Three-dimensional Linear Elasticity Theory Research Articles

Effect of extreme in-plane and transverse stiffness ratios on frequencies and load transfer between face sheets and core of rectangular sandwich plates

Challenging issues for sandwich structures include accurately finding their frequencies and how the load transfers from face sheets to the core when their material stiffness ratio (FCSR) in either the axial or the transverse directions varies from 1 to 107. Here we use both the equivalent single layer and the layer wise shear and normal deformable plate theories to analyze problems for rectangular plates of different thickness/length ratios. The Ritz method with Jacobi polynomials as basis functions is employed to numerically solve three-dimensional linear elasticity theory equations. Transverse stresses are calculated using a one-step stress recovery scheme. It is found that the FCSR in the transverse direction determines the minimum degree of complete polynomials in the thickness coordinate needed to accurately compute the lowest six frequencies and the transverse stresses. New results include the following: (i) the FCSR in the transverse direction has a greater influence than that in the axial direction on the computed frequencies and transverse stresses, and (ii) whereas both transverse normal and shear stresses in the core contribute to transmitting loads between the two face sheets for FCSR < 103, however, only the former is effective for FCSR ≥ 105.

Modularity of the displacement coefficients and complete plate theories in the framework of the consistent-approximation approach

Starting from the three-dimensional theory of linear elasticity, we arrive at the exact plate problem by the use of Taylor series expansions. Applying the consistent-approximation approach to this problem leads to hierarchic generic plate theories. Mathematically, these plate theories are systems of partial-differential equations (PDEs), which contain the coefficients of the series expansions of the displacements (displacement coefficients) as variables. With the pseudo-reduction method, the PDE systems can be reduced to one main PDE, which is entirely written in the main variable, and several reduction PDEs, each written in the main variable and several non-main variables. So, after solving the main PDE, the reduction PDEs can be solved by insertion of the main variable. As a great disadvantage of the generic plate theories, there are fewer reduction PDEs than non-main variables so that not all of the latter can be determined independently. Within this paper, a modular structure of the displacement coefficients is found and proved. Based on it, we define so-called complete plate theories which enable us to determine all non-main variables independently. Also, a scheme to assemble Nth-order complete plate theories with equations from the generic plate theories is found. As it turns out, the governing PDEs from the complete plate theories fulfill the local boundary conditions and the local form of the equilibrium equations a priori. Furthermore, these results are compared with those of the classical theories and recently published papers on the consistent-approximation approach.

Analysis of Longitudinal Guided Wave Propagation in the Functionally Graded Hollow Cylinder Using State-Vector Formalism and Legendre Polynomial Hybrid Approach

A numerical approach for analyzing the guided wave propagation behavior in a functionally graded material (FGM) hollow cylinder is presented. Based on the three-dimensional linear elasticity theory, the state matrix form of displacement and stress component is introduced into the differential governing equation. Combining with boundary conditions, the dispersion characteristic equations can be constructed. Meanwhile, taking the orthogonal completeness of Legendre series into consideration, the exhausting root findings of transcendental equation can be avoided, and it transforms the redundant integral operators into analytical expressions by means of its recursive properties. Moreover, it is not only unnecessary to stratify FGMs, and introducing the univariate nonlinear regression to modelling the arbitrary gradient distribution, and transforming the redundant integral operator into an analytical expression by means of recursive property. It should be pointed out that the coefficient matrix consists only zero and constant terms, which are related to the elastic constants and normalized wave number. And it provides a fast and effectively approach to extracting the dispersion curves, displacement distribution and stress profile, simultaneously. Meanwhile, the proposed method can get rid of the exhausted root-finding algorithm, and accordingly distinguish the different modes and the dispersion curves simultaneously. Finally, the typical non-stratified computation of dispersion curves of longitudinal guided wave for FGM hollow cylinder is realized. The accuracy of the proposed method is further demonstrated through comparison with the available data from Legendre polynomial method. Several numerical cases about iron based alumina FGM cylinder are studied, and the convergence of the present polynomial approach is discussed. Then the effect of the circumferential wave number, gradient variation, diameter-thickness ratio on the dispersion characteristics are illustrated. Moreover, the distribution characteristics of displacement and stress profiles of F(1,1) mode at a given frequency are discussed in detail. It is shown that the state-vector formalism and Legendre polynomial hybrid approach is a powerful tool in solving wave motions in FGM elastic cylinder, which lays a theoretical foundation for the non-destructive evaluation and quantitative characterization of the gradient structure characteristics of functionally graded hollow cylinder.

