Natural frequency analysis of laminated piezoelectric beams with arbitrary polarization directions

Piezoelectric devices exhibit unique properties that vary with different vibration modes,closely influenced by their polarization direction.This paper presents an analysis on the free vibration of laminated piezoelectric beams with varying polarization directions,using a state-space-based differential quadrature method.First,based on the electro-elasticity theory,the state-space method is extended to anisotropic piezoelectric materials,establishing state-space equations for arbitrary polarized piezoelectric beams.A semi-analytical solution for the natural frequency is then obtained via the differential quadrature method.The study commences by examining the impact of a uniform polarization direction,and then proceeds to analyze six polarization schemes relevant to the current research and applications.Additionally,the effects of geometric dimensions and gradient index on the natural frequencies are explored.The numerical results demonstrate that varying the polarization direction can significantly influence the natural frequencies,offering distinct advantages for piezoelectric elements with different polarizations.This research provides both theoretical insights and numerical methods for the structural design of piezoelectric devices.

Force-displacement characteristics of simply supported beam laminated with shape memory alloys

As a preliminary step in the nonlinear design of shape memory alloy（SMA） composite structures,the force-displacement characteristics of the SMA layer are studied.The bilinear hysteretic model is adopted to describe the constitutive relationship of SMA material.Under the assumption that there is no point of SMA layer finishing martensitic phase transformation during the loading and unloading process,the generalized restoring force generated by SMA layer is deduced for the case that the simply supported beam vibrates in its first mode.The generalized force is expressed as piecewise-nonlinear hysteretic function of the beam transverse displacement.Furthermore the energy dissipated by SMA layer during one period is obtained by integration,then its dependencies are discussed on the vibration amplitude and the SMA＇s strain（Ms-Strain） value at the beginning of martensitic phase transformation.It is shown that SMA＇s energy dissipating capacity is proportional to the stiffness difference of bilinear model and nonlinearly dependent on Ms-Strain.The increasing rate of the dissipating capacity gradually reduces with the amplitude increasing.The condition corresponding to the maximum dissipating capacity is deduced for given value of the vibration amplitude.The obtained results are helpful for designing beams laminated with shape memory alloys.

ON THE STABILITY OF A THERMOELASTIC LAMINATED BEAM

In this paper we consider a one-dimensional laminated beam system. The only dissipation in the system is through heat conduction in the interfacial slip equation. We prove that this unique dissipation is strong enough to exponentially stabilize the system provided the wave speeds of the system are equal. This result extends previous works where additional internal or boundary controls were used together with a frictional damping in the interfacial slip.

Elasticity solution of laminated beams with temperature-dependent material properties under a combination of uniform thermo-load and mechanical loads

An exact solution for simply-supported laminated beams with material properties variable with temperature under a combination of uniform thermo-load and mechanical loads was investigated,based on the two-dimensional(2-D)thermo-elasticity theory.Firstly,the beam was divided into a series of layers with uniform material properties along the interfaces of the beam.The uniform thermo-load acted on each layer was transformed into a combination of the normal surface forces acted at the two ends and the transverse thermo-load.Secondly,the state space method was employed to obtain the general solutions of displacements and stresses in an arbitrary layer.Thirdly,based on the interfacial continuity conditions between adjacent layers,the relations of displacement and stress components between the top and bottom layers of the beam were recursively derived by use of the transfer-matrix method.The unknowns in the solutions can be solved by the mechanical loads acted on the top and bottom surfaces.The convergence of the present solutions was checked.The comparative study of the present solutions with the Timoshenko’s solutions and the finite element(FE)solutions was carried out.The effects of material properties variable with temperature on the thermo-elastic behavior of laminated beams were discussed in detail.

Analytical solution of temperature in laminated beams subjected to general thermal boundary conditions

The temperature distribution in laminated beams underging thermal boundary conditions has been investigated.The thermal boundary conditions are general and include various combinations of prescribed heat fluxes and temperatures at the edges.An analytical solution of temperature for the laminated beam is present on the basis of the heat conduction theory in this paper.The proposed method is applicable to the beams with arbitrary thickness and layer numbers.Due to the complexity of the boundary conditions,the temperature field to be determined was considered from two sources.The first part was the temperature field from the complex temperature conditions at two edges of the laminated beam.The solution for the temperature of the first part was constructed to satisfy temperature boundary conditions at two edges.The second part was the temperature field from the upper and lower surface temperatures without taking account of the thermal conditions at two edges.In this part,the exact solution for the temperature was obtained based on the heat conduction theory.The convergence of the solution was examined by analyzing terms of Fourier series.The validity and feasibility of the proposed method was verified by comparing theoretical results with numerical results due to the equivalent single layer approach and the finite element method(FEM).The influences of surface temperatures,beam thicknesses,layer numbers and material properties with respects to the solution of the temperature field of the beam were investigated via a series of parametric studies.

