Line-spring Model Research Articles

Dynamic response and sound radiation of cracked asphalt pavements under moving loads and variable thermal profiles

ABSTRACT This study investigates the acoustic performance and dynamic response of cracked, viscoelastic porous asphalt pavements under moving loads and variable thermal conditions. Realistic pavement cracks are modelled using an advanced line spring model with fracture compliance coefficients. The novelty lies in the synergistic integration of a heterogeneous triple-porosity media model, a non-uniform depth-dependent temperature profile, and an advanced fracture mechanics-based line spring model with arbitrary crack orientation. Variations in temperature that are uniform, linear, and nonlinear with respect to the thickness layer axis are taken into account. The governing equations are derived by Hamilton’s principle and solved analytically via Fourier-Laplace transforms. The acoustic pressure is first-time predicted through Rayleigh integral analysis. Durbin’s numerical inversion validates the model’s accuracy versus sophisticated finite element simulations. Parametric analysis offers new insights into the effects of crack length, pore size distribution, loading frequency, and thermal gradients on noise pollution and fatigue cracking. The integrated chemo-thermo-mechanical modelling enables optimal structural design and accelerated pavement testing to limit acoustic radiation and premature failure in porous asphalt pavements. It allows for the development of noise-reducing porous asphalt mixtures by modifying aggregate gradation, binder content, and air void distribution.

A nonlinear breathing crack model for characterization of chaotic dynamics of damaged large-amplitude vibrating plates

Large-amplitude vibrating plates (LVPs) inherently exhibit intricate chaotic behaviors stemming from geometric nonlinearity. Such chaotic dynamics could be substantially influenced by the occurrence of a breathing crack, characterized by a cyclic opening and closing process. However, current models for breathing cracks primarily focus on beam-like structures and often rely on the assumption of periodic dynamical responses, which is inadequate for depicting the chaotic vibrations experienced by cracked LVPs. To that end, this study presents a novel nonlinear breathing crack model (NBCM) to precisely characterize the breathing effect on chaotic dynamics of large-amplitude vibrating plates containing a parallel part-through breathing crack. By refining the line spring model and devising a mathematical expression for crack parameters and dynamical responses, the NBCM-based governing equation for cracked LVPs is derived with a Duffing type expression having a quadratic term. Through the utilization of the harmonic balance method and the Hilbert–Hughes–Taylor algorithm, both analytical and numerical dynamic responses of the cracked LVP based on the NBCM are obtained. Comprehensive nonlinearity analyses are then conducted, including hardening nonlinearity, breathing nonlinearity, and bifurcation nonlinearity. Comparison results demonstrate the superiority of the presented nonlinear breathing crack model in characterizing the chaotic dynamics of LVPs with breathing cracks. Accordingly, the presented NBCM shows significant potential for identifying abnormalities within practical engineering plate-like structures when considering the effects of crack breathing.

Analytical solution of forced vibration of rectangular plates with part through surface crack based on wave propagation method

An analytical wave propagation framework is developed in this study for steady state forced vibration of plates with part through surface cracks for the first time. Rectangular thin plates with partial-width and full-width cracks are considered. The analytical waves are obtained by solving the dual equations of the plate established considering the line–spring model (LSM) of crack. The dual equations formed here are characterized by simultaneously solving kinematic variables and their dual dynamic variables. Both conventional and refined LSMs are considered. Two ways are proposed to generate the system equation in wave space based on the relationships of wave propagation, wave reflection and wave scattering. Compared with the traditional methods, most of them can only give exact analytical solutions under the condition of opposite-side simply supported boundaries, the developed analysis framework can treat arbitrary combinations of simple boundary condition of the cracked plate. Comparisons of results of the proposed method with those from the literature and finite element method (FEM) validate the effectiveness of the proposed method. A series of parametric studies for investigating the effects of length, position and depth of crack on the forced response of cracked plate are also presented. The proposed framework are effective for predicting the forced responses and natural frequencies of thin plates with part-through surface cracks.

