A new reduced integration solid‐shell element based on EAS and ANS with hourglass stabilization

In this paper, a novel reduced integration eight‐node solid‐shell finite element formulation with hourglass stabilization is proposed. The enhanced assumed strain method is adopted to eliminate the well‐known volumetric and Poisson thickness locking phenomena with only one internal variable required. In order to alleviate the transverse shear and trapezoidal locking and correct rank deficiency simultaneously, the assumed natural strain method is implemented in conjunction with the Taylor expansion of the inverse Jacobian matrix. The projection of the hourglass strain‐displacement matrix and reconstruction of its transverse shear components are further employed to avoid excessive hourglass stiffness. The proposed solid‐shell element formulation successfully passes both the membrane and bending patch tests. Several typical examples are presented to demonstrate the excellent performance and extensive applicability of the proposed element. Copyright © 2015 John Wiley & Sons, Ltd.     

A reduced integration solid‐shell finite element based on the EAS and the ANS concept—Geometrically linear problems

In this paper a new reduced integration eight‐node solid‐shell finite element is presented. The enhanced assumed strain (EAS) concept based on the Hu–Washizu variational principle requires only one EAS degree‐of‐freedom to cure volumetric and Poisson thickness locking. One key point of the derivation is the Taylor expansion of the inverse Jacobian with respect to the element center, which closely approximates the element shape and allows us to implement the assumed natural strain (ANS) concept to eliminate the curvature thickness and the transverse shear locking. The second crucial point is a combined Taylor expansion of the compatible strain with respect to the center of the element and the normal through the element center leading to an efficient and locking‐free hourglass stabilization without rank deficiency. Hence, the element requires only a single integration point in the shell plane and at least two integration points in thickness direction. The formulation fulfills both the membrane and the bending patch test exactly, which has, to the authors' knowledge, not yet been achieved for reduced integration eight‐node solid‐shell elements in the literature. Owing to the three‐dimensional modeling of the structure, fully three‐dimensional material models can be implemented without additional assumptions. Copyright © 2009 John Wiley & Sons, Ltd.     

A reduced integration solid‐shell finite element based on the EAS and the ANS concept—Large deformation problems

In this paper we address the extension of a recently proposed reduced integration eight‐node solid‐shell finite element to large deformations. The element requires only one integration point within the shell plane and at least two integration points over the thickness. The possibility to choose arbitrarily many Gauss points over the shell thickness enables a realistic and efficient modeling of the non‐linear material behavior. Only one enhanced degree‐of‐freedom is needed to avoid volumetric and Poisson thickness locking. One key point of the formulation is the Taylor expansion of the inverse Jacobian matrix with respect to the element center leading to a very accurate modeling of arbitrary element shapes. The transverse shear and curvature thickness locking are cured by means of the assumed natural strain concept. Further crucial points are the Taylor expansion of the compatible cartesian strain with respect to the center of the element as well as the Taylor expansion of the second Piola–Kirchhoff stress tensor with respect to the normal through the center of the element. Copyright © 2010 John Wiley & Sons, Ltd.     

An improved solid‐shell element based on ANS and EAS concepts

This paper presents an eight‐node nonlinear solid‐shell element for static problems. The main goal of this work is to develop a solid‐shell formulation with improved membrane response compared with the previous solid‐shell element (MOS2013), presented in 1 . Assumed natural strain concept is implemented to account for the transverse shear and thickness strains to circumvent the curvature thickness and transverse shear locking problems. The enhanced assumed strain approach based on the Hu–Washizu variational principle with six enhanced assumed strain degrees of freedom is applied. Five extra degrees of freedom are applied on the in‐plane strains to improve the membrane response and one on the thickness strain to alleviate the volumetric and Poisson's thickness locking problems. The ensuing element performs well in both in‐plane and out‐of‐plane responses, besides the simplicity of implementation. The element formulation yields exact solutions for both the membrane and bending patch tests. The formulation is extended to the geometrically nonlinear regime using the corotational approach, explained in 2 . Numerical results from benchmarks show the robustness of the formulation in geometrically linear and nonlinear problems. Copyright © 2016 John Wiley & Sons, Ltd.     

Finite deformation formulation of a shell element for problems of sheet metal forming

An elastic-plastic thin shell finite element suitable for problems of finite deformation in sheet metal forming is formulated. Hill's yield criterion for sheet materials of normal anisotropy is applied. A nonlinear shell theory in a form of an incremental variational principle and a quasi-conforming element technique are employed in the Lagrangian formulation. The shell element fulfills the inter-element C ¹ continuity condition in a variational sense and has a sufficient rank to allow finite stretching, rotation and bending of the shell element. The accuracy and efficiency of the finite element formulation are illustrated by numerical examples.     

