Peridynamic modeling of delaminations in laminated composite beams using refined zigzag theory

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Highlights

  • PeriDynamic Differential Operator is employed to solve the equations of the Refined Zigzag Theory (RZT).

  • The solution to the RZT equilibrium equations is achieved iteratively by using the Newton-Raphson method.

  • A cohesive zone model is used to model delamination initiation and evolvement inlaminated beams.

  • The present approach PD-RZT is computationally efficient and produces accurate stress and failure predictions for delaminated beams.

Abstract

This study aims to model delaminations in laminated composite beams by using the PeriDynamic Differential Operator (PDDO) and Refined Zigzag Theory (RZT). A cohesive zone model is employed to monitor the delamination evolvement in laminated composite beams by embedding an interfacial resin layer between two potentially separable material layers. PDDO calculates the local derivatives in their nonlocal forms and produces highly accurate predictions for the solution of differential equations. RZT eliminates the consideration of shear correction factors and paves the way for modeling both thin and thick laminates. RZT enables computationally efficient analyses since it considers a constant number of kinematic variables regardless of the number of layers in the beam. The principle of virtual work is employed to derive the governing equations and boundary conditions of the RZT. The calculation of transverse normal and shear stresses at critical locations plays an important role in the delamination event. Therefore, this approach is promising for the delamination analysis of laminated composite beams. The present approach performs the nonlocal integration for the approximation of the local derivatives; hence, it reduces the undesirable localized stress peaks. It was demonstrated that the present approach successfully captured the deformation and stress fields as well as the delamination onset and evolvement of the laminated beams.

Introduction

The application of multi-layered composite structures can be found in a wide range of engineering fields such as aerospace and military due to their lightweight, strength, and durability. The presence of multiple materials in these structures may cause damages in the form of intralaminar (matrix breaking and fiber breaking) and interlaminar (delamination). Delamination is a major damage model in the laminated composites [1]. This phenomenon is characterized by the loss of adherence between two different material layers and leads to a considerable degradation in the stiffness of the structure. Since the delamination takes place in the structures, it can be rarely detected on the outer surfaces until the catastrophic failures appear. In order to gain reliable composite models, their stress and strain distributions under loads must be well understood. The experimental investigations may be expensive and time-consuming. Therefore, many studies on the analyses of composite materials have focused on accurate and robust numerical tools.

Damage mechanics and progressive ply failure have been widely used methods for the failure prediction in laminated composite structures. The damage initiation and propagation are modeled by means of physically based equations while considering the microstructure of materials [2], [3] in damage mechanics. These approaches suffer from the determination of damage parameters since they are not feasible in most cases. Progressive failure models pave the way for the identification of the damage modes [4] and degradation of the material stiffness [5], [6], [7], [8]. Numerous studies have investigated the strength of laminated composite materials by employing different progressive failure models.

The virtual crack closure technique (VCCT) [9], [10], [11], [12] and cohesive zone model (CZM) [13], [14], [15], [16] are commonly used approaches for simulating delamination initiation and evolvement. The VCCT was developed by Rybicki and Kanninen [17] for simulating the delamination evolvement. The VCCT predicts the delamination growth when the energy released for the delamination propagation equals the energy required to close the delamination at its original length. The VCCT obtains stresses and displacements from the finite element analysis to determine the energy release rates, GI, GII, and GIII for modes I, II, and III, respectively, considering a self-similar delamination propagation. The VCCT primarily concerns the evolution of pre-existing cracks, rather than the initiation of new cracks. Also, the VCCT predictions are highly sensitive to mesh refinement, and remeshing is inevitable when the crack path is unknown a priori.

In order to overcome the above difficulties, the cohesive zone model (CZM) was introduced by Dugdale [18] and Barenblatt [19]. In CZM, a process zone is modeled by lower and upper cohesive surfaces within the framework of a traction-separation law [20]. The tractions along the cohesive surfaces are zero when the opening displacement reaches a critical value. The cohesive zone models are related to Griffith’s fracture theory. Thus, the fracture toughness of the material is equal to the area under the traction-relative displacement relation.

Finite Element Method (FEM) is mostly employed to examine the stress and failure behaviors of multi-layered composite structures since it provides a way to model complex domains. Generally, laminated composite structures are made of a combination of different types of materials having various layer thicknesses. Traditional three-dimensional finite elements require a highly mesh refinement for the accuracy of the solution. This makes the finite element analysis unfeasible and leads to problems of being computational cumbersome, especially for non-linear and failure analysis.

The equivalent single layer (ESL) and layer-wise (LW) theories have been developed to achieve robust and efficient stress analyses of laminated composite structures. The displacement fields in the ESL theories have constant kinematic variables without considering the number of material layers in laminates. The equilibrium equations of the LW theories are derived for each layer; thus, the number of unknown kinematic variables is proportional to the number of material layers in the laminates. Modeling highly heterogeneous laminates with many material layers by the LW theories may be computationally unfeasible and impractical.

