Carrera Unified Formulation Research Articles

Free vibration analysis of composite beams and laminated reinforced panels by refined dynamic stiffness method and CUF-based component-wise theory

This paper presents a comprehensive analysis of the free vibration behavior of composite beams and laminated reinforced panels. Employing high-order theories with displacement variables only, the investigation combines dynamic stiffness method (DSM) and the Carrera unified formulation (CUF). The 3D displacement field can be expanded in the framework of CUF as any order of generic unknown variables over the cross-section, in the case of beam theories. Specifically, Lagrange expansions (LE) of cross-sectional displacement variables are considered, enabling the refinement modeling of complex cross-sections with different layers (layer-wise) and components (component-wise). The governing differential equations and natural boundary conditions are derived using the principle of virtual displacements. Subsequently, an exact dynamic stiffness matrix is developed by establishing a relationship between the amplitudes of harmonically varying loads and the corresponding responses. The Wittrick–Williams algorithm is employed to solve the transcendental eigenvalue problem resulting from this approach. The Lagrange-based CUF(LE)-DSM outperforms other polynomials (such as Taylor)-based CUF in analyzing composite structures, enabling detailed analysis with various geometries, lamination schemes and boundary conditions. It delivers accurate results in reinforced panel analysis, requiring only 13% of the DOFs compared to 3D FE models, thereby reducing computational costs significantly, as confirmed by comparative studies and validations.

High-order effects on the vibration of membrane structures

This study analyzed pretension membranes, focusing on high-order effects. Using a combination of the Carrera Unified Formulation (CUF) and the finite element method, the research employs the unique framework of CUF to integrate innovative structural theories in a seamlessly scalable way. The analysis methodically explores the complex effects of shear and bending stiffness on crucial factors such as thickness, geometry, and materials. A comprehensive numerical examination of the subject is undertaken on numerous cases, and the assessment includes established analytical solutions as well as findings provided by commercial software tools. Particular emphasis is placed on mode aberration, underscoring the critical need for more refined structural models. Furthermore, the results show the crucial role of adopting high-order theory in the dynamic analysis of membrane structure, especially for application in space.

Transient Thermal Analysis of Space Rectangular Thin-Walled Structures by Carrera Unified Formulation

The accurate prediction of transient thermal fields of space structures is of paramount significance for the analysis of geometrical stability and thermal management systems. This paper proposes a novel method to solve the transient thermal field of the thin-walled cross section of beams in outer space. High-order beam elements are derived by means of the Carrera unified formulation (CUF), where the longitudinal direction of the beam is discretized by one-dimensional two-node and four-node elements, whereas arbitrary high-order expansion functions are used for the description of the temperature expansion within the cross section. The governing differential equations of the transient temperature field are derived according to the transient thermal conduction theory and the weighted residual method. The solution of the initial value problem for ordinary differential equations is obtained through the application of the Adams–Bashforth method, leading to the determination of the transient thermal field. In order to demonstrate the accuracy of the temperature results, a convergence analysis is conducted. Finally, temperature distributions within the cross section and along the longitudinal direction of the model are discussed to show the effects of several factors, including material properties, angle of solar radiation, and shadows induced by the shelter of other components.

Dynamic stiffness method and CUF-based component-wise theories applied to free vibration analysis of solid beams, thin-walled structures and reinforced panels

In this paper, an exact dynamic stiffness formulation using higher-order theories with displacement variables only is presented and subsequently used to investigate the free vibration characteristics of solid beams, thin-walled structures and reinforced panel structures. In essence, higher-order displacement fields are developed by using the Carrera unified formulation (CUF), and by discretizing the cross-section kinematics with bilinear, cubic and fourth-order Lagrange polynomials. In particular, the component-wise (CW) approach based on Lagrange expansion is applied in which the solid part and thin-walled part are considered as two independent components that can be assembled. The principle of virtual displacements is used to derive the governing differential equations and the associated natural boundary conditions. An exact dynamic stiffness matrix is then developed by relating the amplitudes of harmonically varying loads to those of the responses. The explicit terms of the dynamic stiffness matrices are also presented. The Wittrick–Williams algorithm of the dynamic stiffness method (DSM) is applied with the explicit expressions of the J0 count for beam elements under all above support conditions. In return, there is no need to refine the element in the DSM, and thus, it becomes immensely efficient. The accuracy and efficiency of the proposed methodology are extensively assessed for different solid beams, thin-walled structures and reinforced panels and the results are compared with those appearing in published literature and also checked by 3D finite element (FE) solutions.

