Incompatible Displacement Research Articles

Thermal-elastic analysis of laminated plates based on the incompatible generalized partial mixed method

Based on the generalized Hellinger-Reissner (H-R) mixed variational principle, the incompatible generalized partial mixed finite element method (IGPMFEM) is proposed in this paper for the thermomechanical stress analysis of laminated composites and sandwich plates. This method inherits the advantages of the generalized mixed element method, allowing for the simultaneous introduction of displacement and stress boundary conditions, and satisfies the discontinuity of the in-plane stresses at the material interface. Introduction of incompatible displacement mode adds higher-order terms in the polynomial expansion of the exact displacements, effectively improving the bending performance and the element accuracy. Although adding stress components would normally increase the computational effort, we address this issue by the elimination method, considering only the out-of-plane stress variables to reduce the matrix dimensions. The accuracy and effectiveness of the proposed method are evaluated by comparing the numerical results of this method with existing elasticity solutions, while the effect of stress boundary conditions is also investigated.

Two 8‐node quadrilateral unsymmetric elements with different incompatible modes immune to severe distortion

AbstractMost unsymmetric quadrilateral elements have demonstrated a high tolerance to mesh distortion in pure bending problems involving coarse meshes. However, they have difficulty achieving acceptable results in shearing‐dominated problems, and the element's performance is affected by the degree of mesh distortion. In this paper, two 8‐node quadrilateral elements are developed by introducing two different incompatible modes into the unsymmetric formulation. The unsymmetric framework with two different sets of interpolation functions for displacement fields provides the basis for the immunity property. The incompatible displacement mode and the enhanced assumed strain guarantee the completeness of the displacement field interpolation polynomials, and the satisfaction of ‐orthogonality condition ensures the accuracy of the simulations. The resulting elements, respectively, denoted as UQ10 and UQ10/E4, can provide exact solutions for both pure and linear bending problems and exhibit excellent insensitivity in shear‐dominated tests. Furthermore, they are invariant, rapidly converging, and free of various locking problems.

On the higher-order pseudo-continuum characterization of discrete kinematic results from experimental measurement or discrete simulation

The purpose of this work is the development and determination of higher-order continuum-like kinematic measures which characterize discrete kinematic data obtained from experimental measurement (e.g., digital image correlation) or kinematic results from discrete modeling and simulation (e.g., molecular statics, molecular dynamics, or quantum DFT). From a continuum point of view, such data or results are in general non-affine and incompatible, due for example to shear banding, material defects, or microstructure. To characterize such information in a (pseudo-) continuum fashion, the concept of discrete local deformation is introduced and exploited. The corresponding measures are determined in a purely discrete fashion independent of any relation to continuum fields. Demonstration and verification of the approach is carried out with the help of example applications based on non-affine and incompatible displacement information. In particular, for the latter case, molecular statics results for the displacement of atoms in and around a dislocation core in fcc Au are employed. The corresponding characterization of lattice distortion in and around the core in terms of higher-order discrete local deformation measures clearly shows that, even in the simplest case of planar cores, such distortion is only partly characterized by the Nye tensor.

Refined Vehicle‐Bridge Interaction Analysis Using Incompatible Solid Finite Element for Evaluating Stresses and Impact Factors

The vehicle‐bridge interaction can induce bridge vibration and consequently fatigue, durability deterioration, local damage, and even collapse of bridge structure. In this paper, a solid vehicle‐bridge interaction (VBI) analysis method is developed to provide refined analysis on the bridge responses including displacement and local stress under vehicle loads. The incompatible solid finite element (FE) is introduced to model the bridge, where the element shear locking is alleviated by incompatible displacement modes without sacrificing the computational efficiency. Benchmark example shows the incompatible solid element has superior computational efficiency compared to the conventional solid element. By virtue of the mass‐spring‐damper vehicle model, the interaction between vehicle and bridge is simulated with point‐to‐point contact assumption and the coupled dynamic equations are solved via nonlinear iteration. A case study on a simply supported T‐girder bridge is conducted to validate the developed solid VBI analysis method and then the dynamic impact factor (DIF) of the bridge is evaluated based on the computed stress results and compared to code values. Results show that the solid VBI analysis method yields more accurate time‐history bridge responses including displacement and stress under moving vehicles than the grillage method despite higher computational cost. Particularly, it can simulate realistic stress distribution and concentration along any concerned sections as well as in local components, which can provide detail information on the bridge behavior under dynamic loads. On the other hand, the DIF based on the computed stress result generally agrees well with the code values except for heavy vehicles where the stress‐based DIF is slightly higher than the value in Chinese code while lower than that of AASHTO, suggesting the value specified by Chinese code may underestimate the DIF of heavy vehicles in certain circumstances to which more attention should be paid.

