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This paper provides a brief review on modeling of composite structures. Composite structures in this paper refer to any structure featuring anisotropy and heterogeneity, including but not limited to their traditional meaning of composite laminates made of unidirectional fiber-reinforced composites. Common methods used in modeling of composite structures, including the axiomatic method, the formal asymptotic method, and the variational asymptotic method, are illustrated in deriving the classical lamination theory for the composite laminated plates. Future research directions for modeling composite structures are also pointed out. [ABSTRACT FROM AUTHOR]

Equivalent 3D properties are obtained for composite thin-walled 3D structures using the concept of mechanics of structure genome. The adopted homogenization technique interprets the unit cell associated with the thin-walled 3D structures as an assembly of plates, and the overall strain energy density of the unit cell as a summation of the plate strain energies of these individual plates. The concept of mechanics of structure genome is then applied to drop all higher-order terms and the remaining energy is minimized with respect to the unknown fluctuating functions. This has been done by discretizing the two-dimensional unit cell into one-dimensional frame elements in a finite element description. This allows the handling of structures with different levels of complexities and internal geometry within a general framework. Comparisons have been made with existing work to show the advantage the proposed model offers over existing ones. [ABSTRACT FROM AUTHOR]

Equivalent plate properties are obtained for composite corrugated structures using mechanics of structure genome. The method developed in this paper interprets the unit cell associated with the corrugated structure as an assembly of plates, and the overall strain energy density of the unit cell as a summation of the plate strain energies of these individual plates. Mechanics of structure genome is then applied to drop all higher-order terms and the remaining energy is minimized with respect to the unknown fluctuating functions. This has been done by discretizing the two-dimensional unit cell into one-dimensional frame elements in a finite element description. This allows the handling of structures with different levels of complexities and internal geometry within a general framework. Comparisons have been made with existing work to show the advantage the proposed model offers over existing ones. [ABSTRACT FROM AUTHOR]

Damaging constitutive models are incorporated into the variational asymptotic beam sectional analysis method (VABS). Under the governing of the virtual work principle, Newton’s method is applied to solve for the finite cross-sectional warping field and the damage variables according to continuum damage mechanics models from the literature. These solutions result in mesh-insensitive predictions of the damaged beam stiffness and three-dimensional stress and strain fields in the damaged state. The present theory is implemented using the finite element method to capture the high-fidelity elastic-to-damaged beam constitutive behaviors by the cross-sectional analyses. The predicted beam constitutive relations and local fields are compared with results from finite element analyses (FEA) using solid elements and plate elements, and experiments. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with steady state thermoelastic problems in beam structures using the variational asymptotic method. First, we simplify the original three-dimensional heat conduction problem into a two-dimensional thermal cross-sectional analysis along with an optional one-dimensional heat conduction analysis. The refinements up to the second order of the initial curvatures and twist are captured in the model. Having obtained the temperature field, we then construct a generalized Timoshenko model for the one-way coupled thermoelastic analysis. Several examples will be used to demonstrate the application and performance of the present model. The results will be compared with three-dimensional thermoelastic analysis using ANSYS. [Copyright &y& Elsevier]

Abstract: This paper presents an exact solution for effective properties and local fields of periodic layered composites, obtained based on variational asymptotic method for unit cell homogenization, a recently developed micromechanics modeling framework. The layered composites could be made of multiple general anisotropic layers. This solution reproduces some published results when specialized to layered composites made of two orthotropic or isotropic layers and is located within the correctly interpreted bounds of the rules of mixtures according to Voigt and Reuss. It is also emphasized in this paper that the rules of mixtures according to Voigt and Reuss are absolute upper and lower bounds for the effective properties of layered composites if the bounds are interpreted in terms of the stiffness matrix or the fourth-order elasticity tensor. The confusion in a few recent papers regarding effective Young''s modulus of layered isotropic composites could exceed the Voigt and Reuss estimates is clarified. [Copyright &y& Elsevier]

Abstract: The variational asymptotic method is used to construct a thermomechanical model for homogenizing heterogeneous materials made of temperature-dependent constituents subject to finite temperature changes with the restriction that the strain is small. First, we presented the derivation for a Helmholtz free energy suitable for finite temperature changes using basic thermodynamics concepts. Then we used this energy to construct a thermomechanical micromechanics model, extending our previous work which was restricted to small temperature changes. The new model is implemented in the computer code VAMUCH using the finite element method for the purpose of handling real heterogeneous materials with arbitrary periodic microstructures. A few examples including binary composites, fiber reinforced composites, and particle reinforced composites are used to demonstrate the application of this model and the errors introduced by assuming small temperature changes when they are not necessarily small. [Copyright &y& Elsevier]

