Mixed Variational Principle Research Articles

Geotechnical localization analysis based on Cosserat continuum theory and second-order cone programming optimized finite element method

Based on the Hellinger-Reissner mixed variational principle, a second-order cone programming optimized finite element method for elastic and elastoplastic analyses of Cosserat continuum (named CosFEM-SOCP) is proposed. To examine the correctness and effectiveness of CosFEM-SOCP, three well-known numerical examples are revisited. In the plane-strain elastic problem of stress concentration around a circular hole, the stress concentration factors calculated by CosFEM-SOCP are in good agreement with the existing solutions, and the correctness of CosFEM-SOCP can be verified. In the analysis of load-displacement response of a flexible strip footing resting on the soil ground, it is found that load-displacement response may be affected by the internal characteristic length lc, and generally a larger lc may result in a wider and deeper slip surface as well as more elements being included in the shear bands. From the plane-strain soil strip with the Drucker-Prager (DP) failure criterion subjected to the prescribed incremental displacement, it can also be observed that the pathologically mesh-dependent problem can be partially or even completely removed by employing CosFEM-SOCP, as long as the mesh is fine enough so that the average element size in the mesh discretization is smaller than lc.

A mixed variational framework for the design of energy–momentum integration schemes based on convex multi-variable electro-elastodynamics

In Ortigosa et al. (2018), the authors presented a new family of time integrators for large deformation electromechanics. In that paper, definition of appropriate algorithmic expressions for the discrete derivatives of the internal energy and consideration of multi-variable convexity of the internal energy was made. These two ingredients were essential for the definition of a new energy–momentum (EM) time integrator in the context of large deformation electromechanics relying on materially stable (ellipticity compliant) constitutive models. In Betsch et al. (2018), the authors introduced a family of EM time integrators making use of mixed variational principles for large strain mechanics. In addition to the displacement field, the right Cauchy–Green deformation tensor, its co-factor and its Jacobian were introduced as unknown fields in the formulation. An elegant cascade system of kinematic constraints was introduced in this paper, crucial for the satisfaction of the required conservation properties of the new family of EM time integrators. The objective of the present paper is the introduction of new mixed variational principles for EM time integrators in electromechanics, hence bridging the gap between the previous work presented by the authors in Ortigosa et al. (2018) and Betsch et al. (2018), opening up the possibility to a variety of new Finite Element implementations. The following characteristics of the proposed EM time integrator make it very appealing: (i) the new family of time integrators can be shown to be thermodynamically consistent and second order accurate; (ii) piecewise discontinuous interpolation of the unknown fields (except displacements and electric potential) has been carried out, in order to yield a computational cost comparable to that of standard displacement–potential formulations. Finally, a series of numerical examples are included in order to demonstrate the robustness and conservation properties of the proposed scheme, specifically in the case of long-term simulations.

Efficient three-node triangular element based on a new mixed global-local higher-order theory for multilayered composite plates

An accurate and computationally attractive global-local higher-order theory (GLHT) is developed for the linearly elastic analysis of cross-ply multilayered composite plates. The theory is derived using the kinematic assumptions of GLHT in conjunction with the Reissner mixed variational principle. For a low-order linear element, it is difficult to accurately compute the transverse shear stresses even applying the three-dimensional equilibrium equation post-processing technique. The reason for this difficulty is that the higher-order derivatives of displacement variables are included in the transverse shear stress fields after using the post-processing technique. Thus, by employing the Reissner mixed variational principle, the higher-order derivatives of displacement variables have been removed from the transverse shear stress components before the finite element procedure is implemented. Based on the mixed GLHT, a computationally efficient C0-type three-node triangular plate element with linear interpolation function is proposed for the analysis of multilayered composite plates. The advantage of the present formulation is that no post-processing approach is needed to calculate the transverse shear stresses while maintaining the computational accuracy of a linear plate element. Performance of the proposed element is assessed by comparing with several benchmark solutions. Numerical results show that the present elements can robustly and accurately predict the displacements and stresses of multilayered composite plates.

