Quantitative Detection of the Defects in Thin-Walled Pressure Vessels ...

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Quantitative assessment of the coordinates, size, embedding depth, and type of defects of a thin- walled pressure vessel is becoming increasingly important for ...
Journal of Nondestructive Evaluation, Vol. 21, No. 3, September 2002 (䉷 2003)

Quantitative Detection of the Defects in Thin-Walled Pressure Vessels with Holography and Shearing Speckle Interferometry Xide Li,1,* Xingfu Liu,2 and Kai Wang1 Received December 12, 2001; Revised September 21, 2002

Quantitative assessment of the coordinates, size, embedding depth, and type of defects of a thinwalled pressure vessel is becoming increasingly important for both economic and safety reasons. Nondestructive testing methods, holographic interferometry, shearing speckle interferometry, and simple mechanical models are combined to quantitatively estimate these defect characteristic parameters (DCPs). Experimental tests were conducted to demonstrate the efficiency and the accuracy of this combined technique for thin-walled spherical vessels that contain the cavities or cracks. Relationships between the DCPs and partial fringe patterns caused by the local defects are presented, and factors that affect the estimative accuracy of DCPs are discussed.

KEY WORDS: Holography; shearing speckle interferometry; pressure vessels.

1. INTRODUCTION

There are many NDT techniques have been applied to defect detection, such as visual examination, liquid penetration testing, leak testing, vibration monitoring, magnetic particle inspection, ultrasonic testing, eddy current testing, gamma and x-radiography, acoustic emission, magnetic flux leakage method, infrared thermography, optical methods, and others. General speaking, these NDT techniques possess different advantages and are propitious to varied cases. However, with the advantages of having an extremely fast response, no contact, and full field, the two epoch-making optical methods, holographic interferometry (HI) and speckle interferometry (SI), find a wide use in material research, manufacturing, and lifetime inspection of industrial components.(1–3) Shearing speckle interferometry (SSI) is a SI method that allows the direct measurement of displacement gradient.(4–7) Based on the detection of the defect-induced partial fringe patterns from the global fringe pattern in NDT, HI and SSI have had great success in the qualitative determination of the existence of the defects. However, the shortage of quantitative information about the DCPs using HI and SSI detection makes HI and SSI rarely involved with the

It is now widely accepted that all structures, including pressure vessels, possess defects from the start of their service life. Therefore, the defect characteristic parameters (DCPs) of the defect, the stress subjected, and the local properties of the material will play a major role in determining the structural security and reliability. According to fracture mechanics, defects in the materials lead to failure by growing to a critical size. Thus, by knowing the dimensions of the defects present in the structures, it is possible to estimate both the remaining life of the structures and the extent of the degradation. The objective of nondestructive testing (NDT) is to detect these defects, estimate the DCPs of the defects, and assess the structural security, reliability, and integrity under the effects of the defects by combining the failure and fracture mechanics. 1

Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, P.R. China. 2 Institute of Structure Mechanics, Chengdu, 610003, P.R. China. * Corresponding author. E-mail: [email protected]

85 0195-9298/02/0900-0085/0 䉷 2003 Plenum Publishing Corporation

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Quantitative Detection of the Defects in Thin-Walled Pressure Vessels

analysis of fracture mechanics and restricts the applications of these NDT techniques in accurately estimating the remaining life and the extent degradation of the inspected structures. In this report, HI, SSI, and a simple mechanical model have been combined to quantitatively detect the DCPs, that is, the size, coordinates, and embedding depth of the defects. The defects are a finely manufactured cavity and crack on the inner wall of the spherical pressure vessels. Moreover, the factors that affect the estimative accuracy of DCPs, such as load parameters in the model, are also discussed.

