Three-dimensional Theory Of Elasticity Research Articles

Practical parameter tuning toward enhancing thermomechanical shock resistance of the nanocomposite structure

As pioneer research, the authors of this study focused on the analyzing impact of thermomechanical shock load on the behavior of the circumferentially graded single-walled carbon-nanotubes reinforced (CG-SWCNTR) nanocomposite sector circular plates with fully clamped boundaries. To ensure the highest level of accuracy, the three-dimensional theory of elasticity is considered the platform to define the governing equations. To solve the time-dependent equations, the differential quadrature approach (DQA) is combined with the Laplace transform. The construction of such structures will be possible in practice in the near future based on the accurate results obtained in this study.

On the complete solution of the three-dimensional solid space problems based on a novel curvilinear elasticity representation

A solution scheme is presented for the three-dimensional elasticity theory in the curvilinear cylindrical coordinate, without introducing any new physically dubious potential function. Exact analytical solutions for the essential 3-D solid full- and half-space problems are developed based on the introduced elasticity approach. The Kelvin's classical problem of concentrated force acting in the interior of an infinite solid space, as well as the Boussinesq's half-space problem of concentrated force normal to the free surface of a semi-infinite 3-D solid are treated. Furthermore, the three-dimensional non-axisymmetric problem of semi-infinite solid space under concentrated tangential force on its surface (the so-called Cerruti's problem) is revisited. It is demonstrated that unlike most of the conventional solutions in the literature for the aforementioned problems, the present approach is capable of obtaining the complete solution without needing to combine with other approaches. Evidently, the completeness issue associated with any new method of potentials is not present for the established direct elasticity approach. At the end, an extension of the approach to a nonhomogeneous elastic media is presented and, as an application, the Boussinesq's problem for a nonhomogeneous half-space is solved analytically. The developed representation may serve as an efficient replacement of the original three-dimensional Navier's elasticity equations for solving different 3-D elastic problems in the future.

Prediction of machining deformation induced by turning residual stress in thin circular parts using ritz method

Workpiece deformation induced by surface residual stress during turning is a serious concern in the manufacturing of thin-walled parts. Prediction of machining deformations caused by residual stress is essential to explain deformation mechanism and design machining strategies. Although many studies have been focused on calculating those deformations using the finite element method (FEM), it is time-consuming and redundant. Based on energy equations considering residual stress fields developed by the three-dimensional elastic theory and the principle of minimum potential energy, a new analytic model was established to predict machining deformation of the thin circular parts. In this model, the modified Rayleigh-Ritz method and pseudo inverse method were used to solve energy equations under various machining conditions to obtain machining deformation. Corresponding experiments and existing methods were considered to validate the accuracy and efficiency of the proposed model. Based on the quantitative results of the proposed model, the effects of machining conditions on machining deformation were discussed comprehensively, which provides guidance for machining strategies. • The energy equations of the thin circular plates considering residual stress distributions in all directions is developed. • The improved Rayleigh–Ritz method associated with the pseudo inverse method are used to solve the machining deformation . • The influence factors of machining deformation induced by residual stresses are comprehensively analyzed.

Three-dimensional free vibration analysis of functionally graded graphene platelets-reinforced composite toroidal shells

In this work, based on the weak form of three-dimensional (3D) theory of elasticity, we develop a mixed isoparametric finite element method (IFEM) for the analysis of the free vibration behavior of freely-supported, functionally graded (FG) graphene platelets-reinforced composite (GPLRC) toroidal shells. The functions of five distribution patterns of graphene platelets (GPLs) through the thickness direction are considered, for which the effective Young’s modulus is estimated using the Halpin-Tsai model, and the effective Poisson’s ratio and the effective mass density are estimated using the rule of mixtures. By incorporating Hamilton’s principle and Reissner’s strain energy functional, we derive the system equations of the mixed IFEM. Based on the mixed IFEM, the 3D solutions for the natural frequency parameters of the FG-GPLRC toroidal shells converge rapidly, and the convergent solutions are in excellent agreement with the accurate solutions for both homogeneous isotropic toroidal shells and FG transversely isotropic toroidal shells which are reported in the literature. Numerical results show that the impacts of the GPL distribution, the weight fraction of the GPL, and the radius-to-thickness ratio on the lowest natural frequencies of the FG-GPLRC toroidal shells is significant.

