Laminated Beam Model Research Articles

Dynamics of a one-dimensional non-autonomous laminated beam

In this paper is analyzed the pullback dynamics of a laminated beam model subject to fractional Laplacian dissipation, nonlinear source terms and non autonomous external forces. For each gamma exponent of the Laplacian in the open interval with endpoints 0 and 1/2, the model is well-posedness and the evolution process associated to solutions of problem possesses a pullback attractor for a general basin of attraction. To conclude, we prove that the attractors are upper semicontinuous as γ tends to 0. The result is new and it is the first time when the non-autonomous laminated beam is studied.

Asymptotic limits and attractors for a laminated beam model

Asymptotic limits and attractors for a laminated beam model

A Laminated Beam Model for the Effective Tensile and Bending Elastic Moduli of Nanowires

Surface elasticity has a strong influence on the size-dependent mechanical properties of nanowires (NWs). In this work, a laminated beam model is used to predict the effective tensile and bending moduli of a NW. The surface layer is assumed to have a definite thickness and different elastic moduli from the bulk layer. The effective moduli of the NW are expressed as a function of surface thickness and surface constant. The theoretical predictions for stiffening and softening effects are in good agreement with experimental results. High surface elasticity increases the bending modulus more than the tensile modulus. Low surface elasticity decreases the bending modulus more than the tensile modulus. Surface thickness has a considerable influence on the effective modulus of NW and amplifies stiffening and softening effects.

Multiscale modelling of functionally graded porous beams: Buckling and vibration analyses

This paper combines the finite element (FE) homogenisation and structural assessments to conduct the multiscale modelling of laminated functionally graded (FG) porous beams made of closed-cell foams, with a focus on the beam buckling and vibration performances. In the FE homogenisation, representative volume elements (RVEs) are built up according to the relative densities, average cell sizes, and base material properties from foam samples. They are used to predict the foam Young’s modulus and verified against existing studies. A detailed investigation is then carried out to disclose the effects of RVE and porosity features, where the relative density is confirmed as the dominating factor in the calculation of foam modulus. Quantitative relations between pore structure and Young’s modulus are obtained via data fitting.In the subsequent structural assessment, the predicted Young’s modulus is employed in the buckling and vibration analyses of FG porous beams, which are constructed by a six-layer laminated beam model and consist of high-, medium-, and low-density layers with varying designated relative densities. The Timoshenko beam theory and the Ritz method are applied to derive and solve the governing equations. Convergence and validation studies are performed accordingly to ensure the validity of presented results, followed by a parametric study to examine various FG beams concerning different porosity distributions, boundary conditions, and slenderness ratios. The prominence of graded porosities is evidenced by apparently promoted beam stiffness with over 30% and 10% increments in the critical buckling load and fundamental natural frequency, respectively. The graded porosities can be linked to realistic closed-cell Aluminium foams with developed quantitative relations, shedding important and practical guidance into the porous structural design.

Impact response of inclined self-weighted functionally graded porous beams reinforced by graphene platelets

This study presents a theoretical analysis on the low-velocity impact response of inclined porous nanocomposite beams under various impulsive loads. The laminated beam model consists of multiple layers modelled as closed-cell cellular solids with identical thickness, where each layer contains uniformly distributed internal pores and is reinforced by dispersing graphene platelets into the matrix. The layer-wise continuous variations in both internal pore size/density and graphene fraction result in functionally graded lightweight beams with controllable density distributions and varying elastic moduli across the thickness direction. The material properties of each layer are determined according to Halpin–Tsai micromechanics model and the extended rule of mixture. The governing equations of the inclined beam are derived based on Timoshenko beam theory then solved by employing Ritz method for space domain and Newmark method for time domain. The static bending due to the self-weight of the beam is examined first, and then imported into the dynamic analysis as the initial stress state for the beam under impulsive impacts with six different pulse shapes. A comprehensive parametric study is conducted with a special focus on the combined effects of graded material distributions and inclined angle on the beam behaviour. Results show that a larger inclined angle reduces the beam deflection, and the rectangular impulsive load can lead to the largest mid-span deflection of fully clamped graded beams that may reach over 70% more than those under some of other impact load types. This study should provide insights into the design of lighter and stiffer inclined structural components subjected to various impulsive loading conditions.

