Two-dimensional Theory Of Elasticity Research Articles

A computational approach to integrate three-dimensional peridynamics and two-dimensional higher-order classical elasticity theory for fracture analysis

AbstractAs a nonlocal alternative of classical continuum theory, peridynamics (PD) is mathematically compatible to discontinuities, making it particularly attractive for failure prediction. The PD theory on the other side can be computationally demanding due to its nonlocal interactions. A coupling between PD and refined higher-order finite element method (FEM) integrates their salient features. The present study proposes a computational approach to couple three-dimensional peridynamics with two-dimensional higher-order finite elements based on classical elasticity. The bond-based PD modeling is considered in a region where damage might appear while refined finite element modeling is used for the remaining region. The refined finite elements employed in this study are based on the 2D Carrera Unified Formulation (CUF), which provides 3D-like accuracy with optimized computational efficiency. The coupling between PD and FEM is achieved through the Lagrange multiplier method which permits physical consistency and compatibility at the interface domain. An adaptive convergence check algorithm is also proposed to achieve predetermined accuracy in the solution with minimum computational effort. Simulations of quasi-static tension tests, wedge splitting tests and L-plate cracking tests are carried out for verification. In-depth analysis shows that the present approach can reproduce the linear deformation, material degradation and crack propagation in an effective way.

Free vibration of the piezoelectric laminated composite beams under the elastic foundation

Due to its self-adaptability and electric-mechanical interaction, the piezoelectric laminated composite structure has received increased attention in the engineering community, with the intricacy of the actual scenario, the structural stability based on numerous foundations has also been recognized. In this paper, a unified formulation which is based on classical laminated beam is presented to conduct free vibration analysis of piezoelectric laminated beams subjected to classical boundary conditions with the varied elastic foundations. The semi-analytical solutions of layered beams in various boundaries have been obtained based on the two-dimensional piezoelectric elastic theory and state space-differential quadrature method (SS-DQM), the solution of the general structural equation has been generated introducing the electrical boundary conditions and interlayer continuity conditions with this understanding. The different elastic foundations have been used to calculate the frequency equation of the piezoelectric laminated construction. The proposed finite element model is validated through the analysis of the piezoelectric laminated composite beam (PLCB) using the SS-DQM, and the present solutions are compared with those available in the literature to confirm their validity. A systematic parameter study for the piezoelectric laminated beams with various boundary conditions, piezoelectric parameters, lamina numbers and thickness ratios is also performed. The results of the PLCBs with various boundary constraints under the elastic foundation are presented and they may be served as a benchmark for researchers in this field.

Active vibration control of rotating smart bidirectional FGPTC subjected to internal mechanical shock

This study considers the active vibration control of a sandwich truncated cone, whose intermediate layer is made of bidirectional functionally graded (FG) porous material bonded with integrated piezoelectric actuator and sensor layers on its outer and inner surfaces. The material properties of the bidirectional functionally graded porous truncated cone (FGPTC), with even and uneven porosity distributions along the length and thickness directions, are changed using a power law function. The governing motion equations utilize the two-dimensional (2D) axisymmetric theory of elasticity rather than simple shell theories. The solving of the governing motion equations is accomplished through the utilization of graded finite element and Newmark methods. The study investigates the impact of dimensions, parameters, and porosity types on transient response behavior, including the semi-vertex angle and the ratio of truncated cone length to outer radius. Furthermore, the effects of power index, porosity volume fractions, rotational velocities, and boundary conditions are examined. Active vibration control (AVC) of the piezoelectric FGPTC is achieved by employing a closed-loop control algorithm based on velocity feedback. Several numerical simulations are conducted to evaluate the effectiveness of the proposed active vibration control system in reducing structural vibrations. The results demonstrate significant reductions in vibration amplitudes and improved structural stability achieved through the active control approach.

The flexural behavior of cross laminated bamboo and timber (CLBT) and cross laminated timber (CLT) beams

The flexural behavior of cross laminated bamboo and timber (CLBT) and cross laminated timber (CLT) beams are studied to explore the potential of using glued laminated bamboo (glubam) in CLT panels to obtain similar or even better mechanical performance and other constructional and environmental merits. The 4-point bending experiments were conducted for full-scale model CLBT and CLT beams, assisted with using deformation instrumentations including the digital image correlation method (DICM) and conventional methods (LVDT and strain gauges). All the experimental results show that the CLBT beams have the similar even better bending stiffness and capacity than corresponding CLT beams, and the advantage is more significant for beams under in-plane bending condition. Analytical models based on two-dimensional (2D) elasticity theory are also applied to predict the flexural response of CLBT beams. The accuracy of the proposed analytical models can be verified by comparing the results with experimental results. Some parametric studies for CLBT beams with different combination configurations of glubam and wood layers are also conducted to provide several design suggestions to achieve favorite flexural performance for this type of cross laminated bamboo and timber beams.

