Modeling delamination in composite structures by incorporating the Fermi–Dirac distribution function and hybrid damage indicators

Abstract

Conventional finite element approaches for modeling delaminations in laminated composite structures use the Heaviside unit step function at the interfacial nodes in the delaminated zone of the structure to model the possible jumps in the displacement field during “breathing” of the delaminated layers. In quantum mechanics, the Fermi–Dirac distribution applies to Fermion particles whose characteristics are half-integer spins. The present paper uses the Fermi–Dirac distribution function to model a smoother transition in the displacement and the strain fields of the delaminated interfaces during the opening and closing of the delaminated layers under vibratory loads. This paper successfully shows that the Fermi–Dirac distribution function can be used to more accurately model the dynamic effects of delaminations in laminated composite structures. Optimizing the parameters in the Fermi–Dirac distribution function can lead to more accurate modeling of the dynamic and transient behavior of the delaminated zones in laminated composite structures. This paper also effectively demonstrates how hybrid sensors comprising of out of plane displacement sensors and in plane strain sensors can effectively map a composite structure to detect and locate the delaminated zones. It also shows how simple mode shapes can be used to determine the locations of single and multiple delaminations in laminated composite structures.

Introduction

Inter-layer debonding or delamination is a prevalent form of damage phenomenon in laminated composites. Delamination can be often pre-existing or generated during service life. For example, delamination often occur at stress free edges due to the mismatch of properties at ply interfaces and it can also be generated by external forces such as out of plane loading or low speed impact during the service life. The existence of delamination not only alters the load-carrying capacity of the structure, it can also affect its dynamic response. Thus detection and quantification of delamination is an important technology that must be addressed for the successful implementation and improved reliability of such structures.

Modeling and detection of delamination in composite structures has primarily been based on classical laminate theory (CLT) and first-order shear deformation theory (FSDT) in most works [1], [2], [3], [4]. This means that transverse shears are completely ignored (CLT) or are modeled using shear correction factor (FSDT). Although three-dimensional approaches are more accurate than two-dimensional theories, their implementation can be very expensive in practical applications. The higher-order or layerwise approaches are alternatives since they are capable of capturing transverse shear effects. A higher-order theory (HOT), developed by Chattopadhyay and Gu [5], [6], was shown to be both accurate and efficient for modeling delamination in composite plates and shells of moderately thick construction. The theory was further extended to study the effect of delamination on the dynamic response of smart composite structures [7], [8]. Although the higher-order-based theories provide good accuracy at global level, they fail to satisfy stress continuity at ply interfaces. To address this issue, Barbero and Reddy [9] introduced a layerwise approach. Lee [10] analyzed the free vibration characteristics of a delaminated composite laminates using a layerwise theory. However, the computational effort associated with such analysis increases with the increase in number of plies in the laminate. To reduce the number of structural unknowns, Cho and Kim [11], [12] developed a higher-order zig-zag theory for laminated composite plates with multiple delaminations. Similarly, Kim et al. [13], [14] proposed an improved layerwise theory to characterize delamination of laminated structures.

Conventional finite element approaches for modeling delaminations in laminated composite structures use sub laminates technique [15], [16], [17] or the Heaviside unit step function [11], [12], [13], [14], [18] to model possible jumps in the displacement field due to delamination allowing slipping and separation at the delaminated interface. Normally, sub laminates technique or Heaviside unit step function are not easy to model smooth transition in the displacement and the strain field of the delaminated interfaces. In quantum mechanics, the Fermi–Dirac distribution applies to Fermions particles whose characteristics are half-integer spins [19]. The present paper uses Fermi–Dirac distribution function to model a smooth transition in the displacement and the strain fields of the delaminated interfaces. The improved layerwise laminate theory [20] are incorporated into Fermi–Dirac distribution function to account transverse shear effects of anisotropic laminated composites. The effects of delamination are investigated by modal analysis of delaminated composite structures. It is expected that the Fermi–Dirac distribution function can be used to accurately model the effects of delamination in composite laminates.