Robust C0 high-order plate finite element for thin to very thick structures: mechanical and thermo-mechanical analysis

This paper presents a new C⁰ eight-node quadrilateral finite element (FE) for geometrically linear elastic plates. This finite element aims at modeling both thin and thick plates without any pathologies of the classical plate finite elements (shear and Poisson or thickness locking, spurious modes, etc). A C¹ FE was previously developed by the first author based on the kinematics proposed by Touratier. This new FE can be viewed as an evolution towards three directions: (1) use of only C⁰ FE approximations; (2) modeling of thick to thin structures; and (3) capability in multifield problems. The transverse normal stress is included allowing use of the three-dimensional constitutive law. The element performances are evaluated on some standard plate tests, and comparisons are given with exact three-dimensional solutions for plates under mechanical and thermal loads. Comparisons are made with other plate models using C¹ and semi-C¹ FE approximations as well as with an eight node C⁰ FE based on the Reissner–Mindlin model. All results indicate that the present element is highly insensitive to mesh distortion, has very fast convergence properties and gives accurate results for displacements and stresses. Copyright © 2012 John Wiley & Sons, Ltd.     

An area-average approach to peening residual stress under multi-impacts using a three-dimensional symmetry-cell finite element model with plastic shots

We estimate realistic peening residual stress based on area-averaged solution using a 3D multi-impact symmetry-cell finite element (FE) model. The analytical model includes elaborate factors reflecting actual peening phenomena and plastic shot effect. Area-averaged solution is much closer to X-ray diffraction (XRD) experimental solution than four-node-averaged solution in plastic shot FE model. The area-averaged solution, moreover, converges to the perfect equi-biaxial stress state. From this, based on the area-averaged solution, we obtained the FE Almen curve, and then derived related equations among FE arc height, FE coverage and shot velocity. The FE Almen curve corresponds well with experimentally obtained by Kim et al. [Kim T, Lee JH, Lee H. An Effective 2D FE model with plastic shot for evaluation of peening residual stress. J Mater Process Technol, submitted for publication; Kim T, Lee H, Lee JH. A 3D phenomenological FE model for unique solution of peening stress due to multi-impacts. Int J Numer Methods Eng, submitted for publication]. Using the FE Almen curve, we examine the FE area-averaged solution in major peening materials. The FE solutions of surface, maximum compressive residual stress and deformation depth quite reach experimental solutions. The FE Almen curve is thus confirmed to be useful for estimation of residual stress solution. Consequently, we validated that the concept of area-averaged solution is the systematical analytical method for evaluation of real peening residual stress.     

Laminated composite rectangular and annular plates: A GDQ solution for static analysis with a posteriori shear and normal stress recovery

The Generalized Differential Quadrature (GDQ) Method is applied to study laminated composite degenerate shell panels such as rectangular and annular plates. The theoretical treatment is maintained general in order to expose in a unique way the procedure adopted to obtain the stress profiles through the thickness of plates without specifying the equations for rectangular and annular plates. By simply imposing some geometrical relations the equations governing the problem of plates under consideration, that are degenerate shells, are inferred from the theory of shells of revolution. The mechanical model is based on the so called First-order Shear Deformation Theory (FSDT) deduced from the three-dimensional theory in order to analyse the above moderately thick structural elements. The solution is given in terms of generalized displacement components of points lying on the middle surface of the plate. After the solution of the fundamental system of equations in terms of displacements and rotations, the generalized strains and stress resultants are evaluated by applying the Differential Quadrature rule to the generalized displacements. The transverse shear and normal stress profiles through the laminate thickness are reconstructed a posteriori by using local three-dimensional elasticity equilibrium equations. No preliminary recovery or regularization procedure on the extensional and flexural strain fields is needed when the Differential Quadrature technique is used. By using GDQ procedure through the thickness, the reconstruction procedure needs only to be corrected to properly account for the boundary equilibrium conditions. In order to verify the accuracy of the present method, GDQ results are compared with the ones obtained with semi-analytical formulations and with 3D finite element methods. Stresses of several composite plates are evaluated. Very good agreement is observed without using mixed formulations and higher order kinematical models. Various examples of stress profiles for rectangular and annular plate elements are presented to illustrate the validity and the accuracy of GDQ method.     

