelastic properties of thermoplastic polyurethane (TPU) samples through creep experiments is presented. Experiments were conducted at different constant temperature levels (15, 25, and 35 ∘C), for three different tensile stress levels (0.3, 0.5, and 0.7 MPa), and at different physisorbed water contents, providing access to: (i) the temperature dependency of creep parameters and (ii) the assessment, if behavior is indeed viscoelastic. The physisorbed water content was achieved by exposing virgin samples to environments with relative humidity ranging from 0 to 80 percent until mass stability was reached. Creep tests were conducted immediately afterwards with this particular humidity level. The main results of this study are as follows. The temperature dependency of the obtained creep parameters is well described in Arrhenius plots. With regard to water content, two prototype material responses were observed in the experimental program and accurately modeled using the following fractional-type models: (i) Scott Blair-type (i.e., power-law-type) only behavior, pronounced for the combination of low water content/low temperature; (ii) combined Scott Blair plus Lomnitz (i.e., log-type) behavior for high water content/high temperature. This change in behavior associated with certain thresholds for the specified environmental conditions (temperature and relative humidity) may indicate the initiation of hydrogen bond breakage and rearrangement (carbamate H-bonds and physisorbed water H-bonds). Regarding the short-term or quasi-instantaneous behavior, the Scott Blair element seems highly appropriate and may be better suited than the standard elastic model: the Hookean spring. We associated Scott Blair behavior with the load-induced, quasi-instantaneous re-arrangement of polymer network chains. The secondary viscoelastic mechanism associated with the Lomnitz element, hydrogen bond breakage and rearrangement, comes into play for higher temperatures and/or higher physisorbed water contents. In this case, the contribution of the two constitutive elements is well separated due to the large number of the characteristic time of the Lomnitz element, much larger than the respective value for the Scott Blair element.

Dynamic Ramsey Theory of Mechanical Systems Forming a Complete Graph and Vibrations of Cyclic Compounds

Ramsey theory constitutes the dynamics of mechanical systems, which may be described as abstract complete graphs. We address a mechanical system which is completely interconnected by two kinds of ideal Hookean springs. The suggested system mechanically corresponds to cyclic molecules, in which functional groups are interconnected by two kinds of chemical bonds, represented mechanically with two springs k1 and k2. In this paper, we consider a cyclic system (molecule) built of six equal masses m and two kinds of springs. We pose the following question: what is the minimal number of masses in such a system in which three masses are constrained to be connected cyclically with spring k1 or three masses are constrained to be connected cyclically with spring k2? The answer to this question is supplied by the Ramsey theory, formally stated as follows: what is the minimal number R(3,3)? The result emerging from the Ramsey theory is R(3,3)=6. Thus, in the aforementioned interconnected mechanical system at least one triangle, built of masses and springs, must be present. This prediction constitutes the vibrational spectrum of the system. Thus, the Ramsey theory and symmetry considerations supply the selection rules for the vibrational spectra of the cyclic molecules. A symmetrical system built of six vibrating entities is addressed. The Ramsey approach works for 2D and 3D molecules, which may be described as abstract complete graphs. The extension of the proposed Ramsey approach to the systems, partially connected by ideal springs, viscoelastic systems and systems in which elasticity is of an entropic nature is discussed. “Multi-color systems” built of three kinds of ideal springs are addressed. The notion of the inverse Ramsey network is introduced and analyzed.

New perspective on the creep characteristic of fiber–dependent shape memory polymers: variable–order fractional constitutive model

With the aim of describing and explaining the creep tensile property of shape memory polymers (SMPs) better, a variable--order fractional creep constitutive model with fewer parameters and good simulation performance was constructed on the basis of existing constitutive models. The consists of a Hookean spring connected in series with a Newtonian dashpot was employed to propose a new variable--order fractional Maxwell model for SMPs. Drawing support from the Laplace transform, we derive the analytical solution of this novel model. The verification of the obtained theoretical results with the creep test data for different fiber weight fractions and stress levels available from the literature reveals their satisfactory agreement. It can be concluded that the developed variable--order fractional constitutive model can better describe the complex viscoelastic relationship of shape memory polymers which is difficult to be described by the classical integer order derivative, and obtain a more accurate constitutive model of shape memory polymers with fewer parameters.

A Systematical Rheological Study of Maize Kernel.

