Maximum curvatures of 0/90 plates under thermal stress: Modelling and experimental validation

In this paper, special attention is paid to the maximum curvatures induced by uniform thermal fields in 0_m/90_n laminated square plates. The effects of the relative thickness of the 90° ply e₉₀ and of the elastic and thermoelastic anisotropy on the curvature tensor are then studied. The conditions for which the curvatures are maximum are established by adopting the polar method within the context of the classical lamination theory. It is found that the location of the maximum curvatures is influenced by the level of elastic anisotropy, defined by the polar parameter R₁, while the magnitude of the maximum curvatures is driven by the thermoelastic properties, that is, the coefficients of thermal expansion. The capabilities of a geometrical nonlinear model are explored and experimental evidence of maximum curvatures of glass/epoxy composite materials are provided.     

The effect of thermo-oxidation on matrix cracking of cross-ply [0/90]S composite laminates

The present article focuses on the effects of thermo-oxidation on matrix cracking in cross-ply [0/90]_S composite laminates. IM7/977-2 carbon/epoxy samples were firstly aged at 150 °C under 1.7 bars of oxygen for 24 h, 48 h and 96 h, respectively. Quasi-static tensile tests were then carried out on un-aged and aged samples. The number of matrix cracks was counted during the tensile tests in order to establish the evolution of the crack density as a function of the applied stress and a numerical model was employed to evaluate the critical energy release rate of un-aged and aged laminates. A reduction of the critical energy release rate of aged samples was measured compared to un-aged sample. Scanning Electron Microscopy (SEM) observations were carried out by replicas of the sample surfaces in order to identify a possible relationship between the thermo-oxidation induced damage at the local scale and the onset of matrix cracking at ply scale.     

Predicting loss of bifurcation behaviour of 0/90 unsymmetric composite plates subjected to environmental loads

This paper presents an efficient Rayleigh–Ritz numerical model to predict the deformed shape and the multistable behaviour of free-standing square and rectangular 0/90 unsymmetric composite plates subjected to thermal and environmental loads. The out-of-plane displacement functions are generated through proper development of the analytical solution presented by Ashwell [34] for the nonlinear pure bending of isotropic plates; the in-plane displacement functions are formulated following a procedure similar to that employed by Galletly and Guest [32], starting from geometrical considerations about the actual deformations of the distorted structure. The resulting model is characterised by high-order displacement functions and by few unknown terms, it is appropriate to achieve sufficient accuracy, with high efficiency. The performance of the proposed approach is compared to that of some alternative models available in the literature, to FE simulations and to experimental results.     

Advanced calculation of the room-temperature shapes of thin unsymmetric composite laminates

It is well known that the room-temperature shapes of unsymmetric laminates do not always conform to the predictions of classical lamination theory. Instead of being saddle shaped, as classical lamination theory predicts, the room-temperature shapes of unsymmetrically laminated composites are often cylindrical in nature. In addition, a second cylindrical shape can sometimes be obtained from the first by a simple snap-through action. Hyer developed for the class of all square unsymmetric cross-ply laminates which can be fabricated from four layers i.e., [0₃/90], [0₂/90/0], [0/90/0/90], [0₂/90₂], an extended classical lamination theory to predict whether these laminates have a saddle shape or one or two cylindrical shapes. The Finite Element Analysis (FEA) has just recently been used for the calculation of the room-temperature shapes of unsymmetric laminates, because more sophisticated finite element codes are now available and the calculations can be made in an acceptable time. The hope is to get more accurate results for the shape and the stresses and forces that occur during the snap through action. These results are needed for the development of active deformable composite structures based on unsymmetric laminates and incorporated shape memory alloy wires [Schlecht M. & Schulte K., Development of active deformable structures due to thermal residual stresses and incorporating shape memory alloys. In Proc. ECCM Smart Composites Workshop, ECCM6, Bordeaux, 1993, pp. 20–115.] Results for different lay-ups are presented and compared.     

Phenomena in the bifurcation of unsymmetric composite plates

The phenomena observed in the large scale bifurcation of unsymmetric composite plates are characterized. Such plates can exhibit more than one stable geometry and can be ‘snapped’ from one geometry to another. It is convenient to think of this process as providing sufficient strain energy to overcome an energy maximum and is analogous to an activation energy. To describe the snap-through process, 2-ply, [0, 90] unsymmetric plates were forced from one state to the other in rigid test rigs such that great control could be achieved over the process and loads and deformations were monitored throughout the bifurcation. A hitherto unrecognised phenomenon was noted in that the snap-through process was seen to take place not as a single bifurcation but as two closely coupled bifurcations, as first one side snaps through and then the other. The experimental approach taken here, coupled with the simple energy analysis, should be of value in understanding practical structures of multiple stable states, such as might be used in morphing or adaptive aircraft.     