2. PRINCIPLE Quantitative DCPs are difficult to obtain mainly due to the nonunique relation between the DCPs and deformation of the detected defects subjected to loading. Moreover, the complex structures in the shape of the defects in a practical structure prohibit obtainment of the analytical displacement solution. To obtain the quantitative DCPs, we first proximately simplify the defect shapes as two kinds—one a cavity and the other a crack—and then construct their mechanical model as subjected to uniform inner pressure. Then, the fringe equations of HI and SSI, defect displacements, and displacement derivatives obtained from the models are combined to quantitatively obtain the values of DCP. 2.1. Mechanical Models for a Cavity and a Crack on the Thin Wall of the Pressure Vessel Cavities and cracks are two usual types of defects that present in the raw material or are introduced into a structure when it was manufactured, fabricated, heat treated, and assembled. It also may be generated due to deterioration of the material or the structure under working conditions. In our case, we assume that the cavity defect is a circle half through hole with diameter of 2a and embedding depth t, and the crack defect is a half through rectangle with length of 2a, height of 2b (b  a), and embedding depth of t. During testing, the thinwalled pressure vessel was subjected to a uniform load q inside its chamber. Figure 1 shows their loading configurations and structures. If we assume t is smaller than ␦, the wall thickness of the pressure vessel, then cavity and crack defects can be treated as a circle sheet and a rectangle sheet where their edges are clamped, respectively. Thus, when the sheets are subjected to uniform load, their out-of-plane

Fig. 1. Loading configuration and schematic of tested structures.

displacements are obtained easily through the elastic theory: Circle sheet: w(r) ⫽

3q(1 ⫺ v2) 2 (a ⫺ r2)2 16Et3

(1)

where E and v are the material constants, also called Young modulus and Poisson ratio, respectively. Therefore, the gradient of w(r) can be expressed as ⭸w(r) 3q(1 ⫺ v2) 2 ⫽ (a ⫺ r2)r ⭸r 4Et3

(2)

Rectangle sheet: w(x,y) ⫽





21q(1 ⫺ v2)(x2 ⫺ a2)2(y2 ⫺ b2)2 4 32Et3(a4 ⫹ b4 ⫹ a2b2) 7 21q(1 ⫺ v2) 32Et3

冋冢 冣 b a

(3)

册 冋冢 冣 2

2

1 b

x ⫺ b2 1⫹

4

冢冣 b a



2

y2 ⫺ 1



2

2

冢冣

4 b 7 a

Considering b  a, Eq. (3) can be simplified as

Li, Liu, and Wang

87 21q(1 ⫺ v2)(y2 ⫺ b2)2 32Et3

(4)

a⫽

冪N(r )r

Similarly, the displacement gradient along the yaxis is

t⫽



w(y) ⫽

⭸w(y) 21q(1 ⫺ v )(y ⫺ b )y ⫽ ⭸y 8Et3 2

2

2

(5)

Equations (1) to (5) provide the out-of-plane displacements and displacement gradients of cavity defect and crack defect, respectively. Meanwhile, the DCPs, such as defect size 2a, 2b, and embedding depth t, are all included in these equations. Therefore, if fringe equations are combined together with Eqs. (1) to (5), DCP values can be solved quantitatively.

n

3

2 m

⫺ N(rm)r2n ⫹ 冪N(rm) ⭈ N(rn)(r2m ⫺ r2n) N(rn) ⫺ N(m) (10)

3q 1 ⫺ v2 (a2 ⫺ r2m)2 ⭈ ⭈ 8 E␭ N(0)

where m ⫽ n. If we select rn ⫽ 0, Eq. (10) is simplified as a⫽

⫹ 冪N(r ) ⭈ N(0) r 冪N(0)N(0) ⫺ N(r ) m

m

(11)

m

t⫽

⫺v 冪 8 ⭈ 1 E␭ 3

3q

2



a4 N(0)

Another conclusion is N(a) ⫽ 0

(12)

when rm ⫽ a. 2.2. DCP Measurements for a Cavity Defect with HI HI is a three-dimensional displacement-sensitive optical method. In double-exposure HI, the fringe equation can be expressed by(8) →



e ⭈ D ⫽ N␭

(6) →



where e is sensitive vector of the displacement, D is displacement vector of the object, N is bright fringe order, and ␭ is laser wavelength. If illumination and observation are in the same direction as the out-of-plane displacement, Eq. (6) can be simplified to w⫽

N ␭ 2

SSI is an SI method to measure displacement derivatives directly and very sensitive to the partial abruption of the deformations. SSI also eliminates the reference beam of holography and uses the simplified optical setup. Consequently, SSI has already received the wide applications for NDT today. Further details are in the literature.(3–5) If we use the normal illumination and normal viewing in the experiments, the bright fringes are formed wherever(5) ⭸w Nx␭ ⫽ ⭸x 2⌬x

