NDE 2011, December 8-10, 2011
of-sight. Because of this the inspection of material using Lamb wave is easy and fast. However, as Lamb wave is a dispersive wave and can sustain many modes of vibration in the same material depending upon the frequency of excitation, thickness of material, isotropy / anisotropy in material properties. Hence propagation and interaction of Lamb wave is always critical and needs to be understood well about their interactions with the presence of flaws. In this regard, the numerical modelling of Lamb wave propagation in material is of great help. This helps in clear understanding of the propagation of required mode and their interaction with the flaws present in the material.
Finite Element Method (FEM) as a numerical simulation tool is very popular and effective for simulation of similar problems. In this paper, a COMSOL® based FEM model has been developed and numerical simulations were carried out for lamb wave propagation in a composite structure and its interactions with the symmetric delaminations. The results are validated with the reported results.
1. INTRODUCTION
Composite materials [1] are being preferred in many areas over the metallic counterpart because of high specific strength, corrosion resistance. These properties facilitates for light weight components manufactured for high strength applications. However the presence of flaws like delaminations within the layers of composite can drastically reduce the intended performance. In view of that it is necessary to detect the delamination kind of flaws present the in the composite before its end use preferably through the Non-destructive testing methods. Ultrasonic testing methods like c-scan are usually used for detection of these kinds of flaws in composites. However, the method is tedious and time consuming in view of the requirement of physical positioning of probe on each point of interest of testing on the composite.
In contrast to the c-scan methods, Lamb wave also known as guided wave can propagate long distances in plate-like or cylindrical structures and provides information along the line-
COMSOL
®BASED 2-D FEM MODEL FOR ULTRASONIC GUIDED WAVE PROPAGATION IN SYMMETRICALLY DELAMINATED UNIDIRECTIONAL MULTI-
LAYERED COMPOSITE STRUCTURE
Bikash Ghose1, 2*, Krishnan Balasubramaniam2*, C V Krishnamurthy3 and A Subhananda Rao1
1 High Energy Materials Research Laboratory, Sutarwadi, Pune, India
2 CNDE, Mechanical Engg Department, IIT Madras, Chennai, India
3 Department of Physics, IIT Madras, Chennai, India
ABSTRACT
A two dimension Finite Element Method (FEM) based model has been developed for simulation of ultrasonic guided wave propagation in a multi layered composite structure in COMSOL® Multiphysics. The composite structure consists of 8 layers and symmetrically arranged having lay up of [0/+45/-45/90]s. Each ply is made up of unidirectional fibres with the predefined mechanical properties. The individual layers are considered to be made of carbon/epoxy material.
The unidirectional fibres have been rotated at different angles at different layers as in the given lay up. From the basic mechanical properties, the elasticity matrix has been generated for each layer with respect to the global co-ordinate system. The plane strain condition has been assumed to prevail in this model as the dimension of the composite in the z-direction is considerably higher than that of the thickness of the composite.
A hanning windowed cosine signal has been taken as the applied ultrasonic disturbance and applied to the end of the structure. Initially the ultrasonic guided wave has been generated in the complete structure without any debond. Then a symmetric delamination at the middle of the structure has been created and again simulated for the guided wave propagation with the presence of the delamination. The different modes of ultrasonic vibration are seen to be present due to the presence of the symmetric delamination in the structure. The results obtained have been verified with the reported result.
Keywords: Ultrasonic, Guided wave propagation, COMSOL, Composite, FEM, Delamination, Unidirectional fibres
2. TWO DIMENSIONAL FEM MODEL OF COMPOSITE USING COMSOL
A two dimensional COMSOL® [2] based FEM model has been developed for simulation of ultrasonic guided wave propagation within a composite structure.
2.1 Material
The composite laminate that has been modelled for guided wave propagation consists of 8 layers and lay-up is symmetric.
The thickness of the laminate is 2 mm and thickness of each lamina is 250 μm. The modelling has been done on the laminate of the length of 300 mm. The material of the lamina is considered to be of CFRP and has the predefined material properties.