The Euler-Bernoulli beam theory (EBT) and first-order shear deformation theory (FSDT) are the most popular and widely used ESL theories for the stress analysis of laminated composite beam. The EBT disregards the shear and rotational deformation effects; therefore, it becomes challenging when modeling thick beams. It is generally considered that the FSDT is superior to the EBT since it considers the transverse shear deformations which make it suitable for modeling moderately thick beams [21], [22]. However, the FSDT may violate the zero traction boundary conditions along the top and bottom layers of the thick and highly heterogeneous laminates due to the use of an inappropriate shear correction factor [23], [24]. The higher-order shear deformation theories (HSDT) [25], [26] and zigzag (ZZ) [27], [28], [29], [30], [31], [32], [33], [34] theories were developed to improve the computational accuracy and eliminate the shortcomings in the EBT and FSDT. They possess additional kinematic variables in terms of higher-order polynomials. They generally suffer from the incomplete stress field representation for the analysis of thick and heterogeneous laminates.

Alternatively, Tessler et al. [35], [36], [37] proposed a novel Refined Zigzag Theory (RZT) for the analyses of the laminated composite beam, plate, and shell structures [38], [39], [40], [41], [42]. The RZT eliminates the shortcomings arise from the existing ESL theories. The accuracy of the in-plane displacements of the FSDT is improved by defining novel zigzag functions. Thus, more realistic cross-sectional distortions are achieved, especially in moderately thick laminated composites without using shear correction factors. The novel zigzag functions are continuous piecewise linear and disappear along the top and bottom surfaces of the laminate. The RZT has four and seven unknown kinematic variables for the beam and plates, respectively. These kinematic variables are independent of the number of material layers in the laminates. The RZT is a good compromise of the ESL and LW theories as it offers an acceptable accuracy as in the LW and a reduced computational cost as in the ESL. Eijo et al. [43] modeled the delamination damages (mode II) in laminated composite beams by using RZT. They applied a continuum damage model to an interfacial resin layer and employed the modified Newton-Raphson method for the solution of the non-linear problem. The predictions were compared with two-dimensional finite elements under plane stress assumptions and anticipated predictions for both delamination onset and propagation were achieved. Eijo et al. [44] extended their previous study to plate/shell structures for modeling fracture modes II and III. They studied the performance of the four-noded quadrilateral plate finite element based on the RZT by modeling the delamination in a simply supported rectangular plate with a hole under bending. They compared both the onset and evolution of delamination with the reference solutions obtained with three-dimensional finite elements. The predictions were in good agreement with the reference solutions. Groh et al. [45] simulated the delamination onset and growth in laminated beams within the framework of the concept of cohesive zone model and RZT. They inserted a resin layer between individual composite layers with isotropic material properties to track the delamination propagation. Groh and Tessler [46] used mixed form of the RZT for the stress analysis of delaminated sandwich and laminated composite beams.

The high accuracy of the finite element method (FEM) relies on the mesh quality of the model. The equilibrium equations of the FEM are expressed in terms of spatial derivatives. Hence, it fails to approximate the solutions if the domain involves discontinuities such as cracks and sharp gradients in different materials. In order to alleviate the abovementioned problems in a more convenient fashion, mesh-free methods such as radial basis functions (RBF) [47], element free galerkin methods (EFG) [48], smoothed particle hydrodynamics (SPH) [49], reproducing kernel particle method (RKPM) [50], and Peridynamics (PD) [51], [52], [53], [54], [55], [56], [57] have been devised. Meshless methods remove the reliance on the mesh topology and consider the interactions of material points within a finite zone [58], [59]. Meshless computational schemes present superior advantages over the local methods; however, the choice of the selection of shape parameters, requirement of symmetric kernel functions, and the enforcement of the boundary conditions may cause inaccurate approximations.

Madenci et al. [60], [61] developed Peridynamic Differential Operator (PDDO) for the approximation of the local derivatives in their nonlocal forms. Hence, it is applicable to estimate any order of derivatives of functions although the solution domain involves discontinuities such as interfaces and cracks. It paves the way of eliminating the aforementioned deficiencies in the existing methods because it does not require ghost points near the boundaries and use of symmetric kernels. Moreover, the PDDO enables the construction of non-uniformly discretized solution domains with varying horizon sizes, leading to a computationally efficient analysis. Recently, RZT and PDDO are coupled for the stress analysis of beams [24], [62], [63] and plates [64] which is called PD-RZT. The PD-RZT produced highly accurate displacement and stress predictions, especially for the thick and highly heterogeneous laminates.

