Closed-form Elasticity Solution Research Articles

CLOSED-FORM SOLUTION OF A METAL PLATE WITH A CENTRAL CRACK UNDER TENSION

Accurate prediction of stress and displacement in plates with central cracks is vital for engineering design safety. However, the impact of finite plate size on stress and displacement distribution is an important factor that has not been fully addressed in previous studies, highlighting the need for further investigation. Thus, this study presents a closed-form elastic solution for a rectangular plate with a central crack under tension load. The solution uses complex polynomial functions satisfying equilibrium equations and boundary conditions. Numerical finite element models verify the analytical solution's accuracy for plates with varying crack lengths. The presented closed-form solution accounts for the impact of finite plate size on stress and displacement distribution accurately. Developing closed-form solutions for structural mechanics problems enables engineers to optimize designs for safety, efficiency, and cost-effectiveness more accurately.

A Closed-Form Elastic Solution of Ground Response Curve for Noncircular Openings

Abstract Ground response curves (GRCs) can be used to characterize the evolution law of stress and displacement during tunnel excavation. In this study, an attempt is made to obtain the GRC and fin...

Interlaminar stress fields in circular cylindrical cross-ply laminated shells subjected to transverse loadings

This paper aims at contributing to the field of efficient analysis methods by introducing a semi-analytical approach that enables the investigation of the three-dimensional stress fields in cylindrically curved cross-ply laminated shells subjected to different transverse loadings under consideration of the free-edge effect. The novel analysis method incorporates a closed-form elasticity solution that is superimposed with a layerwise approach, in which the three-dimensional displacement field is discretized with respect to the thickness direction. Lagrangian shape functions of various order are used to interpolate between the a priori unknown displacement functions in the discrete grid points of the numerical layers. Employing the principle of minimum total potential energy, yields a system of coupled ordinary differential equations which is solved by utilizing the state-space approach. The accuracy of the developed method is verified by comparison with existing analytical approaches in the literature and detailed three-dimensional finite element computations. In addition to a study of the out-of-plane stresses in a circular cylindrical fibre metal laminate (GLARE®), the effect of different radius to thickness ratios and length to thickness ratios on the distribution of the interlaminar stress field is investigated.

Closed-form elasticity solution for smart curved sandwich panels with soft core

Due to the rapid development of intelligent engineering structure, the demand for high performance smart structures has been increased in recent years. In this paper, an exact elasticity solution for bending analysis of sandwich panel with generally orthotropic facings and core is developed. Two electro-orthotropic piezoelectric layers in the top and bottom surfaces of sandwich panel as sensor and actuator are considered. From practical point of view, an initially curved sectorial geometry is considered for smart sandwich panel. In this regard, the constitutive relations and governing equations are considered in polar coordinate and coupled Euler–Cauchy Equations are derived. The characteristic equations are determined and closed-form basis functions of displacements and stresses are achieved for various material and geometrical conditions. Furthermore, based on classical and first-order shell theories, the governing equations of smart curved sandwich panels are derived. The governing equations are solved analytically and compared with exact elasticity solution. Several parametric studies are performed on both material and geometrical properties such as angular span, facing and core thickness, and external electrical voltage.

Closed-form elastic solution for irregular frozen wall of inclined shaft considering the interaction with ground

An analytical solution is derived that provides a closed-form formulation for stresses and displacements of a deep inclined shaft with irregular frozen wall liner in an infinite elastic medium, subjected to a non-uniform stresses. This solution is based on: (i) the conformal mapping of a doubly connected domain of prescribed shape onto a circular ring by an appropriate numerical optimization scheme which is achieved by MATLAB program, (j) the complex variable method. For a particular case, the distributions of hoop stress and radial displacement inside frozen wall are presented with analytical solution. The accuracy of stress functions is verified by comparison of different number of coefficients in stress functions and it show that the number of coefficients k≥15 is advised. The validity of analytical solution is verified by a comparison between its result and that obtained from the finite element program ANSYS. Noting that the shaft excavation can be regarded as relaxation of initial ground stresses around shaft excavation boundary, this paper presents the mechanical problem as two separate problems: (i) initial stresses and displacements model where stresses keep constant and displacements equal zero around frozen wall and ground, (j) stresses relaxation model with relaxed normal and shear stresses acted on excavated boundary. The solution is worked out by the principle of superposition in elastic medium. The Young's modulus ratio (β) of the frozen wall to the ground reflects the influence of ground to frozen wall, and the influence decreases with the growth of β. The proposed solution can provide analytical basis for frozen wall design and be used as a quick-solver for back-analysis of in situ stresses and displacements.

