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Identification of crack positions and crack loading quantities from strain gauge data by inverse problem solution
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ECF22 - Loading and Environmental effects on Structural Integrity ECF22 - Loading and Environmental effects on Structural Integrity
Identification Identification of of crack crack positions positions and and crack crack loading loading quantities quantities from from gauge data inverse problem XV Portuguesestrain Conference on Fracture, PCF 2016, 10-12 Februarysolution 2016, Paço de Arcos, Portugal strain gauge data by by inverse problem solution a Ramdane Boukellif a,∗ a,∗, Andreas Ricoeura Ramdane Boukellif , Andreas Ricoeur Thermo-mechanical modeling of a high pressure a Institute a Institute
turbine blade of an
of Mechanics, Department of Mechanical Engineering, University of Kassel, 34125 Kassel, Germany of Mechanics, Department of Mechanical Engineering, University of Kassel, 34125 Kassel, Germany
airplane gas turbine engine
Abstract P. Brandãoa, V. Infanteb, A.M. Deusc* Abstract A method a for the detection of cracks in finite and semi-infinite plane structures is presented. This allows both the identification Department of Mechanical Engineering, Superior Técnico, Lisboa, Av.This Rovisco Pais,both 1, 1049-001 Lisboa, A method for the parameters detection ofsuch cracks finiteInstituto and semi-infinite planeUniversidade structures isderespect presented. allows the identification of crack position as in length, location and inclination with to a reference coordinate system and Portugalangles of crack position parameters such as length, location and inclination angles with respect to a reference coordinate system and b the calculation of stressofintensity factors (SIFs). Instituto The method is based onUniversidade strains measured at different locations on the surface IDMEC, Department Mechanical Engineering, Superior Técnico, de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, the calculation of stress intensity factors (SIFs). The method is based on strains measured at different locations on the surface of a structure and the application of the dislocation technique.Portugal Cracks and boundaries are modelled by continuous distributions of dislocation acCeFEMA, structure Department and the application of the dislocation Cracks andUniversidade boundaries are modelled by continuous distributions of Mechanical Engineering, Superior Técnico, de Lisboa, Av. Rovisco Pais, Lisboa, of densities. This approach gives a set Instituto of technique. singular integral equations with Cauchy kernels, which can1,be1049-001 solved using of dislocation densities. This approach gives a set of singular integral equations with Cauchy kernels, which can be solved using Portugal Gauss-Chebyshev numerical quadrature [Erdogan et al. (1973)]. Once knowing the dislocation densities, the strain at an arbitrary Gauss-Chebyshev numerical quadrature [Erdogan et al. Once are knowing the dislocation thebystrain at an point can be calculated. The crack parameters as well as (1973)]. external loads parameters which aredensities, determined solving thearbitrary inverse point canwith be calculated. The crack parameters as well as external loads are parameters which determined by solving the inverse problem a genetic algorithm. Once knowing loading and crack parameters, the SIFs canare be calculated. Abstract problem with a genetic algorithm. Once knowing loading and crack parameters, the SIFs can be calculated. c 2018 � 2018The TheAuthors. Authors. Published by Elsevier © Published by Elsevier B.V. B.V. their operation, modern aircraft engine components are subjected to increasingly demanding operating conditions, c During � 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the ECF22 organizers. Peer-review under responsibility of the ECF22 organizers. especially under the high pressure turbine Such conditions cause these parts to undergo different types of time-dependent Peer-review responsibility of the(HPT) ECF22blades. organizers. degradation, one of which is creep. A model the finite element method (FEM) was developed, in order to be able to predict Keywords: Crack detection; distributed dislocations;using inverse problem; finite plate; strain gauges detection; dislocations; problem;(FDR) finite plate; gaugesaircraft, provided by a commercial aviation Keywords: the creepCrack behaviour of distributed HPT blades. Flight inverse data records for strain a specific