Hu-Washizu Variational Principle Research Articles

Reddy-type zig-zag model for multilayered composite plate based on the HW variational theorem

ABSTRACTReddy's higher-order theory is quite attractive, but it could not describe a zig-zag shape distribution of in-plane displacement through the thickness direction and violates the continuity of transverse shear stresses at interfaces. This is due to neglect of the zig-zag function in the in-plane displacement field. Thus, a Reddy-type higher-order zig-zag theory is developed for analysis of multilayered composite plates. The developed model differs from existing ones by two features. First, a Reddy-type zig-zag function (RZZF) satisfying the bounding surface free traction condition is constructed. By introducing the RZZF into Reddy's model, a Reddy-type higher-order zig-zag model can be obtained. Second, a functional suitable for composite plate has been presented to obtain improved transverse shear stresses by employing the three-field Hu–Washizu (HW) variational principle. It is significant that the higher-order derivatives of displacement parameters in expression of transverse shear stresses have been eliminated, which is convenient for the model's finite element implementation. Equilibrium equations and analytical solution can be also presented by means of the HW variational principle. The performance of the proposed model is tested with different numerical examples, and numerical results show its accuracy and range of applicability.

A refined three-node triangular element based on the HW variational theorem for multilayered composite plates

For the three-node triangular elements, the displacement parameters are linear variation within one element, so that the second and the third derivatives of displacement parameters are close to zero. Therefore, the three-node triangular element will encounter difficulty to accurately predict transverse shear stresses by integrating the three-dimensional equilibrium equation through the thickness direction. Thus, the few three-node triangular plate elements can accurately yield the transverse shear stresses of multilayered composite plates. A higher-order zig-zag model accounting for the zero transverse shear strain conditions at the top and the bottom surfaces is firstly presented in this work. Based on the proposed model, a refined three-node triangular plate element satisfying the requirement of C1 weak-continuity conditions in the inter-element is developed to accurately produce the distributions of displacements and stresses through the thickness of multilayered composite plates. To obtain the improved transverse shear stresses, the simplified three-dimensional equilibrium equation has been a priori used. It is significant that the higher-order derivatives of displacement parameters have been taken out from the expression of transverse shear stresses by employing the three-field Hu-Washizu (HW) variational principle, which is convenient for the finite element implementation. Performance of the present element is tested by comparing with three-dimensional elasticity solutions as well as the reference results evaluated using other models. A detailed study is conducted to highlight the influences of boundary condition, number of layers and aspect ratio on the static response of multilayered composite plates, and numerical results can show the accuracy and range of applicability for the proposed element.

A global higher-order zig-zag model in terms of the HW variational theorem for multilayered composite beams

Accurate prediction of the transverse shear stresses for multilayered composite structures is still the challenging issue for the displacement-based and the mixed equivalent single-layer models. Therefore, an equivalent single-layer model which can predict accurately transverse shear stresses of multilayered composites without using any postprocessing approaches is expected. To this end, a global higher-order zig-zag model satisfying the transverse shear traction-free condition is developed for bending analysis of laminated composite beams. The number of unknown variables in the proposed model does not depend on the number of layers, and only four variables are involved in the displacement field. To obtain the accurate transverse shear stresses, the functional suitable for beams can be obtained by using the three-field Hu-Washizu (HW) variational principle. The equilibrium equations and analytical solution of the present model can be obtained based on the HW variational equation. By analyzing the bending behaviors of composite beams, the performance of the proposed model is verified. In addition, effects of transverse shear stress on the displacements and in-plane stress of composite beams have been detailedly investigated. Differing from previous equivalent single-layer models, the transverse shear stresses can be accurately calculated from the proposed model, and actively impact on displacements and in-plane stresses of multilayered composite beams.

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

SummaryThis 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 . 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 . Numerical results from benchmarks show the robustness of the formulation in geometrically linear and nonlinear problems. Copyright © 2016 John Wiley & Sons, Ltd.

Application of the Timoshenko–Mindlin Theory to the Calculation of Nonlinear Deformation and Stability of Anisotropic Shells

The nonlinear deformation and stability of composite shells are estimated by using the Timoshenko–Mindlin theory of anisotropic shells. The resolving system of equations is presented in a mixed form in displacements, forces, and moments. For its derivation, a modified version of the generalized Hu–Washizu variational principle formulated in rates for a quasi-static problem is used. However, instead of differentiation with respect to time, displacements, stresses, and loads are assumed to depend on a parameter, for which it is advisable to take the length of the arc of equilibrium states, as demonstrated in some studies. On variation of this parameter, the shell-load system can occur either at regular or singular points. A boundary value problem is formulated in the form of a normal system of differential equations in the derivatives of displacements, forces, and moments. In the separation of variables, the Fourier series are used in a complex form. The boundary value problem is solved by the Godunov discrete orthogonalization method in the field of complex numbers. Then, the Cauchy problem is solved by using known methods. Using the methodology developed, an analysis of the influence of composite properties and parameters of the layered structures on the form of the equilibrium curves of cylindrical shells is carried out. The mechanical characteristics of the initial elementary layers of the reinforced material are determined by the micromechanics methods developed by Eshelby, Mori–Tanaka, and Vanin.

