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Ultrasonic Nondestructive Testing of Materials

Theoretical Foundations

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## Book Description

**Ultrasonic Nondestructive Testing of Materials: Theoretical Foundations** explores the mathematical foundations and emerging applications of this testing process, which is based on elastic wave propagation in isotropic and anisotropic solids. In covering ultrasonic nondestructive testing methods, the book emphasizes the engineering point of view, yet it relies on the physics and mathematics aspects involved in elastic wave propagation theory.

As a result, this resource becomes a missing link in the literature by combining coverage of the theoretical aspects of testing and providing intuitive assessments of numerous standard problems to illustrate fundamental assertions. Content includes a brief description of the theory of acoustic and electromagnetic fields to underline the similarities and differences as compared to elastodynamics. It also covers vector algebra and analysis, elastic plane and Rayleigh surface waves, and ultrasonic beams, as well as transducer radiation, inverse scattering, and ultrasonic nondestructive imaging.

__Includes numerical computations to explain wave propagation phenomena and compare results of analytical formulations__

Although ultrasonic nondestructive testing can often be roughly understood in terms of plane waves and beams, this book addresses the key issues of transducer radiation and defect scattering and imaging, respectively. The authors physically formulate point source synthesis, and, in mathematical terms, they use representation integrals with Green functions, always including intuitive interpretations with mathematical evaluations.

Replacing cumbersome index notation with a coordinate-free version, this reference offers step-by-step documentation of relevant tensorial elastodynamic cases involving isotropic and anisotropic materials. It provides all necessary mathematical tools readers require to understand the mathematical and physical basis for ultrasonic nondestructive testing.

## Table of Contents

**Contents **Introduction

Contents Flow Chart

**Mathematical Foundations**Scalar, Vector and Tensor Fields

Vektor and Tensor Analysis

Time and Spatial Spectral Analysis with Fourier Transforms

Delta Function

**Governing Equations of Elastodynamics **Newton-Cauchy Equation of Motion and Deformation Rate Equation in the Time and Frequency Domain

Physical Foundations

Transition and Boundary Conditions

**Constitutive Equations; Governing Equations; Elastodynamic Energy Conservation**Materialgleichungen

Linear Non-Dissipative Materials: Cauchy-Hooke Law

Elastodynamic Energy Conservation Theorem for Non-Dissipative Materials in the Time and Frequency Domain

Linear Dissipative Materials

Piezoelectricity and Magnetostriction

**Acoustics**Governing Equations of Acoustics

Transition and Boundary Conditions

Wave Equations in the Time and Frequency Domain

Solutions of the Homogeneous Acoustic Wave Equations in Homogeneous Materials: Plane Longitudinal Pressure Waves

Acoustic Source Fields in Homogeneous Materials: Point Source Synthesis with Green Functions

Hygens’ Principle for Acoustic Scattered Fields in Homogeneous Materials

**Electromagnetism**Maxwell Equations; Poynting Vector; Lorentz Force

Transition and Boundary Conditions

Constitutive Equations: Permittivity, Permeability; Dissipation: Susceptibility Kernels, Conductivity

Wave Equations in the Time and Frequency Domain

Solutions of Homogeneous Electromagnetic Wave Equations in Homogeneous Isotropic Materials: Plane Transverse Electromagnetic Waves

Electromagnetic Source Fields in Homogeneous Isotropic Materials; Tensor Electromagnetic Green Functions

Electromagnetic Scattered Fields; Electromagnetic Formulation of Huygens’ Principle

Two-Dimensional Electromagnetism: TM- and TE-Decoupling

**Vector Wave Equations**Wave Equations for Anisotropic and Isotropic Non-Dissipative Materials

Helmholtz Decomposition for Homogeneous Isotropic Materials: Pressure and Shear Waves

Decoupling of Scalar SH-Waves for Inhomogeneous Isotropic Two-Dimensional

Materials

Frequency Domain Wave Equations for Non-Dissipative and Dissipative Materials

**Elastic Plane Waves in Homogeneous Materials **Homogeneous Plane Waves in Isotropic Non-Dissipative Materials

Inhomogeneous Plane Waves in Isotropic Non-Dissipative Materials

Plane Waves in Anisotropic Non-Dissipative Materials

Plane Waves in Isotropic Dissipative Materials

**Reflection, Transmission and Mode Conversion of Elastic Plane Waves at Planar Boundaries between Homogeneous Non-Dissipative Materials **Stress-Free Planar Boundary of a Homogeneous Isotropic Non-Dissipative Elastic Half-Space

Planar Boundary between Homogeneous Isotropic Non-Dissipative Elastic HalfSpaces

Planar Boundary between a Homogeneous Isotropic Non-Dissipative and a Homogeneous Transversely Isotropic Non-Dissipative Half-Space and a Homogeneous Transversely Isotropic Non-Dissipative Half Space

**Rayleigh Surface Waves**Planar Surfaces

Slightly Curved Surfaces

**Plane Wave Spatial Spectrum**Acoustic Plane Wave Spatial Spectrum

Elastic Plane Wave Spatial Spectrum

**Ultrasonic Beams and Wave Packets**Gaussian Beams as Paraxial Approximation of a Spatial Plane Wave Spectrum

Pulsed Beams as Exact Solutions of an Approximate Wave Equation

Pulsed Beams as Approximate Solutions of Eikonal and Transport Equations

**Point Sources in Homogeneous Isotropic Infinite Space; Elastodynamic Source Fields **Homogeneous Infinite Space Scalar Green Function

Homogeneous Isotropic Infinite Space Green Tensors of Elastodynamics

Two- and Three-Dimensional Elastodynamic Source Fields

Elementary Spherical Waves and Plane Waves

**Force Density and Dilatation Rate Sources on Surfaces of Homogeneous Isotropic Half-Spaces; Radiation Fields of Piezoelectric Transducers **Acoustic Half-Spaces with Soft or Rigid Surfaces

Strip-Like Normal and Tangential Force Density Distributions on the StressFree Surface of an elastic Half-Space: Plane Wave Spectral DecomposItion of the Two-Dimensional Second Rank Green Tensor

Force Densities on the Surface of a Stress-Free Half-Space

Circular Normal Force Force Density Distribution on the Stress-Free Surface of an Elastic Half-Space: Point Source Characteristic

Radiation Fields of Piezoelectric Transducers

**Scatterers in Homogeneous Isotropic Non-Dissipative Infinite Spaces**Huygens' Principle

Integral Equations for Secondary Surface Deformation Sources on Scatterers with Stress-Free Surfaces: Displacement Field Integral Equation and Stress Field Integral Equation

Integral Equations for the Equivalent Sources of Penetrable Scatterers

Scattering Tensor; Far-Fields

**Inverse Scattering: US-NDT Imaging **SAFT: Synthetic Aperture Focusing Technique

FT-SAFT: Fourier Transform Synthetic Aperture Focusing Technique

## Reviews

"... absolutely a must for every scientist who would like to further evaluate theoretically ultrasonic NDT. The studies described by Langenberg et al. have very strongly enhanced the interpretation of propagation of elastic waves also in anisotropic and inhomogeneous media we have in practice, for instance, in welds of austenitic stainless steels or dissimilar metal (Ni-alloys) welds in the nuclear and chemical industries."

-- Gerd Dobmann, Fraunhofer-IZFP, Saarbrücken, Germany