Michel Castaings

Modal decomposition method for modeling the interaction of Lamb waves with cracks

The interaction of the low-order antisymmetric (a0) and symmetric (s0) Lamb waves with vertical cracks in aluminum plates is studied. Two types of slots are considered: (a) internal crack symmetrical with respect to the middle plane of the plate and (b) opening crack. The modal decomposition method is used to predict the reflection and transmission coefficients and also the through-thickness displacement fields on both sides of slots of various heights. The model assumes strip plates and cracks, thus considering two-dimensional plane strain conditions. However, mode conversion (a0 into s0 and vice versa) that occurs for single opening cracks is considered. The energy balance is always calculated from the reflection and transmission coefficients, in order to check the validity of the results. These coefficients together with the through-thickness displacement fields are also compared to those predicted using a finite element code widely used in the past for modeling Lamb mode diffraction problems. Experiments are also made for measuring the reflection and transmission coefficients for incident a0 or s0 lamb modes on opening cracks, and compared to the numerical predictions.

Finite element model for waves guided along solid systems of arbitrary section coupled to infinite solid media

The Semi-Analytical Finite Element (SAFE) method is becoming established as a convenient method to calculate the properties of waves which may propagate in a waveguide which has arbitrary cross-sectional shape but which is invariant in the propagation direction. A number of researchers have reported work relating to lossless elastic waves, and recently the solutions for nonpropagating waves in elastic guides and for complex waves in viscoelastic guides have been presented. This paper presents a further development, addressing the problem of attenuating waves in which the attenuation is caused by leakage from the waveguide into a surrounding material. This has broad relevance to many practical problems in which a waveguide is immersed in a fluid or embedded in a solid. The paper presents the principles of a procedure and then validates and illustrates its use on some examples. The procedure makes use of absorbing regions of material at the exterior bounds of the discretized domain.

Transfer matrix of multilayered absorbing and anisotropic media. Measurements and simulations of ultrasonic wave propagation through composite materials

This paper presents an adaptation of the well‐known Thomson/Haskell method to introduce anisotropic attenuation into the formulation of the transfer matrix. The expression for the transfer matrix of one elementary ply is modified to take into account the heterogeneity of the generated modes at interfaces between absorbing media. This heterogeneity depends on the direction of propagation and on the anisotropic attenuation of the ply. The transfer matrix of the stratified composite is derived classically from the multiplication of the elementary matrices and leads to the transfer function of the plate. The transmission and reflection coefficients of multilayered viscoelastic anisotropic composite materials can be computed, for any direction of propagation and for any stacking sequence, with a program that requires the viscoelastic characteristics of the ply. The method is tested with PEEK matrix/carbon fibers composites. The viscoelastic properties of the ply were previously measured with a unidirectional c...

Inversion of ultrasonic, plane-wave transmission data in composite plates to infer viscoelastic material properties

Abstract Stiffness and damping properties of viscoelastic materials are given by the real and imaginary components, respectively, of the material constants. A new technique is proposed to experimentally measure the real and imaginary components of anisotropic (and isotropic) viscoelastic plates. Main advantage of this technique is that material properties of thin plates can be measured where many other techniques fail. Material properties are obtained by numerically inverting the transmitted ultrasonic fields, obtained for different incident angles. Simplex inversion algorithm is applied to initial estimates of plate thickness and plate properties. By this iterative technique the values of the unknown parameters (material properties and plate thickness) are continuously modified to give better agreement between the experimental and theoretical transmitted fields. After a certain number of iterations the speed of convergence of the Simplex scheme is significantly reduced. To improve the accuracy of convergence the Newton–Raphson inversion technique is adopted at that point. By this technique material properties of different types of plates are measured. These is a glass plate (isotropic plate with no damping), a polymer plate (isotropic plate with damping), and glass fiber reinforced epoxy plates with different fiber orientations (anisotropic plates with damping). Both real and imaginary components are successfully measured for all these plates. In a relative scale the measurement error for the imaginary components is higher. Reliability of the measured material constants of fiber reinforced epoxy plates is verified by the method of invariance. All experiments are carried out in the frequency range that is appropriate for satisfying two conditions—the specimen homogeneity and the plane wave conditions.

Single Sided Inspection of Composite Materials Using Air Coupled Ultrasound

A single sided, air coupled ultrasonic NDT system based on the generation and reception of the a0 Lamb mode is described. 1–3 composite ultrasonic transducers are employed, transmitting and receiving transducers being oriented at the appropriate coincidence angle for the generation and detection of the mode; they are placed close together and scanned over the surface of the plate to produce a ‘C-scan’ image. Tests have been carried out on carbon fiber composite plates with delamination defects, the damaged areas being readily detected. The measurements have also been compared with numerical predictions based on a finite element model, good agreement being obtained.

