Nonlinear Elasticity

An exponential mapping for the conformation tensor for flow of viscoelastic fluids; application in turbulent channels K. HOUSIADAS, A. BERIS, Univ. of Delaware -The conformation tensor, a quantity that describes the internal microstructure of polymer molecules, is usually being used as the primary variable in viscoelastic flow calculations. Its main property is that is a positive definite, second order, tensor. Unless special care is taken, the conformation tensor may lose this property resulting to instabilities during the calculations and finally either to break-up of the simulations or to non-physical results. This situation is greatly intensified under turbulent flow conditions. In order to resolve these problems we have expressed the conformation tensor, 'c', as the exponential of another tensor 'a', c=exp(a), and we solve for 'a' instead of 'c'. By construction, the positive definite property of 'c' is always preserved since its eigenvalues are the exponential of the eigenvalues of 'a'. The method is illustrated for viscoelastic turbulent channel flow. Direct Numerical Simulations are being performed using spectral spatial approximations and a stabilizing artificial diffusion term in the viscoelastic constitutive model. That term is needed to smooth the solution to be resolvable with the mesh size used due to the very fine scales that are being created in chaotic flow fields. The FENE-P constitutive model is used to represent the effect of polymer molecules in solution. We will offer a comparison of the results, for exactly the same flow, viscoelastic and numerical parameters, using the old and the new formulation of the constitutive model in terms of the conformation tensor and the exponential tensor, respectively.

In this paper, direct variational calculus was put into practical use to analyses the three dimensional (3D) stability of rectangular thick plate which was simply supported at all the four edges (SSSS) under uniformly distributed compressive load. In the analysis, both trigonometric and polynomial displacement functions were used. This was done by formulating the equation of total potential energy for a thick plate using the 3D constitutive relations, from then on, the equation of compatibility was obtained to determine the relationship between the rotations and deflection. In the same way, governing equation was obtained through minimization of the total potential energy functional with respect to deflection. The solution of the governing equation is the function for deflection. Functions for rotations were obtained from deflection function using the solution of compatibility equations. These functions, deflection and rotations were substituted back into the energy functional, from where, through minimizations with respect to displacement coefficients, formulas for analysis were obtained. In the result, the critical buckling loads from the present study are higher than those of refined plate theories with 7.70%, signifying the coarseness of the refined plate theories. This amount of difference cannot be overlooked. However, it is shown that, all the recorded average percentage differences between trigonometric and polynomial approaches used in this work and those of 3D exact elasticity theory is lower than 1.0%, confirming the exactness of the present theory. Thus, the exact 3D plate theory obtained, provides a good solution for the stability analysis of plate and, can be recommended for analysis of any type of rectangular plates under the same loading and boundary condition.

Brownian motion is a phenomenon observed in the late 17th and early 19th centuries, which Albert Einstein's 1905 work helped explain, laying the foundation for atomism and statistical physics. However, the reported magnitudes of Brownian motion do not match theoretical results. In this paper, I propose that the origin of Brownian motion lies in seismic noise-a magnitude that could not be measured at the time. The reported magnitudes of Brownian motion align more closely with seismic noise than with molecular collisions on a pollen grain.

A nonlinear isotropic elastic block is subjected to a homogeneous deformation consisting of simple shear superposed on triaxial extension. Two new relations are established for this deformation which are valid for all nonlinear elastic isotropic materials, and hence are universal relations. The first is a relation between the stretch ratios in the plane of shear and the amount of shear when the deformation is supported only by shear tractions. The second relation is established for a thin-walled cylinder under combined extension, inflation and torsion. Each material element of the cylinder undergoes the same local homogeneous deformation of shear superposed on triaxial extension. The properties of this deformation are used to establish a relation between pressure, twisting moment, angle of twist and current dimensions when no axial force is applied to the cylinder. It is shown that these relations also apply for a mixture of a nonlinear isotropic solid and a fluid.

In his study of combined finite extension and torsion of a nonlinear, incompressible, isotropic elastic circular cylinder, Rivlin [1] established a relation for the torsional stiffness which depends only on the axial force, the axial extension ratio and the radius of the undeformed cylinder, in the case of small twist. The relationship did not depend on the structure of the stored energy function and is hence a "universal relation". In this paper, we extend Rivlin's result to the case of combined extension and torsion of a cylindrical mixture of a nonlinear elastic solid and fluid.

1. EXECUTIVE SUMMARY
Given that South Africa is the most unequal country, voting remain its only equalising system.  The adoption of e-voting must be processed through this understanding. An understanding that will avoid creating unequal access to voting. Unequal access to voting can emerge once cosmetic policy reforms are allowed to contaminate this system. Therefore, the proposed modernisation of the ballot paper or digit must be subjected into a cosmetic policy reform test. The question must be asked whether e-voting has the potential to conceal voter apathy? The same question must be apportioned to paper voting. If e-voting is more vulnerable to concealing voter apathy than the on station paper voting, then e-voting would have failed cosmetic policy reform. Failing cosmetic policy reform test would mean e-voting is cosmetic and has the potential to conceal voter apathy. Otherwise, passing cosmetic policy reform test would mean digitised voting would unlikely conceal voter apathy. To test this, five variables must be measured- Sourcing of ICT Infrastructure, Cybersecurity, ISP Equal Access, Backup Power Security and Apathy-Empathy Elasticity.

