Nonlinear Elastic Effects Research Articles

Spool-Nematic Ordering of dsDNA and dsRNA under Confinement.

The ability of double-stranded DNA or RNA to locally melt and form kinks leads to strong nonlinear elasticity effects that qualitatively affect their packing in confined spaces. Using analytical theory and numerical simulation we show that kink formation entails a mixed spool-nematic ordering of double-stranded DNA or RNA in spherical capsids, consisting of an outer spool domain and an inner, twisted nematic domain. These findings explain the experimentally observed nematic domains in viral capsids and imply that nonlinear elasticity must be considered to predict the configurations and dynamics of double-stranded genomes in viruses, bacterial nucleoids or gene-delivery vehicles. The nonlinear elastic theory suggests that spool-nematic ordering is a general feature of strongly confined kinkable polymers.

Effect of Grain Shape and Relative Humidity on the Nonlinear Elastic Properties of Granular Media

AbstractThis study focuses on unraveling the microphysical origins of the nonlinear elastic effects, which are pervasive in the Earth's crust. Here, we examine the influence of grain shape on the elastic nonlinearity of granular assemblies. We find that the elastic nonlinearity of angular sand particles is of the same order of magnitude as that previously measured in spherical glass beads. However, while the elastic nonlinearity of glass beads increases by an order of magnitude with RH that of sand particles is rather RH independent. We attribute this difference to the angularity of sand particles: absorbed water on the spherical grains weakens the junctions making them more nonlinear, while no such effect occurs in sand due to grain interlocking. Additionally, for one of the nonlinear parameters that likely arises from shearing/partial slip of the grain junctions, we observe a sharp amplitude threshold in sand which is not observed in glass beads.

Control of guided modes propagating in unconsolidated granular layers : Combined effects of local resonators and nonlinear elasticity

In this talk, we analyze the attenuation, by a line of local resonators, of guided surface modes propagating in an unconsolidated granular layer. Local resonators are made of dense solid parts added to the granular layer, mobilizing the elasticity of the medium itself for the restoring force and their own mass for the inertia effects. In addition, their weight induces local elasticity changes thanks to the nonlinear elasticity of such unconsolidated granular layers. We show numerically and experimentally that the induced resonance and elasticity change are key ingredients in describing the observed transmission and reflection effects, including mode conversion, for the different guided modes observed. We demonstrate that all these ingredients can be rationally combined to enhance the shielding effect and could be used as strategies to control seismic waves on a larger scale.

LEFM is agnostic to geometrical nonlinearities arising at atomistic crack tips

Various fields such as mechanical engineering, materials science, etc., have seen a widespread use of linear elastic fracture mechanics (LEFM) at the continuum scale. LEFM is also routinely applied to the atomic scale. However, its applicability at this scale remains less well studied, with most studies focusing on non-linear elastic effects. Using a harmonic “snapping spring” nearest-neighbor potential which provides the closest match to LEFM on a discrete lattice, we show that the discrete nature of an atomic lattice leads to deviations from the LEFM displacement field during energy minimization. We propose that these deviations can be ascribed to geometrical nonlinearities since the material does not have a nonlinear elastic response prior to bond breaking. We demonstrate that crack advance and the critical stress intensity factor in an incremental loading scenario is governed by the collectively loaded region, and can not be determined analytically from the properties (max. elongation, max. sustained force, etc.) of the stressed crack tip bond alone.

Polyisocyanide hydrogels with tunable nonlinear elasticity mediate liver carcinoma cell functional response

Hepatocellular carcinoma development is closely related to the changes in tissue mechanics induced by excess collagen deposition and crosslinking, which leads to liver fibrosis and malignant progression. The role of matrix stiffness has been widely assessed using various linearly elastic materials. However, the liver, like many soft tissues, also exhibits nonlinear elasticity by strain-stiffening, allowing cells to mechanically interact with their micromilieus which has attracted much attention in cellular processes recently. Here, we use a biomimetic hydrogel grafting of GRGDS peptide with tunable nonlinear mechanical properties, polyisocyanides (PIC), to investigate the influence of strain-stiffening on HepG2 liver cancer cell behavior by tuning PIC polymer length. Compared to short PIC polymer with lower critical stress, PIC hydrogels composed of long polymer with higher critical stress promote the motility and invasiveness of HepG2 cells, and induce more actin stress fibers and higher expression level of mechanotransducer YAP and its nuclear translocation. Strikingly, the expression of calcium-activated potassium channel KCa3.1, an important biomarker in hepatocellular carcinoma, is also affected by the mechanical property of PIC hydrogels. It was also shown that downregulating the KCa3.1 channel can be achieved by inhibiting the formation of actin fibers. Our findings imply that the strain-stiffening property of PIC hydrogels affects the expression of KCa3.1 potassium channel via mediating cytoskeletal stress fiber formation, and ultimately influences the liver carcinoma cell functional response. Statement of significanceThe effect of nonlinear elasticity by strain-stiffening, is assessed in HepG2 liver cancer cell behavior by using a biomimetic hydrogel with tunable mechanical properties, polyisocyanides (PIC). PIC gels with higher critical stress promote the motility and invasiveness of HepG2 cells and induce upregulated expression levels of KCa3.1 potassium channel and YAP, but which can be suppressed by inhibiting the formation of actin fibers. Our findings imply that the strain-stiffening property of PIC gels influences the expression of KCa3.1 potassium channel via mediating cytoskeletal stress fiber formation and, ultimately affects the liver carcinoma cell functional response.