Development of 3D Finite Element Code of Incompatible Displacement Mode for Flexural Analysis

For three-dimensional finite element modeling, it is necessary to use a solid element taking into account the flexural response and also should be simple for computer storage. One of the elements that can attain the previous requirements is the incompatible (non-conforming) element. In this paper, formulation and algorithm of the developed 3D finite element code of incompatible displacement mode for flexural analysis are presented using 3D 8-noded isoparametric incompatible brick element. Several comparisons and a parametric study have been carried out to verify the incompatible brick element and the developed code. Accurate results have been obtained by the 3D incompatible element in flexural analysis of beams in comparison with the standard element.

Transition finite element families for adaptive analysis of axisymmetric elasticity problems

In this paper, four transition element families that comprise five- to seven-node quadrilateral elements are developed based on the hybrid-stress and enhanced assumed strain (EAS) formulations for adaptive analyses of axisymmetric elasticity problems. For members in the first hybrid-stress family, a quasi-linear stress field with ten equilibrating stress modes is derived and employed. To study the effect of including more stress modes in the assumed stress field, another family with two additional stress modes is implemented. On the other hand, two EAS element families are constructed with respect to the incompatible displacement modes of two existing incompatible displacement transition element families. Several numerical examples are exercised. It can be seen that the first hybrid-stress family is the most accurate one among the proposed families. Moreover, the EAS families are close to the respective incompatible families in accuracy yet the former families are not only more efficient in computation but also more concise in formulation.

A hybrid-stress element based on Hamilton principle

A novel hybrid-stress finite element method is proposed for constructing simple 4-node quadrilateral plane elements, and the new element is denoted as HH4-3β here. Firstly, the theoretical basis of the traditional hybrid-stress elements, i.e., the Hellinger–Reissner variational principle, is replaced by the Hamilton variational principle, in which the number of the stress variables is reduced from 3 to 2. Secondly, three stress parameters and corresponding trial functions are introduced into the system equations. Thirdly, the displacement fields of the conventional bilinear isoparametric element are employed in the new models. Finally, from the stationary condition, the stress parameters can be expressed in terms of the displacement parameters, and thus the new element stiffness matrices can be obtained. Since the required number of stress variables in the Hamilton variational principle is less than that in the Hellinger–Reissner variational principle, and no additional incompatible displacement modes are considered, the new hybrid-stress element is simpler than the traditional ones. Furthermore, in order to improve the accuracy of the stress solutions, two enhanced post-processing schemes are also proposed for element HH4-3β. Numerical examples show that the proposed model exhibits great improvements in both displacement and stress solutions, implying that the proposed technique is an effective way for developing simple finite element models with high performance.

Combined hybrid method applied in the Reissner–Mindlin plate model

In this paper, the combined hybrid method is applied to the Reissner–Mindlin plate model and a corresponding variational formulation is presented. Based on this combined hybrid variational formulation, we introduce the Wilson incompatible displacement mode, assumed moment modes and two types of assumed shear stress to design two quadrilateral finite elements. A series of standard numerical tests show that two new elements are free of locking and yield high performance on the coarse mesh.

Research of combined hybrid method applied in the Reissner–Mindlin plate model

This paper has introduced two kinds of combined hybrid variational formulations for plate bending finite elements based upon the Reissner–Mindlin theory, according to whether assumed constant moment stress is introduced when assumed constant shear stress has been introduced. Due to two options for incompatible displacement modes, four new types of combined hybrid elements are proposed. A set of standard tests for both thin and thick plates show that new elements perform well.

Adaptive mesh refinement FEM for damage evolution of heterogeneous brittle media

Damage evolution of heterogeneous brittle media involves a wide range of length scales. The coupling between these length scales is the underlying mechanism of damage evolution and rupture. However, few of previous numerical algorithms consider the effects of the trans-scale coupling effectively. In this paper, an adaptive mesh refinement finite element method (FEM) algorithm is developed to simulate this trans-scale coupling. The adaptive serendipity element is implemented in this algorithm, and several special discontinuous base functions are created to avoid the incompatible displacement between the elements. Both the benchmark and a typical numerical example under quasi-static loading are given to justify the effectiveness of this model. The numerical results reproduce a series of characteristics of damage and rupture in heterogeneous brittle media.

Micromechanics of braided composites via multivariable FEM

A new method for micromechanical properties of three-dimensionally braided composite materials via homogenization theory and incompatible multivariable FEM is proposed in this paper. An incompatible displacement element and a hybrid stress element are developed to model the effective mechanical properties of three-dimensional textile composites. Some illustrative applications are presented for a typical class of four-step braided composites. Results of the hybrid stress element approach compare more favorably with the experimental data than other numerical methods widely used.

