Generalized Unified Formulation Research Articles

Finite plate elements with variable kinematics based on sublaminate generalized unified formulation

The Sublaminate Generalized Unified Formulation (SGUF) is extended for the first time to the framework of Finite Element Method (FEM) for both displacement-based and mixed (RMVT) formulation. The variable kinematics approach allows to choose different plate models according to the desired level of accuracy. Furthermore, the mixed ESL/LW approach of SGUF makes the model particularly convenient for sandwich structures analysis. A substitute interpolation for the first-order transverse shear strain field, referred to as QC4 interpolation, makes the developed four-node FE locking free and insensitive to mesh distortion. The complete expression of finite element matrices for the PVD-based and RMVT-based elements is provided. The possibility of exactly satisfying transverse stress boundary conditions for RMVT-based elements is also investigated for the first time. The flexibility and accuracy of the computational approach is demonstrated on linear static problems of sandwich plates and beams ranging from global bending response to local indentation problems. In particular, it is demonstrated that the proposed approach is capable of recovering full three-dimensional response with a 2D FE mesh and with less degrees of freedom than the conventional models available in commercial FE packages.

Extension of an analytical solution of a unified formulation to the frequency response of composite plates with viscoelastic layers

AbstractResulting from their high stiffness and low weight, lightweight structures made from fiber reinforced polymers are usually prone to vibrations. The inclusion of viscoelastic interlayers can provide a viable mean to passively damp these structures due to constrained layer damping. For the modeling of such structures, a layerwise approach based on the Generalized Unified Formulation has previously been proven suitable for static load cases.This contribution presents an analytical procedure to determine the frequency response and vibration characteristics of such heterogeneous laminates incorporating viscoelastic layers. Specifically, simply supported plates modeled by the Generalized Unified Formulation are considered and solved using Navier type solutions. The approach is validated by comparison with 3D finite element models. Particular focus is put on the order of expansion in thickness direction needed to accurately predict the frequency response of the laminate.

Large displacement models for composites based on Murakami's Zig-Zag Function, Green-Lagrange Strain Tensor, and Generalized Unified Formulation

Abstract Murakami's Zig-Zag Function in conjunction of Green-Lagrange Strain- and Second-Piola Kirchhoff Stress-tensors and Generalized Unified Formulation are introduced for the first time to present a class of Zig-Zag theories for composite structures. Transverse strain effects are retained and modeled, making the present approach also suitable for thick structures. Each displacement variable is independently represented with user-selected order of expansion. The stress recovery is achieved with a special procedure in which combination of deformation gradient and Second-Piola Kichhoff transverse stresses are integrated along the thickness of the undeformed geometry. Comparison with available data and commercial codes indicate the effectiveness of the proposed framework for variable kinematics Zig-Zag theories in the presence of large displacements.

Functional Reconstitution of Reissner’s Mixed Variational Theorem for Finite Element Applications

The functional reconstitution technique is used for the first time to theoretically and a priori assess the numerical performance of Reissner’s Mixed Variational Theorem in the case of linear quadrilateral finite element, Generalized Unified Formulation, and composite materials. From the finite element discretization, the semi-complementary energy is reconstructed in integral form. Analytical expressions for the exact and spurious terms are obtained. Using the elasticity solution for a simply supported plate, it is numerically shown that these spurious contributions do not significantly alter the quality of the approximation of the total level of energy, with overall excellent computational performance. On the other hand, when the starting functional is the strain energy and the principle of virtual displacement is adopted, the finite-element-introduced additional terms and in particular the contributions associated to the transverse shear stresses are demonstrated to add a very large error, which is a priori proven to be eliminated with selective integration. The presented technique could be extended to analyze the performance of other finite elements (e.g., higher-order triangular ones) within Reissner’s Mixed Variational Theorem and Generalized Unified Formulation.

An implicit mesh discontinuous Galerkin formulation for higher-order plate theories

In this work, a discontinuous Galerkin formulation for higher-order plate theories is presented. The starting point of the formulation is the strong form of the governing equations, which are derived in the context of the Generalized Unified Formulation and the Equivalent Single Layer approach from the Principle of Virtual Displacements. To express the problem within the discontinuous Galerkin framework, an auxiliary flux variable is introduced and the governing equations are rewritten as a system of first-order partial differential equations, which are weakly stated over each mesh element. The link among neighboring mesh elements is then retrieved by introducing suitably defined numerical fluxes, whose explicit expressions define the proposed Interior Penalty discontinuous Galerkin formulation. Furthermore, to account for the presence of generally curved boundaries of the considered plate domain, the discretisation mesh is built by combining a background grid and an implicit representation of the domain. hp-convergence analyses and a comparison with the results obtained using the Finite Element Method are provided to show the accuracy of the proposed formulation as well as the computational savings in terms of overall degrees of freedom.

