Elastic Solution Research Articles

Analytical solution of circular tunnel-lining interaction with elastic contact

Closed-form solutions for tunnel lining are useful to estimate the member forces in the first stages of lining design. A key element in these solutions is the interaction between the lining and the ground. Existing solutions have limited this interaction to two limited cases: perfect bonding, in which there is no separation between the lining and the ground, and perfect slip in which there is no friction between them. In this study, a new analytical solution is established for an elastic contact between the lining and the ground in which the interaction forces depend on the relative displacement of the two elements. This model can cover more realistic cases of the lining-ground interactions in tunnels, for instance, the existence of tail void partially filled with grouting materials. It has been shown that the new solution extends well the previous ones, which can be considered as its limit cases, and also corrects certain errors inherent in these solutions. The analytical solution and compared between them and also to numerical results obtained by the Finite Element Analysis code Disroc. The differences between the solutions are estimated in some typical cases and explained based on their different simplification assumptions. Finally, a simple linear transformation of stresses around the tunnel is shown to provide an efficient method to assess the frictional slip possibility for the results obtained from an analytical elastic solution.

Three-dimensional analytical elasticity solutions in isosceles-right triangular composite laminates

In this study, analytical solutions are derived for three-dimensional, transversely isotropic and multilayered isosceles-right triangular plates under simply supported edge conditions. For any homogeneous layer, we construct the general solution in terms of a simple formalism that resembles the Stroh formalism and the dual-variable and position method. For multilayered isosceles-right triangular plates, we derive the analytical solution by the new propagating method in terms of the symmetry theory. Numerical results clearly show the influence of load distribution, stacking sequence, and mechanical and geometric parameters on the elastic fields. The proposed solutions can also serve as benchmarks for other numerical methods such as the finite element method.

Pipe geometry effect on fatigue cracking location of 90° elbows under in-plane bending

In this technical note, the criterion for fatigue cracking location for 90° elbows under in-plane bending is presented. The criterion is found using published elastic solutions and elastic-plastic FE analysis results. It is found that the most important parameter affecting fatigue crack location is the mean radius-to-pipe thickness ratio, rm/t. The circumferential crack at outer intrados is predicted for rm/t <4–5, and the axial crack at inner crown for rm/t >4–5. The exact value of rm/t depends slightly on the bend radius-to-mean pipe radius ratio. This conclusion is found to be appropriate from high cycle fatigue to very low cycle fatigue loading, and to be independent on the material.

Elasticity solution for a hollow cylinder under axial end loads: Application to a blister of a stayed bridge

AbstractStarting from an applied problem related to the modeling of an element of a cable‐stayed bridge, we compute the elasticity solution for a hollow circular cylinder under axial end loads. We prove results of symmetry for the solution and we expand it in proper Fourier series; computing the Fourier coefficients in adapted power series, we provide the explicit solution. We consider an engineering case of study, applying the corresponding approximate formula and giving some estimates on the error committed with respect to the truncation of the series.

Elasticity solutions of functionally graded piezoelectric plates under electric fields in cylindrical bending

Elasticity solutions of functionally graded piezoelectric plates under electric fields in cylindrical bending

The Improved Irwin’s Model for Crack Problems in Power-Law Hardening Materials

In this paper, an improved Irwin’s model for power-law hardening materials is established through the static equivalence between the linear elastic stress solution and the stress redistribution in the plasticity-affected zone (PAZ) under small-scale yielding (SSY) conditions. HRR solution is adopted to describe the stress field within the crack-tip plastic zone (CTPZ). Analytical expressions of the effective stress intensity factor (SIF) and the size of PAZ are derived to illustrate the effect of CTPZ on the fracture behavior for power-law hardening materials. Influences of external load and hardening exponent on fracture behavior are explored. The proposed model can assess the toughening effect of CTPZ accurately using the SIF ratio, which is always smaller than 1. This model can be applied to various materials obeying power-law hardening constitutive equations. It will reduce to the extended Irwin’s model for elastic-perfectly plastic materials in the limiting case.

