Mixed Variational Principle Research Articles

Deleterious localized stress fields: the effects of boundaries and stiffness tailoring in anisotropic laminated plates.

The safe design of primary load-bearing structures requires accurate prediction of stresses, especially in the vicinity of geometric discontinuities where deleterious three-dimensional stress fields can be induced. Even for thin-walled structures significant through-thickness stresses arise at edges and boundaries, and this is especially precarious for laminates of advanced fibre-reinforced composites because through-thickness stresses are the predominant drivers in delamination failure. Here, we use a higher-order equivalent single-layer model derived from the Hellinger–Reissner mixed variational principle to examine boundary layer effects in laminated plates comprising constant-stiffness and variable-stiffness laminae and deforming statically in cylindrical bending. The results show that zigzag deformations, which arise due to layerwise differences in the transverse shear moduli, drive boundary layers towards clamped edges and are therefore critically important in quantifying localized stress gradients. The relative significance of the boundary layer scales with the degree of layerwise anisotropy and the thickness to characteristic length ratio. Finally, we demonstrate that the phenomenon of alternating positive and negative transverse shearing deformation through the thickness of composite laminates, previously only observed at clamped boundaries, can also occur at other locations as a result of smoothly varying the material properties over the in-plane dimensions of the laminate.

A computational framework for large strain nearly and truly incompressible electromechanics based on convex multi-variable strain energies

The series of papers published by Gil and Ortigosa (Gil and Ortigosa, 2016; Ortigosa and Gil, 2016, 0000) introduced a new convex multi-variable variational and computational framework for the numerical simulation of Electro Active Polymers (EAPs) in scenarios characterised by extreme deformations and/or extreme electric fields. Building upon this body of work, five key novelties are incorporated in this paper. First, a generalisation of the concept of multi-variable convexity to energy functionals additively decomposed into isochoric and volumetric components. This decomposition is typical of nearly and truly incompressible materials, group which represents the majority of the most relevant EAPs. Second, convexification or regularisation strategies are applied to a priori non-convex multi-variable isochoric functionals to yield physically meaningful convex multi-variable functionals. Third, based on the mixed variational principles introduced in Gil and Ortigosa (2016) in the context of compressible electro-elasticity, a novel extended Hu–Washizu mixed variational principle for nearly and truly incompressible scenarios is presented. From the computational standpoint, a static condensation procedure is applied in order to condense out the element-wise extra fields, the resulting formulation having a comparable cost to the more standard three-field displacement-potential-pressure mixed formulation. Fourth, the computational framework for the three-field mixed variational principle in nearly and truly incompressible scenarios is also presented. In this case, the novelty resides in the consideration of convex multi-variable energy functionals. Ultimately, this leads to the definition of new tangent operators for the Helmholtz’s energy functional in the specific context of incompressible electro-elasticity. Fifth, a Petrov–Galerkin stabilisation technique is applied on the three-field formulation for the circumvention of the Ladyz˘enskaja–Babus˘ka–Brezzi (LBB) condition, enabling the use of linear tetrahedral finite elements for the interpolation of the unknowns of the problem. Finally, a series of challenging numerical examples is presented in order to provide an exhaustive comparison of the different variational formulations presented in this paper.

A high-performance parametrized mixed finite element model for bending and vibration analyses of thick plates

Based on the parametrized mixed variational principle, a high-performance finite element model has been developed for bending and vibration analysis of thick plates. The proposed mixed variational formulation combines the concept of Reissner’s mixed variational theorem and Rong’s generalized mixed variational theorem. In addition to unknown parameters of the displacement fields, the transverse normal component of the stress tensor is also assumed as the independent field variable in the present variational formulation. The latter allows the fulfillment of the boundary conditions of the transverse normal stress at the top and bottom surfaces of the plate in a natural way. The variational constraint of the transverse normal strain–displacement relations is relaxed via the parametrized Lagrange multipliers method. The functional of the proposed variational formulation contains an arbitrary free parameter, called the splitting factor. Simple formulations have been proposed for selecting the splitting factor which leads to the results of higher precision. The numerical results obtained from the proposed mixed formulation have been compared with the results of three-dimensional (3D) theory of elasticity and other plate theories available in the literature. The comparison study shows that the proposed parametrized mixed plate theory is capable of achieving the accuracy of nearly exact 3D solutions.