Carrera Unified Formulation (CUF) for the composite plates and shells of revolution. Layer-wise models

Here, higher order layer-wise models of elastic composite multilayer plates and shells of revolution are developed using the variational principle of virtual power for the 3-D linear anisotropic theory of elasticity and generalized series in the thickness coordinates. Following the Unified Carrera Formulation (CUF), the stress and strain tensors, as well as the displacement vector, were expanded into series in terms of the coordinates of the shell thickness. The higher-order rectangular plate and cylindrical shell supported on the edges under sinusoidal loading, are considered and solved analytically using a Navier close form solution method. Also, composite axisymmetric conical, spherical, elliptical and catenoidal shell fixed at the ends are considered. Numerical calculations were performed using the computer algebra software Mathematica. The resulting equations can be used for theoretical analysis and calculation of the stress–strain state, as well as for modeling thin-walled structures used in science, engineering, and technology. The results of calculation can be used as benchmark examples for finite element analysis of the higher order composite laminate shells.

Rotordynamic analyses of stiffened cylindrical structures using high-fidelity shell models

The dynamic response of various rotating stiffened cylindrical and disk structures has been performed using low- and high-fidelity two-dimensional finite element models. The methodology is based on the Carrera Unified Formulation, in which the shell models are obtained hierarchically and automatically. These theories are formulated by expanding the unknown variables over the shell thickness. Various shell models can be implemented depending on the choice of the polynomial employed in the expansion. Lagrange polynomials are considered for developing different kinematic models and for easily modeling the stiffeners. The stiffener deformation is precisely captured through higher-order models, and the mechanical continuity between the stiffeners and skin is automatically guaranteed at the Lagrange points. The linearized equations of motion include the Coriolis and initial stress contributions. Stiffened cylinders and disks with different boundary conditions are considered. The results are compared with the three-dimensional finite element solutions from the literature or obtained using commercial software.

A machine learning approach to evaluate the influence of higher-order generalized variables on shell free vibrations

This work focuses on deriving guidelines for choosing structural theories for composite shells using Convolutional Neural Networks (CNN). The Axiomatic/Asymptotic Method (AAM) is used to evaluate higher-order structural theories’ accuracy and computational efficiency based on polynomial expansions. AAM exploits the Carrera Unified Formulation to derive the finite element matrices and obtain natural frequencies. The outcomes of AAM concerning the accuracy and computational cost are used to train CNN for various composite shell configurations. The trained network can then be used as a substitute for finite element models to estimate the accuracy of a given structural theory. The results are provided via Best Theory Diagrams (BTD), in which the set of generalized displacement variables to retain the best accuracy can be read for a given amount of nodal degrees of freedom. Verification is carried out using results from FEM. The results proved the computational efficiency of CNN and highlighted the influence of the shell thickness for the proper choice of the structural theory. Third-order terms and transverse stretching are often necessary to obtain acceptable accuracy.

A high‐order shell finite element for the large deformation analysis of soft material structures

SummaryThis work proposes a higher‐order unified shell finite element for the analysis of cylinders made of compressible and nearly incompressible hyperelastic materials. The nonlinear governing equations are derived employing the Carrera unified formulation (CUF), thanks to which it is possible to build shell elements with the capability to capture three‐dimensional (3D) transverse and out‐of‐plane effects. The material and geometric nonlinearities are expressed in an orthogonal curvilinear reference system and the coupled formulation of hyperelastic constitutive law is considered. The principle of virtual work and a total Lagrangian approach is used to derive the nonlinear governing equations, which are solved by a Newton–Raphson scheme. The numerical investigations deal with a curved arch and both thick and thin cylinders subjected to line and point loadings. The obtained results are validated by comparing them with those from the literature. They demonstrate the reliability of the proposed method to analyze compressible and incompressible hyperelastic shell structures.