The Ritz – Sublaminate Generalized Unified Formulation approach for piezoelectric composite plates

ABSTRACTThis paper extends to composite plates including piezoelectric plies the variable kinematics plate modeling approach called Sublaminate Generalized Unified Formulation (SGUF). Two-dimensional plate equations are obtained upon defining a priori the through-thickness distribution of the displacement field and electric potential. According to SGUF, independent approximations can be adopted for the four components of these generalized displacements: an Equivalent Single Layer (ESL) or Layer-Wise (LW) description over an arbitrary group of plies constituting the composite plate (the sublaminate) and the polynomial order employed in each sublaminate. The solution of the two-dimensional equations is sought in weak form by means of a Ritz method. In this work, boundary functions are used in conjunction with the domain approximation expressed by an orthogonal basis spanned by Legendre polynomials. The proposed computational tool is capable to represent electroded surfaces with equipotentiality conditions. Free-vibration problems as well as static problems involving actuator and sensor configurations are addressed. Two case studies are presented, which demonstrate the high accuracy of the proposed Ritz-SGUF approach. A model assessment is proposed for showcasing to which extent the SGUF approach allows a reduction of the number of unknowns with a controlled impact on the accuracy of the result.

Equivalent Single Layer, Zig-Zag, and Layer Wise theories for variable angle tow composites based on the Generalized Unified Formulation

Variable Angle Tow (VAT) composites remove the constraint of having straight fibers, typical of traditional composite structures. This dramatically increases the design space and allows a more effective tailoring of the material properties to minimize the weight and increase structural performance. To provide an accurate prediction of the displacement and stress fields in an efficient computational framework, a multi-theory architecture based on the Generalized Unified Formulation (GUF) is proposed. In particular, Equivalent Single Layer, Zig-Zag, and Layer Wise theories with different orders of expansions for the different variables are generated with a theory-invariant mathematical model. This feature allows the user to tailor the computational accuracy and cost to the needs of the case under investigation and is inherently well suited for optimization and reliability problems. For the in-plane discretization, a fourth-order triangular shell element presenting 15 nodes is adopted. The interlaminar displacement continuity is imposed in the finite element assembling in the thickness direction of the layer stiffness matrices, whereas the inter-element displacement compatibility is enforced with the penalty method. This approach allows one to use independent GUF discretization for any desired direction within each element, providing significant versatility (accuracy vs computational time) in the structural modeling. A transverse stress recovery procedure taking into account the variability of the structural properties due to the fibers’ curvilinear paths is also presented. Results are compared with the literature and a commercial software (NX NASTRAN) featuring 3D finite elements.

Bending analysis of composite laminated and sandwich structures using sublaminate variable-kinematic Ritz models

This paper presents a novel numerical tool for the bending analysis of thin and thick composite plates, including monolithic and sandwich structures. The formulation is developed within a displacement-based approach, where the Principle of Virtual Displacements (PVD) and the method of Ritz are adopted to derive the governing equations. The approach relies upon the Sublaminate Generalized Unified Formulation (S-GUF) as underlying kinematic theory describing the behavior across the plate thickness. Main idea of the S-GUF is to group the plies into a number of smaller units called sublaminates, each of them characterized by an independent, variable-kinematic theory. Continuity conditions between the sublaminates are enforced in strong form during the assembly procedure of the governing equations. The S-GUF appears particularly useful when theories of different order are needed to approximate the displacement field of different portions of the structure, such as in the case of sandwich panels. A number of test cases from the literature is discussed, and results are validated against exact 3D solutions. The results demonstrate the ability of the approach to obtain accurate results, both in terms of deformed shapes, and intra- and inter-laminar stress distributions. A set of novel results is also presented for future benchmarking purposes.

