Classical Nonlinear Theory Research Articles

Theoretical description of the electrical double layer for a mixture of n ionic species with arbitrary size and charge asymmetries. I. Spherical geometry.

In this work, we propose a theoretical finite element description of the ionic profiles of a general mixture of n species of spherical charged particles dissolved in an implicit solvent, with arbitrary size and charge asymmetries, neutralizing a spherical macroion. This approach aims to close the gap between the nano- and the micro-scales in macroion solutions, taking into account the ion correlations and ionic excluded volume effects consistently. When these last two features are neglected, the classical non-linear Poisson-Boltzmann theory for n ionic species-with different ionic closest approach distances to the colloidal surface-is recovered as a limit case. As a proof of concept, we study the electrical double layer of an electroneutral mixture of oppositely charged colloids and small microions, with an asymmetry 1:333 in size and 1:10 in valence, in salt-free and added salt environments. Our theoretical approach displays a good agreement regarding the ionic profiles, the integrated charge, and the mean electrostatic potential obtained from molecular dynamics simulations with explicit-sized microions. Although the non-linear Poisson-Boltzmann colloid-colloid and colloid-microion profiles differ notably from those obtained via molecular dynamics simulations with explicit small-sized ions, the associated mean electrostatic potential agrees well with the corresponding explicit microion simulations.

On the emergence of the second harmonic shear horizontal wave in presence of tangential prestress

Nonlinear ultrasonic techniques have significant advantages in characterizing incipient defects of materials. The classical nonlinear ultrasonic theory indicates that a primary longitudinal wave can generate a second harmonic longitudinal wave due to the material nonlinearity (MN) or contact acoustic nonlinearity (CAN). However, in both MN and CAN, a second harmonic shear horizontal (SH) component generated from a normally-incident SH wave has not been reported. In this work, a time domain spectral finite element method combined with the bi-potential contact theory is extended to simulate the CAN problems with prestresses. Numerical verifications are performed to show the robustness and accuracy of the developed method. After that, the nonlinear interactions between the SH wave and frictional interfaces under complex prestresses are investigated. It is found for the first time that when there are tangential prestresses at the frictional interfaces or crack surfaces, a primary SH wave of normal incidence can generate the second harmonic SH wave. A theoretical model is then proposed to interpret this new phenomenon. It is found that the even SH wave harmonics result from the asymmetric sliding of the frictional interface induced by the tangential prestresses. This asymmetric sliding mechanism can be compared with the breathing behavior of a crack caused by an incident longitudinal wave. The findings of this work could enrich the understanding of contact acoustic nonlinearity behavior of the SH wave and promote the development of related nondestructive evaluation techniques.

STUDY OF THE INFLUENCE OF THE KINETICS OF HYDROGEN SATURATION ON THE STRESS-DEFORMED STATE OF A SPHERICAL SHELL MADE FROM TITANIUM ALLOY

In this paper, a mathematical model is considered that allows one to determine the stress-strain state of a spherical shell made of titanium alloy VT1-0, the external load is assumed to be transverse uniformly distributed, acting on the outer surface, the medium is assumed to act on the inner surface of the shell. For this, a nonlinear model was used, presented in normalized stress spaces. Fastening along the contour of the shell is rigid. Nonlinear resolving equations for calculating a spherical shell are obtained. An algorithm for solving the problem of hydrogenation of titanium alloy shells has been developed. A practical solution was made by a two-step method of sequential perturbations of parameters using the MatLab and Maple software packages. To solve the system of the obtained differential equations, the method of finite differences is applied. The calculation of the stress-strain state of the shell is obtained taking into account the diffusion process of an aggressive hydrogen-containing medium, and the obtained solution is compared with the results of the classical nonlinear theory without taking into account the aggressive medium. The results of comparing these solutions demonstrate quantitative differences in the parameters of the shell deformation process, which are explained by a more accurate account of the influence of the type of stress state. This approach has a rather flexible mechanism for considering the initial and induced differential resistance, demonstrates a high accuracy of matching the obtained theoretical results with empirical data on loading a wide range of materials under complex types of stress state.