Lamb and SH waves generated and detected by air-coupled ultrasonic transducers in composite material plates

Abstract Electrostatic, air-coupled, ultrasonic transducers are used to generate and detect guided waves in anisotropic solid plates. Waves considered in this study are Lamb-type and SH-type, guided modes. If the plane of propagation coincides with a plane of symmetry of the material, then Lamb modes only are launched and detected by the transducers. If the plane of propagation does not coincide with a plane of symmetry of the material, then Lamb modes are still generated and detected, but guided, SH-like modes are, too. The variation of phase velocity with frequency is measured for several modes propagating in different directions along a glass–epoxy composite plate. A numerical model that takes into account the anisotropy of composite materials is developed to predict the dispersion curves (phase velocity, group velocity or wave-number versus frequency) and the displacement fields of plate waves, the plane of propagation being either a plane of symmetry or not. The experimental phase velocities are in good agreement with the predicted dispersion curves, thus showing that the forward problem concerning the propagation of plate waves in anisotropic, homogeneous, composite material plates is properly solved. The dispersion curves associated with the predicted displacement fields show that guided modes in composite plates have different behaviors depending on their direction of propagation.

Delta operator technique to improve the Thomson–Haskell‐method stability for propagation in multilayered anisotropic absorbing plates

A modified version of the transfer‐matrix method that models propagation of heterogeneous plane waves through immersed multilayered plates made of anisotropic absorbing layers is presented. Since this method suffers from numerical instabilities, the so‐called delta matrix operator is applied. As for propagation through isotropic media, the case of propagation in the principal plane of anisotropic media requires us to define sixth‐order delta matrices. This method eliminates numerical difficulties. The same technique is used for propagation out of the principal plane. Fifteenth‐order delta matrices are necessary to get improved results. However, numerical problems persist with computational data corresponding to the usual experimental situations. Then, a modified version of the delta matrix operator is proposed. From the reflection and transmission coefficients expressions, both 15th‐ and 20th‐order delta matrices are necessary to get reliable numerical results, whatever the computational data may be.

Finite element predictions for the dynamic response of thermo-viscoelastic material structures

In this paper, constitutive relations are solved in the Fourier domain using a finite-element-based commercial software. The dynamic responses of viscoelastic bars or plates to either thermal or mechanical loads are predicted by considering complex moduli (Young, Poisson, stiffness moduli) as input data. These moduli are measured in the same frequency domain as that which is chosen for modeling the wave propagation. This approach is simpler since it suppresses the necessity of establishing a rheological model. Specific output processing then allows the numerical predictions to be compared to analytical solutions, in the absence of scatterers. The performances of this technique and its potential for simulating more complicated problems like diffraction of waves or for solving inverse problems are finally discussed.

Guided waves propagating in sandwich structures made of anisotropic, viscoelastic, composite materials

The propagation of Lamb-like waves in sandwich plates made of anisotropic and viscoelastic material layers is studied. A semi-analytical model is described and used for predicting the dispersion curves (phase velocity, energy velocity, and complex wave-number) and the through-thickness distribution fields (displacement, stress, and energy flow). Guided modes propagating along a test-sandwich plate are shown to be quite different than classical Lamb modes, because this structure does not have the mirror symmetry, contrary to most of composite material plates. Moreover, the viscoelastic material properties imply complex roots of the dispersion equation to be found that lead to connections between some of the dispersion curves, meaning that some of the modes get coupled together. Gradual variation from zero to nominal values of the imaginary parts of the viscoelastic moduli shows that the mode coupling depends on the level of material viscoelasticity, except for one particular case where this phenomenon exists whether the medium is viscoelastic or not. The model is used to quantify the sensitivity of both the dispersion curves and the through-thickness mode shapes to the level of material viscoelasticity, and to physically explain the mode-coupling phenomenon. Finite element software is also used to confirm results obtained for the purely elastic structure. Finally, experiments are made using ultrasonic, air-coupled transducers for generating and detecting guided modes in the test-sandwich structure. The mode-coupling phenomenon is then confirmed, and the potential of the air-coupled system for developing single-sided, contactless, NDT applications of such structures is discussed.

Surface impedance matrices to model the propagation in multilayered media

The surface impedance matrices in stratified plates made of fluid layers and/or anisotropic absorbing solid layers link the particle velocity field to the stress field at any interface. A surface impedance matrix represents the impedance at a given interface of all the layers located between that interface and one boundary of the medium. For each interface, there are two surface impedance matrices, each one corresponding to one boundary. This notion simplifies the computations of the modal solutions. The number of elements in the matrices involved in the computations is divided by a factor of four in comparison to usual matrix methods. This paper describes the method and presents examples to illustrate its interests and its efficiency where other techniques fail, for instance in the case of modes possessing energy in layers embedded in the structure.