On the simplified method for evaluating tunnel response due to overlying foundation pit excavation

Foundation pit excavation may cause excessive uplift displacement and structural damage to the underlying shield tunnel. In this paper, the rationality of the excavation-induced inputs in the current force-based and displacement-based methods is analyzed theoretically and verified by comparing the elastic finite element method (FEM) and the elastic continuum solution. Then, the force input is modified by the load amplification factor to obtain a reasonable tunnel response, and the calculation method of the load amplification factor is given. Based on the mechanism of tunnel-soil interaction, the difference between Winkler subgrade moduli under active and passive conditions is analyzed. It is recommended that the active and passive Winkler subgrade moduli should be adopted under the force and displacement inputs, respectively. By comparing the elastic continuum solution and the Winkler solution, the validity of the Winkler subgrade moduli under active and passive conditions is verified. Thus, a force-based Winkler solution with modified force input is proposed. Finally, the proposed method is applied to a case study, and its applicability is verified by comparing the measured results and the FEM analysis with the HS-Small model.

Semi-analytical method for solving stresses in slope under general loading conditions

Abstract Assessing the stress distribution within the slope in geotechnical engineering is critical. Despite the widely available numerical methods, no analytical solutions are available for determining the stress distribution within a slope under general loading conditions. This study presents a method of analytically approximating elastic stresses within a slope of arbitrary inclination subject to general surcharges and supporting forces. The prototype model of this problem is equivalent to a superposition of two sub-models: a half-plane body subjected to an initial earth stress field as well as surcharges on the crest (Model I) and a slope loaded by the release stresses caused by excavation, together with supporting forces on its inclined surface and bottom (Model II). The former stresses can be calculated analytically using Flamant’s solution, and the latter stresses can be further thought of as being composed of two additional components: one in an infinite plane with a half-infinite hole loaded by virtual tractions upon hole’s boundary (Model II1), which can be analytically approximated, and the other in a half-plane subjected to virtual tractions along the ground surface (Model II2), which can be calculated analytically as well. The two sets of virtual tractions that lead to stresses in Model II are calculated using an iterative process. The current approach provides analytical approximations of elastic stress solutions for slopes that are sufficiently close to the exact ones as accurate as much. A case study demonstrates that such solutions are in good agreement with those of the finite-element method’s over the entire region, the stresses within the region up to 10−11 times the slope’s height away from the slope toe can also be accurately determined using the current method. With this method, contour plots of stresses within a slope inclined at various angles are presented, which can be applied directly in practical engineering.

Characterization of elastic modulus at glass/fiber interphase using single fiber composite tensile tests and utilizing DIC and FEM

The bonding performance and formation of the interphase region between fiber and matrix have significant effects on the strength and durability of composites. A new approach, involving combined experimental and numerical analyses, has been developed to avoid the shortcomings and scattering associated with local experimental methods. For this purpose, tensile tests are performed on specimens fabricated from a single glass fiber and epoxy resin. The elongations for specified lengths are measured using the digital image correlation (DIC) technic. The size of the fiber diameter and specimen section are also measured from SEM images. The obtained experimental displacements for micro tensile tests are used in an inverse elastic finite element solution to obtain the interphase elastic modulus. It is shown that, considering the interphase thickness of 1.0 μm that is more realistic, the interphase elastic modulus is in the range of 12 ∼ 19 GPa. However, the performed sensitivity analysis shown that considering interphase thickness ranging from zero to 2.0 μm the interphase elastic modulus varies between 9 and 46 GPa. It is shown that the proposed procedure can be used to obtain the overall mechanical properties of the matrix/fiber interphase in long fiber composites.