Dynamic Response of the Tunnel Lining with a Circumferential Crack Subjected to a Harmonic Point Load

Cracks are one of the most common diseases of tunnel lining, and the structural dynamic response can be used to assess the health of a tunnel. Hence, this paper investigates the dynamic response of shield tunnel lining with a partly circumferential crack. The shield tunnel lining is regarded as a thin cylindrical shell and analyzed independently. The research methodology integrates the wave propagation method, the local flexibility matrix, the line spring model and the wave superposition principle. The results show that the position and depth of a partly circumferential crack can influence the natural frequency of the shield tunnel lining. Under the fixed-position load, as the distance from the monitoring point to the crack increases, the difference in displacement response amplitude between the undamaged and cracked linings diminishes. Moreover, deepening cracks enlarge the magnitude of amplitude differences. When the load approaches the crack, the radial amplitude difference first increases and then decreases as the monitor moves away from the crack. This finding helps determine the required monitor position. The displacement response of the selected monitor indicates that the closer the load position is to the crack, the larger the amplitude difference. The results aid in identifying the crack position and selecting corresponding load and monitor locations.

Surface crack effect on frequency and vibration mode switching of solar-powered aerospace structures

In the engineering design of modern wing panels and their optimal control system, the first step is to find and predict their dynamic characteristics. These characteristics are dependent on material properties, panel geometry, and boundary conditions. In addition, possible fracture existence, imperfections and environmental conditions may unpredictably change the frequency and modes shape of the panel and then the parameters of a control system. In the present paper, the effect of surface fracture depth and orientation on frequency and vibration mode switching of perovskite solar cells-based aircraft wing panels is investigated for the first time. The PSC (perovskite solar cell) panel is made of a polymer substrate reinforced with even-distributed graphene platelets (GPLs) and thin solar layers to enhance the overall electromechanical performance of the whole structure. Equations of motion of the panel are obtained by the Hamilton approach based on the properties of Reddy's third-order shear deformation theory and linear components of the Green-Lagrange strain tensor. The novel aspect is a line-spring model of surface crack by Cartmell et al. adapted to the model of the PSC panel under consideration. The discretization of the established equations of motion for the cracked panel is carried out by the element-free IMLS-Ritz method with mesh-free features. Validation of the obtained solutions is carried out by comparisons with results from previous studies for cracked laminated plates.First, material and geometric properties effect on vibration characteristics of the PSC structure was investigated. Next, partial surface crack depth and orientation angle are examined to show changes in frequency and vibration modes shape of the panel. It is shown, that there is a mode swapping between the second and third modes when transitioning from an intact panel to a cracked one. As the crack length increases, higher-order mode variations become more noticeable, potentially leading to their swapping or the excitation of new ones. Additionally, the findings demonstrate that the surface crack orientation angle significantly impacts the higher-order modes of the panel. When the orientation angle is 90°, the first six modes align with those of the intact plate. Mode shapes and frequency shifts are significant for crack orientation angles between 0° and 90°.

Analytical solutions for buckling analysis of cracked plates under in-plane compressive loads: 2D refined modeling

Two refined two-dimensional (2D) crack models are developed to deal with rectangular isotropic and orthotropic plate problems with all-over part-through surface cracks under in-plane compressive loads. In the refined models, the crack terms are directly incorporated into the governing equations of cracked plates. In particular, the effect of internal in-plane forces can be investigated by theoretical analysis. The critical buckling loads of thin plates with uni-directional or bi-directional part-through cracks under uni-axial or bi-axial compressive loads can be determined analytically. In this work, different material properties, including steel, tempered glass, glass-fibre-reinforced composite, and graphite-epoxy composite, are considered. The influence of crack depth ratio and plate aspect ratio on the critical buckling loads of these structures is also presented. The analytical results are compared with those results obtained by the line-spring model and three-dimensional finite element method to verify their correctness and accuracy. The proposed models are highly effective for the buckling analysis of plates with part-through cracks.