Nonlinear elasticity, yielding, and entropy in amorphous solids.

The holographic duality has proven successful in linking seemingly unrelated problems in physics. Recently, intriguing correspondences between the physics of soft matter and gravity are emerging, including strong similarities between the rheology of amorphous solids, effective field theories for elasticity, and the physics of black holes. However, direct comparisons between theoretical predictions and experimental/simulation observations remain limited. Here, we study the effects of nonlinear elasticity on the mechanical and thermodynamic properties of amorphous materials responding to shear, using effective field and gravitational theories. The predicted correlations among the nonlinear elastic exponent, the yielding strain/stress, and the entropy change due to shear are supported qualitatively by simulations of granular matter models. Our approach opens a path toward understanding the complex mechanical responses of amorphous solids, such as mixed effects of shear softening and shear hardening, and offers the possibility to study the rheology of solid states and black holes in a unified framework.

The Third-Order Elastic Constants and Mechanical Properties of 30° Partial Dislocation in Germanium: A Study from the First-Principles Calculations and the Improved Peierls–Nabarro Model

The elastic constants, core width and Peierls stress of 30° partial dislocation in germanium has been investigated based on the first-principles calculations and the improved Peierls–Nabarro model. Our results suggest that the predictions of lattice constant and elastic constants given by LDA are in better agreement with experiment results. While the lattice constant is overestimated at about 2.4% and most elastic constants are underestimated at about 20% by the GGA method. Furthermore, when the applied deformation is larger than 2%, the nonlinear elastic effects should be considered. And with the Lagrangian strains up to 8%, taking into account the third-order terms in the energy expansion is sufficient. Except the original γ—surface generally used before (given by the first-principles calculations directly), the effective γ—surface proposed by Kamimura et al. derived from the original one is also used to study the Peierls stress. The research results show that when the intrinsic−stacking−fault energy (ISFE) is very low relative to the unstable−stacking−fault energy (USFE), the difference between the original γ—surface and the effective γ—surface is inapparent and there is nearly no difference between the results of Peierls stresses calculated from these two kinds of γ—surfaces. As a result, the original γ—surface can be directly used to study the core width and Peierls stress when the ratio of ISFE to the USFE is small. Since the negligence of the discrete effect and the contribution of strain energy to the dislocation energy, the Peierls stress given by the classical Peierls–Nabarro model is about one order of magnitude larger than that given by the improved Peierls–Nabarro model. The result of Peierls stress estimated by the improved Peierls–Nabarro model agrees well with the 2~3 GPa reported in the book of Solid State Physics edited by F. Seitz and D. Turnbull.

Effect of Nonlinear Elasticity on the Swelling Behaviors of Highly Swollen Polyelectrolyte Gels.

Polyelectrolyte gels exhibit swelling behaviors that are dependent on the external environment. The swelling behaviors of highly charged polyelectrolyte gels can be well explained using the Flory–Rehner model combined with the Gibbs–Donnan effect and Manning’s counterion condensation effect (the FRGDM model). This study investigated the swelling properties of a series of model polyelectrolyte gels, namely tetra-polyacrylic acid-polyethylene glycol gels (Tetra-PAA-PEG gels), and determined the applicability of the FRGDM model. The swelling ratio (Vs/V0) was well reproduced by the FRGDM model in the moderate swelling regime (Vs/V0 < 10). However, in the high swelling regime (Vs/V0 > 10), the FRGDM model is approx. 1.6 times larger than the experimental results. When we introduced the finite extensibility to the elastic free energy in the FRGDM model, the swelling behavior was successfully reproduced even in the high swelling regime. Our results reveal that finite extensibility is one of the factors determining the swelling equilibrium of highly charged polyelectrolyte gels. The modified FRGDM model reproduces well the swelling behavior of a wide range of polyelectrolyte gels.

