Mixed Variational Principle Research Articles

On some energy‐based variational principles in non‐dissipative magneto‐mechanics using a vector potential approach

AbstractThis contribution covers the variational‐based modeling of non‐dissipative magneto‐mechanical systems using a vector potential approach and the thorough analysis and discussion of corresponding conforming finite element methods. Since the construction of divergence‐free finite element spaces explicitly enforcing the Coulomb gauge poses some major challenges, we propose some primal and mixed variational principles that ensure well posedness of the problem and allow to seek the vector potential in unconstrained function spaces. The performance of these methods is assessed in two comparative benchmark studies. The focus of both studies lies on the accurate approximation of field quantities in systems with material discontinuities and re‐entrant corners.

A modified quasi-3D theory and mixed beam element method for static behaviour analysis of functionally graded beams

This paper develops a modified quasi-3D theory and the mixed beam element model for static analysis of functionally graded beams. The modified quasi-3D theory encompasses two innovations. Firstly, the accurate distribution of transverse shear stress is derived from the differential equilibrium equation that describes the relationship between stresses, rather than the traditional one derived from geometric relations and constitutive equations. Secondly, the effect of distributed loads associated with transverse stretching deformation, a factor typically overlooked in the existing quasi-3D theory-based beam models, is involved. In the development of beam finite element model, the mixed variational principle is firstly utilized to formulate the model of a quasi-3D theory-based beam element, where generalized displacements and internal forces are treated as two types of independent field quantities. Two numerical examples are conducted to investigate the validity and accuracy of the proposed theory and beam element. Numerical results indicate that the proposed mixed beam element can accurately predict the stress distributions over the cross-section and produce accurate displacement solutions. Meanwhile, it is noted that the effect of distributed loads related to the transverse stretching deformation should be correctly considered in the analysis of functionally graded beams based on quasi-3D theory.

Thermal buckling analysis of nano composite laminated and sandwich beams based on a refined nonlocal zigzag model

A refined nonlocal zigzag model for thermal buckling analysis of nano composite laminated and sandwich beams is proposed in this study based on a refined zigzag theory and Eringen’s nonlocal theory. Firstly, present model satisfies the stress-free and continuity conditions a priori by introducing the piecewise-linear zigzag functions and a preprocessing, such that the transverse shear correction factors are not needed. In the preprocessing, accurate and continuous transverse shear stresses are obtained with the aid of the general mixed variational principle, which can be solved simultaneously with other stresses in the governing equations. This is quite different from the previous post-processing. Subsequently, thermal buckling problems of nano composite laminated and sandwich beams are analytically solved in simply supported boundary conditions. The degenerated results without small effect indicate that the non-dimensional critical loads and critical temperatures have a good agreement with the 3D elasticity solutions and previous results, which demonstrate the accuracy and reliability of present model. Moreover, it is observed that the small effect of the critical temperatures can be effectively captured by Eringen’s differential constitutive law (EDCL), which shows the small effect decreases the critical temperature by weakening the stiffness of the beam. Finally, the effects of different thermal expansion coefficients, laminations, geometric sizes and beam theories are discussed. The results show that present model is robust in the arbitrary layouts for both of composite and sandwich structures, which may have some referential significance to Micro-Electro-Mechanical Systems (MEMS) sensors and actuators.

Rational generalized mixed finite element method for 2D linear elastic problems

Combining the ideas of rational displacement finite element and generalized mixed finite element, the rational generalized mixed element with 5 nodes is proposed. The finite formulations for the proposed element are derived on the basis of the generalized mixed variational principle, and the two field variables of displacement and stress can be solved simultaneously. Fundamental solutions of planar linear elasticity are adopted as the displacement interpolation function, while polynomials satisfying the equilibrium equation are adopted as the stress interpolation function. Both the displacement and stress boundary conditions can be introduced directly at same time in the proposed method, which contributes to being a more rational model. The solution technique is utilized to analyze a couple of representative static problems, and the results show the properties of excellent accuracy and good convergence. In addition, it is found to be evident for improvement of the accuracy when the stress boundary conditions are applied.