company, were used to obtain thermal and mechanical data for three different flight cycles. In order to create the 3D model needed for the FEM analysis, a HPT blade scrap was scanned, and its chemical composition and material properties were obtained. The data that was gathered was fed into the FEM model and different simulations were run, first with a simplified 3D block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The 1. rectangular Introduction 1. overall Introduction expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a model can be useful in the goal turbine blade life, given a set of FDR data. Engineering structures are of in predicting general exposed to cyclic or stochastic mechanical loading. Exhibiting incipient
Engineering structures are in general exposed to cyclic or stochastic mechanical loading. Exhibiting incipient cracks, particularly light-weight shell and plate structures suffer from fatigue crack growth, limiting the life time of © 2016particularly The Authors. Published byshell Elsevier cracks, light-weight andB.V. plate structures suffer from fatigue crack growth, limiting the life time of thePeer-review structure and supplying theofrisk a fatalCommittee failure. Due to the uncertainty of loading boundary conditions and responsibility the of Scientific of PCF 2016. the structure under and supplying the risk of a fatal failure. Due to the uncertainty of loading boundary conditions and the geometrical complexity of many engineering structures, numerical predictions of fatigue crack growth rates and the geometrical complexity of many engineering structures, numerical predictions of fatigue crack growth rates and residual strength are not reliable. monitoring techniques, Keywords: High Pressure Turbine Blade;Most Creep;experimental Finite Element Method; 3D Model; Simulation.nowadays, are based on the principle residual strength are not reliable. Most experimental monitoring techniques, nowadays, are based on the principle of wave scattering at the free surfaces of cracks. Many of them are working well, supplying information about the of wave scattering at the free surfaces of cracks. Many of them are working well, supplying information about the position of cracks. One disadvantage is that those methods do not provide any information on the loading of the crack position of cracks. One disadvantage is that those methods do not provide any information on the loading of the crack tip. In this work, the development of a concept for the detection of cracks in finite and semi-infinite plate structures tip. In this work, the development of a concept for the detection of cracks in finite and semi-infinite plate structures under mixed mode loading conditions is presented. The concept is based on the measurement of remote strain fields under mixed mode loading conditions is presented. The concept is based on the measurement of remote strain fields by using strain gauges and the application of the distributed dislocation technique of linear elasticity [Bilby et al. by using strain gauges and the application of the distributed dislocation technique of linear elasticity [Bilby et al. (1968); Weertman (1996)]. As an approach, different from the FEM, cracks are modelled with a collocation of discrete (1968); Weertman (1996)]. As an approach, different from the FEM, cracks are modelled with a collocation of discrete Corresponding author. Tel.: +351 218419991. ∗*Corresponding author. Tel.: +49 561 804 2505 ; ∗ Corresponding E-mail address: [email protected] author. Tel.: +49 561 804 2505 ;
fax: +49 561 804 2720. fax: +49 561 804 2720. E-mail address: [email protected] E-mail address: [email protected] 2452-3216 2016 The Authors. Published Elsevier B.V. c © 2210-7843 � 2018 The Authors. Published byby Elsevier B.V. cunder 2210-7843 � 2018 The responsibility Authors.of Published by organizers. Elsevier B.V. Peer-review under of the Scientific Committee of PCF 2016. Peer-review responsibility the ECF22 Peer-review under responsibility of the ECF22 organizers. 2452-3216 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the ECF22 organizers. 10.1016/j.prostr.2018.12.015
Ramdane Boukellif et al. / Procedia Structural Integrity 13 (2018) 85–90 Boukellif et al. / Structural Integrity Procedia 00 (2018) 000–000