The EIC e-voting policy fairly engage four of these test variables- but miss to evaluate the  Apathy-Empathy Elasticity of introducing e-voting. This paper relies on optimisation of five parameters for modelling apathy-empathy elasticity. Targeted five parameters are; ballot paper, electronic ballot, neuro-political framing principles and family institutionalisation, voting culturalisation using family institutionalisation vector. The goal being to assist the electoral custodians to comprehend the system of voting and abstaining. So as to correctly contextualise advances of modernised elections whether through; electronic, internet or mailed voting convenience. Factoring in its discourse the role of neuro circuits of reward, fear, and trust. The core of this position paper is a call for caution. The paper seeks to draw the nation’s attention on a public vote as currently the only equaliser in the Republic of South African. Given South African institutional trust deficit levels and well proven incomparable  national disparities, election is the only remaining system that legitimatise RSA as democratic state. So modelling apathy-empathy elasticity can help the nation decide whether digitising ballot paper and ballot box is consistent with the liberty principles of nation’s constitution or not.

La actual norma peruana de diseño sismo resistente (NPDSR) establece desplazamientos espectrales crecientes con el periodo estructural para estructuras con periodo mayor al de la plataforma. Esto no corresponde a la tendencia natural de los espectros, ya que para periodos largos los desplazamientos deben ser constantes e iguales al desplazamiento máximo del suelo. En la actualidad se están construyendo en nuestro medio, edificios con aislación sísmica cuyos periodos superan los 2.5 segundos y la NPDSR no permite estimar adecuadamente los desplazamientos a los que estarían sometidas. Por esta razón, se hace necesario el ajuste del espectro de la actual norma de diseño sismo resistente. En este trabajo se desarrolló una propuesta para los espectros de diseño en la zona de periodos mayores a 2.5 s. Para construir estos espectros no fue posible emplear el método de Newmark& Hall, ya que actualmente en el país no se cuenta con una estimación apropiada de los valores máximos de aceleración, velocidad y desplazamiento del suelo. El estudio se desarrolló estudiando tendencias y empleando regresiones basadas en promedios espectrales de la zona de periodos largos. Se obtuvieron espectros para los tipos de suelo S1 y S3 de la norma actual, correspondientes a valores medios y promedio más desviación estándar. Del análisis realizado se identificó el periodo a partir de la cual las aceleraciones espectrales se reducen con el cuadrado del periodo. Se espera que con esta propuesta, se logre mejorar la estimación de desplazamientos en las estructuras peruanas de periodos mayores a 2.5 s. Esto contribuirá a mejorar el estudio y diseño de los edificios con aislamiento sísmico en el Perú. Índice Capítulo 1. Introducción 1.1 El diseño sismo resistente y el control de desplazamientos laterales 1.2 Los sistemas de aislamiento sísmico 1.3 Objetivo de la tesis 1.4 Alcance del Trabajo 1.5 Organización del documento Capítulo 2. Espectros de Diseño para fuerzas y Desplazamientos 2.1 Espectros de diseño basados en fuerzas (sa) 2.2 Procedimientos para obtener espectros de diseño 2.3 Espectros de diseño de algunos países sísmicos  Norma Chilena  ASCE  Norma Colombiana Capítulo 3. El espectro de desplazamientos de la actual Norma Peruana de DSR 3.1 Estructura del espectro 3.2 Comparación registros peruanos con la norma 3.3 Espectros de desplazamiento de las normas extranjeras 23 Capítulo 4. Propuesta del espectro de diseño para la zona de desplazamientos. 4.1 Metodología de Trabajo 4.2 Obtención de información 4.3 Procesamiento y Clasificación de Señales 4.4 Obtención de los Espectros de diseño 4.5 Propuesta de espectros de desplazamiento 4.6 Propuesta de espectros de aceleración

We show experimental evidence of extremely low thresholds for subharmonic generation of ultrasonic waves in one-dimensional artificial piezoelectric plates with Cantor-like structure, as compared to the corresponding homogeneous and periodical plates. The origin of this apparent anomaly is theoretically investigated by studying anharmonic coupling between normal modes. We demonstrate that the large enhancement of nonlinear interaction results from the more favorable frequency and spatial matching of coupled modes (fractons and phonons) in the Cantor-like structure, with no need to invoke anomalous modifications of the nonlinear elastic constants.

We study the homogenisation of geometrically nonlinear elastic composites with high contrast. The composites we analyse consist of a perforated matrix material, which we call the "stiff" material, and a "soft" material that fills the pores. We assume that the pores are of size 0 < ε ≪ 1 and are periodically distributed with period ε. We also assume that the stiffness of the soft material degenerates with rate ε 2γ , γ > 0, so that the contrast between the two materials becomes infinite as ε ↓ 0. We study the homogenisation limit ε ↓ 0 in a low energy regime, where the displacement of the stiff component is infinitesimally small. We derive an effective two-scale model, which, depending on the scaling of the energy, is either a quadratic functional or a partially quadratic functional that still allows for large strains in the soft inclusions. In the latter case, averaging out the small scale-term justifies a single-scale model for high-contrast materials, which features a non-linear and non-monotone effect describing a coupling between microscopic and the effective macroscopic displacements.

