This paper presents a new inverse tangent shear deformation theory (ITSDT) for the static, free vibration and buckling analysis of laminated composite and sandwich plates. In the present theory, shear stresses are vanished at the top and bottom surfaces of the plates and shear correction factors are no longer required. A weak form of the static, free vibration and buckling models for laminated composite and sandwich plates based on ITSDT is then derived and is numerically solved using an isogeometric analysis (IGA). The proposed formulation requires -continuity generalized displacements and hence basis functions used in IGA fulfill this requirement. Numerical examples are provided to show high efficiency of the present method compared with other published solutions.
A micro scale Timoshenko beam model is developed based on strain gradient elasticity theory. Governing equations, initial conditions and boundary conditions are derived simultaneously by using Hamilton's principle. The new model incorporated with Poisson effect contains three material length scale parameters and can consequently capture the size effect. This model can degenerate into the modified couple stress Timoshenko beam model or even the classical Timoshenko beam model if two or all material length scale parameters are taken to be zero respectively. In addition, the newly developed model recovers the micro scale Bernoulli–Euler beam model when shear deformation is ignored. To illustrate the new model, the static bending and free vibration problems of a simply supported micro scale Timoshenko beam are solved respectively. Numerical results reveal that the differences in the deflection, rotation and natural frequency predicted by the present model and the other two reduced Timoshenko models are large as the beam thickness is comparable to the material length scale parameter. These differences, however, are decreasing or even diminishing with the increase of the beam thickness. In addition, Poisson effect on the beam deflection, rotation and natural frequency possesses an interesting “extreme point” phenomenon, which is quite different from that predicted by the classical Timoshenko beam model.
In this paper, a new first-order shear deformation theory is presented for functionally graded sandwich plates composed of functionally graded face sheets and an isotropic homogeneous core. By making a further assumption to the existing first-order shear deformation theory, the number of unknowns and governing equations of the present theory is reduced, thereby making it simple to use. In addition, the use of shear correction factor is no longer necessary in the present theory since the transverse shear stresses are directly computed from the transverse shear forces by using equilibrium equations. Equations of motion are derived from Hamilton's principle. Analytical solutions for bending, buckling and free vibration analysis of rectangular plates under various boundary conditions are presented. Verification studies show that the present first-order shear deformation theory is not only more accurate than the conventional one, but also comparable with higher-order shear deformation theories which have a greater number of unknowns.
Recent experimental evidence points to limitations in characterizing the critical strain in ductile fracture solely on the basis of stress triaxiality. A second measure of stress state, such as the Lode parameter, is required to discriminate between axisymmetric and shear-dominated stress states. This is brought into the sharpest relief by the fact that many structural metals have a fracture strain in shear, at zero stress triaxiality, that can be well below fracture strains under axisymmetric stressing at significantly higher triaxiality. Moreover, recent theoretical studies of void growth reveal that triaxiality alone is insufficient to characterize important growth and coalescence features. As currently formulated, the Gurson Model of metal plasticity predicts no damage change with strain under zero mean stress, except when voids are nucleated. Consequently, the model excludes shear softening due to void distortion and inter-void linking. As it stands, the model effectively excludes the possibility of shear localization and fracture under conditions of low triaxiality if void nucleation is not invoked. In this paper, an extension of the Gurson model is proposed that incorporates damage growth under low triaxiality straining for shear-dominated states. The extension retains the isotropy of the original Gurson Model by making use of the third invariant of stress to distinguish shear dominated states. The importance of the extension is illustrated by a study of shear localization over the complete range of applied stress states, clarifying recently reported experimental trends. The extension opens the possibility for computational fracture approaches based on the Gurson Model to be extended to shear-dominated failures such as projectile penetration and shear-off phenomena under impulsive loadings.
The present paper is focused on the size-dependent shear buckling of nanoplates embedded in Winkler-Pasternak foundation and hygrothermal environment. Hence, the refined higher-order plate theories (Polynomial, Exponential, and Hyperbolic) needless of any shear correction factor are used in the formulations. The equations of motion are derived based on the mentioned theories in conjunction with the nonlocal strain gradient theory employing Hamilton's principle. The four unknown functions denoting the buckling load of plates are defined in a modal manner, and Navier solution method is used to find the shear buckling response. Results for the shear buckling and thermal buckling analysis of nanoplates are approved by existing literature to demonstrate the accuracy of present formulation and solution method. From our knowledge, it is the first time that the hygrothermal environment and also the nonlocal strain gradient theory are applied to study on shear buckling of nanoplates. Hence, the influence of nanoplate geometry, various hygrothermal conditions, elastic medium, nonlocal parameter and gradient parameter on the shear buckling load are obtained and discussed using different plate theories. The numerical results indicate that the shear buckling of nanoplate in the absence of strain gradient parameter is significantly affected by the temperature and moisture variations.