Clarification of Faber series and related applications to complex variable methods in two-dimensional elasticity

Faber series are used extensively in the application of complex variable methods to two-dimensional elasticity theory, for example, in the mechanical analysis of composite materials where Faber series representations of complex potentials lead to convenient expressions for the corresponding displacement and stress distributions. In many cases, the use of the Faber series is combined with conformal mapping techniques which “transfer” a boundary value problem defined in the elastic body (physical plane) to a simpler problem posed in an imaginary plane characterized by the conformal mapping. In several instances in the literature, however, little attention has been paid to the domain of definition of the Faber series in the imaginary plane leading often to misunderstandings and erroneous conclusions regarding the concept and feasibility of the use of the Faber series. In this paper, we present a thorough and rigorous examination of the representation of the Faber series in both the physical (occupied by the material) and imaginary (defined by the conformal mapping) plane. In addition, we show that replacing a truncated Faber series by a truncated Taylor series does not induce any additional errors in the numerical analysis of the corresponding boundary value problem. We anticipate that the discussion in this paper will help clarify any existing misinterpretations regarding the application of the Faber series and help further extend their use to a range of problems in composite mechanics.

Analytical solutions and optimization design for bending behavior of cross-laminated timber beams considering orthotropy and interface slip

The aim of this work was to propose analytical solutions and an optimization design method for the bending behavior of cross-laminated timber (CLT) beam, in which the orthotropy of timber and interface slip between the neighboring layers were considered simultaneously. The lamina layers of orthotropic woods were modeled based on the two-dimensional (2D) elasticity theory without transverse shear deformation assumption. By means of series expansion and solving differential equations, the general solutions for stresses and deformations in the beam were obtained with undetermined coefficients. The unknown coefficients were further determined by the transfer matrix method. The comparison study indicated the proposed solutions had higher accuracy than the Euler-Bernoulli beam solution and isotropic solution, since the proposed solutions renounced the transverse shear deformation assumption and considered the orthotropy of wood. Moreover, it was found that the interfacial stiffness has an economical upper bound above which no significant gain on the bending stiffness of CLT beam can be obtained, and an optimal configuration of geometry and material can achieve an increase of bending stiffness. We release our program for the proposed analytical solutions, at https://github.com/Njtpw/CLT-Beam.git.

Analysis of Anisotropic Elastic Medium with Multiple Elliptical Holes Subjected to Uniform Load of in-Plane and Out-of-Plane Shear at Infinity

In this paper, a stress concentration around two holes of a 3D-printed material and stress intensity factors (SIFs) at straight crack tips of a rock are shown graphically as several practical calculation examples. The stress components and SIFs are calculated using the two-dimensional elastic theory. The elastic constitutive equation of materials has been used as isotropy or orthotropy in most previous analyses. Consequently, the mutual influence of the in-plane and out-of-plane shear problem has rarely been discussed. Therefore, at the first in this paper, the solutions that take into consideration the general constitutive equation with such mutual influence, are derived by the modified Lekhnitskii formulation. Next, numerical calculations for a 3D-printed polycarbonate plate subjected to uniform tension are performed in order to evaluate the mutual influence of anisotropy on the stress concentration for two circular holes. By changing the distance between the holes and angle formed by the stack and tensile directions, not only the effect of these parameters on the magnitude of out-of-plane shear stress but also the stress concentration factor for tension are clarified. Further, SIF for fracture mode I is calculated for a straight crack group in granite, and the mutual influence of SIF for fracture mode III is evaluated. As the results, it is found about SIF for fracture mode III that it becomes zero for enough crack spacing, whereas it is increasing continuously with increase of crack density.

