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This paper is concerned with analysis of the time-dependent strain energy release rate for a longitudinal crack in a beam subjected to linear relaxation. A viscoelastic model with an arbitrary number of parallel units is used for treating the relaxation. Each unit has one dashpot and two springs. A stress-strain-time relationship is derived for the case when the coefficient of viscosity in each unit of the viscoelastic model changes continuously with time. The beam exhibits continuous material inhomogeneity along the thickness. Thus, the moduli of elasticity and the coefficients of viscosity vary continuously in the thickness direction. The aim of the present paper is to obtain time-dependent solutions to the strain energy release rate that take into account the relaxation when the coefficient of viscosity changes with time.

Time-dependent solutions to the strain energy release rate are derived by considering the time-dependent strain energy and also by using the compliance method. The two solutions produce identical results.

The variation of the strain energy release rate with time due to the relaxation is analysed. The influence of material inhomogeneity and the crack location along the beam width on the strain energy release rate are evaluated. The effects of change of the coefficients of viscosity with time and the number of units in the viscoelastic model on the strain energy release rate are assessed by applying the solutions derived.

The time-dependent strain energy release rate for a longitudinal vertical crack in a beam under relaxation is analysed for the case when the coefficients of viscosity change with time.

The purpose of this paper is to parametrically investigate the vibration and damping characteristics of a functionally graded (FG) inhomogeneous and porous curved sandwich beam with a frequency-dependent viscoelastic core.

The FG material properties in this study are assumed to vary through the beam thickness by power law distribution. Additionally, FG layers have porosities, which are analyzed individually in terms of even and uneven distributions. First, the equations of motion for the free vibration of the FG curved sandwich beam were derived by Hamilton's principle. Then, the generalized differential quadrature method (GDQM) was used to solve the resulting equations in the frequency domain. Validation of the proposed FG curved beam model and the reliability of the GDQ solution was provided via comparison with the results that already exist in the literature.

A series of studies are carried out to understand the effects on the natural frequencies and modal loss factors of system parameters, i.e. beam thickness, porosity distribution, power law exponent and curvature on the vibration characteristics of an FG curved sandwich beam with a ten-parameter fractional derivative viscoelastic core material model.

This paper focuses on the vibration and damping characteristics of FG inhomogeneous and porous curved sandwich beam with frequency dependent viscoelastic core by GDQM – for the first time, to the best of the authors' knowledge. Moreover, it serves as a reference for future studies, especially as it shows that the effect of porosity distribution on the modal loss factor needs further investigation. GDQM can be useful in dynamic analysis of sandwich structures used in aerospace, automobile, marine and civil engineering applications.

This paper aims to investigate analytical solutions of natural frequencies and mode shapes of Euler-Bernoulli beams with step changes in the stiffness.

In this work, analytical solutions for a beam with a single discontinuity was performed. Subsequently, based on an effective matrix formulation, the closed-form expressions of the single discontinuity beam could be conveniently extended to stepped beams with multiple stiffness discontinuities.

The results of the study show that the natural frequency of the beam can be adjusted by the local stiffness variation, and step location plays a significant role in free vibration responses.

The effects of the stiffness of the segment and step location on the natural frequencies of the stepped beams under different boundary conditions were examined using the proposed analytical scheme. This study provides insights into the design of variable-stiffness beam structures with the capability to adjust natural frequencies.

The purpose of this paper is to study the static buckling and free vibration of continuously graded ceramic-metal beams by employing a refined higher-order shear deformation, which is also the primary goal of this paper.

The proposed model is able to catch both the microstructural and shear deformation impacts without employing any shear correction factors, due to the realistic distribution of transverse shear stresses. The material properties are supposed to vary across the thickness direction in a graded form and are estimated by a power-law model. The equations of motion and related boundary conditions are extracted using Hamilton’s principle and then resolved by analytical solutions for calculating the critical buckling loads and natural frequencies.

The obtained results are checked and compared with those of other theories that exist in the literature. At last, a parametric study is provided to exhibit the influence of different parameters such as the power-law index, beam geometrical parameters, modulus ratio and axial load on the dynamic and buckling characteristics of FG beams.

Searching in the literature and to the best of the authors’ knowledge, there are limited works that consider the coupled effect between the vibration and the axial load of FG beams based on new four-variable refined beam theory. In comparison with a beam model, the number of unknown variables resulting is only four in the general cases, as against five in the case of other shear deformation theories. The actual model represents a real distribution of transverse shear effects besides a parabolic arrangement of the transverse shear strains over the thickness of the beam, so it is needless to use of any shear correction factors.