Vibration of solid and thin-walled slender structures made of soft materials by high-order beam finite elements

In this work, high-order beam (1D) finite element models for the modal analysis of structures made of compressible and nearly-incompressible hyperelastic soft materials are presented in the well-established framework of Carrera Unified Formulation (CUF). In this investigation, the modal behavior of soft structures subjected to progressively increasing loads can be correctly predicted by higher-order structural theories, and the influence of pre-stress conditions applied on the modal response of structures is investigated. The mathematical formulation of hyperelastic isotropic materials is presented in terms of invariants of the right Cauchy–Green strain tensor, obtaining the most general expression of the Piola–Kirchoff 2 stress tensor and tangent elasticity tensor, both independent of the model adopted for the material strain energy function. Governing equations in matrix forms for the static nonlinear analysis and subsequent vibration problem around non-trivial equilibrium states are derived through the Principle of Virtual Displacements (PVD) under a total Lagrangian formulation, defining the fundamental nuclei of stiffness matrices and internal and external forces vector, all independent of expansion theories and kinematic models adopted in the mathematical modeling of finite elements. Actual numerical results are obtained by an iterative Newton–Raphson linearized scheme coupled with line-search algorithms, and they are compared with results obtained by the commercial code ABAQUS. Our proposed models are tested with large strain problems involving hyperelastic slender and thin-walled structures, for which mode aberration such as crossing or bearing are observed.

Legendre-based node-dependent kinematics shell models for the global–local analysis of homogeneous and layered structures

The present work demonstrates the use of the node-dependent kinematics method to derive and compare several two-dimensional shell theories. The three dimensional displacement field is expressed in terms of generalized coordinates, which are subsequently expanded along the shell thickness using arbitrary functions. The in-plane unknowns, are then discretized through classical finite element approximation. Based on the Carrera Unified Formulation, the proposed method combines in a unique manner the theory of structures and the finite element method; thickness interpolation functions are defined node-wise. As a consequence, the resulting finite element model represents diverse approximation theories at each single node. In this work Taylor-based kinematics (including the Murakami Zig-Zag function) and Legendre-type nodal kinematics are incorporated at the element level without adopting mathematical artifices leading to the global–local strategy, where refined theories are selectively employed in specific areas, while maintaining acceptable computational costs. Numerical cases from the existing literature are employed to establish the effectiveness of node-dependent models in bridging a locally refined theories to global kinematics when local effects need to be considered. The analyses focus on localized loads for both homogeneous and multi-layered structures.

A consistent crack bandwidth for higher-order beam theories: Application to concrete

Higher-order theories have a broad range of successful applications but also suffer from localization instability and mesh-size dependency when modeling quasi-brittle materials such as concrete with strain-softening behavior. To overcome the above difficulties, this paper proposes a fracture energy regularization method with a unified, consistent crack bandwidth specifically tailored for higher-order beam theories. The Carrera unified formulation (CUF) is applied to develop scalable structural theories and related finite elements. To evaluate the accuracy of the new crack bandwidth, three typical experimental quasi-static benchmarks of pure concrete structures are utilized. A modified Mazars damage model with tensile and compressive softening laws is implemented in these benchmarks. The comparison between numerical and experimental results demonstrates that the proposed method can accurately determine the correct crack bandwidth and preserve the dissipated energy per unit area of a fracture surface. Moreover, this robust estimation of crack bandwidth reduces the mesh dependency in general, ensuring the high efficiency of the CUF model.

Advancements in manufacturing and vibration analysis of functionally graded polymer composites: A review

Functionally graded materials are examined in-depth in this review study, which covers their classification, effect of environmental factors, cost reduction and efficient use, production processes, numerous theories and methodologies for vibration analysis. Classical lamination theory, First and Higher Order Shear Deformation Theories, and Carrera Unified Formulation. Also, numerical techniques such as Navier’s solution, Ritz and Galerkin’s methods are some of the methodologies that are covered. The article highlights the use of power, exponential, and sigmoidal laws to characterize material qualities as it examines the variety of FGM properties. Additionally, it examines the applications of FGMs and suggests possible future studies.

