Physical Ultrasonics of Composites

Description

Physical Ultrasonics of Composites is a rigorous introduction to the characterization of composite materials by means of ultrasonic waves. Composites are treated here not simply as uniform media, but as inhomogeneous layered anisotropic media with internal structure characteristic of composite laminates. The objective here is to concentrate on exposing the singular behavior of ultrasonic waves as they interact with layered, anisotropic materials, materials which incorporate those structural elements typical of composite laminates. This book provides a synergistic description of both modeling and experimental methods in addressing wave propagation phenomena and composite property measurements. After a brief review of basic composite mechanics, a thorough treatment of ultrasonics in anisotropic media is presented, along with composite characterization methods. The interaction of ultrasonic waves at interfaces of anisotropic materials is discussed, as are guided waves in composite plates and rods. Waves in layered media are developed from the standpoint of the "Stiffness Matrix," a major advance over the conventional, potentially unstable Transfer Matrix approach. Laminated plates are treated both with the stiffness matrix and using Floquet analysis. The important influence on the received electronic signals in ultrasonic materials characterization from transducer geometry and placement are carefully exposed in a dedicated chapter. Ultrasonic wave interactions are especially susceptible to such influences because ultrasonic transducers are seldom more than a dozen or so wavelengths in diameter. The book ends with a chapter devoted to the emerging field of air-coupled ultrasonics. This new technology has come of age with the development of purpose-built transducers and electronics and is finding ever wider applications, particularly in the characterization of composite laminates. Contents Preface Introduction 1 Fundamentals of Composite Elastic Properties 1.1 Introduction 1.2 Mechanical Relationships 1.2.1 Stress 1.2.2 Strain 1.2.3 Constitutive equation 1.3 Generalized Hooke’s Law 1.4 Stress–Strain Relations for Media of Higher Symmetries 1.4.1 Monoclinic symmetry 1.4.2 Orthotropic symmetry 1.4.3 Transversely isotropic symmetry 1.4.4 Cubic symmetry 1.4.5 Isotropic material 1.5 Relation between Stiffness and Compliance 1.5.1 Engineering constants for orthotropic materials 1.6 Determination of the Full Compliance Matrix 1.7 Material Coordinate System Transformations 1.7.1 Property matrices in a rotated coordinate system 1.7.2 Rotation about the x[(sub)3] axis 1.8 Analysis in a Planar Geometry 1.8.1 Two-dimensional stress–strain relations 1.8.2 Rotation about the principal axis 1.8.3 Boron-epoxy composite: An example 1.9 Experimental Determination of Stiffness Bibliography 2 Elastic Waves in Anisotropic Media 2.1 Equations of Motion 2.1.1 Introduction 2.1.2 Differential approach 2.1.3 Integral principle 2.2 Plane Wave Propagation in Bulk Materials 2.3 Determination of the Polarization Vectors 2.4 Example: Plane Waves in an Orthotropic Material 2.5 Energy Flux and Group Velocity 2.6 Relation between the Phase and Group Velocities 2.7 Measurement of the Group Velocity 2.8 Examples 2.9 Phase Velocity, Group Velocity, and Slowness 2.9.1 Slowness 2.9.2 Examples for a graphite-epoxy composite 2.9.3 Phase and group velocities in symmetry planes of orthotropic materials 2.9.4 Phase and group velocities in non-symmetry planes of orthotropic and transversely isotropic materials Bibliography 3 Bulk Ultrasonic Techniques for Evaluation of Elastic Properties 3.1 Bulk Wave Refraction Method for Phase Velocity Measurement 3.1.1 Introduction 3.1.2 Delay times for phase and group velocities in a solid layer of general anisotropy 3.1.3 Phase velocity measurement 3.1.4 Double through-transmission phase velocity measurement 3.1.5 Self-reference method 3.1.6 Multiple reflection method 3.1.7 Comments on accuracy, phase