Numerical and experimental investigation of cold rotary forging of a 20CrMnTi alloy spur bevel gear

Due to the complex three-dimensional (3D) geometry and tooling design, mass production of precision spur bevel gears by machining or conventional forging technology is impeded by numerous stubborn barriers to date. Cold rotary forging, an innovative incremental metal forming process, has great potential to improve the current situation owing to its flexibility and low tool load requirement. In the present study, a sound 3D rigid-plastic finite element (FE) model of cold rotary forging of a 20CrMnTi alloy spur bevel gear is developed under the DEFORM-3D platform. To ensure the precision of the proposed FE model, a series of experiments are well performed for the identification of the mechanical properties of 20CrMnTi alloy and the realistic friction conditions prevailing at the die-workpiece interface. By utilising this FE model, the workpiece geometry is optimised with the intention of achieving a better filling of gear shape and a lower forming load requirement, and then the distribution of different field-variables such as flow velocity and effective strain are thoroughly investigated. For verification purposes, the cold rotary forging experiments of 20CrMnTi alloy spur bevel gears are subsequently conducted. Good agreement between the experimental results and the simulation ones is highlighted by comparing the gear shape after cold rotary forging and the axial forging force, which validates employed model.     

Finite element simulation and experimental study on mechanical behavior of 3D woven glass fiber composite sandwich panels

The results of finite element simulation followed by an experimental study are presented in order to investigate the mechanical behavior of three-dimensional woven glass-fiber sandwich composites using FE method. Experimental load–displacement curves were obtained for flatwise compressive, edgewise compressive, shear, three-point bending and four-point bending loads on the specimens with three different core thicknesses in two principal directions of the sandwich panels, called warp and weft. A 3D finite element model is employed consisting of glass fabric and surrounding epoxy resin matrix in order to predict the mechanical behavior of such complex structures. Comparison between the finite element predictions and experimental data showed good agreement which implies that the FE simulation can be used instead of time-consuming experimental procedures to study the effect of different parameters on mechanical properties of the 3D woven sandwich composites.     

General higher-order layer-wise theory for free vibrations of doubly-curved laminated composite shells and panels

ABSTRACT

The present article illustrates a general formulation for a higher-order layer-wise theory related to the analysis of the free vibrations of thick doubly-curved laminated composite shells and panels. The theoretical framework relates to the dynamic analysis of shell structures by using a general displacement field based on the Carrera Unified Formulation (CUF), including the stretching effect for each layer. The order of the expansion along the thickness direction is taken as a free parameter. The starting point of the present general higher-order layer-wise formulation is to propose a kinematic assumption, with an arbitrary number of degrees of freedom. The main aim of this work is to determine the explicit fundamental operators that can be used for the layer-wise (LW) approach. These fundamental operators are obtained for the first time by the author and are related to motion equations of doubly-curved shells described in an orthogonal curvilinear co-ordinate system. The free vibration shell and panel problems are computationally solved using the generalized differential quadrature (GDQ) and generalized integral quadrature (GIQ) techniques. The numerical results are compared with recent papers in the literature and commercial finite element codes.     

Free vibration analysis of reinforced thin-walled plates and shells through various finite element models

