Stress Continuity Conditions Research Articles

Stability and dynamic analyses of hybrid laminates using refined higher-order zigzag theory

This article presents a comparative study on the stability and dynamic response of hybrid laminated composite plates over monolithically produced laminated composite plates. A Refined Higher-order Zigzag Theory (RHZT) is formulated and implemented using finite element method for the analysis of hybrid laminates. This formulation considers both the interlaminar shear stress continuity and shear-free boundary conditions at surfaces of the plates. The numerical implementation employs C 0 continuous eight-noded isoparametric plate elements considering geometric nonlinearity. The element stiffness matrix is formulated to consider both the linear and nonlinear terms of the strain-displacement equations. Derivation of the consistent mass matrix for a plate element is presented following the approximation of total kinetic energy and utilising Hamilton’s principle. Numerical examples are provided to demonstrate buckling and free vibration analyses (as eigenvalue problems). The results are numerically verified and validated using data from published literature. Finally, parametric studies are presented to demonstrate the effectiveness of fibre hybridisation approach on the buckling and free vibration behaviour of laminated composites.

A novel semi-analytical model for laterally loaded piles

A new model for laterally loaded piles is proposed, consisting of a linear part and a non-linear iterative part. The linear part is based on the Green's function method, which strictly satisfies the displacement and stress continuity conditions at the pile-soil interface and leads to a rigorous solution. The non-linear iterative part assumes that the shear modulus of the soil around the pile decreases with the average strain of the soil layer. Only two soil parameters need to be calibrated, including the initial shear modulus of the soil and the reference shear strain at half the initial modulus. Compared to the finite element model, the advantage of this model is that it converges quickly while maintaining accuracy, which greatly improves computational efficiency. The advantage of this model over the p-y curve model is that it does not have the difficulty of selecting p-y spring parameters, especially for monopiles. The accuracy of this model is verified by comparison with three-dimensional finite element models and nine existing studies, including numerical models, field tests and centrifuge tests. These cases cover various piles with diameters of D∼0.273–10 m and aspect ratios of L/D∼3–40, installed in sand or clay by driving, jacking or vibration.

MICROPOLAR FLUID FLOWS RELATIVE TO A SWARM OF SPHERICAL POROUS SHELLS

This article investigates the creeping axisymmetric flow of an incompressible micropolar fluid past a swarm of porous shells. Employing the Darcy and a transition Brinkman porous layer, the study presents an analytical model that captures the flow behavior by integrating continuity conditions for velocity, normal and tangential stresses, and microrotations at fluid-porous interface regions.Distinct unit cell techniques, including those proposed by Happel, Kuwabara, Kvashnin, and Mehta and Morse, are analyzed to observe the effects of hydraulic resistivity, porous layer thickness, and porosity on the dimensionless drag for a bounded micropolar fluid system. The results, graphically represented in a series of plots, reveal a complex interplay between these parameters, significantly impacting drag forces and providing insight into the hydrodynamics of a swarm of porous particles, akin to that encountered in oral drug delivery systems.The study identifies a general inverse relationship between hydraulic resistivity and drag and highlights the nuanced effects of porous layer thickness and porosity on fluid resistance, with stark contrasts observed among different unit cell models. These findings underscore the importance of the chosen unit cell technique in predicting and optimizing the flow behavior in micropolar fluid systems.

Magnetohydrodynamic Flow of Immiscible Jeffrey Fluids Through a Horizontal Porous Cylinder

This paper investigates the steady and fully developed magnetohydrodynamic (MHD) flow of two nonmiscible Jeffrey fluids in a horizontal cylinder filled with a homogenous porous medium. A constant pressure gradient drives the flow in both regions, and a uniform magnetic field is applied transversely to the flow direction. Instead of the usual no‐slip condition, the linear Navier slip is used as a boundary condition on the cylinder’s surface, whereas the usual continuity conditions of velocity and shear stress are presumed at the fluid–fluid interface. The equations describing the problem under consideration are mathematically formulated and transformed into nondimensional forms with the proper choice of nondimensional variables. Analytical solutions have been obtained by solving the transformed equations of motion. The effects of different nondimensional parameters involved in the flow problem, including the magnetic number, Jeffrey parameter, Darcy number, ratio of viscosities, Reynolds number, slip parameter, and pressure gradient, on the velocity in each region are studied, and the results are exhibited graphically. In addition, numerical values for stress at the wall and volume flow rates are calculated for various fluid parameters and shown in tables. It is noticed that an increase in the values of the magnetic number and viscosity ratio reduces the velocity of the fluid, whereas increasing the Jeffrey parameters, Darcy number, Reynolds number, slip parameter, and pressure gradient enhances fluid velocity. Furthermore, the most favorable agreement was observed between the results of this study and the results of the previous studies.