B-spline wavelet discrete-continual finite element method for the local solution to the two-dimensional problem of the theory of elasticity

Introduction. The distinctive article presents a local semi-analytical solution to the problem of the two-dimensional theory of elasticity. The corresponding structures, featuring the regularity (constancy) of physical and geometric parameters (the modulus of elasticity of the material of the structure, the Poisson’s ratio of the material of the structure, dimensions of the cross section of the structure) along one direction (dimension) are under consideration. This direction is conventionally called the basic direction.
 Materials and methods. The B-spline wavelet discrete-continual finite element method (DCFEM) is used. The initial ope­rational formulation of the problem was constructed using the theory of distribution and the so-called method of extended domain, proposed by Prof. Alexander B. Zolotov.
 Results. Some topical issues of construction of normalized basis functions of a B-spline are considered, the approximation technique for corresponding vector functions and operators within DCFEM is described. Along the basic direction, the problem remains continual and an exact analytical solution can be obtained, while along the non-basic direction the finite element approximation is used in combination with a wavelet analysis apparatus. As a result, we can obtain a discrete-continual formulation of the problem. Thus, we have a multi-point (in particular, a two-point) boundary problem for the first-order system of ordinary differential equations with constant coefficients. A special correct analytical method for the solution of such problems was developed, described and verified in numerous papers written by the authors. In particular, we consider the simplest sample analysis of a deep beam, fixed along the side faces and subjected to the load concentrated in the centre of the structure.
 Conclusions. The solution to the verification problem obtained using the proposed version of the wavelet-based DCFEM was in good agreement with the solution obtained using a conventional finite element method (corresponding solutions were constructed with localization and without localization; these solutions coincide almost completely, while the advantages of the numerical-analytical approach are quite obvious). It is shown that the use of B-splines of various degrees within the wavelet-based DCFEM leads to a significant reduction in the number of unknowns.

Conservation laws and solutions of the first boundary value problem for equations of two-dimensional elasticity theory

Conservation laws and solutions of the first boundary value problem for equations of two-dimensional elasticity theory

An efficient semi-analytical static and free vibration analysis of laminated and sandwich beams based on linear elasticity theory

Based on the two-dimensional (2D) elastic theory without enforcing any beam assumption, an efficient semi-analytical scaled boundary finite element method (SBFEM) is proposed to solve the bending and free vibration responses of composite laminated and sandwich beams under the mechanical load. The scaled center is placed at infinity, which produces the accurate result by discretizing only the longitudinal direction of the beam structure treated as a one-dimensional (1D) discretization problem. A new kind of 1D high-order spectral element shape functions with the advantages of high accuracy and superior convergence is introduced in SBFEM coordinate system to approximate the geometric model and corresponding variables. The principle of weighted residual in conjunction with the Green’s theorem are applied to obtain the SBFEM governing equation of each layer with respect to radial displacement fields. The solution of equation is indicated analytically by a matrix exponential function, which can be accurately solved by using the precise integration technique (PIT). Finally, an effective and simple stiffness matrix is obtained. By comparing two examples with the solutions based on the finite element method (FEM), the results show that the proposed method has good accuracy and rapid convergence with only a few meshes. The numerical examples are given to investigate the parametric effects of the stacking sequence, thickness ratio, boundary condition, and load form on the variation of the displacement, stress and natural frequency. The results validate that the present technique is also applicable to the complex beam structure with softcore layer inside.

Two-dimensional low-velocity impact analysis of curved sandwich beams with FG-CNTRC face sheets and porous core

The dynamic responses of curved sandwich beams with functionally graded carbon nanotube reinforced composite (FG-CNTRC) face sheets and porous core subjected to a low velocity impactor is studied. In order to accurately simulate the dynamic behavior of the sandwich curved beams, a layerwise-finite element approach based on the two-dimensional elasticity theory is employed. The contact force is estimated based on the modified Hertz law. Equivalent mechanical properties of the FG-CNTRC face sheets are determined using a refined rule of mixture approach. After validating the approach, some parametric studies are carried out to examine the influences of carbon nanotube distribution patterns, impactor velocity, the curved beam opening angle, length-to-thickness ratio and core-to-face sheet thickness ratio on the responses of the sandwich curved beams under investigation.

Exact Solutions for Time-Varying Bending Behavior of Multilayer Orthotropic Beams Bonded by Viscoelastic Interlayers

This study proposed exact solutions for the purpose of investigating and understanding the time-varying bending behavior of a multilayer orthotropic beam bonded by viscoelastic interlayers. In the analytical model, each orthotropic layer involved four independent elastic constants and was modeled on the basis of the two-dimensional (2D) elasticity theory. The interlayer exhibited viscoelastic properties and was described by the standard linear solid model. By means of Fourier series expansion, the general solutions of stresses and displacements were derived with unknown coefficients, which were analytically determined by interfacial and loading conditions, in which the convolution was converted by the Laplace transform. The results of numerical examples showed that the present solutions converged rapidly and were consistent with the finite element solutions. Comparison of the present 2D solutions with the one-dimensional (1D) solutions based on the Euler–Bernoulli beam theory indicated that the 1D solutions were inaccurate for thick beams. The effects of viscoelastic material and orthotropic constants on the long-term stresses and displacements in the beam were thoroughly examined. These results provided a reference for the optimization of a multilayer orthotropic beam design.