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dislocations. Within a continuum mechanics framework, these dislocations are no lattice defects but displacement discontinuities describing the local crack opening displacement. Thus, it is not necessary to discretize the domain around the crack, considerably saving computation time and data, which is crucial for an efficient solution of the inverse problem. Solving the inverse problem, e.g. with a genetic algorithm [Holland (1992)] or an adaptive simulated annealing [Metropolis et al. (1953)], this allows the identification of external loading, crack position parameters, such as length, location or inclination angles and the calculation of stress intensity factors for straight and even for curved cracks in partly or fully bounded plane structures. 2. Theoretical Background The stress field in a body with cracks can be calculated by superposition of the stresses σiDj due to dislocations, which are distributed at the positions of the cracks. The total stress field σ∗i j ( xˆ, yˆ) in local coordinates ( xˆ, yˆ ) is as follows:
σ∗i j ( xˆ, yˆ) =
M �
σiD;I ˆ ) + σ∞ ij , j ( xˆ, y
(1)
I=1
where M is the number of cracks and i j = xˆ xˆ, yˆyˆ , xˆyˆ . Using the principle of superposition, the induced stresses in particular at the points on the I-th crack line due to the distribution of dislocations of all the M cracks, can be calculated by: D;I IJ IJ ˆ ˆ � a σxˆ xˆ ( xˆI ) M � J G ˆ xˆ xˆ ( xˆ I ; ξ J ) G yˆ xˆ xˆ ( xˆ I ; ξ J ) B (ξˆ )� 2µ � xIJ D;I xˆ J IJ ˆ ˆ ( x ˆ ) σ d ξˆJ , | xˆI | < aI , yˆ yˆ I = G xˆyˆ yˆ ( xˆI ; ξ J ) Gyˆ yˆ yˆ ( xˆI ; ξ J ) D;I IJ π (κ + 1) J=1 Byˆ (ξˆJ ) IJ ˆ ˆ σ xˆyˆ ( xˆI ) −a J G xˆ xˆyˆ ( xˆ I ; ξ J ) G yˆ xˆyˆ ( xˆ I ; ξ J )
(2)
where κ is Kolosov’s constant and µ is the shear modulus. Eq. (2) is applied to incorporate boundary conditions at crack faces. The dislocation influence functions Gki j are given in Hills et al. (1996); Dundurs et al. (1964). These describe stresses at a field point ( xˆ, yˆ) due to a unit Burgers vector at the source point ξˆ J . The index k = xˆ, yˆ in the influence function indicates the direction of dislocations whereas i j = xˆ xˆ, yˆ yˆ , xˆyˆ denote the components of induced stresses. The Eq. (2) gives a set of singular integral equations with Cauchy kernels, which can be solved using Gauss-Chebyshev numerical quadrature. The dislocation densities Bk (ξˆJ ) are determined accounting for boundary conditions. The first condition is that the crack surfaces are traction free. Secondly, the stresses on the external boundaries are equal to the subjected boundary loads. Finely, the displacement jumps at the crack tips are equal to zero and the gradient fields at this points are singular. Once having computed Bk (ξˆJ ), the strain at arbitrary points is calculated assuming plane stress conditions. 3. Crack detection and parameter identification 3.1. Finite and semi-infinite plate with inclined cracks at arbitrary locations First verifications have been carried out numerically. The strain εi j (Pm ) emerging from the distributed dislocation technique is ”measured” at points Pm representing the positions of virtual strain gauges aligned along the edges of a rectangle with corner coordinates ( x¯, y¯) and ( x˜, y˜ ). First example is a finite plate (20 mm x 20 mm) with two inclined cracks and Pm (m = 1, ..., 12) measuring points, ( x¯(a) ; y¯ (a) ) = (1; 19) mm, ( x˜(a) ; y˜ (a) ) = (19; 1) mm, as shown in Fig. 1(a). The second example is a semi-infinite plate with inclined interior crack and Pm (m = 1, ..., 4) measuring points, ( x¯(b) ; y¯ (b) ) = (2; 20) mm, ( x˜(b) ; y˜ (b) ) = (18; −20) mm, w = 20 mm, as shown in Fig. 1(b). The third example is a semi-infinite plane with inclined edge crack and Pm (m = 1, ..., 4) measuring points, ( x¯(c) ; y¯ (c) ) = (2; 20) mm,
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( x˜(c) ; y˜(c) ) = (18; −20) mm, as shown in Fig. 1(c). The ”unknown” parameters from the inverse problem solution based on a genetic algorithm [Holland (1992)] are given in Tables 1, 2, 3. The parameters have successfully been determined by the solution of the inverse problem. The SIFs KI and KII have been calculated subsequently based on the identified parameters. σ∞ yy
σ∞ xy
σ¯ yy
σ¯ xy
σ∞ yy
σ∞ xy
σ¯ xx ( x¯(b) ; y¯ (b) )
( x¯(a) ; y¯(a) )
2a
α2
2a
measuring point Pm
2
(x2 ; y2 )
α1 2a 1 (x1 ; y1 ) ( x˜(a) ; y˜(a) )
(+
( x¯(c) ; x¯(c) )
)
α (xa ; ya ) y
y x
(
−)
α x
a
w ( x˜(c) ; x˜(c) )
( x˜(b) ; y˜ (b) )
y x (b)
(a)
(c)
Fig. 1: (a): Finite plate with two inclined cracks and Pm (m = 1, ..., 12) measuring points; (b): Semi-infinite plate with inclined interior crack and Pm (m = 1, ..., 4) measuring points; (c): Semi-infinite plane with inclined edge crack and Pm (m = 1, ..., 4) measuring points.
