Variable Angle Tow Research Articles

Geometrically nonlinear analysis and vibration of in-plane-loaded variable angle tow composite plates and shells

This work intends to present a novel numerical approach for studying the vibration behaviours of variable angle tow (VAT) composite structures in their quasi-static nonlinear equilibrium states. This methodology is able to predict the buckling load, to investigate the natural frequencies variation for progressively higher loads, and to provide a means for verifying experimental Vibration Correlation Technique results. The use of VAT composites, in which the fibre orientations are allowed to vary along with a curvilinear pattern within each lamina, dramatically increases the design space and provides a significant improvement in buckling performance and benefits in the postbuckling regime. This study has been performed using an innovative methodology based on the well-established Carrera Unified Formulation able to describe several kinematic models for two-dimensional structures. In detail, layerwise theories are employed to characterize the complex phenomena that may appear in VAT composite structures. All Green-Lagrange strain components are employed because far nonlinear regimes are investigated. Furthermore, the geometrical nonlinear equations are written in a total Lagrangian framework and solved with an opportune Newton–Raphson method along with a path-following approach based on the arc-length constraint. Different VAT composite structures have been analyzed to validate the proposed approach and provide some benchmark solutions. The computed equilibrium paths are compared with results obtained using the commercial code ABAQUS. The results document the good accuracy and reliability of the presented methodology and show this numerical tool’s potentialities.

Global buckling and wrinkling of variable angle tow composite sandwich plates by a modified extended high-order sandwich plate theory

The advanced Automated Fibre Placement (AFP) manufacturing technologies make the synthesis of Variable Angle Tow (VAT) composites, which enable the design of lightweight sandwich structures to possess variable stiffness facesheets. However, both global and local instability phenomena of VAT sandwich plates under in-plane compressive loads are rarely explored till now. The objective of this article is to fill this gap by developing a Rayleigh–Ritz procedure based on a modified extended high-order sandwich plate theory (EHSAPT). The original two-dimensional EHSAPT proposed by Phan et al. (2012) is extended to a three-dimensional case with a minor modification that the first-order shear deformation theory instead of the classical Kirchhoff–Love hypothesis is adopted for each facesheet, which brings several merits such as the conciseness in derivation process and the easiness in program implementation. The Rayleigh–Ritz approach combined with the principle of minimum potential energy is employed to derive the eigenvalue equation that governs the instability problem of VAT sandwich plates under in-plane compressive loads. Both global buckling and wrinkling patterns of VAT sandwich plates can be captured under this proposed analytical model framework. Before instability analysis, the nonuniform prebuckling stresses over the entire sandwich plate are determined under the assumption of membrane prebuckling state. The usage of Lagrange multiplier method in the prebuckling regime removes the restrictions inherent in conventional Rayleigh–Ritz formulation and thus provides a general way to model in-plane boundary conditions. The accuracy and effectiveness of the developed Rayleigh–Ritz procedure are validated by comparing with previously published results and FE solutions given by ABAQUS. Effects of core thickness, core orthotropy, and fibre orientation angle of the facesheets on the instability behaviour of VAT sandwich plates are investigated. Finally, the mechanism of steering the fibre trajectory over the facesheets to improve the buckling resistance for the sandwich plate is studied and discussed.

Recent developments in manufacturing, mechanics, and design optimization of variable stiffness composites

The functional advantages of tailored stiffness, often seen in nature, are also utilized in composite structures. Advancements in the multiaxis tow placement and automated fiber placement (AFP) machines led to the development of variable angle tow (VAT) composites, also referred further as variable stiffness composites (VSC). These composites are shown to effectively enhance the stress distribution and buckling load capacity of structures with greater flexibility on the design space. This review systematically presents the status of recent research on the topic of VSCs. Various manufacturing techniques of VSC are discussed; constraints and the defects associated with the manufacturing processes are enlisted. The review highlights the optimization studies based on the fiber profile and macro-scale stiffness invariants. Several studies existing in the domain of buckling, vibration, and aeroelastic tailoring of angle tow composites are summarized to connect the important aspects of analysis and present a holistic approach for future studies in this area.