Prediction of icing interface normal stress response for thin-walled parts machining with ice-based fixturing (IBF)

Fixtures can maintain stability in the machining of thin-walled parts (TWP), thus avoiding the occurrence of workpiece deformation. The stress at the fixture-workpiece contact interface is a key factor in evaluating the reliability of the fixture’s restraint on the TWP. This paper investigates the prediction of icing interface normal stress during TWP milling based on the ice-based fixturing (IBF) previously proposed by the authors of this paper. The principle of the IBF is first explained. The mechanical properties of the ice-workpiece bonding material (IWBM) are then characterized by Dundurs parameters. Afterwards, an analytical model for the prediction of icing interface normal stress under cutting load is developed using the Goursat formula based on the interface continuity conditions of stress and displacement. The finite element method (FEM) is then used to verify the accuracy of the prediction model, and the results show that the average prediction error of the icing interface normal stress is limited to 13.99%. Finally, milling experiments are carried out to verify the validity of the finite element model, and the experimental results show that the simulation errors of the main cutting force and feeding force are limited to 12.35% and 11.47% respectively, and the FEM can greatly simulate the cutting state of the workpiece clamped by IBF. This paper can provide a valuable theoretical and practical basis for the analysis and prediction of stress at the workpiece-fixture contact interface.

Natural Frequencies of Soft-Core Composite Sandwich Plates Reinforced by FG Graphene Platelets

Superior mechanical properties such as ultrahigh stiffness and strength along with amazing physical properties have led to the use of nano graphene platelets (GPLs) as a reinforcement in polymer composite structures such as sandwich plates. In this research, free vibration analysis of rectangular sandwich plates with functionally graded (FG) GPL-reinforced face sheets and a soft core is investigated. A high-order three-layer accurate sandwich plate theory is employed to model the soft-core sandwich structure. Third order shear deformable plate theory is applied for functionally graded graphene nano-composite face sheets while second and third order polynomials are used for displacements of the soft core. In this theory, the transverse flexibility of the core as well as the continuity conditions of transverse shear stresses are assumed. Hamilton’s principle is applied to extract the motion equations and the corresponding analytical solutions are presented. The effects of materials properties and geometrical parameters on the natural frequencies of the sandwich plates are studied. A comparison of the present results in particular cases with those of published studies confirms the accuracy of the present analysis. Results show that the natural frequencies of the sandwich plates are enhanced by reinforcing the face sheets with GPLs.

Development of analytical and FEM solutions for static and dynamic analysis of smart piezoelectric laminated composite plates on elastic foundation

This paper proposes new analytical and finite element solutions for studying the effects of elastic foundations on the uncontrolled and controlled static and vibration responses of smart multi-layered laminated composite plates with integrated piezoelectric layers, acting as actuators and sensors. A non-polynomial higher-order plate theory with zigzag kinematics involving a trigonometric function and a local segmented zigzag function is adopted for the first time for modeling the deformation of a smart piezoelectric laminated composite plate supported on an elastic foundation. This model has only five independent primary variables like that of the first-order shear deformation theory, yet it considers the realistic parabolic behavior of the transverse shear stresses across the thickness of the laminated composites plates, and also maintains the continuity conditions of transverse shear stresses at the interfaces of the laminated plates. A two-parameter foundation model, namely Pasternak’s foundation, is used to model the deformation and shear interactions of the elastic foundation. The governing set of equations is derived by implementing Hamilton’s principle and variational calculus. Two different solution methods, namely, a generalized closed-form analytical solution of Navier-type, and a C0 isoparametric finite element (FE) formulation, are developed for solving the governing set of equations. The solutions in the time domain are obtained with Newmark’s average acceleration method. Comprehensive parametric studies are presented to investigate the influence of elastic foundation parameters, piezoelectric layers, loading, and boundary conditions on the static and dynamic responses of the smart composite plates with piezoelectric layers. The effects of the elastic foundations on the vibration control of the smart composite plates are also presented by coupling the piezoelectric actuator and sensor with a feedback controller. Several benchmark results are presented to show the influence of the various material and geometrical parameters on the controlled and uncontrolled responses of the smart plates, and also the significant effect of the elastic foundations on the static and dynamic responses of the smart structures. The results obtained are in very good agreement with the available literature, and it can be concluded that the proposed analytical solution and FE formulation can be efficiently used to model the static and dynamic electro-elastic behavior of smart laminated plates supported on elastic foundations.