We study the homogenisation of geometrically nonlinear elastic composites with high contrast. The composites we analyse consist of a perforated matrix material, which we call the "stiff" material, and a "soft" material that fills the remaining pores. We assume that the pores are of size 0 < ε 1 and are periodically distributed with period ε. We also assume that the stiffness of the soft material degenerates with rate ε 2γ , γ > 0, so that the contrast between the two materials becomes infinite as ε ↓ 0. We study the homogenisation limit ε ↓ 0 in a low energy regime, where the displacement of the stiff component is infinitesimally small. We derive an effective two-scale model, which, depending on the scaling of the energy, is either a quadratic functional or a partially quadratic functional that still allows for large strains in the soft inclusions. In the latter case, averaging out the small scale-term justifies a single-scale model for high-contrast materials, which features a non-linear and non-monotone effect describing a coupling between microscopic and the effective macroscopic displacements.

We rigorously derive a homogenized von-Kármán plate theory as a Γ-limit from nonlinear three-dimensional elasticity by combining homogenization and dimension reduction. Our starting point is an energy functional that describes a nonlinear elastic, three-dimensional plate with spatially periodic material properties. The functional features two small length scales: the period ε of the elastic composite material, and the thickness h of the slender plate. We study the behavior as ε and h simultaneously converge to zero in the von-Kármán scaling regime. The obtained limit is a homogenized von-Kármán plate model. Its effective material properties are determined by a relaxation formula that exposes a non-trivial coupling of the behavior of the out-of-plane displacement with the oscillatory behavior in the in-plane directions. In particular, the homogenized coefficients depend on the relative scaling between h and ε, and different values arise for h ≪ ε, ε ~ h and ε ≪ h.

In this paper we derive, by means of Γ -convergence, the homogenized von Kármán plate model starting from three dimensional nonlinear elasticity. We assume that the thickness of the plate is h, the oscillations of the material is ε(h), where h ε(h) → γ ∈ (0, +∞). We also assume that the strain energy of the plate has the order h 4 , which corresponds to the von Kármán model in the case of the ordinary plate. The obtained model mixes the bending part of the energy with the stretching part. Keywords homogenization • Gamma convergence • von Kármán plate • two-scale convergence Mathematics Subject Classification (2010) 74K20 • 74B20

It is often assumed in the literature that the nine classical strain invariants, which are used to characterize the strain energy of a compressible anisotropic elastic solid with two preferred non-orthogonal directions are independent. In this paper, it is shown that only six of the classical strain invariants are independent, and syzygies exist between the classical invariants. Alternatively, using principal axis techniques, it is simply proven that, only six of the classical strain invariants are independent and syzygies exist between the principal axis strain invariants. Consequently, all other sets of strain invariants, proposed in the literature, which are uniquely related to the set of principal axis strain invariants, have only six independent invariants. Due to syzygies, it is shown that the number of ground state constants required to fully describe the quadratic linear strain energy function of two-fibre solids is fourteen, not thirteen, as assumed in the literature.

In this paper we first present the known disagreement between the law of a second grade fluid and the statement concerning the asymptotic stability of the rest state. After that we put into evidence that for a third grade fluid (which is not considered in the framework described by Coleman's and Noll Approximation Theorem (ATh» the rest state is asymptotically stable, for any value of the constitutive modulus (tIl.

In this paper we first present the known disagreement between the law of a second grade fluid and the statement concerning the asymptotic stability of the rest state. After that we put into evidence that for a third grade fluid (which is not considered in the framework described by Coleman's and Noll Approximation Theorem (ATh» the rest state is asymptotically stable, for any value of the constitutive modulus (tIl.

The existence and multiplicity of positive solutions are established to periodic boundary value problems for singular nonlinear second order ordinary differential equations. The arguments are based only upon the positivity of the Green's functions and the Krasnosel'skii fixed point theorem. As an example, a periodic boundary value problem is also considered which comes from the theory of nonlinear elasticity.

In order to reduce the costs related to corrosion damage in aircraft structures, it is vital to develop new robust, accurate and reliable damage detection methods. A possible answer to this problem is offered by newly developed nonlinear ultrasonic techniques, which monitors the nonlinear elastic wave propagation behaviour introduced by damage, to detect its presence and location. In this paper, a new nonlinear time reversal technique is presented for the detection and localization of a scattered zone (damage) in a multi-material medium. In particular, numerical findings on a friction stir-welded aluminium plate-like structure are reported. Damage was introduced in the heat affected zone and modelled using a multi-scale material constitutive model (Preisach-Mayergoyz space). Studies were conducted for two different transducer configurations. Particular attention was devoted to find the optimum time-reversed window to be re-emitted in the structures. The methodology was compared with traditional time-reversal acoustics (TRA), showing significant improvements. While the traditional TRA was not able to clearly localise the damage, the developed technique identified in a clear manner the faulted zone, showing its robustness to locate and characterize nonlinear sources, in presence of a multi-material medium.

Mechanical molecular scission is the main problem of polymeric drag reducers. The ability to reduce the drag is notably decreased as the molecules break down step by step as time goes on. A number of researchers have given a large part of their time to attempts to further understand the role that some important features play in polymer degradation. Until now, all efforts have been in experimental approaches. This paper is the first attempt to take into account the effect of molecular scission on a drag reducing flow by a direct numerical simulation. We analyse a turbulent plane Couette flow of a FENE-P fluid. Our degradation model is based on the maximum polymer extension length L. Unlike the standard FENE-P model, in which L is a constant, the polymer extension here is a spatio-temporal field L(x, y, z, t). When the molecules are highly stretched, which is measured by the trace of the conformation tensor, their maximum length is locally reduced and, consequently, so is their ability to reduce drag. The degraded L spreads within the domain by means of a transport equation. We show here that with such a simple idea we are able to predict the main aspects of mechanical degradation in the flow, such as the change of the turbulent structures and velocity field, and, consequently, the fall of the drag reduction over time.