An iterative simulation algorithm for large oscillation of the applicable 2D-electrical system on a complex nonlinear substrate

An iterative algorithm is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the nth approximation is derived from the previous ones. As a first endeavor, the nonlinear dynamic instability performance of the electrical plate on the nonlinear elastic substrate using an iterative algorithm is scrutinized in this research. A nonlinear elastic foundation is assumed to be in contact with the electrical plate during deformation. The nonlinear coupled dynamic equations governing the transverse and longitudinal motions of the electrical plate are derived using Hamilton’s principle method, the von Karman geometric nonlinearity, and improved higher-order shear deformation theory. The iterative algorithm based generalized differential quadrature method (IAB-GDQM) is applied for solving the nonlinear equations with the aid of nonlinear boundary domains. Parametric studies are implemented to explore the impacts of the applied voltage, geometry of the plate, and nonlinear factor on large amplitude motion, and nonlinear dynamics of the presented system for various boundary domains. Results show that the nonlinear dynamic depends on nonlinear elastic foundation effects and the applied voltage, which can be used to design the electrically structures in different environmental conditions accurately.

First‐Principles Study on the Elastic Constants and Structural and Mechanical Properties of 30° Partial Dislocation in GaAs

The second‐order elastic constants, third‐order elastic constants, and the generalized‐stacking‐fault energy for semiconductor GaAs are investigated using the first‐principles calculations. The predictions of elastic constants are obtained from the coefficients of the fitted polynomials of the energy‐strain functions. It is found that the nonlinear elastic effects must be considered when the applied deformations are larger than approximately 1.5%. With the Lagrangian strains up to 6.4%, the terms included up to third order in energy expansion functions are sufficient. The elastic constants given in this work agree well with the previous results and experimental data except for C144. C144 given by the present paper is a positive value, and the estimated 3 GPa agrees well with the experimental result of 2 GPa. The research results can provide a reference for understanding the elasticity of GaAs. The generalized‐stacking‐fault energy has been calculated without and with structural relaxation, respectively. The unstable stacking fault energy with structural relaxation is about two‐thirds of that without relaxation. The dislocation width and Peierls stress for 30° partial in GaAs have been investigated based on the improved P‐N theory. The dislocation width is very narrow (only about one‐fifth of Burgers vector b), which is reasonable for covalent materials. The Peierls stress is about 4 GPa, in good agreement with the experimental result of 2∼3 GPa.

Modeling and analysis of a coupled novel nonlinear magneto-electro-aeroelastic lumped model for a flutter based energy harvesting system

The present paper investigates the modeling and analysis of a novel aeroelastic energy harvesting system using magneto-electro-elastic (MEE) material considering the nonlinear elastic effects of the structure. The main aim of using this group of composite materials, as compared to common piezoelectric materials used in aeroelastic harvesters, is to increase the harvested electrical power. The proposed model to improve energy harvesting is composed of a rigid airfoil connected to an elastic beam, with the elastic beam covered by one or two layers of MEE material and two electrodes connected to the top and bottom of the MEE layers to harvest the electric potential from the generated electric field. An external coil was also provided around the oscillating beam to induce the electrical energy generated by the magnetic field established within the MEE layer. Firstly, simulating the harvesting system as a discrete model considering bending and torsional cubic nonlinear springs, the governing equations are extracted using the constitutive equations of MEE materials, Gauss’s law, and Faraday’s law of induction through Hamilton’s principle. Next, the effects of different design parameters are evaluated to find their optimal values to maximize the generated power as well as the harvester stability. Finally, a comparison is made between the considered MEE harvester and a piezo-aeroelastic (PA) energy harvester, indicating the superiority of the proposed MEE energy harvester.

Analytical solution for high-pressure torsion in the framework of geometrically nonlinear non-associative plasticity