Hybrid homogenization theory with surface effects: Application to columnar nanoporous materials

Nanocomposites, including nanoporous ceramics, metals and polymers, are being widely used in various engineering applications. At the nanoscale, surface/interface effects have significant influence on the homogenized properties and local stresses. To account for these nanoscale effects, the well-established Gurtin–Murdoch theory of surface elasticity is typically incorporated into classical elasticity-based micromechanics schemes and numerical approaches based on the finite-element and most recently finite-volume methods. Whereas the classical micromechanics models either neglect the interactions of adjacent fibers/pores or account for the interactions in an average sense, the finite-element and finite-volume based approaches require high discretization to capture the large stress/strain gradients around the interface/surface. In contrast, the recently proposed hybrid homogenization theory (HHT) combines elements of locally-exact elasticity and finite-volume approaches and is both accurate and efficient, enabling rapid generation and analysis of ordered and quasi-random fiber or pore arrays. Herein, a continuous bulk matrix layer of arbitrary size endowed with a two-dimensional energetic surface is introduced between fiber/pore and discretized matrix region that enables exact calculation of stress jump conditions in the Young–Laplace equations of the Gurtin–Murdoch model. The enhanced HHT with interface/surface effects is validated by comparison with the stress concentration factors obtained using a classical elasticity solution and point-wise stress fields predicted by the locally-exact and finite-volume homogenization theories, exhibiting stable solutions even for very thin bulk matrix layers. The enhanced theory is then employed to study the influence of random nanopore microstructures with random pore radii on the complete set of homogenized moduli of porous silicon used in the electronics industry.

Isogeometric Analysis for Free Vibration of Functionally Graded Plates Using a New Quasi-3D Spectral Displacement Formulation

This paper presents a novel quasi-three-dimensional shear deformation theory called the spectral displacement formulation (SDF) for analyzing the free vibration of functionally graded plates. The SDF expresses the unknown displacement field as a unique form of the Chebyshev series in the thickness direction. By increasing the truncation number in the Chebyshev series, the bending analysis results can approach the three-dimensional elasticity solution and satisfy the traction-free boundary conditions without requiring a shear correction factor. The SDF is an extension of the classical plate theory, thereby naturally avoiding the shear-locking phenomenon. These characteristics enable the SDF to apply to plates of arbitrary thickness while maintaining accuracy. The nonuniform rational B-spline-based isogeometric approach is employed to enhance the applicability of this theory to free vibration analysis of functionally graded plates with complex geometries and different boundary conditions. Numerical examples are presented to demonstrate the accuracy and reliability of the proposed method in analyzing the free vibration of functionally graded plates.

Mesh-Free MLS-Based Error-Recovery Technique for Finite Element Incompressible Elastic Computations

The finite element error and adaptive analysis are implemented in finite element procedures to increase the reliability of numerical analyses. In this paper, the mesh-free error-recovery technique based on moving least squares (MLS) interpolation is applied to recover the errors in the stresses and displacements of incompressible elastic finite element solutions and errors are estimated in energy norms. The effects of element types (triangular and quadrilateral elements) and the formation of patches (mesh-free patch, mesh-dependent element-based patch, and mesh-dependent node-based patch) for error recovery in MLS and conventional least-square interpolation-error quantification are also assessed in this study. Numerical examples of incompressible elasticity, including a problem with singularity, are studied to display the effectiveness and applicability of the mesh-free MLS interpolation-error recovery technique. The mixed formulation (displacement and pressure) is adopted for a finite element analysis of the incompressible elastic problem. The rate of convergence, the effectivity of the error estimation, and modified meshes for desired accuracy are used to assess the effectiveness of the error estimators. The error-convergence rates are computed in the original FEM solution, in the post-processed solution using mesh-free MLS-based displacement, stress recovery, mesh-dependent patch-based least-square-based displacement, and stress recovery (ZZ) as (0.9777, 2.2501, 2.0012, 1.6710 and 1.5436), and (0.9736, 2.0869, 1.6931, 1.8806 and 1.4973), respectively, for four-node quadrilateral, and six-node triangular meshes. It is concluded that displacement-based recovery was more effective in the finite element incompressible elastic analysis than stress-based recovery using mesh-free and mesh-dependent patches.