Static Elastic Bending Analysis of a Three-Dimensional Clamped Thick Rectangular Plate using Energy Method

Analytical formulations and solutions for the thick rectangular plate static analysis with clamped support based on a three-dimensional (3-D) elasticity theory is developed using the energy method. The theoretical model, whose formulation is based on the static elastic principle as already reported in the literature, is presented herein to obviate the shear correction coefficients while considering shear deformation effect and transverse normal strain/stress in the analysis. The equilibrium equations are obtained using 3-D kinematic and constitutive relations. The deflection and rotation functions, which are the solutions of the equilibrium equation, are obtained in closed form using a general variational technique for solving the boundary value problem. The minimization energy equation yields the general equation which was used to obtain the theoretical model for the deflection and stresses of the plate. The results are compared with the available literature and the results-computed trigonometric displacement function shows that this 3-D predicts the vertical displacement and the stresses more accurately than previous studies considered in this paper. The result showed that the percentage difference between the present work and those of 2-D Mindlin FSDT, 2-D numeric analysis, and 2-D HSDT of polynomial shape functions was about 3.02%, 0.62%, and 0.33%, respectively. It is concluded that the 3-D trigonometric model gives an exact solution, unlike other 2-D theories, and can be used for clamped-supported thick plate analysis. Doi: 10.28991/HIJ-2022-03-03-03 Full Text: PDF

Determination of Theoretical Fracture Criteria of Layered Elastic Composite Material by ANFIS Method from Artificial Intelligence

The theoretical breaking boundary of the composite elastic composite material consisting of two is otropichomogeneous layers was investigated [6,7]. Three-dimensional theory of elasticity and a fragment edhomogeneous body model were used to find the theoretical breaking limit [3,6,7]. The defects of this material, which has local defects, are of the same length hand is changing with the angle β. The purpose of the articles [10,11] was to examine the effect of theoretical refraction criterion limit values on the change of the angle β. It has been observed that the theoretical refraction limit values obtained for four different composite materials increase as the angle art increases. In this study, the change values with the β angle of the theoretical refraction limit obtained in [11] article were recovered from the artificial intelligence of the MATLAB platform by ANFIS (Sugeno) method. It was seen that the values obtained with the Anfis method are the same with the theoretical results.

A non-polynomial framework for describing the bending response of a multi-scale composite initially stressed disk

The present study deals with axisymmetric circular/annular disks made of three-phase multi-scale hybrid laminated nanocomposite under distributed static loadings. The structure is placed on Winkler-Pasternak substrate with friction effects. Initial stress is also considered in the composite structure. The mathematical model is prepared using three-dimensional elasticity theory (3D-elasticity theory). The derived differential equations are solved by employing the semi-analytical method of state-space-based differential quadrature method (SS-DQM) to obtain the stress and displacement fields of the composite structure. In the numerical analyses, a comprehensive parametric study is performed to observe the effects of some geometrical and physical parameters on bending characteristics of composite circular/annular disks.

Effects of electrode size and configuration on sound transmission loss in piezo-laminated thick shell

Sound transmission loss (STL) occurs as waves propagate through a structure. In order to reduce the sound transmission through a cylinder with better STL, an acoustic quieting process needs to be implemented. One efficient ways to minimize sound transmission is using piezoelectric materials, which can be applied to reduce transmitted noise in different structures instead of damping approaches avoiding burdensome to fully dampen structural vibrations via passive methods. This study proposes a model using different sizes and configurations of piezoelectric electrode patches to increase the STL through thick-walled piezo-laminated cylindrical shells. The cylindrical shell is modeled using the three-dimensional theory of elasticity and piezoelasticity, and the Helmholtz equation is used for the wave propagation consideration inside the acoustic cavity as well as through the surrounding fluids. The STL is calculated using an exact integration of acoustic power over the outside surface of the shell. The analytical solution is compared with finite element numerical results to verify its validity. Parametric studies are conducted to investigate the effects of sizes and configurations of electrodes on STL. Results are presented with specific recommendations regarding the electrode size and configuration for active sound absorption. In addition, by proper choice of electrode size, one can avoid the resonance frequencies of the cylinder to provide higher sound transmission loss.

