Analytical Solutions Of Displacements Research Articles

Asymptotic analysis of out-of-plane strain and displacement fields at angular corners

This paper presents three-dimensional analytical solutions for displacement and strain fields near angular corners in plates subjected to in-plane loading. The approach is based on the first-order plate theory, which represents an elementary extension of the classical plane theory of linear elasticity. Utilising the Kontorovich–Lebedev transform, the asymptotic behaviour of the out-of-plane displacement field near the apex is investigated for both Mode I and Mode II loadings. The analytical solutions obtained in the present work correctly predict several three-dimensional effects, such as the order of stress singularity and the intensity of the coupled (local) out-of-plane singular mode under remote in-plane shear loading as well as the scale effect associated with the plate thickness. The developed solutions are more general than many previous analytical results obtained for crack and notch geometries and, essentially, generalise the classical William's solution for angular corners. The analytical predictions compare favourably with three-dimensional finite element modelling and experimental results.

Visco-Elastic Analysis on the Effect of Flexible Layer on Mechanical Behavior of Tunnels

Installing flexible layer is one kind of supporting techniques to deal with the large deformation in tunnels excavated in viscoelastic rocks. The role of flexible layer is to absorb rock deformation due to rock rheology. For further understanding the effect of flexible layer on mechanical behavior of tunnels, a three-layered model is established to study the mechanical behavior of tunnel where flexible layer is installed between surrounding rock and primary support. Visco-elastic analytical solutions for displacements and interaction forces in the rock/flexible layer interface and in the flexible layer/primary support interface are provided. Numerical calculation by use of finite element software Abaqus is carried out to verify the effectiveness and reliability of theoretical analysis. It could be found that flexible layer has a good ability to absorb rock deformation. Compared with rigid support structure, pressure and displacement of primary support in tunnels employing flexible layer could achieve a good improvement. This improvement is dramatically affected by the thickness and deformability of reserved flexible layer.

The J-contour integral in peridynamics via displacements

Peridynamics is a nonlocal formulation of solid mechanics capable of unguided modelling of crack initiation, propagation and fracture. Peridynamics is based upon integral equations, thereby avoiding spatial derivatives, which are not defined at discontinuities, such as crack surfaces. Rice’s J-contour integral is a firmly established expression in classic continuum solid mechanics, used as a fracture characterizing parameter for both linear and nonlinear elastic materials. A corresponding nonlocal J-integral has previously been derived for peridynamic modelling, which is based on the calculation of a set of displacement derivatives and force interactions associated with the contour of the integral. In this paper, we present an alternative calculation of the classical linear elastic J-integral for use in peridynamics, by writing Rice’s J-integral as a function entirely of displacement derivatives. The accuracy of the proposed J-integral on displacement formulation is investigated by applying it to the exact analytical displacement solution of an infinite specimen with a central crack and comparing the exact analytical expression of its J-integral K_I^2/E. Further comparison with a well-known peridynamic crack problem shows very good agreement. The suggested method is computationally efficient and further allows testing of the accuracy of a peridynamic model as such.

Spherical cavity in the layer subjected to dynamic load

The problem of wave motion with spherical symmetry is analysed. For this purpose, a spherical cavity surrounded by layer is considered as a mathematical model which can be used for modelling various phenomena in solid mechanics. The additional layer is also spherical and the outer space is described as unbounded medium. All layers are isotropic, homogeneous and linearly elastic, although the presented formulation allows inclusion of weak nonlinearities. Analytical solutions for displacement and stresses (radial and circumferential) are presented along with some discussion of possible model extensions.

Explicit Form of Exact Analytical Solution for Calculating Ground Displacement and Stress Induced by Shallow Tunneling and Its Application