On a family of numerical models for couple stress based flexoelectricity for continua and beams

A family of numerical models for the phenomenological linear flexoelectric theory for continua and their particularisation to the case of three-dimensional beams based on a skew-symmetric couple stress theory is presented. In contrast to the standard strain gradient flexoelectric models which assume coupling between electric polarisation and strain gradients, we postulate an electric enthalpy in terms of linear invariants of curvature and electric field. This is achieved by introducing the axial (mean) curvature vector as a strain gradient measure. The physical implication of this assumption is many-fold. Firstly, analogous to the standard strain gradient models, for isotropic (non-piezoelectric) materials it allows constructing flexoelectric energies without breaking material’s centrosymmetry. Secondly, unlike the standard strain gradient models, nonuniform distribution of volumetric part of strains (volumetric strain gradients) do not generate electric polarisation, as also confirmed by experimental evidence to be the case for some important classes of flexoelectric materials. Thirdly, a state of plane strain generates out of plane deformation through strain gradient effects. Finally, under this theory, extension and shear coupling modes cannot be characterised individually as they contribute to the generation of electric polarisation as a whole. As a first step, a detailed comparison of the developed couple stress based flexoelectric model with the standard strain gradient flexoelectric models is performed for the case of Barium Titanate where a myriad of simple analytical solutions are assumed in order to quantitatively describe the similarities and dissimilarities in effective electromechanical coupling under these two theories. From a physical point of view, the most notable insight gained is that, if the same experimental flexoelectric constants are fitted in to both theories, the presented theory in general, reports up to 200% stronger electromechanical conversion efficiency. From the formulation point of a view, the presented flexoelectric model is also competitively simpler as it eliminates the need for high order strain gradient and coupling tensors and can be characterised by a single flexoelectric coefficient. In addition, three distinct mixed flexoelectric variational principles are presented for both continuum and beam models that facilitate incorporation of strain gradient measures in to a standard finite element scheme while maintaining the C0 continuity. Consequently, a series of low and high order mixed finite element schemes for couple stress based flexoelectricity are presented and thoroughly benchmarked against available closed form solutions in regards to electromechanical coupling efficiency. Finally, nanocompression of a complex flexoelectric conical pyramid for which analytical solution cannot be established is numerically studied where curvature induced necking of the specimen and vorticity around the frustum generate moderate electric polarisation.

Edge effects in elastic and piezoelectric laminated panels under thermal loading

An analytical solution for accurately predicting the edge effects in elastic and hybrid multilayered panels integrated with piezoelectric actuator and sensor layers under thermal gradient loading is presented. The associated thermal problem is solved independently by performing a heat transfer analysis. The extended Reissner-type mixed variational principle is employed to develop the governing equations of the three dimensional (3D) piezothermoelasticity problem. Their solution is obtained using the mixed-field multiterm extended Kantorovich method, developed recently by the author group. The mixed field solution exactly satisfies all the mechanical and electric boundary conditions pointwise, leading to an accurate solution. The formulation is general, and is applicable for composite, sandwich and hybrid panels with arbitrary lay-up and boundary conditions. The solution is validated against the exact 3D solution available for simply supported panels. The free edge stress field in sandwich and hybrid composite laminates is investigated under uniform as well as gradient thermal loading. The solution successfully captures the singularity in free edge interlaminar stresses. The significance of pyroelectric coupling on the free edge thermal stress field is investigated, and the feasibility of its control through electric field actuation is examined.

Static behavior of arbitrarily supported composite laminated cylindrical shell panels: An analytical 3D elasticity approach

First time, an analytical three-dimensional elasticity solution is proposed for bending analysis of composite laminated cylindrical shell panels having arbitrarily end-support conditions which were unavailable hitherto. Governing partial differential equations are obtained in terms of displacements and stresses by employing Ressiner mixed variational principle in the cylindrical coordinate system. Further employing extended Kantorovich method, governing partial differential equations are reduced to sets of non–homogeneous first order ordinary differential equations (ODEs). The set of ODEs with variable coefficients (along radial direction) is ingeniously solved through a new modified power series method whereas the set with constant coefficients (along circumferential direction) is solved using Pagano’s approach. After thorough validation, some new benchmark results are presented for single layer and multilayered laminated composites under various boundary conditions. This development will help to revisit/develop solutions for many classical problems of arbitrarily supported shell structures.