(7)

(13)

Similarly, in y direction,

Using Eqs. (1) and (7), we have 3q(1 ⫺ v2) (a2 ⫺ r2)2 ⭈ N(r) ⫽ 8E␭ t3

2.3. DCP Measurements for a Cavity Defect and a Crack Defect with SSI

⭸w Ny␭ ⫽ ⭸y 2⌬y

(8)

This equation constructs a relation between the fringe orders and a and t, the DCPs of the cavity defect. Obviously, it describes a circle-shaped fringe pattern around the center and has the highest-order fringe pattern at r ⫽ 0. There is a simple way to obtain the values of a and t using Eq. (8). Assuming rm is the mth position where the fringe order is N(rm), we obtain

(14)

where ⌬x and ⌬y are the amounts of shearing in the object plane, and Nx and Ny are the fringe orders in the x and y directions, respectively. To the cavity defect, we apply Eqs. (2), (13), and (14): Nr ⫽

3q(1 ⫺ v2)⌬r 2 (r ⫺ a2)r 2E␭t3

(15)

Here, r can be in x or y direction. From the simple operation, Eq. (15) produces

3q(1 ⫺ v2) (a2 ⫺ r2m)2 N(rm) ⫽ ⭈ , m ⫽ 1, 2, 3 ⭈⭈⭈ 8E␭ t3

3

(9) Solving Eq. (9), the values a and t can be obtained by

Nr max ⫽

q(1 ⫺ v2)⌬r a a ⭈ , while rN max ⫽ ⫺ t 冪3E␭ 冪3

冢冣

(16)

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Quantitative Detection of the Defects in Thin-Walled Pressure Vessels

where we assume only the positive fringe orders are used. Therefore, the DCP of a cavity defect is a ⫽ 冪3앚rN max앚

(17)

a

t⫽

(18)

冪3E␭N 冪q(1 ⫺ v )⌬ 3

3

r max 2 r

Similarly, one can obtain similar formulas as Eqs. (17) and (18) using Eqs. (5), (13), and (14) for the defect of the crack. They can be expressed as Ny ⫽

21q(1 ⫺ v2)⌬y 2 (y ⫺ b2)y 4E␭t3

(19)

Thus b ⫽ 冪3앚yN max앚

(20)

b

t⫽

(21)

冪7q(1 ⫺ v )⌬ 3

3 3E␭N 2冪

y max 2 y

and 앚Ny min앚 ⫽ 0

(22)

while yn min ⫽ ⫾ b. Equations (20) to (22) are able to measure at an instant the DCP values quantitatively with one tested specimen and one fringe pattern. With series specimens, we use another procedure, a comparison measurement, to measure the embedding depths of defects, where we assume that all of the specimens are the same in material, geometric structure, and loading manner and are tested with the same optical measurement system. Thus, for different specimens, we have Nyi ⫽

21qi(1 ⫺ v2)⌬y 2 (yi ⫺ b2)yi 4E␭t3

(23)

where subscript i ⫽ 1, 2, 3. . . denotes the order of tested specimens. Let Ai equal to Ai ⫽

qi(y2i

Nyi 21(1 ⫺ v2)⌬y ⫽ 2 ⫺ b )yi 4E␭t3

(24)

we have t ⫽ t1

冪A 3

A1

i⫽1

(25)

i





Here we assume yi ⫽ 0 and y2i ⫺ b2 ⫽ 0. Equation

(25) shows that specimen selected as the known defect specimen, and the embedding depths of other specimens can be obtained only if they possess the same material, geometric structure, and loading manner with that of the specimen. Obviously, if one obtains the load qi and the data Nyi and yi along the y-axis, the parameters Ai can be easily calculated using Eq. (24) and then embedding t by Eq. (25).