Frame Element with Mixed Formulations for Composite and RC Members with Bond Slip. I: Theory and Fixed-End Rotation

AbstractThis paper proposes a composite frame element for the simulation of the inelastic response of structural members made up of two or more components with different materials, such as reinforced concrete (RC), steel-concrete composite members, prestressed members, and members with fiber-reinforced polymer (FRP) reinforcement. The element accounts for the relative slip at the interface between the components. Nonlinear geometry effects are accounted for through the corotational formulation, which permits the response simulation of composite frame elements under large displacements. The element formulation enhances the standard Hu-Washizu variational principle with fields describing the bond-slip behavior between components. Three alternatives for the mixed formulation of the element are derived in this paper, which focuses on theory and implementation: mixed-displacement, mixed-force, and mixed-mixed. The paper presents the benefits and shortcomings of the formulation alternatives for modeling the pul...

Finite strain quadrilateral shell using least-squares fit of relative Lagrangian in-plane strains

This work presents a finite strain quadrilateral element with least-squares assumed in-plane shear strains (in covariant/contravariant coordinates) and classical transverse shear assumed strains. It is an alternative to enhanced-assumed-strain (EAS) formulation and, in contrast to this, produces an element satisfying ab initio the Patch-test. No additional degrees-of-freedom are present, unlike EAS. Least-squares fit allows the derivation of invariant finite strain elements which are both in-plane and out-of-plane shear-locking free and amenable to standardization in commercial codes. With that goal, we use automatically generated code produced by AceGen and Mathematica to obtain novel finite element formulations. The corresponding exact linearization of the internal forces was, until recently, a insurmountable task. We use the tangent modulus in the least-squares fit to ensure that stress modes are obtained from a five-parameter strain fitting. This reproduces exactly the in-plane bending modes. The discrete equations are obtained by establishing a four-field variational principle (a direct extension of the Hu–Washizu variational principle). The main achieved goal is coarse-mesh accuracy for distorted meshes, which is adequate for being used in crack propagation problems. In addition, as an alternative to spherical interpolation, a consistent director normalization is performed. Metric components are fully deduced and exact linearization of the shell element is performed. Full linear and nonlinear assessment of the element is performed, showing similar performance to more costly approaches, often on-par with the best available shell elements.

A 3D mixed frame element with multi-axial coupling for thin-walled structures with damage

A 3D mixed beam finite element is presented, modeling the warping of the cross-sections as an independent kinematic field. The beam formulation is derived on the basis of the Hu-Washizu variational principle, expressed as function of four independent fields: the standard displacements, strains and stresses and the additional warping displacement. This is interpolated along the beam axis and on the cross-section, by placing on it a regular grid of interpolation points and adopting Lagrange polynomials. The warping degrees of freedom defined at the cross-section interpolation points are condensed, thus preserving the element matrix and vector sizes. A fiber discretization of the cross-sections is adopted. The constitutive relationship at the midpoint of each fiber is based on an isotropic damage model for brittle-like materials, distinguishing between the damage variables in tension and in compression to properly describe the unilateral effect. An efficient algorithm is formulated for the element state determination, based on a consistent linearization of the governing equations. A simple numerical application on a cantilever beam with torsion in the linear elastic range is presented and two torsion tests on plain concrete beams are performed, by comparing the numerical results with the experimental outcomes.

Quadratically consistent nodal integration for second order meshfree Galerkin methods

Robust and efficient integration of the Galerkin weak form only at the approximation nodes for second order meshfree Galerkin methods is proposed. The starting point of the method is the Hu-Washizu variational principle. The orthogonality condition between stress and strain difference is satisfied by correcting nodal derivatives. The corrected nodal derivatives are essentially linear functions which can exactly reproduce linear strain fields. With the known area moments, the stiffness matrix resulting from these corrected nodal derivatives can be exactly evaluated using only the nodes as quadrature points. The proposed method can exactly pass the quadratic patch test and therefore is named as quadratically consistent nodal integration. In contrast, the stabilized conforming nodal integration (SCNI) which prevails in the nodal integrations for meshfree Galerkin methods fails to pass the quadratic patch test. Better accuracy, convergence, efficiency and stability than SCNI are demonstrated by several elastostatic and elastodynamic examples.