Variational-based locking-free energy–momentum schemes of higher-order for thermo-viscoelastic fiber-reinforced continua

Locking-free finite elements and energy–momentum time integration schemes are two of the best-known algorithmic improvements of finite element methods. Both are developed since the middle of the eighties of the last century, but usually independently from each other. Therefore, a smart interface between both algorithms is rarely a development goal of the researcher. In this paper, we present such a smart interface, namely the Hu–Washizuprocedure applied to the principle of virtual power. By using of the resulting mixed variational principle, we avoid locking in a spatial finite element discretization of non-isothermal inelastic fiber-reinforced materials, and obtain a family of corresponding higher-order accurate energy–momentum schemes. Thereby, we consider volumetric locking in the matrix material and line locking in the fibers. We show that this reduction of locking in the energy–momentum schemes leads to an increase of the maximum time step size, such that the efficiency of the time integration is improved in the sense that less CPU time is required. This could be shown by using an automatic time step size control with the iteration number of the applied Newton–Raphson scheme as target function. As numerical examples, we consider slender fiber-reinforced structures as a turbine rotor and a lightweight beam consisting of fiber-reinforced trusses. Here, we simulate different combinations of mechanical and thermal Dirichlet and Neumann boundary conditions.

An energy–momentum time integration scheme based on a convex multi-variable framework for non-linear electro-elastodynamics

This paper introduces a new one-step second order accurate energy–momentum (EM) preserving time integrator for reversible electro-elastodynamics. The new scheme is shown to be extremely useful for the long-term simulation of electroactive polymers (EAPs) undergoing massive strains and/or electric fields. The paper presents the following main novelties. (1) The formulation of a new energy–momentumtime integrator scheme in the context of nonlinear electro-elastodynamics. (2) The consideration of well-posed ab initio convex multi-variable constitutive models. (3) Based on the use of alternative mixed variational principles, the paper introduces two different EM time integration strategies (one based on the Helmholtz’s and the other based on the internal energy). (4) The new time integrator relies on the definition of four discrete derivatives of the internal/Helmholtz energies representing the algorithmic counterparts of the work conjugates of the right Cauchy–Green deformation tensor, its co-factor, its determinant and the Lagrangian electric displacement field. (6) Proof of thermodynamic consistency and of second order accuracy with respect to time of the resulting algorithm is included. Finally, a series of numerical examples are included in order to demonstrate the robustness and conservation properties of the proposed scheme, specifically in the case of long-term simulations.

A new mixed-field theory for bending and vibration analysis of multi-layered composite plate

A novel mixed-field theory with relatively low number of unknown variables is introduced for bending and vibration analysis of multi-layered composite plates. The presented plate theory is derived from a parametrized mixed variational principle which is introduced recently by the first author. A global-local kinematic with a layer-independent number of variables is assumed for the description of the displacements of the plate. The considered kinematic stratifies the displacement and transverse stress continuity conditions at the mutual interfaces of the layers. It also fulfill the homogenous boundary conditions of the shear stresses on the upper/lower surfaces of the plates without using the shear correction factor. One-unknown variable fields which satisfy a priori the continuity conditions at the adjacent interfaces between layers and the zero boundary conditions on the bounding surfaces are considered for the approximation of the transverse shear stresses. The transverse normal stress along the total thickness of the multi-layered plate is approximated via a quadratic polynomial. The presented mixed-field plate theory has been validated through comparison of the bending and vibration analysis results with those obtained from the three-dimensional (3D) theory of elasticity and the results of the other classical and high-order plate theories.

An efficient partial mixed finite element model for static and free vibration analyses of FGM plates rested on two-parameter elastic foundations

In this study, a four-node quadrilateral partial mixed plate element with low degrees of freedom (dofs) is developed for static and free vibration analysis of functionally graded material (FGM) plates rested on Winkler-Pasternak elastic foundations. The formulation of the presented finite element model is based on a parametrized mixed variational principle which is developed recently by the first author. The presented finite element model considers the effects of shear deformations and normal flexibility of the FGM plates without using any shear correction factor. It also fulfills the boundary conditions of the transverse shear and normal stresses on the top and bottom surfaces of the plate. Beside these capabilities, the number of unknown field variables of the plate is only six. The presented partial mixed finite element model has been validated through comparison with the results of the three-dimensional (3D) theory of elasticity and the results obtained from the classical and high-order plate theories available in the open literature.