We address the cross effects between mechanical strains and magnetic fields on the plastic response of magnetoelastic amorphous solids. It is well known that plasticity in nonmagnetic amorphous solids under external strain γ is dominated by the co-dimension 1 saddle-node bifurcation in which an eigenvalue of the Hessian matrix vanishes at γP like √ γP -γ. This squareroot singularity determines much of the statistical physics of elasto-plasticity, and in particular that of the stress-strain curves under athermal-quasistatic conditions. In this Letter we discuss the much richer physics that can be expected in magnetic amorphous solids. Firstly, magnetic amorphous solids exhibit co-dimension 2 plastic instabilities, when an external strain and an external magnetic field are applied simultaneously. Secondly, the phase diagrams promise a rich array of new effects that have been barely studied; this opens up a novel and extremely rich research program for magnetoplastic materials.

By comparing the response to external strains in metallic glasses and in Lenard-Jones glasses we find a quantitative universality of the fundamental plastic instabilities in the athermal, quasistatic limit. Microscopically these two types of glasses are as different as one can imagine, the latter being determined by binary interactions, whereas the former by multiple interactions due to the effect of the electron gas that cannot be disregarded. In spite of this enormous difference the plastic instability is the same saddle-node bifurcation. As a result the statistics of stress and energy drops in the elasto-plastic steady state are universal, sharing the same system-size exponents.

An effective temperature T eff which differs from the bath temperature is believed to play an essential role in the theory of elasto-plasticity of amorphous solids. The definition of a measurable T eff in the literature on sheared solids suffers however from being connected to a fluctuation-dissipation theorem which is correct only in equilibrium. Here we introduce a natural definition of T eff based on measurable structural features without recourse to any questionable assumption. The value of T eff is connected, using theory and scaling concepts, to the flow stress and the mean energy that characterize the elasto-plastic flow.

Ah&met--The torsion and shear load problems of elastic beam cross-sections of moderate wall thickness are formulated in terms of finite elements with the warping function as primary variable. By introduction of suitable internal degrees of freedom in the shear problem the stiffness matrices of both problems become identical, The method makes no distinction between open and closed cross-sections. An efficient PC implementation is discussed and illustrated by examples.

Liquid crystal elastomers realize a fascinating new form of soft matter that is a composite of a conventional crosslinked polymer gel (rubber) and a liquid crystal. These solid liquid crystal amalgams, quite similarly to their (conventional, fluid) liquid crystal counterparts, can spontaneously partially break translational and/or orientational symmetries, accompanied by novel soft Goldstone modes. As a consequence, these materials can exhibit unconventional elasticity characterized by symmetry-enforced vanishing of some elastic moduli. Thus, a proper description of such solids requires an essential modification of the classical elasticity theory. In this work, we develop a rotationally invariant, nonlinear theory of elasticity for the nematic phase of ideal liquid crystal elastomers. We show that it is characterized by soft modes, corresponding to a combination of long wavelength shear deformations of the solid network and rotations of the nematic director field. We study thermal fluctuations of these soft modes in the presence of network heterogeneities and show that they lead to a large variety of anomalous elastic properties, such as singular length-scale dependent shear elastic moduli, a divergent elastic constant for splay distortion of the nematic director, long-scale incompressibility, universal Poisson ratios and a nonlinear stress-strain relation for arbitrary small strains. These long-scale elastic properties are universal, controlled by a nontrivial zero-temperature fixed point and constitute a qualitative breakdown of the classical elasticity theory in nematic elastomers. Thus, nematic elastomers realize a stable "critical phase", characterized by universal power-law correlations, akin to a critical point of a continuous phase transition, but extending over an entire phase.

Among the main limitations of active contours are their high noise sensitivity and poor capture range from the target object. One of the most promising approaches for addressing these limitations is the concept of Vector Field Convolution (VFC). However, due to its isotropic vector field kernel, VFC does not take full advantage of the underlying image structural characteristics. By specifically addressing this idea, a novel local tensor vector field approach is developed to adaptively account for these structural characteristics. Experimental results demonstrate that the proposed adaptive method leads to more accurate segmentation.

Rheological models for polymer solutions and melts based on the finitely extensible non-linear elastic (FENE) dumbbell theory are reviewed in this study. The FENE-P model that is a well-known Peterlin approximation of the FENE model, indicates noticeable deviation from original FENE predictions and also experimental results, especially in the transient flow. In addition, both FENE and FENE-P models have some shortcomings from the point of view of theory. To overcome these shortcomings, a new approximation of the FENE spring force has been established. It has been used to develop a modified constitutive rheological model for polymeric fluids. In the procedure of modeling, the effect of non-affine deformation is introduced into the new model. Comparison between the model predictions and experimental data presented in the literature for transient and steady shear flow of polystyrene indicates that this modified model can predict the rheological behavior of polymeric fluids with a great...