Dynamic analysis of prestressed variable stiffness composite shell structures

The design space for high-performance lightweight composite structures has grown considerably since the advent of the variable stiffness concept. In fact, variable stiffness composites have been found to improve buckling performance and dynamic stability, and to tune the structure’s dynamic response by tailoring structural stiffness. Thus, in order to exploit this wider design space, efficient linear analysis tools have an important role in preliminary design of variable stiffness structures, enabling designers to find more effective solutions when considering prestressed dynamically excited aerospace components. Considering this, a multi-domain Ritz method for eigenfrequency, transient and dynamic instability analysis of prestressed variable stiffness laminated doubly-curved shell structures is presented. Working within the first-order shear deformation theory, Sanders–Koiter shell kinematics allow general orthogonal surfaces to be modelled without further assumptions on the shallowness or on the thinness of the structure. The efficiency of the proposed Ritz method is granted by using Legendre orthogonal polynomials as displacement trial functions, while flexibility in modelling and design is given by penalty techniques that allow stiffened variable angle tow shell structures to be modelled as an assembly of shell-like domains. The proposed approach is verified by comparison with published benchmark results and finite element solutions. Original solutions are presented for a prestressed stiffened variable angle tow shell structure, which show great accuracy with an order of magnitude fewer variables when compared to standard finite element procedures, proving the reliability and efficiency of the present method in dealing with the dynamic analysis of multi-part aerospace structures.

Buckling Analysis and Optimization of Stiffened Variable Angle Tow Laminates with a Cutout Considering Manufacturing Constraints

Variable angle tow laminates (VAT) and stiffeners are known to redistribute the in-plane load distribution and tailor the buckling mode shapes, respectively, for improving structural performance. To leverage the benefits of using VAT laminates in the practical applications, in the present paper, we discuss buckling load maximization conducted for a stiffened VAT laminated plate with a central cutout considering VAT laminate manufacturing constraints. Three representative boundary conditions as seen in the aerospace structures are considered: in-plane axial displacement, in-plane pure shear, and in-plane pure bending displacements. Two common manufacturing constraints, the one on the automatic fiber placement (AFP) manufacturing head turning radius and the other on the tow gap/overlap, while fabricating VAT laminates are considered in the laminate design. These two manufacturing constraints are modeled by controlling the fiber path radius of curvature and tape parallelism in optimizing the fiber path orientations for the VAT laminates. Stiffener layout and fiber path angle for the VAT laminated plates are both considered in the buckling load maximization study. To avoid using a fine mesh in modeling the stiffened VAT laminates with a cutout when employing the finite element analysis during the optimization, the VAT laminated plate and the stiffeners are modeled independently. The displacement compatibility is enforced at the stiffener–plate interfaces to ensure that the stiffeners move with the plate. Particle swarm optimization is used as the optimization algorithm for the buckling load maximization study. Optimization results show that, without considering AFP manufacturing constraints, the VAT laminates can increase the buckling loads by 21.2% and 12.4%, respectively, comparing to the commonly used quasi-isotropic laminates and traditionally straight fiber path laminates for the structure under the in-plane axial displacement case, 19.7% and 12.5%, respectively, for the in-plane shear displacement case, and 62.1% and 26.6%, respectively, for the in-plane bending displacement case. The AFP manufacturing constraints are found to have different impacts on the buckling responses for the VAT laminates, which cause the maximum buckling load to be 9.3–10.1%, 3.0–3.2%, and 23.2–29.8% less than those obtained without considering AFP manufacturing constraints, respectively, for the present studied model under in-plane axial, shear, and bending displacements.

Partitioned parametrized variational procedure for the generation of theorems of structural analysis

Structural field modeling of displacements, stresses, and strains is conceptually linked with the generation of functionals. In particular, this work introduces for the first time the Partitioned Parametrized Variational Procedure (PPVP). In-plane stresses and strains are mathematically treated differently than the transverse counterparts. This allows one to generate all the classical variational theorems of elasticity (such as the Principles of Virtual Displacements and Complementary Energy, Hellinger-Reissner, and the Veubeke-Hu-Washizu theorems) and additional new types of variational theorems which can be exploited for the solution of structural problems in various areas, such as reinforced panels and variable angle tow composites. With the proposed variational procedure energetically consistent constraints naturally arise directly from the formulation. PPVP could also be used to formulate new enhancements of the finite element or meshless methods and to quantify functional errors not related to the numerical discretization.