In urban environment, it is often unavoidable for shallow tunnels to be constructed adjacent to existing pile foundations. To obtain the ground displacements and stresses induced by shallow tunneling and existing pile foundation loads, the key procedure involves superimposing the analytical solution for shallow tunneling in green‐field with the analytical solution for existing structure loads. In green‐field, the complex variable method provides exact analytical solutions of ground displacements and stresses caused by shallow tunneling. However, the exact analytical solutions are not directly expressed as explicit functions of the coordinates (x, y) in the physical plane (called implicit form of exact analytical solutions), whereas the displacements and stresses induced by existing structure loads are explicit functions of the coordinates (x, y) in the physical plane, which makes it difficult to superpose the displacements and stresses induced by existing structure loads. In this paper, explicit form of exact analytical solutions of displacements and stresses induced by shallow tunneling in green‐field is obtained by using the inverse conformal transformation and the Cauchy–Riemann equations. Comparison with implicit form of exact analytical solutions shows that the explicit form of exact analytical solutions is intuitional and easily used by engineers, and moreover, the calculation amount is much smaller than that for the implicit form of exact analytical solutions. Then, an application involving superimposing the explicit form of exact analytical solutions with Mindlin’s solution is implemented to analyze the secondary stress field and the related potential plastic zone caused by shallow tunneling adjacent to pile foundations. Moreover, the influences of pile foundation parameters on the ranges and shapes of the potential plastic zones induced by nearby tunneling are analyzed.

Reprogramming Static Deformation Patterns in Mechanical Metamaterials.

This paper discusses an x-braced metamaterial lattice with the unusual property of exhibiting bandgaps in their deformation decay spectrum, and, hence, the capacity for reprogramming deformation patterns. The design of polarizing non-local lattice arising from the scenario of repeated zero eigenvalues of a system transfer matrix is also introduced. We develop a single mode fundamental solution for lattices with multiple degrees of freedom per node in the form of static Raleigh waves. These waves can be blocked at the material boundary when the solution is constructed with the polarization vectors of the bandgap. This single mode solution is used as a basis to build analytical displacement solutions for any applied essential and natural boundary condition. Subsequently, we address the bandgap design, leading to a comprehensive approach for predicting deformation pattern behavior within the interior of an x-braced plane lattice. Overall, we show that the stiffness parameter and unit-cell aspect ratio of the x-braced lattice can be tuned to completely block or filter static boundary deformations, and to reverse the dependence of deformation or strain energy decay parameter on the Raleigh wavenumber, a behavior known as the reverse Saint Venant’s edge effect (RSV). These findings could guide future research in engineering smart materials and structures with interesting functionalities, such as load pattern recognition, strain energy redistribution, and stress alleviation.

Three-dimensional analytical model for exploration of the block cracking phenomenon in asphalt pavements

Block cracking is a common type of distress in asphalt pavements, which can be found in a variety of climatic and geographical locations. The extensive nature of this cracking form often leads to significant maintenance costs and reduces the ride quality and service life of the pavement surface. Although this deterioration mode is covered in many pavement evaluation guides and condition rating systems, the underlying mechanisms of block cracking have not been fully investigated. This is likely because the phenomenon requires a three-dimensional (3D) fracture analysis to be conducted to capture the structural response associated with this complex cracking form. Computational fracture modelling and simulation will ultimately be needed to explore the detailed thermal and material gradients, time dependencies and nonlinearities associated with this 3D cracking phenomenon. However, it is first necessary to develop rigorous closed-form, analytical solutions to serve as benchmarks for the verification of future 3D numerical simulation results. In addition, the reference analytical solutions provide preliminary new insights into the mechanisms of block cracking. This study presents a 3D, analytical elastic model of a two-layer pavement system subjected to constant thermal stresses. Analytical solutions of displacement and stress fields are presented in the equation and graphical form and new insights towards the mechanisms of block cracking are discussed. The use of the model as a tool to verify numerical solutions, such as finite element analyses is also discussed. Illustrative examples a field validation example and a summary of required model extensions are also provided.

Dynamic response of rectangular membrane excited by heavy rainfall

This paper investigates the dynamic response problem of an orthotropic rectangular membrane structure excited by the heavy rainfall. Firstly, a determined model is proposed to express the load of heavy rainfall imposed on the membrane structure. Then, the geometrically nonlinear governing motion equations of the membrane structure are established based on the von Kármán’s theory, and solved by the perturbation method. Consequently, the approximated analytical solution of displacement can be obtained. Secondly, the experimental study of this dynamic response problem is carried out. The theoretical model is validated by the experimental data based on the developed artificial simulated rainfall impact system. Moreover, the effects of adding weight from the accumulated rain on structural characteristics are analyzed. Additionally, the parameter discussions varying with rainfall intensity, material properties, pre-tension, structural size, and inclined angle are made in detail. This work provides a reliable theoretical model and experimental system to study the dynamic response problem of membrane structures excited by heavy rainfall and the guidance for the dynamic design, construction and maintenance of membrane structures in practical engineering as well.