Natural vibrations and stability of loaded cylindrical shells partially filled with fluid, taking into account gravitational effects

The paper presents the results of studying circular cylindrical shells partially filled with an ideal liquid and subjected to uniform external and internal hydrostatic pressure. The behavior of an elastic structure and a compressible fluid is described in the framework of the classical nonlinear theory of shells, based on the Kirchhoff – Love hypotheses, and the Euler equations. The problem is solved using a semi-analytical version of the finite element method. The influence of the level of fluid in the shell on the critical values of external pressure is analyzed with and without consideration of gravitational effects on the free and lateral surfaces of the fluid. Shells with different boundary conditions and linear dimensions are considered. It has been shown that for certain geometrical parameters the gravitational field can significantly affect the dynamic characteristics of the structure.

Forced Nonlinear vibration and bifurcation analysis of circular cylindrical nanocomposite shells using the normal form

The current paper focuses on investigating the effect of different distribution types of single-walled carbon nanotube (SWCNT) reinforcement and the volume fraction of CNTs on the nonlinear vibration of simply supported nanocomposite circular cylindrical shells. The governing equations are derived for uniform and three kinds of FG distribution of CNTs utilizing the extended mixture rule and Hamilton principle in cylindrical coordinate employing a classical nonlinear shell theory. The considered 1:1 internal resonance between asymmetric driven and companion vibration modes profoundly influences the frequency responses’ nonlinear characteristics. Then by the aids of the Airy stress function and the Galerkin method, the motion equations turn into several nonlinear nonhomogeneous ODEs. Finally, the bifurcation analysis is carried out based on various parameters using the normal form method. Three stable, unstable, and quasi-period regions concerning the locations of saddle nodes, pitchfork bifurcation, and torus points at the limit cycle around the resonance condition would differ due to change the structure of the nanocomposite shells, viscous damping coefficient, and the amplitude of the concentrated external force. The perturbation method’s accuracy is compared against the Runge–Kutta 4th order method’s results validated by available data.

Simulation of Three Constitutive Behaviors Based on Nonlinear Ultrasound

Nonlinear ultrasound has attracted more and more attention. In classical acoustic nonlinear theory, the source of nonlinearity is the change of constitutive relation of materials. Structure response that distorts after a single tone ultrasound wave is important to detect imperfection. This is rarely found in current simulations. The current simulation always introduces defects which do not match to the classical acoustic nonlinear theory. In this manuscript, the recurrence expressions of three kinds of imperfect materials for subroutine are given. The verifying simulation model that is used for verifying recurrence equations and wave propagation model that are used for analysing the process of ultrasonic propagation are established. The results show that the two constitutive models are effective in the verifying simulation and the hysteresis material has some special characteristics. Finally, ultrasonic propagation in two types of materials produce the expected harmonics, which build foundations for simulations of nonlinear ultrasound.

Numerical study of nonlinear elastic wave propagation in locally damaged engineering structures

AbstractIn this paper, the two‐dimensional (2‐D) ultrasonic wave propagation problem in an elastic half‐space with a localized damaged zone is numerically simulated by a mapped Chebyshev pseudo‐spectral collocation method [1]. The considered damaged zone, which is embedded in an undamaged host medium, is modelled by a nonlinear elastic and viscoelastic constitutive law. Classical nonlinear elastic and nonclassical hysteretic material behaviors are separately considered and evaluated. In particular, the classical nonlinear elasticity theory of Murnaghan [2] is implemented, while the hysteretic material behavior is modelled by the Duhem‐model. To transfer the constitutive relations from the one‐dimensional (1‐D) to the 2‐D case the Kelvin decomposition method is used. Furthermore, Convolutional Perfectly Matched Layers (CPML's) are used to simulate the semi‐infinite elastic half‐space. The computed time‐domain signals are transformed to the frequency‐domain and subsequently analyzed for different degrees of the wave attenuation and nonlinearity in the damage zone. The applications of the nonlinear ultrasonic technique are discussed based on the numerical results.