Abstract: This paper is concerned with modeling composite beams with spanwise heterogeneity. We first formulate the original three-dimensional problem in an intrinsic form which admits a geometrically exact formulation. Taking advantage of slenderness of beam structure and smallness of heterogeneity, we use the variational asymptotic method to systematically obtain an effective beam model through simultaneous homogenization and dimensional reduction. This approach is implemented in the commercial code VABS using the finite element technique for the purpose of dealing with composite beams with spanwise heterogeneity in real applications. A few examples are used to demonstrate the capability of this new model. [Copyright &y& Elsevier]

Abstract: The variational asymptotic method is used to construct a new model for composite plates which could have in-plane heterogeneity due to both geometry and material. We first formulate the original three-dimensional problem in an intrinsic form which is suitable for geometrically nonlinear analysis. Taking advantage of smallness of the plate thickness and heterogeneity, we use the variational asymptotic method to rigorously construct an effective plate model unifying a homogenization process and a dimensional reduction process. This approach is implemented in the computer code VAPAS using the finite element technique for the purpose of dealing with realistic heterogeneous plates. A few examples are used to demonstrate the capability of this new model. [Copyright &y& Elsevier]

Abstract: In this paper, a generalized Timoshenko model has been developed for prismatic, beam-like slender structures with embedded or surface mounted piezoelectric type smart materials. Starting from a geometrically exact formulation of the original, three-dimensional electromechanical problem, we apply the variational asymptotic method to carry out a systematic dimensional reduction. In the process, the three-dimensional electromechanical enthalpy functional is approximated asymptotically using the slenderness as the small parameter to find out an equivalent one-dimensional electromechanical enthalpy functional. For Timoshenko-like refinement over the Euler–Bernoulli beam model, terms up to the second order of the slenderness are kept in the enthalpy expression. As an unified analysis tool, the present model can analyze embedded or surface mounted active layer with arbitrary cross-sectional geometry as two cases of a general one, no special assumptions or modifications need to be made for these two separate types of active inclusions. [Copyright &y& Elsevier]

Abstract: An electromechanical Reissner–Mindlin model is constructed for laminated piezoelectric plates using the variational asymptotic method. This model is applicable to laminates without prescribed electric potential through the thickness. Taking advantage of the smallness of the plate thickness, we rigorously split the original 3D piezoelectricity problem into a 1D through-the-thickness analysis and a 2D plate analysis, and both are fully-coupled electromechanical analyses. Examples of single layer and multi-layer plates have been used to demonstrate the accuracy and application of this model. [Copyright &y& Elsevier]

Abstract: In the present paper, a new generalized Timoshenko model is constructed for a composite rod with embedded or attached piezoelectric materials. This model is applicable to composite rods without prescribed electric potential along the lateral surfaces. The Variational-Asymptotic Method (VAM) is applied as a mathematical tool to carry out the dimensional reduction process. The present reduced model captured the effects of dielectric as well as the polarization of the piezoelectric material, which justifies its coupled electromechanical nature. First, the three-dimensional electromechanical enthalpy is asymptotically approximated by VAM using the slenderness of the rod as the small parameter and subsequently an equivalent one-dimensional electromechanical enthalpy is developed. Energy terms, which are asymptotically correct up to the second order are kept in the approximate enthalpy expression. For engineering applications, the approximate enthalpy is then transformed into a generalized Timoshenko model which has the traditional six mechanical degrees of freedom along with an extra one-dimensional electric degree of freedom. [Copyright &y& Elsevier]

Abstract: In this paper, a new micromechanics model is developed to predict the effective properties and local fields of heterogeneous piezoelectric materials using the variational asymptotic method for unit cell homogenization (VAMUCH), a recently developed micromechanics modeling technique. Starting from the total electric enthalpy of the heterogenous continuum, we formulate the micromechanics model as a constrained minimization problem using the variational asymptotic method. To handle realistic microstructures in engineering applications, we implement this new model using the finite element method. For validation, a few examples are used to demonstrate the application and accuracy of this theory and the companion computer program – VAMUCH. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to critically evaluate two Reissner–Mindlin type theories developed recently for composite laminated plates, namely, VAPAS (Variational Asymptotic Plate And Shell analysis) and EFSDT (Enhanced First-Order Shear-Deformation Theory). The fundamentals of both models are briefly summarized along with their unique features in comparison to most other existing models. The similarities and differences between VAPAS and EFSDT are also examined. Exact solutions of three-dimensional elasticity theory for the cylindrical bending problems are used as the arbiter to assess the accuracy of both models. Such a systematic assessment demonstrates that both models have achieved an excellent compromise between the layer-wise theories, which are accurate but computationally demanding, and the first-order shear deformation theories, which are computationally cheap but not accurate. [Copyright &y& Elsevier]