3. EXPERIMENTAL MEASUREMENTS Two kinds of experiments have been carried out to demonstrate the ability and accuracy of the proposed method for detecting the DCPs. The specimens are halfspherical thin-walled pressure vessels, which were prepared by the steel 45 with Young modulus E ⫽ 2.1 ⫻ 105 MPa and Poisson ratio v ⫽ 0.3. The defects for cavities and cracks were manufactured inside the walls of the vessels with different sizes and depths. The specimens were clamped and sealed onto a heavy base, and their half-spherical shell surfaces are the measured areas. The load was exerted with high-pressure oil in the chambers of the vessels. In the experiments, a He-Ne laser with 30 mW and wavelength ␭ ⫽ 0.6328 ␮m was used as an illuminating source. The shearing amounts are ⌬r ⫽ 5.26 mm and ⌬y ⫽ 5.00 mm, respectively. 3.1. DCP Measurements of a Cavity Defect with HI and SSI In this experiment, the wall thickness of the vessel is 5 mm. There are four cavities located on the internal wall of the vessel. The diameters of the preprepared defects are 7.80, 8.00, 8.50, and 9.00 mm, respectively, from number 1 to number 4. The corresponding embedding depths are all 1.0 mm. Figure 2 shows the holographic fringe patterns with the different loads. The defect numbers and locations are also shown on the images. Using Figure 2c and Eq. (12), we obtained the measured diameters of these defects: 7.89, 8.16, 8.68, and 9.12 mm, respectively. These results are in very good agreement with the prepared defects. Figure 3 shows the defect sizes and embedding depths for number 2 vs. the changes in the pressure loads using Eq. (11). Similar results measured by using SSI are shown in Figures 4 and 5, where Figure 4 shows shearing fringe patterns of defect 2 during the loads changing, and Figure 5 shows the defect sizes and embedding depths vs. the changes of the pressure loads using Eqs. (17) and (18). Comparing the calculative DCP values obtained with simple mechanical models and the

Li, Liu, and Wang

89

Fig. 2. Holographic fringe patterns produced with different loads.

practical DCP values of the defect, considerable perfect results are obtained. 3.2. The Embedding Depth Measurements of the Serial Crack Defects with SSI Twenty-two specimens have been prepared to measure the defect embedding depths. The cracks were

equal-width groove-scored inside walls of the thinwalled pressure vessels along their zero degree latitude with b ⫽ 2.0 mm and a  b. The parameters of the 22 specimens are shown in Table I. Figure 6 shows some SSI fringe patterns and the corresponding loads. Table II shows the results of the defect embedding depths. In this table, specimens 1, 4, 7, 10, 13, 15, 18, and 21 are

90

Quantitative Detection of the Defects in Thin-Walled Pressure Vessels

Fig. 3. Defect sizes and embedding depths for number 2 vs. the changes of the pressure loads with HI.

Fig. 5. Defect sizes and embedding depths for number 2 vs. the changes of the pressure loads with SSI.

the known depth specimens in their groups and the others are detected specimens. Indeed, one can select any one as the known embedding depth specimen and

then use Eq. (25) to obtain the other embedding depths only if they have the same structural parameters. For different structural parameters, Eq. (25) must be prop-

Fig. 4. Shearing fringe patterns of the defect number 2 during the loads changing.

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91

Table I. Parameter for 22 Specimens Used in the Serial SSI Method R (mm) ␦ (mm) t (mm)

80 1.0 0.5 0.6 0.7

100 2.0 1.2 1.4 1.6

1.0 0.5 0.6 0.7

2.0 1.2 1.4 1.6

120 2.5 1.5 2.0

1.0 0.5 0.6 0.7

2.0 1.2 1.4 1.6

3.0 1.8 2.4

erly amended. The detailed descriptions are presented in the next section.

