Displacement Continuity Research Articles

A coupled SBFETI-BDEs method for solving 2D static and dynamic contact problem with friction

In this study, we propose a coupled approach called the SBFETI-BDEs method for solving the 2D elastic frictional contact problem. The contact system is decomposed into multiple subdomains, similar to the finite-element tearing and interconnecting (FETI) algorithm. Each subdomain can be regarded as a scaled boundary finite element (SBFE), where only the boundaries are discretized. To avoid computing the global contact flexibility matrix using the force method, we consider the generalized inverse of the stiffness matrix for each subdomain as a subcontact-flexibility matrix. We uniformly formulate the contact equations, displacement continuity equations, and the equilibrium equations of force for the floating subdomains, as a B-differentiable equations (BDEs) system, which we solve using the B-differentiable damped Newton-Raphson method that possesses a global convergence property. The proposed method offers increased computational efficiency without compromising accuracy when compared to the original BDEs algorithm without domain decomposition. We present the dynamic scheme of the SBFETI-BDEs method within the framework of the HHT time-integration scheme, where the equilibrium equations of force for the floating subdomains disappear due to their nonsingular equivalent stiffness matrices. Numerical examples are provided to demonstrate the convergence, accuracy, efficiency, and capability in solving engineering problems of the proposed method.

Dynamic Analysis of FGM Plates with Variable Delamination Parameters by the Chebyshev–Ritz Method

Delamination is one of the failure modes of functionally graded materials (FGM), resulting in the critical change of vibration characteristics. The Chebyshev polynomials are commonly used as admissible functions to improve the computational efficiency of numerical algorithms and avoid the occurrence of ill-conditioned problems. This paper extends the Chebyshev–Ritz method to the free vibration analysis of delaminated FGM plates, in which the material variation through the plate thickness follows the exponential-law distribution. A plane crack that is considered to be perpendicular to the thickness direction penetrates through the width direction. Based on the region approach, the analysis of FGM plates with a mid-plane delamination is divided into four sub-regions. The kinetic energy and potential energy of each sub-region are derived by the thin plate theory and von Kármán nonlinear strain–displacement relation. The modal functions of the displacement fields of FGM plates can be constructed in accordance with the displacement continuity conditions of the delamination interface and the boundary conditions of such plates. The effects of asymmetric material distribution, delamination length ratio, Young’s modulus ratio, and boundary support on the vibration behavior of FGM plates are investigated. This semi-analytical study provides a reasonable theoretical basis for the behavior prediction and delamination identification of composite structures.

A monolithic, finite element-based strategy for solving fluid structure interaction problems coupled with electrostatics

In this work, we present a monolithic finite-element-based strategy for problems involving the deformation of a hyperelastic solid, an incompressible fluid and electrostatics. We use a two-field hybrid Lagrangian formulation for the structure, and a velocity-based ALE mixed formulation for the fluid with appropriately chosen interpolations for the various field variables to ensure stability of the resulting numerical procedure. The equations of electrostatics are solved on the reference configuration over both the solid and fluid domains, with voltage and electric displacement continuity imposed at the interface. Keeping in view that the thickness of typical MEMS structures is small, a stress-based hybrid formulation is used to prevent locking. The use of a monolithic strategy provides a robust and stable algorithm, and allows the user to take large time steps. The consistent linearization ensures a quadratic rate of convergence of the non-linear iterations at each time step. Detailed expressions for the tangent stiffness and associated matrices for the three-way coupled problem are provided. The robustness and accuracy of the proposed method are demonstrated by solving several benchmark problems from the literature.