Abstract: An asymptotically correct classical beam model has been developed for smart slender structures using the variational asymptotic method. Taking advantage of the slenderness of the structure, we asymptotically split the original three-dimensional electromechanical problem into a two-dimensional electromechanical cross-sectional analysis and a one-dimensional beam analysis. The one-dimensional beam analysis could be geometrically nonlinear or linear depending whether the original three-dimensional analysis is geometrically nonlinear or linear. The cross-sectional analysis, implemented using the finite element method, provides an asymptotically correct, one-dimensional constitutive model for smart slender structures without a priori assumptions regarding the geometry of the cross section, the distribution of the electric field, and the location of smart materials, such as embedded or surface mounted. Several examples are used to validate the accuracy of the present theory with available results in the literature and three-dimensional commercial finite element packages. [Copyright &y& Elsevier]

Abstract: A variational asymptotic micromechanics model has been developed for predicting effective thermoelastic properties of composite materials, and recover the local fields within the unit cell. This theory adopts essential assumptions within the concept of micromechanics, achieves an excellent accuracy, and provides a unified treatment for 1D, 2D, and 3D unit cells. This theory is implemented using the finite element method into the computer program, VAMUCH, a general-purpose micromechanics analysis code. Several examples are used to validate the theory and the code. The results are compared with those available in the literature and those produced by a commercial finite element package. [Copyright &y& Elsevier]

An analytic method is presented to study the secondary instability (bifurcation) and mode jumping of thermomechanically deep buckled composite laminates. Unlike most ad hoc approaches, the governing partial differential equations (PDEs) and constitutive relations are rigorously derived from an asymptotically correct, geometrically nonlinear theory. A novel and simpler solution approach is developed to solve the two coupled fourth-order PDEs, namely, the compatibility equation and the dynamic governing equation. The generalized Galerkin method is used to solve boundary value problems corresponding to antisymmetric angle-ply and cross-ply composite plates. The variety of possible modal interactions is expressed in an explicit and concise form by transforming the coupled nonlinear governing equations into a system of nonlinear ordinary differential equations (ODEs). The comparison between the present method and the finite element analysis (FEA) shows a pretty good match in their numerical results in the primary postbuckling region. While the FEA may lose its convergence when solution comes close the secondary bifurcation point, the analytic approach has the capability of exploring deeply into the post-secondary buckling realm and capture the mode jumping phenomenon for various combinations of plate configurations and in-plane boundary conditions. Qualitatively different propagations of buckling pattern are observed before and after mode jumping. Free vibration along the stable primary postbuckling and the jumped equilibrium paths are also studied. [ABSTRACT FROM AUTHOR]

Abstract: A new micromechanics model, namely, the variational asymptotic method for unit cell homogenization (VAMUCH), is developed to predict the effective properties of periodically heterogeneous materials and recover the local fields. Considering the periodicity as a small parameter, we can formulate a variational statement of the unit cell through an asymptotic expansion of the energy functional. It is shown that the governing differential equations and periodic boundary conditions of mathematical homogenization theories (MHT) can be reproduced from this variational statement. In comparison to other approaches, VAMUCH does not rely on ad hoc assumptions, has the same rigor as MHT, has a straightforward numerical implementation, and can calculate the complete set of properties simultaneously without using multiple loadings. This theory is implemented using the finite element method and an engineering program, VAMUCH, is developed for micromechanical analysis of unit cells. Many examples of binary composites, fiber reinforced composites, and particle reinforced composites are used to demonstrate the application, power, and accuracy of the theory and the code of VAMUCH. [Copyright &y& Elsevier]

A rigorous and systematic dimensional reduction of a shell-like structure is undertaken. It starts with geometrically nonlinear, three-dimensional (3-D), anisotropic elasticity theory and takes advantage of small parameters associated with the geometry. This reduction is carried out using the variational asymptotic method and splits the 3-D problem into a linear, one-dimensional (1-D), through-the-thickness analysis and a nonlinear, two-dimensional (2-D), shell analysis. The 2-D equations are put into the form of a nonlinear Reissner–Mindlin shell theory, details of which are dealt with in a separate paper. The focus of this paper is on the through-the-thickness analysis, which is solved by a 1-D finite element method and which provides two useful pieces of information: a generalized 2-D constitutive law for the shell equations, and a set of recovery relations that can be used to express the 3-D field variables through the thickness in terms of 2-D shell variables calculated in the shell analysis. The resulting analysis can be incorporated into standard Reissner–Mindlin shell finite element codes. Numerical results are compared with the exact solution, and the excellent agreement validates the fidelity of this modeling approach. [Copyright &y& Elsevier]

