An overview of layerwise theories for composite laminates and structures: Development, numerical implementation and application

Abstract

Over the past decades, a vast number of theories for numerical modeling of laminated composite plates and shells has been developed by various researchers and for diverse reasons. Three-dimensional elasticity theory, equivalent single-layer theories, zig-zag theories and layerwise theories are notable examples. In general, computing 3D elasticity solutions require huge computational time, the ESL theories cannot furnish satisfying results for thick laminates or laminates with distinct properties between layers, and the zig-zag theories cannot directly obtain the transverse stress fields from the constitutive model. The layerwise theory treats each layer individually and $C_z^0$ continuity is satisfied from the beginning; therefore, it yields results comparable to 3D elasticity solutions. These attributes and advantages have driven the prosperity of layerwise theories for analysis of composite laminates and structures. The main aim of this review is to provide the recent development of layerwise theories, their numerical implementation, and application in the analysis of composite laminated structures. The main conclusions and possible future research trends are presented. We expect this review will provide a clear picture of layerwise theory for modeling of composite laminated structures and serve as a useful resource and guide to researchers who intend to extend their work into these research areas.

Introduction

Laminated composite materials and structures, consisting stacks of laminae with different material properties, having many outstanding properties such as high specific stiffness and strength, high toughness, low specific density, large space of flexible design [1]. In addition, laminated composites can become smart structures through the inclusion of piezoelectric layers. Due to these unique properties, laminated composite structures have been widely used in many fields, such as civil engineering, aerospace engineering, automotive industries, electromechanical systems, and smart structures [2], [3].

The appearance of the laminated composites provides a good opportunity for unique applications. Simultaneously, it also leads to great challenges in the design process. Because of the mismatch of material properties and existence of an interface between interlayers, the laminated composites exhibit very complex behavior under external loads. This phenomenon can be easily observed from the three stress-induced failure models which are caused by different mechanisms [4], [5]. The first one is the breakage of fibers or the yielding of matrix caused by large in-plane stress; the second one is the interlayer slip resulting from the interlaminar shear stress; and the third one is the delamination failure due to the transverse normal stress. Therefore, it is essential and indispensable to determine accurately the strain and stress fields in the laminated composites. To obtain the deformation behavior of laminated composite plate and shells, many theories have been proposed by researchers and they are mainly classified into four categories: there dimensional (3D) elasticity theory, equivalent single-layer (ESL) theories, zig-zag (ZZ) theories and Layerwise (LW) theories [5], [6], [7], [8].

The 3D theories based on elasticity theory treat the laminated composite structure as a general 3D solid without any consideration for the special layered configuration of the laminated composites and they are highly computational expensive. The ESL theories include the classical laminated theory, first-order shear deformation theories, and higher-order shear deformation theories. The ESL theories assume the displacement has the form of

$$\begin{array}{c}u\left(x,y,z,t\right)=u_0\left(x,y,0,t\right)+{\sum }_{i=1}^N{\left(z\right)}^i{\varphi }_x^i\left(x,y,t\right)\hfill \\ v\left(x,y,z,t\right)=v_0\left(x,y,0,t\right)+{\sum }_{i=1}^N{\left(z\right)}^i{\varphi }_y^i\left(x,y,t\right)\hfill \\ w\left(x,y,z,t\right)=w_0\left(x,y,0,t\right)\hfill \end{array}$$

where u, v and w are the displacement components along the x, y and z directions, respectively, u₀, v_0, and w₀ denote the values in the midplane, t represents the time, and ${\varphi }_x,{\varphi }_y$ are functions to be determined. Essentially, the ESL theories reduce the 3D problems into 2D problems. Owing to this assumption (or simplification), the ESL theories cannot satisfy the piecewise continuous displacement (or ZZ continuity, or $C_z^0$ continuity) requirement and are not able to accurately represent the transverse stress fields. To overcome the limitations of ESL theories and 3D models, several refined theories such as the ZZ theories and LW theories that requires a compromise between accuracy and efficiency have been proposed. The fundamental thought behind the ZZ theories is assuming that the displacement field is a superposition of a global first-order, second-order or higher-order displacement field and a local ZZ function [9], [10]. Then imposing the displacement and/or stress continuity conditions at the interfaces and free surface requirements. An important advantage of ZZ theories is that the number of variables in the kinematic equation does not increase with the increase in the number of layers. However, accurate transverse stresses cannot be obtained directly from solving the constitutive equations. To obtain satisfactory results, 3D equilibrium equations must be adopted [11].

The LW theories can be classified into two groups: (1) displacement-based LW theories with the displacements being the only variables, and (2) mixed LW theories with both displacement and transverse stress variables. The displacement-based LW theories treat independent displacement fields in every single layer and then impose compatibility conditions at the interfaces between laminae to decrease the number of unknown variables [7], [12], [13]. This kind of theories can automatically represent the ZZ continuity through the thickness and reasonable transverse stresses can be obtained directly from the constitutive equations [7]. Unlike the displacement-based LW theories, the mixed LW theories satisfies the interlayer displacement fields and transverse stresses spontaneously and derived usually within the framework of Reissner’s mixed variational principle. The main limitation of the LW theories is the variables are related to the number of layers, thus the computation cost is high. The deformation profiles of a three-layered composite structure according to ESL theories, ZZ theories, and LW theory are shown in Fig. 1 as examples to illustrate their intrinsic mechanisms. It is noteworthy to mention the fact that some researchers do not strictly distinguish between ZZ theories and LW theories [5], [10], [14], [15]. They regarded that the ZZ theories as a special case of the LW theory [15], [16]. However, in this review, the authors consider that the LW theory must treat the lamina independently and the number of the unknown is dependent on the number of layers. Otherwise, any refinement of the ESL theory by adding LW terms or of the discrete-layer theory by enforcing additional conditions at the interface will lead to a ZZ theory.

A generic description of these theories for plates and shells analysis has been given above. For more detailed description of these theories, the following textbooks, reviews, and papers are recommended for further reading. The detailed development of ESL theories can be found in many textbooks such as [7], [17] and reviews such as [5], [6], [14], [18], [19]. A historical review of ZZ theories was conducted by Carrera [10] and the recent development of ZZ theories can be seen in references [11], [20], [21]. Due to the numerous advantages of the LW theories, they have been widely used to solve several simple and complicated research problems in composite structures based on the number of yearly publications (Fig. 2). This review article is timely and essential as it helps to collate recent research efforts on the use of LW theories for numerical modeling of composite structures, provides a concise presentation of their main contributions and discusses possible future development trends. Furthermore, books and review papers on the theories of plates and shells often contain limited discussion about LW theories [7], [11], [14], [18], [21]. To the authors’ best knowledge, a thorough and specific review on the developments, numerical methods and applications of the LW theory, especially in the past decade, cannot be found [22].

In this review, we present the recent developments of LW theories, their numerical treatments and applications. The remainder of this article is as follows. Section 2 first presents a very brief introduction of the pioneering works on LW theories and several typical theories, follows by the recent development of the LW theories in terms of remedying the limitation of the early theories as well as the extension for modeling multiphysics problems. Section 3 discusses the numerical treatment of the LW theories covering the finite element method (FEM), meshfree method, and isogeometric analysis. In Section 4, the application of LW theories for static and vibration analysis, buckling and post-buckling analysis, as well as the delamination analysis of composite plates and shells is presented. The paper closes with some concluding remarks.