This study investigates the response of fiber-reinforced polymer (FRP) reinforced engineered cementitious composite (ECC) members with a focus on their flexural load-deformation behavior, residual deflection, damage evolution, and failure mode. Critical aspects of conventional FRP-reinforced concrete members-such as interfacial bond strength, flexural crack formation, composite deformation behavior, and brittle failure mode-are briefly reviewed and compared to FRP reinforced ECC. The interaction of linear elastic FRP reinforcement and ECC matrix with ductile stress-strain behavior in tension results in nonlinear elastic flexural response characteristics with stable hysteretic behavior, small residual deflection, and ultimately gradual compression failure. Compatible deformations of reinforcement and matrix lead to low interfacial bond stress and prevent composite disintegration by bond splitting and cover spalling. Furthermore, flexural stiffness and strength as well as crack formation and widths in FRP-reinforced ECC members are found effectively independent of interfacial bond properties due to the tensile deformation characteristics of the cementitious matrix. A model for the load-deflection envelope based on a nonlinear moment-curvature relationship is suggested.

One approach to solve a nonlinear boundary value problem for the Fredholm integro-differential equation A quasilinear boundary value problem for a Fredholm integro-differential equation is considered. The interval is divided into N parts and the values of the solution to the equation at the left end points of the subintervals are introduced as additional parameters. New unknown functions are introduced on the subintervals and a special Cauchy problem with parameters is solved with respect to a system of such functions. By means of the solution to this problem, a new general solution to the quasilinear Fredholm integro-differential equation is constructed. The conditions of the existence of a unique new general solution to the equation under consideration are obtained. A new general solution is used to create a system of nonlinear algebraic equations in parameters introduced. The conditions for the existence of a unique solution to this system are established.

In this paper, we propose to study some nonlinear boundary problems for the dynamically modified operator by adding a viscosity term -𝛼𝛥𝑢 ′′ to the nonlinear vibrations of the plates. The field of application for vibrating plates is extensive. To meet user needs, we have considered the geometric shape, the density of the material constituting the plate, the plate thickness, and Poisson's ratio. Once the problems have been posed, our approach then consists of transforming them into nonlinear problems of the hyperbolic type. In this work, we study six boundary value problems and we prove for each problem an existence and uniqueness theorem. Finally, we demonstrate the existence of a solution to the stationary problem using a variant of Brouwer's fixed point theorem.

In the present paper we intend to examine in detail the formation of localized modes and waves mediated by modulational instability in an elastic structure. The elastic composite structure consists of a nonlinear foundation coated with an elastic thin plate. The problem deals with flexural waves traveling on the plate. The attention is devoted to the behavior of nonlinear waves in the small-amplitude limit in view of deducing criteria of instability which produce localized waves. it is shown that, in the small-amplitude limit, the basic equation which governs the plate deflection is approximated by a two-dimensional nonlinear SchrGdinger equation. The latter equation allows us to study the modulational instability conditions leading to diflerent zones of instability. The examination of the instability provides useful information about the possible selection mechanism of the modulus of the carrier wave vector and growth rate of the instabilities taking place in both (longitudinal and transverse) directions of the plate. The mechanism of the self-generated nonlinear waves on the plate beyond the birth of modulational instability is numerically investigated. The numerics show that an initial plane wave is then transformed, through the instability process, into nonlinear localized waves which turn out to be particularly stable. In addition, the influence of the prestress on the nature of localized structures is also examined. At length, in the conclusion some other wave problems and extensions of the work are evoked.

Among a lot of fascinating nonlinear effects, there is a great deal of interest in self modulation of a plane wave, or modulational instability (MI), which occurs in nonlinear dispersive media. Qualitatively a MI is the tendency for amplitude of a modulated carrier wave to break into isolated structures or solitons. Energy originally resident in the carrier of wave packets or long pulse is gradually transferred by nonlinear interaction, in the medium, into the spectrum side bands. As the side-band energy grows in amplitude, the modulated wave breaks up into a series of localized objects or envelope solitons. Composite structures formed by a singly or doubly-curved shallow shell resting on a nonlinear elastic foundation are wave guides that enable one to focus a high energy density in order that nonlinearities be excited. This class of elastic structures is an interesting candidate for the real observation of two-dimensional localized modes in elastodynamics. The purpose of the present work is to study the influences of the geometric dispersion, the prestress on the shallow shell and the material nonlinearity of the elastic foundation on the vibrations modes of the structure. The basic equations which govern the dynamics of the elastic structure are deduced from a variational principle. The analysis is restricted to signals which consist of a slowly varying envelope in space and time modulating a harmonic carrier wave. In the limit of low amplitude the coupled equations are solved by means of a reductive perturbation method. It is shown that the complex amplitude of the envelope is governed by a two-dimensional nonlinear Schrödinger equation. This equation allows us to study the modulational instability conditions leading to different zones of instability. The mechanism of the self generated nonlinear waves in the elastic structure beyond the birth of modulational instability is numerically investigated on the original equations.

High damping rubber bearings (HDRB) is a useful seismic resistant device that is widely used in buildings due to many outstanding advantages. However, the mechanical characteristics of HDRB are complicated such as temperature-dependent, rate-dependent. This paper is devoted to investigate the low temperature dependence of HDRB's mechanical behavior. For this purpose, the experiments on HDRB conducted at -30 °C, -10 °C, and 23 °C is to investigate the low temperature dependence of mechanical behavior of HDRB. A numerical calculation to determine the viscosity of HDRB was conducted to investigate the low temperature dependence of HDRB's viscosity. Experimental results show that a significant increase in the rate-dependent overstress at low temperatures. This behavior can be ascribed to the temperature dependence of HDRB's viscosity. It means that the viscosity effect of the material increases when temperature decreases. In addition, the experimental observations indicate that the rate sensitivity of HDRB depends on the low temperatures. On the contrary, the dependence of the rate-independent equilibrium stress at the end of the relaxation process is quite weaker. Furthermore, the results of the numerical calculation to identify the viscosity of HDRB also agree with the conclusions obtained from the experimental observations.