Fundamental Frequency Optimization of Variable Angle Tow Laminates with Embedded Gap Defects

Variable stiffness composite laminates can improve the structural performance of composite structures by expanding the design space. This work explores the application of variable stiffness laminated composite structures to maximize the fundamental frequency by optimizing the tow angle. To this end, an optimization framework is developed to design the fiber angle for each layer based on the maximization of the fundamental frequency. It is assumed that the design process includes the manufacturing constraints encountered in the automated fiber placement process and a linear fiber angle variation. The current study improves existing results by considering embedded gap defects within the optimization framework. The plates are assumed symmetric, with clamped and simply supported boundary conditions. The optimal results and a comparison between the non-steered and steered plates with and without gaps are presented. Results show that, although gaps deteriorate the structural performance, fiber steering can still lead to an increase in the fundamental frequency depending on the plate’s geometry and boundary conditions.

Steering of carbon fiber/PEEK tapes using Hot Gas Torch-assisted automated fiber placement.

In-situ manufacturing of thermoplastic composites using Hot Gas Torch (HGT)-assisted Automated Fiber Placement (AFP) has the potential to produce laminates in an efficient manner by avoiding a secondary process, like autoclave consolidation. One of the advantages of AFP technique is its capability to steer fiber path and to manufacture Variable Angle Tow (VAT) laminates which have shown to have improved mechanical performance. This study investigates the process parameters that affect steering of carbon fiber reinforced thermoplastic tapes (AS4/polyether ether ketone) using an HGT-assisted AFP machine. The effect of the steering radius, laydown speed, number of repasses, and substrate angle on the geometry and bond strength of steered tape was investigated through observation and testing. A modified lap shear test was devised and used to study the bond strength between the steered tape and the substrate and the results were compared with autoclave treated samples which served as a reference. It was found that with a decrease in the steering radius of the tape, there was a decrease in the tape width and an increase in the tape thickness. A significant reduction in the steering-induced defects was observed at higher laydown speeds where the defects were intermittent unlike in the case of lower laydown speeds. Performing a repass over the steered tape smoothed some of the tape defects caused by steering. Furthermore, the lap shear strengths of the steered tapes were found to be functions of laydown speed and substrate angle.

Free vibrations analysis of cracked variable stiffness composite plates by the eXtended Ritz method

Variable stiffness composite laminates show advantageous structural features related to their enlarged design space. They are attractive candidates for advanced engineering applications where the assessment of static and dynamic behavior and strength in the presence of cracks is often required. In the present work, a single-domain extended Ritz formulation is proposed to study the free vibrations of cracked variable stiffness composite plates. The plate model is based on the first-order shear deformation theory whose primary variable, i.e. displacements and rotations, are approximated via a set of orthogonal polynomial trial functions enriched with a set of special crack functions. These functions are able to inherently account for crack opening and crack tip singular fields. The plate governing equations are deduced by the stationarity of the energy functional and the formulation has been implemented in a computer code. The method has been validated by comparing the present results with literature solutions for cracked isotropic plates and uncracked variable stiffness plates as, to the best of author’s knowledge, no data on cracked variable stiffness plates free vibrations are available. An explicative and representative study on the free vibrations of variable angle tow composite laminates is finally presented with the aim of illustrating the approach capabilities, providing benchmarck results and identifying distinctive features and opportunities of the variable stiffness concept for the design of advanced damage tolerant structures.

GA Optimization of Variable Angle Tow Composites in Buckling and Free Vibration Analysis through Layerwise Theory

In the current research, variable angle tow composites are used to improve the buckling and free vibration behavior of a structure. A one-dimensional (1D) Carrera Unified Formulation (CUF) is employed to determine the buckling loads and natural frequencies in Variable Angle Tow (VAT) square plates by taking advantage of the layerwise theory (LW). Subsequently, the Genetic Algorithm (GA) optimization method is applied to maximize the first critical buckling load and first natural frequency using the definition of linear fiber orientation angles. To show the power of the genetic algorithm for the VAT structure, a surrogate model using Response Surface (RS) method was used to demonstrate the convergence of the GA approach. The results showed the cost reduction for optimized VAT performance through GA optimization in combination with the 1D CUF procedure. Additionally, a Latin hypercube sampling (LHS) method with RS was used for buckling analysis. The capability of LHS sampling confirmed that it could be employed for the next stages of research along with GA.