1D higher-order theories for quasi-static progressive failure analysis of composites based on a full 3D Hashin orthotropic damage model

This work investigates the crack propagation in composites by adopting a novel full three-dimensional (3D) Hashin-based orthotropic damage model combined with higher-order one-dimensional (1D) finite elements based on the Carrera Unified Formulation (CUF). Previous literature has proven that CUF provides structural formulations with great accuracy and improved computational efficiency. Moreover, a Layer-Wise (LW) formulation can be implemented within the CUF framework, allowing an accurate description of the 3D stress state in composite laminate, representing crucial information for progressive failure analysis. A Newton–Raphson predictor–corrector algorithm is used for the numerical solution of classical case tests, i.e., compact tension and three-point bending tests. The obtained results are compared with experimental outcomes and with solutions from well-established 2D damage models and a 3D Abaqus numerical model, demonstrating the capability of the proposed method to efficiently capture both the failure load and shape of the crack pattern.

Boundary Discontinuous Fourier analysis of clamped isotropic and cross-ply laminated plates via Unified Formulation

This paper presents an analytical solution for the static analysis of plates with clamped boundary conditions prescribed at the edges. The displacement field is expressed via the Carrera Unified Formulation (CUF) where an Equivalent-Single-Layer (ESL) approach is adopted. The governing equations are obtained by employing the principle of virtual displacements (PVD) statement. The main novelty is the use of the boundary-discontinuous Fourier-based approach to provide accurate numerical solutions. From thick to thin isotropic, cross-ply laminated and sandwich plates with different side-to-thickness ratios and stacking sequences are studied. Furthermore, the out-of-plane stresses are calculated via both the constitutive relation and the stress recovery technique. The accuracy of the proposed solution is verified by comparing the numerical results obtained with those from the literature and 3D FEM solutions. The present approach seems capable of handling not just fully clamped conditions but also mixed external conditions, which may include clamped and simply-supported edges. The solution approach provided in this article is unique, hence the proposed results might be useful as a benchmark for validating new plate theories and finite elements.

Analysis of three-dimensional thermal vibration coupling characteristics of composite shaft under axial load

The Carrera unified formulation (CUF) is employed to simplify the complete three-dimensional dynamic model based on the three-dimensional elasticity theory into a hierarchical one-dimensional dynamic model that maintains three-dimensional solution accuracy. The displacement function is formulated using Taylor polynomials and the improved Fourier series method (IFSM). Subsequently, the study solves the thermal vibration characteristics of the composite shaft under axial load through the application of energy functional and Hamilton's principle. The accuracy and stability of the present method are verified by comparing the present results with simulation. This study investigates the influence of thermal effects, structural parameters, and axial load on the vibrational characteristics of the composite shaft. The analysis results indicate that temperature, structural parameters, and axial load all significantly affect the fundamental properties of the composite shaft. As temperature and length-radius ratio increase, the natural frequency of the composite shaft gradually decreases. Regarding axial load, axial tension exhibits a positive correlation with the structure's natural frequency, whereas axial pressure shows a negative correlation. These findings hold significant engineering implications for utilizing composite shaft in the marine engineering sector.

On the coupling of data-driven computing and model-driven computing with CUF-based beam models

In this article, Model-driven (MD) computing and Data-driven (DD) computing mechanics are coupled under the framework of Carrera’s Unified Formulation (CUF) for beam structures A DD CUF-based computing is applied for the sub-domains of the beam under investigation where the material constitutive models are challenging to be obtained, whereas a MD CUF-based computing is used to deal with the remaining sub-domains where a constitutive model is available. By using CUF, under the premise of sufficient accuracy, the computational cost of DD-MD CUF-based models could be reduced compared with DD-MD FEM models where solid elements are used. Several static cases are analysed to demonstrate the accuracy and reliability of the proposed DD-MD CUF-based model by comparing the obtained results with those from the finite element commercial software tools.