STRESS-STREIN STATE OF THE ROCK MASS AROUND THE VERTICAL MINE

The article deals with the problem of the stress-strain state of a rock mass around a vertical working of a circular cross-section. An exact formulation of the three-dimensional problem of the deformation of a half-space weakened by a deep cylindrical cavity is used. The stress-strain state of a half-space, as a three-dimensional body, strictly obeys the basic requirements of the three-dimensional linear theory of elasticity and is described by its corresponding equations and relations in a cylindrical coordinate system. The specific problem of rock mechanics has been solved, i.e. the considered rock mass works only in compression. The deformation process and stress state around vertical shafts of circular cross-section are expressed in terms of stress functions. Calculation formulas are derived for all nonzero components of the strain and stress tensors, taking into account the axisymmetry of the problem under consideration, represented in terms of stress functions.

Lowest Twelve Frequencies of Sandwich Plates Using Third-Order Shear-Normal Deformation Theory

We delineate effects of 1) face-to-core stiffness ratio (FCSR) and mass density ratio (FCDR) in square laminates and sandwich plates and 2) the fiber and transverse direction elastic modulus ratio in cross-ply laminates on the first 12 distinct frequencies found by analytically solving the three-dimensional linear elasticity theory equations and using an equivalent single-layer third-order shear and normal deformation plate theory (TSNDT). Whereas the elasticity equations for simply supported plates are solved by using Srivinas and Rao’s approach, the TSNDT equations are numerically solved by using weighted Jacobi polynomials and the Ritz method. We note that previous studies have generally compared only the fundamental frequency. For square laminates with a side-length/plate-thickness ratio of 100 (10), the maximum error in the TSNDT predicted first 12 frequencies is less than 0.4% (7%). For a FCSR less than 20, the maximum difference in predictions from the two approaches for the first six distinct frequencies is 5.4%. However, the FCDR has little effect on the difference between frequencies from the two methods. The frequencies and mode shapes presented herein should help us better understand the dynamic behavior of laminated and sandwich plates and provide benchmark results for others to assess their theories.

Own Weight Effect on Static Analysis of Pre-Stretched Plate-Strip Containing Twin Circular Inclusions Under Bending

Bu çalışma, eğilme etkisi altında içerisinde aynı malzemeden yapılmış ve merkezleri serbest yüzeye paralel bir doğrultuda olan ikiz dairesel şekilli dolgular içeren öngerilmeli bir kompozit şerit-plağın statik analizine kendi ağırlığının etkisini ele almaktadır. Dolgular içeren şerit-plak üzerindeki hacimsel kuvvetlerin (kendi ağırlığı) ve yüzeysel kuvvetlerin (öngerilme yüklemesi) etkisi birlikte öngerilmeler olarak düşünülmüştür. Bu ön​​gerilmelerinin, ek bir eğilme yükü altında bir şerit-plak içindeki dolgular etrafındaki gerilme dağılımlarının analizleri üzerindeki etkileri, düzlem şekil değiştirme durumu altında Lineerize Edilmiş Üç Boyutlu Elastisite Teorisi (LEÜBET) ile araştırılmaktadır. Ayrıca, ele alınan sınır değer problemlerinin çözümleri Sonlu Elemanlar Yöntemi kullanılarak gerçekleştirilmiştir. Sonuçlara göre, eğilme etkisi altındaki ikiz dairesel dolgular içeren şerit-plağın kendi ağırlığının plağın statik analizi üzerinde önemli bir etkisi olduğu ortaya çıkmıştır.

The Effect of Own Weight on Dynamic Analysis of a Pre-Stretched Composite Plate-Strip Containing Twin Circular Inclusions Under Bending Using Finite Element Method

This manuscript describes how the weight of a simply supported composite plate-strip (containing twin circular inclusions) affects its dynamic behaviors when exposed to bending load as analyzed via the finite element method (FEM). The centers of the twin circular inclusions are on a line parallel to the free surface, and the materials of both inclusions are the same. First, the effects of body forces (the plate-strip’s own weight) and surface forces (pre-stretching load) on the plate-strip (both are considered to be initial stresses) are identified using the classical linear theory of elasticity. Next, the consequences of these stressors (identified in the first step) under additional time harmonic bending load on the forced vibration around the inclusions are determined using the three-dimensional linearized theory of elasticity (TDLTE) under the plane-strain state. The data herein suggest that the plate-strip’s weight can significantly effect dynamic characteristics of the considered plate-strip.