correction, and applicability of the plane wave approximation in angle-beam, self-referenced, through-transmission method 3.2 Examples of Bulk Wave Refraction Velocity Measurements 3.2.1 Experimental apparatus 3.2.2 Double through-transmission method for graphite/epoxy composites 3.3 Critical Angle Measurement of Elastic Constants 3.3.1 Introduction 3.3.2 Concept of the method 3.3.3 Critical angle measurements 3.3.4 Experimental results for graphite/epoxy composites 3.4 Determination of Elastic Constants from Phase Velocity Data 3.4.1 Reconstruction of elastic constants 3.4.2 Stability of the reconstruction algorithm 3.5 Group Velocity Measurements 3.5.1 Reconstruction of elastic constants from group velocity data 3.5.2 Determination of elastic constants from group velocity 3.5.3 Comparison of the reconstruction results from group and phase velocity data in symmetry planes 3.5.4 Reconstruction from group velocity data in non-symmetry planes Bibliography 4 Reflection and Refraction of Waves at a Planar Composite Interface 4.1 Introduction 4.2 Background 4.2.1 Snell’s law 4.2.2 Isotropic media analysis using slowness surfaces 4.3 Scattering at an Interface between Two Generally Anisotropic Media 4.3.1 Introduction 4.3.2 Snell’s law (anisotropic case) 4.3.3 Determination of the slowness vectors of reflected and refracted waves 4.3.4 Calculation of the wave and polarization vectors 4.3.5 Example: Refracted waves in isotropic solids with incident waves from a fluid 4.4 Determination of Reflection and Transmission Coefficients 4.4.1 Boundary conditions, amplitude coefficients 4.4.2 Energy conversion coefficients 4.5 Examples for Graphite/Epoxy Composite 4.5.1 Fluid/composite interface 4.5.2 Isotropic wedge/composite interface 4.5.3 Composite/composite interface 4.6 Geometrical Considerations on Reflection and Refraction, Grazing and Critical Angles Bibliography 5 Guided Waves in Plates and Rods 5.1 Introduction 5.2 Guided Waves in a Uniaxial Laminate 5.2.1 Preliminaries 5.2.2 Plate Wave Solutions 5.3 Leaky Guided Waves in a Fluid-Loaded Plate 5.4 Fluid–Solid Plate Reflection Coefficient 5.5 Waves in Composite Rods Bibliography 6 Elastic Waves in Multilayer Composites 6.1 Introduction 6.2 Transfer Matrix 6.3 Stiffness Matrix 6.3.1 General formulation of the stiffness matrix for an anisotropic layer 6.3.2 Global stiffness matrix 6.3.3 Relation between transfer and stiffness matrices 6.4 Scattering Coefficients for a Fluid-Loaded Composite Laminate 6.5 Experimental Phenomenology 6.6 Extensions of the Stiffness Matrix 6.6.1 Higher symmetry lamina 6.6.2 Stiffness matrix for a fluid 6.6.3 Stiffness matrix for imperfect boundary conditions 6.6.4 Stiffness matrix computations for a monoclinic lamina 6.6.5 Simple asymptotic method to compute stiffness matrix Bibliography 7 Waves in Periodically Layered Composites 7.1 Introduction and Background 7.2 A Simple Illustration 7.3 Floquet Analysis for Anisotropic Periodic Plates 7.4 Homogenization of Periodically Layered Composites 7.4.1 Floquet wave spectrum and signal distortion 7.4.2 Homogenization approach 7.4.3 Homogenization domain estimation for an anisotropic cell 7.5 Lamina Moduli Measurement by Floquet waves 7.5.1 Problem statement 7.5.2 Floquet wave lamina moduli determination 7.5.3 Effective property determination: Static limit 7.6 Computation of the Guided Wave Spectrum Bibliography 8 Measurement of Scattering Coefficients 8.1 Introduction 8.2 Scattering Coefficient Integral 8.2.1 Uniform asymptotics 8.3 Computational Results 8.4 Complex Transducer Points 8.4.1 Two-dimensional voltage calculation 8.4.2 Synthetic aperture scanning 8.4.3 Focused beams 8.4.4 Three-dimensional effects on the receiver voltage 8.5 Experimental Results 8.6 Elastic Stiffness Reconstruction Bibliography 9 Air-Coupled Ultrasonics 9.1 Introduction 9.2 Transduction and Other Challenges 9.3 Material Characterization in Air 9.4 Focusing 9.5 Techniques and Applications Bibliography Index A B C D E F G H I K L M N O P Q R S T U V W X Y Z