A Koiter reduction technique for the nonlinear thermoelastic analysis of shell structures prone to buckling

AbstractThe multi‐modal Koiter method is a reduction technique for estimating quickly the nonlinear buckling response of structures under mechanical loads requiring a fine discretization. The reduced model is based on a quadratic approximation of the full model using a few linear buckling modes and their second order corrections, followed by the projection of the equilibrium equations onto the modal subspace. In this article, the method is reformulated for geometrically nonlinear thermoelastic analyses of shell structures. The starting point is an isogeometric solid‐shell discretization with an accurate modeling of thermal strains, temperature‐dependent materials and general temperature distributions. The equilibrium path is defined as a displacement versus temperature amplifier curve, while mechanical loads are kept constant. The strain energy nonlinearity with respect to the temperature amplifier is the main obstacle in the definition of an accurate reduced model. This task is achieved by coherent asymptotic expansions in mixed form using independent stress variables at the integration points and accurate linear buckling modes obtained by a two‐point mixed eigenvalue problem. Structures made of isotropic and multi‐layered composite materials, including variable angle tow laminates, are considered to show the accuracy of the proposed method.

Accurate Stress Analysis of Variable Angle Tow Shells by High-Order Equivalent-Single-Layer and Layer-Wise Finite Element Models.

New concepts of lightweight components are conceived nowadays thanks to the advances in the manufacture of composite structures. For instance, mature technologies such as Automatic Fibre Placement (AFP) are employed in the fabrication of structural parts where fibres are steered along curvilinear paths, namely variable angle tow (VAT), which can enhance the mechanical performance and alleviate the structural weight. This is of utmost importance in the aerospace field, where weight savings are one of the main goals. For that reason, shell structures are commonly found in the aerospace industry because of their capabilities of supporting external loadings. Straight-fibre composite shell structures have been studied in recent decades and, now, spatially varying composite shells are attracting the attention of manufacturers. This work analyses the mechanical behaviour of VAT composite shells subjected to different external loadings and boundary conditions. The Carrera Unified Formulation (CUF) is employed to obtain the different structural models in a systematic and hierarchic manner. The outcomes of such numerical models are discussed and compared with commercial software Abaqus.

A complex Fourier series solution for buckling of VAT composite laminates with elastically restrained edges

Variable Angle Tow (VAT) composites have general anisotropic characteristics, and their fiber orientation angle varies continuously along an arbitrary direction. Therefore, the investigation of their buckling performance is quite challenging. Different from the commonly used numerical methods, a new method called complex Fourier series solution is proposed to study the buckling of VAT laminates with elastically restrained edges. First the prebuckling analysis is achieved by introducing Airy’s stress function and expanding it into a complex Fourier series form. And the corresponding mid-plane internal forces with non-uniform distribution are determined. Based on classical laminated plate theory, the buckling analysis of VAT laminate is carried out. The transverse displacement is also expressed as a complex Fourier series. And the critical buckling load and buckling mode shape of VAT laminate are obtained. The accuracy of the presented complex Fourier series solution is verified by comparing with the existing results. The influence of fiber orientation angles, boundary stiffness and boundary conditions on the critical buckling load and buckling mode shape is investigated in numerical examples. Results show that boundary stiffness as well as fiber path have great influence on the critical buckling load and buckling mode shape of VAT laminate. By adjusting the fiber orientation angles and the translational stiffness of the elastically restrained boundary can effectively improve the critical buckling load of VAT laminate. This paper provides a new method for the analysis of general anisotropic laminates with in-plane variable stiffness and the present complex Fourier series solution can also be used as a benchmark for the verification of other numerical results.

Nonlinear thermal flutter analysis of variable angle tow composite curved panels in supersonic airflow