Transient analysis of FGM and laminated composite structures using a refined 8-node ANS shell element

In this paper, we investigate the vibration analysis of functionally graded material (FGM) and laminated composite structures, using a refined 8-node shell element that allows for the effects of transverse shear deformation and rotary inertia. The properties of FGM vary continuously through the thickness direction according to the volume fraction of constituents defined by sigmoid function, but in this method, their Poisson’s ratios of the FGM plates and shells are assumed to be constant. The finite element, based on a first-order shear deformation theory, is further improved by the combined use of assumed natural strains and different sets of collocation points for interpolation the different strain components. We analyze the influence of the shell element with the various location and number of enhanced membrane and shear interpolation. Using the assumed natural strain method with proper interpolation functions the present shell element generates neither membrane nor shear locking behavior even when full integration is used in the formulation. The natural frequencies of plates and shells are presented, and the forced vibration analysis of FGM and laminated composite plates and shells subjected to arbitrary loading is carried out. In order to overcome membrane and shear locking phenomena, the assumed natural strain method is used. To validate and compare the finite element numerical solutions, the reference solutions of plates based on the Navier’s method, the series solutions of sigmoid FGM (S-FGM) plates are obtained. Results of the present theory show good agreement with the reference solutions. In addition the effect of damping is investigated on the forced vibration analysis of FGM plates and shells.     

Understanding and improving the reduced integration of Mindlin shell elements

The reason for looking in some depth at locking of Mindlin shell elements is introduced. A simple example of straight beam locking is examined in detail. The observations about reduced integration that emerge are subsequently unified with some other facts the writer has come across. A simple curved beam element is then formulated and examined under inextensional bending. The performance of this element under reduced integration is predicted for both the ‘undistorted’ and ‘distorted’ configurations (see text) and for a selective-very reduced integration scheme. The previous predictions are then verified through the performance of an 8 node Mindlin shell element in a suitable cylindrical shell test. The paper finally concludes with a set of important observations that so easily could be drawn from such a simple analysis.     

On implementation of a nonlinear four node shell finite element for thin multilayered elastic shells

A simple non-linear stress resultant four node shell finite element is presented. The underlying shell theory is developed from the three dimensional continuum theory via standard assumptions on the displacement field. A model for thin shells is obtained by approximating terms describing the shell geometry. In this work the rotation of the shell director is parameterized by the two Euler angles, although other approaches can be easily accomodated. A procedure is provided to extend the presented approach by including the through-thickness variable material properties. These may include a general non-linear elastic material with varied degree of orthotropy, which is typical for fibre reinforced composites. Thus a simple and efficient model suitable for analysis of multilayered composite shells is attained. Shell kinematics is consistently linearized, leading to the Newton-Raphson numerical procedure, which preserves quadratic rate of asymptotic convergence. A range of linear and non-linear tests is provided and compared with available solutions to illustrate the approach.The work has been financially supported from Joint Europian Project TEMPUS-ACEM No. 2246-91 and the Ministry of Science and Technology of Slovenia.     

A new one‐point quadrature enhanced assumed strain (EAS) solid‐shell element with multiple integration points along thickness—part II: nonlinear applications

In this work the recently proposed Reduced Enhanced Solid‐Shell (RESS) finite element, based on the enhanced assumed strain (EAS) method and a one‐point quadrature integration scheme, is extended in order to account for large deformation elastoplastic thin‐shell problems. One of the main features of this finite element consists in its minimal number of enhancing parameters (one), sufficient to circumvent the well‐known Poisson and volumetric locking phenomena, leading to a computationally efficient performance when compared to other 3D or solid‐shell enhanced strain elements. Furthermore, the employed numerical integration accounts for an arbitrary number of integration points through the thickness direction within a single layer of elements. The EAS formulation comprises an additive split of the Green–Lagrange material strain tensor, making the inclusion of nonlinear kinematics a straightforward task. A corotational coordinate system is used to integrate the constitutive law and to ensure incremental objectivity. A physical stabilization procedure is implemented in order to correct the element's rank deficiencies. A variety of shell‐type numerical benchmarks including plasticity, large deformations and contact are carried out, and good results are obtained when compared to well‐established formulations in the literature. Copyright © 2006 John Wiley & Sons, Ltd.     