In this study, the rheological behavior of maize kernel was systematically investigated using a dynamic mechanical analyzer. The loss in toughness caused by drying resulted in a downward shift in the relaxation curve and an upward shift in the creep curve. The long relaxation behavior became obvious when the temperature was above 45 °C, resulting from the weakening of hydrogen bonds with temperature. The maize kernel relaxed more rapidly at high temperatures, caused by a reduction in the cell wall viscosity and polysaccharide tangles. The Deborah numbers were all much smaller than one, suggesting that the Maxwell elements showed viscous behavior. Maize kernel, as a viscoelastic material, showed a dominant viscous property at high temperatures. The decline in β with increasing drying temperature indicated an increase in the width of the relaxation spectrum. A Hookean spring elastic portion made up the majority of the maize kernel creep strain. The order-disorder transformation zone of maize kernel was about 50-60 °C. Due to the complexity of maize kernel, the William-Landel-Ferry constants differed from the universal values; these constants should be ascertained through experiments. Time-temperature superposition was successfully used to describe the rheological behavior. The results show that maize kernel is a thermorheologically simple material. The data acquired in this study can be used for maize processing and storage.

Helical Molecular Springs with Varying Spring Constants

AbstractWe report a general synthetic route toward helical ladder polymers with varying spring constants, built with chirality‐assisted synthesis (CAS). Under tension and compression, these shape‐persistent structures do not unfold, but rather stretch and compress akin classical Hookean springs. Our synthesis is adaptable to helices with different pitch and diameter, which allowed us to investigate how molecular flexibility in solution depends on the exact geometry of the ladder polymers. Specifically, we showed with molecular dynamic simulations and by measuring the longitudinal 1H NMR relaxation times (T1) for our polymers at different Larmor frequencies, that increasing the helix diameter leads to increased flexibility. Our results present initial design rules for tuning the mechanical properties of intrinsically helical ladder polymers in solution, which will help inspire a new class of robust, spring‐like molecular materials with varying mechanical properties.

Helical Molecular Springs with Varying Spring Constants.

We report a general synthetic route toward helical ladder polymers with varying spring constants, built with chirality-assisted synthesis (CAS). Under tension and compression, these shape-persistent structures do not unfold, but rather stretch and compress akin classical Hookean springs. Our synthesis is adaptable to helices with different pitch and diameter, which allowed us to investigate how molecular flexibility in solution depends on the exact geometry of the ladder polymers. Specifically, we showed with molecular dynamic simulations and by measuring the longitudinal 1 H NMR relaxation times (T1 ) for our polymers at different Larmor frequencies, that increasing the helix diameter leads to increased flexibility. Our results present initial design rules for tuning the mechanical properties of intrinsically helical ladder polymers in solution, which will help inspire a new class of robust, spring-like molecular materials with varying mechanical properties.

Simulation and Experimental Verification of Dynamic Characteristics on Gas Foil Thrust Bearings Based on Multi-Physics Three-Dimensional Computer Aided Engineering Methods

This paper presents a method to simulate the dynamic operating characteristics of a gas foil thrust bearing based on linear elastic support and constant ambient temperature to mimic the transient structure–fluid interactions. In the physical model, the top and bump foils are simply represented by an infinite number of Hookean springs attached to a solid wall with a small amount of deformation, whereas the gas film in the bearing is under quasi-steady lubrication flow conditions with hydrodynamic pressure distributed on the little-deformed top foil. A three-dimensional multi-physics model in a cylindrical coordinate system is established via a commercial computer-aided engineering software package to predict the nominal dynamic characteristics of the gas foil thrust bearing. To verify the multi-physics model, an experimental bench was built in-house to measure the thrust force on the support of the bearing. With the pertinent bearing parameters being entered into the package, the simulations agree well with the experimental thrust forces. As a further step, a simulation model of a clamped-rotor gas foil thrust bearing design was thoroughly investigated under nominal operating conditions, resulting in predictions of underdamped oscillations in rotor motions. The phenomenon could be described using a linear mass–spring–damper model that is dependent on the gas film thickness. The stiffness and damping coefficients could serve as a base reference for rotor dynamics analysis. This concludes the potential development of a digital twin for gas foil thrust bearing systems.