Application of the piezoelectricity as the effective solution to energy management of the composite structures exposed to abrupt thermal load

This study pays special attention to practical solutions for declining the harmful impacts of thermal shock loading applied on the nanocomposite rectangular plate. The governing equations are obtained in the content of three-dimensional elasticity theory to establish an appropriate approach to achieving reliable results. Due to the time-dependent nature of the applied shock, it is essential to analyze the system’s transient response. Accordingly, the Laplace transform is used to set out the state-space form of the governing equations from time to the Laplace domain. The reliability of the results is verified by comparing them with those recorded in the literature.

ВИЗНАЧЕННЯ НАПРУЖЕНЬ У ТОВСТИХ ПЛИТАХ ПРИ ЛОКАЛІЗОВАНИХ НАВАНТАЖЕННЯХ

The problem of invesigating the stresses in the plates, which are subjected to the action of concentrated forces and locally distributed load is considered. The stresses were determined on the basis of the relations of the three-dimensional theory of elasticity using two methods. The first uses the symbolic Lurie method with the additional application of Vashchenko-Zakharchenko expansion formulas. The solution is constructed in the form of series, which exponentially converge at points distant from the applied forces. However, these series converge slowly in the vicinity of the applied load. In this regard, the solution of the problem is constructed by another method, using the Hankel integral transformation. The approach to find special integrals appearing in problems of the theory of elasticity for plates is proposed. After transformations, the relations for determining the stresses that include the integrals of the functions exponentially attenuating at infinity are obtained. The relations for determining the stresses in the plates on the basis of approximate equations using the Kirchhoff-Lev hypotheses are also given. For this case, the solutions for stresses subjected to the action of concentrated forces and under locally distributed load are given in analytical form. There are cases when stresses found by the elementary formulas, obtained on the basis of Kirchhoff-Lev equations with sufficient accuracy for practice, can be applied in plate bending problems. In particular, it is found that the determined stresses at the boundary opposite to the applied local load turn out to be tensile and at the same time practically accurate if the force is greater than the thickness of the plate. In this case the stresses found under the applied load are determined with larger errors. However, these stresses are compressive and are not essential in the calculation of strength of concrete materials.

Bending and free vibration analyses of multi-directional functionally graded plates in thermal environment: A three-dimensional Isogeometric Analysis approach

This study presents a three-dimensional numerical approach to analyze the bending and free vibration behavior of multi-directional functionally-graded plates under the effect of thermal environment. The effective material properties are temperature dependent and the temperature field within the plate’s volume are determined directly based on a generalized heat transfer equation. The governing equations for static bending and free vibration analyses are developed based on the three-dimensional elasticity theory. The Isogeometric Analysis approach is utilized to discretize the weak forms of heat transfer equation, static bending equation and free vibration equation. A iterative scheme is proposed to heat transfer problems with material being dependent on temperature. Rectangular and circular plates are investigated to validate the efficiency of Isogeometric Analysis approach on modeling and analyzing aspects. Various numerical examples are also presented to validate the accuracy of the proposed approach and investigate the influences of material variation profiles, thickness ratios, temperature rise, and boundary conditions on thermal responses of the plates.

Static and free vibration analyses of functionally graded porous skew plates reinforced by graphene platelet based on three-dimensional elasticity theory

This paper deals with the static and free vibration analyses of FG porous skew plates reinforced by graphene platelets (GPL). The plate comprises of a layered model with uniform or non-uniform dispersion of graphene platelets in a metallic matrix including open-cell interior pores. The extended rule of mixture in conjunction with the modified Halpin–Tsai approach is used to obtain the equivalent properties of the porous nanocomposite plate. Three various types of porosity distributions are assumed through the plate thickness including uniform and two kinds of symmetric functionally graded distributions. Besides, five various kinds of GPL dispersion patterns are assumed along the plate thickness. By substituting in the expression for functional of element and Rayleigh–Ritz method, the equations of motion are derived and a 3D FEM approach is applied to solve the governing equation of motions. The influences of different parameters including various porosity coefficients, porosity distribution in conjunction with GPL pattern and the weight fraction of GPL as nanofiller, different boundary conditions and various angles of skew plates on static response, and free vibration behavior of the skew nanocomposite plate have been surveyed.

Static Electro-Mechanical Response of Axisymmetric One-Dimensional Piezoelectric Quasicrystal Circular Actuator.