Table 1: Results of the crack detection and parameter identification, see Fig. 1(a). parameters
given
identified
σ ¯ xx [MPa] σ ¯ yy [MPa] σ ¯ xy [MPa] a1 [mm] a2 [mm] α1 [◦ ] α2 [◦ ] x1 [mm] x2 [mm] y1 [mm] y2 [mm]
25 50 10 3 2 30 150 8 13 6 13
24, 99 49, 99 10, 0 3, 0 2, 0 30, 0 150, 0 7, 99 13, 0 6, 0 12, 99
Table 2: Results of the crack detection and parameter identification, see Fig. 1(b); √ KI , KII [MPa mm].
Table 3: Results of the parameter identification, see Fig. 1(c); √ KI , KII [MPa mm].
parameters
given
identified
parameters
given
identified
σ∞ yy [MPa] σ∞ xy [MPa] xa [mm] ya [mm] a [mm] α [◦ ] KI (+) KI (−) KII (+) KII (−)
50 30 12 4 6 40 176, 54 168, 55 133, 85 114, 40
50, 00 29, 99 11, 99 3, 99 6, 00 39, 99 176, 62 168, 62 133, 86 114, 40
σ∞ yy [MPa] σ∞ xy [MPa] a [mm] α [◦ ] KI KII
50 30 6 40 −21, 95 156, 71
50, 00 29, 99 5, 99 39, 99 −21, 86 156, 60
3.2. Sensitivity analysis for a finite plate 3.2.1. Crack detection and parameter identification depending on number of strain gauges To study the influence of the number of strain gauges on the accuracy of parameter identification for an inclined center crack in a finite plate (20 mm x 20 mm), the configurations depicted in Fig. 2 have been investigated with ( x¯; y¯ ) = (1; 19) mm, ( x˜; y˜ ) = (19; 1) mm. Four, eight and twelve virtual strain gauges are used to solve the inverse problem, see Fig. 2(a), (b), (c). Tables 4, 5, 6 show the error between the given problem and obtained results by using four gauges, eight gauges and twelve gauges for different crack lengths. Tab. 7 shows the error between the given problem and obtained results for different inclination angles of a center crack a = 6 mm in a plate with twelve strain gauges, see Fig. 2(c). From these results, it appears that four strain gauges may be sufficient for the identification of crack parameters only for the smallest crack. Eight strain gauges are enough to identify the small and medium crack lengths. With twelve strain gauges, all parameters could be identified very well. Also for different inclinations, the parameters were successfully determined by using twelve strain gauges, see, Tab. 7.
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σ¯ xy
σ¯ xy
σ¯ yy
σ¯ yy
σ¯ xx
σ¯ xy
σ¯ yy
(xc ; yc )
σ¯ xx
( x¯ ; y¯ )
(xa ; ya ) measuring point Pm
y
(+) α
2a
α
2a
2a
α
(xa ; ya )
(xa ; ya )
( x˜ ; y˜ ) y
x
y x
x (b)
(a)
(c)
Fig. 2: Inclined center crack in finite plate (xc ; yc ) = (20; 20) mm under boundary loads σ ¯ xx = 30 MPa, σ¯ yy = 90 MPa, σ¯ xy = 20 MPa with: (a): Pm (m = 1, ..., 4) measuring points; (b): Pm (m = 1, ..., 8) measuring points; (c): Pm (m = 1, ..., 12) measuring points. Table 4: Results of the crack detection and parameter identification by using four strain gauges (α = 60◦ ), see Fig. 2(a).