Nonlinear dynamic response of sandwich wide panels

This work focuses on the general nonlinear dynamic formulation of sandwich panels based on the Extended High-order Sandwich Panel Theory. The faces and the core are both considered undergoing large deformation with moderate rotation. In the literature, when it comes to nonlinear dynamic response, various simplifying assumptions are adopted in the nonlinear kinematic relations. A critical assessment of these simplifications, as well as a comprehensive investigation of the geometric nonlinearity effects on the dynamic response of sandwich beams/wide panels is presented. The Extended High-order Sandwich Panel Theory (EHSAPT)-based finite element is used to formulate the equations of motion, and the time response is obtained by the central difference method. The transient response of a sandwich panel subjected to a distributed blast loading is studied as a numerical example. By neglecting the nonlinear terms, the nonlinear dynamic analysis is reduced to the linear dynamic small deformation analysis, for which a closed-form elasticity solution exists. This, the linear response is first compared with the closed-form linear dynamic elasticity solution. It is shown that the linear EHSAPT yields an identical response to the dynamic elasticity analysis. Subsequently, the dynamic response is evaluated by considering the geometric nonlinearities. Several simplified nonlinear models with partial nonlinear terms included are considered. These models reflect the assumptions commonly adopted in the literature. Numerical results are presented to investigate the effects of geometric nonlinearities on the displacements and stresses. It is shown that the geometric nonlinearities have a significant effect on the dynamic response of sandwich panels and, in particular, the nonlinear terms in the core kinematic equations are especially important.

Elasticity Solution for Curved Sandwich Beams/Panels and Comparison with Structural Theories

The linear elasticity problem formulation and solution for a generally asymmetric sandwich curved beam/panel consisting of orthotropic core and face sheets, which is subjected to a top face sinusoidal distributed transverse loading, is presented. The displacement approach is used along with a plane strain assumption, and the panel is assumed to be simply supported at the ends. Closed-form solutions for the displacements and stresses are derived, and results for realistic material systems are presented for the transverse and hoop displacements as well as the core transverse normal and shear stresses and the face hoop stresses. A comparison of the results between the elasticity, the first-order shear deformation theory with an equivalent shear modulus, and the classical sandwich theories is presented. In addition, a comparison is presented with results from the compressive high-order sandwich panel theory. This closed-form elasticity solution can be used as a benchmark in assessing the accuracy of the various sandwich panel theories or numerical approaches.

Extended High-Order Theory for Curved Sandwich Panels and Comparison With Elasticity

A new one-dimensional high-order sandwich panel theory for curved panels is presented and compared with the theory of elasticity. The theory accounts for the sandwich core compressibility in the radial direction as well as the core circumferential rigidity. Two distinct core displacement fields are proposed and investigated. One is a logarithmic (it includes terms that are linear, inverse, and logarithmic functions of the radial coordinate). The other is a polynomial (it consists of second and third-order polynomials of the radial coordinate), and it is an extension of the corresponding field for the flat panel. In both formulations, the two thin curved face sheets are assumed to be perfectly bonded to the core and follow the classical Euler–Bernoulli beam assumptions. The relative merits of these two approaches are assessed by comparing the results to an elasticity solution. The case examined is a simply supported curved sandwich panel subjected to a distributed transverse load, for which a closed-form elasticity solution can be formulated. It is shown that the logarithmic formulation is more accurate than the polynomial especially for the stiffer cores and for curved panels of smaller radius.