Cooperative Dynamics in the Fiber Bundle Model

We discuss the cooperative failure dynamics in the fiber bundle model where the individual elements or fibers are Hookean springs that have identical spring constants but different breaking strengths. When the bundle is stressed or strained, especially in the equal-load-sharing scheme, the load supported by the failed fiber gets shared equally by the rest of the surviving fibers. This mean-field-type statistical feature (absence of fluctuations) in the load-sharing mechanism helped major analytical developments in the study of breaking dynamics in the model and precise comparisons with simulation results. We intend to present a brief review on these developments.

Rheological studies of SBS/EVA blends modified with bio-based cashew nut shell liquid

Research on bio-plasticizers is a topic of strategic interest in polymer blends. A bio-plasticizer, cashew nut shell liquid (CNSL), was studied in blends of ethylene-vinyl acetate copolymer (EVA) and styrene-butadiene-styrene copolymer (SBS). In the literature does not report the addition of plasticizers to SBS/EVA blend. Statistical analyses showed that there was a significant difference in mechanical properties (tension at break, hardness and elongation at break) vs. the unplasticized blend. The minimum CNSL concentration required for a statistical difference was 10 phr. The Carreau-Yasuda rheological model was used to obtain rheological parameters in these blends. The plasticizing influence of CNSL was confirmed by rheology. The effects of CNSL on creep and recovery were evaluated for the SBS/EBA blends. Burger´s model explained well SBS/EVA creep compliance. Moreover, its parameters (Newtonian dashpots and Hookean springs) were evaluated as a function of the CNSL concentrations. The bio-plasticizer concentration influenced significant correlations among the rheological creep-recovery tests, thus enabling a considerable increase in the elastic phase. Experimental creep-recovery data and curve fit were in good agreement.

Modeling and analysing of spring-loaded double parallelogram mechanism using moment balance

Double Parallelogram Lifting Mechanism (DPLM) is a multi-linkage mechanism adopted as the lifting mechanism of the designed robot in this paper. DPLM is designed based on Double Parallelogram Mechanism (DPM), making it a compact and stable multi-linkage mechanism with a low retracted height and a great working height. This paper investigates the condition in which DPLM obtains static equilibrium at any height using Hookean springs and the minimization of the motor’s required torque exertion against the gravitational moment of the weight of the mechanism. A mathematical model describing all the factors involved in moment balancing of DPLM is derived and established in MATLAB and is verified through experiments with a virtual prototype conducted in SOLIDWORKS. The obtained results show that the magnitude of the moment exerted by the tension of a spring can be determined not only by its exact installation positions but also by its D value, which is the difference in the distances between the two installation points of a spring and their respective rotation axes. An optimization method based on minimizing the RMSE of the unbalanced moment for DPLM is introduced.

Dynamics of undulatory fluctuations of semiflexible filaments in a network.

We study the dynamics of a single semiflexible filament coupled to a Hookean spring at its boundary. The spring produces a fluctuating tensile force on the filament, the value of which depends on the filament's instantaneous end-to-end length. The spring thereby introduces a nonlinearity, which mixes the undulatory normal modes of the filament and changes their dynamics. We study these dynamics using the Martin-Siggia-Rose-Janssen-De Dominicis formalism, and compute the time-dependent correlation functions of transverse undulations and of the filament's end-to-end distance. The relaxational dynamics of the modes below a characteristic wavelength sqrt[κ/τ_{R}], set by the filament's bending modulus κ and spring-renormalized tension τ_{R}, are changed by the boundary spring. This occurs near the crossover frequency between tension- and bending-dominated modes of the system. The boundary spring can be used to represent the linear elastic compliance of the rest of the filament network to which the filament is cross linked. As a result, we predict that this nonlinear effect will be observable in the dynamical correlations of constituent filaments of networks and in the networks' collective shear response. The system's dynamic shear modulus is predicted to exhibit the well-known crossover with increasing frequency from ω^{1/2} to ω^{3/4}, but the inclusion of the network's compliance in the analysis of the individual filament dynamics shifts this transition to a higher frequency.