Table 5: Results of the crack detection and parameter identification by using eight strain gauges (α = 60◦ ), see Fig. 2(b).
parameters
error: given problem/obtained results (4 gauges) crack half-length a [mm] 1 3 5 7 9
parameters
error: given problem/obtained results (8 gauges) crack half-length a [mm] 1 3 5 7 9
σ ¯ xx [%] σ ¯ yy [%] σ ¯ xy [%] a [%] α [%] x0 [%] y0 [%]
0, 03 0, 011 0 8, 5 12, 11 0, 1 0, 1
σ ¯ xx [%] σ ¯ yy [%] σ ¯ xy [%] a [%] α [%] x0 [%] y0 [%]
0, 03 0, 011 0 0, 1 0, 05 0 0, 1
0, 23 0, 077 2 69, 8 53, 81 92, 14 12, 4
0, 06 0, 011 0, 15 11 15, 21 0, 1 0
1, 76 0, 188 9, 15 63, 42 8, 98 73, 8 91, 8
2, 83 0, 255 14, 45 79, 33 132, 25 81, 7 45.9
0, 03 0, 011 0, 05 0 0 0 0, 1
0 0 0 0, 2 0 0 0, 1
0 0, 011 0 0 0 0 0, 1
7, 13 5, 41 16, 55 9, 22 34, 85 0 0, 1
Table 6: Results of the crack detection and parameter identification by using twelve strain gauges (α = 60◦ ), see Fig. 2(c).
Table 7: Results of crack detection and parameter identification for different inclination angles α and a = 6 mm , see Fig. 2(c).
parameters
error: given problem/obtained results (12 gauges) crack half-length a [mm] 1 3 5 7 9
parameters
0, 033 0, 011 0 0 0 0 0 0, 036 0
σ ¯ xx [%] σ ¯ yy [%] σ ¯ xy [%] a [%] α [%] x0 [%] y0 [%] KI (+) [%] KII (+) [%]
σ ¯ xx [%] σ ¯ yy [%] σ ¯ xy [%] a [%] α [%] x0 [%] y0 [%] KI (+) [%] KII (+) [%]
0 0 0, 05 0, 033 0, 016 0, 01 0 0, 187 0, 11
0 0 0, 05 0 0, 016 0, 1 0 0, 046 0, 07
0 0 0, 05 0, 14 0, 016 0 0 0, 184 0, 07
1, 93 0, 56 4, 55 0, 07 0, 5 0, 1 0 1, 66 4, 55
0
error: given problem/obtained results inclination α [◦ ] 40 80 120 160
0 0, 02 0, 1 0 1 0 0 0, 030 0, 050
0 0 0, 05 0, 16 0, 025 0 0 0, 16 0, 159
0, 033 0 0, 05 0 0 0 0, 1 0, 03 0, 150
0, 033 0, 011 0, 05 0 0 0 0 0, 029 0, 013
0, 033 0, 011 0 0 0 0, 1 0 0, 008 0.018
3.2.2. Parameter identification depending on locations of strain gauges To study the influence of the distance of strain gauges on the accuracy of the solution of the inverse problem for an inclined center crack (a = 1 mm) in a finite plate (30 mm x 30 mm), the configurations depicted in Fig. 3 have been investigated. Five different squares indicate the positions of virtual strain gauges. The dimensions of these squares are given in the caption of Fig. 3. Along each of the squares, 12 gauges are distributed just as shown in Fig. 3. Tab. 8 shows the given problem and results of parameter identification depending on locations of strain gauges. The parameters have been determined very accurately at all strain gauge positions except for the smallest square (1).