Abstract

This article deals with free vibration analysis of thin-walled structures reinforced by longitudinal stiffeners using refined one-dimensional (1D) models.The 1D theory, which is used in the present article, has hierarchical features and it is based on the Carrera Unified Formulation (CUF). The displacement field over the cross section is obtained by means of Taylor (TE) or Lagrange (LE) expansions. Finite element (FE) method is applied along the beam axis to obtain weak form solutions of the related governing equations. The obtained results are compared with those from classical finite element formulations based on plate and shell (2D), beam (1D), and solid (3D) elements that are available in commercial software. When solid formulation is used to build the FE solutions, stringers and skin are modeled with only 3D elements while, in the 2D-1D FE models, shell and beam elements are used for skin and stringers, respectively. Three benchmark problems are analyzed: a flat plate, a curved panel, and a thin-walled cylinder. When TE models are used, different orders of expansion, N, are considered, where N is a free parameter of the formulation. As far as Lagrange expansions are concerned, four-node (LE 4) and nine-node (LE 9) elements are used to build different meshes on the cross section. The results show that the present 1D models are able to analyze the dynamic behavior of complex structures and can detect 3D effects as well as very complex shell-like modes typical of thin-walled structures. Moreover, the 1D-CUF elements yield accurate results with a low number of degrees of freedom.     

Modelling delamination onset and growth in pin loaded composite laminates

A three-dimensional (3D) finite element (FE) model is created with cohesive zone elements (CZE) to simulate a mechanically fastened [0°/90°]_s pin-loaded joint in a composite laminate. The model incorporates fully integrated solid elements in the pin-loaded area to accurately capture the high stress gradients. Contact based cohesive elements with a bilinear traction–separation law are inserted between the layers to capture the onset and growth of delamination. The stress distribution around the pin-loaded hole was verified with the widely used cosine stress distribution model. Results from the FE model show that delamination damage initiated at the point of maximum average shear stress at the 0°/90° interface. The delaminated area develops an elliptical shape which grows in a non-self similar manner with increasing pin displacement. It is concluded that a progressive damage model should be included to provide a full understanding of the failure sequence, work that the authors are currently engaged with.     

A residual a posteriori error estimate for partition of unity finite elements for three-dimensional transient heat diffusion problems using multiple global enrichment functions

In this article, a study of residual based a posteriori error estimation is presented for the partition of unity finite element method (PUFEM) for three-dimensional (3D) transient heat diffusion problems. The proposed error estimate is independent of the heuristically selected enrichment functions and provides a useful and reliable upper bound for the discretization errors of the PUFEM solutions. Numerical results show that the presented error estimate efficiently captures the effect of h-refinement and q-refinement on the performance of PUFEM solutions. It also efficiently reflects the effect of ill-conditioning of the stiffness matrix that is typically experienced in the partition of unity based finite element methods. For a problem with a known exact solution, the error estimate is shown to capture the same solution trends as obtained by the classical L₂ norm error. For problems with no known analytical solutions, the proposed estimate is shown to be used as a reliable and efficient tool to predict the numerical errors in the PUFEM solutions of 3D transient heat diffusion problems.     

Two solution strategies to improve the computational performance of sequentially linear analysis for quasi-brittle structures

Sequentially linear analysis (SLA), an event-by-event procedure for finite element (FE) simulation of quasi-brittle materials, is based on sequentially identifying a critical integration point in the FE model, to reduce its strength and stiffness, and the corresponding critical load multiplier (λ_crit), to scale the linear analysis results. In this article, two strategies are proposed to efficiently reuse previous stiffness matrix factorisations and their corresponding solutions in subsequent linear analyses, since the global system of linear equations representing the FE model changes only locally. The first is based on a direct solution method in combination with the Woodbury matrix identity, to compute the inverse of a low-rank corrected stiffness matrix relatively cheaply. The second is a variation of the traditional incomplete LU preconditioned conjugate gradient method, wherein the preconditioner is the complete factorisation of a previous analysis step's stiffness matrix. For both the approaches, optimal points at which the factorisation is recomputed are determined such that the total analysis time is minimised. Comparison and validation against a traditional parallel direct sparse solver, with regard to a two-dimensional (2D) and three-dimensional (3D) benchmark study, illustrates the improved performance of the Woodbury-based direct solver over its counterparts, especially for large 3D problems.     