Two-dimensional orthotropic plate problems in a thermal environment: Refined crack modelling

Three two-dimensional (2D) refined models are devised for the analysis of orthotropic thin plates with part-through surface and internal cracks under in-plane forces due to temperature differences. The refined models are used to formulate the governing equations for cracked orthotropic thin plates in a thermal environment. Subsequently, the refined models and their ensuing modified forms can be directly applied to all-over and finite-length cracks, respectively. Using an interface-fitted numerical scheme (i.e., matched interface and boundary technique), the resulting equations of motion for the buckling and free vibration of cracked orthotropic thin plates are solved. The effects of various crack types and dimensions on the critical buckling temperatures and vibration frequencies of orthotropic thin plates are investigated. The present models are also validated by comparing their numerical results with the line-spring model and three-dimensional finite-element solutions. Illustrative examples, including boron fiber/epoxy and glass-fibre-reinforced composite materials, are considered to show the superiority of the proposed models over the conventional one, especially in terms of understanding the effect of in-plane forces arising from temperature differences at the cracked location.

Magnetic field and thermal environment effects on sound transmission loss of double-walled cracked piezoelectric cylindrical nanoshells

Based on a non-local strain gradient theory (NSGT), this article analytically studies the sound transmission loss (STL) performance of a special type of double-walled partially cracked piezoelectric nanoshell structures of cylindrical shape undergoing initial electric potential and longitudinal magnetic field. The structure, which includes a specific number of axial and circumferential surface cracks, is struck by a plane wave in a thermal setting. Such cracks have minimal size compared to the nano-shell structure as a whole, minimizing the effects of curvature for this problem. The first-order shear deformation theory (FSDT) is utilized to explain the displacement components, and Hamilton’s concept is applied to obtain the coupled vibroacoustic equations that are used to generate the governing equations of the system for the two crack types, giving us an idea of the resulting structural stiffness in the presence of cracks using coefficients of crack compliance that are derived from the line spring model (LSM). The mechanical and acoustic properties are then checked to see their effect on the system response followed by a series of validation case studies. This evaluation also allows one to optimize a similar structure by selecting appropriate external parameters such as electric voltage, magnetic field intensity, different crack types, nonlocal and strain gradient parameters, variations of temperature, and incident angle of sound waves.

Size-dependent nonlinear vibration analysis of cracked graphene-platelets-reinforced-composites (GPLRC) plate under parametric excitation

The response of a size-dependent randomly distributed graphene-platelet-reinforced-composite (GPLRC) plates, with an inclined part-through surface crack, resting on viscoelastic foundation under the action of periodic parametric excitation with quadratic and cubic nonlinearities are studied in the present study. The crack is centrally located at the center of the plate while inclined at an angle β concerning the X-axis. The angled crack’s mathematical formulation is based on a line-spring model. Indeed, the assumption of the problem is considered as the crack remains a continuous straight line. Nonlinearity is considered based on Berger’s formulation and Von-Karman’s geometric relations. Using classical plate theory, Eringen’s theory, micromechanical models, and equilibrium equation, the governing equation of motion of the problem have been derived. Then the Galerkin method was applied to transform the partial differential equation into the ordinary one. Eventually, the solution procedure of the cubic–quadratic nonlinear equation was followed by the multiple-scale perturbation method. Steady-state response and nonlinear softening or hardening treatment of the problem have been presented. Significant results of the study can be emphasized as GPLs, crack’s dimension and inclination, nonlocal parameter, damping, and detuning parameter effects on nonlinear behavior and stability of the structure. Dispersing GPLs as reinforcement material in the plate layer caused improvement in structural behavior, as the amplitude response of the system with the presence of a crack is decreased, comparing to a similar system without any reinforcement and crack. In the end, current work will help in the development of nonlinear behavior, stability analysis and critical bifurcation points of size-dependent cracked GPLRC plate exposed to external parametric loading.