Stresses of orthotropic laminated beams subjected to high temperature and mechanical load

Thermo-elastic analysis of simply-supported orthotropic laminated beams subjected to high temperature and mechanical load is presented on the basis of the exact two-dimensional thermoelasticity theory.The beam is composed of several orthotropic layers,each with temperaturedependent material properties.The governing equation for each layer is analytically solved using the state space method.The displacement and stress solutions of the beam are obtained using the transfer-matrix method.A numerical example is included to study the effects of temperature on the mechanical responses of a sandwich beam.The results reveal two main effects of temperature:(i)inducing deformations and stresses by itself;(ii)affecting the deformations and stresses induced by the mechanical load.

Dynamic and electrical responses of a curved sandwich beam with glass reinforced laminate layers and a pliable core in the presence of a piezoelectric layer under low-velocity impact

The dynamic responses and generated voltage in a curved sandwich beam with glass reinforced laminate(GRL)layers and a pliable core in the presence of a piezoelectric layer under low-velocity impact(LVI)are investigated.The current study aims to carry out a dynamic analysis on the sandwich beam when the impactor hits the top face sheet with an initial velocity.For the layer analysis,the high-order shear deformation theory(HSDT)and Frostig's second model for the displacement fields of the core layer are used.The classical non-adhesive elastic contact theory and Hunter's principle are used to calculate the dynamic responses in terms of time.In order to validate the analytical method,the outcomes of the current investigation are compared with those gained by the experimental tests carried out by other researchers for a rectangular composite plate subject to the LVI.Finite element(FE)simulations are conducted by means of the ABAQUS software.The effects of the parameters such as foam modulus,layer material,fiber angle,impactor mass,and its velocity on the generated voltage are reviewed.

Elasticity solution of laminated beams subjected to thermo-loads

According to the two-dimensional(2-D) thermo-elasticity theory, the exact elasticity solution of the simply supported laminated beams subjected to thermo-loads was studied. An analytical method was presented to obtain the temperature, displacement and stress fields in the beam. Firstly, the general solutions of temperature, displacements and stresses for a single-layered simply supported beam were obtained by solving the 2-D heat conduction equation and the 2-D elasticity equations, respectively. Then, based on the continuity of temperature, heat flux, displacements and stresses on the interface of two adjacent layers, the formulae of temperature, displacements and stresses between the lowest layer and the top layer of the beam were derived out in a recurrent manner. Finally, the unknown coefficients in the solutions were determined by the use of the upper surface and lower surface conditions of the beam. The distributions of temperature, displacement and stress in the beam were obtained by substituting these coefficients back to the recurrence formulae and the solutions. The excellent convergence of the present method has been demonstrated and the results obtained by the present method agree well with those from the finite element method. The effects of surface temperatures, thickness, layer number and material properties of the plate on the temperature distribution were discussed in detail. Numerical results reveal that the displacements and stresses monotonically increase with the increase of surface temperatures. In particular, the horizontal stresses are discontinuous at the interface.

The analysis of composite laminated beams using a 2D interpolating meshless technique

Laminated composite materials are widely implemented in several engineering constructions. For its relative light weight, these materials are suitable for aerospace, military, marine, and automotive structural applications. To obtain safe and economical structures, the modelling analysis accuracy is highly relevant. Since meshless methods in the recent years achieved a remarkable progress in computational mechanics, the present work uses one of the most flexible and stable interpolation meshless technique available in the literature—the Radial Point Interpolation Method(RPIM).Here, a 2 D approach is considered to numerically analyse composite laminated beams. Both the meshless formulation and the equilibrium equations ruling the studied physical phenomenon are presented with detail. Several benchmark beam examples are studied and the results are compared with exact solutions available in the literature and the results obtained from a commercial finite element software. The results show the efficiency and accuracy of the proposed numeric technique.

Vibration suppression of magnetostrictive laminated beams resting on viscoelastic foundation

The vibration suppression analysis of a simply-supported laminated composite beam with magnetostrictive layers resting on visco-Pasternak’s foundation is presented.The constant gain distributed controller of the velocity feedback is utilized for the purpose of vibration damping.The formulation of displacement field is proposed according to Euler-Bernoulli’s classical beam theory(ECBT),Timoshenko’s first-order beam theory(TFBT),Reddy’s third-order shear deformation beam theory,and the simple sinusoidal shear deformation beam theory.Hamilton’s principle is utilized to give the equations of motion and then to describe the vibration of the current beam.Based on Navier’s approach,the solution of the dynamic system is obtained.The effects of the material properties,the modes,the thickness ratios,the lamination schemes,the magnitudes of the feedback coefficient,the position of magnetostrictive layers at the structure,and the foundation modules are extensively studied and discussed.

