Nonlinear Elastic Plate Research Articles

Frequency sweep study on the generation of dual-mode second harmonics (DMSH) on an isotropic nonlinear elastic cylindrical rod by T(0,1) mode

Nonlinear ultrasonic (NLU) guided wave (GW) measurements are more sensitive to microscale defects than linear measurements. This nonlinear phenomenon is helpful for early-stage damage detection in material for nondestructive evaluation (NDE) and structural health monitoring (SHM) applications. The amplitude of the higher harmonics generated in the material serves as the basis for the NLU measurement. Recent studies have reported the presence of dual-mode second harmonic (DMSH) on an isotropic nonlinear elastic plate and cylindrical structures. The fundamental non-dispersive mode shear horizontal SH0 and torsional T(0,1) modes can generate simultaneously propagating dual-mode second harmonic(DMSH) on plate and cylindrical guided media respectively. For the case of cylinders, at phase matching condition two dominant second harmonic fundamental longitudinal (l(0,1)) and orthogonal torsional (t(0,1)┴) mode are generated with significant amplitude. The particle vibration of the t(0,1)┴ mode was present in both orthogonal directions compared to conventional T(0,1) mode and hence called orthogonal torsional mode. The present work aims to study the behaviour of DMSH at different frequencies through a frequency sweep study on a weakly nonlinear elastic cylinder of circular cross-section via numerical simulations and validated experimentally for some selected cases. The wave propagation of the T(0,1) mode at several selected frequencies from 300 kHz to 2.2 MHz is chosen to explore the second harmonic mode for better understanding. At frequencies below 1.25 MHz, l(0,1) mode is faster than t(0,1)┴ mode; above 1.25 MHz, it was the other way around. The behaviour of harmonics was observed to be in accordance with the group and phase velocities at the corresponding frequencies in the dispersion curves. The evidence for the presence of DMSH was validated experimentally for some selected frequency cases at different propagation distances. The significance of DMSH is explored via numerical simulations and found that t(0,1)┴ mode is sensitive towards material degradation, while all other generated wave modes were not affected. Research shows that enough energy is available in the generated DMSH to be noticed experimentally and in numerical simulations. This improved understanding of the generation of DMSH across different frequencies would be helpful in new advanced NLU-based NDE and SHM applications.

Analysis of Berger nonlinear elastic static plate bending of rectangular plates

This paper deals with a nonlinear static plate model based on Berger theory, which is a specific case of a generalization of the Woinowsky-Krieger mathematical model of beam bending. It is considered a plate bending with forces acting in the middle plane of the plate and a contact problem with an elastic foundation, where the normal compliance condition is employed. A variational equation of the problem and a functional of the total potential energy corresponding to the variational equation are derived. Under additional assumptions on the data (e.g., clamped plate), the existence and uniqueness of the solution are proved. A numerical solution is based on the Galerkin method and Courant approximation. The theory is illustrated by a numerical example.

Mutual Interactions of Lamb Waves in Nonlinear Elastic Plates

The mutual interactions of Lamb waves in nonlinear elastic plates are studied in this article. Many researchers have investigated the interactions of Lamb wave modes at nonlinear higher harmonics. However, little is known about nonlinearity-driven Lamb modulations from two primary modes with different frequencies. In this study, the existence of symmetric or antisymmetric mode due to Lamb wave mutual interactions is firstly theoretically formulated. Then, an approach is proposed to evaluate the intensity of phase velocity matching for selecting primary modes. Finally, the characteristics of the modulated wave generation are investigated and demonstrated. The generation of modulated waves in an aluminum plate and fatigue crack can be detected by mutual interactions of Lamb waves. The main contribution of this work is the proposed mutual interaction theory of Lamb waves in fatigue plates, which can guide fatigue detection in the metal plate.

Intrinsic formulation of the Kirchhoff–Love theory of nonlinearly elastic plates

The classical Kirchhoff–Love theory of a nonlinearly elastic plate provides a way to compute the deformation of such a plate subjected to given applied forces and boundary conditions by solving the Euler–Lagrange equation associated with a specific minimization problem, whose unknown is the displacement field of the middle surface of the plate. We show that this Euler–Lagrange equation is equivalent to a boundary value problem whose sole unknowns are the bending moments and stress resultants inside the middle surface of the plate, without any reference to the displacement field in its formulation. As a result, computing the stresses inside deformed plate can be done without using the derivatives of the corresponding displacement field, usually a source of numerical instabilities.

Sobolev gradient type iterative solution methods for a nonlinear 4th order elastic plate equation

This paper considers the numerical solution of an elastic bending model of a thin plate, based on material nonlinearity, which leads to a nonlinear elliptic 4th order boundary value problem. Our goal is to summarize possible efficient iterative solvers, adapted to this problem, based on a Sobolev space background. It is proved that the proposed methods exhibit robust behaviour, that is, convergence rates are bounded independently of the considered Galerkin discretization subspace.