Vibrational response of a MRI gradient coil cylinder to time-harmonic Lorentz-force excitations: An exact linear elastodynamic model for shielded longitudinal gradient coils

• A simple linear elastodynamic model of a shielded longitudinal gradient coil (GC) is introduced. • The model is used to investigate the frequency response of a typical whole-body GC cylinder. • The frequency response is governed by the Fourier decomposition of the spatial current density. • The widths of all structural resonances are decreased when shielding currents are present. • Numerical results are consistent with available in-situ experimental data and numerical models. The reduction of gradient coil (GC) vibration continues to be a challenging problem in the optimization of magnetic resonance imaging systems. A key deficiency for passive reduction strategies is that, under realistic thick-shell conditions, there are no existing mathematical models that can provide reliable theoretical predictions about the parametrical behaviors of the vibration response. In this paper, we introduce a simple linear elastodynamic model of a shielded longitudinal GC that can serve as a baseline theoretical model for studying the steady-state linear vibration response of an undamped GC cylinder under the condition that only the Z-coil windings are excited by Lorentz forces . The exact three-dimensional theory of linear elasticity is used to formulate the model, and hence, there are no built-in geometrical constraints. An exact closed-form solution for the steady-state displacement field of the GC cylinder is given, and the solution is then used to numerically investigate the frequency response of a typical whole-body GC cylinder. A core prediction of the model is that the frequency response is essentially governed by the Fourier decomposition of a dimensionless “profile function” that specifies how the current density varies along the GC cylinder axis. An interesting corollary is that generally the same set of resonant modes are excited independent of how the currents are spatially distributed. The model also predicts that the widths of all resonances are substantially decreased when shielding currents are present. Numerical results obtained using generic profile functions are found to be remarkably consistent with available in-situ experimental data and existing numerical models. The model is therefore well-suited for parametric studies of the steady-state linear vibration response assuming that Z-coil windings are excited exclusively.

Investigation of energy release rate under mode II loading in a rectangular orthotropic plate with a band crack using 3D finite element analysis

A rectangular plate containing a band crack and made of orthotropic elastic material is considered and the energy release rate (ERR) component in mode II at this crack front is investigated. It is assumed that the crack’s edge planes are parallel to the upper and lower face planes of the plate and the edge planes of the crack are loaded with uniformly distributed sliding forces. The corresponding boundary value problem is formulated within the scope of the three-dimensional linear theory of elasticity and the solution to this problem is found by employing 3D finite element method (3D FEM) modeling. Numerical results on the ERR for mode II and on the influence of the mechanical-orthotropic properties of the plate material are presented and discussed. In particular, it is established that the influence of the shear modulus in a plane which is simultaneously perpendicular to the crack’s front and crack’s plane on the ERR is more significant than that caused by the change of another modulus of the orthotropic material.

Analytic solutions for free vibration analysis of laminated beams in three-dimensional statement

In this research we consider free vibrations of laminated beams in terms of three-dimensional linear theory of elasticity. Analytic solutions for natural frequencies of laminated beams are obtained by using an asymptotic splitting method. The results were compared with classical Euler“Bernoulli beam theory and Timoshenko beam theory.

Dispersion Waves in a Submerged Thermo-Piezoelectric Membrane

Abstract Wave propagation in a thermo piezoelectric membrane immersed in an infinite fluid medium is discussed using three-dimensional linear theory of elasticity and thermos piezoelectricity. Three displacement potential functions are introduced to uncouple the equations of motion, heat and electric conduction equations. The frequency equations are obtained for longitudinal and flexural modes at the solid fluid interfacial boundary conditions. The numerical results are analyzed for PZT-4 material and the computed stress, strain, electric displacement and temperature distribution are presented in the form of dispersion curves and its characteristics are studied.