Table 1 : Material properties of CFRP used in the model
Properties Unit Value
E11 Pa 1.2849×10
11
E22 Pa 0.0698×10
11
ν13 - 0.288
ν23 - 0.55
G13 Pa 0.0352
ρ Kg/m3 1700
The detail of the dimension of the laminate is mentioned below.
Thickness of each lamina : 0.25 mm
Number of layers : 8
Thickness of laminate : 2 mm
Length of laminate plate : 300 mm
Lay up : [0/+45/-45/90]s
The lay-up of lamina of [0/+45/-45/90]s is illustrated in Figure 1. The global co-ordinate system considered for the model is also given in the figure.
The material properties of the CFRP material used in the model is mentioned in Table 1. The properties are defined for the ply in which the unidirectional fibres are laid along the x-direction.
2.2 Incident ultrasonic wave
A 200 kHz frequency signal has been applied to the composite laminate. The signal is chosen as the 7 cycles of cosine function and operated with a hanning window. The applied disturbance as the input signal is applied to the left side of the material with displacements along the y-direction. The input signal is provided from a data file and the linear interpolation is considered for the value of displacement at the undefined time in the data file. The time domain representation of input displacement pulse used in the model is shown in the Figure 2.
2.3 Model Geometry
The laminate of length 300 mm has been considered to avoid the reflections from the sides. Eight numbers of laminas of thickness 250 μm each has been stacked to form the main laminate. The initial disturbances is applied to the left hand side of the laminate along the +ve y direction. The interior boundaries were not defined separately. The length 300 mm of the laminate is aligned along the x-axis of the global co- ordinate system whereas the thickness of 2 mm is along the y- axis of the co-ordinate system. The length of the laminate along the z-axis is assumed to be large enough for approximation of
Fig. 1 : Composite lamina shown with the lay-up of the layers and the global co-ordinate system used in the model
Fig. 2 : Time domain representation of the input displacement signal
Fig. 3 : A portion of the model geometry
the model to follow the plain strain condition. All observations were made on the top surface of the laminate. Figure 3 shows a portion of the model geometry.
2.4 Meshing
The quadrilateral (QUAD) elements available in COMSOL are used to mesh the domain of each lamina. Size of each element is chosen as 25 m × 25 μm. The minimum element quality is 0.99 and the minimum element area ratio is about 0.96.
2.5 Application mode
For modeling of the two dimensional geometry, the case of plane strain was considered because of the dimension of the material considered to be large enough in the third dimension (z-direction) as compared to the x and y directions. Time dependent analysis, Lagrange quadratic type of element and time dependent solver was used for the solution. The direct linear solver UMFPACK available in COMSOL is used for obtaining the solution. The duration of time span for the solution was chosen to be 200 μs such that the complete signal is captured at the point of observations. The time steps chosen in the model is 100 ns. BDF scheme of order 2 has been used as the integration scheme for time marching solution.
All the simulations were carried out in the 64-bit XP environment.
2.6 Global co-ordinate system of the model
As mentioned in section 2.1, the material properties are defined for the ply in which the unidirectional fibres are laid along the x-direction. The x-direction of the global co-ordinate system of the model is chosen to be the direction of the unidirectional fibres laid in the ply for the 0° orientation.
2.7 Input of material properties to the model
Although the CFRP material is orthotropic in nature but complete un-isotropy has been modelled in the simulation.
The elasticity matrix for the complete un-isotropy has been evaluated and given as an input to the model.
As per lay up of the composite [0/+45/-45/90]s, the fibre lay up direction of the uni-directional composite layer is rotated about +45, -45 and +90 degree to the global co-ordinate system
correspondingly in two layers each. The rotation is about the y-axis i.e., on x-z plane. In a 2D model (x-y plane) there is no provision (in COMSOL) of rotation of local co-ordinate about y-axis (on x-z plane). So off-axis behaviour of the composite has been studied. The material properties have been given to the model as the stiffness or elasticity matrix in 2D (4×4 matrix) as a complete anisotropic material. The elasticity matrices have been evaluated for the plies rotated by +45°, -45°and 90° about y-axis of the global co-ordinate system. The formulation used for evaluation of elasticity matrix for global and rotated co- ordinate system (about y-axis) is as described by Berthelot [1]. The elasticity matrices as evaluated for individual plies are used by the model for necessary evaluation.