The finite element square reduced (FE2R) method with GPU acceleration: towards three‐dimensional two‐scale simulations

SummaryThe FE2 method is a renown computational multiscale simulation technique for solid materials with fine‐scale microstructure. It allows for the accurate prediction of the mechanical behavior of structures made of heterogeneous materials with nonlinear material behavior. However, the FE2 method leads to excessive CPU time and storage requirements, even for academic two‐dimensional problems. In order to allow for realistic three‐dimensional two‐scale simulations, a significant reduction of the CPU and memory usage is required. For this purpose, the authors have recently proposed a reduced basis homogenization scheme based on a mixed incremental variational principle. The approach exploits the potential structure of generalized standard materials. Thereby, important speed‐ups and memory savings can be achieved. Using high‐performance GPUs, the reduced‐basis method can be further accelerated. In the present contribution, our previous works are combined and extended to form the FE2‐reduced method: the FE2R. The FE2R can be used to simulate three‐dimensional structural problems with consideration of the nonlinearity and microstructure of the underlying material at acceptable computational cost. Thereby, it allows for a new level of complexity in nonlinear multiscale simulations. Numerical examples illustrate the capabilities of the chosen approach. Copyright © 2016 John Wiley & Sons, Ltd.

A new framework for large strain electromechanics based on convex multi-variable strain energies: Variational formulation and material characterisation

Following the recent work of Bonet et al. (2015), this paper postulates a new convex multi-variable variational framework for the analysis of Electro Active Polymers (EAPs) in the context of reversible nonlinear electro-elasticity. This extends the concept of polyconvexity (Ball, 1976) to strain energies which depend on non-strain based variables introducing other physical measures such as the electric displacement. Six key novelties are incorporated in this work. First, a new definition of the electro-mechanical internal energy is introduced expressed as a convex multi-variable function of a new extended set of electromechanical arguments. Crucially, this new definition of the internal energy enables the most accepted constitutive inequality, namely ellipticity, to be extended to the entire range of deformations and electric fields and, in addition, to incorporate the electro-mechanical energy of the vacuum, and hence that for ideal dielectric elastomers, as a degenerate case. Second, a new extended set of variables, work conjugate to those characterising the new definition of multi-variable convexity, is introduced in this paper. Third, both new sets of variables enable the definition of novel extended Hu–Washizu type of mixed variational principles which are presented in this paper for the first time in the context of nonlinear electro-elasticity. Fourth, some simple strategies to create appropriate convex multi-variable energy functionals (in terms of convex multi-variable invariants) by incorporating minor modifications to a priori non-convex multi-variable functionals are also presented. Fifth, a tensor cross product operation (de Boer, 1982) used in Bonet et al. (2015) to facilitate the algebra associated with the adjoint of the deformation gradient tensor is incorporated in the proposed variational electro-mechanical framework, leading to insightful representations of otherwise complex algebraic expressions. Finally, under a characteristic experimental setup in dielectric elastomers, the behaviour of a convex multi-variable constitutive model capturing some intrinsic nonlinear effects such as electrostriction, is numerically studied.

A new framework for large strain electromechanics based on convex multi-variable strain energies: Finite Element discretisation and computational implementation

In Gil and Ortigosa (2016), Gil and Ortigosa introduced a new convex multi-variable framework for the numerical simulation of Electro Active Polymers (EAPs) in the presence of extreme deformations and electric fields. This extends the concept of polyconvexity to strain energies which depend on non-strain based variables. The consideration of the new concept of multi-variable convexity guarantees the well posedness of generalised Gibbs’ energy density functionals and, hence, opens up the possibility of a new family of mixed variational principles. The aim of this paper is to present, as an example, the Finite Element implementation of two of these mixed variational principles. These types of enhanced methodologies are known to be necessary in scenarios in which the simpler displacement-potential based formulation yields non-physical results, such as volumetric locking, bending and shear locking, pressure oscillations and electro-mechanical locking, to name but a few. Crucially, the use of interpolation spaces in which some of the unknown fields are described as piecewise discontinuous across elements can be used in order to efficiently condense these fields out. This results in mixed formulations with a computational cost comparable to that of the displacement-potential based approach, yet far more accurate. Finally, a series of very challenging numerical examples are presented in order to demonstrate the accuracy, robustness and efficiency of the proposed methodology.