In this paper, the free vibration characteristics of embedded functionally graded carbon nanotube-reinforced composite (FG-CNTRC) spherical shells are studied based on a numerical approach. The elastic foundation is considered to be Pasternak-type. Moreover, the extended rule of mixture is used so as to obtain the material properties of FG-CNTRC. The shell is also modeled according to the first-order shear deformation shell theory. The energy functional of the structure is obtained first. Using differential operators, the discretized form of the energy functional is derived. By means of the variational differential quadrature (VDQ) method, the reduced forms of mass and stiffness matrices are then obtained. Selected numerical results are given to investigate the effects of different parameters such as elastic foundation coefficients, boundary conditions, CNT volume fraction, thickness-to-radius ratio and type of distribution of CNT on the vibrations of FG-CNTRC spherical shells.
Fracture toughness is an important material property used to perform the integrity assessment of engineering components containing cracks. Due to the difference in crack tip constraint, specimens may show different fracture toughness. The constraint difference for cruciform specimen with shallow crack, compact tension (CT) specimen and three point bending specimen with shallow and deep cracks are investigated. Both linear elastic and elastic-plastic fracture mechanics are applied to study the constraint effect based on two-parameter fracture criterion. Crack tip constraint depends on the applied loading. J-A method is used to precisely capture the crack tip constraint and crack tip stress distributions. Local approach to fracture can be applied to transfer the fracture toughness among different specimens under uniaxial and biaxial loadings. In case of positive T , T increases with K . In the case of negative T , T decreases with K . Q generally decreases with applied loading for both deep crack and shallow crack cases. Loss of constraint occurs for the single-edged bending (SEB) specimen with deep crack and thus raises the question whether the SEB specimen is proper to be used to obtain material toughness. For the cruciform bending (CRB) specimen, the constraint at the crack tip surface shows a least constraint while the deepest point has a relatively higher constraint. At a fracture probability of 10%, the fracture toughness difference between CT specimen and CRB specimen is about 50 MPa m , i.e 200% of the fracture toughness. This big difference demonstrates the importance of considering the constraint effects in the integrity analysis.
The central focus of the paper is set on modelling of bending of armchair carbon nanotubes by means of the gradient elasticity theory. Influence of small-size effects on the Young's modulus is investigated. An attempt to determine small size (or nonlocal) parameter employed in the Bernoulli-Euler and Timoshenko gradient formulations is presented. To obtain such a goal, the paper provides an extensive set of molecular structural mechanics simulations of armchair nanotubes with different loading and kinematic boundary conditions. Dependence of the Young's modulus on small size effects is clearly noticed. Based on these results, small scale parameters for the gradient model are identified and limits of the method are pointed out. Results of the study indicate that the widely used theory should be modified to obtain a physically justified and reliable nanobeam model based on Bernoulli-Euler or Timoshenko kinematic assumptions.
The heart is not only our most vital, but also our most complex organ: Precisely controlled by the interplay of electrical and mechanical fields, it consists of four chambers and four valves, which act in concert to regulate its filling, ejection, and overall pump function. While numerous computational models exist to study either the electrical or the mechanical response of its individual chambers, the integrative electro-mechanical response of the whole heart remains poorly understood. Here we present a proof-of-concept simulator for a four-chamber human heart model created from computer topography and magnetic resonance images. We illustrate the governing equations of excitation–contraction coupling and discretize them using a single, unified finite element environment. To illustrate the basic features of our model, we visualize the electrical potential and the mechanical deformation across the human heart throughout its cardiac cycle. To compare our simulation against common metrics of cardiac function, we extract the pressure–volume relationship and show that it agrees well with clinical observations. Our prototype model allows us to explore and understand the key features, physics, and technologies to create an integrative, predictive model of the living human heart. Ultimately, our simulator will open opportunities to probe landscapes of clinical parameters, and guide device design and treatment planning in cardiac diseases such as stenosis, regurgitation, or prolapse of the aortic, pulmonary, tricuspid, or mitral valve.
Due to excessive service load, inappropriate operating conditions or simply end of life fatigue, damage can occur in gears. When a fault, either distributed or localised, is incurred by gears, the stiffness and consequently vibration characteristics of the damaged tooth will change. In this work an analytical formulation of the time varying gearmesh stiffness was derived. An original analytical modelling of tooth cracks is presented and the gearmesh stiffness reduction due to this fault is quantified. A comparison with finite element model is presented in order to validate the analytical formulation.
