Simulation of Propagating Ultrasonic Waves in Complex Composite Material

Transcription

1 ENDT Mo..3. Simulation o Propagating Ultrasonic Waves in omplex omposite Material Julia HEGEMANN, Björn AN DEN BROUKE, Alexander PEIFFER, Rudol BILGRAM, EADS Deutschland, Munich, Germany Horst BAIER; Technische Universität München, Munich, Germany Abstract. The Elastodynamic Finite Integration Technique (EFIT) is a numerical method to simulate the propagation o ultrasonic waves through complex material structures. In this paper the simulation method is applied to composite laminate materials. The detailed microstructure models o the laminate specimens are generated by the sotware module WiseTex rom the Katholieke Universiteit Leuven in Belgium. The simulation results o ultrasonic wave propagation in multi-axial multi-ply laminates are presented. Introduction In aerospace industry high quality standards or surveillance and maintenance o composite structures require requent testing or instance by using ultrasonic inspection techniques. Due to interactions between ultrasonic waves and complex structures there is still the need o simulation tools in order to better the understanding o the underlying phenomena. This enables an optimising o test parameters and the reliable interpretation o results The Elastodynamic Finite Integration Technique EFIT is a promising numerical simulation method which supports this objective. It has its origin in the work o Weiland [] or the calculation o electromagnetic waves and is transerable to various wave propagation phenomena. The application o EFIT to elastodynamic ultrasonic waves in the ield o non-destructive testing o concrete structures was introduced by Marklein et al. [] [3] and Schubert [4]. In this paper the simulation is perormed in complex composite material in order to study the interactions o ultrasonic waves with the (micro-) structure. In addition to the heterogeneity o the material consisting o carbon ibres, resin, voids etc. the carbon ibres possess orthotropic material properties. For the generation o this sophisticated geometrical arrangement the sotware module WiseTex was chosen. The EFIT- code and urther relevant aspects are implemented in ++. The ++code provides an interace or the user and enables the output o dierent quantities that help to analyse the simulation results. The combination o EFIT and WiseTex is described and irst results or selected examples are presented. The results are visualised by the sotware GStudio Max.. o olume Graphics.

2 . Elastodynamic Finite Integration Technique. Equations o Motion and Hooke s Law EFIT is based upon the basic elastodynamic equations, the Newton-auchy equations o motion and Hooke s law. These equations connect the most important elastodynamic ield variables, the stress tensor T and the particle velocity vector v. The integral orm o the equation o motion is: and Hooke s law: ( ) v& ( R, d = nt ( R, ds ρ + R ( R, d S () ( R, d = nv( R, ds + T & g( R, d. () S ρ stands or the equilibrium mass density, or the volume orce density vector, is the elasticity tensor and g the source excitation. The derivative with respect to time o one ield variable in a control volume is given by the surace integral S (with the normal n) o the complementary ield variable. The transormations are done using Gauss law. In the presented examples the excitation out o the passive state is always realised through the volume orce density in the longitudinal direction (x- direction). Thereore all other components o the volume orce density and the surace excitation are neglected in the ollowing.. Discretisation or Transversal-Isotropic Heterogeneous Material The discretisation o the equations yields a spatial and temporal discrete orm. Thereore the model is embedded in a three-dimensional cubic staggered grid with the edge length d. The components o the stress tensor are located on the grid nodes and the components o the velocity vector in the middle o the grid edges, as shown in Figure. y z x d n+m v T T T 33 T v T 3 n+m let (l) ront () top ( back (ba) right (r) T 3 v 3 bottom (b) n+m 3 Figure : Location o the Discrete Field ariables in the Staggered Grid

3 Respectively, the discretisation in time is also staggered. The time derivatives o the ield variables are calculated iteratively at alternating hal time steps (leap-rog method). Equations (3) and (4) exempliy the discrete orms whereas Equations (5) and (6) present the dierence quotient discrete in time or time step l: (r) (n ) (n ) (ba ) (n ) ( ρ v & d = T T + T T + T T + d, (3) 3 3 T& ( n) d = + 5 v ( ) ( ( ) ( )) ( ( ) ( ) ( ) ( ) 3 v n ba b n n ) 3 v3 + 4 v v + v3 v3, (4) ( ) ( ( ) ( n ) ( r ) ( n )) ( n ) ( l) ( n ) ( ba ) v + v v + ( u ) ( n ) ( r ) ( n ) 6 v + v3 v3 + v v v + v v v [l] i All constraints or providing stability and consistency are adhered. For the boundaries in longitudinal direction (x-direction) stress-ree boundary conditions are chosen. In order to avoid relections in the two other directions periodic boundary conditions are implemented. i ij [l-] [l- ] = v + Δt v&, (5) i [l T + ] [l- ] [l] = T + Δt T& ij. (6) ij. Modelling with WiseTex WiseTex is a sotware module o the Department o Metallurgy and Materials Engineering (MTM) o the Katholieke Universiteit Leuven (K.U.Leuven) in Belgium. It serves as an integrated textile pre-processor and aords three-dimensional modelling o the internal geometry and microstructure o undeormed and deormed material respectively. For example, it provides lexible modelling o multi-axial multi-ply abrics which include cracks, channels, and others [5]. For this work Proessor Lomov rom the K.U.Leuven provided validated models o unit cells. The models represented here are stitched multi-axial multi-ply abrics. For a quadriaxial one o them, the geometric data and the inormation about the dierent layers are shown in Table and Table. Table : Geometric Data o the Unit ell Size 8, x 5, x 0,93 mm Areal density 667,9 g/m² Porosity (inter-yarn) 99,4 % Fibre olume Fraction 4, % Ply Fibre Orientation grad Table : Data o the Fibre in the Plies, average % average % hannel/rack Width mm Length mm 0 45,0 5,7 hannel 0,66 n/a 45 45,0 45,8 rack 0,8, ,0 45,8 rack 0,8, ,0 5,5 rack 0,48 7,3 3