4. DISCUSSION Using simple mechanical models and interferometry fringe patterns, we obtain DCP values quantitatively. However, there are several factors affecting the accuracy of the results, such as the magnitude of the load, the material parameters, the type of defects and DCP of the defects, the testing methods, and the parameters selection of the experimental system. Because these factors are coupling together and controlling the defect deformations and fringe pattern structures, proper selection of the changeable parameters of these factors is important to obtain the correct DCP values. For example, load is an easily changing parameter in the experiment; if its value is too small, the testing system cannot detect the out-ofplane displacements near the defect edge. Similarly, if the load value is large, the support boundary condition is not correct. The results in Figure 3 showed this conclusion, where the small and large loads corresponded to the obvious errors. Another factor is the structure and geometric parameters in experiments, the wall thickness ␦, and the radius R of the vessel. We cannot obtain correct DCP values from that of a known defect. This shortage is due to the absent effects of structure geometric parameters in the mechanical models. A modification can change the limitation of Eq. (25); that is, we introduce a structural impact factor ␰(R1, ␦1; R, ␦) into Eq. (25), obtaining t ⫽ t1

冪A 3

A1

␰(R1, ␦1; R, ␦)

i⫽1

(26)

1

Obviously, ␰(R1, ␦1; R, ␦) is dimensionless and can be defined as ␰1(R, ␦) ␰(R1, ␦1; R, ␦) ⫽ ␰k(R1, ␦1)

(27)

where ␰1(R, ␦) is the structural impact factor of the tested

specimen, and ␰k(R1,␦1) is that of the known specimen. Applying Eqs. (24), (26), and (27), the general form of impact factor can be obtained by ␰(R1, ␦1; R, ␦) ⫽

Ny(y) E␭t3 ⭈ 2 4(1 ⫺ v )⌬y q ⭈ y(b2 ⫺ y2)

(28)

Equation (28) provides a relationship between the structural impact factor and the experimental data and some known parameters. Figure 7 shows the serial detection for DCP values where the DCP values of specimen 1 are the known values used to calculate the other defect DCP values. The corresponding structural impact factors are ␰(80, 1.0; 80, 1.0), ␰(80, 1.0; 100, 1.0) ⫽ 2.09, ␰(80, 1.0; 120, 1.0) ⫽ 2.66, ␰(80, 1.0; 80, 2.0) ⫽ 5.09, ␰(80, 1.0; 100, 2.0) ⫽ 13.22, ␰(80, 1.0; 120, 2.0) ⫽ 24.19, ␰(80, 1.0; 100, 2.5) ⫽ 20.42, and ␰(80, 1.0; 120, 3.0) ⫽ 52.11. We obtained very good agreement with the practical DCP values. Moreover, we find that the wall thickness has more effects than its radius, even though we do not know what form it is for the structural impact factor ␰(R1␦1;R,␦). Finally, we should point out that the derivatives of the out-of-plane displacements given by Eqs. (2), (5), (13), and (14) are different. In fact, Eqs. (2) and (5) represent the actual surface slopes of the tested specimens, and Eqs. (13) and (14) are approximations. We assume that these derivatives are equivalent for calculating the DCP values. This equivalence will introduce some errors into prediction of the DCP values when Eqs. (17), (18), (20), and (21) are applied. The shearogram is interpreted as the derivative of the displacement along the shear direction under small shear in the SSI measurement. In our case, the amount of the shear applied in the experiements is about 5 mm. It posesses the same order dimension as that of the defects and rather larger than the deformation values of the tested pressure vessels, but the prodicted DCP values given by our combined technique are considerably accurate compared with the preprepared DCP values. We consider the reason to be that the used amount of the shear is appropriate for the view field in shear measurement. That is, the amount of the shear is too small to compare with the size of the view field in the experiment (the view dimension is about 40 times of the used shear), so that the approximation of ⌬wx ⬵

⭸w ⭸w ⭸w ⭸w ⌬x ⫽ ⌬ or ⌬wy ⬵ ⌬y ⫽ ⌬ ⭸x ⭸x x ⭸y ⭸y y

is correct for the measured view field in SSI measurement. Therefore, the displacement field encoded by shear fringe patterns including that produced by the defects contained in the measured view field can be approximative as its displacement derivative field. Here ⌬wx and ⌬wy are

92

Quantitative Detection of the Defects in Thin-Walled Pressure Vessels

Fig. 6. Shearing fringe patterns and the corresponding loads in serial crack defect detections.

Li, Liu, and Wang

93 Table II. Embedding Depth Detections with the Serial SSI Method

No.

R (mm)

␦ (mm)

t (mm)

A (MPa-mm)

No.