The major objective of this article is to present recent advances in the methodology to fine tune the adhesive performance of a PSA. In addition to the so-called Dahlquist criterion requiring a low modulus, we propose two additional rheological predictors of the adhesive properties. The first one is derived from the description of the detachment of a linear elastic layer from a rigid substrate. We made an approximate extension of this analysis to the viscoelastic regime and showed that the transition from interfacial cracks to cavitation and fibrillation can be quantitatively predicted from the easily measurable ratio tan(d)=G 0 (x). If a fibrillar structure is formed, the nonlinear large strain properties become important. We showed that the ability of the fibrils to be stretched before final debonding can be predicted from the analysis of simple tensile tests. The softening, which occurs at intermediate strains, and, more importantly, the hardening which occurs at large strains, can be used to predict the mode of failure and the energy of adhesion. The use of this methodology to tune the PSA structure for a specific application has been illustrated for the special case of wb-PSA made of core-shell particles, and improved adhesive properties on polyethylene surfaces have been obtained.

Experimental tensile tests were performed on polyamide-based (PA66) woven strap samples. A strain measuring device was used to measure the strain in the middle and effective part of the woven tensile sample. The tests were performed, on the one hand under monotonous tension at different strain rates and on the other hand under sophisticated cyclic loading histories, including relaxation and creep sequences. The analysis of experimental results was made through a viscoelasto-hysteresis model, based on the superimposition of three stress components. The proposed method allows for characterizing the steady state viscous stress as a function of strain and strain rate, the time-independent irreversible behaviour and the instantaneous modulus increasing with the strain. Based on the visco-elasto-hysteresis model, an analysis enabled us to understand and predict the change in relaxation and creep orientations during complex loading histories.

It is known that the standard full integration ten node tetrahedral element is inaccurate for thin nearly incompressible structures. The objective of this work is to develop a ten node tetrahedral Cosserat Point Element (CP E) for nonlinear isotropic hyperelastic materials. Hyperelastic constitutive equations for the CPE are developed by treating the element as a structure with a strain energy function that is restricted to satisfy a nonlinear form of the patch test. A number of examples are considered which demonstrate that the resulting CPE is accurate and robust.

A generalized form for the strain energy of inhomogeneous deformations is developed for a 3-D brick Cosserat Point Element (CPE) which includes full coupling of bending and torsional modes of deformation. The constitutive coefficients, which depend on the reference geometry of the element, are determined by solving eighteen bending problems and six torsion problems on special elements that are parallelepipeds with two right angles. The resulting constitutive coefficients ensure that the strain energy for inhomogeneous deformations remains a positive definite function of the inhomogeneous strain measures for all reference element shapes. A number of example problems are considered which show that the generalized CPE produces results as accurate as enhanced strain and incompatible elements for thin structures and is free of hourglass instabilities typically predicted by these enhanced elements in regions experiencing combined high compression with bending.

A generalized form for the strain energy of inhomogeneous deformations is developed for a 3-D brick Cosserat Point Element (CPE) which includes full coupling of bending and torsional modes of deformation. The constitutive coefficients, which depend on the reference geometry of the element, are determined by solving eighteen bending problems and six torsion problems on special elements that are parallelepipeds with two right angles. The resulting constitutive coefficients ensure that the strain energy for inhomogeneous deformations remains a positive definite function of the inhomogeneous strain measures for all reference element shapes. A number of example problems are considered which show that the generalized CPE produces results as accurate as enhanced strain and incompatible elements for thin structures and is free of hourglass instabilities typically predicted by these enhanced elements in regions experiencing combined high compression with bending.

The shear properties of pultruded composite plates are especially needed for the structural design of composite load bearing elements when one considers the shear capacity, the local buckling of plate type elements and some types of joints. The paper presents experimental results obtained on Iosipescu type samples cut from glass fibre reinforced polyester (GFRP) composite strips fabricated by pultrusion. It has been found out that the pultrusion process leads to composite products with uniform and controllable properties, and the fibre orientation is decisive in establishing the shear structural response.

This paper studies the evolution mechanism of surface rippling in polymer nanofibers under axial stretching. Such rippling phenomenon has been detected in as-electrospun polyacrylonitrile (PAN) in recent single-fiber tension tests and in electrospun polyimide (PI) nanofibers after imdization. We herein propose a one-dimensional (1D) nonlinear elastic model that takes into account the combined effect of surface tension and nonlinear elasticity during the rippling initiation and its evolution in compliant polymer nanofibers. The polymer nanofiber is modeled as incompressible, isotropically hyperelastic Mooney-Rivlin solid. The fiber geometry prior to rippling is considered as a long circular cylinder. The governing equation of surface rippling is established through linear perturbation of the static equilibrium state of the nanofiber subjected to finite axial pre-stretching. Critical stretch and ripple wavelength are determined in terms of surface tension, elastic property, and fiber radius. Numerical examples are demonstrated to examine these dependencies. Besides, a critical fiber radius is determined, below which the polymer nanofibers are intrinsically unstable. The present model, therefore, is capable of predicting the rippling condition in compliant nanofibers, and can be further used as continuum mechanics approach for the study of surface instability and nonlinear wave propagation in compliant fibers and wires at nanoscale.