Calculation Derivation and Test Verification of Indirect Tensile Strength of Asphalt Pavement Interlayers at Low Temperatures.

When the direct tensile test is adopted to determine the interlayer tensile strength of the asphalt pavements, specimen separation or internal cracking often occurs at the bonding area of the loading head, rather than at the interlaminar bonding interface. In view of the tedious and discrete data of the direct tensile test, this paper attempts to introduce an indirect tensile test to determine the interlayer bond strength of asphalt pavement to solve this problem. However, the indirect tensile test method of a binder lacks the corresponding mechanical theory. This paper deduces the calculation formula of the indirect tensile strength of a binder based on elastic theory. A mechanical model of the test was established with the finite element method. In accordance with the two-dimensional elastic theory and the Flamant solution, an analytical solution of tensile stress in the indirect tensile test is proposed through the stress superposition. On this basis, the calculation formula for the indirect tensile strength of the interlaminar bonding is derived according to Tresca’s law. A low-temperature indirect tensile test was designed and conducted to verify the correctness of the formula. By comparing the results of the indirect tensile test and direct tensile test, it is found that the interlaminar strength of the mixture measured by them is similar, and the dispersion of indirect tensile test results is small. The results show that the indirect tensile test can replace the direct tensile test to evaluate the interlaminar tensile strength.

Free vibration analysis of nonlocal nanobeams: a comparison of the one-dimensional nonlocal integral Timoshenko beam theory with the two-dimensional nonlocal integral elasticity theory

Beam theories such as the Timoshenko beam theory are in agreement with the elasticity theory. However, due to the different nonlocal averaging processes, they are expected to yield different results in their nonlocal forms. In the present work, the free vibration behavior of nonlocal nanobeams is studied using the nonlocal integral Timoshenko beam theory (NITBT) and two-dimensional nonlocal integral elasticity theory (2D-NIET) with different kernels and their results are compared. A new kernel, termed the compensated two-phase (CTP) kernel, is introduced, which entirely compensates for the boundary effects and does not suffer from the ill-posedness of previous kernels. Using the finite element method, the free vibration analysis is performed for different boundary conditions based on the first three natural frequencies. For both the NITBT and 2D-NIET with both the two-phase (TP) and CTP kernels, the nonlocal parameter has a softening effect on the natural frequencies for all the boundary conditions, without observing the paradoxical behaviors of the nonlocal differential theory. For both theories, the softening effect of the nonlocal parameter is more pronounced for the TP kernel compared to the CTP kernel. The sensitivity of the 2D-NIET to the nonlocal parameter is found to be higher than that of the NITBT. Also, the softening effects for different vibration modes are compared to each other for both theories and both kernels. The obtained results can be extended for various important beam problems with nonlocal effects and help obtain a better understanding of applicable nonlocal theories.

LOCALIZATION OF SOLUTION OF THE PROBLEM OF TWO-DIMENSIONAL THEORY OF ELASTICITY WITH THE USE OF B-SPLINE DISCRETE-CONTINUAL FINITE ELEMENT METHOD

Localization of solution of the problem of two-dimensional theory of elasticity with the use of B-spline discrete-continual finite element method (specific version of wavelet-based discrete-continual finite element method) is under consideration in the distinctive paper. The original operational continual and discrete-continual formulations of the problem are given, some actual aspects of construction of normalized basis functions of a B-spline are considered, the corresponding local constructions for an arbitrary discrete-continual finite element are described, some information about the numerical implementation and an example of analysis are presented.

Free Vibration of Functionally Graded Graphene Platelets Reinforced Magnetic Nanocomposite Beams Resting on Elastic Foundation.

The vibrational characteristics of multilayer magnetic nanocomposite beams reinforced by graphene nanoplatelets (GPLs) are analytically investigated in this paper. The effects of an elastic foundation are also studied. The material properties of piece-wise GPL-reinforced nanocomposites (GPLRCs) are assumed to be graded in the thickness direction of the beams and can be estimated by using the modified Halpin–Tsai model and rules of mixtures. The two-dimensional elasticity theory is adopted to derive the governing equation combined with the state space method, and the analytical frequency equations for simply supported beams are obtained. In addition, the effects of a magnetic field are involved via Maxwell’s equation, and the corresponding Lorentz forces are considered in this work. Numerical examples are carried out to examine the effects of magnetic fields in various directions, the GPL distribution pattern, the scale parameter and weight function of GPLs, as well as an elastic foundation, on the vibration behaviors of functionally graded (FG)-GPLRC beams.