Closed-form elasticity solution for three-dimensional deformation of functionally graded micro/nano plates on elastic foundation

This paper addresses the static deformation of simply supported rectangular micro/nano plates made of functionally graded (FG) materials based on the three-dimensional nonlocal elasticity theory of Eringen. The plates are assumed to be simply supported and rested on a Winkler-Pasternak elastic foundation. Elasticity modulus is assumed to obey an exponential law along the thickness direction of the micro/nano plate. Using the Fourier series, a displacement field is defined that satisfies simply supported boundary condition and reduces three elasticity equations to two independent equations. The closed-form bending response is achieved by exerting boundary conditions of the lateral surfaces. Numerical results are presented to investigate the influences of the gradient index of the material properties, nonlocal parameter and stiffness of elastic foundation on the mechanical behavior of the plates.

Analytical solutions for multilayered composite cylinders with harmonic quadratic eigenstrain in arbitrary layers

The eigenstrain problem of multilayered hollow and solid composite cylinders working in a constant magnetic field is investigated analytically in this paper. Each layer of the composite cylinder can undergo a harmonic and spatially varying eigenstrain. The eigenstrain is assumed to be a quadratic polynomial function of the radial coordinate. The closed-form elastic solutions are obtained by solving the inhomogeneous governing Bessel differential equations. Then, the effects of eigenstrain distribution, angular frequency and the intensity of the magnetic field on the radial displacement, radial stress, hoop stress, and axial stress are presented graphically.

Free vibration of unidirectional sandwich panels, Part I: Compressible core

The free vibration response of a unidirectional sandwich panel with a compressible and incompressible core using the various computational models is presented and compared with closed-form elasticity solutions and finite element results. The mathematical formulations for various models along with the numerical investigation are presented in two parts. In the first part compressible core are considered using the elasticity closed-form solution and various high-order computational models such as the high-order sandwich panel theory (HSAPT) and the extended HSAPT model (EHSAPT). The second part is dedicated to incompressible cores and includes classical models, first-order and high-order shear deformable models, and zig-zag displacement pattern model, ordinary sandwich panel theory. The elasticity-based model serving as the benchmark solution (in first part) assumes isotropic, orthotropic, as well as layered core types. The mathematical formulation utilizes Hamilton’s principle to derive the general equations of motion. A closed-form solution of the elasticity model is available only for a simply-supported panel and it is compared with all various models numerically. The numerical investigation includes: eigenfrequencies, displacement modes along the length and through the depth of panel, as well as stress modes through the depth of panel. The results of the various models are compared with the 2D elasticity solution and finite element results of ADINA. In general, the lower mode correlates well for all models while for the higher modes only the EHSAPT and the HSAPT with displacement formulation compared well.

Study on the effect of functionally graded coating layers on elastic deformation of thick circular plates: A closed-form elasticity solution

In this paper, a closed-form solution is presented for displacements and stresses in a thick homogeneous circular plate coated by functionally grade (FG) layers subjected to transverse loading. The solution procedure is in the framework of infinitesimal theory of elasticity. The elasticity modulus of the FG coating layers is assumed to vary exponentially through the thickness, whereas Poisson’s ratio remains constant. The solution procedure is on the basis of a Plevako’s representation, which reformulates the elasticity equations into some uncoupled fourth-order partial differential equations with respect to some potential functions. We explicitly obtain the analytical solution by writing the potential functions as Fourier–Bessel series expansions with respect to the radial coordinate. The solution is validated by comparing the numerical results with their counterparts reported in literature for the conventional system of thick plates coated by homogeneous layers, as well as the 3-D finite element analysis. A comparative study is presented between a thick circular plate coated by functionally graded layers and the one with homogeneous coating layers.