In this paper, the dispersion characteristics of elastic waves propagating in a monolayer piezoelectric nanoplate is investigated with consideration of the surface piezoelectricity as well as the nonlocal small-scale effect. Nonlocal electroelasticity theory is used to derive the general governing equations by introducing an intrinsic length, and the surface effects exerting on the boundary conditions of the piezoelectric nanoplate are taken into account through incorporation of the surface piezoelectricity model and the generalized Young–Laplace equations. The dispersion relations of elastic waves based on the current formulation are obtained in an explicit closed form. Numerical results show that both the nonlocal scale parameter and surface piezoelectricity have significant influence on the size-dependent properties of dispersion behaviors. It is also found that there exists an escape frequency above which the waves may not propagate in the piezoelectric plate with nanoscale thickness.
A multitude of composite materials ranging from polycrystals to rocks, concrete, and masonry overwhelmingly display random morphologies. While it is known that a Cosserat (micropolar) medium model of such materials is superior to a Cauchy model, the size of the Representative Volume Element (RVE) of the effective homogeneous Cosserat continuum has so far been unknown. Moreover, the determination of RVE properties has always been based on the periodic cell concept. This study presents a homogenization procedure for disordered Cosserat-type materials without assuming any spatial periodicity of the microstructures. The setting is one of linear elasticity of statistically homogeneous and ergodic two-phase (matrix-inclusion) random microstructures. The homogenization is carried out according to a generalized Hill–Mandel type condition applied on mesoscales, accounting for non-symmetric strain and stress as well as couple-stress and curvature tensors. In the setting of a two-dimensional elastic medium made of a base matrix and a random distribution of disk-shaped inclusions of given density, using Dirichlet-type and Neumann-type loadings, two hierarchies of scale-dependent bounds on classical and micropolar elastic moduli are obtained. The characteristic length scales of approximating micropolar continua are then determined. Two material cases of inclusions, either stiffer or softer than the matrix, are studied and it is found that, independent of the contrast in moduli, the RVE size for the bending micropolar moduli is smaller than that obtained for the classical moduli. The results point to the need of accounting for: spatial randomness of the medium, the presence of inclusions intersecting the edges of test windows, and the importance of additional degrees of freedom of the Cosserat continuum.
This paper presents an analytical investigation on nonlinear thermal dynamic behavior of imperfect functionally graded circular cylindrical shells eccentrically reinforced by outside stiffeners and surrounded on elastic foundations using the Reddy's third order shear deformation shell theory in thermal environment. Material properties are graded in the thickness direction according to Sigmoid power law distribution (S-FGM) in terms of the volume fractions of constituents with metal–ceramic–metal layers. The shells are affected by mechanical, damping loads and temperature. The stress function and the Bubnov–Galerkin method are applied. Unlike previous publications, we propose a general formulation for forces and moments which allow the non-linear dynamic of shear deformable eccentrically stiffened shell to be studied taking into account the thermal stress in both the shells and the stiffeners. Numerical results are given for evaluating effects of temperature, material and geometrical properties, elastic foundation and eccentrically outside stiffeners on nonlinear dynamic of the shear deformable S-FGM shells. A good agreement is obtained by comparing the present analysis with other available in the literature.
This paper deals with free vibration problems of functionally graded shells. The analysis is performed by radial basis functions collocation, according to a higher-order shear deformation theory that accounts for through-the-thickness deformation. The equations of motion and the boundary conditions are obtained by Carrera’s Unified Formulation resting upon the principle of virtual work, and further interpolated by collocation with radial basis functions. Numerical results include spherical as well as cylindrical shell panels with all edges clamped or simply supported and demonstrate the accuracy of the present approach. ► Functionally graded shell panels are analyzed. ► Carrera’s Unified Formulation and collocation with radial basis functions are used. ► A higher-order shear deformation theory allowing thickness-stretching is employed. ► The fundamental frequency decreases as the radius of curvature increases. ► The thickness-stretching effect is independent of curvature radius.
A Mindlin microplate model based on the modified strain gradient elasticity theory is developed to predict axisymmetric bending, buckling, and free vibration characteristics of circular/annular microplates made of functionally graded materials (FGMs). The material properties of functionally graded (FG) microplates are assumed to vary in the thickness direction. In the present non-classical plate model, the size effects are captured through using three higher-order material constants. By using Hamilton's principle, the higher-order equations of motion and related boundary conditions are derived. Afterward, the generalized differential quadrature (GDQ) method is employed to discretize the governing differential equations along with various types of edge supports. Selected numerical results are given to indicate the influences of dimensionless length scale parameter, material index and radius-to-thickness ratio on the deflection, critical buckling load and natural frequency of FG circular/annular microplates.