Numerical Analysis of the Nonlinear Interactions Between Lamb Waves and Microcracks in Plate

In this paper, three-dimensional finite-element modeling is conducted to investigate the nonlinear interactions between Lamb waves and microcracks. The simulation research focuses on the influence of microcrack orientation on the propagation direction of generated sum-frequency Lamb waves. The simulation results show that the resonant conditions based on classical nonlinear theory are valid for such interactions, leading to the generation of transmitted and reflected sum-frequency S0 waves (SFSWs). Moreover, the propagation directions of these two SFSWs exhibit different trends with respect to the orientations of microcracks. The transmitted SFSW can be used to detect microcracks, whereas the reflected one can be used to measure their orientations.

Curvature Expressions for the Large Displacement Analysis of Planar Beam Motions

While several curvature expressions have been used in the literature, some of these expressions differ from basic geometry definitions and lead to kinematic coupling between bending and shear deformations. This paper uses three different elastic force formulations in order to examine the effect of the curvature definition in the large displacement analysis of beams. In the first elastic force formulation, a general continuum mechanics approach (method 1) based on the nonlinear strain–displacement relationship is used. The second approach (method 2) is based on a classical nonlinear beam theory, in which a curvature expression consistent with differential geometry and independent of the shear deformation is used. The third elastic force formulation (method 3) employs a curvature expression that depends on the shear angle. In order to examine numerically the effect of using different curvature definitions, three different planar beam elements are used. The first element (element I) is the fully parameterized absolute nodal coordinate formulation (ANCF) shear deformable beam element. The second element (element II) is an ANCF consistent rotation-based formulation (CRBF) shear deformable beam element obtained from element I by consistently replacing the position gradient vectors by rotation parameters. The third element (element III) is a low-order bilinear ANCF/CRBF finite element in which nonzero differential geometry-based curvature definition cannot be obtained because of the low order of interpolation. Numerical results are obtained using the three elastic force formulations and the three finite elements in order to shed light on the definition of bending and shear in the large displacement analysis of beams. The results obtained in this investigation show that the use of method 2, with a penalty formulation that restricts the excessive cross section deformation, can improve significantly the convergence of the ANCF finite element.

Nonlinear Lamb waves in plate/shell structures

In view of the fact that conventional ultrasonic detection technology is not effective for the detection of distributed fne damage and contact-type structural damage in materials, nonlinear ultrasound technology has been attracting more and more attention in recent years. Ultrasonic waves propagate in the form of Lamb wave modes in the structure of plates and shells. However, due to the dispersion and multimode characteristics of Lamb waves, the progress of theoretical and experimental research on nonlinear Lamb waves is slow. Based on the classical nonlinear theory, this paper summarizes the theoretical and experimental research progress of nonlinear Lamb waves induced by the inherent nonlinear-ity of material, and summarizes the application of the second harmonics of Lamb waves in damage evaluation of material. The current status of the research on nonlinear Lamb waves induced by contact-type structural damage is discussed based on contact acoustic nonlin-earity. At the end, this paper envisions the future research topics and trends in nonlinear Lamb waves.

Large transformations with moderate strains of tensile membrane structures

Using a classical non-linear theory, we analytically investigate possible ways for transforming the shape of a curved elastic membrane while keeping it tensioned and moderately strained. This is a critical issue because, as a rule, membranes must be considerably stretched in order to avoid wrinkling and slackening. If the final configuration is fixed, the membrane can be cut and formed according to the final shape, but this cannot be done if more configurations, considerably distant from one another, have to be achieved. Nevertheless, we propose large transformation movements that can be obtained starting from flat membranes while maintaining their strain as limited. We discuss in detail the paradigmatic example of the hyperbolic-paraboloid-shaped membrane. These opportunities are suitable for applications of transformable architecture because they do not require excessive tensioning, compatible with the strength of materials used for this kind of structures.