Ground Vibration Generated by an Underground Moving Load

The three-dimensional transmission of ground vibration subjected to a point load moving along a beam is investigated theoretically, where the beam is located inside a layer and parallel to both the ground surface and bottom of the layer. According to the specified boundary conditions along the beam and layer, analytical solutions of displacement are obtained in the wave number-frequency domain employing a Fourier transform, and the space-time domain response is obtained employing a fast inverse Fourier transform. Numerical results show that, for a constant moving load, the ground vibration has a typical low-frequency feature, and vibration in the horizontal direction should be considered when studying the effect of an underground moving load on environmental vibration.

Revisiting the elastic solution for an inner-pressured functionally graded thick-walled tube within a uniform magnetic field

In this paper, the mechanical responses of a thick-walled functionally graded hollow cylinder subjected to uniform magnetic field and inner-pressurized loads are studied. Rather than directly assuming the material constants into some certain function forms as displayed in previous researches, we firstly give the volume fractions of different constituents of the FGM cylinder and then determine the expressions of material constants. By the use of the Voigt method the corresponding analytical solutions of displacements in radical direction, the strain and stress components and the perturbation magnetic field vector are derived by the following. In numerical part, the influences of volume fraction on displacement, stain and stress components and the magnetic perturbation filed vector are investigated; indicated by the results, we can conclude that the Poisson's ratio has a significant effect on the FGM cylinder's mechanical behaviors. Moreover, by some appropriate choices of the material constants it can be found that the obtained results in this paper can reduce to some special cases given in previous literatures.

Analytical estimation of fracture toughness for circumferential cracks in thin hard films on ductile substrates

AbstractIn this study, the fracture toughness of circumferential crack caused by indentation effect of a rigid indenter on a thin and elastic coating deposited on the elastic substrate was calculated. In the coating and substrate, the analytical solution of displacement and stress field was used. The complete adhesion was considered for the coating on the substrate. The location of maximum circumferential stress was investigated using the analytical calculation of the stress and it was found that this place was located at a distance away from the center of the indenter. Then, the stress intensity factor and energy release rate for plane strain state was determined, and consequently, the energy release rate for a channel crack was calculated. Finally, the fracture toughness was calculated with energy release rate curves for plane strain crack and crack channeling. This method was used to calculate the fracture toughness of TiN/TiCrN ceramic multilayer coating which was deposited on the GTD450 substrate using the Cathodic Arc PVD method. To validate the results, the analytically calculated crack radius was compared with the experimental crack radius in the fracture load and the difference between the radiuses was in the acceptable range.

Analytical displacement solutions for statically determinate beams based on a trilinear moment–curvature model

A standard approach is presented to obtain analytical solutions for deflection field of determinate beams subjected to conventional loading patterns. The solutions are based on a trilinear moment–curvature response using a deflection hardening behavior characterized by flexural crack initiation, inelastic response due to crack extension, and full plastic hinge formation. Methodology for full span deflection and rotation distributions are presented for multiple cases that include three‐ and four‐point bending, uniform load, concentrated moment, and cantilever beams. The proposed approach provides analytical expressions for the curvature, rotation, and deflection at any point along the beam, and correlated to stress or strain distribution. The procedure can therefore be integrated into a serviceability‐based design approach. A parametric study of the effects of model parameters on the stages of the response is addressed. Several case studies involving steel fiber reinforced concrete (SFRC), textile reinforced concrete (TRC) and ultra‐high performance concrete (UHPC) are conducted and the simulated load‐deflection responses are verified against the experimental data from several published experiments. Size effect on the serviceability limits of beams with spans ranging from 0.22 to 8.6 m is studied by tracking the full‐range moment–curvature and load‐deflection responses.