Vibration analysis of cracked plates resting on elastic foundation via moving least squares differential quadrature method

In this work, moving least squares differential quadrature scheme is employed to formulate and solve the free vibration analysis of cracked plates resting on the Winkler–Pasternak elastic foundation. Two types of cracks as a part-through surface and internal crack are studied in a rectangular plate. Formulation of the problem is dependent on transverse shear deformation theory and line spring model. Also, continuous weight functions around the crack are produced by diffraction method. Boundary conditions are augmented in the equation of motion and the problem is converted to a discrete eigenvalue problem. The accuracy and efficiency of the present results is validated by comparing with exact, experimental and numerical results. Moreover, the effects of different boundary conditions, crack length, crack depth and foundation parameters on the vibration frequencies and mode shapes are examined in detail. The obtained results may be used for structural health monitoring.

Non-linear vibration analysis of specially orthotropic tapered micro-plates with arbitrary located crack: A non-classical analytical approach

The present work aims to study the non-linear vibrations in a cracked orthotropic tapered micro-plate. Linear and parabolic variation in the plate thickness is assumed in one as well as two directions. The partial crack is located in the centre, and it is continuous; this crack’s location is arbitrary and can be varied within the centre-line. Based on classical plate theory, the equilibrium principle is applied, and the governing equation of tapered orthotropic plate is derived. Additionally, the microstructure’s effect has been included in the governing equation using the non-classical modified couple stress theory. The simplified line spring model is used to consider the impact of partial crack on the plate dynamics and is incorporated using in-plane forces and bending moments. The introduction of Berger’s formulation brings the non-linearity in the model in terms of in-plane forces. Here, Galerkin’s method has been chosen for converting the derived governing equation into time-dependent modal coordinates, which uses an approximate solution technique to solve the non-linear Duffing equation. The crack is considered along the fibres and across the fibres to show the effect of orthotropy. Results are presented for an orthotropic cracked plate with non-uniform thickness. The effects of the variation of taper constants, crack location, crack length, internal material length scale parameter on the fundamental frequency are obtained for two different boundary conditions. The non-linear frequency response curves are plotted to show the effect of non-linearity on the system dynamics using the method of multiple scales, and the contribution of taper constants and crack parameters on non-linearity is shown with bending-hardening and bending-softening phenomenon. It has been found that vibration characteristics are affected by the taper parameters and fibre direction for a cracked orthotropic plate.

A Non-Classical Analytical Approach for Vibration Analysis of Isotropic and Fgm Plate Containing a Star Shaped Crack

ABSTRACTA non-classical analytical model for vibration analysis of thin isotropic and FGM plate containing multiple part-through cracks (star shaped) of arbitrary orientation is proposed. A plate containing four concentric cracks of arbitrary orientation in the form of continuous line is considered for analysis. The proposed governing equation is derived based on classical plate theory and modified couple stress theory. Line spring model is modified to accommodate all the crack terms. The application of Berger’s formulation introduces nonlinearities in the governing equation and then the Galerkin’s method is applied for solving final governing equation. Results for fundamental frequencies for different values of crack length, crack orientation, gradient index and material length scale parameters are presented for two different boundary conditions. Furthermore, to study the phenomenon of bending hardening/softening in a cracked plate, the frequency response curves are plotted for the parameters stated above. Based on the outcomes of this study, it can be concluded that stiffness of the plate is severely affected by the presence of multiple cracks and the stiffness goes on decreasing with increase in number of cracks thereby affecting the fundamental frequency.

Vibration and buckling characteristics of cracked natural fiber reinforced composite plates with corner point-supports

This paper is concerned with the accurate free vibration and buckling analysis of a partially or internally cracked natural fiber reinforced composite (NFRC) plate with corner point-supports by symplectic elasticity method. Mechanical degradation of NFRC plate during the hygrothermal ageing is described by using a theoretical damage model. Crack effects on the plate derived by using the line spring model (LSM) are considered in the governing equation of motion in the form of crack compliance coefficients. In the framework of symplectic elasticity, free vibration and buckling eigenvalue problems for three types of boundary conditions can be reduced into a set of algebraic equations by a superposition of boundaries. Highly accurate critical buckling loads, fundamental frequencies and corresponding mode shapes are solved simultaneously. A comparison of the new proposed model with finite element method is presented to validate the accuracy and effectiveness of the present method. Subsequently, the influence of hygrothermal ageing and crack parameters on the free vibration and buckling behaviors of cracked NFRC plates is rigorously presented. The research results conclude that both length and crack location are important factors that affect the hygrothermal ageing of NFRC plates.