4 The single plies and the 3D image o this unit cell are shown in Figure. racks and channels that are illed with resin in the simulation can be seen very well. Figure : The Four Plies and the 3D Image o the Unit ell The next step or realising a model as realistic as possible is to compress the unit cell to increase the ibre volume raction up to 60 %, beore building a laminate o our unit cells. By mirroring a symmetrical composition is implemented. In the same way a second laminate is build up by ten biaxial unit cells (ibre orientation 0 and 90 ) [6]. The laminates (see Figure 3) are then discretised and used as input or the simulations. z x y Figure 3: Laminates o Four Quadriaxial Unit ells and o Ten Biaxial Unit ells 3. First Results For both models a raised cosine signal with two cycles R serves as excitation pulse. The centre requency is MHz. A maximum occurring requency o 4 MHz is assumed. The excitation is introduced on the surace o the laminate (in x-direction). The R - signal is described by the ollowing equation: (x= 0) R = ( cos( π ) cos( π 0 ür ür 0 t t >. (7) The irst discrete model consists o 0 x 3 x 34 cubic cells. For the interpretation o the results the wave propagation is considered in relation to the density o the model. The density model is shown on the let hand side in Figure 4. The plies with the 4

5 dierent orientations are in the colours green and red, resin is presented in blue. The turquoise material is a stitching yarn. On the right hand side a cross-section through a threedimensional snapshot o the simulated wave propagation is illustrated. The excitation pulse is nearly totally introduced. The particle velocity is blue or negative and red or positive velocity. Figure 4: Density Model and ut o a Snapshot through the Simulated Wave Propagation In Figure 5 there are our consecutive cuts both through the density model and the snapshot o the simulated wave propagation. The cuts are made normal to the longitudinal direction o the wave propagation. In the ourth cut a channel illed with resin can be seen. But already in the irst cut the eects caused by the channel are visible. In the orange circle there is a brighter square inside. The particle velocity is lower there caused by the lower density o the resin compared to the density o the carbon ibres. The wave is scattered at the contact surace. Inside the resin the wave is slowed down. 5

6 Figure 5: uts through the Density Model and the Snapshot o the Simulated Wave Propagation o the Quadriaxial Model The second laminate consists o 08 x 06 x 43 cubic cells in its discrete orm. In Figure 6 an axial, a sagittal, and a rontal cut through the wave propagation snapshot in relation to the respective density cut are presented. It is visible that the wave propagation is not symmetrical in both transversal directions due to the ibre orientation in the layers. Some eects caused by resin accumulations and the stitching yarn are marked by circles and arrows. There the scattering and the dispersion can be connected directly to the (micro-) structure. Figure 6: uts through the Density Model and the Snapshot o the Simulated Wave Propagation o the Biaxial Model 4. onclusions First results clearly demonstrate scattering, relections, and other eects on the received modelled ultrasonic signals produced by interactions with the (micro-) structure. Further simulations will analyse eects caused by porosity and undulation. It is also planned to parallelise the ++-code to perorm the simulations on a cluster. This oers the possibility o simulating larger models. Moreover the excitation signals shall be optimised. 6

7 Acknowledgements The authors would like to thank the Department o Metallurgy and Materials Engineering o the Katholieke Universiteit Leuven or providing the models generated with WiseTex and supporting the work with them. Reerences [] T. Weiland, Eine Methode zur Lösung der Maxwellschen Gleichungen ür sechskomponentige Felder au diskreter Basis, AEÜ, Bd. 3, Issue 3, P. 6 0, 997 [] R. Marklein, K. Mayer, P. Ampha, K.-J. Langenberg, omputer Simulation o Ultrasonic Non- Destructive Testing o oncrete Using the Elastodynamic Finite Integration Technique (EFIT), Proc. O the NDT-E, Berlin, 003 [3] R. Marklein, Numerische erahren zur Modellierung von akustischen, elektromagnetischen, elastischen und piezoelektrischen Wellenausbreitungsproblemen im Zeitbereich basierend au der Finiten Integrationstechnik, Shaker erlag Aachen, 997 [4] F. Schubert, Ausbreitungsverhalten von Ultraschallimpulsen in Beton und Schlussolgerungen ür die zerstörungsreie Prüung, Dissertation, Technische Universität Dresden, 999/000 [5] S. Lomov, T. Peeters, Integrated Textile Preprocessor WiseTex - User s guide, K.U.Leuven, Leuven, 00 [6] J. Hegemann, Simulation der Ultraschallwellenausbreitung in komplexen Faserverbundwerkstoen, Diploma Thesis, Technische Universität München, 005 7