Elastic Potential Research Articles

Molecular dynamics simulations of the monomer density profiles of knotted ring polymer chains confined in a slit of two parallel walls with one attractive and another repulsive surface.

We have used Molecular Dynamics simulations to obtain the monomer density profiles for real linear and ring polymer chains of 360 monomers length with different topological structures such as simple knots: 31, 61, 91, 10124, complex knots 313151 and twisted knots with n = 10 and n = 20 in a slit geometry of two parallel walls with one attractive and another repulsive surface. We have used Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) software to perform simulations with the Verlet integration algorithm. The interactions between monomers were simulated as Lennard-Jones 12-6 potential, for bonds we have used Finitely Extensible Nonlinear Elastic (FENE) potential and the interaction with the walls was taken into account via Lennard-Jones 9-3 potential. We observed that topologically complex polymers have lower monomer density profiles near the attractive wall, but at some distance in the direction to the repulsive wall this tendency changes to the opposite. We showed that most complex twisted knots have two maxima in narrow slits. In the wide slits we do not observe such relation for twisted knots at higher temperatures. These results are important for better understanding the nature of the depletion forces which arise in a slit geometry of two parallel walls with one attractive and one repulsive wall.

AN ALGEBRAIC DECOMPOSITION OF HEMITROPIC PSEUDOTENSORS IN N DIMENSIONS AND APPLICATIONS TO MICROPOLAR CONTINUUM THEORIES

The paper is devoted to problems related to algebraic decompositions and coordinate representations of tensors with constant components, hemitropic tensors and pseudotensors. Such objects of tensor algebra are interesting from the micropolar continuum mechanics viewpoint. Algebraic properties and coordinate representations of tensors and pseudotensors with constant components are discussed. The base examples of tensors with constant components usable in continuum mechanics are given. An algebraic algorithm for coordinate representations of tensors and pseudotensors with constant components proposed by prof. G.B. Gurevich is highlighted and employed.The notions of fully isotropic, conventionally isotropic, unconventionally isotropic, semi-isotropic (demitropic, hemitropic, chiral) fourth rank tensors and pseudotensors are proposed. The coordinate representations of a Cartesian hemitropic fourth rank tensor in three-dimensional space are obtained in terms of the Kronecker delta and metric tensor products. Based on an unconventional definition of a hemitropic fourth rank tensor, general coordinate representations in dimensions in terms of the Kronecker deltas and metric tensors are given. A comparison of an arbitrary hemitropic fourth rank tensor and a tensor with constant components is carried out. A general form of the elastic potential of a linear anisotropic micropolar elastic continuum is obtained by the pseudotensor technique. Obtained anisotropic micropolar potential is reduced to a hemitropic one by proposed coordinate representations of fourth rank tensors. Coordinate representations for constitutive tensors and pseudotensors usable in mathematical modeling of linear hemitropic micropolarelastic continua are obtained as a modification of pseudotensors with constant components in three-dimensional space.

Exact large strain analysis for the Poynting effect of freely twisted thin-walled tubes made of highly elastic soft materials

Toward handling nonlinear coupling effects at large elastic deformations, a decoupled form of the multi-axial elastic potential for highly elastic soft materials is first constructed based on Hencky’s logarithmic strain, and this new potential is then used to obtain analytic results for the uniaxial extension and compression of a bar, the equi-biaxial extension of a plate, the plane-strain extension of a strip and, in particular, the large torsion of a thin-walled tube with free ends. Novelties are included in the following three respects: (a) Experimental data for responses of the uniaxial extension, equi-biaxial extension (uniaxial compression) and plane-strain extension (simple shear) may be accurately matched up to large strain with strain-stiffening effects; (b) exact responses for the Poynting effect of freely twisted thin-walled tubes may be derived for changes in the axial length and the wall thickness as well as the averaged radius; and (c) the coupling nonlinear complexities involved in all such responses may be worked out in a decoupled sense.