An energy-balanced method for determining the optimized parameter of the incompatible generalized mixed element

A novel method for determining the optimized parameter of the four-node incompatible generalized mixed element is presented based on the equilibrium between strain energy and complementary energy. The presented energy formulations are derived from the generalized mixed variational principle, which contains an arbitrary additional parameter. The initial solutions expressed by the displacement field are firstly assumed for the description of the energy of each generalized mixed element. Then, the identical relationship between strain energy and complementary energy is subsequently expressed at element level, which includes the arbitrary parameter. At the same time, a formulation for determining the optimized parameter at element level is proposed. Several representative examples with varying geometrical parameters, boundary and loading conditions are used to validate this method. By contrasting with the results of generalized mixed elements with different parameter values and other traditional finite elements. The effectiveness of the presented method has been demonstrated. On one hand, by ensuring the strain energy and complementary energy remain consistent under both coarse and fine meshes, the optimized parameter can adjust the stiffness of the generalized mixed element, thereby enhancing its resemblance to the real elastic body. On the other hand, the generalized mixed element has the additional advantage of conveniently introducing stress boundary conditions, thereby satisfying the requirement for zero-conditions of shear stresses on the exterior surfaces of beams. The numerical results obtained by the proposed method are accurate and stable.

Mixed variational principle for shape memory solids

Mixed variational principle for shape memory solids

Variational free energy based macroscopical modeling of ferroelectroelasticity

In this paper, a thermodynamically consistent minimum-type variational model for ferroelectric materials in a macroscopical continuum approach is presented. The motivation for this results from the lack of models in the literature that have on the one hand a Helmholtz free energy based variational structure and on the other hand are able to represent all important characteristic phenomena of ferroelectrics under quasi-static conditions. First of all, a unified variational theory for the material response of dissipative electro-mechanical solids in line with the framework of the generalized standard materials (GSM) is outlined. A macroscopic ferroelectric model with microscopically motivated internal state variables representing the switching processes taking place at the material microscale is adapted to the above mentioned variational structure. Additionally, a mixed variational principle for the global electro-mechanical boundary value problem is introduced in order to embed the Helmholtz free energy based local theory in a suitable finite element formulation. The solution processes for the resulting local and global variational problems is described in detail to enable easy implementation. The capability of the presented methods to reproduce the real behavior of ferroelectric systems is demonstrated by numerical examples. Here, a comparison to experimental results from the literature is a particular focus.

An implicit 3D nodal integration based PFEM (N-PFEM) of natural temporal stability for dynamic analysis of granular flow and landslide problems

The particle finite element method (PFEM) is a robust approach for modelling large deformation problems with free surface evolution. The classical PFEM, however, requires variable mapping from old to new quadrature points when adopting history-dependent material models in granular flow and landslide problems. Although the nodal integration technique circumvents this issue, it makes the PFEM temporal instable in dynamic analysis when using a displacement-based formulation. In this study, we developed a new version of a three-dimensional (3D) Nodal integration based PFEM (N-PFEM) using a mixed variational principle with the final problem resolved in mathematical programming. The proposed N-PFEM not only inherits the benefit from the nodal integration scheme that no variable mapping is required for handling history-dependent models but also is naturally temporal stable requiring no ad-hoc stabilization technique. We simulated a series of benchmark problems to demonstrate its nature of temporal stability as well as other admirable features such as the volumetric-locking free property and capability for tackling extreme configuration changes. Additionally, its application to a 3D landslide with a sensitive clay layer is shown to highlight its robustness.

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.

Incorporation of thermal effects into the energy‐momentum consistent elastic model for fiber‐bending stiffness in composites