General analytical solution to bending of composite laminated beams with delaminations

Based on the first-order shear deformable beam theory, a refined model for composite beams containing a through-the-width delamination is presented, and the deformation at the delamination front is considered. Different from the ordinary delami- nated beam theory, each of the perfectly bonded portions of the new model is constructed as two separated beams along the interface without assuming a plane section at the de- lamination front. The governing equations of the delaminated portions and bonded ones are established, combined with continuity conditions of displacements and internal forces. Solutions of delaminated composite beams with different boundary conditions, delamina- tion locations and sizes axe shown in excellent agreement with the finite element results, showing efficiency and applicability of the present model.

Experimental Vibration Analysis of a Cracked Laminated Glass Beam with Bolted Joint

In this study, in order to determine the very early load transfer behavior in the bolted joint connection, experimental dynamic analysis of different laminated glass beams including two surface cracks is considered. For this purpose, both three different plastic interlayers （i.e., three types of polyvinyl butyral--PVB） and three different glass-lamina thicknesses are taken into account. Effects of the plastic interlayer, thickness of the glass-lamina, number of surface cracks and their locations on the vibration characteristics/structural performances are examined experimentally. Vibration tests are performed to present free vibration characteristics of the laminated glass beams under clamped-free boundary conditions. Experimental dynamic analysis consists of six parts： （I） vibration analysis with no-crack and no-hole with a bolted joint; （I1） vibration analysis with a surface crack and no-hole with a bolted joint; （III） vibration analysis with two surface cracks and no-hole with a bolted joint; （IV） vibration analysis with no-crack and a hole with a bolted joint; （V） vibration analysis with a surface crack and a hole with a bolted joint; （VI） vibration analysis with two surface cracks and a hole with a bolted joint. For these experimental steps, an impact hammer with a force transducer is used to excite the uncracked or cracked composite beams through the selected points. After the excitation, the responses are obtained by an accelerometer. The vibration measurements are completed using a microprocessor-based data acquisition system and nCode GlyphWorks software. Results are given in tabular and graphical forms.

Fabrication, microstructure and properties of electron beam-physical vapor deposited TiAl sheet and TiAl/Nb laminated composites

The TiAl-based alloys sheet with 150 mm×100 mm×0.4 mm and the TiAl/Nb laminated composites with 150 mm×100 mm×0.2 mm were fabricated by using electron beam-physical vapor deposition(EB-PVD) method, respectively. The microstructure and properties of the sheet were investigated by AFM, SEM and EDS. The results show that the TiAl based alloys sheet has a good surface quality, and its microstructure is columnar crystal. The component of the alloys indicates a regular and periodical gradient change which leads to the spontaneous delamination along the normal direction of substrate. In the TiAl/Nb laminated composites alternating overlaid by TiAl of 24 layers and Nb of 23 layers, the interface of each layer evenly distributed throughout the cross-section is transparent, and the interlayer spacing is about 8μm. The component of TiAl layers also changes regularly along the normal direction of substrate, but no delamination phenomenon is found. The TiAl/Nb laminated composites have better ductility than the TiAl-based alloys sheet.

A new higher-order shear deformation theory and refined beam element of composite laminates

A new higher-order shear deformation theory based on global-local superposition technique is developed. The theory satisfies the free surface conditions and the geometric and stress continuity conditions at interfaces. The global displacement components are of the Reddy theory and local components are of the internal first to third-order terms in each layer. A two-node beam element based on this theory is proposed. The solutions are compared with 3D-elasticity solutions. Numerical results show that present beam element has higher computational efficiency and higher accuracy.

Optimized design and preparation of Ti/TiAl laminated composite

Laminated composite is a new type of composite structure which is used to improve the fracture toughness and flexure strength and is good for optimizing the mechanical properties of intermetallics. On the basis of bionic principle, the optimized design (via establishing the mathematical model, stress intensity factor K_Ⅰ was computed by the finite element method) of Ti/TiAl laminated composite was studied by varying the thickness ratio and layer amounts, then the raw materials of Ti and TiAl were evaporated and deposited alternatively to form laminated metal/intermetallic composites in vacuum chamber by electron beam physical vapor deposition method. The results show that the toughness of TiAl is improved and agrees well with theoretical analysis.

