Effect of matrix cracking and material uncertainty on composite plates

https://doi.org/10.1016/j.ress.2010.02.004Get rights and content

Abstract

A laminated composite plate model based on first order shear deformation theory is implemented using the finite element method. Matrix cracks are introduced into the finite element model by considering changes in the A, B and D matrices of composites. The effects of different boundary conditions, laminate types and ply angles on the behavior of composite plates with matrix cracks are studied. Finally, the effect of material property uncertainty, which is important for composite material on the composite plate, is investigated using Monte Carlo simulations. Probabilistic estimates of damage detection reliability in composite plates are made for static and dynamic measurements. It is found that the effect of uncertainty must be considered for accurate damage detection in composite structures. The estimates of variance obtained for observable system properties due to uncertainty can be used for developing more robust damage detection algorithms.

Introduction

Composites play an important role in modern industry, especially in aerospace structures because of their high specific strength and specific stiffness values [1]. Moreover, composites can be aeroelastically tailored to meet different design requirements. Despite their significant advantages, composites suffer from complicated damage mechanisms compared to metals, because of their heterogeneous composition and directional properties. The primary physical mechanisms of damage in composites are matrix cracking, fiber failure, fiber-matrix debonding and delamination. For composites subjected to quasi-static or cyclic tensile load, matrix cracking is often the first defect to occur, which generally triggers the other failure modes. Though matrix cracking is not so important from a structural failure point of view, it is indicative of the initiation of the other severe modes of failure such as delamination and fiber breakage. Structural health monitoring of composites is therefore an important area of current research [2].

Since matrix cracking is an important problem in composites, researchers have proposed various methods to calculate the stiffness degradation of composites containing matrix cracks. Some of these methods are discussed next. The ply discount method assumes that the damaged plies cannot take transverse loads and hence underestimates the stiffness of the cracked laminate [3], [4]. Reifsnider [5] predicted stiffness reduction in the cracked laminate with the simple shear lag method. However, the shear lag method requires experimental data for its successful application [6], [7], [8]. Hashin [9] suggested a variational approach for crack detection in composite laminates with the assumption that the axial normal stresses are constant across the thickness of each layer. However, damage accumulation and its progression cannot be calculated by this method. Gottesman and Hashin [10] derived the expressions for the overall stiffness of an infinite orthotropic composite laminate with a single crack, using expressions for the associated elastic energy, which does not consider the crack interaction effect. Gudmundson and Zang [11] formulated an analytical model for tensors of extensional stiffness and thermal expansion for a composite laminate with matrix cracks including the effect of crack interaction. Gudmundson and Adolfson [12]proposed close form expressions for the reduction in thermo-elastic properties of a cracked laminate in combined extension and bending, using the elastic energy expressions of the damaged composite and applied them for a composite plate modeled based on classical plate theory.

The methods discussed in [3], [4], [5], [6], [7], [8], [9], [10], [11], [12] focus on physical modeling of matrix cracks in composite structures. Several researchers have investigated the detection and propagation of matrix cracks in practice. Wang et al. [13] explained how polarimetric optical fiber sensors can be used for the detection of transverse matrix cracks in a composite cross-ply laminate. Kramb et al. [14] studied damage propagation in notches and explained the effective use of ultrasonic C-scan in matrix cracking monitoring. Akira et al. [15] found the best position for integrated probes to be placed in a CFRP laminate, for matrix crack detection. They considered the change in electrical resistance of the laminate as damage indicator. We see that most matrix crack detection work has focussed on experimental approaches based on non-destructive testing (NDT). However, it is also possible to detect matrix cracking using model based damage detection methods such as changes in the structural properties. For example, Pawar and Ganguli [16], [17] addressed the problem of matrix crack detection in rotor blades and composite tubular beams, respectively. While these studies showed that matrix cracking could be detected from changes in structural response and frequencies, they did not account for uncertainty in composite materials. Furthermore, they addressed beam structures and not the more general plate structures which are more effective models of aerospace structures such as aircraft wings and fins.

Besides the complex damage mechanisms, composites are prone to relatively high material property uncertainties compared to isotropic materials. The scatter in material properties depends on a wide variety of variables at the micro-level. For example, the fiber and matrix properties, fabrication variables at all stages of the fabrication processes (such as fiber volume ratio, misalignment of ply orientation, fiber waviness or undulation), inter-lamina voids, incomplete curing of resin, excess resin between plies, and variation in ply thickness are cause of uncertainties [18]. The uncertainties in composite material properties cause uncertainties in their structural response. The analysis and reliable design of a composite structure should therefore consider the randomness of the material properties.