Performance of a shock isolator inspired by skeletal muscles

Nonlinear vibration isolators with low dynamic stiffness have advantages compared to linear isolators, but they also have limitations in terms of the maximum allowable displacement around a chosen equilibrium position. To overcome this issue, a novel nonlinear isolator is proposed that outperforms the classical quasi-zero stiffness nonlinear isolator when large inputs are applied. In this paper, the response of the isolator when subject to shock excitation is investigated. The isolator suspension is designed to exhibit a quasi-static force-deflection curve with sigmoidal shape, which has been observed in many studies reporting the characteristics of skeletal muscles. The shape of the force-deflection curve is such that the suspension system can store greater elastic potential energy. Simulations show that the proposed four-spring system outperforms the classical nonlinear three-spring configuration, in terms of maximum displacement of a suspended mass, when the shock amplitude is relatively large. This is achieved by exploiting the softening effect for large deformations.

ON THE CONVEXITY OF POTENTIAL OF NONLINEAR ELASTIC MEDIA MODEL WITH TENSOR DAMAGE PARAMETER

For solving problems of the theory of elasticity using nonlinear models, the question of the convexity of the potential used and the proof of the uniqueness of the solution are of decisive importance. The present work is devoted to the determination of conditions for the local strict convexity of the potential for the nonlinear elasticity model, which ensure the uniqueness of the solution to the problem in a sufficiently small neighborhood of the desired solution. The considered model is a generalization of a scalar nonlinear rheological model of brittle solid deformation for the case of the tensor damage parameter, the principal values of which describe the reduction in the cross-sectional area of the material in three orthogonal directions. An additional term of the second order in deformation in the elastic potential makes it possible to describe the dependence of the elastic moduli on the type of the stress-strain state, the dilatancy of the material under shear deformation, as well as the nonlinear deformation response even at low loads. The introduced damage tensor of the second rank makes it possible to describe the damage-induced anisotropy of the elastic properties of the material. Conditions of local strict convexity in the principal axes of the strain tensor are obtained in this work for the general case of misaligned strain and damage tensors. To illustrate the obtained convexity conditions, two special cases of the damage tensor type are considered: a transversely isotropic fractured medium with coaxial strain and damage tensors, and a transversely isotropic medium with obliquely oriented fracturing. For both cases, the dependence of the limiting values of damage on the parameter of the degree of anisotropy is shown. It is shown that in the case of weak damage anisotropy, the potential convexity conditions for the scalar damage parameter give a minorant estimate of the maximum allowable damage for various types of stress-strain state. For obliquely oriented fracturing, the dependences of the maximum permissible damage on the degree of anisotropy, the angle of inclination and the type of stress-strain state are plotted.

Elastic potentials as yield surfaces for isotropic materials.

This paper proposes that elastic potentials, which may be rigorously formulated using the negative Gibbs free energy or the complementary strain energy density, may be used as the yield surface of elasto-plastic constitutive models. Thus, the yield surface may be assumed in some materials as an elastic potential surface for a specific level of critical complementary strain energy density. Traditional approaches, such as the total strain energy criterion, only consider second order terms, i.e., the elastic potential is centred at the origin of the current stress state. Here, first order terms are considered, and consequently, the elastic potential may be translated, which allows to reproduce the desired level of tension-compression asymmetry. The proposed approach only adds two additional parameters, e.g., uniaxial compressive and tensile yield limits, to the elastic ones. For linear elasticity, the proposed approach provides elliptical yield surfaces and shows a correlation between the shape of the ellipse and the Poisson’s ratio, which agree with published experimental data for soils and metallic glasses. This elliptical yield surface also fits well experimental values of amorphous polymers and some rocks. Besides, the proposed approach automatically considers the influence of the intermediate stress. For non-linear elasticity, a wider range of elastic potentials, i.e., yield surfaces, are possible, such as distorted ellipsoids. For the case of incompressible non-linear materials, the yield surfaces are between von Mises and Tresca ones.