Polymer curing assisted formation of optically visible sub-micron blisters of multilayer graphene for local strain engineering

The local or global straining techniques are used to modulate the electronic, vibrational and optical properties of the two-dimensional (2D) materials. However, manipulating the physical properties of a 2D material under a local strain is comparatively more challenging. In this work, we demonstrate an easy and efficient polymer curing assisted technique for the formation of optically visible multilayer graphene (MLG) blisters of different shapes and sizes. The detailed spectroscopic and morphological analyses have been employed for exploring the dynamics of the confined matter inside the sub-micron blisters, which confirms that the confined matter inside the blister is liquid (water). From further analyses, we find the nonlinear elastic plate model as an acceptable model under certain limits for the mechanical analyses of the MLG blisters over the (poly)vinyl alcohol (PVA) polymer film to estimate the MLG-substrate interfacial adhesion energy and confinement pressure inside the blisters. The findings open new pathways for exploiting the technique for the formation of sub-micron blisters of the 2D materials for local strain-engineering applications, as well as the temperature-controlled release of the confined matter.

Natural Vibrations of Micropolar Elastic Flexible Plates and Shallow Shells

A mathematical model representing the dynamics of geometrically nonlinear (flexible) micropolar elastic thin plates in Cartesian and curvilinear coordinates is constructed (the approach is generalized to the case of micropolar flexible shallow shells as well). The model is developed under the assumption that the elastic deflection of a plate is comparable with the plate thickness but is small compared to the characteristic plate size in plan. Based on the given model of micropolar elastic flexible plates, the problem on free vibrations is solved for rectangular and circular plates and shallow shells. Effective manifestations of characteristic features of a micropolar material are considered in comparison with the corresponding classical material.

Cubic nonlinearity parameter measurement and material degradation detection using nonlinear ultrasonic three-wave mixing

Early detection of accumulated damage or material degradation in structures is important to ensure their structural safety. Nonlinear ultrasonic techniques are widely used to measure the quadratic nonlinearity parameter that represents the third-order elastic constants of materials for material degradation detection. In addition, there are ongoing efforts to exploit both the third- and fourth-order elastic constants that describe the cubic nonlinearity parameter to detect material degradation. This study develops a nonlinear ultrasonic three-wave mixing technique to generate and measure third-order combined harmonics (TOCH) in plate-like structures and to measure cubic nonlinearity parameter. The proposed three-wave mixing technique generates three primary Lamb waves in the target structure and measures the TOCH produced by nonlinear cross-interaction of the primary Lamb waves. A theoretical model is developed to describe the generation of TOCH in a nonlinear elastic and homogeneous plate, and the effectiveness of the theoretical model is validated experimentally. Measurements of the TOCH are conducted for intact and degraded aluminum specimens with different degradation levels. Because inherent material nonlinearity and material degradation alter the third- and fourth-order elastic constants of a structure, the three-wave mixing technique for measuring the TOCH can be used to identify the inherent material nonlinearity and material degradation. In particular, the experimental results indicate that the proposed technique is more sensitive to early-stage material degradation than existing nonlinear ultrasonic techniques such as two-wave mixing, and third harmonic generation techniques.

Observing dual-mode second-harmonic generation in an isotropic nonlinear elastic plate

In this paper, we discuss the observation of simultaneously propagating dual-mode second-harmonic fundamental symmetric (s0) mode and a vertical-shear horizontal (sh0⊥) mode generated on an aluminium plate of 1 mm thickness while excited with a fundamental shear horizontal (SH0) mode in a large domain. The experimental observations, and numerical models, were used to confirm the presence of dual-mode second-harmonics. The two wavemodes would separate from each other in the time-domain, wherein the travel length exceeded a minimum cut-off length of 40λ. This approach can provide important implications for more versatile early state defect detection.

Method of fundamental solutions and a high order continuation for bifurcation analysis within Föppl-von Karman plate theory

An efficient numerical model is developed in this work for the bifurcation analysis within Föppl-von Karman plate theory. This procedure is based on the use of the method of fundamental solutions in a high order continuation method that relies on Taylor series expansion with respect to a path parameter. This numerical model has an adaptive step length, which is effective especially for solving nonlinear problems and detecting bifurcation points. Despite of the huge application field of nonlinear elasticity, the method of fundamental solution was very rarely applied in this field and never to a nonlinear elastic plate model. The governing equations are strongly formulated in terms of the two unknowns: the deflection and the stress function. The singular value decomposition regularization is used to overcome the difficulty of the resulting system ill-conditioning. The accuracy and efficiency of the numerical model are illustrated on buckling numerical examples.