The piezoelectric effect of piezoelectric quasicrystalline materials is coexcited by phonon and phason fields. Piezoelectric quasicrystalline materials have excellent properties of both piezoelectric materials and quasicrystalline materials, which are expected to be used as actuators in the fields of aerospace, automotive, and intelligent manufacturing. Based on the three-dimensional elastic theory of piezoelectric quasicrystals, the state space equation for axisymmetric piezoelectric quasicrystal circular plate actuators is derived by using the state space method. Afterwards, the finite Hankel transformation is performed on the state equation, and a system of ordinary differential equations and corresponding boundary conditions are obtained. Finally, the exact solution of axisymmetric bending of one-dimensional hexagonal piezoelectric quasicrystal circular actuators under generalized elastic simply supported boundary conditions is obtained by using the propagator matrix method. Numerical results are given to compare the degradation results in this paper with those in the literature, and present the influences of the thickness-to-span ratio and stacking sequence on the phonon, phason, and electric fields when the surface of the laminated circular actuators is subjected to mechanical load. The exact solution obtained does not introduce any deformation assumption; therefore, the exact solution can provide references for numerical calculations of the mechanical behavior of piezoelectric quasicrystals.

Three-Dimensional Response of Multilayer FG-GPLRC Spherical Panels Under Blast Loading

In the present research, the dynamic responses of the multilayer functionally graded graphene platelets reinforced composite (FG-GPLRC) spherical panels under blast loading are studied. Three-dimensional elasticity theory is employed to derive the governing equations. The distribution of graphene platelets (GPLs) in each layer is uniform and random with a constant weight fraction. GPLs concentration across the panel thickness may be uniform or graded. The layerwise-differential quadrature method (LW-DQM) together with a non-uniform rational B-spline-based multi-step time integration scheme is used to discretize the motion equations. The convergence behavior of the method is examined numerically. Further, to assure its accuracy, the results in the limit cases are compared with those available in the literature. Finally, through the parametric studies, the effects of material and geometrical parameters such as GPLs distribution patterns, GPLs weight fraction and dimension ratios on the transient responses of the FG-GPLRC spherical panels subjected to blast loading are investigated.

Static and Buckling Analysis of a Three-Dimensional (3-D) Rectangular Thick Plates Using Exact Polynomial Displacement Function

This paper is devoted to study the buckling response of axially compressed rectangular thick plate based on the exact polynomial potential functional. The governing and equilibrium equation of an isotropic plate was derived based on the three-dimensional (3-D) static theory of elasticity, to get the relations between the rotations and deflection. These equations are solved in the form of polynomial analytically to obtain the exact displacements and stresses that are induced due to uniaxial compressive load action on the plate. By incorporating deflection and rotation function into the fundamental equation and minimized with respect to deflection coefficient, a new expression of the determination of the critical buckling load was established. This expression was applied to solve the buckling problem of a clamped thick rectangular plate which was simply supported at the first and freely supported at the third edge (SCFC). A graphic representation of results showed that, as the aspect ratio of the plate increases, the value of critical buckling load decreases while as critical buckling load increases as the length to breadth ratio increases. This implies that an increase in plate width increases the chance of failure in a plate structure. This theory obviates the numerical approximations in the thickness direction thereby guaranteeing accuracy in the solution of the displacement along the direction of thickness axis of the plate, hence, a significant lessening of the cost of computation.

ЛИНЕЙНАЯ ТЕОРИЯ УПРУГОСТИ С УЧEТОМ КВАДРАТИЧНЫХ СЛАГАЕМЫХ

We suggest a linear theory version based on Taylor decompositions for stresses and power-series for quadratic summand deformations. Thus, static equations of equilibrium in stresses are written in the form of the second-order partial derivatives differential equations. The resolving equations of equilibrium in displacements are represented in the form of the third order partial derivatives differential equations. The physical equations in this version of the linear theory of elasticity are written in the same way as in the classical linear theory of elasticity. Equilibrium equations, along with other parameters – physical constants of the medium – contain minor parameters dx, dy, dz, the value of which, as shown by numerical modelling, has little effect on the nature of the stress-strain state. It is suggested to use experimental data to determine them. Along with the formulating of the basic equations of the three-dimensional theory of elasticity, particular cases of the stress-strain state of elastic continuous medium are considered: uniaxial stressed state; uniaxial deformed state; flat deformation; generalized plane stress state. Determination of the stressed and deformed state of a thin elastic bar by integrating the resolving equations in stresses and displacements is considered as examples. The suggested version of the linear theory of elasticity, due to the quadratic summand in Taylor decompositions for stresses and in power-series for deformations, expands the classical linear theory of elasticity and, with an appropriate experimental justification, can lead to new qualitative effects in the calculation of elastic deformable bodies.