An accurate beam theory and its first-order approximation in free vibration analysis

An infinite system of one-dimensional differential equations is derived from the two-dimensional theory of elasticity by expanding the displacement field in a series of trigonometrical functions together with a linear term. Since the trigonometrical functions are pure thickness-vibration modes of infinite plates or beams with the top and bottom surfaces being free, the differential equations and the corresponding boundary conditions serve as the basis of an accurate beam theory for vibration analysis, named Lee's beam theory (LBT). Naturally, a high-order set of the infinite system should be quite useful in the analysis of beams at high frequencies. With the objective of vibration analysis of beams, this paper focuses on the first-order approximation, which leads to a first-order shear deformation beam theory for flexural vibrations (LBT1st). The differential equations in LBT1st are equivalent to those in Timoshenko's beam theory (TBT). The most important difference between LBT1st and TBT is the different field displacements. For the assessment of the accuracy of LBT1st, the numerical results of frequencies of free vibrations, frequency spectra and mode shapes of beams with classical boundary conditions are obtained and compared with those by TBT and plane stress problem of elasticity. Considering the plane stress problem as a reference, LBT1st is slightly more accurate in describing the field shapes of beams than TBT. Therefore, LBT1st, as well as LBT, is an addition to the existing beam theories with improved accuracy for the vibration analysis of beams and their combinations.

Elasticity Solutions for In-Plane Free Vibration of FG-GPLRC Circular Arches with Various End Conditions

In-plane free vibration of functionally graded graphene platelets reinforced nanocomposites (FG-GPLRCs) circular arches is investigated by using the two-dimensional theory of elasticity. The graphene platelets (GPLs) are dispersed along the thickness direction non-uniformly, and the material properties of the nanocomposites are evaluated by the modified Halpin-Tsai multi-scaled model and the rule of mixtures. A state-space method combined with differential quadrature technique is employed to derive the governing equation for in-plane free vibration of FG-GPLRCs circular arch, the semi-analytical solutions are obtained for various end conditions. An exact solution of FG-GPLRCs circular arch with simply-supported ends is also presented as a benchmark to valid the present numerical method. Numerical examples are performed to study the effects of GPL distribution patterns, weight fraction and dimensions, geometric parameters and boundary conditions of the circular arch on the natural frequency in details.

Accurate solution for functionally graded beams with arbitrarily varying thicknesses resting on a two-parameter elastic foundation

This work presents analytical solutions for bending deformation and stress distributions in functionally graded beams with arbitrarily and continuously variable thicknesses and resting on a two-parameter Pasternak elastic foundation. Based on two-dimensional elasticity theory directly, the general solutions of displacements and stresses which completely satisfy the differential equations governing the equilibrium for arbitrarily varying thickness functionally graded beams are derived for the first time. The undetermined coefficients in the general solution are obtained using Fourier series expansion along the upper and lower surfaces. The accuracy and efficiency of the proposed method are verified through several typical examples. The effects of mechanical and geometry parameters on the stress and displacement distributions of varying thickness functionally graded beams resting on a two-parameter Pasternak elastic foundation are discussed further.

Exact solutions for multilayer functionally graded beams bonded by viscoelastic interlayer considering memory effect

This work proposes exact solutions for simply supported multilayer functionally graded (FG) beams with viscoelastic interlayer to predict its time-dependent mechanical behavior. The stresses and deformations of FG layers are described by the two-dimensional (2-D) elasticity theory. The viscoelastic property of the interlayer is simulated by the standard linear solid model, and its constitutive relation is expressed in the convolution form which considers the strain memory effect. The general solutions of FG layers involving undetermined coefficients are obtained by virtue of the Fourier series expansion. By the use of the Laplace transform method, the solutions of stresses and deformations are determined analytically. The present solutions including Fourier series possess good convergence property. The comparison studies indicate that the finite element (FE) solutions approach the present ones when each FG layer is divided into many isotropic sub-layers in FE model and the gradient factor is taken as small value. The relative error of the one-dimensional (1-D) Euler-Bernoulli solutions is slight for slender beam with small gradient factor, but it significantly increases as the beam thickness or the gradient factor grows. Finally, the influences of the material constants on the time-dependent behavior in the beam are investigated in detail.