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σ ¯ xy
σ ¯ yy
(3)
(5) (4)
(30;30) σ ¯ xx
(2) 2a α (1) (xa , ya )
y x Fig. 3: Inclined center crack (xa ; ya ) = (15; 15) mm, 2a = 2 mm, α = 60◦ in finite plate 30 mm x 30 mm under boundary loads σ¯ xx = 30 MPa, σ¯ yy = 90 MPa, σ¯ xy = 20 MPa. Different locations of strain gauges: square (1): 6 mm x 6 mm; square (2): 12 mm x 12 mm; square (3): 18 mm x 18 mm; square (4): 24 mm x 24 mm; square (5): 28 mm x 28 mm. Table 8: Results of parameter identification depending on locations of strain gauges, see Fig. 3. parameters
given
σ¯ xx [MPa] σ¯ yy [MPa] σ¯ xy [MPa] a [mm] α [◦ ] x0 [mm] y0 [mm]
30 90 20 1 60 15 15
square (1) 29, 94 89, 973 20, 0 1, 006 66, 337 5, 59 11, 87
square (2) 30, 0 90, 0 19, 99 0, 999 60, 0 14, 99 14, 99
identified square (3) 29, 99 89, 99 20, 0 1, 0 59, 986 15, 0 15, 0
square (4) 29, 99 89, 99 20, 0 0, 999 59, 986 15, 0 14, 99
square (5) 29, 99 90, 0 20, 0 0, 99 60, 0 15, 0 15.0
3.2.3. Parameter identification depending on the size of the plate and related gauge positions To determine the maximum size (xc , yc ) of the plate at which a small crack is detected accurately, different plate sizes have been examined, where the strain gauges are always close to the edges. Hereby a center crack length of 2a = 2 mm was assumed, see Fig. 2(c). Tab. 9 shows the given problems and results of parameter identification depending on the dimensions of the plate. It is very interesting to see that the small crack and all parameters could be identified on large plates up to 60 mm x 60 mm. Table 9: Results of parameter identification depending on the size of the plate and gauge positions. parameters (xc , yc ) [mm] σ¯ xx [MPa] σ¯ yy [MPa] σ¯ xy [MPa] α [◦ ] a [mm] x0 [mm] y0 [mm]
given 30 90 20 60 1 10 10
identified (20; 20) 29, 99 89, 99 20, 0 60, 0 1, 0 10, 0 10, 0
given 30 90 20 60 1 15 15
identified (30; 30) 29, 99 90, 0 20, 0 60, 0 0, 99 15, 00 15.00
given 30 90 20 60 1 20 20
identified (40; 40) 29, 99 89, 99 20, 00 59, 99 0, 999 19, 99 19, 99
given 30 90 20 60 1 25 25
identified (50; 50) 29, 99 89, 99 19, 99 60, 35 1, 000 24, 99 25, 00
given 30 90 20 60 1 30 30
identified (60; 60) 30, 00 90, 00 20, 00 59, 92 0, 999 29, 98 29, 99
given 30 90 20 60 1 35 35
identified (70; 70) − − − − − − −
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3.2.4. Parameter identification depending on strain measurements with presumed errors In order to investigate the sensitivity of the results to disturbances in the measured strains, random errors in the ”measured” strain components have been simulated. An inclined center crack in a finite plate (20 mm x 20 mm) is used for this investigation, see Fig. 2(c). Tab. 10 shows the results of the sensitivity analysis based on presumed measurement errors and demonstrates the sensitivity towards inaccurate strain determination. The results show that the external loads are not sensitive to inaccurate strain measurements. In contrast, crack length, inclination and position show non-negligible deviations starting from ±5% assumed error in strains. Stress intensity factors are well reproduced up to ±10% of assumed errors, then particularly KII becomes useless. Table 10: Results of parameter identification depending on strain measurement with presumed errors. parameters
given 0%
±1%
identified presumed error in measured strain ±3% ±5% ±7% ±9%
30 90 19, 999 59, 999 2, 99 9, 99 10 94, 63 52, 59
29, 87 89, 83 20, 03 59, 30 3, 06 10, 05 10, 04 96, 90 55, 90
30, 32 90, 81 19, 98 61, 18 3, 03 9, 99 10, 46 95, 01 49, 18
(xc , yc ) = (20; 20) mm σ ¯ xx [MPa] σ ¯ yy [MPa] σ ¯ xy [MPa] α [◦ ] a [mm] x0 [mm] y0 [mm] KI (+) [%] KII (+) [%]
30 90 20 60 3 10 10 94, 63 52, 60
29, 38 88, 83 19, 65 61, 47 3, 5 10, 85 9, 21 101, 42 52, 55
29, 25 87, 62 19, 61 62, 8 3, 42 11, 52 9, 62 97, 55 46, 26
28, 89 87, 09 19, 46 59, 17 3, 38 12, 35 9, 21 102, 94 60, 14
±11%
±13%
28, 83 85, 48 19, 35 67, 39 4, 07 11, 35 10, 04 103, 97 27, 19
30, 41 89, 14 19, 96 92, 60 0, 52 5.97 6, 0 41, 61 −28, 96