1405 静水圧力を受けるカーボンナノチューブの力学特性(OS-14B ナノ組織,OS-14 電子・原子・マルチスケールシミュレーションに基づく材料特性評価)

A frame structural model is presented to study the elastic properties of carbon nanotubes (CNTs) under hydrostatic pressure. Simple formulae are given for the bulk modulus of CNTs under hydrostatic pressure as a function of chiral structures. The double-walled nanotubes are coupled together through the van der Waals force between the inner and outer nanotubes. According to this model, the effects of the CNT structures with the armchair and zigzag on the pressure behavior are investigated based on closed-form elastic solution. The computational result indicates that the bulk modulus is less sensitive to the nanotube structures except for very small tube diameter.

Effects of carbon nanotube structures on mechanical properties

This paper describes a structural mechanics approach to modelling the mechanical properties of carbon nanotubes (CNTs). Based on a model of truss structures linked by inter-atomic potentials, a closed-form elastic solution is obtained to predict the mechanical properties of single-walled carbon nanotubes (SWNTs). Moreover, the elastic modulus of multi-walled carbon nanotubes (MWNTs) is also predicted for a group of the above mentioned SWNTs with uniform interval spacing. Following the structural mechanics approach, the elastic modulus, Poisson’s ratio, and the deformation behaviors of SWNTs were investigated as a function of the nanotube size and structure. Poisson’s ratio of SWNTs shows a chirality dependence, while the elastic modulus is insensitive to the chirality. The disposition of the strain energy of bonds shows quite a difference between the zigzag and armchair tubes subjected to axial loading. A zigzag tube is predicted to have a lower elongation property than an armchair tube.

Caustic study on stress singularities in laminated composites under concentrated loads

The local stress singularities of laminated composite materials under concentrated loads are studied by the optical method of caustics. The parametric equation of the caustics and its initial curve surrounding the stress concentration zone are derived from the theoretical closed-form elastic solutions and the principle of reflective caustics. Theoretical formulae of determining the relation between the local concentrated load and the geometric characteristic size of caustic curve are obtained. Theoretical caustics and initial curves for laminate orthotropic composites are simulated and analysed. To compare with the numerical results, a dynamic caustic experiment by means of multi-spark high-speed camera is performed to investigate the dynamic evolution of local stress singularities in laminated carbon fibre/epoxy composites subjected to impact-concentrated loading.

Pipe-anchor discontinuity analysis utilizing power series solutions, Bessel functions, and Fourier series

One of the paradigmatic classes of problems that frequently arise in piping stress analysis discipline is the effect of local stresses created by supports and restraints attachments. Over the past 20 years, concerns have been identified by both regulatory agencies in the nuclear power industry and others in the process and chemicals industries concerning the effect of various stiff clamping arrangements on the expected life of the pipe and its various piping components. In many of the commonly utilized geometries and arrangements of pipe clamps, the elasticity problem becomes the axisymmetric stress and deformation determination in a hollow cylinder (pipe) subjected to the appropriate boundary conditions and respective loads per se. One of the geometries that serve as a pipe anchor is comprised of two pipe clamps that are bolted tightly to the pipe and affixed to a modified shoe-type arrangement. The shoe is employed for the purpose of providing an immovable base that can be easily attached either by bolting or welding to a structural steel pipe rack. Over the past 50 years, the computational tools available to the piping analyst have changed dramatically and thereby have caused the implementation of solutions to the basic problems of elasticity to change likewise. The need to obtain closed form elasticity solutions, however, has always been a driving force in engineering. The employment of symbolic calculus that is currently available through numerous software packages makes closed form solutions very economical. This paper briefly traces the solutions over the past 50 years to a variety of axisymmetric stress problems involving hollow circular cylinders employing a Fourier series representation. In the present example, a properly chosen Fourier series represent the mathematical simulation of the imposed axial displacements on the outside diametrical surface. A general solution technique is introduced for the axisymmetric discontinuity stresses resulting from an anchor restraint on a selected of pipe geometry. These solutions can be economically implemented on today's symbolic calculus software packages with no loss in solution accuracy when compared to often more expensive techniques such as the finite element method. Verification of the axisymmetric solution technique is illustrated by the comparison of results for the closed form solutions versus those approximated by the finite element technique. Extensions of the general axisymmetric solution technique to other geometries and applied loads are also discussed while the numerical and graphical results are tendered.