AbstractOwing to the wide acceptance of fiber reinforced composites in the lightweight structure industry, the need to study its thermo‐elastic behavior in dynamical multi‐body systems still remains an open question. Standard continuum models, by virtue of first‐order gradient of deformation, cannot capture curvature effects and hence appropriate extensions of the standard modeling techniques are indispensable to model the accurate physical stiffness. The fiber‐bending curvature are captured by introducing a higher‐order gradient of the deformation as an independent field in the standard continuum. This work explores in detail, a further extension of this continuum to understand the thermal effects on the bending stiffness of fiber bundle in fiber‐reinforced composites. Our proposed model incorporates the temperature of the composite and its bidirectional dependency with the higher‐order gradients of the deformation into the extended mechanical continuum model. The implementation of the model is pursued in the context of multi‐field mixed finite element method and dynamic variational principle. The principle of virtual power enables our extended non‐isothermal continuum to derive an energy‐momentum scheme of higher order. The resulting time‐integrator exhibits physically consistent locking‐ free numerical behavior in dynamical simulations. Our novel approach is illustrated by simple transient dynamic simulations with a hyperelastic, transversely isotropic, polyconvex material behavior. We investigate the conservation of total energy and total momentum in the framework of thermo‐elasticity and study the spatial and temporal convergence for long‐term dynamic simulations with coarser grids and increased accurate numerical solutions.

Three-dimensional analytical elasticity solution for the mechanical analysis of arbitrarily-supported, cross and angle-ply composite plates under patch loads

Patch-type discontinuous loads lead to highly localized stresses which are three-dimensional (3D) in nature and can initiate failures. Their accurate characterization is of primal importance for composite and sandwich structures. The present contribution elaborates a 3D analytical elasticity solution for the mechanical analysis of cross and angle-ply composite plates under transverse patch loads and arbitrary support conditions. For the solution, the Reissner-type, mixed variational principle is used to derive the governing equations. Analytical solutions are obtained using the multiterm, extended Kantorovich method, by dividing the plate into loaded and unloaded parts. The present analytical solution exactly satisfies inter-segment, inter-layer continuity and edge support conditions, treating stresses and displacements as primal variables. Its accuracy is established by direct comparisons with existing analytical – where available – and 3D finite element results, both for cross-ply and angle-ply designs, for different boundary conditions. Extensive analysis is performed to quantify the effect of thickness, angle-ply and patch load position on the composite mechanical performance. Through thickness stress asymmetries are identified and quantified, along with angle-ply designs that can smoothen stress concentrations in the vicinity of patch loads. The methodology can be used as a reference solution in the design of multilayer anisotropic plates under discontinuous loads.

A mixed finite element method for large deformation of flexoelectric materials

Flexoelectricity is a universal and size-dependent electromechanical phenomenon coupling the electric field with strain gradient in dielectric materials. we establishe a mixed variational principle and develop a mixed finite element method (FEM). This allows us to use C0-continuous interpolation functions in FEM instead of C1 that is difficult to construct but has to be used in conventional FEM to investigate flexoelectricity. A good agreement between the analytical solutions and our numerical results of an infinite cylinder problem validates the present developed mixed FEM. We further investigate the effects of geometrically nonlinearity on flexoelectricity in soft dielectrics under large deformation. It is found that the flexoelectricity can be greatly enhanced due to the large deformation of soft dielectrics even with much smaller flexoelectric coefficients than popular ceramics. We also demonstrate the necessarity of adopting finite strain model or considering geometrical nonlinearity to precisely describing flexoelectric response in soft dielectrics. The findings in this work also suggest tremendous potential application of soft materials in the field of electromechanical sensors via flexoelectricity.

Two-dimensional model for composite beams and its state-space based mixed finite element solution by DQM

Most one-dimensional composite beam theories are based on simplified shear deformation assumptions, or even ignore shear deformation, which have limitations in analyzing multilayer beams with significant material differences. Therefore, this paper firstly proposed a two-dimensional analytical model for composite beams through the equivalent transformation of cross section. Based on the mixed variational principle, the state equations are then derived by finite element meshing and interpolation along the length of the beam, with nodal displacements and their energy-conjugated stresses as state variables. Subsequently, the differential quadrature method (DQM) is introduced to solve the equations and a two-dimensional analysis method for composite beams is established. Due to the use of displacements and stresses as fundamental variables in the state equations, various transfer characteristics of displacements and stresses at the interlayer interfaces of composite beams can be conveniently handled. This method is finally verified by numerical examples of steel-concrete composite beams and concrete beams with corrugated steel webs. Since no assumptions of displacement and stress distributions along the thickness of beams are required, the present method can provide benchmarks for various one-dimensional beam theories.