Niniejszy dokument stanowi szkic koncepcyjnej teorii, nazwanej Teorią Dwóch Energii (TDE 3.0), której celem nie jest przedstawienie gotowej „teorii wszystkiego”, lecz intuicyjnej próby połączenia istniejących modeli fizycznych – od mechaniki kwantowej, przez teorię względności, aż po Model Standardowy – poprzez elegancki mechanizm rezonansu dwóch pól falowych: Cienia (ψₙ) i Światła (ψₛ).
Model powstał jako eksperyment intelektualny wynikający z poszukiwania przyczyn masy i grawitacji, rozwijany przeze mnie bez wcześniejszego przygotowania fizycznego i matematycznego, przy wsparciu sztucznej inteligencji, która umożliwiła przełożenie intuicji na formalizm matematyczny: od równań Lagrange’a i Hamiltona, przez kwantyzację, po zastosowania technologiczne.

Mam nadzieję, że ten koncept — nawet jako niedokończony szkic roboczy — zainspiruje innych badaczy do:
• poszukiwania prostszych modeli fizycznych opartych na falach i rezonansie,
• prób technologicznej implementacji nowych koncepcji (np. napęd rezonansowy, zmienna masa, fale informacyjne),
• wyznaczenia nowych ścieżek dla badań nad naturą masy, pola, światła i struktury Wszechświata.
Zapraszam do dalszego rozwijania tej idei — jako otwartej platformy do eksperymentów intelektualnych i naukowych.

W niniejszej pracy przedstawiam Teorię Dwóch Energii (TDE) 2.0 – koncepcyjny model, w którym masa cząstek elementarnych jawi się jako energia rezonansowej interferencji dwóch pól skalarnych: globalnego pola tła ψ_c oraz lokalnego oscylatora ψ_s. Model zakłada możliwość istnienia konfiguracji solitonowych oraz zależność wartości masy od względnej fazy Δϕ między polami, co otwiera drogę do potencjalnych stanów o masie dodatniej i ujemnej. Koncepcja została wypracowana przy aktywnym wsparciu narzędzi sztucznej inteligencji (w szczególności ChatGPT), co umożliwiło szybkie iteracje pomysłów i gruntowną analizę formalną.

Uwaga: przedstawiona wersja TDE 2.0 pozostaje szkicem roboczym. Dotychczasowa analiza wykazała liczne luki formalne (m.in. brak pełnego rozwiązania solitonowego, kwestie symetrii cechowania oraz problem ujemnej masy), które wymagają dalszych badań i korekty. Pomimo pierwotnej atrakcyjności koncepcji, kolejne prace ujawniły szereg błędów i nieścisłości. Planuje sukcesywne dopracowanie modelu oraz uzupełnianie brakujących elementów teoretycznych i fenomenologicznych.

The equations of polyconvex elastodynamics can be embedded to an augmented symmetric hyperbolic system. This property provides a stability framework between solutions of the viscosity approximation of polyconvex elastodynamics and smooth solutions of polyconvex elastodynamics. We devise here a model of stress relaxation motivated by the format of the enlargement process which formally approximates the equations of polyconvex elastodynamics. The model is endowed with an entropy function which is not convex but rather of polyconvex type. Using the relative entropy we prove a stability estimate and convergence of the stress relaxation model to polyconvex elastodynamics in the smooth regime.

Streszczenie. artykuł jest kontynuacją oraz uzupełnieniem badań prezentowanych w . na podstawie macierzy indukowanych przez schematy różnicowe aproksymujące operator laplace'a wyprowadzono i opisano w nim widmowe operatory przejścia masek konwolucyjnych filtrów krawędziowych laplace'a trzeciego stopnia wykorzystywanych w procedurach przetwarzania obrazów cyfrowych. oznacza to, że macierze prezentowane w niniejszej pracy odpowiadają maskom krawędziowych filtrów laplace'a -szeroko omówionych w [6] -z dokładnością do znaku (-) (porównaj ). widmowy (spektralny) operator przejścia, operator (funkcja) wzmocnienia, funkcja transformacji (lub przejścia) (transfer function) -krótko transformata f(k, l) -ma szczególne znaczenie w przetwarzaniu zdjęć cyfrowych, bowiem opisuje własności spektralne masek liniowych filtrów konwolucyjnych. wartości tej funkcji tworzą macierz mnożników zależnych tylko od liczb falowych k i l w taki sposób, że każdej parze (k, l) funkcja ta przypisuje liczbę f p (k, l), przez którą w procesie konwolucji jest mnożony filtrowany obraz (indeks j wprowadzono w celu identyfikacji filtru). funkcje wzmocnienia dwóch masek operatora laplace'a przedstawia Jähne w [4]. analizowane maski pochodzą bezpośrednio od schematów różnicowych operatora ∇ 2 konstruowanych różnymi metodami (różnic skończonych i elementu skończonego). na zamieszczone tu wyniki można więc również spojrzeć od strony najważniejszych własności metod numerycznych (w odniesieniu do zagadnień stacjonarnych): zgodności i dokładności aproksymacji różnicowej (patrz np. [3, 11, 13]). w celu ułatwienia wspomnianej analizy w pracy są zamieszczone tzw. Π-formy pierwszego przybliżenia różniczkowego. Podstawowym celem prezentowanej pracy jest, przede wszystkim, poszerzenie i uporządkowanie wiedzy na temat filtrów laplace'a trzeciego stopnia, a przy okazji wyjaśnienie ich własności spektralnych, które najwięcej mówią o cechach filtrów w obszarze dużych nieregularności, czyli znacznych różnic jasności w blisko położonych punktach zdjęcia cyfrowego. w tej publikacji nie będziemy się jednak