Quantum filtering for multiple input multiple output systems driven by arbitrary zero-mean jointly Gaussian input fields

In this paper, we treat the quantum filtering problem for multiple input multiple output (MIMO) Markovian open quantum systems coupled to multiple boson fields in an arbitrary zero-mean jointly Gaussian state, using the reference probability approach formulated by Bouten and van Handel as a quantum version of a well-known method of the same name from classical nonlinear filtering theory, and exploiting the generalized Araki-Woods representation of Gough. This includes Gaussian field states such as vacuum, squeezed vacuum, thermal, and squeezed thermal states as special cases. The contribution is a derivation of the general quantum filtering equation (or stochastic master equation as they are known in the quantum optics community) in the full MIMO setup for any zero-mean jointy Gaussian input field states, up to some mild rank assumptions on certain matrices relating to the measurement vector.

Nonlinear ultrasonic waves for monitoring thermal stresses in solids

It is known that nonlinear ultrasonic waves in solids are sensitive to quasi-static stresses. The stress sensitivity of elastic waves is typically associated to finite strains (e.g., theory of acoustoelasticity). In the case of waveguides, classical nonlinear theories for guided waves are still based on the assumption of finite strains. In the case of constrained solids subjected to thermal excursions, however, there are theoretically no finite strains (for perfectly constrained solids) associated with thermal stresses. A new model is therefore needed to justify the existence of wave nonlinearities in this case of stress without strain. This problem is solved on the basis of the interatomic potential of the solid that indicates a “residual” strain energy, due to the prevented thermal expansion, which is at least cubic as a function of strain. Consequently, a nonlinear wave equation can be derived. The solution to this equation leads to a new nonlinear parameter for double harmonic generation that is directly related to the thermal stresses in the structure. This study finds applications in the monitoring of thermal stresses in buckling-prone structures, such as continuously welded railroad tracks and pipelines. Experimental tests conducted on railroad tracks with realistic support will be also presented.

Nonlinear theory of intense laser-plasma interactions modified by vacuum polarization effects

The classical nonlinear theory of laser-plasma interactions is corrected by taking account of the vacuum polarization effects. A set of wave equations are obtained by using the Heisenberg-Euler Lagrangian density and the derivative correction with the first-order quantum electrodynamic effects. A model more suitable to formulate the interactions of ultra-strong lasers and high-energy-density plasmas is developed. In the result, some environments in which the effects of vacuum polarization will be enhanced are discussed.

A Study on Flexural Properties of Sandwich Structures with Fiber/Metal Laminate Face Sheets

In this work, a new family of sandwich structures with fiber metal laminate (FML) faces is investigated. FMLs have benefits over both metal and fiber reinforced composites. To investigate the bending properties of sandwich beams with FML faces and compare with similar sandwich beams with fibrous composite faces, 6 groups of specimen with different layer arrangements were made and tested. Results show that FML faces have good resistance against transverse local loads and minimize stress concentration and local deformations of skin and core under the loading tip. In addition, FML faces have a good integrity even after plateau region of foam cores and prevent from catastrophic failures, which cannot be seen in fibrous composite faces. Also, FML faces are lighter than metal faces and have better connection with foam cores. Sandwich beams with FML faces have a larger elastic region because of simultaneous deformation of top and bottom faces and larger failure strain thanks to good durability of FMLs. A geometrical nonlinear classical theory is used to predict force-deflection behavior. In this model an explicit formula between symmetrical sandwich beams deflections and applied force which can be useful for designers, is derived. Good agreement is obtained between the analytical predictions and experimental results. Also, analytical results are compared with small deformation solution in a parametric study, and the effects of geometric parameters on difference between linear and nonlinear results are discussed.

Orientation-preserving condition and polyconvexity on a surface: Application to nonlinear shell theory

We propose in this paper a definition of a “polyconvex function on a surface”, inspired by the definitions set forth in other contexts by J. Ball (1977) [3] and by J. Ball, J.C. Currie, and P.J. Olver (1981) [5]. When the surface is thought of as the middle surface of a nonlinearly elastic shell and the function as its stored energy function, we show that it is possible to assume in addition that this function is coercive for appropriate Sobolev norms and that it satisfies specific growth conditions that prevent the vectors of the covariant bases along the deformed middle surface to become linearly dependent, a condition that is the “surface analogue” of the orientation-preserving condition of J. Ball. We then show that a functional with such a polyconvex integrand is weakly lower semi-continuous, a property which eventually allows to establish the existence of minimizers. We also indicate how this new approach compares with the classical nonlinear shell theories, such as those of W.T. Koiter and P.M. Naghdi.

