Stochastic nonlinear failure analysis of laminated composite plates under compressive transverse loading

This paper present the second ordered statistics of first-ply failure response of laminated composite plate with random material properties under random loading. The basic formulation is based on higher order shear deformation plate theory (HSDT) with the geometrically nonlinearity in the von-Karman. The direct iterative based C⁰ nonlinear finite element method combined with mean centered first order perturbation technique developed by the authors are extended and successfully applied nonlinearity for failure problem with a reasonable accuracy to predict the second order statistics (standard deviation) of first-ply failure response using Tsai-Wu and Hoffman failure criterion with macroscopic analysis. Typical numerical results for various combinations of boundary conditions, plate thickness ratios, aspect ratios, laminates scheme and layers, elastic modulus ratios have been presented to illustrate the application of developed procedure. Some new results are presented and examined which clearly demonstrated the importance of the randomness in the system parameters in the failure response of the structures subjected to transverse loadings.     

Free vibration of composite cylindrical panels with random material properties

Virtually in all structural systems, and in particular composites, there are uncertainties in the system parameters because of practical bounds on the quality control. In the present work the effect of variations in the mechanical properties of laminated composite cylindrical panels on its natural frequency has been obtained by modeling these as random variables. The transverse shear and rotatory inertia effects have been included in the governing equations. A perturbation approach is presented to obtain the mean and variance of the random natural frequencies. The effects of thickness ratios, edge support conditions and standard deviation of material properties on response of shallow square panels have been investigated. Results have been obtained by employing the finite element method. The approach has been validated by comparison of results with other approaches.     

Free vibration of composite circular cylindrical shells with random material properties. Part II: Applications

In the second part of this study the approach developed in Part I has been used to analyse free vibration of three composite circular cylindrical shells with random scatter in the material properties. The cases considered are – specially orthotropic symmetric shells in axisymmetric and asymmetric oscillations, and antisymmetric cross ply laminated shell in axisymmetric oscillations. With known statistics of the material properties the mean and the variance of the natural frequencies have been obtained. Numerical results have been presented for graphite–epoxy composite shells.     

Stochastic nonlinear free vibration of laminated composite plates resting on elastic foundation in thermal environments

This paper presents the nonlinear free vibration analysis of laminated composite plates resting on elastic foundation with random system properties in thermal environments. System parameters are modeled as basic random variables for accurate prediction of system behavior. A C ⁰ nonlinear finite element based on HSDT in von Karman sense is used to descretize the laminate. A direct iterative method in conjunction with first-order perturbation technique is outlined and applied to solve the stochastic nonlinear generalized eigenvalue problem. The developed stochastic procedure is successfully used for thermally induced nonlinear free vibration problem with a reasonable accuracy. Numerical results for various combinations of boundary conditions, geometric parameters, amplitude ratios, foundation parameters and thermal loading have been compared with those available in literature and an independent MCS. Some new results are also presented which clearly demonstrate the importance of the randomness in the system parameters on the response of the structures.     

A further discussion of nonlinear mechanical behavior for FGM beams under in-plane thermal loading

Due to the variation in material properties through the thickness, bifurcation buckling cannot generally occur for plates or beams made of functionally graded materials (FGM) with simply supported edges. Further investigation in this paper indicates that FGM beams subjected to an in-plane thermal loading do exhibit some unique and interesting characteristics in both static and dynamic behaviors, particularly when effects of transverse shear deformation and the temperature-dependent material properties are simultaneously taken into account. In the analysis, based on the nonlinear first-order shear deformation beam theory (FBT) and the physical neutral surface concept, governing equations for both the static behavior and the dynamic response of FGM beams subjected to uniform in-plane thermal loading are derived. Then, a shooting method is employed to numerically solve the resulting equations. The material properties of the beams are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents, and to be temperature-dependent. The effects of material constants, transverse shear deformation, temperature-dependent material properties, in-plane loading and boundary conditions on the nonlinear behavior of FGM beams are discussed in detail.     