A new one‐point quadrature enhanced assumed strain (EAS) solid‐shell element with multiple integration points along thickness: Part I—geometrically linear applications

Accuracy and efficiency are the main features expected in finite element method. In the field of low‐order formulations, the treatment of locking phenomena is crucial to prevent poor results. For three‐dimensional analysis, the development of efficient and accurate eight‐node solid‐shell finite elements has been the principal goal of a number of recent published works. When modelling thin‐ and thick‐walled applications, the well‐known transverse shear and volumetric locking phenomena should be conveniently circumvented. In this work, the enhanced assumed strain method and a reduced in‐plane integration scheme are combined to produce a new eight‐node solid‐shell element, accommodating the use of any number of integration points along thickness direction. Furthermore, a physical stabilization procedure is employed in order to correct the element's rank deficiency. Several factors contribute to the high computational efficiency of the formulation, namely: (i) the use of only one internal variable per element for the enhanced part of the strain field; (ii) the reduced integration scheme; (iii) the prevention of using multiple elements' layers along thickness, which can be simply replaced by any number of integration points within a single element layer. Implementation guidelines and numerical results confirm the robustness and efficiency of the proposed approach when compared to conventional elements well‐established in the literature. Copyright © 2004 John Wiley & Sons, Ltd.     

Finite element analysis of the superplastic forming of thick sheet using the incremental flow formulation

The paper discusses the finite element analysis of the superplastic forming of thick sheet components. The incremental formulation proposed is based on a geometrical approximation of the flow type of constitutive equations that describe the behaviour of the alloy during forming. The spatial discretization is achieved using eight-noded finite elements. An algorithm capable of predicting the correct forming pressure is also presented in a form consistent with the incremental flow formulation. Some experimental validation of these techniques will be shown together with a number of more realistic applications which will illustrate the generality of these techniques and their ability to simulate the forming of complex components. Most of the material in this section is standard but has been included for the purpose of completeness and to introduce the reader to the notation used in the paper. © 1997 John Wiley & Sons, Ltd.     

A simplified approach for incorporating thickness stress in the analysis of sheet metal forming using shell elements

A simplified scheme for considering the thickness stress of shell elements induced by contact is presented which improves the accuracy of sheet metal forming analysis. The yield function formulated on the basis of plane stress conditions is modified to incorporate the effect of transverse normal stress induced by contact forces acting on shell elements and return mapping routine is used to update in‐plane stresses at each time step. The transverse normal stress distributions in the thickness direction are determined using the analytic solution of the cylindrical tube under the internal pressure. As numerical examples, uni‐axial compression, bi‐axial tension and bending tests are treated. The problem of cylindrical cup drawing is also calculated. Each result is compared with the results obtained by the analysis using ABAQUS continuum elements. Copyright © 2002 John Wiley & Sons, Ltd.     

Coupled use of reduced integration and non-conforming modes in quadratic Mindlin plate element

The applications of the reduced integration technique, and the addition of non-conforming modes and their coupling to Mindlin plate elements to improve their basic behaviour are reviewed and the establishment of a series of new plate elements by combined use of these schemes is presented in this paper. The element formulation is based upon the quadratic Mindlin plate concept. The results obtained by new elements converge to the exact solutions very rapidly as the mesh is refined and show reliable solutions even for severely distorted meshes. The new elements have the requisite numbers of zero eigenvalues associated with rigid body modes to avoid spurious zero energy modes. These elements are shown to solve the shear locking problem completely so that they are applicable to a wide range of plate problems, giving a high accuracy for both thick and thin plates.     

A continuum sensitivity method for finite thermo‐inelastic deformations with applications to the design of hot forming processes

A computational framework is presented to evaluate the shape as well as non‐shape (parameter) sensitivity of finite thermo‐inelastic deformations using the continuum sensitivity method (CSM). Weak sensitivity equations are developed for the large thermo‐mechanical deformation of hyperelastic thermo‐viscoplastic materials that are consistent with the kinematic, constitutive, contact and thermal analyses used in the solution of the direct deformation problem. The sensitivities are defined in a rigorous sense and the sensitivity analysis is performed in an infinite‐dimensional continuum framework. The effects of perturbation in the preform, die surface, or other process parameters are carefully considered in the CSM development for the computation of the die temperature sensitivity fields. The direct deformation and sensitivity deformation problems are solved using the finite element method. The results of the continuum sensitivity analysis are validated extensively by a comparison with those obtained by finite difference approximations (i.e. using the solution of a deformation problem with perturbed design variables). The effectiveness of the method is demonstrated with a number of applications in the design optimization of metal forming processes. Copyright © 2002 John Wiley & Sons, Ltd.     

