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1 CAVITY INSPECTION NDT&E Methods: UT VJ Technologies

2 NDT&E Methods: UT 6. NDT&E: Introduction to Methods 6.1. Ultrasonic Testing: Basics of Elasto-Dynamics 6.2. Principles of Measurement 6.3. The Pulse-Echo Method 6.4. UT-Systems: Transducer, Instrument, Manipulator 6.5. Current Developments 6.6. Case Studies by Movies

3 NDT&E Methods: UT Low Frequency Airborne High Power Conventional Industrial High Frequency Acoustic Microscopy Sub Sonic Range Audible Range Ultrasonic Range

4 In the 1870s: UT BASICS Lord Rayleigh: Theory of Sound Nature and Properties of Sound Result from Mechanical Vibrations in Solids, Liquids, and Gases Spring Model of Vibrating Particle Masses

5 NDT&E Methods: UT Sound Generation by mechanical Impact

6 NDT&E Methods: UT Constitutive Equations of Sound Waves In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities that is specific to a material, and approximates the response of that material to external stimuli, usually as applied fields or forces. They are combined with other equations governing physical laws to solve physical problems; for example in fluid mechanics the flow of a fluid in a pipe, in solid state physics the response of a crystal to an electric field, or in structural analysis, the connection between applied stresses or forces to strains or deformations. Some constitutive equations are simply phenomenological; others are derived from first principles (From WIKIPEDIA)

7 NDT&E Methods: UT Constitutive Equations of Sound Waves Stress and Strain - The roots for elastic waves The stress-strain constitutive relation for linear materials is commonly known as Hooke's law. In its simplest form, the law defines the spring constant (or elasticity constant) k in a scalar equation, stating the tensile/compressive force is proportional to the extended (or contracted) displacement x: meaning the material responds linearly. Equivalently, in terms of the stress σ, Young's modulus E, and strain ε (dimensionless):

8 NDT&E Methods: UT Constitutive Equations of Sound Waves Stress and Strain The roots for elastic waves In general, forces which deform solids can be normal to a surface of the material (normal forces), or tangential (shear forces), this can be described mathematically using the stress tensor: where C is the elasticity tensor and S is the compliance tensor

9 NDT&E Methods: UT the spring model Constitutive Equations of Sound Waves Hooke's law is only a first order linear approximation to the real response of springs and other elastic bodies to applied forces Hook s Law: F = -kδx k: stiffness (positive constant) Simplifications: 1-dimensional Linear Newton s Law: F = m d2 x dt 2 m: mass

10 NDT&E Methods: UT the spring model Constitutive Equations of Sound Waves Harmonic Oscillator: F = m d2 x dt2 = -kx Solving this differential equation, we find that the motion is described by the function:

11 NDT&E Methods: UT the spring model Constitutive Equations of Sound Waves Mass Displacement Wave propagation It is already a 1-dimensional undamped harmonic wave

12 NDT&E Methods: Elastic Waves in Solids Constitutive Equations of Sound Waves ASSUMING: Linear Hook s Law, 1-dimensional case, isotropic material F E L L L F L E Hook Law: Stress = E * Strain E A Little Experiment The stress created by the hammer blow, causes one side of the rod to be displaced by an amount h x and h x+dx on the right due to the wave taking time to travel along dx and to the elastic properties of the material

13 NDT&E Methods: Elastic Waves in Solids Constitutive Equations of Sound Waves The force on the left face is given as (using the definition of stress): F left A x AE AE h x x While the force on the right side is: F right A x dx AE AE h x x dx The net force is the difference between the left and right sides and of course is equal to the mass times the acceleration of the segment. F F right F left m 2 h t 2 Or, more explicitly AE h h Adx 2 h x x dx x x t 2

14 NDT&E Methods: Elastic Waves in Solids Constitutive Equations of Sound Waves Rearranging the previous equation we have: h h E x x dx x x 2 h dx t 2 This represents a 1-dimensional wave equation for the propagation of a longitudinal wave through an elastic homogeneous medium as a function of position and time. With v representing the wave speed through the medium and comparing this to the standard form of the wave equation we have 2 h x 1 2 h 2 v 2 t 2 E 2 h x 2 E 2 h t 2 2 h t 2 Remember: m d2 x dt2 = -kx