Prediction of the dynamic response of composite sandwich beams under shock loading

Finite element calculations are reported for the dynamic shock response of fully clamped monolithic and sandwich beams, with elastic face sheets and a compressible elastic–plastic core. Predictions of the peak mid-span deflections and deflected shapes of the beams are compared with the previously reported measured response of end-clamped sandwich beams, made from face sheets of glass fibre reinforced vinyl ester and a core of PVC foam or balsa wood [1]. Good agreement is observed, and the maximum sustainable impulse is also predicted adequately upon assuming a tensile failure criterion for the face sheets. The finite element calculations can also be used to bound the response by considering the extremes of a fully intact core and a fully damaged core. It is concluded that the shock resistance of a composite sandwich beam is maximised by selecting a composite with fibres of high failure strain.     

Evaluation of the residual strength degradation in composite laminates under fatigue loading

Residual strength degradation in graphite/epoxy composite laminates is evaluated and a model proposed relating the residual strength to the applied fatigue cycles and the maximum applied stress. Based on this model, the statistical distribution of the residual strength is derived and compared with available experimental data. Good agreement is observed between the proposed model and experimental results.     

Low-velocity impact analysis of composite laminates under initial in-plane load

In this paper, low-velocity impact response and damage of composite laminates is analytically investigated. A modified displacement field of the plate considering initially loaded in-plane strain is proposed. From the displacement field, a finite element equation on the structural behavior of composite laminate is newly induced and a computational program is coded. Numerical results using the FEM code is compared with the numerical ones from reference. Additional numerical analysis is performed on another impact condition, and the effect of initial in-plane load is investigated. Potential delamination damage area in the first inter-ply surface from the bottom of the laminate is approximately estimated, and the effect of the initial in-plane load and the impact condition are also investigated. Consequently, it may be concluded that the initial in-plane load of the laminate does not affect so much on the impact damage area of the laminate.     

Dynamic response of curved composite panels to underwater explosive loading: Experimental and computational comparisons

The response of E-Glass/Vinyl ester curved composite panels subjected to underwater explosive loading has been studied. The work consists of experimental testing utilizing a water filled conical shock tube facility and computational simulations with the commercially available Ls-Dyna finite element code. The composite specimens are 0/90 biaxial laminates with a thickness of approximately 1.3 mm. The samples are round panels with curved midsections. The transient response of the plates is measured using a three-dimensional (3D) Digital Image Correlation (DIC) system, along with high speed photography. This ultra high speed system records full field shape and displacement profiles in real time. The DIC data and the computational results show a high level of correlation using the Russell error measure.     

Intersonic delamination in curved thick composite laminates under quasi-static loading

Dynamic delamination in curved composite laminates is investigated experimentally and numerically. The laminate is 12-ply graphite/epoxy woven fabric L-shaped laminate subject to quasi-static loading perpendicular to one arm. Delamination initiation and propagation are observed using high speed camera and load–displacement data is recorded. The quasi-static shear loading initiates delamination at the curved region which propagates faster than the shear wave speed of the material, leading to intersonic delamination in the arms. In the numerical part, the experiments are simulated with finite element analysis and a bilinear cohesive zone model. Cohesive interface elements are used between all plies with the interface properties obtained from tests. The simulations predict a single delamination initiating at the corner under pure mode-I stress field propagating to the arms under pure mode-II stress field. The crack tip speeds transition from sub-Rayleigh to intersonic in conjunction with mode change. In addition to intersonic mode-II delamination, shear Mach waves emanating from the crack tips in the arms are observed. The simulations and experiments are found to be in good agreement at the macro-scale, in terms of load-displacement behavior and failure load, and at the meso-scale, in terms of delamination initiation location and crack propagation speeds. Finally, a mode dependent crack tip definition is proposed and observation of vibrations during delamination is presented. This paper presents the first conclusive evidence of intersonic delamination in composite laminates triggered under quasi-static loading.     