Stability of a nonlinear elastic plate under lateral compression

Introduction . Loss of stability and buckling of a round plate may be observed if the plate is loaded on the lateral surface. The solution to this problem is based on a bifurcation approach. In this case, a plate is considered as a nonlinear elastic body. It is important to choose the relation between stresses and deformations in sustainability problems of nonlinear elasticity. Simple laws of state (constitutive equations) were considered in early works devoted to this problem, for example, material of the “harmonic type” (Sensenig). Materials and Methods . Equations of neutral equilibrium for round plates made of Murnaghan and Blatz-Ko materials are obtained. Assuming a uniform initial deformation on the plate, the stability problem is considered. Strict threedimensional neutral equilibrium equations provide exploring related forms of equilibrium taking into account physical and geometric nonlinearity. Derivation of these equations is based on the application of the theory of superposition of small deformation on the final one. Results . Progress in solution to the corresponding secular equation (with non-linear parameter entry) for practically important laws of elasticity of Murnaghan and Blatz-Ko is possible using the numerical methods only. The developed method for calculating bifurcation values of loading parameters makes it possible to analyze the effect of nonlinearity. Discussion and Conclusions. The influence of physical and geometric nonlinearity on the upper critical value of the initial deformation parameter is explored. The results obtained can be used under the assessment of reliability of elastic third-order moduli for various physical materials. Data on these moduli is still scarce. The numerical research has shown that the constants given in some sources should be treated with caution. The use of elasticity moduli in the law of state of Blatz-Ko is also discussed.

Correction to: An improved quadrilateral finite element for nonlinear second-order strain gradient elastic Kirchhoff plates

This article was published with an erroneous version of Eq. 6 and therefore has been corrected.

On material optimisation for nonlinearly elastic plates and shells

This paper investigates the optimal distribution of hard and soft material on elastic plates. In the class of isometric deformations stationary points of a Kirchhoff plate functional with incorporated material hardness function are investigated and a compliance cost functional is taken into account. Under symmetry assumptions on the material distribution and the load it is shown that cylindrical solutions are stationary points. Furthermore, it is demonstrated that the optimal design of cylindrically deforming, clamped rectangular plates is non trivial, i.e. with a material distribution which is not just depending on one axial direction on the plate. Analytical results are complemented with numerical optimization results using a suitable finite element discretization and a phase field description of the material phases. Finally, using numerical methods an outlook on the optimal design of non isometrically deforming plates and shells is given.

Determination of the equivalent properties of periodic heterogeneous composite plates using the structural genome approach

A structural genome (SG) is defined as the smallest structural building block of composite materials. In this study, we use the concept of structure genome to propose a simple approach for obtaining the equivalent mechanical properties of periodic heterogeneous composite plates. Firstly, an SG from the structural model of a periodic heterogeneous composite plate is extracted. Secondly, a python script is developed in commercial software ABAQUS to conveniently impose periodic boundary conditions and apply different loads on the SG as well as obtain the equivalent mechanical properties of the structural genome. To verify the proposed SG approach, numerical examples are presented and discussed for both linear and nonlinear elastic cases. The results show that this approach is efficient and the results are of sufficient precision to calculate the mechanical properties of both linear and nonlinear elastic periodic heterogeneous composite plates.

Stability of a nonlinear elastic plate under uniaxial loading with respect to finite perturbations

The stability of a plate made of nonlinear elastic material with respect to finite perturbations is considered. The analysis of the main process of plate deformation is reduced to the solution of the nonlinear boundary value problem with respect to finite perturbations. Solutions for perturbations of displacements are chosen in the form of eigenfunction series, that are solutions of the corresponding linearized problems and satisfy the geometric boundary conditions. After applying the principle of possible displacements, the question on the stability of the ground state of the nonlinear problem is reduced to the study of the stability of zero solution of an infinite system of ordinary differential equations with constant coefficients, the number of terms in which is specified by the elastic potential. For the obtained system of equations under certain restrictions on initial perturbations, a Lyapunov function is constructed. The dimension of the strange attractor of the dynamical system is found which allows to limit the number of terms in the Bubnov-Galerkin series.

Transverse to longitudinal mode conversion via nonlinear inertial effects in a locally resonant metamaterial plate

Acoustic and elastic metamaterials have potential for tailoring wave propagation in ways beyond the capabilities of conventional materials and have been shown to exhibit a myriad of rich dynamical effects, such as negative effective properties, cloaking, extreme damping, and non-reciprocity. However, most of the existing literature has been focused on linear dynamics in which these behaviors are restricted to narrow frequency bands. More recently, nonlinearity has been investigated as a method of increasing this bandwidth, as well as a means to broaden the palette of available dynamics. Past studies on nonlinear acoustic and elastic metamaterials have demonstrated effects not easily accessible in the linear regime, including harmonic generation, mode hopping and conversion, tunable band gaps, chaos, and intrinsic localized modes. In this work, we present a model for a nonlinear elastic plate metamaterial that exhibits mode conversion between transverse and longitudinal waves. The mode conversion is enabled by cantilevered beam attachments undergoing large-angle out-of-plane oscillations. By analyzing the unit cell of this locally-resonant plate, we interpret the transverse-longitudinal coupling as a nonlinear, anisotropic effective mass density. The mode conversion effect is demonstrated via direct numerical simulations. [Work supported by NSF and ARL:UT.]