Mixed variational principle for micropolar elasticity and an accurate two-dimensional plate model

A mixed variational principle for the micropolar elasticity theory is established in which the displacements, microrotations, force stresses and couple stresses are considered as independent fields. The presented mixed variational principle not only collects all the intrinsic features of the model in one statement, but, it helps to overcome some inconsistencies between introduced simple kinematic assumptions and the boundary conditions. The mixed variational statement is used to present improved 2D model for micropolar plates. A simple displacement vector accounting for shear deformation and normal stain effects is introduced, as well as, microrotations are taken varying linearly through the plate thickness. Furthermore, force stresses and couple stresses are assumed in complete consistent with the loading conditions on the external faces of the plates. Bending and free vibration applications are presented for micropolar homogenous and functionally graded (FG) plates with different edges conditions. The validity of the presented model is examined by comparing some present numerical results with other published results due to 2D and 3D micropolar plate models. Numerical and graphical analyses are presented to demonstrate the importance of the normal strain and microrotation effects on the bending and free vibration responses of the micropolar plates.

A mixed method for 3D nonlinear elasticity using finite element exterior calculus

AbstractIn this study, a mixed finite element model for 3D nonlinear elasticity using a Hu–Washizu (HW) type variational principle is presented. This mixed variational principle takes the deformed configuration and sections from its cotangent bundle as the input arguments. The critical points of the proposed HW functional enforce compatibility of these sections with the configuration, in addition to mechanical equilibrium and constitutive relations. Using this variational principle we construct a mixed FE approximation that distinguishes a vector from a 1‐form, a feature not commonly found in FE approximations for nonlinear elasticity. This distinction plays a pivotal role in identifying suitable FE spaces for approximating the 1‐forms appearing in the variational principle. These discrete approximations are constructed using ideas borrowed from finite element exterior calculus, which are in turn used to construct a discrete approximation to our HW functional. The discrete equations describing mechanical equilibrium, compatibility, and constitutive rule, are obtained by seeking extremum of the discrete functional with respect to the respective degrees of freedom. The discrete extremum problem is then solved numerically; we use Newton's method for this purpose. This mixed FE technique is then applied to a few benchmark problems wherein conventional displacement based approximations encounter locking and checker boarding. These studies help establish that our mixed FE approximation, which requires no artificial stabilizing terms, is free of these numerical bottlenecks.

A novel four-field mixed FE approximation for Kirchhoff rods using Cartan’s moving frames

A four-field mixed variational principle is proposed for large deformation analysis of Kirchhoff rods with the lowest order C0 mixed FE approximations. The core idea behind the approach is to introduce a one-parameter family of points (the centerline) and a separate one-parameter family of orthonormal frames (the Cartan moving frame) that are specified independently. The curvature and torsion of the curve are then related to the relative rotation of neighboring frames. The relationship between the frame and the centerline is then enforced at the solution step using a Lagrange multiplier (which plays the role of section force). In case of Kirchhoff rods, the cross sectional orientation can be described using frames like the Frenet–Serret, which are defined only using the centerline, thereby demanding higher-order smoothness for the centerline approximation. Decoupling the frame from the position vector of the base curve leads to a description of torsion and curvature that is independent of the position information, thus allowing for simpler interpolations. The four-field mixed variational principle we propose has the frame, section force, extension strain and position vector as input arguments. We discretize the position vector using linear Lagrange shape functions, while the frames are interpolated as piecewise geodesics on the rotation group. Finite element approximations for extensional strain and section force are constructed using constant shape functions. Using these discrete approximations, a discrete mixed variational principle is laid out which is then numerically extremized. The discrete approximation is then applied to a few benchmark problems. Vis-á-vis most available approaches, our numerical studies reveal an impressive performance of the proposed method without numerical instabilities or locking.

