6th Australasian Congress on Applied Mechanics, ACAM 6

12-15 December 2010, Perth, Australia

New damage detection technique based on governing differential equations of continuum mechanics. Part II: in-plane loading Stuart J. Wildy*, Ben S. Cazzolato, Andrei G. Kotousov

School of Mechanical Engineering, The University of Adelaide, SA 5005 Australia *Email: stuart.wildy@adelaide.edu.au

Abstract: This paper discusses a new method for detection of crack damage in plate structures based on direct application of with a 3D scanning laser vibrometer. The recent advances in scanning laser technology now allow a very accurate measurement of the displacement field, which in the case of undamaged structure must satisfy the fundamental equations of continuum mechanics everywhere. A violation of these equations at specific locations indicates the presence of the damage, which can be detected and sized by utilising a defect search algorithm.

Keywords: crack, equilibrium, governing differential equation, laser vibrometry, Savitzky-Golay filter, strain compatibility.

1 Introduction

This is the second paper in the series on new damage detection techniques based on governing differential equations of continuum mechanics (e.g. the principles strain compatibility and equations of equilibrium) [1]. The first paper in the series [2] presented an algorithm based on the governing differential equation for beam deflection, which was derived from the two principles. An experimental study was conducted using a 1D scanning laser vibrometer to determine whether the detection of delamination damage in a composite beam was possible. By employing a Savitzky-Golay differentiating filter to the out-of-plane displacements measured by a 1D scanning laser vibrometer, the residual of the governing differential equation of beam deflection was acquired. The study showed that the delamination damage was could be successfully detected.

In this paper, an algorithm is derived from the two principles and will be used for the detect crack damage in plate structures with in-plane displacements. The algorithm utilises the non-contact three-dimensional displacement measurements from a 3D scanning Doppler laser vibrometer [3,4,5]. Essentially, the damage is detected by identifying a violation of the measured in-plane displacement field within the strain compatibility and equilibrium conditions.

An experimental investigation of the newly proposed algorithm derived from strain compatibility and equations of equilibrium will be presented. Using a Polytec PSV-3D scanning laser vibrometer, the in-plane displacements of a plate with various damage scenarios was measured and a Savitzky-Golay filter was utilised to evaluate the algorithm.

2 Structure of the document

A general governing differential equation of surface strains (�� and ��) can be evaluated for a thin plate with extensional deformations (�� and ��) by equating the shear strain components ���of the strain compatibility equation for plate structures [6] and generalised equilibrium equation [7]. The result is found to be,

���� �� � 0 (1)

By introducing the strain-displacement relationship into (1), the governing differential equation can be rearranged in terms of in-plane displacement components (�� and ��) and expressed as

��� ��

��� � ��

��� ��

��� �� � � 0 (2)

The strategy used to detect the presence of damage is based on the validity of (2) to the experimentally measured displacement fields (�� and ��). By applying the experimentally measured displacement field to (2) a residual term (�) can be determined. A non-zero residual term (� � 0) will

signify a violation of (2) and, therefore, indicate the presence of damage. The residual term can be represented by

� � ��� ��

��� � ��

��� ��

��� �� � (3)

4 Experimental Set Up

In order to investigate the effectiveness of the proposed algorithm, an experiment was conducted to identify crack damage in plate specimens. The plates were made from 2mm thick transparent acyclic (PMMA) and were 215mm long by 100mm wide, with a slit at either end to accommodate the centre clamp bolts. A detailed drawing can be seen in Figure 1. A number of specimens were created for the in-plane experiments. These included a plate with an edge crack, centre crack and notch with a protruding crack. The notch utilised was of ninety degrees and 20mm long, and cracks investigated were 5mm, 10mm, 15mm, 20mm and 25mm in length.

Figure 1: General test specimen design.

To provide purely in-plane displacements a test rig was designed for uni-axial loading of plate specimens, as seen in Figure 2(a). The rig consists of two clamps to fix the specimen, a 15kN load-cell (Novatech F204) and a piezo-stack actuator (Physik Instrumente P-235.40 PICA) all connected in series.

The plate specimens were oscillated a 1Hz with an approximate peak to peak load of 200N and the resulting three dimensional displacement fields were measured using a 3D scanning Doppler laser vibrometer (PSV-400-3D), seen in Figure 2(b). The 3D scanning Doppler Laser Vibrometer operates on the Doppler principle and measure the vibratory displacement in the direction of the three laser beams. By aligning the lasers it is possible to focus all three laser beams at a point on an object and acquire the displacement components in three orthogonal directions via an orthogonal decomposition.

For each test, three dimensional surface displacements were measured on a square mesh of 39 × 39 points with a spatial mesh interval of approximately 2mm, thus covering an area of approximately 76mm x 76mm.

y

x

(3)

(a) (b)

Figure 2: Experimental Equipment (a) In-plane displacement rig and (b) 3D scanning Doppler laser vibrometer mounted on a fixed support

4 Experimental Study

The application of the proposed algorithm for detection of an edge crack (Figure 3), centre crack (Figure 4) and a crack propagating from a 90-degree notch (Figure 5) was conducted, where the cracks lengths investigated were 5mm, 10mm, 15mm, 20mm and 25mm in length.