Head-on collision between two hydroelastic solitary waves with Plotnikov-Toland's plate model

ABSTRACT Head-on collision between two hydroelastic solitary waves propagating at the surface of an incompressible and ideal fluid covered by a thin ice sheet is analytically studied by means of a singular perturbation method. The ice sheet is represented by the Plotnikov-Toland model with the help of the special Cosserat theory of hyperelastic shells and the Kirchhoff-Love plate theory, which yields the nonlinear and conservative expression for the bending forces. The shallow water assumption is taken for the fluid motion with the Boussinesq approximation. The resulting governing equations are solved asymptotically with the aid of the Poincare-Lighthill-Kuo method, and the solutions up to the third order are explicitly presented. It is observed that solitary waves after collision do not change their shapes and amplitudes. The wave profile is symmetric before collision, and it becomes, after collision, unsymmetric and titled backward in the direction of wave propagation. The wave profile significantly reduces due to greater impacts of elastic plate and surface tension. A graphical comparison is presented with published results, and the graphical comparison between linear and nonlinear elastic plate models is also shown as a special case of our study.

Dispersion curves for Lamb wave propagation in prestressed plates using a semi-analytical finite element analysis.

Acoustoelastic techniques have been recently used to characterize the state of prestress in structures such as plates. The velocity of guided wave modes propagating through plates is sensitive to the magnitude and orientation of the initial state of stress. Dispersion curves for phase velocities of plate guided waves can be computed using the superposition of partial bulk waves (SPBW) method. Here, a semi-analytical finite element (SAFE) method is formulated for the acoustoelastic problem of guided waves in weakly nonlinear elastic plates. The SAFE formulation is shown to provide phase velocity dispersion curve results identical with those provided by the SPBW method for the problem of a plate under a uniaxial and uniform tensile stress. Analytical phase and group velocity dispersion curves are also obtained for a plate with an initial prestress gradient through its thickness using the SAFE method. The magnitude of the prestress gradient is shown to have a significant effect on phase and group velocities of the fundamental and first order Lamb modes, only in certain frequency-thickness regimes.

Dynamic response of geometrically nonlinear, elastic rectangular plates under a moving mass loading by inclusion of all inertial components

Dynamic deformations of beams and plates under moving objects have extensively been studied in the past. In this work, the dynamic response of geometrically nonlinear rectangular elastic plates subjected to moving mass loading is numerically investigated. A rectangular von Karman plate with various boundary conditions is modeled using specifically developed geometrically nonlinear plate elements. In the available finite element (FE) codes the only way to distinguish between moving masses from moving loads is to model the moving mass as a separate entity. However, these procedures still do not guarantee the inclusion of all inertial effects associated with the moving mass. In a prepared finite element code, the plate elements are developed using the conventional nonlinear methods, i.e., Total Lagrangian technique, but all inertial components associated with the travelling mass are taken into account. Since inertial components affect the mass, damping, and stiffness matrices of the system as the moving mass traverses the plate, appropriate time increments shall be selected to avoid numerical instability. The dynamic response of the plate induced by the moving mass is evaluated and compared to previous studies. Also, unlike the existing FE programs, the different inertial components of the normal contact force between the moving mass and the plate are computed separately to substantiate the no-separation assumption made for the moving mass. Also, it is observed that for large moving mass velocities, the peak plate deformation occurs somewhere away from the plate center point. Under the two extreme in-plane boundary conditions considered in this study, it is shown that if the geometrical nonlinearity of plate is accounted for, the deformations obtained would be less than the case with classical linear plate theory.

Normal form analysis of a forced aeroelastic plate

A nonlinear elastic plate in a supersonic unsteady flow forced by a dynamic excitation and a biaxial compressive load is studied. The physical behavior of the plate is modelized by the Von Kármán equations and the aerodynamic loads are modeled by using the piston theory including nonlinearities up to the third order. The space-continuum model is space-discretized by a Galerkin projection and then studied by a perturbation approach based on the Normal Form method in order to reduce the system to a simpler and essential form defined by its resonance conditions. A physical interpretation of the involved small divisors is given by analyzing how different equation parameters influence the reduced normal form model in the neighborhood of both static and dynamic bifurcation points.