Uniform stability for laminated beams with time‐dependent boundary conditions

In this paper we consider a laminated beam model. This structure is given by two identical uniform layers on top of each other, taking into account that an adhesive of small thickness is bonding the two surfaces and produces an interfacial slip. In the presence of nonlinear boundary controls modulated by time‐dependent coefficients, we use the multiplier method and overcome the difficulty of finding suitable multipliers to establish a uniform stability result. This result provides an explicit formula for the energy decay rates without imposing any conditions on the coefficients of the system which is more meaningful and realistic from the physical point of view. Our result improves the earlier related results in the literature.

Nonlinear Vibration of Functionally Graded Graphene Nanoplatelets Polymer Nanocomposite Sandwich Beams

We provide an analytical investigation of the nonlinear vibration behavior of thick sandwich nanocomposite beams reinforced by functionally graded (FG) graphene nanoplatelet (GPL) sheets, with a power-law-based distribution throughout the thickness. We assume the total amount of the reinforcement phase to remain constant in the beam, while defining a relationship between the GPL maximum weight fraction, the power-law parameter, and the thickness of the face sheets. The shear and rotation effects are here considered using a higher-order laminated beam model. The nonlinear partial differential equations (PDEs) of motion are derived from the Von Kármán strain-displacement relationships, here solved by applying an expansion of free vibration modes. The numerical results demonstrate the key role of the amplitudes on the vibration response of GPL-reinforced sandwich beams, whose nonlinear oscillation behavior is very important in the physical science, mechanical structures and other mathematical analyses. The sensitivity of the response to the total amount of GPLs is explored herein, along with the possible effects related to the power-law parameter, the structural geometry, and the environmental conditions. The results indicate that changing the nanofiller distribution patterns with the proposed model can remarkably increase or decrease the effective stiffness of laminated composite beams.

Laminated Timoshenko beams with viscoelastic damping

In this paper we consider a viscoelastic laminated beam model. This structure is given by two identical uniform layers on top of each other, taking into account that an adhesive of small thickness is bonding the two surfaces and produces an interfacial slip. We use minimal and general conditions on the relaxation function and establish explicit energy decay formula which gives the best decay rates expected under this level of generality. Our new result generalizes the earlier related results in the literature.

Boundary control of laminated beams with interfacial slip

In this paper, we consider a laminated beam model. This structure is given by two identical uniform layers on top of each other, taking into account that an adhesive of small thickness is bonding the two surfaces and produces an interfacial slip. We use boundary feedback control and establish an exponential energy decay result. Our result improves the earlier related results in the literature.

On the stabilization of viscoelastic laminated beams with interfacial slip

In this paper, we consider a viscoelastic laminated beam model. This structure is given by two identical uniform layers on top of each other, taking into account that an adhesive of small thickness is bonding the two surfaces and produces an interfacial slip. We use viscoelastic damping with general assumptions on the relaxation function and establish explicit energy decay result from which we can recover the optimal exponential and polynomial rates. Our result generalizes the earlier related results in the literature.