Numerical solution of the two-dimensional Steigmann–Ogden model of material surface with a boundary

In this paper, we consider an infinite isotropic elastic matrix that contains a material surface described by the Steigmann–Ogden theory, where the material surface possesses a boundary. In two-dimensional setting, the surface represents a material curve with endpoints. The presence of the surface tension introduces non-linearity in the problem since a superposition with respect to the far-field load is not allowed. We review the governing equations, the jump conditions, and the conditions at the surface tips. For solving the problem in the plane strain setting, the displacements in the matrix are sought in the complex variables form of a single layer elastic potential. The density of the potential represents the jump in complex tractions across the surface. Exact expressions for the elastic fields in the matrix are provided in terms of complex integral representations. The problem of the surface located along a straight segment is considered in detail. Its solution is reduced to that of the system of boundary integral equations using the approximations of the boundary data that involve the series of Chebyshev’s polynomials. We present numerical examples to illustrate the influence of dimensionless parameters on the solution.

Excluded Volume Effect on the Extensional Rheology of Carbon Nanotubes: A Mesoscopic Theory

We present a mesoscopic theory of the extensional rheology of liquid crystalline carbon nanotubes (CNTs). We implement the mesoscopic Leslie–Ericksen theory, considering the repulsive excluded volume potential acting in the direction opposite to the Frank elastic potential. Given the volume fraction, diameter, and length of the CNTs, our model predicts the extensional viscosity. The liquid crystalline phase of the CNTs exhibited extensional thinning behavior. In our theory, the rotational relaxation time is proportional to the reciprocal excluded volume, particularly for long CNTs with waviness. Our model also predicts the minimum draw ratio, below which the nanotube orientation cannot reach the target orientation. The existence of a minimum draw ratio is due to the large excluded volume potential of the long CNTs.

Local postbuckling of omega-stringer-stiffened composite panels

In order to utilize the full lightweight potential of thin-walled stringer-stiffened composite panels, a new closed-form analytical solution for the buckling and postbuckling of omega-stringer-stiffened composite panels is introduced. Because of the high computational efficiency, it allows the analysis of the early postbuckling behaviour in the framework of preliminary design. The solution is derived based on the principle of the minimum of the total elastic potential. A suitable set of buckling shape functions models the buckling pattern and imperfection. The load is applied via prescribed displacements. The results of the closed-form model are compared and verified with finite element simulations and the implementation of another closed-form model found in the literature. An exemplary cross-ply laminated panel under uniaxial compression is investigated. The results show that the proposed model achieves excellent general agreement in the practically most relevant design space.

Experimental Verification of Theoretical Stress-Strain Model for Compressed Concrete Considering Post-Peak Stage.

The theoretical stress-strain model for compressed composite cement materials’ behavior without empirical coefficients was proposed by Iskhakov in 2018. This model includes the following main parameters describing concrete behavior: stresses and strains corresponding to the border between the elastic and non-elastic behavior stages of a concrete specimen, ultimate elastic strains, and stresses and strains at the end of the post-peak region. Particular attention is focused on the descending branch of the stress-strain diagram, as well as on the analysis of concrete elastic and plastic potentials. These potentials are important for assessing the dynamic response of the concrete element section, as well as for concrete creep analysis. The present research is aimed at experimental verification of the above-mentioned theoretical model. The obtained experimental results are in good agreement with the theoretical ones, which confirms the model’s accuracy and enables a significant reduction in the empirical coefficients number in compressed reinforced concrete elements design. This, in turn, represents the scientific novelty of this study.

The Ogden and the extended tube model as backbone in describing electroactive polymers: advancements in modelling nonlinear behaviour and fracture.

Hyperelastic constitutive relations form the basis of advanced models for novel materials. Such elastic deformation potentials are the backbone for complex material formulations at elastic and inelastic deformations, especially when embedded into powerful frameworks like generalized standard materials, as well as multiphysical and multiscale formulations. With the focus on electroactive polymers, the article at hand demonstrates the derivation of a variational, rate-dependent electromechanical model for quasi-incompressible polymers and the derivation of an electromechanical model for regularized fracture mechanics by means of the phase-field method. Starting at the prominent Ogden and the extended tube model, some developments from the last decades are revisited and presented via the principle of virtual power, for instance, the established mixed element formulation, nonlinear viscoelasticity and electromechanical coupling. An electromechanically fully coupled representative crack element is used to derive a novel phase-field model for fracture. A key property of the proposed model is the ability to describe the electrical free-space behaviour inside the crack gap, which is demonstrated by adopting three common crack-face conditions. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'.