The ballistic performance of clamped circular carbon fibre reinforced polymer (CFRP) and Ultra High Molecular Weight Polyethylene (UHMWPE) fibre composite plates of equal areal mass and 0/90° lay-up were measured and compared with that of monolithic 304 stainless steel plates. The effect of matrix shear strength upon the dynamic response was explored by testing: (i) CFRP plates with both a cured and uncured matrix and (ii) UHMWPE laminates with identical fibres but with two matrices of different shear strength. The response of these plates when subjected to mid-span, normal impact by a steel ball was measured via a dynamic high speed shadow moiré technique. Travelling hinges emanate from the impact location and travel towards the supports. The anisotropic nature of the composite plate results in the hinges travelling fastest along the fibre directions and this results in square-shaped moiré fringes in the 0/90° plates. Projectile penetration of the UHMWPE and the uncured CFRP plates occurs in a progressive manner, such that the number of failed plies increases with increasing velocity. The cured CFRP plate, of high matrix shear strength, fails by cone-crack formation at low velocities, and at higher velocities by a combination of cone-crack formation and communition of plies beneath the projectile. On an equal areal mass basis, the low shear strength UHMWPE plate has the highest ballistic limit followed by the high matrix shear strength UHMWPE plate, the uncured CFRP, the steel plate and finally the cured CFRP plate. We demonstrate that the high shear strength UHMWPE plate exhibits Cunniff-type ballistic limit scaling. However, the observed Cunniff velocity is significantly lower than that estimated from the laminate properties. The data presented here reveals that the Cunniff velocity is limited in its ability to characterise the ballistic performance of fibre composite plates as this velocity is independent of the shear properties of the composites: the ballistic limit of fibre composite plates increases with decreasing matrix shear strength for both CFRP and UHMWPE plates.
In this paper free vibration behavior of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) cylindrical panel embedded in piezoelectric layers with simply supported boundary conditions is investigated by using three-dimensional theory of elasticity. By using Fourier series expansion along the longitudinal and latitudinal directions and state space technique across the thickness direction, state space differential equations are solved analytically. The traction-free surface conditions then give rise to the characteristic equation for natural frequencies. Accuracy and convergence of the present approach are validated by comparing the numerical results with those found in literature. In addition, the effects of volume fraction of CNT, four cases of FG-CNTRC, piezoelectric layer thickness, mid radius to thickness ration and modes number on the vibration behavior of the hybrid cylindrical panel are also examined.
A number of refined beam theories are discussed in this paper. These theories were obtained by expanding the unknown displacement variables over the beam section axes by adopting Taylor's polynomials, trigonometric series, exponential, hyperbolic and zig-zag functions. The Finite Element method is used to derive governing equations in weak form. By using the Unified Formulation introduced by the first author, these equations are written in terms of a small number of fundamental nuclei, whose forms do not depend on the expansions used. The results from the different models considered are compared in terms of displacements, stress and degrees of freedom (DOFs). Mechanical tests for thick laminated beams are presented in order to evaluate the capability of the finite elements. They show that the use of various different functions can improve the performance of the higher-order theories by yielding satisfactory results with a low computational cost.
Finite element models of microstructure-dependent geometrically nonlinear theories for axisymmetric bending of circular plates, which accounts for through-thickness power-law variation of a two-constituent material, the von Kármán nonlinearity, and the strain gradient effects are developed for the classical and first-order plate theories. The strain gradient effects are included through the modified couple stress theory that contains a single material length scale parameter which can capture the size effect in a functionally graded material plate. The developed finite element models are used to determine the effect of the geometric nonlinearity, power-law index, and microstructure-dependent constitutive relations on the bending response of functionally graded circular plates with different boundary conditions.
In the present article the mechanical instability and free vibration of FGM micro-plate based on the modified strain gradient theory were studied using the spline finite strip method. By daily increase in the application of micro-scale structures, developing theories were become essential to account in a way for the size-reduction effect. The modified strain gradient theory based on three length-scale parameters, has the capability of evaluating structures at the micro size level. Considering the obtained results, it was clear that increasing the length-scale parameter would increase the critical buckling load and the vibration frequency, similar to the macroscopic case. In addition, increasing the power of volume fraction module decreases the critical load and the natural frequency of micro plate. Finally, the effect of length-scale parameter, boundary conditions, volume fraction module and dimensions of the micro-plate on critical loading and natural frequency of micro-plate were studied. (C) 2016 Elsevier Masson SAS. All rights reserved.