In this paper an approximate analytical solution for constrained high-pressure torsion problem is proposed. The finite-strain elastic–plastic model based on multiplicative split of deformation gradient tensor is utilized. Plastic deformation is described by non-associative flow rule. Thus, the dependence of the plastic properties of the material on pressure is taken into account. The problem is reduced to a boundary value problem with a moving boundary for a coupled nonlinear hyperbolic system. The obtained approximate solution is constructed under the assumption of a stationary pressure field in the plastic region, which agrees with the experimental results and also with the exact solution for pressure insensitive material; this analytical solution is very close to the numerical one. The closed-form expressions are obtained for stress tensor and accumulated plastic strain. The effects of the material properties, as well as the value of the initial pressure on the distribution of local mean stress and on the accumulated plastic strain are investigated. Simple analytical condition for cohesion-to-slip transition on the contact surfaces between sample and anvils is obtained. At the same time, this transition sets a limit on the applicability of the obtained solution. With the pressure sensitivity coefficient tends to zero, the approximate solution tends to an exact analytical elastic–plastic solution in the framework of the associative plasticity. Stress inhomogeneity in the presented solution is a consequence of nonlinear elastic effects in the main. For hard materials with small elastic deformations the presented solution predicts almost uniform stress distribution. At the same time, stress inhomogeneity in HPT of hard materials can be significant, and is a consequence of inhomogeneity of the initial pressure, strain inhomogeneity due to deformation of the gasket, as well as phase transformations of the material. For this reason the presented solution utility is restricted to soft hyperelastic materials.

Circumferential wrinkling of polymer nanofibers.

Surface wrinkles are commonly observed in soft polymer nanofibers produced in electrospinning. This paper studies the conditions of circumferential wrinkling in polymer nanofibers under axial stretching. A nonlinear continuum mechanics model is formulated to take into account the combined effects of surface energy and nonlinear elasticity of the nanofibers on wrinkling initiation, in which the soft nanofibers are treated as incompressible, isotropically hyperelastic neo-Hookean solid. The critical condition to trigger circumferential wrinkling is determined and its dependencies upon the surface energy, mechanical properties, and geometries of the nanofibers are examined. In the limiting case of spontaneous circumferential wrinkling, the theoretical minimum radius of soft nanofibers producible in electrospinning is determined, which is related closely to the intrinsic length l_{0}=γ/E of the polymer (γ: the surface energy; E: a measure of the elastic modulus), and compared with that of spontaneous longitudinal wrinkling in polymer nanofibers. The present study provides a rational understanding of surface wrinkling in polymer nanofibers and a technical approach for actively tuning the surface morphologies of polymer nanofibers for applications, e.g., high-grade filtration, oil-water separation, tissue scaffolding, etc.

Elastic softening of sandstone due to a wideband acoustic pulse.

Elastic wave propagation experiments were performed on a thin bar sample composed of Texas "moss" sandstone in order to study nonlinear elastic effects in the time domain. The present experiments utilized a pendulous hammer to produce axially propagating transient signals with strain amplitude between 15 and 130 microstrain in the mid-audio band. Particle velocity along the bar axis was measured with a laser Doppler vibrometer, focused at various locations along the bar. Nonequilibrium dynamics and nonlinear elasticity effects were observed on the propagating pulse as it reflected between the ends of the bar. The same effects were also observed at a single location along the mid-length of the bar using a continuous-wave 1 MHz probe signal, propagating transverse to the bar axis. The results demonstrate larger strain amplitudes and greater reduction in the Young's modulus than in previously reported measurements that employed narrowband excitation. Nonlinear attenuation of the axial pulse is also observed, which increases with excitation amplitude. The present results indicate significant conditioning of the sandstone, particularly "softening" of the Young's modulus of up to 20%, primarily during the tensile phase of propagation, with a "slow dynamic" memory that is similar to that reported in previous investigations.

Non-linear elasticity effects and stratification in brushes of branched polyelectrolytes.

Brushes formed by arm-tethered starlike polyelectrolytes may exhibit internal segregation into weakly and strongly extended populations (stratified two-layer structure) when strong ionic intermolecular repulsions induce stretching of the tethers up to the limit of their extensibility. We propose an approximate Poisson-Boltzmann theory for analysis of the structure of the stratified brush and compare it with results of numerical self-consistent field modeling. Both analytical and numerical models point to the formation of a narrow cloud of counterions (internal double electrical layer) localized inside a stratified brush at the boundary between the layers.

Nonlinear Elastic Effects on High-Strength Concrete Stress Invariants

Abstract Computational modeling of concrete behavior and most failure criteria for concrete strongly rely on the use of invariants of stress. These invariants allow a straightforward geometric interpretation of stress states and concrete failure surfaces in the three-dimensional Haigh-Westergaard stress space. Although second-order linear elastic relations have been traditionally applied to evaluate stresses in concrete, the magnitudes of concrete third-order parameters based on finite deformation theory—Murnaghan parameters l, m, n—are such that, if nonlinear stress-strain relations are considered, the stress invariants’ magnitudes are directly affected. In this work, acoustoelastic stress-strain relations for a general triaxial stress state and ultrasonically calculated second- and third-order elastic parameters are used in the evaluation of three high-strength concrete mixtures with compressive strengths: f′c = 43.0, 44.8, and 56.0 MPa (6.2, 6.5, and 8.1 ksi, respectively). The results demonstrate that nonlinear elastic effects introduce anisotropy when uniaxial stresses are applied and the magnitudes of concrete elastic and shear moduli change because of the applied stress field. When triaxial loads are considered, all stress invariants—first (I1), second (I2), and third (I3)—also undergo variations in their magnitudes. The stress-induced anisotropy due to uniaxial loading application and also the changes in the stress invariants’ magnitudes calls attention to the importance of considering nonlinear third-order effects in cement-based materials’ stress-strain relations and failure criteria.