Study of stress wave propagation across a non-persistent joint based on a boundary integral equation method

Studies on stress wave propagation across persistent joints have been conducted extensively. Nevertheless, there exists a consensus that non-persistent joints are widely and densely distributed, which have a profound impact on wave propagation in jointed rock masses. A boundary integral equation method is suggested in this paper to investigate the characteristics of transmitted wave field for the case of stress wave propagation across a single non-persistent joint. The displacement continuity and discontinuity boundaries are combined in the method. The method presented in the current study is applicable to the analysis of wave propagation across non-persistent joints with arbitrary incident angles. Then, taking a single non-persistent joint arranged with only one joint segment as an example, the applicability of the method in dealing with the problems of wave propagation is verified by comparing the results with those from the discrete element method and analytical methods. Subsequently, parametric studies are carried out, including the effects of joint-segment length, rock-bridge length, wave frequency and incident angle on the transmitted wave. The result indicates that the existence of non-persistent joint makes the transmitted displacement field different from that of persistent joint, because the scattered wave is produced during the process of wave propagation. The displacement amplitude may be amplified evidently in some regions and the spatial distribution pattern of the transmission coefficient is closely related to the joint-segment length, rock-bridge length and incident wavelength.

Several cracks in a functionally graded piezoelectric plane: In-plane steady harmonic loading case solved by dislocation method

In this article, the problem of fracture mechanics in an infinite plane made of piezoelectric materials with functionally graded properties containing straight cracks with different orientations and arrangements as well as a curved crack under steady harmonic loading is presented. First, the in-plane dislocation solution in the functionally graded piezoelectric plane (FGPP) under investigation is solved according to the multi-valued conditions of displacement and electric potential, as well as considering the continuity of stress and electric displacement on the dislocation line and using Fourier transform. Then, linear constitutive equations are used and stress and electric displacement fields are presented. In this study, based on a logical assumption and to get closer to simplifying the solution of the equations governing the problem, the gradual and continuous changes of the functional material are considered exponentially and in the x-axis direction. In the following, to analyze the problem of multiple cracks, the dislocation distribution method is used to find singular integral equations. Finally, several graphs are presented to investigate the effect of material properties, electromechanical coupling coefficient, loading conditions and cracks arrangement on the field intensity factors and the results are compared and confirmed by the findings of other researchers.

Multipatch isogeometric analysis for geometrically exact shell elements using B-bar method and Bézier extraction

In the present study, a multipatch isogeometric analysis method for geometrically exact shell problems is proposed. Generalized curvilinear coordinates are used for the first-order shear-deformable geometrically exact shell elements. This general tensor-based shell formulation can be directly linked to a computer-aided design system such as nonuniform rational basis spline (NURBS), which is represented by two parameters. Nitsche’s method for patch-to-patch connections in conforming or nonconforming meshes is adopted. The compatibility conditions between NURBS patches in Nitsche’s coupling are weakly imposed using the continuity of displacements, rotations, and stress resultants. The final algebraic equation for the solution fields has a simple symmetric form with previously derived strain–displacement relation matrices and material modulus for membrane, bending, and transverse shear terms. To alleviate the locking phenomenon, the B-bar method, which utilizes the strain projected onto a lower-order field, is used in the isogeometric shell element. Reduced-order B-splines for strain fields are constructed based on tying points, which eased the locking phenomenon compared with the classical B-bar method. To further alleviate additional shell locking by a higher-order regularity, the Bézier extraction method is adopted to reduce the element regularity. The proposed combination of Nitsche’s coupling and B-bar projection demonstrates the superior accuracy and robustness in the multipatch isogeometric shell examples with a high convergence rate.

Three-Dimensional Vibration Model of Cylindrical Shells via Carrera Unified Formulation.

In this paper, we present a novel and unified model for studying the vibration of cylindrical shells based on the three-dimensional (3D) elastic theory and the Carrera Unified Formulation. Our approach represents a significant advancement in the field, as it enables us to accurately predict the vibrational behavior of cylindrical shells under arbitrary boundary conditions. To accomplish this, we expand the axial, circumferential, and radial displacements of the shell using Chebyshev polynomials and Taylor series, thereby reducing the dimensionality of the expansion and ensuring the precision and rigor of our results. In addition, we introduce three groups of artificial boundary surface springs to simulate the general end boundary conditions of the cylindrical shell and coupling springs to strongly couple the two surfaces of the cylindrical shell φ = 0 and φ = 2π to ensure continuity of displacements on these faces. Using the energy function of the entire cylindrical shell model, we obtain the characteristic equation of the system by finding the partial derivatives of the unknown coefficients of displacement in the energy function. By solving this equation, we can directly obtain the vibration characteristics of the cylindrical shell. We demonstrate the convergence, accuracy, and reliability of our approach by comparing our computational results with existing results in the literature and finite element results. Finally, we present simulation results of the frequency features of cylindrical shells with various geometrical and boundary parameters in the form of tables and figures. Overall, we believe that our novel approach has the potential to greatly enhance our understanding of cylindrical shells and pave the way for further advancements in the field of structural engineering. Our comprehensive model and simulation results contribute to the ongoing efforts to develop efficient and reliable techniques for analyzing the vibrational behavior of cylindrical shells.