Variational sensitivity analysis of a nonlinear solid shell element

SUMMARYThe paper is concerned with variational sensitivity analysis of a nonlinear solid shell element, which is based on the Hu–Washizu variational principle. The sensitivity information is derived on the continuous level and discretized to yield the analytical expressions on the computational level. Especially, the pseudo load matrix and the sensitivity matrix, which dominate design sensitivity analysis of shape optimization problems, are derived. Because of the mixed formulation, condensation of the pseudo load matrix on the element level is performed to compute the sensitivity matrix. An illustrative example from the field of geometry‐based shape optimization demonstrates the possible application of the presented formulation. Copyright © 2013 John Wiley & Sons, Ltd.

Computational multiscale methods for granular materials

The fine-scale heterogeneity of granular material is characterized by its polydisperse microstructure with randomness and no periodicity. To predict the mechanical response of the material as the microstructure evolves, it is demonstrated to develop computational multiscale methods using discrete particle assembly-Cosserat continuum modeling in micro- and macro- scales, respectively. The computational homogenization method and the bridge scale method along the concurrent scale linking approach are briefly introduced. Based on the weak form of the Hu-Washizu variational principle, the mixed finite element procedure of gradient Cosserat continuum in the frame of the second-order homogenization scheme is developed. The meso-mechanically informed anisotropic damage of effective Cosserat continuum is characterized and identified and the microscopic mechanisms of macroscopic damage phenomenon are revealed.

A meshfree‐enriched finite element method for compressible and near‐incompressible elasticity

SUMMARYIn this paper, a two‐dimensional displacement‐based meshfree‐enriched FEM (ME‐FEM) is presented for the linear analysis of compressible and near‐incompressible planar elasticity. The ME‐FEM element is established by injecting a first‐order convex meshfree approximation into a low‐order finite element with an additional node. The convex meshfree approximation is constructed using the generalized meshfree approximation method and it possesses the Kronecker‐delta property on the element boundaries. The gradient matrix of ME‐FEM element satisfies the integration constraint for nodal integration and the resultant ME‐FEM formulation is shown to pass the constant stress test for the compressible media. The ME‐FEM interpolation is an element‐wise meshfree interpolation and is proven to be discrete divergence‐free in the incompressible limit. To prevent possible pressure oscillation in the near‐incompressible problems, an area‐weighted strain smoothing scheme incorporated with the divergence‐free ME‐FEM interpolation is introduced to provide the smoothing on strains and pressure. With this smoothed strain field, the discrete equations are derived based on a modified Hu–Washizu variational principle. Several numerical examples are presented to demonstrate the effectiveness of the proposed method for the compressible and near‐incompressible problems. Copyright © 2012 John Wiley & Sons, Ltd.

Meshfree-enriched simplex elements with strain smoothing for the finite element analysis of compressible and nearly incompressible solids

This paper presents a meshfree-enriched finite element formulation for triangular and tetrahedral elements in the analysis of two and three-dimensional compressible and nearly incompressible solids. The new formulation is first established in two-dimensional case by introducing a meshfree approximation into a linear triangular finite element with an enriched node. The interpolation functions of the four-noded triangular element are constructed by the meshfree convex approximations and are completed to a polynomial of degree one. The reference mapping using the constructed interpolation functions is shown to be invertible everywhere in the element and the global element area is proven to be conserved under a standard three-point integration rule. The triangular element formulation is extendable to the tetrahedral element in three-dimensional case. To provide a locking-free analysis for the nearly incompressible materials, an area-weighted strain smoothing is developed in conjunction with the enriched interpolation functions to yield a discrete divergence-free property at the integration point. The resultant element formulation with strain smoothing is shown to pass the patch test. To introduce the smoothed strain into Galerkin formulation, a modified Hu–Washizu variational principle is adopted to formulate the discrete equations. Since the Kronecker-delta property in element interpolation is held along the element boundary using meshfree convex approximation, boundary conditions can be treated in a standard way. Several numerical benchmarks are provided to demonstrate the effectiveness and accuracy of the proposed method.