A Sublaminate Generalized Unified Formulation for the analysis of composite structures

This paper presents a very flexible variable kinematics modeling technique for composite structures. The key point is the subdivision of the multilayered cross-section into numerical sublaminates, each consisting of one or more physical plies, and the formulation of model assumptions independently in each sublaminate according to the compact index notation known as Generalized Unified Formulation. Reference is made to the classical displacement-based and the mixed RMVT variational frameworks. For each sublaminate, the user may thus freely choose the order of the through-thickness polynomial distribution of each unknown as well as whether the unknown is described in an Equivalent Single Layer (ESL) or Layer-Wise (LW) approach. Sublaminates are always assembled in a Layer-Wise manner. This Sublaminate-GUF (S-GUF) is generally applicable to multilayered structures, but is particularly useful for sandwich panels, in which different models may be used for the thick, soft core and the thin, stiff skins. A Navier-type solution is used for demonstrating the validity of this approach by means of benchmark problems concerning the static response of sandwich plates.

Benchmark solutions and assessment of variable kinematics models for global and local buckling of sandwich struts

This paper investigates the periodic global buckling and wrinkling of sandwich struts subjected to uniaxial loads. An exact elasticity solution is obtained in the plane strain setting for selected benchmark problems and serves as reference for a systematic assessment of several displacement-based sandwich plate models. The Sublaminate Generalized Unified Formulation is used to generate the algebraic two-dimensional governing equations of the variable kinematics models, for which a Navier-type solution is adopted.

Generalized Unified Formulation shell element for functionally graded Variable-Stiffness Composite Laminates and aeroelastic applications

Composite materials have been extensively used in engineering thanks to their lightweight, superior mechanical performances and possibility to tailor the structural behavior, increasing the available design space. Variable Angle Tow (VAT) structures exploits this advantage by adopting a curvilinear patterns for the fibers constituting the lamina.This work, for the first time, extends the Generalized Unified Formulation (GUF) to the case of fourth-order triangular shell elements and VAT composites. Functionally graded material properties in both the thickness and in-plane directions are also possible. The finite element has been formulated with layers of variable thickness with respect to the in-plane coordinates.GUF is a very versatile tool for the analysis of Variable Stiffness Composite Laminates (VSCLs): it is possible to select generic element coordinate systems and define different types of axiomatic descriptions (Equivalent Single Layer, Layer Wise, and Zig-Zag enhanced formulations) and orders of the thickness expansions. Each displacement is independently treated from the others. All the infinite number of theories that can be generated with GUF are obtained by expanding six theory-invariant kernels (formally identical for all the elements), allowing a very general implementation. Finally, the possibility of tailoring the theory/order to increase the accuracy in desired directions makes the GUF VAT capability a very powerful tool for the design of aerospace structures.

Partially Layer Wise advanced Zig Zag and HSDT models based on the Generalized Unified Formulation

The Generalized Unified Formulation (GUF) for composites and sandwich structures is a computational multi-theory architecture which can generate a large variety of types of theories with any combination of the orders of expansion for the different primary variables. All the theories are generated from the expansion of six theory-invariant 1×1 arrays (kernels). For the first time, an extension and further generalization of the GUF is proposed: each displacement unknown is independently described (in an axiomatic sense) with respect the other primary variables. In particular, each displacement can present a Layer Wise or an Equivalent Single Layer types of discretizations with or without the Zig Zag enhancement which takes into account the displacements’ thickness derivative interlaminar discontinuity.The Partially Zig Zag Advanced Higher Order Shear Deformation Theories (PZZAHSDTs), Partially Layer Wise Higher Order Shear Deformation Theories (PLHSDTs), Partially Layer Wise Advanced Higher Order Shear Deformation Theories (PLAHSDTs), Partially Layer Wise Advanced Zig Zag and Higher Order Shear Deformation Theories (PLAZZHSDTs) are introduced and discussed for the first time. These types of theories are compared with the advanced Equivalent Single Layer, Zig Zag, and Layer Wise approaches. In the framework of displacement-based models, a Finite Element implementation of the Generalized Unified Formulation is presented for the first time and numerical evaluations of sandwich structures subjected to various boundary conditions are discussed. A challenging high Face-to-Core Stiffness Ratio (FCSR) is also adopted to asses the performances of this large set of new approaches. It is demonstrated that the Partially Layer Wise Theories provide a computationally efficient alternative to standard Layer Wise models.