Variability response functions for apparent material properties

The computation of apparent material properties for a random heterogeneous material requires the assumption of a solution field on a finite domain over which the apparent properties are to be computed. In this paper the assumed solution field is taken to be that defined by the shape functions that underpin the finite element method and it is shown that the variance of the apparent properties calculated using the shape functions to define the solution field can be expressed in terms of a variability response function (VRF) that is independent of the marginal distribution and spectral density function of the underlying random heterogeneous material property field. The variance of apparent material properties can be an important consideration in problems where the domain over which the apparent properties are computed is smaller than the representative volume element and the approach introduced here provides an efficient means of calculating that variance and performing sensitivity studies with respect to the characteristics of the material property field. The approach is illustrated using examples involving heat transfer problems and finite elements with linear and nonlinear shape functions and in one and two dimensions. Features of the VRF are described, including dependency on shape and scale of the finite element and the order of the shape functions.     

On random characteristics of composite materials

Abstract

Theoretical formulation is described, which presents random characteristics in composite materials. The presented method is based on Poisson processes. Examples of random characteristics of composite materials are shown. The method presented can be used for evaluation of composite materials and the subsequent quantitative stochastic design as well as stochastic optimisation of composite materials.     

Fatigue lifetime under uniaxial random loading with different mean values according to some selected models

In the paper, the experimental fatigue lives of plane specimens made of 10HNAP steel have been compared with the calculated lives for random tension loading with non-zero mean value. The cycle amplitude transformation methods proposed by Goodman and Gerber, the Van Dang criterion, averaging with the Svenson–Lipp method and the strain energy density parameter was used. Cycles were counted by the rain flow algorithm; in the case of the Svenson–Lipp method the fatigue cycles were counted by the rain flow algorithm and the level crossing algorithm. Damages were accumulated according to the Palmgren–Miner hypothesis.     

Variability response functions for effective material properties

The variability response function (VRF) is a well-established concept for efficient evaluation of the variance and sensitivity of the response of stochastic systems where properties are modeled by random fields that circumvents the need for computationally expensive Monte Carlo (MC) simulations. Homogenization of material properties is an important procedure in the analysis of structural mechanics problems in which the material properties fluctuate randomly, yet no method other than MC simulation exists for evaluating the variability of the effective material properties. The concept of a VRF for effective material properties is introduced in this paper based on the equivalence of elastic strain energy in the heterogeneous and equivalent homogeneous bodies. It is shown that such a VRF exists for the effective material properties of statically determinate structures. The VRF for effective material properties can be calculated exactly or by Fast MC simulation and depends on extending the classical displacement VRF to consider the covariance of the response displacement at two points in a statically determinate beam with randomly fluctuating material properties modeled using random fields. Two numerical examples are presented that demonstrate the character of the VRF for effective material properties, the method of calculation, and results that can be obtained from it.     

Scale effects on the response of composite structures under impact loading

For several years, composite materials have taken a significant part in the realization of structures designed for transport (aeronautical, nautical, automotive…). In order to qualify the behavior of such structures, preliminary validation tests have to be done. These specific tests are often very expensive and difficult to set up, especially when the structure dimensions are large (fuselages of aircraft, ship hulls…). An alternative way is then to employ small-scale models.The use of these reduced scale structures requires the identification of similitude models allowing the extrapolation of the small-scale model behavior to the real structure. Although largely developed in the case of homogeneous materials, such similitude techniques are not clearly identified for composite materials taking into account the damage evolution during an impact.The purpose of this article is firstly to present existing similitude techniques making it possible to predict the composite structure behaviour from the knowledge of small-scale model response. Secondly, experiments were done on two scale of samples carried out by stratification of unidirectional carbon/epoxy plies. These results were finally compared with the analytical predictions of similitude laws currently used.The aim of this paper is to contribute to similitude laws development applied to composite structures. These laws permit to extrapolate the small-scale model behavior to the real scale one. Existing approaches have been established following two different methods. They are summarized in this paper and applied to impact loadings on two laminated plate scales. In order to complete data collected by “conventional” instrumentation (force transducer, displacement sensor, accelerometer…), optical device such as an high-velocity CCD camera, associated with optical techniques for the monitoring of markers, were used. These techniques make possible to compare displacement lines corresponding to each scale. It is shown that existing similitude laws, used for elastic materials, do not allow to simulate the behavior of the real scale when this one is damaged.     