The advanced Automated Fibre Placement (AFP) technologies make the synthesis of Variable Angle Tow (VAT) composites possible, which provides an extended space for the design of lightweight structures. At the same time, the variable stiffness feature inherent in tow-steered composite panels also poses a huge challenge to solve the mechanical problems including the panel flutter problem. In this article, the nonlinear thermal flutter characteristics of an infinitely long VAT composite curved panel in supersonic airflow is investigated for the first time. The proposed panel model is based on the Donnell’s shallow shell theory and accounts for the von Kármán’s geometrical nonlinearity. The aerodynamic load induced by the supersonic airflow passing over the panel surface is constructed from the first-order piston theory, whilst the thermal load imposed on the curved panel follows the quasi-steady thermal stress theory. The aerothermoelastic equations of motion that govern the nonlinear thermal flutter problem are determined via Hamilton’s variational principle. The Galerkin’s method is applied to convert the partial differential governing equation with variable coefficients into a set of ordinary differential equations, and the resulting nonlinear equations dependent upon time are solved through the fourth order Runge–Kutta method. Several qualitative tools, including the time history, phase portrait, Poincaré map and bifurcation diagram, are employed to study the dynamic behaviours, and a variety of attractors, both regular and chaotic, are observed. The accuracy and effectiveness of the proposed Galerkin procedure are validated by comparing with those available in literature. Effects of height-rise parameter, temperature parameter and fibre orientation angle on the nonlinear flutter responses of VAT composite curved panels in thermal environments are discussed in details.

A non-linear Ritz method for the analysis of low velocity impact induced dynamics in variable angle tow composite laminates

Variable angle tow (VAT) laminates feature composite layers reinforced by fibres following continuous curved paths and offer a wide structural design space for the manufacturing of composite components. In this work, a formulation for the analysis of the impact-induced dynamics in VAT laminated plates is proposed, implemented and tested in this work. The method is based on the adoption of first order shear deformation kinematics and includes von Karman non-linear strains. The discrete system is obtained by employing a pb-2 Ritz series expansion into the Hamilton’s variational statement, while the impact loading is modelled through Hertzian contact law. The resulting non-linear governing equations are solved using a Newmark time step integration algorithm, coupled with an iterative solution strategy. After validation against available literature data, several tests on different VAT configurations are performed, highlighting analogies and differences between different layups in terms of impact-induced dynamic response.

Overlap-stiffened panels for optimized buckling performance under minimum steering radius constraints

Recent research on variable stiffness laminates have shown both numerically and experimentally that further improvement on the buckling performance is possible by incorporating overlaps that result in variable thickness profiles. We present the concept of overlap-stiffened designs that take advantage of the non-linear coupling between the tow steering and the local thickness, allowing embedded regions of higher stiffness into individual plies of a variable-angle tow (VAT) laminate. The proposed method naturally copes with minimum steering radius constraints of different manufacturing processes by connecting transition regions by means of fiber tow arcs, such that the radius of curvature always cope with a desired minimum radius constraint. The present study focuses on two tow-steering processes: automated fiber placement (AFP) and continuous tow shearing (CTS). Each individual ply exploring the overlap-stiffened design is described using 5 design variables, producing a straight stiffener. A first benchmark study compares overlap-stiffened laminates optimized for a maximum volume-normalized buckling performance under bi-axial compression against a reference straight-fiber laminate. In a second benchmark, overlap-stiffened panels were optimized for minimum weight under a design buckling load constraint, and compared against a reference straight-fiber laminate. For both AFP and CTS, is verified that overlap-stiffened VAT panels can achieve at least the double of the volume-normalized buckling performance of an optimized straight-fiber panel. Moreover, the proposed design method can at least achieve the same weight and buckling load carrying capacity of an optimal straight-fiber panel, demonstrating the potential of the proposed design method to include embedded regions of higher thickness.

Thermal postbuckling analysis of variable angle tow composite cylindrical panels

The advanced Automated Fiber Placement (AFP) technologies allow the fiber trajectories to be curvilinearly steered over the panel platform, which brings out a novel type of cylindrical panel structure, termed as Variable Angle Tow (VAT) composite cylindrical panel. So far, the thermal postbuckling problem of VAT composite cylindrical panels under temperature loadings remain unexplored. The aim of this article is to cover this lack by developing a methodology based on a generalized Rayleigh–Ritz formulation in the framework of the principle of thermoelastic complementary energy. The proposed panel model is based on Donnell’s shallow shell theory and accounts for von Kármán’s geometrical nonlinearity. With the aid of the Lagrange multiplier method and the introduction of Airy’s stress function, a mixed variational formula that governs the nonlinear thermoelastic problem of VAT composite cylindrical panels is derived from the principle of thermoelastic potential energy. The Rayleigh-Ritz formulation combined with the mixed variational formula is then applied to derive the simultaneous nonlinear thermoelastic equations, and the Crisfield’s arc-length method, which can cope well with snap-through or snap-back problems, is employed to trace the equilibrium path that characterizes the thermal postbuckling features. Two different modeling strategies, namely, semi-inverse method and Lagrangian multiplier method, are adopted to formulate the nonlinear thermoelastic problem of VAT composite cylindrical panels undergoing temperature change. The application of Lagrangian multiplier method overcomes some limitations inherent in semi-inverse method in terms of constructing Rayleigh-Ritz formulation, and thus provides generality to model general in-plane boundary constraint. The accuracy and robustness of the proposed Rayleigh-Ritz model are validated against finite element solutions and previously published results. Effects of geometric imperfection, curvature radius, aspect ratio and fiber orientation angle on thermal postbuckling responses of VAT cylindrical plates are discussed by several examples. The study presented here may be helpful for the preliminary design of VAT curved panels in thermal environments.