15 NDT&E Methods: Elastic Waves in Solids Constitutive Equations of Sound Waves We ve determined the speed of a sound wave in a 1-dimensional medium by applying Newton s laws of to a small segment of material and we find that that the speed of sound depends further on its density, and the elastic properties of the medium, or equivalently in terms of the compressibility, K, through The compressibility is defined as the fractional change in volume of material per unit increase in pressure Thus a large compressibility, K, means that it is easier to squeeze something. Again, E is called the elastic (or Young s) modulus of the material and is a measure of the stiffness of the material. v E 1 K K 1 V dv dp

16 NDT&E Methods: UT Waves in Liquids Direction of propagation Particle Motion Wavelength λ

17 NDT&E Methods: UT Waves in Liquids

18 NDT&E Methods: UT Waves in Liquids

19 NDT&E Methods: UT Waves in Liquids Constitutive Equations of Sound Waves Euler s identity is an equality found in mathematics that has described as "the most beautiful equation." It is a special case of a foundational equation in complex arithmetic called Euler s Formula In Fluid Dynamics, the constitutive equation is a Euler equation

20 NDT&E Methods: Elastic Waves in Solids Constitutive Equations of Sound Waves In fluid dynamics, the Euler equations are a set of quasilinear hyperbolic equations governing adiabatic and inviscid flow. They are named after Leonhard Euler (St. Peterburg) The equations represent Cauchy equations of conservation of mass (continuity), and balance of momentum and energy, and can be seen as particular Navier Stokes equations with zero viscosity and zero thermal conductivity

21 NDT&E Methods: UT Waves in Liquids

22 NDT&E Methods: UT

23 NDT&E Methods: Elastic Waves in Solids Constitutive Equations of Sound Waves However, The World, Material Properties and Applied Stresses are 3-Dimensional Makes it much more difficult

24 NDT&E Methods: UT Plane Body Waves The 3-D Constitutive Equation for an infinite medium containing a linear, elastic, homogeneous isotropic material (Hook s Law) Cauchy STRESS-Tensor Σ: σ xx σ yx σ zx σ xy σ yy σ zy σ xz σ yz σ zz The Stress tensor is symmetric due to the conservation of linear and angular momentum σ xy = σ yx σ xz = σ zx σ yz = σ zy

25 NDT&E Methods: UT Plane Body Waves The 3-D Constitutive Equation for an infinite medium containing a linear, elastic, homogeneous isotropic material (Hook s Law) Cauchy STRESS-Tensor Σ: σ xx σ yx σ xy σ yy σ xz σ yz σ zx σ zy σ zz The Stress tensor is symmetric due to the conservation of linear and angular momentum Euler Cauchy stress principle σ xy = σ yx σ xz = σ zx σ yz = σ zy

26 NDT&E Methods: UT Waves in Solids Constitutive Equations of Sound Waves Deformation and Strain We distinguish Displacement & Deformation Deformation: An alteration of shape, as by pressure or stress. Displacement: A vector or the magnitude of a vector from the initial position to a subsequent position assumed by a body

27 NDT&E Methods: UT Waves in Solids Constitutive Equations of Sound Waves Strain characterizes a deformation Example: 1D strain ε 1 = L L 0 L 0 L 0 : L:

28 NDT&E Methods: Strain & Stress Kinematics of Continuous Body

29 NDT&E Methods: Strain & Stress Kinematics of Continuous Body THERE ARE TWO POSSIBLE DESCRIPTIONS: LAGRANGIAN: The motion is described by the material coordinate and time EULERIAN: The motion is described by the spatial coordinate and time