爆炸荷载作用下复合材料加筋板的动力响应

为了减轻抗爆结构质量,采用玻璃纤维增强聚合物基复合材料（SMC）与碳纤维增强聚合物基复合材料（CFRP）预浸料,通过数值模拟和等效计算理论,对传统加筋抗爆板结构进行轻质高强设计。利用LS-DYNA有限元数值模拟软件进行分析,发现在爆炸荷载作用下加筋板的运动以弹性运动为主,该种复合材料具有较好的抗爆性能。对复合材料加筋板结构进行参数化分析,发现在爆炸荷载作用下横筋对加筋板结构最大位移值影响最大,纵筋和面板对加筋板的影响依次减小。结合刚度折算方法,建立了爆炸荷载作用下正交异性加筋板结构动力响应分析理论。利用该理论计算得到板结构在爆炸荷载作用下的最大位移,与数值模拟对比发现两者结果较为接近,为加筋抗爆板的设计提供了一种简化有效的计算方法。     

Compressive strain limits of composite repaired pipelines under combined loading states

Recent literature and the industry have widely accepted the benefits of strain-based design in regards to steel pipelines. In accordance with the recent increase in the use of strain-based design for pipelines, in this paper, the authors investigate composite repaired pipelines and its applicability in reference to the strain-based design. In an earlier work, the authors found that under combined loading conditions, a repaired pipe would tend to buckle locally in a location adjacent to the composite repaired wrap. In an attempt to limit the initiation of local buckling response in composite repaired pipelines, the compressive strain limit was investigated. For that, a finite element study was conducted. The results clearly showed that the maximum strains would not occur in the post-repaired defect region. The local buckling would however occur in the unrepaired (undamaged) section of the steel pipelines. An experimental study was also conducted on Fibre Reinforced Polymers (FRP) repaired pipelines to verify the finite element results. A parametric computational investigation was also conducted to understand the limits of composite repair wraps.     

Transient response of composite sandwich plates

The transient response of orthotropic, layered composite sandwich plates is investigated by using two new C⁰ four and nine node finite element formulations of a refined form of Reddy's higher-order theory. This refined third order theory accounts for parabolic variation of the transverse shear stresses, and requires no shear correction factors. The assumed strain approach is employed to model both thin and thick plates without any major defects like shear locking and parasitic spurious zero energy modes. A consistent mass matrix formulation is adopted. The Newmark direct integration scheme is used to solve the governing equilibrium equations. The parametric effects of plate aspect ratio, length to thickness ratio, boundary conditions and lamination scheme on the transient response are investigated. The present results are in very close agreement with earlier published results in the literature and can serve as a benchmark for future investigators.     

On numerical evaluation of effective material properties for composite structures with rhombic fiber arrangements

This paper deals with unidirectional fiber reinforced composites with rhombic fiber arrangements. It is assumed, that there is a periodic structure on micro level, which can be taken by homogenization as a representative volume element (RVE) for the composite, where the composite phases have isotropic or transversely isotropic material characterizations. A special procedure is developed to handle the primary non-rectangular periodicity with common numerical homogenization techniques based on FE-models. Due to appropriate boundary conditions applied to the RVE elastic effective macroscopic coefficients are derived. Results are listed and compared with other publications and good agreements are shown. Furthermore new results are presented, which exhibit the special orthotropic behavior of such composites caused by the rhombic fiber arrangement.     

Nonlinear static response of laminated composite plates by discrete singular convolution method

In this paper, large deflection analysis of laminated composite plates is analysed. Nonlinear governing equation for bending based on first-order shear deformation theory (FSDT) in the von Karman sense is presented. These equations have been solved by the method of discrete singular convolution (DSC). Regularized Shannon’s delta (RSD) kernel and Lagrange delta sequence (LDS) kernel are selected as singular convolution to illustrate the present algorithm. The effects of plate aspect ratio, fiber orientation, boundary conditions, thickness-to-side ratio, and applied load on the nonlinear static response of the laminated plate are investigated.