On fractional and fractal formulations of gradient linear and nonlinear elasticity

In this paper, we consider extensions of the gradient elasticity models proposed earlier by the second author to describe materials with fractional non-locality and fractality using the techniques developed recently by the first author. We derive a generalization of three-dimensional continuum gradient elasticity theory, starting from integral relations and assuming a weak non-locality of power-law type that gives constitutive relations with fractional Laplacian terms, by utilizing the fractional Taylor series in wave-vector space. In the sequel, we consider more general field equations with fractional derivatives of non-integer order to describe nonlinear elastic effects for gradient materials with power-law long-range interactions in the framework of weak non-locality approximation. The special constitutive relation that we elaborate upon can form the basis for developing a fractional extension of deformation theory of gradient plasticity. Using the perturbation method, we obtain corrections to the constitutive relations of linear fractional gradient elasticity, when the perturbations are caused by weak deviations from linear elasticity or by fractional gradient non-locality. Finally, we discuss fractal materials described by continuum models in non-integer dimensional spaces. Using a recently suggested vector calculus for non-integer dimensional spaces, we consider problems of fractal gradient elasticity.

Damping effects of linear and nonlinear tuned mass dampers on nonlinear hinged-hinged beam

This study analyzes the vibration reduction of a tuned mass damper (TMD) applied to a hinged-hinged nonlinear Euler-Bernoulli beam. We compare the effects of linear and nonlinear tuned mass dampers (LTMD/NLTMD) and obtain results undiscovered in previous studies. We analyze the frequency responses (fixed points) of the system using the method of multiple scales (MOMS) and obtain analytical solutions for the NLTMD in the frequency and time domains using Dimensional Analysis and the Perturbation Method. The present study shows that when the damping coefficient (λ), TMD mass (mD), TMD initial displacement (A) and nonlinear spring constant (β) of the NLTMD fulfill a function relationshipλ=3mDβA, the damping and nonlinear elastic effects cancel each other out, and the nonlinear damping frequency coincides with the linear natural frequency. With a fixed damping coefficient and an LTMD or NLTMD at a fixed position, optimal damping is achieved when the product of the TMD mass and spring constant are at fixed values. Finally, we use the Floquet transition matrix of the system and Floquet multipliers (F.M.) to create the basin of attraction (BOA) and analyze the stability of the system under aerodynamic forces.

Ultrafast dynamic response of single-crystal β-HMX (octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine)

We report experimental and computational studies of shock wave dynamics in single-crystal β-HMX on an ultrafast time scale. Here, a laser-based compression drive (∼1 ns in duration; stresses of up to ∼40 GPa) is used to propagate shock waves normal to the (110) and (010) lattice planes. Ultrafast time-domain interferometry measurements reveal distinct, time-dependent relationships between the shock wave velocity and particle velocity for each crystal orientation, which suggest evolving physical processes on a sub-nanosecond time scale. To help interpret the experimental data, elastic shock wave response was simulated using a finite-strain model of crystal thermoelasticity. At early propagation times (&amp;lt;500 ps), the model is in agreement with the data, which indicates that the mechanical response is dominated by thermoelastic deformation. The model agreement depends on the inclusion of nonlinear elastic effects in both the spherical and deviatoric stress-strain responses. This is achieved by employing an equation-of-state and a pressure-dependent stiffness tensor, which was computed via atomistic simulation. At later times (&amp;gt;500 ps), the crystal samples exhibit signatures of inelastic deformation, structural phase transformation, or chemical reaction, depending on the direction of wave propagation.

Modeling of Phase Equilibria in Ni-H: Bridging the Atomistic with the Continuum Scale

In this paper, we present a model which allows bridging the atomistic description of two-phase systems to the continuum level, using Ni-H as a model system. Considering configurational entropy, an attractive hydrogen–hydrogen interaction, mechanical deformations and interfacial effects, we obtained a fully quantitative agreement in the chemical potential, without the need for any additional adjustable parameter. We find that nonlinear elastic effects are crucial for a complete understanding of constant volume phase coexistence, and predict the phase diagram with and without elastic effects.