Assess the Accuracy of the Variational Asymptotic Plate and Shell Analysis Using the Generalized Unified Formulation

The accuracy of the Variational Asymptotic Plate and Shell Analysis (VAPAS) is assessed against several higher order, zig zag, and layerwise theories generated by using the invariant axiomatic framework denoted as Generalized Unified Formulation (GUF). These theories are also compared against the elasticity solution developed for the case of a sandwich structure with high Face to Core Stiffness Ratio. GUF allows to use an infinite number of axiomatic theories (Equivalent Single Layer theories with or without zig-zag effects and Layerwise theories as well) with any combination of orders of the displacements and it is an ideal tool to precisely assess the range of applicability of the Variational Asymptotic Plate and Shell Analysis or other theories in general. In fact, all the axiomatic theories generated by GUF are obtained from the kernels or fundamental nuclei of the Generalized Unified Formulation and changing the order of the variables is “naturally” and systematically done with GUF. It is demonstrated that VAPAS achieves accuracy comparable to a fourth (or higher) order zig-zag theory or lower-order layerwise theories with the least number of Degrees of Freedom. The differences between the axiomatic zig-zag models and VAPAS are also assessed. Range of applicability of VAPAS will be discussed in detail and guidelines for new developments based on GUF and VAPAS are provided.

Partially Zig-Zag Advanced Higher Order Shear Deformation Theories Based on the Generalized Unified Formulation

The Generalized Unified Formulation (GUF) is a multi-theory and a multi-fidelity architecture for the generation of a virtually infinite class of Advanced Higher Order Shear Deformation Theories (AHSDT) or Zig-Zag theories or Layer-Wise (LW) theories with any order of expansion for each of the primary variables. This work will present, for the first time in the literature, an extension of GUF to address problems in which every single variable can have either an Equivalent Single Layer (ESL) or a Zig-Zag-enhanced ESL description [Partially Zig-Zag Advanced Higher Order Shear Deformation Theories (PZZAHSDT)]. Applications to the case of thick sandwich structures are presented: starting from a baseline fourth-order AHSDT which also includes the transverse strain effects, all the possible types of PZZAHSDT are generated and compared with the baseline and with a fourth-order fully Zig-Zag theories.

Three-Dimensional Closed Form Solutions and ∞3 Theories for Orthotropic Plates

A Navier-type method for finding the exact three-dimensional solution for orthotropic rectangular plates is presented. The present procedure is an extension of a previous author's work (which was valid for isotropic plates) and uses the Mixed Form of Hooke's Law (MFHL) which leads to the formulation of the boundary conditions on the top and bottom surfaces of the plate directly in terms of transverse stresses. The paper also extends the Generalized Unified Formulation (GUF) to the case of orthotropic materials. ∞3 theories are presented. The unknown displacements are expanded along the thickness of the plate by using a powerful compact formulation which leads to the writing, with a single formal theory, of a class of ∞3 theories. This formal technique allows the user to treat each unknown independently and, therefore, different numerical approaches can be used in the FEM codes based on this Generalized Unified Formulation. This technique can be very useful in multifield problems such as thermoelastic applications and multilayered plates embedding piezo-layers. The paper analyzes the effects of the different orders of expansion for the displacement variables for various material properties. It is shown that when the anisotropy of the mechanical properties is strong the results are more sensitive to the order used to approximate the displacements. Thirty different theories are assessed for both thick and thin plates and the results are compared against the present exact solution.

Mixed plate theories based on the Generalized Unified Formulation. Part I: Governing equations

Abstract The generalized unified formulation (GUF) is a formal technique which was introduced in the framework of displacement-based theories. GUF is extended for the first time to the case in which a mixed variational statement (Reissner’s mixed variational theorem) is used. Each of the displacement variables and out-of-plane stresses is independently treated and different orders of expansions for the different unknowns can be chosen. Since infinite combinations can be freely chosen for the displacements u x , u y , u z and for the stresses σ xz , σ yz , σ zz , the generalized unified formulation allows the user to write, with a single formal theory, ∞ 6 theories which can be successfully implemented in a single FEM code. In addition, this formulation allows the user to treat each unknown independently and, therefore, different numerical approaches can be used in the FEM codes based on this generalized unified formulation. All the theories are originated from 13 independent fundamental nuclei (kernels of the generalized unified formulation) which are formally invariant and the layerwise mixed theories (analyzed in Part II), mixed higher order shear deformation theories (analyzed in Part III) and advanced mixed zig-zag models (analyzed in Part IV) can be studied without extra implementations or theoretical developments. Numerical performances and convergence properties of a very large amount of new mixed theories are discussed (Part V) with particular focus on the effects of the orders of expansion in the thickness direction of the displacements and modeled stresses. Multilayered composite plates will be analyzed. Different mixed variational statements could be used and the formulation could be easily adopted for multifield problems such as thermoelastic applications and multilayered plates embedding piezo-layers.