Energy extraction via Vortex-Induced Vibrations: The effect of spring bistability

We study Vortex-Induced Vibration (VIV) of an elastically mounted cylinder placed in uniform flow using numerical simulations based on Immersed Boundary Method and theoretical analysis. We characterize the effect of bistability in elastic potential on energy harvesting from fluid flow, in which the energy generator is modeled as structural damping. A non-dimensional distance of 2β separates the two minima of the bistable spring potential. An increase in the peak energy extraction efficiency and widening of lock-in range is observed for VIV with bistable springs compared to VIV with linear springs, especially for low values of β. We report peak efficiency of 10% for VIV with bistable springs, whereas the peak efficiency for VIV with linear spring is 4%. These results can be explained by the presence of an “Equilibrium Constraint (EC)”, between oscillation amplitude and structure frequency, obtained by balancing the rate of energy production and the rate of energy dissipation due to either structural damping or electrical generators. The EC-based theory predicts that the maximum displacement amplitude depends on damping parameter b∗=2π2m∗ξ/Ur as suggested by Klamo et al. (2005), where m∗ is the solid-to-fluid mass ratio, ξ is the damping ratio and Ur is the reduced velocity of the incident fluid flow. We briefly discuss the wake structures for bistable springs for different β and b∗. We also derive a theoretical expression for oscillation amplitude and energy extraction efficiency for VIV with bistable springs, which agrees well with data from high fidelity numerical simulations, carried out in the present work.

Defect absorption and emission for p-atic liquid crystals on cones.

We investigate the ground-state configurations of two-dimensional liquid crystals with p-fold rotational symmetry (p-atics) on fixed curved surfaces. We focus on the intrinsic geometry and show that isothermal coordinates are particularly convenient as they explicitly encode a geometric contribution to the elastic potential. In the special case of a cone with half-angle β, the apex develops an effective topological charge of -χ, where 2πχ=2π(1-sinβ) is the deficit angle of the cone, and a topological defect of charge σ behaves as if it had an effective topological charge Q_{eff}=(σ-σ^{2}/2) when interacting with the apex. The effective charge of the apex leads to defect absorption and emission at the cone apex as the deficit angle of the cone is varied. For total topological defect charge 1, e.g., imposed by tangential boundary conditions at the edge, we find that for a disk the ground-state configuration consists of p defects each of charge +1/p lying equally spaced on a concentric ring of radius d=(p-1/3p-1)^{1/2p}R, where R is the radius of the disk. In the case of a cone with tangential boundary conditions at the base, we find three types of ground-state configurations as a function of cone angle: (i) for sharp cones, all of the +1/p defects are absorbed by the apex; (ii) at intermediate cone angles, some of the +1/p defects are absorbed by the apex and the rest lie equally spaced along a concentric ring on the flank; and (iii) for nearly flat cones, all of the +1/p defects lie equally spaced along a concentric ring on the flank. Here the defect positions and the absorption transitions depend intricately on p and the deficit angle, which we analytically compute. We check these results with numerical simulations for a set of commensurate cone angles and find excellent agreement.

Prediction Model of Elastic Load Resource Regulation Potential under Hierarchical and Partitioned Dynamic Control Architecture

In order to deal with the problem of energy shortage, demand side management has attracted more and more attention. Taking demand side resources as an example to understand the regulation potential of demand side resources, this paper designs a hierarchical and partitioned dynamic regulation architecture for demand side adjustable resources, which can carry out the interconnection between multi-level subjects and efficient information interaction. The regulation potential prediction model of elastic load is established through the hierarchical and partitioned architecture, and the credible and adjustable potential of resident users under two regulation scenarios is obtained through calculation: participating in emergency load reduction and peak valley difference, which proves the effectiveness of the architecture.

Development of a Mucoadhesive Liquid Crystal System for the Administration of Rifampicin Applicable in Tuberculosis Therapy.