Viscometric functions and rheo-optical properties of dilute polymer solutions: Comparison of FENE-Fraenkel dumbbells with rodlike models

Rigid macromolecules or polymer chains with persistence length on the order of the contour length (or greater) have traditionally been modelled as rods or very stiff springs. The FENE-Fraenkel-spring dumbbell, which is finitely extensible about a non-zero natural length with tunable harmonic stiffness, is one such model which has previously been shown to reproduce bead–rod behaviour in the absence of hydrodynamic interactions. The force law for the FENE-Fraenkel spring reduces to the Hookean or FENE spring force law for appropriately chosen values of the spring parameters. It is consequently possible to explore the crossover region between the limits of bead–spring and bead–rod behaviour by varying the parameters suitably. In this study, using a semi-implicit predictor–corrector Brownian dynamics algorithm, the FENE-Fraenkel spring is shown to imitate a rod with hydrodynamic interactions when spring stiffness, extensibility and simulation timestep are chosen carefully. By relaxing the spring stiffness and extensibility, the FENE-Fraenkel spring can also reproduce spring-like behaviour, such as a crossover from −1∕3 to −2∕3 power-law scaling in the viscosity with shear rate, and a change from positive to negative second normal stress difference. Furthermore, comparisons with experimental data on the viscosity and linear dichroism of high aspect ratio, rigid macromolecules shows that the extensibility and stiffness of the FENE-Fraenkel spring allows for equal or improved accuracy in modelling inflexible molecules compared to rodlike models.

Continual Learning of Multiple Memories in Mechanical Networks

We contrast the distinct frameworks of materials design and physical learning in creating elastic networks with desired stable states. In design, the desired states are specified in advance and material parameters can be optimized on a computer with this knowledge. In learning, the material physically experiences the desired stable states in sequence, changing the material so as to stabilize each additional state. We show that while designed states are stable in networks of linear Hookean springs, sequential learning requires specific non-linear elasticity. We find that such non-linearity stabilizes states in which strain is zero in some springs and large in others, thus playing the role of Bayesian priors used in sparse statistical regression. Our model shows how specific material properties allow continuous learning of new functions through deployment of the material itself.

A slip-spring simulation model for predicting linear and nonlinear rheology of entangled wormlike micellar solutions

We extend the single-chain slip-spring model developed by Likhtman [Macromolecules 38, 6128 (2005)] to describe the dynamics and rheology of entangled polymers to wormlike micellar solutions by incorporating chain breakage and rejoining, which are the key additional dynamics present in wormlike micellar solutions. We show that the linear rheological properties obtained from this micelle slip-spring model are in good agreement with mesoscopic simulations using the “pointer algorithm” [W. Zou and R. G. Larson, J. Rheol. 58, 681 (2014)] and can be fit to experimental results after an adjustment to correct for the too-high flexibility of the micelles assumed in the slip-spring model. Finally, we use this model to predict the nonlinear rheological properties of entangled wormlike micelles, which are the first predictions that include the effects of entanglements, breakage and rejoining, Rouse modes, and stretch of bead-spring micellar chains with Hookean springs.

Zero-shear viscosity of Fraenkel dumbbell suspensions

Whereas rigid dumbbell suspensions predict, at least qualitatively, most of the viscoelastic material functions measured in the laboratory, Hookean dumbbells predict few of these. For instance, whereas rigid dumbbells predict a shear-thinning viscosity curve, as they should, Hookean dumbbells yield a constant for the steady shear viscosity. In this paper, we explore the addition of a Hookean spring to the end of a rigid rod, a dumbbell attributed to Fraenkel. In this way, we focus our attention on how macromolecular extensibility affects the configuration distribution in steady shear flow. We arrive at the exact solution to this configuration distribution in steady shear flow at low shear rate and then insert it into the Giesekus expression for the stress tensor to arrive at an exact solution for the zero-shear viscosity and the zero-shear values of the normal stress differences.

Use of loading devices with low stiffness may cause uncertainty in measuring the strength of cellular adhesion

ABSTRACT Cellular adhesion via the formation of molecular bond clusters is a fundamental biological phenomenon. Cell adhesion forces can be measured by various advanced techniques with adjustable probe stiffness. To quantitatively understand how the stiffness of a loading device may influence the measurement of these forces, we consider an idealized theoretical model of a cluster of molecular bonds that are subjected to an applied tensile load by a Hookean spring. In this model, the rebinding and rupture of individual molecular bonds are governed by stochastic master equations. By deriving an exact expression of the rebinding rate and directly solving the master equation, we find that the conventional moment method provides incorrect predictions for this problem. Moreover, the measured adhesion strength of a molecular bond cluster exhibits strong dependence on the stiffness of the loading device and the bond number when the stiffness value is close to or smaller than the effective stiffness of the bond cluster. Thus, an improper selection of the stiffness value of the loading device may lead to measurement uncertainty. Monte Carlo simulations are performed to verify the proposed analytical results.