Quick and accurate predictions of cross-sectional properties are required in the preliminary design of thin-walled composite beams. Existing literature provides accurate analytical solutions to the problem, albeit limited to open and single cell sections with stress continuity constraints imposed. Mechanics of structure genome has been used to develop an asymptotically correct model for thin-walled open, closed, single and multi-cell cross sections consisting of any arbitrary anisotropic material without imposing any additional conditions. In this paper, Euler-Bernoulli thin-walled beam cross-sectional model is developed in a finite element framework and demonstrated using several examples, followed by a comparison with existing approaches. • Applied MSG to model thin-walled beams for the first time using degenerated elements. • Addressed multi-cell problem with no additional constraints. • Improved accuracy of results by considering off-centric reference axis of plate segments. • Reduced computational power for calculating thin-walled cross-sectional properties. • Generated beam properties for realistic wind turbine blade with more than 50 layers. [ABSTRACT FROM AUTHOR]

Accurate predictions of physically nonlinear elastic behaviors of a material point in the structure are essential to the further analyses which are beyond the linear elasticity regime, for example, the progressive damage and the failure. In light of substantial experimental evidence of nonlinear shear stress-strain responses in composites, it is necessary to consider them in the structure-level simulations rigorously. A variational asymptotic beam model is developed for this purpose. The three-dimensional continuum is rigorously reduced to a two-dimensional cross-sectional analysis and a one-dimensional Euler-Bernoulli beam analysis. The original three-dimensional continuum features material nonlinearities in longitudinal shear. The unknown cross-sectional warping is solved by finite element method using the principle of virtual work. Nonlinear beam constitutive relation and three-dimensional stress and strain fields are obtained. [ABSTRACT FROM AUTHOR]

The effective properties as well as local fields of heterogenous magneto-electro-elastic (MEE) materials with coated fibers are investigated by using the variational asymptotic homogenization method. Starting from the total electromagnetic enthalpy of the heterogenous continuum, the multiphysics micromechanics model is formulated as a constrained minimization problem taking advantage of the face that the size of the microstructure is small compared to the macroscopic size of the material. To treat general microstructure of smart material in engineering applications, this new model is implemented using finite element technique. A comparison with the finite element analysis (FEA) results demonstrates the application and accuracy of the proposed model for prediction of multiphysical behavior. The effects of the coating with different thickness and stiffness on the stress concentration factor and effective MEE properties are discussed and a lot of interesting MEE interaction phenomena are revealed, which are useful to estimate and optimize the performance of such MEE composites. [ABSTRACT FROM AUTHOR]

An analytical investigation is performed to study the secondary instability and dynamic aspects of the mode jumping in hygrothermally buckled cross-ply laminated plates. The governing partial differential equations (PDEs) and constitution relations are derived rigorously from an asymptotically correct, geometrically nonlinear theory. A novel and relatively simpler solution approach is developed to solve the two coupled fourth-order PDEs, namely, the compatibility equation and the dynamic governing equation. The von Kàmàn plate equation, namely, the coupled nonlinear governing equation, is reduced to a system of nonlinear ordinary differential equations (ODEs) by expressing the transverse deflection as a series of linear buckling modes. The ODEs, combined with the non-linear algebraic constraint equations obtained from in-plane boundary conditions, are then solved numerically under the parametric variation of the temperature and moisture. The comparison between the present method and the FEA shows that the secondary bifurcation point of the hygrothermally loaded plate is far beyond the primary buckling point, and the jump behavior cannot be predicted correctly without sufficient assumed modes, while the present method has the capability of exploring deeply into the post-secondary buckling realm and capturing the mode jumping phenomenon for various combinations of plate configurations boundary conditions. Furthermore, by monitoring free vibration along the stable primary postbuckling and the jumped equilibrium paths, we find that a mode shifting phenomenon (the exchange of vibration modes) exists in the primary post-buckling regime. [ABSTRACT FROM AUTHOR]

Abstract: An equivalent classical plate model of corrugated structures is derived using the variational asymptotic method. Starting from a thin shell theory, we carry out an asymptotic analysis of the strain energy in terms of the smallness of a single corrugation with respect to the characteristic length of macroscopic deformation of the corrugated structure. We obtained the complete set of analytical formulas for effective plate stiffnesses valid for both shallow and deep corrugations. These formulas can reproduce the well-known classical plate stiffnesses when the corrugated structure is degenerated to a flat plate. The extension–bending coupling stiffnesses are obtained the first time. The complete set of relations are also derived for recovering the local fields of corrugated structures. [Copyright &y& Elsevier]