A vibrational approach to determine the elastic modulus of individual thin films in multilayers

A vibrational approach is presented to determine the elastic modulus of individual thin films deposited over a thicker substrate in multilayered systems. The approach requires measurement of the fundamental frequency of the multilayer and a laminated beam model for data reduction. A one-dimensional model based on classical laminated beam theory is introduced to provide a simple analytical approximation of the natural frequency of thin multilayered materials deposited over a significantly thicker substrate in cantilever beam configuration. The model has the advantage of providing an easy-to-use analytical expression for the natural frequency of a multilayered beam in terms of the elastic moduli of each layer, which can be inverted to calculate the elastic modulus of any individual layer if the elastic modulus of the remaining layers is known, and the natural frequency of the multilayered beam is measured. The limits of applicability of the proposed model are investigated by comparing its predictions of the fundamental frequency to those of an existent analytical model for bilayers and finite element analysis of materials comprising two and three dissimilar layers. The proposed model is applied to obtain the elastic modulus of Al and Au thin films in an Al/Au/Kapton multilayer.

Nonlinear vibration of a post-buckled giant-magnetostrictive thin film laminated beam

One buckled simply supported giant magnetostrictive material (GMM) thin film laminated beam model subject to axial magnetic excitations is proposed. Themagnetostrictive force issimplified as a harmonic excitation. Applying the Hamilton principle and Galerkin approach the model is expressed as a one dimensional vibrating model with parametrical excitation. The multi-scales method is used to obtain the approximation of the solution when the vibration is around the buckled position and the amplitude of the oscillation is small. With the help of the undermined fundamental frequency method and normal form theory the Melnikov approach is improved to calculate more precise threshold value of the amplitude of excitation leading to chaos. The numerical simulation shows the performance of the new Melnikov approach is much better than the traditional way.

Effects of initial stress on wave propagation in multi-walled carbon nanotubes

This paper reports the result of an investigation on the effect of initial stress on wave propagation in multi-walled carbon nanotubes (MWNTs). The initial stress in MWNTs often occurs due to thermal or lattice mismatch between different materials. Using a laminated beam model and considering the interaction of van der Waals forces between two adjacent nanotubes, the effects of initial stress in MWNTs on the number of wave speeds and the vibration amplitude ratio between the two adjacent tubes are described. It is noticed that the number of terahertz waves in MWNTs with initial stress not only depends on the response frequency and the noncoaxial vibrational modes, but also depends on the initial stress in the MWNTs.

Validation of a Laminated Beam Model of LIPCA Piezoelectric Actuators

This paper presents the results obtained by using a laminated beam model to predict the displacement of lightweight piezoceramic composite actuators (LIPCA) due to an applied input voltage and an applied transverse load for developmental purposes. The laminated beam model results in two primary design parameters, the actuation coefficient of a laminated beam Culb and the bending stiffness of a laminated beam EIlb The parameter Culb controls maximum displacement when the actuators are subjected to an applied voltage, whereas the parameter EIlb affects the displacement performance when a transverse load is applied to the actuators. The result of the laminated beam model indicates that designing to maximize Culb and EIlb could generate LIPCA-type actuators with larger displacement and higher force. This laminated beam result is validated numerically by using a NASTRAN finite element model result and is also validated experimentally.

Modeling and analysis for the development of Lightweight Piezoceramic Composite Actuators (LIPCA)

This paper focuses on the modeling and analysis of Lightweight Piezoceramic Composite Actuators (LIPCA) for developmental purposes. A simple analytical model and a numerical model for predicting the displacement of the actuators due to both an applied input voltage and transverse load to the piezoelectric ceramic layer are presented. The analytical model describing the laminated beam actuation results in two primary design parameters, an actuation coefficient of a laminated beam C ulb and the bending stiffness of a laminated beam EI lb. The C ulb is a parameter when maximum displacement is required. The bending stiffness affects the displacement performance when a transverse load is applied to the actuators. An experiment was performed to verify the proposed laminated beam model. The numerical model, a NASTRAN finite element model, is used to assess the effect of initial dome heights of actuators on the displacement. An increase in the initial dome height produces a positive effect for the low anisotropic actuators and gives a negative effect for the high anisotropic actuators. In conclusion, the results indicate that designing to maximize the C ulb, the EI lb, and the initial dome height of the low anisotropic actuators could generate LIPCA-type actuators with larger displacement and higher force.