复合材料非对称正交薄层板的固化变形

利用Rayleigh-Ritz 法研究了复合材料非对称正交薄层板的固化变形。建立了考虑几何非线性的固化变形分析模型, 预报了其固化后的变形形状及变形量。利用热压釜工艺进行了实验研究。实验发现, 方板边缘发生了较大的向内卷曲变形, 板边缘附近理论预报值与实验结果差别较大, 在距板边缘一定距离内理论预报值和实验结果吻合较好, 矩形板实验结果与理论预报值吻合良好。      

Effect of microstructure on the thermo-oxidation of solid isotactic polypropylene-based polyolefins

In the present work we aim to clarify the role of the microstructure and the crystalline distribution from the thermo-oxidation of solid isotactic PP (iPP) and ethylene-propylene (EP) copolymers. The effects of the content and quality of the isotacticity interruptions, together with the associated average isotactic length, on the induction time (t_i) as well as on the activation energy (E_act) of the thermo-oxidation are analysed. Both parameters have been found to change markedly at an average isotactic length (n₁) of 30 propylene units. While t_i reaches a minimum when n₁ is approximately 30 units, E_act increases quasi-exponentially as the number of units decreases from 30. This variation can be explained in terms of changes induced in the crystalline interphase, i.e. local molecular dynamics, which are closely linked to the initiation of the thermo-oxidation of isotactic PP-based polyolefins.     

Effect of microstructure on the thermo-oxidation of solid isotactic polypropylene-based polyolefins

Abstract

In the present work we aim to clarify the role of the microstructure and the crystalline distribution from the thermo-oxidation of solid isotactic PP (iPP) and ethylene-propylene (EP) copolymers. The effects of the content and quality of the isotacticity interruptions, together with the associated average isotactic length, on the induction time (t_i) as well as on the activation energy (E_act) of the thermo-oxidation are analysed. Both parameters have been found to change markedly at an average isotactic length (n₁) of 30 propylene units. While t_i reaches a minimum when n₁ is approximately 30 units, E_act increases quasi-exponentially as the number of units decreases from 30. This variation can be explained in terms of changes induced in the crystalline interphase, i.e. local molecular dynamics, which are closely linked to the initiation of the thermo-oxidation of isotactic PP-based polyolefins.     

An enhanced FSDT model for the calculation of interlaminar shear stresses in composite plate structures

In this paper a procedure is proposed to calculate the interlaminar shear stresses in layered composite plates. The transverse shear stresses are obtained via the constitutive law and derivatives of some warping functions. For 4-node elements the derivatives of curvatures and strains of the reference surface with respect to the in-plane coordinates are determined through a system of four equations. Hence the equilibrium equations lead to a coupled system of ordinary differential equations, which are solved applying a displacement method. The resulting interlaminar shear stresses are continuous at the layer boundaries. The quality of the obtained results is demonstrated within several plate examples with symmetric and unsymmetric lay-ups. Comparisons with two other approaches using 9-node elements and a solid shell formulation together with a three-dimensional material law show good accuracy and efficiency of the proposed algorithm.     

Influence of neutral surface position on the nonlinear stability behavior of functionally graded plates

Nonlinear behavior of functionally graded material (FGM) skew plates under in-plane load is investigated here using a shear deformable finite element method. The material is graded in the thickness direction and a simple power law based on the rule of mixture is used to estimate the effective material properties. The neutral surface position for such FGM plates is determined and the first order shear deformation theory based on exact neutral surface position is employed here. The present model is compared with the conventional mid-surface based formulation, which uses extension-bending coupling matrix to include the noncoincidence of neutral surface with the geometric mid-surface for unsymmetric plates. The nonlinear governing equations are solved through Newton–Raphson technique. The nonlinear behavior of FGM skew plates under compressive and tensile in-plane load are examined considering different system parameters such as constituent gradient index, boundary condition, thickness-to-span ratio and skew angle. An erratum to this article can be found at     

Three-dimensional simulations of impact induced damage in composite structures using the parallelized SPH method

To address the strain-rate dependent behavior of unidirectional composites in Air Force and Navy military systems subjected to impact loading, a one-parameter visco-plasticity composite material model was developed and incorporated into the MAGI code which was parallelized for this study. This code is based on the smoothed particle hydrodynamics method. The strain-rate dependent composite model is applied here to investigate the high-velocity impact induced damage of armored composite plates which consisted of eight graphite/epoxy (Gr/Ep) layers with a lay-up of [±45/0/90]_s. The face sheets consisted of two different materials (either aluminum or boron carbide) of variable thickness. The effects of face sheet position, face sheet material types, and impact velocity on the detailed damage of the laminate are presented.     