On non-linear flexure of beams based on non-local elasticity theory

Reissner mixed variational principle is employed for establishment of the nonlinear differential and boundary conditions of dynamic equilibrium governing the flexure of beams when the effects of true shear stresses are included. Based on the Reissner mixed variational principle, the nonlinear size-dependent model of the Reissner nano-beam is derived in the framework of the nonlocal strain gradient elasticity theory. Furthermore, the closed form analytical solutions for the geometrically nonlinear flexural equations are derived and compared to the nonlinear flexural results of the Timoshenko size-dependent beam theory. The profound differences in the assumptions and formulations between the Timoshenko and the Reissner beam theory are also comprehensively discussed. The Reissner beam model is shown not to be a first-order shear deformation theory while comprising the influences of the true transverse shearing stress and the applied normal stress. Moreover, it is exhibited that the linear and nonlinear deflections obtained based on the Reissner beam theory are consistently lower than their Timoshenko counterparts for various gradient theories of elasticity.

Shakedown analysis for porous materials

Porous materials have numerous applications in industry. Elements such as gears, pins, filters, bearings, and connecting rods, among others, are examples of parts that can be manufactured from these materials, where the presence of varying loads may cause a catastrophic failure. The shakedown analysis is a useful tool to establish the safe conditions for the operation of a structural or mechanical component subjected to varying loads. However, the shakedown analysis of porous materials has received little attention and a general formulation for this problem is not available. This paper presents a static and a mixed variational principle for the shakedown analysis of porous materials and the corresponding numerical solution via the finite element method. A yield function for porous material is obtained from the specific strain energy combined with a relationship between the relative density and the applied hydrostatic pressure. The shakedown analysis performed in this paper is based on a mixed variational principle which gives rise to an optimization problem. The solution provides, not only the largest amplification factor for the domain of load variation, but also the residual stresses and velocities fields. The discretization of the mixed variational principle is performed by the finite element method and uses a triangle with six nodes which permits a quadratic interpolation of velocities together with a linear interpolation of stresses. The validation of the numerical procedure is accomplished by comparing the present simulation with previously published results and analytical solutions for the residual stresses and amplification factor. The finite element method provided good solutions for all the presented examples. The presented variational formulation may be used to obtain the safe condition for mechanical parts made of porous materials and subjected to variable loads. Moreover, it can be used to estimate the relative density distribution in the parts.

Highly accurate symplectic element based on two variational principles

For the stability requirement of numerical resultants, the mathematical theory of classical mixed methods are relatively complex. However, generalized mixed methods are automatically stable, and their building process is simple and straightforward. In this paper, based on the seminal idea of the generalized mixed methods, a simple, stable, and highly accurate 8-node noncompatible symplectic element (NCSE8) was developed by the combination of the modified Hellinger-Reissner mixed variational principle and the minimum energy principle. To ensure the accuracy of in-plane stress results, a simultaneous equation approach was also suggested. Numerical experimentation shows that the accuracy of stress results of NCSE8 are nearly the same as that of displacement methods, and they are in good agreement with the exact solutions when the mesh is relatively fine. NCSE8 has advantages of the clearing concept, easy calculation by a finite element computer program, higher accuracy and wide applicability for various linear elasticity compressible and nearly incompressible material problems. It is possible that NCSE8 becomes even more advantageous for the fracture problems due to its better accuracy of stresses.

A curvilinear high order finite element framework for electromechanics: From linearised electro-elasticity to massively deformable dielectric elastomers

This paper presents a high order finite element implementation of the convex multi-variable electro-elasticity for large deformations large electric fields analyses and its particularisation to the case of small strains through a staggered scheme. With an emphasis on accurate geometrical representation, a high performance curvilinear finite element framework based on an a posteriori mesh deformation technique is developed to accurately discretise the underlying displacement-potential variational formulation. The performance of the method under near incompressibility and bending actuation scenarios is analysed with extremely thin and highly stretched components and compared to the performance of mixed variational principles recently reported by Gil and Ortigosa (2016) and Ortigosa and Gil (2016). Although convex multi-variable constitutive models are elliptic hence, materially stable for the entire range of deformations and electric fields, other forms of physical instabilities are not precluded in these models. In particular, physical instabilities present in dielectric elastomers such as pull-in instability, snap-through and the formation, propagation and nucleation of wrinkles and folds are numerically studied with a detailed precision in this paper, verifying experimental findings. For the case of small strains, the essence of the approach taken lies in guaranteeing the objectivity of the resulting work conjugates, by starting from the underlying convex multi-variable internal energy, whence avoiding the need for further symmetrisation of the resulting Maxwell and Minkowski-type stresses at small strain regime. In this context, the nonlinearity with respect to electrostatic counterparts such as electric displacements is still retained, hence resulting in a formulation similar but more competitive with the existing linearised electro-elasticity approaches. Virtual prototyping of many application-oriented dielectric elastomers are carried out with an eye on pattern forming in soft robotics and other potential medical applications.