The residual term is evaluated from measured in-plane displacement fields via a two-dimensional Savitzky-Golay differentiating filter. A detailed explaination of the Savitzky-Golay differentiating filter can be found in [2,8,9]. The Savitzky-Golay differentiating filter essentially performs a local two-dimensional polynomial regression (of degree �) on a distribution of 2� 1 x 2� 1 points to determine the smoothed value of the �� and �� derivatives with respect to the �- and �-axis, respectively, for each point. The main advantage of this filter is that it generally preserves features of the distribution such as relative maxima, minima and width. In the following experimental results a Savitzky-Golay differentiating filter utilising a 3 order polynomial (� � 3) over a distribution of 5x5 points (� � 2) was employed. A crack is then detectable if the evaluated residual term satisfies the following condition

|�| � � ��� ��

��� � ��

��� ��

��� �� �� � Ψ (4)

where Ψ is a threshold value that is larger than the noise floor. In Figures 3, 4 and 5 a threshold value of 200m-3 was selected.

(a) 5mm crack (b) 10mm crack

(c) 15mm crack (d) 20mm crack

(g) 25mm crack

Figure 3: Acquired residual of governing differential equation (�) for various edge crack lengths. The dark blue and dark red areas on the graph represent the location where � has exceed the threshold Ψ � 200. At � � 0mm and � � 100mm represents the edges of the plate and the pink line defines the

location of the edge crack

(a) 5mm crack (b) 10mm crack

(c) 15mm crack (d) 20mm crack

(h) 25 mm crack

Figure 4: Acquired residual of governing differential equation (�) for various centre crack lengths. The dark blue and dark red areas on the graph represent the location where � has exceed the threshold Ψ � 200m-3

. At � � 0mm and � � 100mm represents the edges of the plate and the pink line defines the location of the centre crack

(a) No crack (b) 5mm crack

(c) 10mm crack (d) 15mm crack

(e) 20mm crack (f) 25mm crack

Figure 5: Acquired residual of governing differential equation (�) for a notch with various crack lengths. The dark blue and dark red areas on the graph represent the location where � has exceed

the threshold Ψ � 200m-3. At � � 0mm and � � 100mm represents the edges of the plate and the pink line defines the location of the notch and crack

5 Discussion

To evaluate the governing differential equation (2) at a point on a structure from measurements of displacement, a number of neighbouring measurement points are required within the Savitzky Golay differentiating filter. The filter is constructed under the assumption that there are no cracks, voids or other discontinuities in the vicinity or between these measurement points. Therefore, if a crack intersects any two neighbouring points used within filter, the large displacements that occur between the faces of the crack will violate the governing differential equation (2). It should be noted that as the neighbouring measurement points used to evaluate the governing differential equation get closer together, the governing differential equation gets closer to being satisfied (equal to zero).

Figures 4, 5 and 6 show that with the defined threshold, all the cracks investigated were easily detectable and that the lengths of the cracks were able to be determined successfully. Figure 5a showed that the notch without a crack did not significantly affect the residual of the governing differential equation. However, when the crack was within the measurement grid (Figure 5b), the residual term of the governing differential equation at the crack location was significantly influenced. This suggests that the structural geometry of the specimen has little influence on the residual term and that the residual term is only influenced by the damage within the measurement grid.

6 Conclusion

A new technique for the detection of crack damage in plates, which based on the governing differential equation for in-plane displacements, is presented. This technique, which utilised the strain compatibility conditions and equilibrium equations, can determine a violation of these equations for a localized area, indicating the presence of cracks.

An experimental investigation was conducted to evaluate the implementation of the algorithm for the detection of crack damage in plate structures. The in-plane displacement fields of the acrylic plate specimens where measured by a 3D scanning Doppler laser vibrometer. The accurate detection and localisation of the crack damage, verifies the validity of the technique which needs no a-priori information about the specimen geometry or construction.

References

1. S.J. Wildy, A.G. Kotousov, and J.D. Codrington, 2008, “A new passive defect detection technique based on the principle of strain compatibility”, Smart Materials and Structures, vol. 17.

2. S.J. Wildy, B.S. Cazzolato, A.G. Kotousov and J.D. Codrington, 2010, “New damage detection technique based on governing differential equations of continuum mechanics. part I: out-of-plane loading", Proceedings of 6

th Australasian Congress on Applied Mechanics, Perth, Australia, Dec 12-15.

3. H. Weisbecker, B.S. Cazzolato, S.J. Wildy, S. Marburg, J.D. Codrington, A.G. Kotousov, 2010, “Surface strain measurements using a 3D scanning laser vibrometer”, Experimental Mechanics, under review.

4. B.S. Cazzolato, S.J. Wildy, J.D. Codrington, A.G. Kotousov and M. Schuessler, 2008, “Scanning laser vibrometer for non-contact three-dimensional displacement and strain measurements”, Proceedings of the Australian Acoustical Society Conference, Geelong, Australia, Nov 24-26.

5. S.J. Wildy, B.S. Cazzolato, A.G. Kotousov and H. Weisbecker, 2010, “New method for accurate strain measurements utilising a 3D scanning laser Doppler vibrometer ", Proceedings of 6

th Australasian Congress

on Applied Mechanics, Perth, Australia, Dec 12-15.

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