The nonlinear response of unsymmetric multilayered plates using smeared laminate and layerwise models

A continually growing interest in the response of unsymmetric multilayered plates is apparent. Analyses were recently completed addressing the load-deflection behaviour of these plate geometries. The characteristic feature of the analyses is the use of nonlinear strain-displacement relations, even at low loading levels, in reaction to the large-deflection effect enhanced by the bending-extension and twisting-shearing coupling. Approaches where use is made of Higher Order Shear Deformation Theories (HSDT) for predicting global quantities, such as deflections and critical loads, are not found in the open literature. Such modelling approaches, in particular those of the layerwise type, are reserved to predict distributions across the thickness. Thus, a further assessment of the influence of the transverse shear effect on global quantities should be required. To give some preliminary contributions on this subject, the load-deflection behaviour of a [90₄/0₄] cross-ply plate with pinned edges, subjected to cylindrical bending under uniform transverse pressure, is investigated. Use is made of the Layerwise Higher Order Shear Deformation Layerwise Theory (RHSD) to serve this purpose. From the numerical results presented, the influence of modelling is enhanced or reduced, depending on the sign of loading. It is concluded that, depending on the loading, boundary conditions and lay-up, higher-order approaches can be used for predicting global quantities in unsymmetric multilayered plates. In order to investigate stability, nonlinear equations are developed where critical points are located under boundary and combined loading conditions which vary during perturbation.     

A simple non-layered finite element for the elasto-plastic analysis of shear flexible plates

A simple and accurate non-layered plastic element which is capable of capturing the development of plastic curvatures across the thickness of the plates is presented in this paper. The influence of transverse shear forces on the plastic behaviour of plates is also investigated here. This non-layered plastic element is based on: (i) the four-node quadrilateral strain element for shear flexible elastic plates given by the authors; (ii) the plastic hinge formulation outlined by Shi and Atluri; and (iii) the modified Ilyushin's yield function which can account for both the development of plastic deformations across the thickness of plates and the effect of transverse shear forces. The numerical examples given here demonstrate that the present non-layered plastic element can achieve the results obtained by the layered element model and the transverse shear forces considerably affect the plastic behaviour of plates under certain load conditions.     

Impact analysis of laminated composite plates and shells by super finite elements

A super finite element method that exhibits coarse-mesh accuracy is used to predict the transient response of laminated composite plates and cylindrical shells subjected to non-penetrating impact by projectiles. The governing equations are based on the classical theories of thin laminated plates and shells taking into account the von Karman kinematics assumptions for moderately large deflections. A non-linear Hertzian-type contact law accounting for curvatures of the colliding bodies is adopted to calculate the impact force . The theoretical basis of the present finite element model is verified by analysing impact-loaded laminated composite plate and shell structures that have previously been studied through analytical or other numerical procedures. The predictive capability of the present numerical approach is successfully demonstrated through comparisons between experimentally-measured and computed force-time histories for impact of carbon fibre-reinforced plastic (CFRP) plates. The current computational model offers a relatively simple and efficient means of predicting the structural impact response of laminated composite plates and shells.     

Post-buckling behavior of composite laminated plates under end shortening and pressure loading, using two versions of finite strip method

Two different versions of finite strip method, namely spline and semi-analytical methods, are developed for analyzing the geometrically non-linear response of rectangular composite laminated plates of arbitrary lay-up to progressive end-shortening in their plane and to pressure loading. The plates are assumed to be thin so that the analysis can be carried out based on the classical plate theory. The in-plane lateral deflection υ is allowed at the loaded ends of the plate, whilst the lateral expansion of the unloaded edges is either free or completely prevented. Geometric non-linearity is introduced in the strain–displacement equations in the manner of the von Karman assumptions. The formulations of the finite strip methods are based on the concept of the principle of the minimum potential energy. A number of applications involving isotropic plates, symmetric and unsymmetric cross-ply laminates are described to investigate the effects of pressure loading. The comparison between the two sets of results obtained by different finite strip methods is very good. The study of the results revealed that the response of the laminates is significantly influenced by the application of the normal pressure loading. Particularly, the response of unsymmetric laminates is strongly affected by the sign of the normal pressure loading.     