Static Analysis of Shear Actuated Piezo-Electric Beams via Hierarchical FEM Theories

Several one-dimensional finite elements for the static analysis of shear actuated piezo-electric three-dimensional beams are presented. A generic expression of stiffness and mass matrices is obtained through a Unified Formulation. The derivation is general regardless of the approximation order of the displacements and the electric potential over the cross-section and the number of nodes along the axial direction. A Lagrange’s polynomials based layer-wise approximation is used. Several mechanical boundary conditions and sensor and actuator configurations are investigated. Results are assessed towards three-dimensional finite element solutions. It is demonstrated that the proposed class of finite elements is able to yield very accurate results.     

Closed-form plastic collapse loads of pipe bends under combined pressure and in-plane bending

Based on three-dimensional (3-D) FE limit analyses, this paper provides plastic limit, collapse and instability load solutions for pipe bends under combined pressure and in-plane bending. The plastic limit loads are determined from FE limit analyses based on elastic-perfectly-plastic materials using the small geometry change option, and the FE limit analyses using the large geometry change option provide plastic collapse loads (using the twice-elastic-slope method) and instability loads. For the bending mode, both closing bending and opening bending are considered, and a wide range of parameters related to the bend geometry is considered. Based on the FE results, closed-form approximations of plastic limit and collapse load solutions for pipe bends under combined pressure and bending are proposed.     

A variable kinematic doubly-curved MITC9 shell element for the analysis of laminated composites

The present article considers the linear static analysis of composite shell structures with double-curvature geometry by means of a shell finite element with variable through-the-thickness kinematic. The refined models used are grouped in the Unified Formulation by Carrera (CUF) and they permit the distribution of displacements and stresses along the thickness of the multilayered shell to be accurately described. The shell element has nine nodes and the mixed interpolation of tensorial components (MITC) method is used to contrast the membrane and shear locking phenomenon. The governing equations are derived from the principle of virtual displacement (PVD) and the finite element method (FEM) is employed to solve them. Cross-ply spherical shells with simply-supported edges and subjected to bi-sinusoidal pressure are analyzed. Various laminations, thickness ratios, and curvature ratios are considered. The results, obtained with different theories contained in the CUF, are compared with both the elasticity solutions given in the literature and the analytical solutions obtained using the CUF and the Navier’s method. From the analysis, one can conclude that the shell element based on the CUF is very efficient and its use is mandatory with respect to the classical models in the study of composite structures. Finally, shells with different lamination, boundary conditions, and loads are also analyzed using high-order layer-wise theories in order to provide FEM benchmark solutions.     

Accurate calculation of transverse shear stresses for soft-core sandwich laminates

Accurate evaluation of transverse stresses in soft-core sandwich laminates using the existing 2D finite element (FE) models involves cumbersome post-processing techniques. In this paper, a simple and robust method is proposed for accurate evaluation of through-the-thickness distribution of transverse stresses in soft-core sandwich laminates by using a displacement-based C⁰ continuous 2D FE model derived from refined higher-order shear deformation theory (RHSDT) and a least square error (LSE) method. In this refined higher-order shear deformation theory (RHSDT), the in-plane displacement field for the face sheets and the core is obtained by superposing a global cubically varying displacement field on a zigzag linearly early varying displacement field. The transverse displacement is assumed to have a quadratic variation within the core, and it remains constant in the faces beyond the core. The proposed C⁰ FE model satisfies the condition of transverse shear stress continuity at the layer interfaces and the zero transverse shear stress condition at the top and bottom of the sandwich plate. The nodal field variables are chosen in an efficient manner to circumvent the problem of C¹ continuity requirement of the transverse displacements associated with the RHSDT. The LSE method is applied to the 3D equilibrium equations of the plate problem at the post-processing stage, after in-plane stresses are calculated by using the above FE model based on RHSDT. Thus, the proposed method is quite simple and elegant compared to the usual method of integrating the 3D equilibrium equations at the post-processing stage for the calculation of transverse stresses in a sandwich laminates. The accuracy of the proposed method is demonstrated in the numerical examples through the comparison of the present results with those obtained from different models based on HSDT and 3D elasticity solutions.     