A scaled boundary finite element method for static and dynamic analyses of cylindrical shells

A modeling technique based on the scaled boundary finite element method is developed for both static and dynamic analyses of cylindrical shells. A new scaling strategy is employed to ensure that the shell boundaries and the cross sections at the element inferfaces are accurately represented through the scaling process. The formulation starts directly from the three-dimensional linear elasticity theory for cylindrical shells. The principle of virtual work involving the inertial force is applied to derive the scaled boundary finite element equation. Only the in-plane dimensions of the structure are discretized with finite elements while the solution through the thickness is expressed analytically as a Padé expansion. A variable transformation procedure facilitates the development of the dynamic stiffness matrix, which leads to the static stiffness matrix and mass matrix naturally. A laminate model with arbitrary number of layers can readily be constructed. Numerical examples demonstrate the accuracy, applicability and efficiency of the two-layer model.

Wave propagation in a generalized piezothermoelastic rotating bar of circular cross section

Purpose – The purpose of this paper is to study the wave propagation in a generalized piezothermoelastic rotating bar of circular cross-section using three-dimensional linear theory of elasticity. Design/methodology/approach – A mathematical model is developed to study the wave propagation in a generalized piezothermelastic rotating bar of circular cross-section by using Lord-Shulman (LS) and Green-Lindsay (GL) theory of thermoelasticity. After developing the formal solution of the mathematical model consisting of partial differential equations, the frequency equations have been derived by using the thermally insulated/isothermal and electrically shorted/charge free boundary conditions prevailing at the surface of the circular cross-sectional bar. The roots of the frequency equation are obtained by using the secant method, applicable for complex roots. Findings – In order to include the time requirement for the acceleration of the heat flow and the coupling between the temperature and strain fields, the analytical terms have been derived for the non-classical thermo-elastic theories, LS and GL theory. The computed physical quantities such as thermo-mechanical coupling, electro-mechanical coupling, frequency shift, specific loss and frequency have been presented in the form of dispersion curves. From the graphical patterns of the structure, the effect of thermal relaxation times and the rotational speed as well as the anisotropy of the of the material on the various considered wave characteristics is more significant and dominant in the flexural modes of vibration. The effect of such physical quantities provides the foundation for the construction of temperature sensors, acoustic sensor and rotating gyroscope. Originality/value – In this paper, the influence of thermal relaxation times and rotational speed on the wave number with thermo-mechanical coupling, electro-mechanical coupling, frequency shift, specific loss and frequency has been observed and are presented as dispersion curves. The effect of thermal relaxation time and rotational speed on wave number for the case of generalized piezothermoelastic material of circular cross-section was never reported in the literature. These results are new and original.

On exact rod/beam/shaft-theories and the coupling among them due to arbitrary material anisotropies

We prove the existence of an exact one-dimensional representation of the three-dimensional theory of linear elasticity for a structural member with constant, two-fold symmetric cross-section, without the use of any a priori assumptions. We show that the general problem of three-dimensional elasticity decouples into four independent one-dimensional subproblems for isotropic material: a rod-, a shaft- and two orthogonal beam-problems. A unique decomposition of any three-dimensional load case with respect to the direction and the symmetries of the load is introduced. It allows us to identify each part of the decomposition as a driving force for one of the four one-dimensional subproblems. Furthermore, we show how the coupling behavior of the four subproblems can be derived directly from the sparsity scheme of the stiffness tensor for general anisotropic materials.

Continuum-based free vibration of circular trigonal and isotropic plates

The free vibration behavior of completely unrestrained elastic circular plates with trigonal and isotropic material symmetry is studied with an approach involving approximate continuum solutions to the three-dimensional theory of linear elasticity. Of primary interest are (1) the influence of trigonal material symmetry on the modes of free vibration and (2) the accuracy of thin plate theory relative to the more exact three-dimensional theory. Resonant frequencies are calculated from the weak form of the equations of motion for the plate through the use of approximation functions and the Ritz method formulated in cylindrical coordinates. This approach enables the resulting eigenvalue problem to be split through group-theoretical symmetry analysis. Representative examples are given and quantitative limits are discussed.

Three-dimensional elasticity solution for vibration analysis of composite rectangular parallelepipeds