An analytical solution for displacements due to reservoir compaction under arbitrary pressure changes

• An analytical solution for displacements due to arbitrary pressure changes in reservoirs is proposed. • The solution is obtained through the three-dimensional integration of the nucleus of strain approach solution. • The solution allows the calculation of displacements inside and outside the reservoir. • The results agree well with the solution obtained via FEM. Production of fluids in reservoirs, such as oil, gas and water, leads to changes in the stress and strain fields, generating compaction in the reservoir and subsidence of the ground surface. Some of the severe consequences of these displacements are rock fracture, induced seismicity, fault reactivation, wellbore failure, and reduction in the storage capacity of aquifer systems. Analytic methods based on the nucleus of strain approach with simplifying assumptions can be useful to estimate compaction and subsidence, and to indicate when more detailed analyses are required. However, these methods may present problems by the evaluation of the displacement response inside and in the vicinity of the reservoir due to the presence of singularities in the solution. In this work, an analytical displacement solution is obtained for reservoirs of arbitrary shape in a linear elastic semi-infinite medium under arbitrary distribution of pressure changes. For this, the reservoir is discretized in parallelepiped cells. A three-dimensional integration of the solution is carried out over each cell for an infinitesimal reservoir based on the nucleus of strain approach. The final displacement field is given by the superposition of the solutions of individual cells. The three-dimensional integration eliminates the singularities and allows the evaluation of displacements throughout the entire model, including the reservoir. The solutions presented can be easily implemented in a computational procedure, maintaining the low effort of similar methods. Finite element models are employed to verify the accuracy of the proposed solution.

Elastic displacement fields of multi-layered transversely isotropic materials under rectangular loads

Based on the precise integration algorithm (PIA) and the technique of dual vector forms, analytical solutions of displacements for the complete elastic field induced by the vertical loadings uniformly distributed over a rectangular area are deduced. The rectangular area is assumed horizontally resting on the surface or embedded in the interior of a linearly elastic, homogeneous transversely isotropic continuum. The planes of transverse isotropy are chosen to be parallel to the horizontal boundary surface. Associated with the approach of dual vector, the classical integral transform method is adopted to convert the partial differential governing equations into the concise first-order ordinary differential matrix equation. As a highly accurate method to solve sets of the first-order ordinary differential equations, the procedure of PIA is introduced to evaluate the key matrix equation. As a result, any desired accuracy of the elastic solutions can be achieved. Additionally, dual vector forms of equations in the transformed domain make the assembly of the global stiffness matrix simple and convenient. Finally, some illustrative examples are analysed to verify the proposed solutions and elucidate the influences of the dimensions of the loading area, the type and degree of material anisotropy and the stratified characters on the load–displacement relationship in the transversely isotropic media.

Visco-elastic imperfect bonding effect on dynamic response of a non-circular lined tunnel subjected to P and SV waves

A visco-elastic interface model is proposed to investigate the dynamic stress around a non-circular lined tunnel subjected to P and SV waves, and the analytical solutions of displacements and stresses are obtained. The wave function expansion method combining with the conformal transformation method is applied to express the wave fields around the tunnel. To obtain the analytical solution, the non-circular tunnel is mapped into an annular region. A visco-elastic interface model with elastic and viscosity coefficients is introduced to analyze the interface effect. In numerical examples, the effects of stiffness parameter and viscosity coefficient of interface on the dynamic stress under different wave frequencies are discussed. Comparison with existing numerical results is given to validate this dynamic model.

A 4-node quadrilateral flat shell element formulated by the shape-free HDF plate and HSF membrane elements

Purpose The purpose of this paper is to propose an efficient low-order quadrilateral flat shell element that possesses all outstanding advantages of novel shape-free plate bending and plane membrane elements proposed recently. Design/methodology/approach By assembling a shape-free quadrilateral hybrid displacement-function (HDF) plate bending element HDF-P4-11β (Cen et al. 2014) and a shape-free quadrilateral hybrid stress-function (HSF) plane membrane element HSF-Q4θ-7β (Cen et al. 2011b) with drilling degrees of freedom, a new 4-node, 24-DOF (6 DOFs per node) quadrilateral flat shell element is successfully constructed. The trial functions for resultant/stress fields within the element are derived from the analytical solutions of displacement and stress functions for plate bending and plane problems, respectively, so that they can a priori satisfy the related governing equations. Furthermore, in order to take the influences of moderately warping geometry into consideration, the rigid link correction strategy (Taylor 1987) is also employed. Findings The element stiffness matrix of a new simple 4-node, 24-DOF quadrilateral flat shell element is obtained. The resulting models, denoted as HDF-SH4, not only possesses all advantages of original HDF plate and HSF plane elements when analyzing plate and plane structures, but also exhibits good performances for analyses of complicated spatial shell structures. Especially, it is quite insensitive to mesh distortions. Originality/value This work presents a new scheme, which can take the advantages of both analytical and discrete methods, to develop low-order mesh-distortion resistant flat shell elements.