Several works have addressed the reliability of composite structures. Boyer et al. [19] investigated the effect of material uncertainty in reliability of composite structures where Tsai-Hill failure criteria and first order reliability methods are used. Optimum design of multiaxial fiber reinforced laminate systems under probabilistic conditions of loads and material properties is addressed by Miki et al. [20]. Guedes Soares [21] has described about the different probabilistic approaches that have been used to represent the strength of fiber reinforced composite materials and to assess the reliability of laminated components. Sentler [22] has considered size dependence of fibers and time dependence of failure stresses for studying the reliability of composite structures. A state-of-the art survey of the probabilistic formulations available and a critical evaluation of their potential for practical assessments of uncertainty in strength predictions for composite materials is given in the review paper by Sutherland and Guedes Soares [23].

Most of the studies focus on the fiber breakage and laminate failure and the reliability of composite structures and not much attention is given to the specific damage mechanism such as matrix cracks. Minnetyan et al. discussed about the material uncertainties in progressive fracture and evolution of matrix cracking and probability of failure in composite structures [24]. Lamon investigated the effect of uncertainty on prediction of stress–strain behavior of microcomposites with multiple cracking [25]. The initiation and saturation of matrix cracking provides an indicator for the development of thresholds on measured system parameters beyond which the structure should be watched and monitored carefully [26].

There have been several works in recent years which address uncertainty in damage detection of isotropic materials. Yong and Hong [27] analyzed the reliability of damage detection in an isotropic member using frequency changes. They considered the effects of random noise in the vibration data and finite element model. Kim et al. [28] introduced a vibration based method to estimate the probability of damage detection in isotropic plates, considering the uncertainties in mass, forces and foundation stiffness, etc. These studies clearly indicate that for damage to be detectable, the change in the system behavior due to damage must be more than that due to uncertainty. In addition, practical damage detection should be expressed in probabilistic terms as the effect of uncertainty is always present. However, to the best of authors knowledge, the problem of quantifying the effect of uncertainties in the composite materials on matrix crack detection has not been addressed in the literature.

A probabilistic structural integrity analysis is needed due to deviations of the structural response of laminated composite structures produced by existing uncertainties in physical properties at the layer level. Several approaches have been developed to take into account uncertainty in the field of structural analysis. In the context of finite element (FE) modeling, the stochastic finite element modeling (SFEM) based on perturbation, spectral Neumann expansion, Karhunen–Loeve expansion, projection on a polynomial chaos [29], [30], [31], [32] or direct Monte-Carlo simulation (MCS) methods [33], [34] are the most popular approaches widely used. Methods based on non-determinist arithmetic operations or on response surface are also interesting alternatives to SFEM [35]. The non-intrusive methods like Monte-Carlo simulation are applied outside the finite element code and do not necessitate any modification of the FE code [36], [37], [38]. Monte Carlo simulation is highly computational expensive due to the large number of function evaluations needed and is often used to verify the other approaches [39], [40]. Furthermore, the growing power of computer has made MCS possible for many engineering problems [41], [42], [43]. Methods addressing uncertainty are playing an important role in fault diagnosis [44], [45], [46], [47].

In the present study, the Gudmundson matrix crack model [11] is integrated into a first order shear deformation theory (FSDT) model to model the damaged laminated composite plate. Monte Carlo simulation is conducted to study the effects of material uncertainty on matrix cracking and the probability of damage detection.

Section snippets

Composite plate model

A well established composite plate model is used in this study, details of which can be found in the book by Reddy [48]. For a laminated composite plate, the kinematics is governed by the midplane displacements uo, vo, the transverse displacement wo and the rotations ψx and ψy about y- and x-axes, respectively, as shown in Fig. 1: u(x,y,z)=uo(x,y)zψx(x,y)v(x,y,z)=vo(x,y)zψy(x,y)w(x,y,z)=wo(x,y)Then, the displacement components u, v and w along x, y and z directions, in terms of midplane nodal

Baseline model validation

In order to assess the accuracy of the finite element calculations, a laminated composite plate made of AS4/3501-6 graphite epoxy material with mean values of properties mentioned in Table 2 is considered. Uniformly distributed transverse load is applied on the plate with three different boundary conditions namely, (a) cantilever (clamped–free), (b) fixed (clamped–clamped) on two opposite sides and (c) simply supported on two opposite sides.

