Giurgiutiu Structural Health Monitoring of Aerospace Composites

1. Auflage 2015
ISBN: 978-0-12-410441-9
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 470 Seiten

Structural Health Monitoring of Aerospace Composite Structures offers a comprehensive review of established and promising technologies under development in the emerging area of structural health monitoring (SHM) of aerospace composite structures. Beginning with a description of the different types of composite damage, which differ fundamentally from the damage states encountered in metallic airframes, the book moves on to describe the SHM methods and sensors currently under consideration before considering application examples related to specific composites, SHM sensors, and detection methods. Expert author Victor Giurgiutiu closes with a valuable discussion of the advantages and limitations of various sensors and methods, helping you to make informed choices in your structure research and development. - The first comprehensive review of one of the most ardent research areas in aerospace structures, providing breadth and detail to bring engineers and researchers up to speed on this rapidly developing field - Covers the main classes of SHM sensors, including fiber optic sensors, piezoelectric wafer active sensors, electrical properties sensors and conventional resistance strain gauges, and considers their applications and limitation - Includes details of active approaches, including acousto-ultrasonics, vibration, frequency transfer function, guided-wave tomography, phased arrays, and electrochemical impedance spectroscopy (ECIS), among other emerging methods

Dr. Giurgiutiu is an expert in the field of Structural Health Monitoring (SHM). He leads the Laboratory for Active Materials and Smart Structures at the University of South Carolina. He received the award Structural Health Monitoring Person of the Year 2003 and is Associate Editor of the international journal Structural Health Monitoring.

Autoren/Hrsg.

Giurgiutiu, Victor

Leseproben

Chapter 2

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 26

2.2 Anisotropic Elasticity 27

2.2.1 Basic Notations 28

2.2.2 Stresses—The Stress Tensor 28

2.2.3 Strain–Displacement Relations—The Strain Tensor 29

2.2.4 Stress–Strain Relations 29

2.2.4.1 Stiffness Tensor; Compliance Tensor 29

2.2.4.2 From Tensor Notations to Voigt Matrix Notation 30

2.2.4.3 Stiffness Matrix 32

2.2.4.4 Compliance Matrix 33

2.2.4.5 Stress–Strain Relations for an Isotropic Material 34

2.2.5 Equation of Motion in Terms of Stresses 35

2.2.6 Equation of Motion in Terms of Displacements 35

2.3 Unidirectional Composite Properties 37

2.3.1 Elastic Constants of a Unidirectional Composite 37

2.3.2 Compliance Matrix of a Unidirectional Composite 38

2.3.3 Stiffness Matrix of a Unidirectional Composite 40

2.3.4 Estimation of Elastic Constants from the Constituent Properties 41

2.3.4.1 Estimation of the Longitudinal Modulus  41

2.3.4.2 Estimation of the Transverse Modulus  42

2.3.4.3 Estimation of Poisson Ratio  44

2.3.4.4 Estimation of the LT Shear Modulus  45

2.3.4.5 Estimation of Transverse Shear Modulus 23 46

2.3.4.6 Matrix-Dominated Approximations 47

2.4 Plane-Stress 2D Elastic Properties of a Composite Layer 47

2.4.1 Plane-Stress 2D Compliance Matrix 47

2.4.2 Plane-Stress 2D Stiffness Matrix 48

2.4.3 Rotated 2D Stiffness Matrix 49

2.4.4 Rotated 2D Compliance Matrix 52

2.4.5 Proof of RTR-1=T- 53

2.5 Fully 3D Elastic Properties of a Composite Layer 54

2.5.1 Orthotropic Stiffness Matrix 55

2.5.2 Rotated Stiffness Matrix 56

2.5.3 Equations of Motion for a Monoclinic Composite Layer 61

2.5.4 Rotated Compliance Matrix 62

2.5.5 Note on the Use of Closed-Form Expression in the C and S matrices 63

2.5.6 Proof of RTR-1=T- in 3D 63

2.6 Problems and Exercises 65

References 65

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

?]=[0°90°+45°-45°-45°+45°90°0°]t (1)

(1)

Figure 1 Composite laminates: (a) layup made up of a stack of composite laminae (a.k.a. plies) with various orientations ; (b) longitudinal, transverse, and shear definitions in a lamina (ply) [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)

(2)

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

(3)

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

(4)

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

(5)

2.2.2 Stresses—The Stress Tensor

In 1x2x3 notations, the stress tensor is defined as

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

(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=siji,j=1,2,3(symmetryofstresstensorinx1x2x3notations) (7)

(7)

The stress tensor can be represented in an array form as

sij]=[s11s12s13s13s22s23s13s23s33] (8)

(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)

(9)

Hence, Eq. (8) becomes

sij]=[sxxsxysxzsxysyysyzsxzsyzszz] (10)

(10)

The stress symmetry in yz notations is expressed as

yx=sxyszy=syzszx=sxz(symmetryofstresstensorinxyznotations) (11)

(11)

2.2.3 Strain–Displacement Relations—The...