A new method for the determination of the flexural rigidity of an orthotropic plate

In this paper a new method for the determination of flexural rigidities in orthotropic plate bending problems is presented. Boundary integral equations are established for the curvatures and the deflections inside the domain. By a simple discretization of the boundary and the inside plate, the elimination of curvatures is possible. If the fundamental solution of isotropic plates is chosen, then a linear system of n equations with three unknowns is obtained. These equations are provided by the knowledge of the deflections inside the plates, and the unknowns are the flexural rigidities. By using the least square method, the computation of these rigidities becomes easy.     

Nonlinear vibration and dynamic response of shear deformable laminated plates in hygrothermal environments

This paper deals with hygrothermal effects on the nonlinear vibration and dynamic response of shear deformable laminated plates. The temperature field considered is assumed to be a uniform distribution over the plate surface and through the plate thickness. The material properties of the composite are affected by the variation of temperature and moisture, and based on a micro-mechanical model. The formulations are based on higher-order shear deformation plate theory and general von Kármán-type equation of motion, which includes hygrothermal effects. The equations of motion are solved by an improved perturbation technique to determine nonlinear frequencies and dynamic responses of shear deformable antisymmetric angle-ply and unsymmetric cross-ply laminated plates. The numerical illustrations concern the nonlinear vibration and dynamic response of the shear deformable laminated plates under different sets of hygrothermal environmental conditions. Effects of temperature rise, the degree of moisture concentration, and fiber volume fraction on natural frequencies, nonlinear to linear frequency ratios and dynamic responses are studied.     

Vibration analysis of laminated composite plates with damage using the perturbation method

The present work focuses on vibration characteristics of damaged laminated composite plates. Damage is considered as a local reduction of anisotropic plate stiffness, and three damage factors (representing the damage severity, damage anisotropy, and damage location/area, respectively) are defined to describe damage status in the laminated composite plates. The analytical solutions are obtained by the perturbation method. A numerical analysis is conducted on the vibration of damaged laminated composited plates, and the effect of damage factors on the vibration characteristics is discussed. Results indicate that three damage factors have different influences on the vibration characteristics. Also, the modal curvatures and strain energy show higher damage sensitivity than the natural frequencies and displacement mode shapes. The perturbation-based vibration analysis developed in this study can be used to effectively evaluate the effect of damage on the vibration behavior of anisotropic plates and potentially identify the damage in the laminated plates.     

Improvement of the mode II interface fracture toughness of glass fiber reinforced plastics/aluminum laminates through vapor grown carbon fiber interleaves

The effects of acid treatment, vapor grown carbon fiber (VGCF) interlayer and the angle, i.e., 0° and 90°, between the rolling stripes of an aluminum (Al) plate and the fiber direction of glass fiber reinforced plastics (GFRP) on the mode II interlaminar mechanical properties of GFRP/Al laminates were investigated. The experimental results of an end notched flexure test demonstrate that the acid treatment and the proper addition of VGCF can effectively improve the critical load and mode II fracture toughness of GFRP/Al laminates. The specimens with acid treatment and 10 g m⁻² VGCF addition possess the highest mode II fracture toughness, i.e., 269% and 385% increases in the 0° and 90° specimens, respectively compared to those corresponding pristine ones. Due to the induced anisotropy by the rolling stripes on the aluminum plate, the 90° specimens possess 15.3%–73.6% higher mode II fracture toughness compared to the 0° specimens. The improvement mechanisms were explored by the observation of crack propagation path and fracture surface with optical, laser scanning and scanning electron microscopies. Moreover, finite element analyses were carried out based on the cohesive zone model to verify the experimental fracture toughness and to predict the interface shear strength between the aluminum plates and GFRP laminates.     

Low velocity impact analysis of sandwich plates with functionally graded face sheets

This article deals with the study of low velocity impact response on sandwich plates with functionally graded face sheets. High-order sandwich plate theory is improved by considering the in-plane stresses of the core that usually are ignored in the analysis of sandwich structures. A new approach is used to reduce the equations of motion from 27 equations to 15 equations and then solving them for both unsymmetric and symmetric sandwich plates. The model is also checked by finite element simulation and by comparing with other references for validation. A parametric study is done for various geometrical and mechanical properties.