Three dimensional free vibration analysis of functionally graded graphene reinforced composite laminated cylindrical panel

In this paper, natural frequency analysis of functionally graded graphene platelet (GPL) reinforced composite cylindrical panel is investigated. Halpin-Tsai equations are employed to obtain the mechanical properties of the structure. The linear three-dimensional elasticity theory based on the Hamilton principle and the numerical finite element method are used for obtaining the governing equations of motion. Four patterns of GPL distributions such as: FG-X, FG-V, FG-O and UD are assumed through the thickness of shell. The effects of various parameters such as different distributions and weight fractions of GPL, zigzag and armchair lay-up, geometry and different boundary conditions on the natural frequencies and mode shapes of shell have been investigated. The results indicate that the maximum and minimum natural frequencies belong to FG-X pattern and FG-O pattern, respectively.

A novel stiffness prediction method with constructed microscopic displacement field for periodic composite plates

A novel method for predicting effective stiffnesses of composite plates with in-plane periodicity is proposed according to two-scale strain energy equivalence. The macroscale energy is reached with the Reissner-Mindlin plate theory, while the microscale energy is obtained through the newly constructed displacement functions of a unit cell within the framework of the three-dimensional elasticity theory. The new displacement functions consist of homogenized displacements, two decoupled expansion terms, and a higher-order perturbed displacement term. The elaborate finite element formulations are presented for realizing the present method in Comsol Multiphysics. Several numerical experiments demonstrate the effectiveness of the new stiffness prediction method.

Some improvements on the energy absorbed in composite plate resting on the viscoelastic substrate

The vibration, as well as energy of a reinforced three-phase hybrid composite plate resting on a viscoelastic foundation subjected to initial stress, is examined with the aid of three-dimensional elasticity theory. The two-parameter viscoelastic Winkler–Pasternak foundation is formulated. The multilayer plate with three, five, and seven layers are modeled with the aid of compatibility conditions. The state-space-based analytical method is utilized as the solution procedure to extract the results. To attain the composite material properties, the rule of the mixture alongside modified Halpin–Tsai model is incorporated. The distribution of the carbon nanotubes is supposed to be nonuniform. The parametric examination on the impacts of the sequence of the composite layers, initial stresses, foundation coefficients, and nanocomposite related constants on the vibration frequency as well as energy stored in the system is carried out. The extracted results indicate that the initial stresses and foundation parameters play a significant role in determining the frequency value along with the energy of the system.

NUMERICAL SOLUTION OF THE DYNAMIC PROBLEM OF AXISYMMETRIC VIBRATIONS OF REINFORCED SHELLS

The reliability of the results obtained in the work is determined by the rigor and correctness of the statements of the initial problems; theoretical substantiation of the finite-difference schemes used; controlled accuracy of numerical calculations; conducting test calculations; compliance of the established regularities with the general properties of oscillations of thin-walled structural elements. The correctness of the formulation of the problems is achieved by using the well-known equations of the theory of shells and rods of the Tymoshenko type, which are approximations of the original equations of the three-dimensional theory of elasticity. When deriving the equations, the equations of oscillations of the multilayer shell in the smooth region and the equations of oscillations of reinforcing ribbed elements (transverse ribs) were obtained. It is not difficult to show that the indicated equations by the classification of equations in partial derivatives are equations of the hyperbolic type, which are an approximation of the oscillating equations of three-dimensional elastic bodies and sufficiently correctly reproduce wave processes in non-homogeneous shell structures, taking into account spatial gaps. Numerical algorithms for approximate solutions of the original equations are based on the use of the integro-interpolation method of constructing difference schemes. When constructing difference schemes, kinematic quantities refer to difference points with integer indices, and the values of deformations and moment forces refer to difference points with half-integer indices. This approximation of the initial kinematic and static values allows the fulfillment of the law of conservation of the total mechanical energy of the elastic structure at the difference level. The numerical algorithm is based on the use of separate finite-difference relations in the smooth domain and on the lines of spatial discontinuities with the second order of accuracy in spatial and temporal coordinates.