Vertical Dynamic Response of a Concrete Filled Steel Tube due to Transient Impact Load: Analytical Solution

This paper presents an analytical solution of vertical dynamic response of a concrete-filled steel tube (CFST) due to transient impact loading. Both the concrete and steel are modeled by linear elastic material. The impact load is simulated by a semisinusoidal impulse. Three-dimensional (3D) wave equations those considering the vertical displacement are established. By combining the initial and boundary conditions, the frequency-domain analytical solution of displacement is deduced by Laplace transformation and separation of variables methods. The time-domain dynamic response is then obtained by numerical inverse Fourier transformation (IFT). Numerical examples are presented to verify the validity of the analytical solution developed in this study. The results indicate that the analytical solution proposed in this study shows good consistence with the existing solutions.

Magneto-thermo-elastic response of a rotating functionally graded cylinder

In this paper, an analytical solution of displacement, strain and stress field for rotating thickwalled cylinder made of functionally graded material subjected to the uniform external magnetic field and thermal field in plane strain state has been studied. Stress, strain and displacement field as a function of radial coordinates considering magneto-thermo-elasticity are derived analytically. According to the Maxwell electro-dynamic equations, Lorentz force in term of displacement is obtained in cylindrical coordinates. Also, symmetric temperature distribution along the thickness of hollow cylinder is obtained by solving Fourier heat transfer equation in cylindrical coordinates. Using equation of equilibrium and thermomechanical constitutive equations associated with Lorentz force, a second-order inhomogeneous differential equation in term of displacement is obtained and will be solved analytically. Except Poisson\'s ratio, other mechanical properties such as elasticity modulus, density, magnetic permeability coefficient, heat conduction coefficient and thermal expansion coefficient are assumed to vary through the thickness according to a power law. In results analysis, non-homogeneity parameter has been chosen arbitrary and inner and outer surface of cylinder are assumed to be rich metal and rich ceramic, respectively. The effect of rotation, thermal, magnetic field and non-homogeneity parameter of functionally graded material which indicates percentages of cylinder\'s constituents are studied on displacement, Von Mises equivalent stress and Von Mises equivalent strain fields.

Mechanical study on initial profile solution for inflatable reflector made of orthotropic material

The woven fabric membrane materials are widely used in space and terrestrial inflatable reflectors. However the material׳s anisotropy makes the design analysis more complex. The deviation from the desired shape, so-called “W-profile error”, influences the precision of the membrane surface significantly. In this study, a model of an axisymmetric paraboloid surface using membrane theory is established for the purpose of facilitating the surface precision optimization. Analytical solutions for displacements of the reflector are derived. An iteration method of initial reflector profile solution is stated and a finite element (FE) software employed in the solution is presented. A case study is illustrated to make a comparison between numerical and theoretical analyses. Finally, the conclusions are drawn that the analytical method and the FE iterative method for initial profile solution are feasible and efficient.

A physically and geometrically nonlinear scaled‐boundary‐based finite element formulation for fracture in elastomers

SUMMARYThis paper is devoted to the formulation of a plane scaled boundary finite element with initially constant thickness for physically and geometrically nonlinear material behavior. Special two‐dimensional element shape functions are derived by using the analytical displacement solution of the standard scaled boundary finite element method, which is originally based on linear material behavior and small strains. These 2D shape functions can be constructed for an arbitrary number of element nodes and allow to capture singularities (e.g., at a plane crack tip) analytically, without extensive mesh refinement. Mapping these proposed 2D shape functions to the 3D case, a formulation that is compatible with standard finite elements is obtained. The resulting physically and geometrically nonlinear scaled boundary finite element formulation is implemented into the framework of the finite element method for bounded plane domains with and without geometrical singularities. The numerical realization is shown in detail. To represent the physically and geometrically nonlinear material and structural behavior of elastomer specimens, the extended tube model and the Yeoh model are used. Numerical studies on the convergence behavior and comparisons with standard Q1P0 finite elements demonstrate the correct implementation and the advantages of the developed scaled boundary finite element. Copyright © 2014 John Wiley & Sons, Ltd.