Analytical elasticity solution for accurate prediction of localized stresses in laminated composites under patch loading

Discontinuous patch-type loads cause highly localized stresses which can initiate failures like crack and delamination, and thus must be predicted accurately for efficient design of structures. This paper presents an iterative analytical elasticity solution for accurate stress analysis of laminated composite beams with arbitrary support conditions under a transverse patch loading. The beam is divided into segments with and without the patch loading. The governing equations are derived in the mixed form by employing the Reissner-type mixed variational principle and solved analytically for each segment using the multiterm extended Kantorovich approach. As both displacements and stresses are primary variables, the solution ensures the point-wise satisfaction of inter-layer and inter-segment continuity and boundary conditions. The accuracy of the solution is established by comparing the present numerical results with exact elasticity solutions available in literature for simply-supported edge conditions and compared with finite element solutions of Abaqus for other boundary conditions. An extensive numerical study is performed, and benchmark results are provided for various composite and sandwich laminates, thickness ratios, and boundary conditions under different cases of distributed patch load. The behavior of laminated metal-composite beams is also studied for patch loading, which reveals that thin aluminum layers on the composite beam are highly efficient and effective in reducing the stress concentration in the vicinity of patch load. The presented benchmark results will be beneficial to assess the accuracy and efficacy of different beam theories and elasticity-based approximate models for the analysis of multilayered anisotropic beams subjected to discontinuous loads.

A mixed variational principle in nonlinear elasticity using Cartan’s moving frames and implementation with finite element exterior calculus

This article is an attempt at offering a new perspective for the mechanics of solids using Cartan’s moving frames, specifically discussing a mixed variational principle in non-linear elasticity. We treat quantities defined on the co-tangent bundles of reference and deformed configurations as primary unknowns along with deformation. Such a treatment invites compatibility of the fields (defined on the co-tangent bundle) with the base-space (configurations of the solid body) so that the configuration can be realized as a subset of the Euclidean space. Using the moving frame, we rewrite the metric and connection through differential forms. These quantities are further utilized to write the deformation gradient and Cauchy–Green deformation tensor in terms of frame and co-frame fields. The geometric understanding of stress as a co-vector valued 2-form fits squarely within our overall program. We also show that, for a hyperelastic solid, an equation similar to the Doyle–Ericksen formula may be written for the co-vector part of the stress 2-form. Using this kinetic and kinematic understanding, we rewrite a mixed functional in terms of differential forms, whose extremum leads to the compatibility of deformation, constitutive relations, and equations of equilibrium. Finite element exterior calculus is then utilized to construct a finite-dimensional approximation for the differential forms appearing in the variational principle. These approximations are then used to construct a discrete functional which can be numerically extremized. The discretization leads to a mixed model as it involves independent approximations of differential forms related to stress and deformation gradient. The mixed variational principle is then specialized for the 2D case, whose discrete approximation is applied to problems in nonlinear elasticity. From the numerical study, it is found that the present discretization does not suffer from locking and related convergence issues.

Thermo-electro-mechanical dynamic fracture analysis on laminated piezoelectric plates

A thermo-electro-mechanical extended layerwise method (TEM-XLWM) is established for the multi-physical dynamic problems of piezoelectric laminates with delaminations or/and transverse cracks. The proposed method is deduced from the thermo-electro-mechanical mixed variational principle. The strong discontinuous functions and classical the extended finite element method (XFEM) are used to model delaminations and the transverse cracks, respectively. The Newmark integration algorithm and Crank–Nicolson scheme are combined to solve the transient multi-physical fields simultaneously. In the numerical example analysis, the proposed dynamic fracture analysis method is verified by a three-dimensional model. The laminated piezoelectric plates with delamination and/or transverse cracks are analyzed and the strain energy release rate (SERR) is calculated based on the virtual crack closure technique (VCCT).

Second-order phase-field formulations for anisotropic brittle fracture

We address brittle fracture in anisotropic materials featuring two-fold and four-fold symmetric fracture toughness. For these two classes, we develop two variational phase-field models based on the family of regularizations proposed by Focardi (2001), for which Γ-convergence results hold. Since both models are of second order, as opposed to the previously available fourth-order models for four-fold symmetric fracture toughness, they do not require basis functions of C1-continuity nor mixed variational principles for finite element discretization. For the four-fold symmetric formulation we show that the standard quadratic degradation function is unsuitable and devise a procedure to derive a suitable one. The performance of the new models is assessed via several numerical examples that simulate anisotropic fracture under anti-plane shear loading. For both formulations at fixed displacements (i.e. within an alternate minimization procedure), we also provide some existence and uniqueness results for the phase-field solution.