On General Representations of Papkovich–Neuber Solutions in Gradient Elasticity

On General Representations of Papkovich–Neuber Solutions in Gradient Elasticity

The effective crack extension and its implications on the single cantilever beam test for sandwich composites

The response of Single Cantilever Beam sandwich specimens may be analyzed effectively through one-dimensional theories accounting for the actual deformation of the core and its effects on fracture parameters and compliance. The model of a beam on an elastic Winkler foundation describes the detaching facesheet as an Euler–Bernoulli beam resting, in the bonded region, on an elastic foundation which generates oscillating deflections with characteristic length [Formula: see text]. Modified beam theory (MBT) describes the debonded part of the face sheet as a cantilever beam, built-in at the debond tip, with an effective length which is longer than the actual detached length, by a quantity [Formula: see text]. If [Formula: see text] the two models predict the same energy release rate and similar load-point displacements. Effective crack extension and characteristic length depend on the thicknesses and elastic constants of facesheets and core and may be derived through experimental measurements of the specimen compliance at various increments of crack growth, as prescribed by experimental protocols. In the theoretical analyses, [Formula: see text] is typically assumed, either by directly using formulas proposed for other specimens or by calibrating their parameters to case specific finite element results. In this paper, we investigate the applicability of such formulas to a wide selection of sandwich composites for naval and aeronautical applications having thicknesses or materials other than those used for the original fitting; we then highlight their limitations and effects on the design of experimental protocols for material characterization. The applicability of a recent isotropic-elasticity based formula for [Formula: see text] is then verified using extensive data from the literature and novel finite element (FE) results for typical sandwich composites. Finally, a novel elasticity solution for the load-point displacement of a layer on a half plane is derived which supports the test data reduction procedure for [Formula: see text].

Investigation of The Effect of Knitting Row on Vibration Structure for A Wide Chimney Made With Brick Bar

Building systems have unique structural behaviour. In addition to the obvious material properties used in the formation of this behaviour, the geometric shape is also of great importance. In this study, the mechanical effects on the chimney-shaped structure built with bar bricks for product storage were investigated under compression and vibration conditions. Finite element analysis was used as a research method. Solid bar bricks and cement with standard specifications were used as the intermediate filling material for the construction system, which was formed in a wide circular array and short knitting height. Considering the ideal behaviour of the materials, a fully elastic solution was made. As a result of the examination, the effect of the knitting order on the natural frequency was found to be low, as the reason for this, it was seen that the geometric structure exhibited a very tough behaviour against deformation. The stress concentration was observed at the fillings between the bricks laid in the same layer. The dimensionless deformation at natural frequencies was 1.4 times the diameter of the chimney. The knitting order affected stress values and symmetrical distribution formed in the brick. The results were discussed over the findings.

High-Precision Isogeometric Static Bending Analysis of Functionally Graded Plates Using a New Quasi-3D Spectral Displacement Formulation

A new quasi-three-dimensional (3D) shear deformation theory, called the spectral displacement formulation (SDF), is proposed for high-precision static bending analyses of functionally graded plates. The main idea is to expand unknown displacement fields into Chebyshev series of a unique form in the thickness direction; the truncation numbers are set to be adjustable to meet various application requirements. Specifically, 3D elasticity solutions and traction-free boundary conditions can be approached by increasing the number of Chebyshev bases. The SDF is also an extension of the classical plate theory and naturally avoids the shear locking problem, making it versatile for functionally graded material (FGM) plates of arbitrary thicknesses. The C1 continuity requirement for the discretization of the generalized displacements is conveniently fulfilled by the nonuniform rational B-splines (NURBS)-based isogeometric method. Numerical examples demonstrate the excellent performance of the proposed method for the displacement and stress analyses of functionally graded plates. The high precision and versatility of the present method have manifested its great potential applications in strain-based or stress-based reliability analysis, optimization design, fatigue analysis, and fracture analysis of FGM plates, and other related fields.