Interlaminar stress analysis of multilayered composites based on the Hu-Washizu variational theorem

Up to date, accurate prediction of interlaminar stresses is still a challenging issue for two-node beam elements. The postprocessing approaches by integrating the three-dimensional equilibrium equation have to be used to obtain improved transverse shear stresses, whereas the equilibrium approach requires the first-order derivatives of in-plane stresses. In-plane stresses within two-node beam element are constant, so the first-derivatives of in-plane stresses are close to zero. Thus, two-node beam elements encounter difficulties for accurate prediction of transverse shear stresses by the constitutive equation or the equilibrium equation, so a robust two-node beam element is expected. A two-node beam element in terms of the global higher-order zig-zag model is firstly developed by employing the three-field Hu-Washizu mixed variational principle. By studying the effects of different boundary conditions, stacking sequence and loading on interlaminar stresses of multilayered composite beams, it is shown that the proposed two-node beam element yields more accurate results with lesser computational cost compared to various higher-order models. It is more important that accurate transverse shear stress has active impact on displacements and in-plane stresses of multilayered composite beams.

Simple and efficient analyses of micro-architected cellular elastic-plastic materials with tubular members

In a novel way, nature evolves cellular structures to obtain mechanically efficient materials. Natural cellular materials combine low weight with superior mechanical properties to provide optimum strength and stiffness at low density such as trabecular bone, hornbill bird beaks, and bird wing bones. Inspired by these naturally architected cellular materials, humankind has also developed lightweight cellular materials for a broad range of applications consisting of structural components, energy absorption, heat exchange, and biomaterials. In this paper we present a highly efficient computational method to predict the bulk elastic-plastic homogenized mechanical properties of low-mass metallic systems with architected cellular microstructures. The proposed methodology provides a computational framework for the analysis, design, and topology optimization of such cellular materials. With a view for Direct Numerical Simulation of a cellular solid or structure with millions of cellular members, and considering the plausible deformations in each member, each such member is sought to be modeled by using only one or only a very few nonlinear three-dimensional (3D) beam elements with 6 degrees of freedom (DOF) at each of the 2 nodes of the element, and the nonlinear coupling of axial-torsional-bidirectional bending deformations is considered for each element. The effect of plasticity in each member is included using the mechanism of plastic hinges which may form at any point(s) along the length of each element or member. To make Direct Numerical Simulation of a micro-latticed cellular solid possible for its eventual applications, the tangent stiffness matrix for a spatial beam element undergoing large elastic-plastic deformation is explicitly derived using a Reissner-type mixed variational principle in the co-rotational updated Lagrangian reference frame. In order to avoid the inversion of the Jacobian matrix, a Newton homotopy method is employed to solve the tangent-stiffness equations. We are developing a code called CELLS/LIDS [CELLular Structures/Large Inelastic DeformationS] providing the capabilities to study the variation of the mechanical properties of the low-mass metallic cellular structures by changing their topology. Thus, due to the efficiency of this method we propose to employ it for topology optimization design, and for impact/energy absorption analyses of elastic-plastic micro-cellular structures, and then micro-architecting them for desired elastic-plastic properties.