This study aimed to solve the engineering problem of free vibration disturbance and local mesh refinement induced by microcrack damage in circularly curved beams. The accurate identification of the crack damage depth, number and location depends on high-precision frequency and vibration mode solutions; therefore, it is critical to obtain these reliable solutions. The high-precision finite element method for the free vibration of cracked beams needs to be developed to grasp and control error information in the conventional solutions and the non-uniform mesh generation near the cracks. Moreover, the influence of multi-crack damage on the natural frequency and vibration mode of a circularly curved beam needs to be detected.

A scheme for cross-sectional damage defects in a circularly curved beam was established to simulate the depth, location and the number of multiple cracks by implementing cross-section reduction induced by microcrack damage. In addition, the h-version finite element mesh adaptive analysis method of the Timoshenko beam was developed. The superconvergent solution of the vibration mode of the cracked curved beam was obtained using the superconvergent patch recovery displacement method to determine the finite element solution. The superconvergent solution of the frequency was obtained by computing the Rayleigh quotient. The superconvergent solution of the eigenfunction was used to estimate the error of the finite element solution in the energy norm. The mesh was then subdivided to generate an improved mesh based on the error. Accordingly, the final optimised meshes and high-precision solution of natural frequency and mode shape satisfying the preset error tolerance can be obtained. Lastly, the disturbance behaviour of multi-crack damage on the vibration mode of a circularly curved beam was also studied.

Numerical results of the free vibration and damage disturbance of cracked curved beams with cracks were obtained. The influences of crack damage depth, crack damage number and crack damage distribution on the natural frequency and mode of vibration of a circularly curved beam were quantitatively analysed. Numerical examples indicate that the vibration mode and frequency of the beam would be disturbed in the region close to the crack damage, and a greater crack depth translates to a larger frequency change. For multi-crack beams, the number and distribution of cracks also affect the vibration mode and natural frequency. The adaptive method can use a relatively dense mesh near the crack to adapt to the change in the vibration mode near the crack, thus verifying the efficacy, accuracy and reliability of the method.

The proposed combination of methodologies provides an extremely robust approach for free vibration of beams with cracks. The non-uniform mesh refinement in the adaptive method can adapt to changes in the vibration mode caused by crack damage. Moreover, the proposed method can adaptively divide a relatively fine mesh at the crack, which is applied to investigating free vibration under various curved beam angles and crack damage distribution conditions. The proposed method can be extended to crack damage detection of 2D plate and shell structures and three-dimensional structures with cracks.

The purpose of this paper is to deal with large deformation analysis of plane beams composed of functionally graded (FG) elastic material with a variable Poisson’s ratio.

The material is assumed to be linear elastic, with a Poisson’s ratio varying according to a power law along the thickness direction. The finite element used is a plane beam of any-order of approximation along the axis, and with four transverse enrichment schemes, which can describe constant, linear, quadratic and cubic variation of the strain along the thickness direction. Regarding the constitutive law, five materials are adopted: two homogeneous limiting cases, and three intermediate FG cases. The effect of both finite element kinematics and distribution of Poisson’s ratio on the mechanical response of a cantilever is investigated.

In accordance with the scientific literature, the second scheme, in which the transverse strain is linearly variable, is sufficient for homogeneous long (or thin) beams under bending. However, for FG short (or moderate thick) beams, the third scheme, in which the transverse strain variation is quadratic, is needed for a reliable strain or stress distribution.

In the scientific literature, there are several studies regarding nonlinear analysis of functionally graded materials (FGMs) via finite elements, analysis of FGMs with constant Poisson’s ratio, and geometrically linear problems with gradually variable Poisson’s ratio. However, very few deal with finite element analysis of flexible beams with gradually variable Poisson’s ratio. In the present study, a reliable formulation for such beams is presented.

The purpose of this paper is to investigate the response of viscoelastic beam resting on a Winkler’s foundation and subjected to an axial initial stress, thermal load and an ultra-fast laser heating.

In this introduced model, the authors considered the interaction design between the vertical springs only. The beam is considered as an Euler–Bernoulli beam exposed to sinusoidal varying heat.

The deflection and the temperature response of the beam are obtained using Laplace transform and its numerical inversion method. In the numerical example, the effect of the laser pulse duration and viscous damping coefficient on the transverse displacement response of the beam is discussed. The thermoelastic interactions of the beam due to the axial load are also illustrated.

Physical views of this paper may be useful for the design and vibration analysis of micro-resonators and micro-sensors applications. In addition, the utilization of laser-ultrasonic technology has found wide applications in lab environments, and in an expanding number of cases, it is extending to the industrial field and realm application.