Abstract A three-dimensional elasticity solution is proposed for determining the free and transient vibrations of composite rectangular parallelepipeds with arbitrary combinations of boundary constraints. The theoretical model is formulated by means of a modified variational principle in conjunction with a multi-segment partitioning procedure based on the three-dimensional linear elasticity theory. The displacement components of each parallelepiped segment are expanded by a triplicate series of orthogonally polynomial functions i.e., the Chebyshev orthogonal polynomials of first and second kind and the Legendre orthogonal polynomials of first kind. To demonstrate the reliability and accuracy of the present methodology, a considerable number of free vibration solutions are given for isotropic and composite rectangular parallelepipeds (including beams, plates, and solids) with different combinations of free, simply-supported, clamped and elastic-supported boundary conditions. The present results are compared with existing experimental and analytical results published in the literature as well as those solutions obtained from finite element analyses. New benchmark solutions are also obtained for composite parallelepipeds. With regard to the transient vibration analyses, composite rectangular parallelepipeds subjected to several time-dependent impulsive loads, including a rectangular pulse, a triangular pulse, a half-sine pulse and an exponential pulse, are investigated. The effects of the fiber orientation, boundary condition as well as the type of impulsive load on the transient responses of composite parallelepipeds are also discussed in detail.

Three dimensional shear buckling of FG plates with various boundary conditions

In this paper, the buckling of functionally graded plates under shear are considered for various boundary conditions. The buckling results obtained by using Ritz method based on the three-dimensional linear elasticity theory and the higher order shear deformation theories. The material properties of the plate vary through the thickness direction according to a simple power law. The comparison results are presented and the variation of the critical shear buckling load and the critical shear buckling modes with the shear buckling characteristics like the material composition, the plate geometry parameters (side-side ratio, side-thickness ratio) are investigated.

Three-Dimensional Free Vibration of Functionally Graded Material Plates on Different Boundary Conditions

Free vibration of rectangular functionally graded material plates on different support conditions is studied based on the three-dimensional linear theory of elasticity. The three displacements of the plates are expanded by a series of Chebyshev polynomials multiplied by appropriate functions to satisfy the essential boundary conditions. The natural frequencies are obtained by the Ritz method. The present approach is validated by comparing the results with those from other theories and methods. The effects of volume fraction index, aspect ratios, and thickness-side ratios are discussed in the parametric study.

Wall thickness and elastic moduli of single-walled carbon nanotubes from frequencies of axial, torsional and inextensional modes of vibration

Abstract Free vibrations of thirty-three armchair, zigzag and chiral single-walled carbon nanotubes (SWCNTs) of aspect ratios (length/diameter) between ∼3 and ∼15 and having free ends have been studied using the MM3 potential. It is found that these tubes exhibit Rayleigh and Love inextensional modes of vibration. The lowest natural frequency of a mode of vibration corresponds to a circumferential wave number greater than one. Recall that a cylindrical shell of small aspect ratio and comprised of a linear elastic and isotropic material also exhibits the Rayleigh and the Love inextensional modes of vibration. In order to quantitatively compare frequencies of a shell with those of a SWCNT, we find geometric and material parameters for the shell in two ways. In the first approach, we require that the lowest Rayleigh and the radial breathing mode frequencies and the lowest frequencies of the axial and the torsional modes of vibration of a SWCNT match with the corresponding ones of the shell having length and mean diameter equal to those of the SWCNT. In the second technique, we account for the transverse inertia effects, and equate frequencies of the lowest Love, axial and torsional modes of vibration of a SWCNT to that of a shell. Each one of these two methods determines Young’s modulus and Poisson’s ratio of the material of the shell and its thickness, and enables us to explore similarities and differences between vibrations of a shell and of a SWCNT. It is found that the two techniques give very close results for the material and the geometric parameters of the shell, and hence of the SWCNT. The SWCNT thickness increases from ∼0.88 A to 1.37 A when the tube radius is increased from ∼3.6 A to 15 A and stays at 1.37 A for further increases in the tube radius. The wall thickness is essentially independent of the tube chirality. We use these results to provide an expression for the wall thickness in terms of the tube radius and the bond length in the initial relaxed configuration of a SWCNT. We also compare higher vibrational modes of shells and hollow cylinders with those of the corresponding SWCNTs. For a shell we use a first-order shear deformable shell theory (FSDST). Frequencies of a shell using the FSDST and of the hollow cylinder using the three-dimensional linear elasticity theory are computed with the finite element method. It is found that for low to moderate circumferential and axial wave numbers frequencies and mode shapes of the shell and of the hollow cylinder agree well with those of the corresponding SWCNT computed with the MM simulations.

Saint-Venant's theory for beams with multi-connected cross-section: Justification and error estimate

The aim of the paper is twofold. First, starting from the threedimensional theory of linear elasticity, we give a simple justification of Saint-Venant theory for beams with multi-connected cross-section by means of Γ-convergence. Second, we estimate the error between the three-dimensional problem and the limit problem.