A coupled FETI-BDNM for solving 3D elastic frictional contact problem

Based on the B-differentiable Newton method (BDNM) and finite element tearing and interconnecting (FETI) algorithm, a coupled approach named FETI-BDNM is developed to solve the 3D elastic contact problem with friction. In this method, the contact bodies are decomposed into a number of subdomains. The interfaces between the two adjacent substructures will be skillfully considered as the special contact surfaces that are always in the stick state. By resorting to the generalized inverse of stiffness matrices of the subdomains, the relationships between the relative displacements and forces on contact/interface, i.e., contact flexibility matrices will be computed efficiently. The contact equations, displacement continuity equations and equilibrium equations for each subdomain are expressed in the form of B-differentiable equations (BDEs) uniformly, which can be solved by the damped Newton-Raphson method with 1D Armijo type linear search. The proposed method makes it possible to solve large-scale contact problems without compromising the solution accuracy compared to the original BDNM, and the good convergence of BDEs and parallel scalability of the FETI algorithm will be preserved. Numerical examples validate the accuracy, efficiency, robustness and effects of the number of subdomains on the computational time of the proposed method.

Closed-form asymptotic micromechanics model of fiber reinforced polymer and metal matrix composites

This work presents an analytical asymptotically-correct micromechanics model that helps to predict the effective material properties of a unidirectional composite material. The conventional and numerical approaches estimate the homogenized material properties of composites for their defined component volume fractions, their constituent properties and configurational geometry. Presently these approaches are based on kinematic assumptions such as having displacement or stress components vary through the cross section for beam like structures or through the thickness for plate like structures according to certain predefined functions that doesn’t always logically follow the 3D analysis. In the present formulation, the micromechanics model is developed by accommodating all possible deformations without assuming the displacement function or stress components. These are derived by minimizing the potential energy in terms of generalized strain measures. In the present formulation, Berdichevsky’s Variational Asymptotic Method (VAM) is employed as a mathematical tool to accomplish the homogenization procedure. The Hashin-Rosen model popularly referred to as the Concentric Cylinder Model (CCM) serves as the framework to estimate all the relevant homogenized elastic moduli and coupling coefficients. The derived quantities of interest are obtained as closed form expressions which are functions of the properties of the reinforcement material, the matrix material, their volumes fraction and the geometry of their relative arrangement. These expressions are arrived following the 3D elasticity governing rules by satisfying the interfacial displacement continuity and transverse stress equilibria conditions at the reinforcement and matrix materials interface. The developed expressions for the elastic moduli, shear moduli and Poisson’s ratios of few typical polymer and metal matrix composite materials are validated with some of the relevant results available in the literature.

A close-formed solution for the horizontal vibration of a pipe pile in saturated soils considering the radial heterogeneity effect

Based on the theory of Biot’s two-phase medium and Novak’s thin layer, a coupled mechanical model of pipe pile and radially inhomogeneous saturated soil is established by comprehensively taking into account the saturated surrounding soil’s construction disturbance effect and the influence of the pipe pile’s saturated inner soil. First, the displacements and horizontal impedances of the surrounding and inner soil are derived using the potential function, Laplace transformation and variable separation methods. Second, the analytical solution for the horizontal impedance of pile head is obtained by combining the conditions of the displacement continuity and stress equilibrium on the soil-pile interface. Finally, the obtained solution for the pile head’s horizontal impedance is reduced to validate the rationality of the proposed solution by making comparisons with existing solutions. On this basis, a detailed parametric analysis is given to explore the influence of the pile parameters and construction disturbance effect on the horizontal vibration characteristics of pipe pile.