A mechanical analysis of variable angle-tow composite plates through variable kinematics models based on Carrera’s unified formulation

Variable Angle-Tow (VAT) laminates offer a promising alternative to straight fiber composites. By varying fibers orientation within the structure plane, ambitious design and performance goals can be achieved. However, the wider design space results in a more complex problem with more parameters to consider. Carrera’s Unified Formulation (CUF) has been used in previous works performing buckling, vibrational and stress analyses of VAT plates. Usually, one-dimensional (1D) CUF beam models are used, while two-dimensional (2D) plate models are obtained as a particular case of shells by considering a null curvature. In most cases, a linear law is considered to describe the variation of fibers orientation in the main plane of the structure. The purpose of this article is to extend the CUF 2D plate finite elements family to the mechanical analysis of composite laminated plate structures with curvilinear fibers. The main contribute consists in the development of a CUF FE model within the Reissner’s Mixed Variational Theorem (RMVT) context for an improved calculation of the out-of-plane stress components. Results show that RMVT can predict in-plane stresses and satisfy the though-the-thickness transverse stresses continuity due to inter-layer equilibrium. The accuracy of RMVT-based models is also investigated using two different approximation points distributions along the plates thickness.

Thermal stress analysis of variable angle tow composite plates through high-order structural models

This work focuses on thermo-elastic analysis of Variable Angle Tow (VAT) composite plates. Emphasis is given to the through-the-thickness 3D stress component distributions as a consequence of constant temperature profiles. High-order structural models are used, in order to take into account the volumetric and the distortion contributions. The last contribute arise from boundary conditions or asymmetric staking sequences. Governing equations of the uncoupled thermo-elastic problem are derived in the domain of the Carrera Unified Formulation (CUF), which in turn is coupled with a classical finite element method to obtain opportune numerical solutions. The advantage of using CUF lies in the fact that different theories of structures can be implemented automatically and a critical study on the use of standard to high-order plate finite elements can be performed with ease. In this manner we classify models based on their efficiency and depending on the degree of accuracy needed, both in terms of displacements and stresses. Different numerical problems are considered and it is demonstrated that layerwise approximations are needed whenever shear stresses trends are of interest. Boundary Conditions influence more the choice of the adequate expansion theory otherwise.

Analysis of transversely isotropic compressible and nearly-incompressible soft material structures by high order unified finite elements

This work proposes high-order beam (1D) and plate (2D) finite element models for the large strain analysis of compressible and incompressible transversely isotropic hyperelastic media, defined within the Carrera Unified Formulation (CUF) framework. The strain energy density function adopted in fiber-reinforced hyperelastic materials modeling is presented and expressed in terms of invariants and pseudo-invariants of the right Cauchy-Green strain tensor. The explicit expression of the tangent elasticity tensor is derived through the assumption of coupled formulation of strain energy functions. Refined fully nonlinear beam and plate models are defined in a total Lagrangian formulation, deriving the governing equations of the nonlinear static analysis through the Principle of Virtual Displacements in terms of fundamental nuclei, in resulting expressions of internal and external forces vectors, and tangent stiffness matrix independent of kinematic models and approximation theories adopted. The iterative Newton-Raphson linearization scheme coupled with the arc-length constraint is adopted to obtain actual numerical solutions. Different benchmark analyses in hyperelasticity are performed to assess the capabilities of our proposed model, analyzing the three-dimensional stress field for moderate to large strain states and comparing actual numerical results with exact closed-form solutions or results available in the literature, demonstrating the capabilities and reliability of CUF models in the analysis of fiber-reinforced soft materials and structures.

Thermomechanical analysis of carbon fiber-reinforced polymer representative volume elements with voids

This work presents results of numerical simulations to investigate the effect of different void percentages on composite materials’ Coefficient of Thermal Expansion (CTE) and local stress fields. A random distribution of voids is considered within the Representative Volume Element (RVE) matrix, and different types of microstructures are considered, including square-packed and randomly distributed fibers. The use of a higher-order beam model within the framework of Carrera Unified Formulation (CUF) leads to a Component-Wise (CW) approach, resulting in an accurate, 3D description of the cross-section although using a 1D formulation. Numerical results for different fiber volume fractions and void concentration percentages demonstrate the agreement of the computed effective coefficients of thermal expansion of the present micromechanical thermoelastic model with references from the literature. The local stress fields are affected by voids, with higher effects over the matrix. Furthermore, higher void fractions lead to higher variability of stress peaks.