2.8 Symmetric delamination within the laminate A delamintion of length 60 mm has been created at the half of the length i.e., at 150 mm (centre of the delamination is at 150 mm) and located at the mid portion within the laminate as shown in Figure 4 by indication of thick black line.
The delamination is basically created in the COMSOL model by de-merging the nodes of the elements at the delamination.
The ultrasonic signal is observed at four points on the top of the laminates which are 50 mm, 100 mm, 200 mm and 230 mm from the left end side of the composite laminate.
3. RESULTS
The A-scan signals at the four locations are simulated with and without the presence of the symmetric delamination of length 60 mm.
3.1 A-scan signals without delamination
A scan signal has been taken at a distance of 50 mm and 100 mm from the left end of the material and shown in Figure 5 &
Figure 6.
Figure 7 & Figure 8 shows the A scan signals at 200 mm and 230 mm from the left end of the material.
3.2 A-scan signals with delamination
A-scans in Pulse-echo mode from the numerical simulations taken at (a) 50 mm and (b) 100 mm for de-lamination size of 60 mm as simulated by COMSOL is shown in Figure 9 &
Figure 10.
Fig. 4 : Indicates the location of symmetric de-lamination present in the composite laminate
A-scans from the numerical simulations in pitch-catch mode, taken at (a) 200 mm and (b) 230 mm for a delamination size of 60 mm are shown in Figure 11 & Figure 12 respectively 4. DISCUSSION
The model developed in COMSOL multiphysics for numerical simulation clearly matches with the results already reported [3]. It is clear from the signal obtained without any delaminations that there is no generation of any other mode of lamb wave in the main laminate. Only incident A0 mode
travels in the main laminate and observed at all points of observations. However due to presence of delamination exactly at the mid plane (symmetric delamination) there is a generation of S0 mode at the entry edge of the delamination which travels within the sub laminate along with the incident A0 mode. When the S0 mode interacts with the exit end of the delamination again there is mode conversion and new A0 mode gets generated which propagates in the sub laminate as well as in main laminate. Along with this when the transmitted A0 mode interacts with the exit of the delamination it generates another S0 mode which propagates towards the entrance of the
Fig. 7 : A scan at 200 mm Fig. 8 : A scan at 230 mm
Fig. 5 : A scan at 50 mm Fig. 6 : A scan at 100 mm
Fig. 9 : Signal at 50 mm from left hand end Fig. 10 : Signal at 100 mm from left hand end
COMSOL® Multiphysics using the time domain analysis with proper selection of modelling parameters.
REFERENCES
1. “Composite Materials, Mechanical Behaviour and Structural Analysis”, Jean-Marie Berthelot, Springer, 1998
2. COMSOL Multiphysics User Guide
3. Ramdas C. et al, “Interaction of the primary anti- symmetric Lamb mode (Ao) with symmetric delaminations: numerical and experimental studies”, Smart Materials and Structures, 18, 2009, pp 1-7 4. Ramdas C, “Interaction of Lamb Wave with
Discontinuities in Composite Structures”, Chapter - 2, PhD Thesis, IIT Madras, 2010
Fig. 11 : A scan at 200 mm Fig. 12 : A scan at 230 mm
delamination. The S0 mode is confined to the sub laminate and undergoes multiple reflections at both ends of delamination whereas the A0 mode propagates both in main and sub laminate.
5. CONCLUSION
The COMSOL based implicit FEM code is capable of correctly simulate the propagation of A0 Lamb mode propagation in a composite structure. The mode conversion of A0 mode at the entry and exit of the delamination to generate the S0 mode is simulated correctly. The mode conversion of generated S0 mode when interacted with the entry or exit point of delamination was simulated correctly by COMSOL by the time domain analysis. The solution converges correctly for the chosen parameters of modelling. Hence it is possible to simulate the ultrasonic guided wave problems with