Investigation of buckling characteristics of cracked variable stiffness composite plates by an eXtended Ritz approach

Variable Angle Tow (VAT) composite plates are characterized by in-plane variable stiffness properties, which opens to new concepts of stiffness tailoring and optimization to achieve higher structural performance for advanced lightweight structures where damage tolerance consideration are often mandatory. In this paper, a single-domain eXtended Ritz formulation is proposed to study the buckling behaviour of variable stiffness laminated cracked plates. The plate behaviour is described by the first order shear deformation theory whose generalized displacements, namely reference plane translations and rotations, are expressed via suitable admissible trial functions. These consist of a set of regular terms, built using orthogonal polynomials, augmented with special functions able to describe the crack opening and the singular behaviour at the crack tips; boundary functions are used to ensure the required homogeneous essential boundary conditions. Governing equations are inferred via the stationarity of the energy functional an solved to carry out an extensive study on the buckling behaviour of variable angle tow homogeneous and layered composite plates. The results obtained for homogeneous plates evidence that the crack presence strongly influences the buckling behaviour depending on its length and inclination and plate boundary conditions with a meaningful variability with respect to the fibre paths configuration, which can arrange for better performances with respect to the straight fibre case. Also for cracked laminates the results show that there are several fibre path able to provide higher buckling loads and higher overall axial stiffness with respect to the straight fibre case. In the framework of damage tolerant engineering applications, this allows to select fibre paths that guarantee predefined design levels of buckling load and axial stiffness even in presence of cracks. Finally, this study highlights the potential of the proposed approach for the analysis of the buckling behaviour of cracked composite variable stiffness plates, which provides an efficient analysis tool for the damage tolerant design and optimization of advanced variable stiffness structures.

Material Design for Optimal Postbuckling Behaviour of Composite Shells.

Lightweight thin-walled structures are crucial for many engineering applications. Advanced manufacturing methods are enabling the realization of composite materials with spatially varying material properties. Variable angle tow fibre composites are a representative example, but also nanocomposites are opening new interesting possibilities. Taking advantage of these tunable materials requires the development of computational design methods. The failure of such structures is often dominated by buckling and can be very sensitive to material configuration and geometrical imperfections. This work is a review of the recent computational developments concerning the optimisation of the response of composite thin-walled structures prone to buckling, showing how baseline products with unstable behaviour can be transformed in stable ones operating safely in the post-buckling range. Four main aspects are discussed: mechanical and discrete models for composite shells, material parametrization and objective function definition, solution methods for tracing the load-displacement path and assessing the imperfection sensitivity, structural optimisation algorithms. A numerical example of optimal material design for a curved panel is also illustrated.

Nonlinear thermoelastic analysis of shell structures: solid-shell modelling and high-performing continuation method

This work presents a cost-effective and reliable numerical framework for geometrically nonlinear thermoelastic analyses of thin-walled structures. Firstly, the structure is discretised using an isogeometric solid-shell formulation, that allows an accurate approximation of geometry and kinematics avoiding the parameterisation of finite rotations. Focus is given to an efficient modelling of thermal strains, temperature-dependent materials and general temperature profiles. Then, a generalised arc-length method is developed for solving the discrete equations with the temperature amplifier as additional unknown.A consistent definition of the tangent operators and a mixed integration point strategy lead to a robust analysis with a reduced iterative burden, even in case of complex nonlinear behaviours where standard simulation tools are outperformed or even fail. Finally, an accurate eigenvalue analysis is derived for a quick estimate of critical temperatures. The overall framework is validated by a number of applications with isotropic and multi-layered composite materials, including variable angle tow laminates.