Since 1966, rifampicin (RIF) has been considered one of the most potent drugs in the treatment of tuberculosis (TB), which is caused by infection with M. tuberculosis (Mtb). New nanostructured formulations for RIF delivery and alternative routes of administration have been studied as potential forms of treatment. This study evaluates a liquid crystal system for RIF delivery, using alternative drug delivery routes. The systems developed are composed of surfactant, oleylamine, and soy phosphatidylcholine. With the aid of polarized light microscopy, it was possible to determine that the developed systems had a hexagonal mesophase. All systems developed showed non-Newtonian pseudoplasticity and a high degree of thixotropy. Liquid crystal systems with RIF showed an increase in elastic potential, indicating greater mu-coadhesiveness. The evaluation of mucoadhesive forces revealed an increase in the mucoadhesive potential in the presence of mucus, indicating the presence of satisfactory mucoadhesive forces. The 9DR and 10DR liquid crystal systems, when submitted to Differential Scanning Calorimetry analysis, remained structured even at temperatures above 100 °C, showing excellent stability. The developed liquid crystal systems showed a tolerable degree of cytotoxicity and bactericidal potential, for example, the 9DR system demonstrated a reduction in bacterial load after the third day and reached zero CFU on the seventh day of the test. The developed systems were also evaluated in the preclinical model of Mtb-infected mice, using the nasal, sublingual, and cutaneous route for the delivery of RIF associated with a nanostructured liquid crystal system as a possible tool in the treatment of TB.

Relative growth rate for trees at the growth stage is coordinated with leaf bulk modulus of elasticity and osmotic potential in a subtropical forest of China

Current theoretical models for tree relative growth rate (RGR) have considered physiological processes based on carbon economy and nutrient-productivity relations, but have not taken water relations into account. According to the ecological energetics and Lockhart equation theories, if more energy is needed to maintain water balance in plants, less energy should be invested towards growth, resulting in a trade-off between growth and drought tolerance. Leaf pressure–volume (PV) curve parameters are the best recognised classical indicators of plant drought tolerance, however, the relationships between tree growth and leaf PV parameters have rarely been investigated in forest communities. In the study, we selected two evergreen (Lithocarpus glaber, Cyclobalanopsis oxyodon) and two deciduous species (Quercus serrata, Platycarya strobilacea) that dominate northern subtropical forests on Mt. Shennongjia, central China. For each species, we selected individuals at different developmental stages (growth vs. mature stage), and measured the relative growth rate of diameter at breast height (RGRDBH) and leaf PV curve parameters, including leaf saturated osmotic potential (Ψsat), maximum bulk elastic modulus (εmax), turgor loss osmotic potential (Ψtlp) and leaf water capacity (Cleaf). Correlation analysis and structural equation models (SEM) were applied to explore the relationships between RGRDBH and PV parameters. The results showed that: 1) at both stages, RGRDBH was positively correlated with Ψsat, but not with Ψtlp and Cleaf; 2) εmax was negatively related to RGRDBH at the growth stage; 3) the relationship between PV parameters and RGRDBH was determined mainly by inter-specific difference; 4) SEM showed that both Ψtlp and Ψsat were affected by εmax; and 5) due to the interaction between the PV parameters, Ψsat and εmax had a direct effect on RGRDBH only during the growth stage. Our results showed that the functional/structural coordination between PV parameters adjusted the RGRDBH of trees, and among different species, there exists a tradeoff between energy allocation for drought tolerance and growth. These findings can be applied to improve RGR models, and show how forests respond to warm-induced drought, which is a major threat to global forests in a future warmer climate.

Analytical model for viscous and elastic Rayleigh–Taylor instabilities in convergent geometries at static interfaces