Highlights: [•] VAM is adopted as mathematical foundation without any additional ad hoc assumptions. [•] All types of UCs are handled uniformly and numerical implemented straightforwardly. [•] Different material properties in different directions are obtained simultaneously. [•] The effective properties and the local fields of magnetostrictive composites have been accurately predicted. [•] Both the multiscale asymptotic series and the periodic boundary conditions are derived. [Copyright &y& Elsevier]

Abstract: A magneto-electro-elastic Reissner–Mindlin model is developed for heterogeneous multilayer laminates made of functionally graded magneto-electro-elastic material using the variational asymptotical method. This model is applicable to laminates without prescribed electric and magnetic potential though the thickness. Taking advantage of the smallness of the plate thickness, we asymptotically decouple the original 3D magneto-electro-elastic problem into a 1D through-the-thickness analysis and a 2D plate analysis, and both are fully-coupled magneto-electro-elastic analyses. Interface continuity conditions are explicitly derived and solved to obtain the multilayer solutions. Various examples have been used to demonstrate the accuracy and application of this model. [Copyright &y& Elsevier]

Abstract: The variational asymptotic method is used to construct a fully coupled Reissner–Mindlin model for piezoelectric and piezomagnetic laminates with some surfaces parallel to the reference surface coated with electrodes and magnetism. Taking advantage of the smallness of the plate thickness, we asymptotically split the original 3D electromagneto-mechanical problem into a 1D through-the-thickness analysis and a 2D plate analysis, and both are fully-coupled multiphysics analyses. The through-the-thickness analysis serves as a link between the original 3D analysis and the plate analysis by providing a constitutive model for the plate analysis and recovering the 3D field variables in terms of global responses calculated by the plate analysis. The present theory is implemented into the computer program VAPAS (Variational Asymptotic Plate and Shell Analysis). A numerical example of three-layer sandwich plate has been used to validate the present model. [Copyright &y& Elsevier]

Abstract: An efficient shell model is developed for analyzing one-way coupled thermomechanical behavior of composite cylindrical shells by using the variational asymptotic method (VAM). Taking advantage of the smallness parameter inherent in the shell structure, the VAM is applied to rigorously decouple the 3-D, thermoelasticity problem into a 1-D through-the-thickness analysis and a 2-D shell analysis. The through-the-thickness analysis servers as a link between the original 3-D analysis and the shell analysis by providing a constitutive model for the shell analysis and recovering the 3-D field variables in terms of global responses calculated by the shell analysis. The present model is valid for large displacements and global rotations and can capture all the geometric nonlinearity of a shell when the strains are small. A few examples are used to validate this model. [Copyright &y& Elsevier]

Abstract: An efficient high-fidelity shell model is developed for heterogeneous multilayer cylindrical shells made of functionally graded material by using the variational asymptotic method (VAM). Taking advantage of the smallness parameters inherent in the shell structure, the VAM is applied to rigorously decouple the 3-D, anisotropic elasticity problem into a 1-D through-the-thickness analysis and a 2-D shell analysis. The through-the-thickness analysis servers as a link between the original 3-D analysis and the shell analysis by providing a constitutive model for the shell analysis and recovering the 3-D field variables in terms of global responses calculated by the shell analysis. The present model is valid for large displacements and global rotations and can capture all the geometric nonlinearity of a shell when the strains are small. A cylindrical bending example of a homogeneous substrate with a thin SiC–Al functionally graded coating under sinusoidal pressure on the top surface is used to validate this model. [Copyright &y& Elsevier]

Abstract: The variational asymptotic method (VAM) is used to develop a beam theory for the planar deformation of isotropic strips with initial in-plane curvature. This VAM-based theory provides asymptotically exact expressions for the strain energy per unit length and the cross-sectional stress and strain in terms of 1D generalized strain variables. It is first linearized and successfully verified by comparison with elasticity solutions in the literature. It is then used to carry out verification for certain aspects of the computer program VABS (variational asymptotic beam section). In particular, the comparison shows that variations of cross-sectional stiffnesses vs. initial curvature obtained from VABS 3.3 are in error, and that corrections embodied in VABS 3.4 capture the correct behavior. Finally, two different methodologies are proposed, both of which can be implemented in VABS in the future to improve the results: a partial transformation to generalized Timoshenko theory and evaluation of the second-order warping. [Copyright &y& Elsevier]