R (mm)

␦ (mm)

t (mm)

A (MPa-mm)

1 2 3 4 5 6 7 8 9 10 11

80 80 80 80 80 80 100 100 100 100 100

1.0 1.0 1.0 2.0 2.0 2.0 1.0 1.0 1.0 2.0 2.0

0.5 0.6 0.7 1.2 1.4 1.6 0.5 0.6 0.7 1.2 1.4

0.365 0.211 0.133 0.134 0.084 0.056 0.759 0.440 0.277 0.348 0.219

12 13 14 15 16 17 18 19 20 21 22

100 100 100 120 120 120 120 120 120 120 120

2.0 2.5 2.5 1.0 1.0 1.0 2.0 2.0 2.0 3.0 3.0

1.6 1.5 2.0 0.5 0.6 0.7 1.2 1.4 1.6 1.8 2.4

0.147 0.275 0.116 0.965 0.558 0.351 0.635 0.400 0.268 0.407 0.172

displacement increments in x direction and y direction, respectively. To investigate this idea, we calculated the ⭸w ⌬ about a half-spherical thinvalues of ⌬wy and ⭸y y walled pressure vessel, that was subjected to the same load and boundary conditions as our experiment. The used parameters are q ⫽ 1.5 MPa, R ⫽ 100 mm, ⌬y ⫽ 5.0 mm, ␦ ⫽ 1.0 mm, E ⫽ 2.1 ⫻ 105 MPa, and ⭸w ⌬ at the position ␭ ⫽ 0.6328. The values of ⌬wy and ⭸y y of the first fringe order are 0.3164 and 0.3509 ␮m, respectively. Therefore, their absolute difference is 0.0345 ␮m, nearly a tenth of the value of the displacement increment. Similarly, at the position of the second fringe order, we ⭸w obtained ⌬wy ⫽ 0.6328 ␮m, ⌬y ⫽ 0.6774 ␮m, and ⭸y the corresponding absolute difference, 0.04465 ␮m, about

one fourteen of the value of the displacement increment. ⭸w It is obvious that even the values of ⌬ are slightly ⭸y y greater than that of the ⌬wy owing to the approximations used in SSI; good approximate results are obtained when the amount of the shear is considerably smaller than the dimension of the view field. Moreover, the proper magnitude of the shear can enhance the sensitivity of the SSI and improve the measurement accuracy.

5. CONCLUSION Based on the simple mechanical models and HI and SSI fringe patterns, DCPs were investigated quantitatively. Experimental results show that the estimation of the size and embedding depth of the cavity and the crack defects are considerably accurate in thin-walled pressure vessels NDT by this combined technique. This is a hybrid and quantitative analysis method to provide defect information. Therefore, the forecast and assessment criteria for the remaining life and the degradation of tested structures can be established based on the fracture mechanics. We provided advice for the practical defect analysis even though the simple mechanical models have been used and possess the potential of being developed into a practical NDT tool for quantitative measurements in the case of complex shape of the defects by combining the HI, SSI, and FEM.

ACKNOWLEDGMENTS

Fig. 7. Embedding depth measurements with serial detection method where specimen number 1 as the known specimen.

We gratefully acknowledge financial support (project 10072031) by NSFC, Visiting Scholar Foundation of Key Lab in University of the Ministry of Education of

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Quantitative Detection of the Defects in Thin-Walled Pressure Vessels

China, and the Basic Research Foundation of Tsinghua University.

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3. R.S. Sirohi, Ed., Speckle metrology, (Marcel Dekker, New York, 1993). 4. Y.Y. Hung, Shearography: a novel and practical approach for nondestructive testing, J. Nondestru. Testing 8 (2); pp. 55–67 (1998). 5. Y.Y. Hung, Shearography: a new optical method for strain measurement and nondestructive testing, Opt. Eng. 21 (3), pp. 391–395 (1982). 6. Y.Y. Hung, Applications of digital shearography for testing of composite structures, Composites Part B, 30, pp. 765–773 (1999). 7. X. Li, X. Liu, and Z. Chen, On-site detection of seam weld defects by using electronic speckle pattern interferometry, Acta Photon. Sini. 27 (10), 911–918 (1998). 8. T. Kreis, Holographic interferometry, (Springer-Verlag, Berlin, 1996).