A new mixed finite-element approach for the elastoplastic analysis of Mindlin plates

The objective of this paper is to develop an accurate and efficient solution procedure for elastoplastic problems in structural mechanics in the framework of a two-field mixed variational principle. A novel solution algorithm is proposed and applied to the elastoplastic analysis of Mindlin plates. The Hellinger–Reissner principle is adopted to obtain the global finite-element equations of the problem. Instead of a static condensation, the stress-type field variables are preserved during the solution. According to the proposed approach, the strain increments within a nonlinear solution step are obtained directly at the nodal points from matrix operations instead of gradients of a displacement field. In the present implementation, the von Mises yield criterion with linear hardening is adopted. For the integration of the elastoplastic constitutive rate equations at the nodal points, a 3D fully implicit algorithm is employed. A layered approach is followed to enable the resolution of the plastic strains through the plate thickness. The mixed formulation of the Mindlin plate theory is shear-locking free by construction. The proposed solution strategy is verified by solving several benchmark problems that demonstrate the high accuracy and convergence rate of the presented layered mixed formulation for elastoplastic analyses.

A state space augmented generalised RKPM for three-dimensional analysis of thick and laminated composite plates

In this paper, a hybrid numerical method which combines the states space approach with the generalised Reproducing Kernel Particle Method (RKPM) is proposed for the three-dimensional analysis of thick and laminated composite plates. Employing the state space approach in the thickness direction, the governing partial differential equation is transformed into a state equation. The meshless RKPM is utilised to present the in-plane distribution of the displacement and stress components while their variation in the transverse direction of the plate and lamina is captured by adopting the state space method. This results in the formation of an algebraic system of equations where the number of unknowns is independent of the number of material layers of the laminate. Moreover, the solution provides a continuous transverse stress and displacement field which is not generally achievable using conventional numerical methods. Because the method is based on a mixed variational principle, and a meshless method is incorporated as the numerical tool, enforcement of the essential boundary conditions requires special treatment and a detailed description is given of the mixed method developed for the enforcement of the essential boundary conditions. A series of numerical examples is performed to illustrate the application of the proposed method and the results obtained are compared with known analytical and numerical solutions.

State space finite element analysis for piezoelectric precision positioning considering secondary converse piezoelectric effect

A microgripper component which is made of piezo-cantilever is presented in this paper. The modified mixed variational principle including piezoelectric materials is established and then the state-vector equations of laminated piezoelectric beams are derived directly from the principle. Finally the exact expressions of mechanical and electrical variables along the thickness direction based on the state-vector equation are simply given. Based on the semi-analytical solution of the state-vector equation, a state space finite element model is established. The static analysis is carried out for a laminated beam fully covered with piezoelectric actuating layers on the top and bottom surfaces of the substrate. Both the substrate and piezoelectricity of the laminated beam are considered as one-dimensional line solids and the same linear element is used to discretize the whole beam. This semi-analytical technique can guarantee the continuity of generalized displacements and generalized stresses inter-layers and the transfer matrix method can be applied. Furthermore the exact solution in one direction ensures there is no restriction on the thickness for the smart structures. At the end of this paper the influence of secondary converse piezoelectric effect (SCPE) on the deflection of the laminated beam covered with piezoelectric driving layers for static precision displacement control is investigated. The above mentioned state space finite element method is used to predict the deflection of the beam with and without SCPE. If the influence of SCPE is eliminated, the deflections of the clamped beam will increase by ~2.7%. In addition, the driving characteristic with different PZT/steel thickness ratio considering SCPE are also studied. The maximum influence of SCPE on deflection can reach ~3% for the beam fully covered with piezoelectric active layers, which should not be neglected in practical microoperation applications.

Mixed moving least-squares method for shakedown analysis

The elastic shakedown theory can be used to analyze and design structures under loads that change over time. Methods that use mixed interpolations provide simultaneous information regarding the kinematics and the equilibrium without the necessity of postprocessing. Meshless methods, as suggested by the name, do not require mesh generation or node connectivity and are an alternative to classical numerical methods. However, no current method exists that combines a mixed approximation of the kinematics and equilibrium with meshless methods for an elastic shakedown problem. In this paper, a discretization of a mixed variational principle for the elastic shakedown is presented based on the moving least-squares method, where stress and velocity approximation functions are used. The optimization problem, which comes from the mixed variational principle, determines the load factor and the residual stress and velocity fields. The stresses and velocities are approximated using linear and quadratic polynomial functions, respectively. The solution to the problem is obtained through a nonlinear algorithm based on Newton’s method. The results obtained for the load factor and the residual stress field approach those obtained from the analytical solution for the tested examples.