A closed-form elastic solution for stresses and displacements around tunnels

Abstract A closed-form plane strain solution is presented for stresses and displacements around tunnels based on the complex potential functions and conformal mapping representation. The tunnel is assumed to be driven in a homogeneous, isotropic, linear elastic and pre-stressed geomaterial. Further, the tunnel is considered to be deep enough such that the stress distribution before the excavation is homogeneous. Needless to say that tunnels of semi-circular or “D” cross-section, double-arch cross-section, or tunnels with arched roof and parabolic floor, have a great number of applications in soil/rock underground engineering practice. For the specific type of semi-circular tunnel the distribution of stresses and displacements around the tunnel periphery predicted by the analytical model are compared with those of the FLAC 2D numerical model, as well as, with Kirsch's “circular” solution. Finally, a methodology is proposed for the estimation of conformal mapping coefficients for a given cross-sectional shape of the tunnel.

An Experimental and Numerical Investigation of the Free Edge Problem in Composite Laminates

A quasi-three-dimensional finite element analysis of a symmetric angle-ply [+454/-454]slaminate subjected to a uniform axial extension was conducted and the results compared with experimental data generated for this purpose. Singular finite elements were embedded around the interface/free edge and eight-noded quadrilateral elements were used elsewhere. The order of the singularity was obtained from a closed form elasticity solution. The order to the singularity compared well will those in previous published numerical analyses. The analyses predicted the presence of high interlaminar normal and shear strain concentrations along the free edge. The results were compared to experimental data obtained using the moiré interferometric technique. The experimental data consist of displacement patterns and an interlaminar normal strain concentration along the free edge. Favorable agreement between analysis and experiment is obtained only if the singular element is embedded in a small region around the singular point. Supported by a detailed literature review, the agreement between the analysis and the experimental data shows that a quasi-three-dimensional analysis is sufficiently accurate.

Two-dimensional elastic defects in orthotropic bicrystals

A method is presented for obtaining the closed form elastic solutions of two-dimensional inclusion and dislocation problems in orthotropic bicrystals. The bicrystal consists of two semi-infinite single crystals with a planar interface. The interface is a plane of symmetry for both crystals. The two single crystals are either perfectly bonded together or in frictionless contact with each other. Both inplane and antiplane deformations that include plane strain and plain stress conditions are considered. The method is based on Eshelby's method for inclusion problems in infinite isotropic solids. The analytic results for point forces, inclusions and dislocations are given. The elastic fields of the defects are represented by quasi-harmonic potentials that are associated with the defects. Explicit solutions for elliptic inclusions and dislocation lines are given as examples. Results for elliptic inclusions and dislocation lines of different orientations in cubic structures are also presented.

Micromechanics of defects in transversely isotropic elastic solids with interfaces

The response of microstructural defects to internal and external fields is an important subject in materials science and engineering. The knowledge of elastic deformations due to defects is the first step towards understanding the fundamental effects of defects on the mechanical and physical properties of materials. A recently developed method for obtaining the explicit three- dimensional closed form elastic solution in solids with defects in the presence of an interface is reviewed. The interface is a planar boundary between two semi-infinite solids. The solids are either perfectly bonded together or in frictionless contact with each other at the planar interface. The defects are point defects, dislocations, disclinations, inclusions, elastic inhomogeneities, and thermal inhomogeneities. The method is a modified point force method. The elastic solution is expressed in terms of force double Green's functions in a simple form. The force double Green's functions are the partial derivatives of point force solutions. The only task in finding the solution is to find the appropriate surface or volume integral of functions of the distance from the defects to the point of interest. The integrals are the Newtonian potential of the defects with appropriate mass density and other biharmonic and harmonic potentials that are associated with the Newtonian potentials. The same method applies to both isotropic and transversely isotropic solids.