Hybrid method for modelling anti-plane vibrations of layered waveguides with bonded composite joints

The use of mesh-based methods for modeling elongated composite structures with inhomogeneities leads to an increase in computational costs when discretizing the waveguide part of greater linear dimensions, while the semi-analytical numerical methods do not allow one to directly describe the structures with local inhomogeneities of an arbitrary shape. To compensate for the shortcomings of these two classes of numerical methods, we propose a hybrid scheme based on the spectral element method (SEM) and the semi-analytical finite element method (SAFEM) for studying the anti-plane vibrations of a composite structure in the frequency domain. Thus, for a waveguide, this scheme makes it possible to represent the solution via the SAFEM as a sum of modes or guided waves, and the adjacent regions are discretized using the SEM. The displacement and stress continuity conditions are imposed on the common boundary of two domains. To couple the solutions, we introduce an auxiliary function of displacement, which is approximated by applying the same basis functions as those used in SEM and SAFEM (Lagrange interpolation polynomials on the Gauss–Legendre–Lobatto nodal points). The unknown expansion coefficients of this function are determined by the Galerkin and collocation methods. It has been established that both methods provide the same accuracy. The results obtained by the hybrid approach employing the Galerkin and collocation methods are compared with the results calculated in the standard finite element software. It is shown that they are in good agreement. The presented hybrid approach can be straightforwardly extended to the case of in-plane motion, but it requires significant refinement for a three-dimensional case.

Application of UPV-Instrument in Health Monitoring of Indian Rail Section Using AE Technique

In present work, buckling analysis of laminated composite plates is carried out using recently proposed higher-order zigzag theory (HOZT). Third order variation of in-plane displacement field is taken across the thickness of the plate. For predicting the behavior efficiently for thick plates, quadratic variation of transverse displacement field is assumed for core layer and constant for face layers. The present theory satisfies inter-laminar transverse shear stress continuity condition at interface along with zero value at top and bottom surface of the plate. In present study, nine-noded finite element having eleven degrees of freedom per node is used. Present model is free from requirement of any kind of penalty function or C-1 condition.

Buckling Analysis of Laminated Composite Plates under Thermal Conditions

In present work, buckling analysis of laminated composite plates is carried out using recently proposed higher-order zigzag theory (HOZT). Third order variation of in-plane displacement field is taken across the thickness of the plate. For predicting the behavior efficiently for thick plates, quadratic variation of transverse displacement field is assumed for core layer and constant for face layers. The present theory satisfies inter-laminar transverse shear stress continuity condition at interface along with zero value at top and bottom surface of the plate. In present study, nine-noded finite element having eleven degrees of freedom per node is used. Present model is free from requirement of any kind of penalty function or C-1 condition.

Exact artificial boundary condition and 1D time-domain method for seismic response analysis of layered saturated poroelastic half-space

This study proposed a one-dimensional (1D) time-domain finite element method (FEM) for seismic response analysis of a layered saturated soil half-space under obliquely incident plane waves, based on Biot's u-w wave equations. The spatially two-dimensional problem was transformed into an equivalent 1D time-domain problem along the depth according to Snell's law. An exact artificial boundary condition for saturated poroelastic media was developed to simulate the wave absorption and input effects of the truncated half-space. Using the displacement and stress continuity conditions between the adjacent saturated poroelastic-medium layers and combining all finite elements and the exact artificial boundary condition, the total dynamic equation for the spatially 1D problem was obtained. The 1D dynamic equation was solved using the FEM with explicit time integration. The proposed method was verified by analyzing two engineering sites in the time domain and a comparison with the frequency-domain transfer matrix method with fast Fourier transform.