Niniejszy artykuł przedstawia matematyczny model powstawania masy w ramach koncepcyjnej Teorii Dwóch Energii (TDE) — koncepcji zakładającej, że cała materia wyłania się z napięcia rezonansowego między dwiema współistniejącymi falami: Energią Cienia i Energią Światła. W przeciwieństwie do klasycznych i kwantowych modeli pola, w których masa traktowana jest jako cecha wewnętrzna lub wynik oddziaływania z polem (np. polem Higgsa), model TDE opisuje masę jako wielkość emergentną — bezpośrednio wynikającą z różnicy fazowej pomiędzy falami.

Model wprowadza całkę rezonansową, która określa lokalną gęstość energetyczną powstałą na skutek interferencji falowej. Kluczowym wynikiem jest identyfikacja maksimum masy przy przesunięciu fazowym δ = 0,5 oraz jej zaniku przy δ = 0 lub 1. Opracowane równania umożliwiają zarówno obliczenia bezpośrednie (masa z przesunięcia i długości rezonansowej), jak i odwrotne (wyznaczenie przesunięcia fazowego z masy i długości). Zastosowanie modelu do rzeczywistych mas atomowych pozwoliło wyznaczyć spójne długości rezonansowe i potwierdzić wartość predykcyjną teorii. Przykładowo, przesunięcie fazowe wyliczone dla atomu tlenu wynosi δ ≈ 0,5029 — co wspiera poprawność modelu.

Formalizm ten oferuje nowe podejście do opisu materii, w którym masa nie jest stałą właściwością, lecz zjawiskiem rezonansowym zakorzenionym w dynamice falowej. Model ten może stanowić podstawę dalszych badań w dziedzinie fizyki cząstek, inżynierii energetycznej oraz w kierunku koncepcyjnego połączenia fizyki z filozoficznymi refleksjami nad dualizmem i strukturą rzeczywistości.

Dark energy is a fundamental yet poorly understood component of the universe, responsible for its accelerating expansion. Observations from the Wilkinson Microwave Anisotropy Probe (WMAP) suggest that dark energy constitutes approximately 71.35% of the total energy density of the universe. This study explores a theoretical framework that derives the dark energy contribution using general relativity and Newtonian mechanics without modifying existing gravitational laws. By modelling gravitational interactions as an elastic system, we propose that potential energy stored in space transforms into kinetic energy during cosmic collapse. A novel equation, incorporating an inverse fourth power force law derived from relativistic effects, is used to compute the kinetic energy fraction in a collapsing universe. The resulting dark energy to total energy ratio is calculated as 71.5%, in close agreement with observational data. This finding suggests that dark energy can be interpreted as the stored potential energy of gravitational interactions, offering a theoretical basis within classical general relativity. These results provide a new perspective on dark energy without requiring exotic modifications to fundamental physics.

The physical mechanism responsible for nonlinear elastic, hysteretic, and discrete memory response of nonlinear mesoscopic elastic solids has to date not been identified. We show, by nanoindenting mica single crystals, that this response is most likely due to the formation of dissipative and fully reversible, dislocation-based kink bands. We further claim that solids with high c=a ratios, which per force are plastically anisotropic, should deform by kinking, provided they do not twin. These kinking nonlinear elastic solids include layered ternary carbides, nitrides, oxides, and semiconductors, graphite, and the layered phases, such as mica, present in nonlinear mesoscopic elastic solids.

Recently we identified and classified a class of solids as kinking nonlinear elastic ͑KNE͒ because they deform by the formation of kink bands. KNE solids represent a large family that include, among others, layered ternary carbides and nitrides, layered oxides and semiconductors, zinc, cadmium, graphite, ice, and the layered silicates, such as mica, present in nonlinear mesoscopic elastic solids. Herein we present a microscale model that accounts for the mechanical response of KNE solids to compressive stresses and apply it to two very different solids: Ti 3 SiC 2 and graphite. Building on the Frank and Stroh model put forth in the 1950's for the formation of kink bands, we developed a comprehensive theory that accounts for the contributions of incipient kink bands ͑IKBs͒ and dislocations pile-ups produced by normal glide processes to the nonlinear strains and stored strain energies. The theory provides estimates for the densities of IKBs, the dislocation densities, both from the IKBs and dislocation pileups, as well as the energy dissipated by the motion of the dislocations.

Recent direct numerical simulations of the finite-extensibility nonlinear elastic dumbbell model with the Peterlin approximation of non-Newtonian hydrodynamics revealed that the phenomenon of drag reduction by polymer additives exists (albeit in reduced form) also in homogeneous turbulence. We use here a simple shell model for homogeneous viscoelastic flows, which recaptures the essential observations of the full simulations. The simplicity of the shell model allows us to offer a transparent explanation of the main observations. It is shown that the mechanism for drag reduction operates mainly on large scales. Understanding the mechanism allows us to predict how the amount of drag reduction depends on the various parameters in the model. The main conclusion is that drag reduction is not a universal phenomenon; it peaks in a window of parameters such as the Reynolds number and the relaxation rate of the polymer.