Fundamentals of Aerospace Composite Materials

This chapter is dedicated to the discussion of fundamental aspect of composite materials. The basic principles and notations of anisotropic elasticity theory are reviewed in tensor notations and then converted to Voigt matrix notations. Strain–displacement and stress–strain relations, equation of motion in terms of stresses, and the equation of motion in terms of displacements are introduced. Attention is next focused on the unidirectional composite lamina: formulae for the estimation of lamina elastic properties from the properties of the constituent fiber and matrix were presented in principal axes. The elastic constants in the longitudinal (L), transverse (T), in-plane shear (LT), and transverse shear (23) directions are deduced and the corresponding stiffness and compliance matrices are presented and related to the constitutive fiber and matrix properties. The properties of the rotated unidirectional lamina are considered next, first under the plane-stress (2D) assumption, and then in the fully 3D case. The 2D and 3D rotation matrices are deduced and applied to obtain the rotated 2D and 3D compliance and stiffness matrices. The proof of some of the more intricate steps during this process is also given as separate sections.

Keywords

Composites; aerospace composites; stiffness matrix; compliance matrix; stress tensor; strain tensor; Voigt notations; strain–stress relations; strain–displacement relations; equation of motion; longitudinal modulus; transverse modulus; in-plane shear modulus; transverse-shear modulus; rotation matrix; rotated compliance matrix; rotated stiffness matrix

Outline

2.1 Introduction

Aerospace composite materials are made of high-strength fibers embedded in a polymeric matrix. Glass-fiber-reinforced polymer (GFRP), carbon-fiber-reinforced polymer (CFRP), and Kevlar-fiber-reinforced polymer (KFRP) are among the most common aerospace composite materials.

Aerospace composite structures are obtained through the overlapping of several unidirectional layers with various angle orientations as required by the stacking sequence. Thus, we distinguish a stack of laminae (a.k.a. plies) bonded together to act as an integral structural element. Each ply (a.k.a. lamina) may have its own orientation with respect to a global system of axes −y (Figure 1). The information about the orientation of all the plies in the laminate is contained in the stacking sequence. For example, 0/90/45/−45]s signifies a laminate made of °, °, and 45° plies placed in a sequence that is symmetric about the laminate mid-surface, i.e., °, °, 45°, 45°, 45°, 45°, °, °. This laminate has =8 plies and its stacking vector is

θ]=[0°90°+45°−45°−45°+45°90°0°]t (1)

The plies in the stacking sequence may be of same composite material (e.g., CFRP) or of different materials (e.g., some CFRP, some GFRP, others KFRP, etc.).

The question that composites lamination theory has to answer could be stated as follows: “Given a certain stacking sequence and a set of external loads, what is the structural response of the composite laminate?” In order to address this question, we need to analyze the mechanics of the composite laminate: first we would analyze the local mechanics of an individual layer (a.k.a. lamina) and then apply a stacking analysis (lamination theory) to determine the global properties of the laminated composite and its response under load.

2.2 Anisotropic Elasticity

This section recalls some basic definitions and relations that are essential for the analysis of anisotropic elastic structures such as aerospace composites.

2.2.1 Basic Notations

∂x()=(·)′and∂∂t()=(·) (2)

ij={1ifi=j0otherwise(Kroneckerdelta) (3)

)ii=()11+()22+()33(Einsteinimpliedsummation) (4)

)i,j=∂()i∂xj(differentiationshorthand) (5)

2.2.2 Stresses—The Stress Tensor

In 1x2x3 notations, the stress tensor is defined as

iji,j=1,2,3(stresstensor) (6)

where the first index indicates the surface on which the stress acts and the second index indicates the direction of the stress; thus, ij signifies the stress on the surface of normal →i acting in the direction →j. The strain tensor is symmetric, i.e.,

ji=σiji,j=1,2,3(symmetryofstresstensorinx1x2x3notations) (7)

The stress tensor can be represented in an array form as

σij]=[σ11σ12σ13σ13σ22σ23σ13σ23σ33] (8)

The array in Eq. (8) was written with the symmetry properties of Eq. (7) already included. In yz notations, Eq. (6) is written as

iji,j=x,y,z (9)

Hence, Eq. (8) becomes

σij]=[σxxσxyσxzσxyσyyσyzσxzσyzσzz] (10)

The stress symmetry in yz notations is expressed as

yx=σxyσzy=σyzσzx=σxz(symmetryofstresstensorinxyznotations) (11)

2.2.3 Strain–Displacement Relations—The...