THE ANALYSIS OF THIN WALLED COMPOSITE LAMINATED HELICOPTER ROTOR WITH HIERARCHICAL WARPING FUNCTIONS AND FINITE ELEMENT METHOD

In the present paper, a series of hierarchical warping functions is developed to analyze the static and dynamic problems of thin walled composite laminated helicopter rotors composed of several layers with single closed cell. This method is the development and extension of the traditional constrained warping theory of thin walled metallic beams, which had been proved very successful since 1940s. The warping distribution along the perimeter of each layer is expanded into a series of successively corrective warping functions with the traditional warping function caused by free torsion or free beading as the first term, and is assumed to be piecewise linear along the thickness direction of layers. The governing equations are derived based upon the variational principle of minimum potential energy for static analysis and Rayleigh Quotient for free vibration analysis. Then the hierarchical finite element method. is introduced to form a,. numerical algorithm. Both static and natural vibration problems of sample box beams axe analyzed with the present method to show the main mechanical behavior of the thin walled composite laminated helicopter rotor.

Lagged strain of laminates in RC beams strengthened with fiber-reinforced polymer

Based on the theory of concrete structure, a new expression was derived for lagged strain of fiber-reinforced polymer （FLIP） laminates in reinforced concrete （RC） beams strengthened with FRP. The influence of different preloaded states and nonlinear stress-strain relationship of compressed concrete were both taken into account in this approach. Then a simplified expression was given by ignoring tensile resistance of concrete. Comparison of analytical predictions with experimental results indicates satisfactory accuracy of the procedures. The errors are less than 8% and 10% respectively when the tensile resistance of concrete is or not considered. While the maximum error of existing procedures is up to 60%.

Preparation and mechanical properties of NiCoCrAlY/NiCr laminates by electron beam physical vapor deposition

Laminates with alternating layers of NiCoCrAlY and NiCr were fabricated by using electron beam physical vapor deposition (EB-PVD) method. The influence of the substrate temperature on morphology of the laminates was investigated. The results show that in order to produce NiCoCrAlY/NiCr laminates with lower porosity, higher substrate temperature is required. The mechanical properties of the as-deposited samples and heat-treated samples were examined using tensile tests. The stress-strain curve of the as-deposited laminate shows a typical characteristic of multilayered materials and the fracture behavior is improved by annealing the samples at high temperatures. The tensile strength of the samples annealed at 760℃ is 658.4MPa, and the elongation reaches 6.2%.

ANALYTICAL－GENERALIZED VARIATIONAL METHOD ABOUT SOLUTION OF SIF AND CONSERVATIVE INTEGRAL FOR DELAMINATlONS OF LAMINATE BEAM

AbstractIn this paper,first of all,the eigen expansions of stress and displacement fields satis-fying all governing equations, boundary conditions along crack surfaces and conditions ofcontinuity along hgament of two dissimilar media are derived.Next, the formula of con-servative integral J1 for the above media with interface crack is established and applied tothe case of detamination of composite laminate beam stacked orthogonally.Furtbermore,the coefficients of the above expansions are determined by generalized variational method,then the values of J1 can be obtained.The stress intensity factor Kn found directly fromthe leading term of the above expansions agrees with that indirectly from J1 satisfactorily,the convergency of results with increase of number of terms in the above expansions is verynice and values of J1 obtained from different paths of integration keep constantaccurately.

Flexural rigidity evolvement laws of reinforced concrete beams strengthened with carbon fiber laminates

Extensive research has shown that externally bonded carbon fiber reinforced polymer （CFRP） laminates are particularly suitable for improving the fatigue behavior of reinforced concrete （RC） beams. This paper presents the research on flexural ngidity evolvement laws by testing 14 simple-supported RC beams strengthened with carbon fiber laminates （CFL） under cyclic load, and 2 under monotone load as a reference. The cyclic load tests revealed the peak load applied onto the surface of a supported RC beam strengthened with CFL is linear to the logarithm of its fatigue life, and the flexural rigidity evolvement undergoes three distinct phases： a rapid decrease from the start to about 5% of the fatigue life; an even development from .5% to about 99% of the fatigue life; and a succedent rapid decrease to failure. When the ratio of fatigue ＂cycles to the fatigue life is within 0.0.5 to 0.99, the flexural rigidity varies linearly with the ratio. The peak load does not affect the flexural rigidity evolvement if it is not high enough to make the main reinforcements yield. The dependences of the flexural rigidity of specimens formed in the same group upon their fatigue cycles normalized by fatigue life are almost coincident. This implies the flexural rigidity may be a material parameter independent of the stress level. These relationships of flexural rigidity to fatigue cycles, and fatigue life may be able to provide some hints for fatigue design and fatigue life evaluation of RC member strengthened with CFL; nevertheless the findings still need verifying by more experiments.