Free edge stress field in smart piezoelectric composite structures and its control: An accurate multiphysics solution

A three-dimensional piezoelasticity based iterative analytical solution is presented for the multiphysics problem of free edge stresses in hybrid composite panels integrated with piezoelectric transducer layers. The formulation, applicable for general laminate configurations with arbitrary layup and materials properties, considers extension, bending, twisting and electric field actuation loadings with full electromechanical coupling. A mixed formulation approach based on the Reissner-type mixed variational principle is adopted to ensure exact point-wise satisfaction of all boundary and interlaminar continuity conditions, which is key to obtaining accurate results for the localized stresses near free edges. The solution is obtained using the recently developed mixed-field multiterm extended Kantorovich method, whose robustness, convergence and accuracy are illustrated for various laminates and loadings. The solution successfully captures singular nature of free edge interlaminar stresses under different loadings. The results show a significant effect of electromechanical coupling on the free edge stresses in hybrid laminates under mechanical loading. The effect of actuator thickness on the interfacial stress distributions under electric load is studied. Finally, it is shown how the free edge stresses due to extension, bending and twisting loads can be controlled by applying appropriate actuation potential.

A state-space finite element method for laminated composite plates under thermal loading

ABSTRACTOn the basis of the mixed variational principle, a state equation for the thermal elasticity in a laminated composite plate is yielded. A state-space finite element method is proposed to solve the established state equation, which is a nonhomogenous one. The horizontal plane of the laminated plate is modeled by traditional finite elements, while the variables along the thickness direction are obtained by the precise integration method. Because the discretization is implemented only on the horizontal plane, fewer elements are needed comparing with the traditional finite element method, in which the mesh is discretized in the whole structure. The different constraint conditions on the plate boundary are discussed. The calculated transverse thermal stresses by the present method satisfy the surface/interface conditions exactly. However, the numerical results by the traditional finite element method only approximately satisfy the surface/interface conditions. The transverse thermal stress distribution along the thickness direction evaluated by the present method is continuous across the interfaces of different layers in the laminated plate, whereas the traditional finite element method fails to provide the continuous transverse thermal stresses at the interfaces.

Investigation of failure initiation in curved composite laminates using a higher-order beam model

Curved laminates are prone to delamination failure from applied bending moments that straighten out the laminate and induce tensile stresses in the unreinforced radial direction. These non-classical through-thickness stresses are important even for thinner configurations and need to be accounted for in the design of lightweight composite structures, preferably in a computationally efficient manner. Here, we investigate failure-inducing critical stresses for a number of curved laminates using a higher-order beam model derived from the Hellinger–Reissner mixed variational principle, which guarantees that the hoop, interlaminar shear and radial stresses are equilibrated. By solving the governing equations of the theory in the strong form using the pseudo-spectral differential quadrature method, the model is capable of predicting accurate 3D stress fields in curved laminates, even in the vicinity of localised features such as supported edges. The model is used alongside commonly used failure criteria to reproduce experimental failure initiation results found in the literature, and the comparison suggests that failure mode, location and load are all predicted accurately. Finally, failure maps that highlight the critical stress component for failure initiation are constructed. As the thickness of the curved laminate increases, the critical stress component transitions from intralaminar hoop stress to interlaminar shear or radial stress depending on the specific laminate configuration. These findings and failure maps collectively provide insights into the mechanics of failure initiation that should also prove useful for design purposes.

Reduced order homogenization for viscoplastic composite materials including dissipative imperfect interfaces

The computational effort of homogenization schemes based on full-field simulations is prohibitively high for many applications and motivates the use of model order reduction techniques. A reduced order homogenization scheme is presented which allows for a coupled treatment of dissipative mechanisms both in the bulk and on imperfect interfaces within the microstructure. Extending the key idea of the nonuniform transformation field analysis (NTFA), reduced bases are introduced for all fields of internal variables and also for the displacement discontinuities at the interfaces. In the online phase, an initial value problem is solved, which is derived from a mixed incremental variational principle. The variational formulation requires that the underlying constitutive models are given by two potentials, as known from the framework of generalized standard materials. As an example, a unidirectional fiber composite with a viscoplastic matrix and a viscoelastic interface is studied. It is demonstrated that the reduced order model can be used to analyze the rate-dependence of the overall behavior as well as the interface-induced size effect and the impact of varying interface parameters. At a significantly reduced computational effort, good quantitative agreement with finite element reference solutions is found for predictions of global quantities such as the effective stress and the effective mechanical work. For this is a rather simple training strategy sufficient. It is found that the reduced model provides good approximations of the local stress field and of the internal variables within the microstructure, which can benefit from a more comprehensive training.