A variational iteration method (VIM) for nonlinear dynamic response of a cracked plate interacting with a fluid media

This paper deals with analyzing the nonlinear vibration of an isotropic cracked plate interacting with an air cavity. A part-through surface crack with variable orientations and positions is considered and modeled using the modified line spring model. In the first step, based on the Von Karman theory, the governing equation of the nonlinear vibration related to the cracked plate–cavity is presented. Then, by employing the Euler equation along with the Galerkin method, the coupling effect between the fluid–solid media inside the enclosure is eliminated. In the next step, the variational iteration method (VIM) is introduced as an appropriate method for nonlinear analysis of the mentioned system. To this end, the convergence of the nonlinear coupled natural frequencies with high precision is proved by performing four iterations of VIM. Finally, the effect of the length, angle, and position corresponding to the crack as well as the cavity depth on the frequency ratio is inspected for various boundary conditions by plotting three and four-dimensional backbone curves. It is revealed that the crack angle is the most effective parameter on the frequency ratio.

Analytical treatment for vibration analysis of partially cracked orthotropic and FGM submerged cylindrical shell with consideration of fluid-structure interaction

Analytical solution for vibration analysis of orthotropic and FGM submerged cylindrical shell containing a surface crack of variable angular orientation is presented in this work. The governing equations in terms of transverse deflection of cracked-submerged shell have been derived using classical shell theory. The fluid forces associated with its inertial effects are added in the governing differential equation to incorporate the fluid–structure interaction effect. The line spring model (LSM) is used to formulate the crack coefficients to accommodate the effect of crack in the governing equation. Furthermore, the governing equation is solved using Donell–Mushtari–Vlasov (DMV) theory to get the fundamental frequency. The results are presented for frequency by giving the input parameters as crack length, crack orientation, and shell’s physical properties such as radius, thickness, and length of the shell. Communicated by Wei-Chau Xie

Analysis of free vibration of an isotropic plate with surface or internal long crack using generalized differential quadrature method

This article proposes a new method for the analysis of free vibration of a cracked isotropic plate with various boundary conditions based on Kirchhoff’s theory. The isotropic plate is assumed to have a part-through surface or internal crack. The crack is considered parallel to one of the plate edges. Existence of the crack modified the governing differential equations which were formulated based on the line-spring model. Generalized differential quadrature method discretizes the obtained governing differential equations and converts them into an algebraic system of equations. Then, an eigenvalue analysis was used to determine the natural frequencies of the cracked plates. Some numerical results are given to demonstrate the accuracy and convergence of the obtained results. To demonstrate the efficiency of the method, the results were compared with finite element solutions and available literature. Also, effects of the crack depth, its location along the thickness, the length of the crack and different boundary conditions on the natural frequencies were investigated.

Analytical Modeling on Vibration Analysis of Cracked Functionally Graded Plate Submerged in Fluid

Background:It is established that the vibration response of submerged structures is quite different than that calculated in vacuum. Therefore, the study of vibration characteristics of submerged plate structures is important for safety and its designing purpose.Objective:To investigate the fundamental frequency of partially cracked Functionally Graded (FG) submerged plate based on analytical approach.Methods:The governing differential equation of the cracked-submerged plate is derived based on Kirchhoff’s thin classical plate theory in conjunction with the potential flow theory. The line spring model is used to incorporate the effect of crack in the form of additional bending whereas the effect of fluid medium is incorporated in form fluids forces associated with inertial effects of its surrounding fluids. The Bernoulli’s equation and velocity potential function are used to define the fluid forces acting on plate surface.Results:An approximate solution for governing equation of coupled fluid-plate system is obtained by using the Galerkin’s method. For validation of the present results, they are compared with the existing results of the previous published work, which are in good agreements. New results for natural frequencies as affected by gradient index, crack length, level of submergence and immersed depth of plate are presented for Simply Supported (SSSS) boundary condition.Conclusion:It has been concluded that the presence of crack and fluidic medium significantly affect the natural frequencies of the plate. It is observed that the increase in the length of crack and level of submergence decreases the fundamental frequency. In this paper, few patents have been discussed.