Effect of the electric field on the sound transmission loss of double-walled electro-rheological fluid sandwich plates with functionally graded carbon nanotube reinforced composite facesheets

Numerous studies have been conducted to examine the vibroacoustic characteristics of lightweight double-walled structures due to the anticipated applications of these structures in noise reduction engineering. In this study, third-order shear deformation theory (TSDT) is used to develop a theoretical model that predicts sound transmission loss (STL) across double-walled electro-rheological fluid (ERF) sandwich plates with functionally graded carbon nanotube reinforced composite (FG-CNTRC) facesheets. The extended rule of mixture is utilized to evaluate the properties of the FG-CNTRC material in the thickness. Depending on the four FG models, the CNT volume fraction varies. A sufficient displacement continuity condition is taken into account between layers. Furthermore, it should be noted that changing the electric field affects the pre-yield zone’s ERF features. The vibroacoustic equations are derived using Hamilton’s principle and solved utilizing weighted residual (Galerkin) technique considering simply supported and clamped boundary conditions. Several studies are done to compare the results of the suggested model with other results found in the literature. An extensive numerical study is conducted to examine the dependence of STL on several parameters, including electric field strength, volume percentage of the CNTs, CNTs distribution, depth of acoustic cavity.

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.

The Use of Asymmetric Polynomial Profiles for Planning a Smooth Trajectory

This paper presents planning of smooth trajectories using the asymmetric profiles of displacement, velocity, acceleration and jerk of described higher degree polynomials. The presented way of determination of polynomial and asymmetry coefficients as well as motion time includes the procedure at both constraints of single kinematic quantities, i.e., velocity, acceleration, jerk and introducing a few constraints simultaneously. Moreover, the paper presents a method of planning trajectories with intermediate points consisting in concatenation of the analyzed polynomials. The proposed method of trajectory planning was compared to that of using the S-curve. The results in the form of kinematic quantity courses are presented graphically. The discussed method guarantees continuity of displacement, velocity, acceleration and jerk. It can be used for planning trajectories of various technical objects (e.g., manipulators, mobile robots, CNC machine tools, optical disk drives, autonomous vehicles, etc.).

Higher-order vibration of thick composite and sandwich plates based on an alternative higher-order model

The transverse stretching vibration of thick sandwich plates, which is attributed to largely different stiffness at the adjacent layers, is a challenging issue, and efficient approach for such issue is less reported in the published literature. Thus, natural frequencies corresponding to stretching vibration modes are generally neglected in engineering design, which might impact structural safety as frequencies of the exciting force are close to transverse stretching vibration frequencies. This paper proposes an alternative higher-order model for dynamic analysis corresponding to the higher-order vibration modes. The proposed model is classified in the displacement-based equivalent single-layer theory, as the number of displacement parameters in the proposed model is independent of the layer number. The continuity of displacements and transverse shear stresses can be fulfilled at the interfaces between the adjacent layers of structures. To demonstrate the capability of the proposed model, typical examples are analyzed by utilizing the proposed model, the three-dimensional finite element method and the chosen higher-order models. By comparing with the exact three-dimensional elasticity solutions, it is found that the proposed model can yield more accurate natural frequencies corresponding to the higher-order displacement modes than the selected models. Moreover, the factors influencing reasonable prediction of the higher-order frequencies are investigated in detail, which can provide a reference for the accurate prediction of the higher-order frequencies.