This study aims to model delaminations in laminated composite beams by using the concept of cohesive zone model developed by Groh et al. [45] within the framework of PDDO and RZT which is called PD-RZT. As explained above, the calculation of transverse normal and shear stresses at critical locations plays an important role in the delamination event. It was demonstrated that the PD-RZT successfully captured the deformation and stress fields of the laminated beams and plates, especially for moderately thick and highly heterogeneous laminates. The PD-RZT approach performs the nonlocal integration for the approximation of the local derivatives; hence, it reduces the undesirable localized stress peaks. Thus, this approach is promising for the delamination analysis of laminated composite beams. The PD-RZT beam formulations are implemented in FORTRAN language using an in-house code.

This paper is organized as follows: the concept of the PD theory and PDDO are introduced in Section 2. Section 3 focuses on the derivation of the RZT equilibrium equations. Section 4 describes the concept of cohesive zone model. The numerical implementation of the present approach is described in Section 5. The solution to the RZT equilibrium equations is achieved iteratively by using the Newton-Raphson method. Section 6 demonstrates the applicability of the PD-RZT for modeling delaminations in laminated composite beams within the framework of cohesive zone model. Section 7 concludes the work with some remarks.

Section snippets

Concept of peridynamics

Peridynamic (PD) theory is a mesh-free nonlocal theory [51], [65], [66]. Unlike the local theories such as FDM and FEM, PD theory considers interactions between material points in a finite region (horizon). Fig. 1 shows a solution domain of D for the PD analyses. Each point in the solution domain has their own domain of interaction which is also called horizon (Hx). The PD point x interacts with many PD points, x in its domain of interaction. In the undeformed state, the material points x and x

Refined zigzag theory for beams

As shown in Fig. 2, the length, width, and total thickness of a beam are denoted by L, b, and 2h, respectively. The beam has N material layers, and each layer has an arbitrary thickness of 2h(k). The Refined Zigzag Beam Theory possesses four kinematic variables through the thickness of the beam in a Cartesian coordinates system (x,z). The displacement field in the kth layer can be expressed in the form ofux(k)(x,z)=u(x)+zθ(x)+ϕ(k)(z)ψ(x)uz(k)(x,z)=w(x)where x[0,L] and z[-h,h] represent the

Cohesive zone model

In this study, the cohesive zone model is employed to monitor the delamination evolvement in laminated composite beams by embedding an interfacial resin layer between two potentially separable material layers. As shown in Fig. 3, a cohesive zone is defined by lower and upper cohesive surfaces modeled using a traction-separation law [20]. The tractions along the cohesive surfaces are zero when the opening displacement reaches a threshold level. The cohesive constitutive law is established in

Numerical implementation

Stiffness coefficients defined in Eq. (24) change after the failure initiates and propagates. Therefore, the solution of the equilibrium equations can be solved in an iterative form by updating the material properties at each iteration. Furthermore, the PD interactions between the material points result in the discrete form of the RZT equilibrium equations asF(u)-b=0where F(u) is the coefficient matrix, u contains the PD unknowns (u,w,θ, and ψ) at each point, and b includes the known values of

Numerical results

In this section, the validity and capability of the PD-RZT for capturing the displacement and stress fields as well as the failure behavior in the delaminated beams are studied. The efficiency and robustness of the PD-RZT were examined by Dorduncu [24] by considering laminated composite beams having thin/thick cross-sections under various loading conditions. Although there is no limitation on the use of weight function defined in Eq. (5), a Gaussian distribution, w(ξ)=e-(2ξ/δ)2 is specified in

Conclusion

This study investigated the deformation and failure behaviors of the laminated composite beams by using PDDO and RZT. The delamination initiation and evolvement in laminates were monitored by means of a cohesive zone model. This was achieved by embedding an interfacial resin layer between two potentially separable material layers. The PDDO expresses the local derivatives in their integral forms. Therefore, the solution of differential equations valid regardless of singularities such as cracks

CRediT authorship contribution statement

Mehmet Dorduncu: Conceptualization, Methodology, Software, Formal analysis, Validation, Visualization, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This study has been supported by The Scientific and Technological Research Council of Turkey (TUBITAK) with the project number: 219M207. This support is gratefully acknowledged. The author also would like to thank Prof. Erdogan Madenci of University of Arizona, Dr. Alexander Tessler, NASA Langley Research Center, VA, and Dr. Rainer Groh, University of Bristol, UK, for pertinent discussions.

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      The RZT describes the shear deformation of each layer through piecewise linear warping functions in a consistent manner without requiring any shear correction factor. The RZT can successfully predict the deformation and stress fields of the laminated structures with general boundary conditions [45,46]. Barut et al. [47] improved the RZT by introducing the transverse stretching effects for the static analyses of laminated plates.

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