For the validation and convergence analysis, the plate

Numerical results

The matrix crack model is integrated into the finite element model of the composite plate. The numerical model of the damaged composite plate is first used to study the effect of ply angles, different laminate types and boundary conditions on the deflection of the composite plate with embedded matrix cracks. These parametric studies are done using a deterministic analysis with the mean values of the composite material properties given in Table 2. As a second result, Monte Carlo simulations are

Conclusion

A finite element model based on FSDT for the analysis of a composite laminated plate in combined bending and extension is integrated with a matrix crack model. The effect of matrix cracking on the stiffness matrix and hence tip deflection of a composite plate is obtained. Parametric studies are performed to investigate the effects of laminate type, ply angle and boundary conditions on matrix crack saturation. The results show that with the increase in the ply orientation angle from 30° to 60°,

References (50)

  • M. Miki et al.

    Reliability-based optimization of fibrous laminated composites

    Reliability Engineering and System Safety

    (1997)
  • L. Sentler

    Reliability of high performance fibre composites

    Reliability Engineering and System Safety

    (1997)
  • L.S. Sutherland et al.

    Review of probabilistic models of the strength of composite materials

    Reliability Engineering and System Safety

    (1997)
  • J. Lamon

    Stochastic approach to multiple cracking in composite systems based on the extreme-values theory

    Composites Science and Technology

    (2009)
  • P.M. Pawar et al.

    On the effect of matrix cracks in composite helicopter rotor blades

    Composites Science and Technology

    (2005)
  • B.H. Kim et al.

    Nondestructive damage evaluation of plates using the multi-resolution analysis of two-dimensional Haar wavelet

    Journal of Sound and Vibration

    (2006)
  • R. Ghanem

    Ingredients for a general purpose stochastic finite elements implementation

    Computer Methods in Applied Mechanics and Engineering

    (1999)
  • N.-Z. Chen et al.

    Spectral stochastic finite element analysis for laminated composite plates

    Computer Methods in Applied Mechanics and Engineering

    (2008)
  • G. Stefanou

    The stochastic finite element method: past, present and future

    Computer Methods in Applied Mechanics and Engineering

    (2009)
  • T. Crestaux et al.

    Polynomial chaos expansion for sensitivity analysis

    Reliability Engineering and System Safety

    (2009)
  • N. Carrere et al.

    Efficient structural computations with parameters uncertainty for composite applications

    Composite Science and Technology

    (2009)
  • M. Marseguerra et al.

    Monte Carlo simulation for model-based fault diagnosis in dynamic systems

    Reliability Engineering and System Safety

    (2009)
  • F. Cadini et al.

    Model-based Monte Carlo state estimation for condition-based component replacement

    Reliability Engineering and System Safety

    (2009)
  • D.H. Oh et al.

    Free vibration and reliability of composite cantilevers featuring-uncertain properties

    Reliability Engineering and System Safety

    (1997)
  • J.B. Weathers et al.

    An exercise in model validation: Comparing univariate statistics and Monte-Carlo based multivariate statistics

    Reliability Engineering and System Safety

    (2009)
  • Cited by (60)

    • Damage identification under uncertain mass density distributions

      2021, Computer Methods in Applied Mechanics and Engineering
      Citation Excerpt :

      In damage identification, the sources of modeling errors include unknown or partially known environmental conditions, input excitations, structural connections, boundary conditions, energy dissipation mechanisms and material properties [6–12]. The uncertainties associated with material properties are of particular interest, for example, with composite materials since many different fabrication aspects, such as the fiber/matrix ratio, the orientation of fibers, the completeness of the curing which cannot be completely controlled, thus resulting in uncertainty in the structural properties of the manufactured materials [6,9,11,13,14]. This is also a problem in civil engineering, for example, where live loads may change from floor to floor and within a floor in a building due to machine layout, see, for example [15,16].

    • Free vibration of a composite plate with spatially varying Gaussian properties under uncertain thermal field using assumed mode method

      2020, Physica A: Statistical Mechanics and its Applications
      Citation Excerpt :

      The stochastic response of a quarter composite cylinder was investigated and sensitivity of structural response due to changes in fiber angle was studied. Gayathri et al. in [13] investigated the effects of crack width (in resin domain) as well as the material uncertainties in a composite beam plate response under static loading. The employed method for solving the equations was SFEM.

    • Identification of matrix cracking in cross-ply laminated composites using Lamb wave propagation

      2020, Composite Structures
      Citation Excerpt :

      Therefore, it is necessary to monitor the condition of composite structures [2]. The matrix cracking is often the first damage that occurs in composites exposed to quasi-static or oscillating tensile loading in the through-thickness and transverse directions [3]. The matrix cracking may lead to other types of damage in composite laminates such as delamination and fiber breakage [4].

    View all citing articles on Scopus
    View full text