Analytical solution for horizontal vibration of end-bearing single pile in radially heterogeneous saturated soil

Employing Biot's dynamic equations and Novak's thin-layer theory, an analytical model for the horizontal vibration of a single pile embedded in the radially heterogeneous saturated soil is established. Firstly, the frequency-domain response of surrounding soil is derived by means of the potential function, Laplace transformation and variable separation methods. Secondly, the analytical solution for the horizontal impedance of pile head is deduced by combining the conditions of the displacement continuity and stress equilibrium on the soil-pile interface. Finally, the obtained solution is degenerated to validate its accuracy via independent comparisons with different existing solutions. On this basis, the effect of soil radial heterogeneity on the horizontal vibration of an end bearing pile is further discussed by conducting an extensive parametric analysis.

Self-consistent homogenization-based parametrically upscaled continuum damage mechanics model for composites subjected to high strain-rate loading

This paper develops a Parametrically Upscaled Continuum Damage Mechanics (PUCDM) model for carbon fiber/epoxy matrix composites subjected to high strain-rate loading. The PUCDM model for predicting high strain-rate induced deformation and damage evolution at the macroscopic scale, explicitly incorporates microstructural morphology and micro-inertia in its constitutive coefficients. It is developed using self-consistent homogenization that uses a concurrent multiscale model to account for the effect of micro-inertia and stress wave interaction with the microstructure. The concurrent model embeds a statistically equivalent representative volume element (SERVE) of the microstructure in an exterior domain, whose constitutive and damage response are described by the PUCDM model. Micromechanical modeling for the composite microstructure involves a unified cohesive zone enhanced phase-field damage model for fiber-matrix interface debonding, fiber breakage, and matrix cracking. The PUCDM model parameters for the upscaled domain are determined through micro-macro-scale energy equivalence, along with traction reciprocity and displacement continuity at the SERVE-exterior domain interface that are enforced using a Lagrange multiplier-based approach. Simulations using the concurrent model analyze stress wave propagation and damage evolution in composite microstructures with three different fiber volume fractions at multiple strain rates in the range of 102 − 105 s−1. The fiber volume fraction and spatial distributions affect the overall stiffness and damage evolution in the PUCDM model. Results of the analysis demonstrate the need for representation of micro-inertia, strain rates and microstructure morphology in the PUCDM-based stiffness and damage model parameters for high strain-rates above [Formula: see text].

2‐D Analytical solutions for the multisegmented panel subjected to arbitrary boundary condition

AbstractA closed‐form bending solution of the multisegmented panels is presented for different support conditions. The panels can have multiple no. of segments along the x‐axis with varying material properties. The governing equations are formulated using the extended Kantorovich method (EKM) in the mixed‐form, and a set of ordinary differential equations have been obtained. The continuity of displacement and stresses are satisfied at the interface of each segment. The panels can have simply‐support, clamp, and free types of support conditions. The panel interface at each segment along the x‐axis is assumed to be perfectly bonded. In this work, the stress and deformation variations for four types of panel configuration have been studied. Two segmented panels having aluminum and Graphite–Epoxy with equal and unequal segments are considered. A four‐segmented Al/SiC panel having gradual material variation is also analyzed. The effect of ply‐angles on the longitudinal variables is also studied for the equal segmented Al/GrEp panel. The deflection and stresses are compared with the finite element solution and found in good agreement with the EKM results. The current development will lead to developing the solution for stepped panels, dissimilar plates, and more complex cases used in aerospace applications.

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.

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.

An exact solution for vibration analysis of pipe coupled with conical-ring stiffened cylindrical shells with arbitrary boundary condition

In this paper, an analytical impedance synthesis method (ISM) is proposed to analyze the vibration characteristics of a pipe-hull coupled system, in which the hull of a submarine can be simplified to a combined conical-stiffened cylindrical-hemisphere shell. Wave solution and power series solution are used to describe the displacement of cylindrical and conical shell respectively. Rib stiffened and bulkhead are modeled as annular plate and circular plate. Based on exact general solutions, the impedance matrix of each segment is established separately and assembled in the global coordinate by displacement continuity and force balance boundary conditions. Natural frequency and forced responses are compared with open literature and FEM results to verify the correctness of the method. In the numerical examples, the effects of direction of external force and stiffness of support on the forced vibration characteristics are discussed, which can provide some guides for acoustic design of pipe in submarine.