The 2011 Koiter Lecture: The Simple Logic of Classical Nonlinear Thermodynamic Shell Theory

A classical nonlinear thermodynamic theory of elastic shells is derived by specializing the three-dimensional equations of motion and the second law of thermodynamics to a very general, shell-like body. No assumptions are made on how unknowns vary through the thickness. Extensional and bending strains are derived from the equations of motion via the principle of virtual power. The Coleman-Noll procedure plus the second law applied to an assumed form of the first law leads to constitutive relations plus reduced forms of the first and second laws. To avoid potential ill conditioning, a Legendre-Fenchel transformation is used to define a mixed-energy density, the logical place to impose the constitutive Kirchhoff hypothesis, if desired, because such an energy density rests, ultimately, on experiments. The Ladevèze-Pécastaings treatment of three-dimensional edge effects to obtain accurate two-dimensional solutions is discussed.

A new locking-free formulation for planar, shear deformable, linear and quadratic beam finite elements based on the absolute nodal coordinate formulation

Many widely used beam finite element formulations are based either on Reissner’s classical nonlinear rod theory or the absolute nodal coordinate formulation (ANCF). Advantages of the second method have been pointed out by several authors; among the benefits are the constant mass matrix of ANCF elements, the isoparametric approach and the existence of a consistent displacement field along the whole cross section. Consistency of the displacement field allows simpler, alternative formulations for contact problems or inelastic materials. Despite conceptional differences of the two formulations, the two models are unified in the present paper. In many applications, a nonlinear large deformation beam element with bending, axial and shear deformation properties is needed. In the present paper, linear and quadratic ANCF shear deformable beam finite elements are presented. A new locking-free continuum mechanics based formulation is compared to the classical Simo and Vu-Quoc formulation based on Reissner’s virtual work of internal forces. Additionally, the introduced linear and quadratic ANCF elements are compared to a fully parameterized ANCF element from the literature. The performance of the respective elements is evaluated through analysis of conventional static and dynamic example problems. The investigation shows that the obtained linear and quadratic ANCF elements are advantageous compared to the original fully parameterized ANCF element.

Quality analysis of nonlinear elasticity theory for the stability problems of planar laminated curved beams. Algorithm and results of numerical study

The algorithm of numerical solution for the linearized equations of the neutral equilibrium earlier constructed for the planar laminated curved beams is presented. The equations are based on the relations of the classical nonlinear theory of elasticity and the similar relations previously proposed in the consistent variant. We analyzed the results of numerical quality studies of the mentioned variants of nonlinear equations used when the linearized problems on possible buckling modes under the action of conservative forces and a stationary temperature field are formulated.

On the lightest baryon and its excitations in large-N (1 + 1)-dimensional QCD

We study baryons in multicolour QCD1 + 1 via Rajeev's gauge-invariant reformulation as a nonlinear classical theory of a bilocal meson field constrained to lie on a Grassmannian. It is known to reproduce 't Hooft's meson spectrum via small oscillations around the vacuum, while baryons arise as topological solitons. The lightest baryon has zero mass per colour in the chiral limit; we find its form factor. It moves at the speed of light through a family of massless states. To model excitations of this baryon, we linearize equations for motion in the tangent space to the Grassmannian, parameterized by a bilocal field U. A redundancy in U is removed and an approximation is made in lieu of a consistency condition on U. The baryon spectrum is given by an eigenvalue problem for a Hermitian singular integral operator on such tangent vectors. Excited baryons are like bound states of the lightest one with a meson. Using a rank-1 ansatz for U in a variational formulation, we estimate the mass and form factor of the first excitation.