Mixed Variational Principles in Nondissipative Coupled Thermoelasticity

Mixed Variational Principles in Nondissipative Coupled Thermoelasticity

Reissner’s mixed variational theorem-based nonlocal Timoshenko beam theory for a single-walled carbon nanotube embedded in an elastic medium and with various boundary conditions

The nonlocal Timoshenko beam theories (TBTs), based on the Reissner mixed variational theorem (RMVT) and principle of virtual displacement (PVD), are derived for the analysis of a single-walled carbon nanotube (SWCNT) embedded in an elastic medium and with various boundary conditions, in which the Eringen nonlocal elasticity theory is used. The strong formulations of RMVT- and PVD-based nonlocal TBTs and their associated possible boundary conditions are presented. The interaction between the SWCNT and its surrounding elastic medium is simulated using the Pasternak foundation model. The generalized displacement and force resultant components induced in the loaded SWCNT are obtained using the meshless collocation method, in which the shape functions of the primary variables are constructed using the differential reproducing kernel (DRK) method. The results show that RMVT-based nonlocal TBT is superior to its PVD-based counterpart in that the convergent rate of the RMVT-based nonlocal TBT is faster than that of the PVD-based one, that the predictions of generalized force resultants obtained using the RMVT-based nonlocal TBT are more accurate than those obtained using the PVD-based nonlocal one, and that the highest order of the base functions used in the RMVT-based nonlocal TBT is lower than that used in the PVD-based one.

Variational Phase Field Modeling of Fracture Caused by Fluid Transport in Poroelastic Solids

AbstractThe numerical modeling of failure mechanisms due to fracture based on sharp crack discontinuities is extremely demanding and suffers in situations with complex crack topologies. This drawback can be overcome by recently developed diffusive crack modeling concepts, which are based on the introduction of a crack phase field. Such an approach is conceptually in line with gradient‐extended continuum damage models which include internal length scales. In this paper, we extend our recently outlined mechanical framework [1–3] towards the phase field modeling of fracture in the coupled problem of fluid transport in deforming porous media. Here, extremely complex crack patterns may occur due to drying or hydraulic induced fracture, the so called fracking. We develop new variational potentials for Biot‐type fluid transport in porous media at finite deformations coupled with phase field fracture. It is shown, that this complex coupled multi‐field problem is related to an intrinsic mixed variational principle for the evolution problem. This principle determines the rates of deformation, fracture phase field and fluid content along with the fluid potential. We develop a robust computational implementation of the coupled problem based on the potentials mentioned above and demonstrate its performance by the numerical simulation of complex fracture patterns. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Three-Dimensional Extended Kantorovich Solution for Accurate Prediction of Interlaminar Stresses in Composite Laminated Panels with Interfacial Imperfections

AbstractAn accurate three-dimensional (3D) elasticity solution is presented for static analysis of flat laminated panels with interlaminar bonding imperfection under arbitrary boundary conditions exhibiting edge effects. The recently developed mixed-field multiterm extended Kantorovich method (MMEKM) for 3D solution of perfectly bonded laminates is generalized to include the interfacial compliance characterized by displacement jumps. A general variationally consistent framework using the Reissner-type mixed variational principle is proposed to treat the imperfect interfacial conditions. It is shown through numerical studies on composite and soft-core sandwich panels under different boundary conditions that, similar to the perfect bonding case, the MMEKM yields accurate results with just two or three terms and in two or three iterations for the weakly bonded laminates also. The roles that the boundary conditions, locations of the imperfect interfaces, and span-to-thickness ratios play on the effect of weak...

Formulation and numerical exploitation of mixed variational principles for coupled problems of Cahn–Hilliard‐type and standard diffusion in elastic solids

SUMMARYThis work develops variational principles for the coupled problem of standard and extended Cahn–Hilliard‐type species diffusion in solids undergoing finite elastic deformations. It shows that the coupled problem of diffusion in deforming solids, accounting for phenomena like swelling, diffusion‐induced stress generation and possible phase segregation caused by the diffusing species, is related to an intrinsic mixed variational principle. It determines the rates of deformation and concentration along with the chemical potential, where the latter plays the role of a mixed variable. The principle characterizes a canonically compact model structure, where the three governing equations involved, that is, the mechanical equilibrium condition, the mass balance for the species content and a microforce balance that determines the chemical potential, appear as the Euler equation of a variational statement. The existence of the variational principle underlines an inherent symmetry of the coupled deformation–diffusion problem. This can be exploited in the numerical implementation by the construction of time‐discrete and space‐discrete incremental potentials, which fully determine the update problems of typical time stepping procedures. The mixed variational principles provide the most fundamental approach to the monolithic finite element solution of the coupled deformation–diffusion problem based on low‐order basis functions. They induce in a natural format the choice of symmetric solvers for Newton‐type iterative updates, providing a speedup and reduction of data storage when compared with non‐symmetric implementations. This is a strong argument for the use of the developed variational principles in the computational design of deformation–diffusion problems. Copyright © 2014 John Wiley & Sons, Ltd.