Steady-State Crack Growth in Nanostructured Quasi-Brittle Materials Governed by Second Gradient Elastodynamics

The elastodynamic stress field near a crack tip propagating at a constant speed in isotropic quasi-brittle material was investigated, taking into account the strain gradient and inertia gradient effects. An asymptotic solution for a steady-state Mode-I crack was developed within the simplified strain gradient elasticity by using a representation of the general solution in terms of Lamé potentials in the moving framework. It was shown that the derived solution predicts the nonsingular stress state and smooth opening profile for the growing cracks that can be related to the presence of the fracture process zone in the micro-/nanostructured quasi-brittle materials. Note that similar asymptotic solutions have been derived previously only for Mode-III cracks (under antiplane shear loading). Thus, the aim of this study is to show the possibility of analytical assessments on the elastodynamic crack tip fields for in-plane loading within gradient theories. By using the derived solution, we also performed analysis of the angular distribution of stresses and tractions for the moderate speed of cracks. It was shown that the usage of the maximum principal stress criterion within second gradient elastodynamics allows us to describe a directional stability of Mode-I crack growth and an increase in the dynamic fracture toughness with the crack propagation speed that were observed in the experiments with quasi-brittle materials. Therefore, the possibility of the effective application of regularized solutions of strain gradient elasticity for the refined analysis of dynamic fracture processes in the quasi-brittle materials with phenomenological assessments on the cohesive zone effects is shown.

Analysis of instantaneous surface settlement of tunnel construction in composite stratum based on semi-analytical method

It is important for shield tunneling safety to predict and control the instantaneous surface settlement. Particularly, the complex geological condition can make accurate prediction more difficult. To adapt to the stratum distribution, a semi-analytical method is established by introducing the finite layer method into tunnel mechanical analysis. Firstly, based on the elastic solution of the finite layer method, the horizontal flexibility coefficient of tunneling and the surface settlement are obtained by applying effective thrust to the excavation face. Secondly, two influencing factors are analyzed, including the penetration of shield tunneling and the Poisson ratio of geotechnical parameter. The relationship formula between the measured value and the theoretical value of the maximum settlement is derived through data fitting. Finally, the settlement prediction method of tunneling is proved to be practical, which provide a new ideal for the mechanical analysis of tunnels in composite strata.

On the time decay for an elastic problem with three porous structures

In this paper, we study the three-dimensional porous elastic problem in the case that three dissipative mechanisms act on the three porosity structures (one in each component). It is important to remark that we consider the case when the material is not centrosymmetric, and therefore, some coupling, not previously considered in the literature concerning the time decay of solutions in porous elasticity, can appear in the system of field equations. The new couplings provided in this situation show a strong relationship between the elastic and the porous components of the material. In this situation, we obtain an existence and uniqueness result for the solutions to the problem using the Lumer-Phillips corollary to the Hille-Yosida theorem. Later, assuming a certain condition determining a “very strong” coupling between the material components, we can use the well-known arguments for dissipative semigroups to prove the exponential stability of the solutions to the problem. It is worth emphasizing that the proposed condition allows bringing the decay of the dissipative porous structure of the problem to the macroscopic elastic structure.

Simplified method for evaluating mechanical interactions between tunnel and soil due to adjacent excavation

Excavations disrupt the original stress field and may have harmful effects on in-service shield tunnels. Therefore, a rational assessment of the responses of existing shield tunnels to construction of excavation is a significant challenge. Current analytical methods for evaluating the mechanical interactions between the excavation and the tunnel generally utilize Winkler’s model, which considers only the plane strain and ignores the spatial effect between the tunnel and the excavation. In this analysis, a new elastic analytical method that uses the Pasternak elastic model is presented to simulate mechanical interactions between the lateral soil and the tunnel when subjected to the unloading pressure of the bottom and surrounding walls of the excavation. The space-effect of excavation and spatial effects of tunnel-soil interaction are both under consideration. The responses of the tunnel to the construction of excavation were assessed by a two-stage analysis method. First, the imposed unloading load at the tunnel location was calculated based on elastic solutions. Second, vertical displacement of the tunnel caused by the corresponding load was evaluated using the finite difference method. In addition to the proposed analytical approach, six analytical approaches were developed for the elastic model. The applicability of the proposed method was validated by comparisons with numerical simulations and field monitoring data. Furthermore, the predictions of the six analytical approaches for the elastic model were compared with those of the proposed method.