The purpose of this study is to investigate the bending analysis of metal (Ti-6Al-4V)-ceramic (ZrO₂) functionally graded material (FGM) sandwich plate with material property gradation along length and thickness direction under thermo-mechanical loading using inverse trigonometric shear deformation theory (ITSDT). FGM sandwich plate with a ceramic core and continuous variation of material properties has been modelled using Voigt’s micro-mechanical model following the power law distribution method. The impact of bi-directional gradation of material properties over the bending response of FGM plate under thermo-mechanical loading has been investigated in this work.

In this study, gradation of material properties for FGM plates is considered along length and thickness directions using Voigt’s micromechanical model following the power law distribution method. This type of FGM is called bi-directional FGMs (BDFGM). Mechanical and thermal properties of BDFGM sandwich plates are considered temperature-dependent in the present study. ITSDT is a non-polynomial shear deformation theory which requires a smaller number of field variables for modelling of displacement function in comparison to poly-nominal shear deformation theories which lead to a reduction in the complexity of the problem. In the present study, ITSDT has been utilized to obtain the governing equations for thermo-mechanical bending of simply supported uni-directional FGM (UDFGM) and BDFGM sandwich plates. Analytical solution for bending analysis of rectangular UDFGM and BDFGM sandwich plates has been carried out using Hamilton’s principle.

The bending response of the BDFGM sandwich plate under thermo-mechanical loading has been analysed and discussed. The present study shows that centre deflection, normal stress and shear stress are significantly influenced by temperature-dependent material properties, bi-directional gradation exponents along length and thickness directions, geometrical parameters, sandwich plate layer thickness, etc. The present investigation also reveals that bi-directional FGM sandwich plates can be designed to obtain thermo-mechanical bending response with an appropriate selection of gradation exponents along length and thickness direction. Non-dimensional centre deflection of BDFGM sandwich plates decreases with increasing gradation exponents in length and thickness directions. However, the non-dimensional centre deflection of BDFGM sandwich plates increases with increasing temperature differences.

For the first time, the FGM sandwich plate with the bi-directional gradation of material properties has been considered to investigate the bending response under thermo-mechanical loading. In the literature, various polynomial shear deformation theories like first-order shear deformation theory (FSDT), third-order shear deformation theory (TSDT) and higher-order shear deformation theory (HSDT) have been utilized to obtain the governing equation for bending response under thermo-mechanical loading; however, non-polynomial shear deformation theory like ITSDT has been used for the first time to obtain the governing equation to investigate the bending response of BDFGM. The impact of bi-directional gradation and temperature-dependent material properties over centre deflection, normal stress and shear stress has been analysed and discussed.

This paper aims to propose numerical-based and experiment-based identification processes, accounting for uncertainties to identify structural parameters, in a wave propagation framework.

A variant of the inhomogeneous wave correlation (IWC) method is proposed. It consists on identifying the propagation parameters, such as the wavenumber and the wave attenuation, from the frequency response functions. The latters can be computed numerically or experimentally. The identification process is thus called numerical-based or experiment-based, respectively. The proposed variant of the IWC method is then combined with the Latin hypercube sampling method for uncertainty propagation. Stochastic processes are consequently proposed allowing more realistic identification.

The proposed variant of the IWC method permits to identify accurately the propagation parameters of isotropic and composite beams, whatever the type of the identification process in which it is included: numerical-based or experiment-based. Its efficiency is proved with respect to an analytical model and the Mc Daniel method, considered as reference. The application of the stochastic identification processes shows good agreement between simulation and experiment-based results and that all identified parameters are affected by uncertainties, except damping.

The proposed variant of the IWC method is an accurate alternative for structural identification on wide frequency ranges. Numerical-based identification process can reduce experiments’ cost without significant loss of accuracy. Statistical investigations of the randomness of identified parameters illustrate the robustness of identification against uncertainties.

The work is aimed at studying the electromagnetic interaction between a homogeneous, isotropic single-negative (SNG) slab and an inhomogeneous, anisotropic double-positive (DPS) slab.

The approach is based on the transformation optics framework, which allows systematic design and modelling of anisotropic, inhomogeneous metamaterials with inherent field-manipulation capabilities.

The paper finds that a transformation-optics-based DPS slab can compensate the inherent opaqueness to the electromagnetic radiation of a SNG slab. Here, “compensation” means that the resulting bi-layer may give rise to zero-reflection for a normally-incident plane wave at a given frequency. Such phenomenon is inherently accompanied by (de)funneling effects for collimated-beam illumination, and it turns out to be quite robust to material losses as well as geometrical and constitutive-parameter truncation.

The results provide further evidence and insight in how SNG-like responses may be emulated (within narrow parametric ranges) by suitably-engineered spatial inhomogeneity and anisotropy in DPS media. Moreover, they also show that resonant transmission phenomena through SNG materials may be engineered via the powerful framework of transformation optics.