A new zigzag shell model in terms of Reissner’s mixed variational theorem for composite laminates with curvature

In this study, a new mixed-field zigzag shell model is proposed in terms of Reissner’s Mixed Variational Theorem (RMVT) for the composite laminates with curvature. In this model, refined zigzag theory (RZT) is employed to improve the accuracy and efficiency by introducing through-the-thickness piecewise linear warping functions. Moreover, the shear correction factors are eliminated. It is also different from the published models for composite laminated shells, the continuity conditions of interlaminar transverse shear stresses (TSSs) are satisfied a priori based on RMVT. The equilibrium equations and related boundary conditions are also derived from RMVT. Subsequently, the static performance, natural frequencies, and critical loads of composite laminates with curvature are verified by comparing them with the 3D elastic solutions and other existing ones. It is observed that the present model accounts for not only the efficiency of the low-order models but also the accuracy of the high-order models. Finally, it is shown that the ply stacking, side-to-thickness ratios, and side-to-radius ratios play an important role in percentage errors between the present model and the previous ones.

Wellbore Integrity After a Blowout: Stress Evolution Within the Casing-Cement Sheath-Rock Formation System

A robust evaluation of the casing-cement sheath-rock formation system is foundational in ensuring wellbore integrity throughout a well's lifecycle. Analytical theory for a multi-layer, thick-walled cylinder is used to evaluate the aggregate stress distributions within the casing-cement sheath-rock formation system, honoring boundary conditions of radial stress and displacement continuity. These stress distributions are adjusted for scenarios that a well may experience in its lifetime. Each layer of the casing-cement sheath-rock formation system is evaluated separately against various possible mechanical failure mechanisms, all of which can compromise wellbore integrity.A blowout scenario after a mismanaged loss-of-well-control situation induces high stress loads in the casing-cement sheath-rock formation system, with the wellbore pressure rapidly decreasing during post-blowout discharge, followed by a rapid increase following successful well capping. Casing and cement sheath failures can expose the surrounding rock formation to the pressurized fluid inside the wellbore, risking hydrocarbons broaching the surface, or the seafloor. The aggregate stress distribution model is applied on a case study performed using parameters from the MC 252–1 “Macondo Well” blowout from April 20, 2010 in the deepwater Gulf of Mexico. The stress evolution suggests stability against mechanical failure mechanisms within the casing (collapse/burst and tensile/compressive), cement-sheath (inner or outer debonding, and shear cracking), and rock-formation layers (longitudinal or transverse tensile fracture initiation, along with shear-slippage along pre-existing faults in the near-well vicinity), throughout the blowout aftermath. Nevertheless, tendencies towards radial cracking and disking (tensile) failures were indicated for the cement-sheath layer as the system reaches radial stress and displacement continuity, after cement setting.

Solution of the Thermoelastic Problem for a Two-Dimensional Curved Beam with Bimodular Effects

In classical thermoelasticity, the bimodular effect of materials is rarely considered. However, all materials will present, in essence, different properties in tension and compression, more or less. The bimodular effect is generally ignored only for simple analysis. In this study, we theoretically analyze a two-dimensional curved beam with a bimodular effect and under mechanical and thermal loads. We first establish a simplified model on a subarea in tension and compression. On the basis of this model, we adopt the Duhamel similarity theorem to change the initial thermoelastic problem as an elasticity problem without the thermal effect. The superposition of the special solution and supplement solution of the Lamé displacement equation enables us to satisfy the boundary conditions and stress continuity conditions of the bimodular curved beam, thus obtaining a two-dimensional thermoelastic solution. The results indicate that the solution obtained can reduce to bimodular curved beam problems without thermal loads and to the classical Golovin solution. In addition, the bimodular effect on thermal stresses is discussed under linear and non-linear temperature rise modes. Specially, when the compressive modulus is far greater than the tensile modulus, a large compressive stress will occur at the inner edge of the curved beam, which should be paid with more attention in the design of the curved beams in a thermal environment.

Parallel Flow of a Pressure-Dependent Viscosity Fluid through Composite Porous Layers

Flow of a fluid with pressure-dependent viscosity through a composite of two porous layers is considered in this work in an attempt to validate velocity and shear stress continuity conditions at the interface, and are popular in the study of flow over porous layers and through composite layers when viscosity of the fluid is constant. For the current problem, conditions at the interface between the porous layers reflect continuity assumptions of velocity and shear stress, with additional continuity assumptions on pressure and viscosity. Viscosity is assumed to vary continuously and exponentially across the layers as a function of pressure. Analytical solutions are obtained to illustrate the effects of flow and media parameters (Darcy numbers, layer thicknesses, angle of inclination, and viscosity adjustment parameter) on the dynamic behaviour of pressure-dependent viscosity fluids in porous structures. All computations, simulations and graphs in this work have been carried out and obtained using Maple 2020 software package.