Mixed variational potentials and inherent symmetries of the Cahn–Hilliard theory of diffusive phase separation

This work shows that the Cahn-Hilliard theory of diffusive phase separation is related to an intrinsic mixed variational principle that determines the rate of concentration and the chemical potential. The principle characterizes a canonically compact model structure, where the two balances involved for the species content and microforce appear as the Euler equations of a variational statement. The existence of the variational principle underlines an inherent symmetry in the two-field representation of the Cahn-Hilliard theory. This can be exploited in the numerical implementation by the construction of time- and space-discrete incremental potentials, which fully determine the update problems of typical time-stepping procedures. The mixed variational principles provide the most fundamental approach to the finite-element solution of the Cahn-Hilliard equation based on low-order basis functions, leading to monolithic symmetric algebraic systems of iterative update procedures based on a linearization of the nonlinear problem. They induce in a natural format the choice of symmetric solvers for Newton-type iterative updates, providing a speed-up and reduction of data storage when compared with non-symmetric implementations. In this sense, the potentials developed are believed to be fundamental ingredients to a deeper understanding of the Cahn-Hilliard theory.

A geometrically exact beam model with non-uniform warping coherently derived from the Saint Venant rod

A new geometrically nonlinear model for homogeneous and isotropic beams with generic section including non-uniform warping due to torsion and shear is derived. Each section is endowed with a corotational frame where statics and kinematics are described using a 3D linear elastic model which extends the Saint-Venant solution to non-uniform warping cases. The algebra of change of observer and a mixed variational principle give the model in terms of generalized parameters. Using a mixed interpolation the model is implemented within a FEM Koiter analysis highly sensitive to the geometrical coherence of the formulation.

Corotational nonlinear analyses of laminated shell structures using a 4-node quadrilateral flat shell element with drilling stiffness

A new 4-node quadrilateral flat shell element is developed for geometrically nonlinear analyses of thin and moderately thick laminated shell structures. The flat shell element is constructed by combining a quadrilateral area coordinate method (QAC) based membrane element AGQ6-II, and a Timoshenko beam function (TBF) method based shear deformable plate bending element ARS-Q12. In order to model folded plates and connect with beam elements, the drilling stiffness is added to the element stiffness matrix based on the mixed variational principle. The transverse shear rigidity matrix, based on the first-order shear deformation theory (FSDT), for the laminated composite plate is evaluated using the transverse equilibrium conditions, while the shear correction factors are not needed. The conventional TBF methods are also modified to efficiently calculate the element stiffness for laminate. The new shell element is extended to large deflection and post-buckling analyses of isotropic and laminated composite shells based on the element independent corotational formulation. Numerical results show that the present shell element has an excellent numerical performance for the test examples, and is applicable to stiffened plates.

A meshless sub-region radial point interpolation method for accurate calculation of crack tip fields

A new meshless sub-region radial point interpolation method (MS-RPIM) is proposed for linear elastic fracture mechanics. The Williams expansions of stress field for mode I/II crack is used as the trial functions in crack tip region, the meshless radial point interpolation is used for the rest of domain, and a mixed variational principle is used for discretisation. In contrast to existing meshless formulations, the present MS-RPIM requires only very few nodes around the crack tip to obtain smooth stress and accurate results and the SIFs can be directly obtained as part of the solution and no additional effort via post-processing.

Mixed Variational Principles and Robust Finite Element Design of Gradient Plasticity at Finite Strains

AbstractThe modeling of size effects in elastic‐plastic solids, such as the width of shear bands or the grain size dependence in polycrystals, must be based on non‐standard theories which incorporate length‐scales. This is achieved by models of strain gradient plasticity, incorporating spatial gradients of selected micro‐structural fields which describe the evolving dissipative mechanisms. The key aspect of this work is to provide a rigorous incremental variational formulation and mixed finite element design of additive finite gradient plasticity in the logarithmic strain space. We start from a mixed saddle point principle for metric‐type plasticity, which is specified for the important model problem of isochoric plasticity with gradient‐extended hardening/softening response. To this end, we propose a novel finite element design of the coupled problem incorporating a local‐global solution strategy of short‐ and long‐range fields. This includes several new aspects, such as extended Q1P0‐type and MINI‐type finite elements for gradient plasticity [4]. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)