Algorithmic formulation and numerical implementation of coupled electromagnetic‐inelastic continuum models for electromagnetic metal forming

The purpose of this work is the algorithmic formulation and implementation of a recent coupled electromagnetic‐inelastic continuum field model (Continuum Mech. Thermodyn. 2005; 17 :1–16) for a class of engineering materials, which can be dynamically formed using strong magnetic fields. Although in general relevant, temperature effects are for the applications of interest here minimal and are neglected for simplicity. In this case, the coupling is due, on the one hand, to the Lorentz force acting as an additional body force in the material. On the other hand, the spatio‐temporal development of the magnetic field is very sensitive to changes in the shape of the workpiece, resulting in additional coupling. The algorithmic formulation and numerical implementation of this coupled model is based on mixed‐element discretization of the deformation and electromagnetic fields combined with an implicit, staggered numerical solution scheme on two meshes. In particular, the mechanical degrees of freedom are solved on a Lagrangian mesh and the electromagnetic ones on an Eulerian one. The issues of the convergence behaviour of the staggered algorithm and the influence of data transfer between the meshes on the solution is discussed in detail. Finally, the numerical implementation of the model is applied to the modelling and simulation of electromagnetic sheet forming. Copyright © 2006 John Wiley & Sons, Ltd.     

Applications of linear inverse finite element method in prediction of the optimum blank in sheet metal forming

A linear inverse finite element method has been developed and investigated to predict the optimum blank. To reduce the computation time, the part is unfolded properly on the flat sheet and treated as a 2D problem. This approach is employed primarily to design the optimum blank shape from the desired final shape with the linear formulations. The procedure is based on the minimization of energy for the unfolded elements. Two solution methods, Direct and Newton–Raphson methods have been examined for the solution of nodal displacements in the equilibrium equations. The convergence show high sensitivity to the initial guess for the strain path when assumed to be linear at the first step. Two applied examples are implemented to show the efficiency of this method. In S rail example, the thickness distributions have been compared with experimental analysis after obtaining the optimum blank with Linear IFEM. In circular cup example, the results have been compared with conventional forward incremental method. New calculation of the external forces vector has been displayed. In this calculation, both blank holder force (BHF) vector and in-plane force vector have been shown. Finally, in this approach good agreement was found between the forward incremental and Linear IFEM results.     

Characterization of temperature-dependent tensile and flexural rigidities of a cross-ply thermoplastic lamina with implementation into a forming model

This paper discusses the characterization of temperature-dependent tensile and flexural rigidities for Dyneema® HB80, a cross-ply thermoplastic lamina. The low coefficient of friction of this material posed a challenge to securing specimens during tensile testing. Therefore, modification to the standard gripping method was implemented to facilitate the collection of meaningful test data. Furthermore, a long gauge length was selected to moderate the influence of slippage on the measure of the elastic modulus. A new experimental setup is presented to characterize the bending behavior at elevated-temperature conditions based on the vertical cantilever method. The material properties derived from the test data were implemented in a finite element model of the cross-ply lamina. The finite element model is generated using a hybrid discrete mesoscopic approach, and deep-draw forming of the material is simulated to investigate its formability. Simulation results are compared with an experimental forming trial to demonstrate the capabilities of the model to predict the development of out-of-plane waves during preform manufacturing.     

On the use of effective limit strains to evaluate the forming severity of sheet metal parts after nonlinear loading

The conventional forming limit curve (FLC) is significantly strain path-dependent and therefore is not valid for formability evaluation of sheet metal parts that undergo nonlinear loading paths during the forming process. The stress-based forming limit curve (SFLC) is path-independent for all but very large prestrains and is a promising tool for formability evaluation. The SFLC is an ideal failure criterion for virtual forming simulations but it cannot be easily used on the shop floor as there is no straightforward experimental method to measure stresses in stamped parts. This paper presents a theoretical basis for predicting the effective limit strain curve (ELSC) using the Marciniak and Kuczynski (MK) analysis (Int J Mech Sci 9:609–620, 1967, Int J Mech Sci 15:789–805, 1973). Since the in-plane strain components are sufficient to calculate the effective strain, the ELSC can easily be determined from strains measured in the stamping plant, and therefore it is a better alternative to the SFLC for formability evaluation. This model was validated using experimental data for AISI-1012 steel (Molaei 1999) and AA-2008-T4 aluminum alloys Graf and Hosford (Metall Trans 24A:2503–2512, 1993). Predicted results showed that, similar to SFLC, the ELSC remains practically unchanged for a significant range of prestrain values under various bilinear loading paths, but some strain-path dependence can be observed for significant magnitudes of the effective prestrain (ε _e ?≥?0.37 for AISI-1012 steel and ε _e ?≥?0.25 for AA-2008-T4 aluminum).