A stabilized underintegrated enhanced assumed strain solid-shell element for geometrically nonlinear plate/shell analysis

In this paper, an underintegrated solid-shell element is developed in the framework of explicit finite element based on the Hu-Washizu variational principle. The reduced in-plane integration with a physical stabilization procedure scheme is adopted to improve computational efficiency. For the underintegrated element, volumetric locking is avoided with the B-bar approach, while thickness locking is eliminated by adopting only one enhancing parameter. In the previous literatures, the assumed natural strain method or artificial stabilization parameters are usually employed to remedy transverse shell locking for the underintegrated element, here a six-parameter enhanced assumed strain method is used, leading to a more straightforward formulation. Several standard benchmark examples including linear and nonlinear problems are calculated. The solid-shell element is proved to be accurate in resolving plate/shell problems. At the end, an in-house program that incorporates the solid-shell element into the explicit finite element method is developed. An S-rail forming process from the NUMISHEET conference is analyzed. The final results are in good accordance with the experiment.

Multiscale modeling of elasto-viscoplastic polycrystals subjected to finite deformations

In the present work, the elasto-viscoplastic behavior, interactions between grains, and the texture evolution in polycrystalline materials subjected to finite deformations are modeled using a multiscale analysis procedure within a finite element framework. Computational homogenization is used to relate the grain (meso) scale to the macroscale. Specifically, a polycrystal is modeled by a material representative volume element (RVE) consisting of an aggregate of grains, and a periodic distribution of such unit cells is considered to describe material behavior locally on the macroscale. The elastic behavior is defined by a hyperelastic potential, and the viscoplastic response is modeled by a simple power law complemented by a work hardening equation. The finite element framework is based on a Lagrangian formulation, where a kinematic split of the deformation gradient into volume preserving and volumetric parts together with a three-field form of the Hu-Washizu variational principle is adopted to create a stable finite element method. Examples involving simple deformations of an aluminum alloy are modeled to predict inhomogeneous fields on the grain scale, and the macroscopic effective stress-strain curve and texture evolution are compared to those obtained using both upper and lower bound models.

Comparison of Three-Dimensional Flexible Beam Elements for Dynamic Analysis: Classical Finite Element Formulation and Absolute Nodal Coordinate Formulation

Three formulations for a flexible spatial beam element for dynamic analysis are compared: a Timoshenko beam with large displacements and rotations, a fully parametrized element according to the absolute nodal coordinate formulation (ANCF), and an ANCF element based on an elastic line approach. In the last formulation, the shear locking of the antisymmetric bending mode is avoided by the application of either the two-field Hellinger–Reissner or the three-field Hu–Washizu variational principle. The comparison is made by means of linear static deflection and eigenfrequency analyses on stylized problems. It is shown that the ANCF fully parametrized element yields too large torsional and flexural rigidities, and shear locking effectively suppresses the antisymmetric bending mode. The presented ANCF formulation with the elastic line approach resolves most of these problems.

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

AbstractIn 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.

Modeling of inelastic behavior of curved members with a mixed formulation beam element

The curved beam element in this paper is based on Hu-Washizu variational principle. The nonlinear response of the element arises from the integration of stress–strain relations over several control sections along the element length. The finite element approximation for the beam uses shape functions for stress resultants that satisfy equilibrium and discontinuous strains along the beam. No approximation for the beam displacement field is necessary in the formulation. The proposed element is free from membrane and shear locking. Examples verify the superior performance of the model under linear and nonlinear material conditions.

The Hu–Washizu variational principle for the identification of imperfections in beams

AbstractThis paper presents a procedure for the identification of imperfections of structural parameters based on displacement measurements by static tests. The proposed procedure is based on the well‐known Hu–Washizu variational principle, suitably modified to account for the response measurements, which is able to provide closed‐form solutions to some inverse problems for the identification of structural parameter imperfections in beams. Copyright © 2008 John Wiley & Sons, Ltd.

Certain Hygrothermopiezoelectric Multi-Field Variational Principles for Smart Elastic Laminae

In this study Hamilton's principle is stated for the motion and deformation of a regular region of hygrothermopiezoelectric materials in the elastic range, and a four-field variational principle is obtained under the quasi-static approximation of the electric field. The variational principle is modified through an involuntary transformation and a generalized variational principle is derived in analogy to the Hu-Washizu variational principle of elasticity. Likewise, a variational principle is formulated for the region with a fixed internal surface of discontinuity, and another one for the region containing any number of perfectly bonded dissimilar materials. The variational principles are shown to generate, as its Euler-Lagrange equations, all the divergence and gradient equations, the constitutive relations and the natural mixed boundary and continuity (jump) conditions for each region of the dissimilar materials. The admissible states of the variational principles have Cauchy's second law of motion and the initial conditions for each region, as its constraint conditions. Certain special cases of the variational principles operating on all the field variables and their reciprocals are indicated. The multi-field variational principles with their well-known features are applicable to an analysis of the physical response of smart laminae under the coupled piezoelectric and hygrothermal effects.