Features of selecting boundary conditions when describing flows of stratified fluids

The article discusses the selection of boundary conditions at the boundary between adjacent layers in a stratified viscous incompressible fluid. It is shown that the "continuity condition + differentiability condition" pair traditionally used in many disciplines gives physically unjustified properties of the resulting exact solution for the velocity field. Although the condition for the differentiability of velocities is close in mathematical form to the stress continuity condition (by virtue of Newton’s law), in terms of physics, taking these conditions into account gives fundamentally different properties of the exact solution of the Navier-Stokes system of equations. It is shown that the consideration of the "the velocity field continuity condition + the stress field continuity condition" pair is more adequate to the physics of the process, which is consistent with the hypothesis of continuity.

A semi-analytical method for elastic-plastic analysis of a deep-buried elliptical tunnel

The elastic–plastic problem of an elliptical tunnel is of great significance in engineering. Previous studies were mainly focused on the circular tunnel. An elastic–plastic analysis is performed for determining the plastic zone around a deep-buried elliptical tunnel subjected to far-field uniform in-situ stresses. The yielding of the rock mass obeys the Mohr-coulomb criterion. The elastic stress field is given by the complex variable method and the plastic stress field is numerically determined by the slip line theory. The solution is obtained according to the stress continuity condition along the elastic–plastic interface. Correspondingly, a set of nonlinear equations involves the mapping function coefficients is established and solved with the help of differential-evolution algorithm. The validity of the method is verified by comparing the results with the numerical simulation of FLAC3D software. Estimates of the size and shape of the plastic zone are then presented for varying parameters including in-situ stress, cohesion, internal friction angle and axis ratio.

An enhanced higher order zigzag theory for laminated composite plates under mechanical/thermal loading

In this paper, an enhanced higher order zigzag theory is developed for laminated composite plates. By satisfying inter-laminar stress continuity conditions and free surfaces conditions, the final displacement field of the present theory can be obtained from Cho’s the initial displacement field [27]. The displacement variables in the present theory are independent on the number of layer. In contrast to Cho’s higher order zigzag theory [27], only one transverse displacement variable is introduced into in-plane displacement field in the proposed theory. To assess the accuracy of this model, simply-supported laminated composite plates subjected to mechanical or thermal loading are analyzed respectively by using the Navier method. Numerical results show that the present model can accurately predict behaviors of laminated composite plates and are more suitable for multilayered plates in comparison with Cho’s higher order zigzag theory.

Analytical solution for determining the plastic zones around two unequal circular tunnels

An analytical solution is proposed for the plastic zones developed around two circular tunnels with unequal size excavated in intact rock which is governed by the Mohr-Coulomb yield criterion in failure. The tunnels are deep buried and subjected to remote uniform in-situ stresses. The elastic–plastic interfaces on the physical plane are mapped onto an annulus on the image plane by using the conformal transformation. The solution is obtained by the stress continuity condition along the interfaces. Correspondingly, the stress field in elastic zone is given by the complex variable method and the stress field in plastic zone is statically determined. Then a set of nonlinear equations involving the mapping function coefficients is constructed, which reduces the problem of finding the elastic–plastic interfaces to be the problem of solving the mapping function coefficients. The differential-evolution algorithm with the coefficients as design variables is used to solve the nonlinear equations. Numerical solution is made to verify the presented solution. Parametric investigation is performed for varying relative size of tunnels and in-situ stresses.

Bending and free vibration analysis of symmetric and unsymmetric functionally graded CNT reinforced sandwich beams containing softcore

In the present work, bending and free vibration analyses of functionally graded carbon nanotube-reinforced (FG-CNTR) sandwich beams are carried out using finite element-based higher-order zigzag theory. Face sheets are assumed to be made up of FG-CNTR composite, and the core is assumed to be made up of balsa wood (softcore). The present formulation also takes into account transverse normal stresses. The computational model incorporates transverse shear stress and transverse normal stress continuity condition at interfaces. Zero transverse shear stress condition at the bottom and top surfaces of the beam is also satisfied. The principle of minimum potential energy is employed for carrying out bending analysis, while Hamilton’s principle is adopted for free vibration analysis. The investigation is carried out for different gradation laws which govern the distribution of CNTs across the thickness of face sheets. The influence of the core’s thickness on stresses and displacements is also critically analyzed in the present work. It has been observed that the thickness of the core and CNT gradation law significantly affect the mechanical behavior of the sandwich FG-CNTRC beam.