Application of finite element methods to thermohydrodynamic lubrication

A finite element formulation is presented for the equations governing the steady thermohydrodynamic behaviour of liquid lubricated bearings. This formulation permits application of the iterative solution scheme to bearings of arbitrary geometry. A generalized Reynolds equation resulting from the combination of the mass and momentum conservation equations is cast into variational form and used to derive general finite element equations. The method of weighted residuals with Galerkin's criterion is used to generate finite element matrix equations for the thermal energy equation. In addition to the finite element formulation, a discussion of appropriate finite difference techniques is also given for problems without complex geometry. As an example, the formulations are applied to obtain numerical solutions for a three-dimensional sector thrust bearing operating in the thermohydrodynamic regime. Pressure, velocity and temperature distributions are give, and the thermohydrodynamic solutions are compared with the results of classical isothermal theory.     

Postbuckling characteristics of nanobeams based on the surface elasticity theory

In the present paper, an attempt is made to numerically investigate the postbuckling response of nanobeams with the consideration of the surface stress effect. To accomplish this, the Gurtin–Murdoch elasticity theory is exploited to incorporate surface stress effect into the classical Euler–Bernoulli beam theory. The size-dependent governing differential equations are derived and discretized along with various end supports by employing the principle of virtual work and the generalized differential quadrature (GDQ) method. Newton’s method is applied to solve the discretized nonlinear equations with the aid of an auxiliary normalizing equation. After solving the governing equations linearly, to obtain each eigenpair in the nonlinear model, the linear response is used as the initial value in Newton’s method. Selected numerical results are given to show the surface stress effect on the postbuckling characteristics of nanobeams. It is found that by increasing the thickness of nanobeams, the postbuckling equilibrium path obtained by the developed non-classical beam model tends to the one predicted by the classical beam theory and this anticipation is the same for all selected boundary conditions.     

Strict error bounds for linear solid mechanics problems using a subdomain-based flux-free method

We discuss, in this paper, a flux-free method for the computation of strict upper bounds of the energy norm of the error in a Finite Element (FE) computation. The bounds are strict in the sense that they refer to the difference between the displacement computed on the FE mesh and the exact displacement, solution of the continuous equations, rather than to the difference between the displacements computed on two FE meshes, one coarse and one refined. This method is based on the resolution of a series of local problems on patches of elements and does not require the resolution of a previous problem of flux equilibration, as happens with other methods. The paper concentrates more specifically on linear solid mechanics issues, and on the assessment of the energy norm of the error, seen as a necessary tool for the estimation of the error in arbitrary quantities of interest (linear functional outputs). Applications in both 2D and 3D are presented.     

Static and free vibration analysis of functionally graded plates based on a new quasi-3D and 2D shear deformation theories

In this study, two dimensional (2D) and quasi three-dimensional (quasi-3D) shear deformation theories are presented for static and free vibration analysis of single-layer functionally graded (FG) plates using a new hyperbolic shape function. The material of the plate is inhomogeneous and the material properties assumed to vary continuously in the thickness direction by three different distributions; power-law, exponential and Mori–Tanaka model, in terms of the volume fractions of the constituents. The fundamental governing equations which take into account the effects of both transverse shear and normal stresses are derived through the Hamilton's principle. The closed form solutions are obtained by using Navier technique and then fundamental frequencies are found by solving the results of eigenvalue problems. In-plane stress components have been obtained by the constitutive equations of composite plates. The transverse stress components have been obtained by integrating the three-dimensional stress equilibrium equations in the thickness direction of the plate. The accuracy of the present method is demonstrated by comparisons with the different 2D, 3D and quasi-3D solutions available in the literature.     

Modelling of mode-I stable crack growth under hydrogen assisted stress corrosion cracking

The paper presents a new strategy based on combined analytical and finite element (FE) solution to hydrogen assisted stress corrosion crack growth. The diffusion process is solved analytically through both one-and two-dimensional modelling. These solutions are adopted with two-dimensional FE based cohesive zone model of crack extension study. The results fit well with published experimental data and show improvement over the predictions by full FE approach. The new solution approach helps to reduce time required for simulation/computation. The study has produced a relationship between concentration dependent reduction in cohesive strength and plastic strain rate.