Mixed plate theories based on the Generalized Unified Formulation. Part III: Advanced mixed high order shear deformation theories

Abstract The Generalized Unified Formulation was introduced in Part I in the case of plate theories based upon Reissner’s Mixed Variational Theorem. Part II analyzed the case of layerwise theories. In this work ( Part III ) the Generalized Unified Formulation is applied, for the first time in the literature, to the case of mixed higher order shear deformation theories. The displacements u x , u y , u z have an equivalent single layer description, whereas the stresses σ zx , σ zy , σ zz have a layerwise description. The compatibility of the displacements and the equilibrium of the transverse stresses between two adjacent layers are enforced a priori . If the out-of-plane stresses are eliminated using the Static Condensation Technique the resulting theories are formally identical to the displacement-based “classical” higher order shear deformation theories. If the static condensation is not applied then a quasi-layerwise higher order theory is obtained. ∞ 6 Mixed higher order shear deformation theories are therefore presented. All ∞ 6 theories are generated by expanding thirteen 1 × 1 invariant matrices (the kernels of the Generalized Unified Formulation). The kernels have the same formal expressions as the ones used for the layerwise theories analyzed in Part II . This makes the generation of the present mixed higher order shear deformation theories particularly effective and easy to implement in a computer code.

Mixed plate theories based on the Generalized Unified Formulation. Part IV: Zig-zag theories

Abstract The generalized unified formulation was introduced in Part I for the case of plate theories based upon Reissner’s mixed variational theorem. Part II analyzed the case of layerwise theories and Part III studied advanced mixed higher order shear deformation theories. In this work the generalized unified formulation is applied, for the first time in the literature, to the case of advanced mixed higher order zig-zag theories. The so called zig-zag form of the displacements is enforced a priori by the adoption of Murakami’s zig-zag function. An equivalent single layer description of the displacements u x , u y and u z is adopted. The out-of-plane stresses σ zx , σ zy and σ zz have a layerwise description. The compatibility of the displacements and the equilibrium of the transverse stresses between two adjacent layers are enforced a priori. ∞ 6 mixed higher order zig-zag theories are therefore presented. The kernels have the same formal expressions as the ones used in the layerwise theories analyzed in Part II and in the higher order shear deformation theories presented in Part III.

Mixed plate theories based on the Generalized Unified Formulation.

The generalized unified formulation introduced in Part I for the case of composite plates and Reissner’s Mixed variational theorem is, for the first time in the literature, applied to the case of layerwise theories. Each layer is independently modeled. The compatibility of the displacements and the equilibrium of the transverse stresses between two adjacent layers are enforced a priori. Infinite combinations of the orders used for displacements ux, uy, uz and out-of-plane stresses σzx, σzy, σzz can be freely chosen. ∞6 layerwise theories are therefore presented. The code based on this capability can have all the possible ∞6 theories built-in, thus, making the code a powerful and versatile tool to analyze different geometries, boundary conditions and applied loads. All ∞6 theories are generated by expanding 13 1×1 invariant matrices (the kernels of the Generalized Unified Formulation). How the kernels are expanded and the theories generated is explained. Details of the assembling in the thickness direction and the generation of the matrices are provided.

Mixed plate theories based on the Generalized Unified Formulation. Part V: Results

Parts I, II, III and IV presented the Generalized Unified Formulation in the framework of Reissner’s Mixed Variational Theorem. Layerwise theories, mixed higher order shear deformation theories and zig-zag models were introduced. In all these types of theories the displacement variables and out-of-plane stresses are independently treated and different orders of expansion for the different unknowns can be chosen. All the possible ∞6 theories are generated by expanding 13 invariant fundamental nuclei. The relative orders used for the expansion of the stresses and displacements are important and can be the source of numerical instabilities. How the instabilities are eliminated is discussed. In the case of layerwise theories and the examined problems, it is shown that there is no numerical instability if the order of displacement uz is the same as the order of stress σzz. New light is also shed on the mixed equivalent single layer theories. It is shown that the poor representation of the a priori calculated transverse stresses is due to the above mentioned numerical instabilities and not only to the insufficient representation of the effects of σzz. Finally, for the mixed case, it is demonstrated that the addition of Murakami’z zig-zag function (MZZF) is convenient but this is not a general property, as was believed in the literature before this work. The convenience of the addition of MZZF is linked to the relative orders of the starting theory that is being improved with the zig-zag term. Several new layerwise and equivalent single layer theories are introduced for the first time in the literature and an assessment is given with new cases compared against the elasticity solution.