Author's Reply to Discussion by George Mylonakis and Costis Syngros of ‘Response spectrum of incompatible acceleration, velocity and displacement histories’ Earthquake Engng Struct. Dyn. 2001; 30(2):279–286

Author's Reply to Discussion by George Mylonakis and Costis Syngros of ‘Response spectrum of incompatible acceleration, velocity and displacement histories’ Earthquake Engng Struct. Dyn. 2001; 30(2):279–286

Response spectrum of incompatible acceleration, velocity and displacement histories by Praveen K. Malhotra, Earthquake Engng Struct. Dyn. 2001; 30(2):279–286

Response spectrum of incompatible acceleration, velocity and displacement histories by Praveen K. Malhotra, Earthquake Engng Struct. Dyn. 2001; 30(2):279–286

Effective Hybrid/Mixed Finite Elements for Folded-Plate Structures

A new formulation in which the assumed stress field is divided into membrane and bending stress fields is developed. A family of effective shell elements of hybrid/mixed finite element method that accounts for transverse shear deformations is proposed based on a modified Hellinger-Reissner variational principle. In this paper, the present shell elements possess different three-field assumptions, which are described by compatible displacement, incompatible displacement, and stress fields. Several benchmark problems are conducted to test both the accuracy and reliability of the proposed shell elements. Numerical analyses show that the present shell elements pass all the patch tests, do not contain any spurious zero energy modes, and have no lock phenomena even for very thin plate and shell. Further insight into the performance of the present shell elements is gained to demonstrate the high accuracy of the elements by several folded-plate structure problems. The numerical results demonstrate the validity, reliability, and applicability of the present elements. The present elements are valid for thick as well as thin shell structures.

Response spectrum of incompatible acceleration, velocity and displacement histories by Praveen K. Malhotra, Earthquake Engng Struct. Dyn. 2001; 30(2):279–286

The Discussers would like to congratulate the Author for his contribution to the analysis of response spectra. As mentioned in the Original Paper, engineers often pay little attention to incompatibilities in the histories of ground acceleration, displacement, and velocity. As a result, computed response spectra may exhibit erroneous trends at short and long periods. Different techniques have been proposed to treat the problem, focusing primarily on the initial conditions of the oscillator [5,6,8,9,10,11]. The Author proposes an alternative technique: determining response spectra by combining the incompatible time histories using a 'mixing rule' (Equations (6)-(8) in the Original Paper, which allows for the proper asymptotic behaviour to be obtained at extreme periods. The scope of this discussion is three-fold: (i) to present a simplified version of the method, by applying the Author's idea only to initial conditions; (ii) to discuss a more general mixing rule; (iii) to propose alternative ways for estimating the effective frequency of ground motion. These items are presented to further validate the results provided by the Author and offer additional insight into the derivations presented in the Original Paper.

Author's Reply to Discussion by George Mylonakis and Costis Syngros of ‘Response spectrum of incompatible acceleration, velocity and displacement histories’ Earthquake Engng Struct. Dyn. 2001; 30(2):279–286

Author's Reply to Discussion by George Mylonakis and Costis Syngros of ‘Response spectrum of incompatible acceleration, velocity and displacement histories’ <i>Earthquake Engng Struct. Dyn.</i> 2001; <b>30</b>(2):279–286

A study of three-dimensional four-step braided piezo-ceramic composites by the homogenization method

Piezoelectric media are intrinsically anisotropic. The electric field and the elastic field are coupled in this type of material, which renders the mathematical or theoretical analysis rather difficult. In this paper, we develop an incompatible displacement finite-element method (FEM) using homogenization theory to examine the effective elastic, piezoelectric, and dielectric properties of three-dimensional four-step piezoelectric braided composites. The variation of the mechanical and electric properties as a function of the fiber volume fraction has been obtained for different numerical tests. The present study provides a useful local/global method for the analysis of 3-D braided composites.

A further study on incompatible models: revise-stiffness approach and completeness of trial functions

The paper concerns the incompatible finite element methodology. First, the patch test condition (PTC) for axisymmetric problems is studied. For the implementation of PTC, a post-treatment approach, the revise-stiffness (R-s) method is suggested to construct incompatible elements. A quadratic-completeness incompatible displacement mode with the fewest element parameters is formulated. In terms of the R-s approach and the general revise-displacement (R-u) method, a set of four-node incompatible elements is constructed for plane and axisymmetric problems. Numerical tests show that the suggested approaches in this paper can be used to produce various incompatible models with high performance.

Micromechanics of composite materials using multivariable finite element method and homogenization theory

A new method for evaluation of the micro-mechanical properties of composite materials via incompatible multivariable FEM and homogenization theory is proposed in this paper. An incompatible displacement element and a hybrid stress element are developed and extended to predict the effective mechanical properties of composite materials. The mechanical performances obtained by these elements are compared with the experimental data as well as the results by Mori–Tanaka theory and traditional rule-of-mixtures. It turns out that the proposed new method provides with the best results comparing to the experimental results.

Response spectrum of incompatible acceleration, velocity and displacement histories

Response spectrum of incompatible acceleration, velocity and displacement histories