30 NDT&E Methods: Strain & Stress Kinematics of Continuous Body

31 NDT&E Methods: Strain & Stress Kinematics of Continuous Body

32 NDT&E Methods: Strain & Stress Kinematics of Continuous Body

33 NDT&E Methods: Strain & Stress Kinematics of Continuous Body

34 NDT&E Methods: Strain & Stress Kinematics of Continuous Body

35 NDT&E Methods: Strain & Stress Kinematics of Continuous Body

36 NDT&E Methods: Strain & Stress Kinematics of Continuous Body

37 NDT&E Methods: Strain & Stress Kinematics of Continuous Body

38 NDT&E Methods: UT the spring model

39 NDT&E Methods: UT the spring model Engineering Strains Coordinates: x,y,z Displacements: x,y,z Normal Strains ε x = u x = e 11 ε y = u y = e 22 ε z = u z = e 33 Shear Strains ε xy = u y + v x = 2e 12 ε xy = v z + w y = 2e 23 ε xy = u z + w x = 2e 13

40 NDT&E Methods: UT the spring model Integrability Condition related to Compatibility of Strain Fields: Integration of Strain Fields Yields Unique Displacement Components Characteristic for Material Constants

41 NDT&E Methods: UT Material Constants The strain tensor is a field tensor it depends on external factors. The compliance tensor is a matter tensor it is a property of the material and does not change with external factors. Let us consider the material by introducing the Lamé Constants: Young Modulus E λ(first parameter) Shear Modulus Poisson s Ratio G μ(second parameter) The Lamé constants are material-dependent quantities denoted by λ and μ that arise in strain-stress relationships.

42 NDT&E Methods: UT Material Constants The Constitutive Equation between Stresses and Strains for an Infinite Medium containing a linear, homogeneous isotropic Material (HOOK S LAW) σ xx = λ + 2μ ε xx + λ ε yy + ε zz σ yz = σ zy = 2με yz = 2με zy σ yy = λ + 2μ ε yy + λ ε xx + ε zz σ zx = σ xz = 2με zx = 2με xz σ zz = λ + 2μ ε zz + λ ε xx + ε yy σ xy = σ yx = 2με xy = 2με yx ε ij : Components of Strains; σ ij : Components of Stresses

43 NDT&E Methods: UT Material Constants Lamé s Constants are related to Young s Modulus E and Poisson s Ratio ν λ = 1+ν Eν 1 2ν μ = E 2 1+ν Governing Differential Equation of Motion for a Continuum: σ xx x + σ xy y + σ xz z = ρ 2 u x 2 t σ xy x + σ yy y + σ yz z = ρ 2 u y 2 t σ xz x + σ yz y + σ zz z = ρ 2 u z 2 t u i : Components of the Particle Displacement Vector

44 NDT&E Methods: UT Material Constants Some Mathematics Later: The Three-Dimensional Wave Equation for Linear, Elastic, Homogeneous Isotropic Material (λ+μ) Δ + x μ 2 u x = ρ 2 u x 2 t (λ+μ) Δ + z μ 2 u z = ρ 2 u z 2 t (λ+μ) Δ + y μ 2 u y = ρ 2 u y 2 t Volume Dilation of the Displacement Δ: Δ = ε xx + ε yy + ε zz

45 NDT&E Methods: UT Modes & Velocities For Isotropic Materials, There are only two Distinct Values for Wave Velocities: v L = λ + 2μ ρ v T = μ ρ Longitudinal Mode Transversal Mode

46 GROUP VELOCITY NDT&E Methods: UT PHASE VELOCITY Speed of Energy Transport Speed of Phase Propagation For isotropic, homogeneous, linear materials due to linearity and causality principles. A propagating medium is said to be dispersive if the phase velocity is a function of frequency, which is the case for attenuating materials, for example. For dispersive and anisotropic materials group and phase vectors are different

47 NDT&E Methods: UT Literature 1. R. Kienzler, G. Herrmann. On conservation laws in elastodynamics, International Journal of Solids and Structures 41, Elsevier (2004) 2. N.S. Ottosen, M. Ristinmaa. The Mechanics of Constitutive Modeling, Elsevier Science (2005), ISBN E. Henneke II, D. E. Chimenti. Ultrasonic Wave Propagation, Nondestructive Handbook, Volume 7 Ultrasonic Testing, Chapter 2, ASNT 3 rd edition

48 NDT&E Methods: Elastic Waves in Solids Constitutive Equations of Sound Waves The strain tensor is a field tensor it depends on external factors. The compliance tensor is a matter tensor it is a property of the material and does not change with external factors.

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