Great attention has been attracted to study the viscous and elastic Rayleigh–Taylor instability in convergent geometries, especially for their low mode asymmetries that behave distinctively from the planar counterparts. However, most analyses have focused on the instability at static interfaces that excludes the studies of the Bell–Plesset effects and the elastic–plastic transition since they involve too complex mathematics. Herein, we perform detailed analyses on the dispersion relations by applying the viscous and elastic potential flow method to obtain their approximate growth rates compared with the exact ones to demonstrate: (i) The approximate growth rates based on potential flow method generally coincide with the exact ones. (ii) An alternative expression is proposed to overcome the discrepancy for the low mode asymmetries at fluid/fluid interface. (iii) Extra care must be taken in solids since the maximum discrepancies occur at the n = 1 mode and at the mode proximate to the cutoff. This analytical method of great simplicity is essential to describe the dynamic interface by including the overall motion of the interface based on the static construction, while the exact analysis involves too complex mathematics to be extended by including the Bell–Plesset effects and the elastic–plastic properties. To sum up, the approximate analytical dispersion relations derived in convergent geometries, have the potential for dealing with dynamic interfaces where Bell–Plesset effects are combined with elastic–plastic transition.

Creep relaxation in nonlinear viscoelastic twisted rods

AbstractThe solution to the problem of stress relaxation in a twisted nonlinear viscoelastic isotropic incompressible rod is presented. We address the multiplicative decomposition of the deformation gradient into reversible (elastic) and irreversible (creep) parts. The elastic potential and the creep potential can be chosen arbitrarily. In particular, the creep law can take into account both the nonlinearity of the relationship between the strain rate and the effective stress, and time‐hardening or softening. We consider two variants of creep constitutive relations. One is based on the Tresca equivalent stress, the other is based on the von Mises equivalent stress. The first of them leads to a significant simplification of the governing equations because in this case a radial elastic strain vanishes. In this framework, we obtain a closed‐form solution in elementary functions for the coupling of Mooney–Rivlin elastic model and linear creep law. For the von Mises material, numerical‐analytical results are obtained. The results are compared with the known small‐strain solutions.

Revisiting DNA Sequence-Dependent Deformability in High-Resolution Structures: Effects of Flanking Base Pairs on Dinucleotide Morphology and Global Chain Configuration.

DNA carries more than the list of biochemical ingredients that drive the basic functions of living systems. The sequence of base pairs includes a multitude of structural and energetic signals, which determine the degree to which the long, threadlike molecule moves and how it responds to proteins and other molecules that control its processing and govern its packaging. The chemical composition of base pairs directs the spatial disposition and fluctuations of successive residues. The observed arrangements of these moieties in high-resolution protein–DNA crystal structures provide one of the best available estimates of the natural, sequence-dependent structure and deformability of the double-helical molecule. Here, we update the set of knowledge-based elastic potentials designed to describe the observed equilibrium structures and configurational fluctuations of the ten unique base-pair steps. The large number of currently available structures makes it possible to characterize the configurational preferences of the DNA base-pair steps within the context of their immediate neighbors, i.e., tetrameric context. Use of these knowledge-based potentials shows promise in accounting for known effects of sequence in long chain molecules, e.g., the degree of curvature reported in classic gel mobility studies and the recently reported sequence-dependent responses of supercoiled minicircles to nuclease cleavage.

Stress State in an Eccentric Elastic Ring Loaded Symmetrically by Concentrated Forces

The stress state from an eccentric ring made of an elastic material symmetrically loaded on the outer boundary by concentrated forces is deduced. The analytical results are obtained using the Airy stress function expressed in bipolar coordinates. The elastic potential corresponding to the same loading but for a compact disk is first written in bipolar coordinates, then expanded in Fourier series, and after that, an auxiliary potential of a convenient form is added to it in order to impose boundary conditions. Since the inner boundary is unloaded, boundary conditions may be applied directly to the total potential. A special focus is on the number of terms from Fourier expansion of the potential in bipolar coordinates corresponding to the compact disk as this number influences the sudden increase if the coefficients from the final form of the total potential. Theoretical results are validated both by using finite element software and experimentally through the photoelastic method, for which a device for sample loading was designed and constructed. Isochromatic fields were considered for the photoelastic method. Six loading cases for two different geometries of the ring were studied. For all the analysed cases, an excellent agreement between the analytical, numerical and experimental results was achieved. Finally, for all the situations considered, the stress concentration effect of the inner hole was analytically determined. It should be mentioned that as the eccentricity of the inner hole decreases, the integrals from the relations of the total elastic potential present a diminishing convergence in the vicinity of the inner boundary.