Vibration, Buckling and Deflection Analysis of Cracked Thin Magneto Electro Elastic Plate Under Thermal Environment

The Magneto-Electro-Elastic (MEE) material exhibits pyroelectric and pyromagnetic effects under thermal environment. The effects of such pyroelectric and pyromagnetic behavior on vibration, buckling and deflection analysis of partially cracked thin MEE plate is presented and discussed in this paper. The aim of the study is to develop an analytical model for the vibration and geometrically linear thermal buckling analysis of cracked MEE plate based on the classical plate theory (CPT). The line spring model (LSM) is modified for the crack terms to accommodate the effect of electric and magnetic field rigidities, whereas the effect of thermal environment is accommodated in the form of thermal moment and in-plane forces. A classical relation for thermal buckling phenomenon of cracked MEE plate is also proposed. The governing equation for cracked MEE plate has also been solved to get central deflection which shows an important phenomenon of shift in primary resonance due to crack and temperature rise. The results evaluated for natural frequencies as affected by crack length, plate aspect ratio and critical buckling temperature are presented for first four modes of vibration. The obtained results reveal that the fundamental frequency of the cracked plate decreases with increase in temperature and crack length. Furthermore the variation of the critical buckling temperature with plate aspect ratio and crack length is also established for different modes of vibration.

Stability and dynamic analysis of partially cracked thin orthotropic microplates under thermal environment: an analytical approach

An analytical model is proposed to analyze the vibration and buckling problem of partially cracked thin orthotropic microplate in the presence of thermal environment. The differential governing equation for the cracked plate is derived using the classical plate theory in conjunction with the strain gradient theory of elasticity. The crack is modeled using appropriate crack compliance coefficients based on the simplified line spring model. The influence of thermal environment is incorporated in governing equation in form thermal moments and in-plane compressive forces. The governing equation for cracked plate has been solved analytically to get fundamental frequency and central deflection of plate. To demonstrate the accuracy of the present model, few comparison studies are carried out with the published literature. The stability and dynamic characteristics of the cracked plate are studied considering various parameters such as crack length, plate thickness, change in temperature, and internal length scale of microstructure. It has been concluded that the frequency and deflection are affected by crack length, temperature, and internal length scale of microstructure. Furthermore, to study the buckling behavior of cracked plate, the classical relations for critical buckling load and critical buckling temperature is also proposed considering the effect of crack length, temperature, and internal length scale of microstructure.

Effect of Fluid–Structure Interaction on Vibration and Deflection Analysis of Generally Orthotropic Submerged Micro-plate with Crack Under Thermal Environment: An Analytical Approach

To develop a new analytical model for vibration analysis of cracked-submerged orthotropic micro-plate affected by fibre orientation and thermal environment. The proposed analytical model is based on Kirchhoff’s classical thin plate theory and the size effect is introduced using the modified couple stress theory. Effect of crack is deduced using appropriate crack compliance coefficients based on line spring model while the effect of thermal environment is introduced in terms of thermal in-plane moments and forces. The coupling of shear and normal stresses for fibre orientation is represented using the coefficient of mutual influence. The fluid forces associated with its inertial effects are added in the governing differential equation to incorporate the fluid–structure interaction effect. The results are presented for frequency response as affected by different fibre orientation, crack length, crack location, level of submergence, temperature variation and material length-scale parameter for simply supported boundary condition. Furthermore, to study the phenomenon of shifting of primary resonance in a cracked micro-plate, the classical relations for central deflection of plate is also proposed. The results show that the fundamental frequency of micro-plate decreases by the presence of crack and thermal environment and this decrease in frequency is further intensified by the presence of surrounding fluid medium in present study. Another important conclusion is that with increase in temperature variation the reduction in frequency at 45° of fibre orientation is less when compared to 0 and 90° for both intact and cracked orthotropic plates.