4. Conclusions In this work, the concept of distributed dislocations is applied for the detection of cracks and the calculation of SIFs in finite and semi-infinite plane structures. The crack and loading parameters could be successfully determined by the solution of the inverse problem. The sensitivity analysis has shown that the positions of the strain gauges have no significant influence on the solution of the inverse problem, as long as they are not too close to the crack. The deviation of the results when the strain gauges are close to the crack may be due i.a. to the genetic algorithm, since the number of collocation points in the genetic algorithm is chosen as small as possible to minimize the time for identification. On the other hand, increasing the number of measurement points could improve the results. The assumed errors in the strains had almost no influence on the identified boundary loads. In contrast, the crack parameters are more sensitive. The quality of parameter identification thus depends on the accuracy of the measured strains where errors should not considerably exceed ±5%. For this reason it is very important in real experiments to reduce errors in measurements of strains. It is interesting to see in the presented approach that even very small cracks in large plates with remote gauges can be identified accurately. References Bilby, B. A. , Eshelby, J. D., 1968. Dislocations and the Theory of Fracture, in Fracture, Ed. H. Liebowitz. Academic Press, New York, 1968. Dundurs, J., Mura, T., 1964. Interaction between an edge dislocation and a circular inclusion. Journal of the Mechanics and Physics of Solids 12, 177–189. Erdogan, F., Gupta, G. D., Cook, T. S., 1973. Numerical solution of singular integral equations. In Methods of Analysis and Solutions of Crack Problems, (1973, Ed. G.C. Sih). Noordhoff, Leyden, 1973. Hills, D. A., Kelly, P. A., Dai, D. N., Korsunsky, A. M., 1996. Solution of Crack Problems. Kluwer Academic, Netherlands, 1996. Holland, J. H., 1992, Adaptation in Natural and Artificial Systems. MIT Press, 1992. Metropolis, N. Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., Teller, E., 1953. Equation of state calculations by fast computing machines, Journal of Chemical Physics, 21(6), 1087–1092. Weertman, J., 1996. Dislocation based fracture mechanics. World scientific publishing Co. Pte. Ltd, 1996.
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ECF22 - Loading and Environmental effects on Structural Integrity ECF22 - Loading and Environmental effects on Structural Integrity
Identification Identification of of crack crack positions positions and and crack crack loading loading quantities quantities from from gauge data inverse problem XV Portuguesestrain Conference on Fracture, PCF 2016, 10-12 Februarysolution 2016, Paço de Arcos, Portugal strain gauge data by by inverse problem solution a Ramdane Boukellif a,∗ a,∗, Andreas Ricoeura Ramdane Boukellif , Andreas Ricoeur Thermo-mechanical modeling of a high pressure a Institute a Institute
turbine blade of an
of Mechanics, Department of Mechanical Engineering, University of Kassel, 34125 Kassel, Germany of Mechanics, Department of Mechanical Engineering, University of Kassel, 34125 Kassel, Germany
airplane gas turbine engine
Abstract P. Brandãoa, V. Infanteb, A.M. Deusc* Abstract A method a for the detection of cracks in finite and semi-infinite plane structures is presented. This allows both the identification Department of Mechanical Engineering, Superior Técnico, Lisboa, Av.This Rovisco Pais,both 1, 1049-001 Lisboa, A method for the parameters detection ofsuch cracks finiteInstituto and semi-infinite planeUniversidade structures isderespect presented. allows the identification of crack position as in length, location and inclination with to a reference coordinate system and Portugalangles of crack position parameters such as length, location and inclination angles with respect to a reference coordinate system and b the calculation of stressofintensity factors (SIFs). Instituto The method is based onUniversidade strains measured at different locations on the surface IDMEC, Department Mechanical Engineering, Superior Técnico, de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, the calculation of stress intensity factors (SIFs). The method is based on strains measured at different locations on the surface of a structure and the application of the dislocation technique.Portugal Cracks and boundaries are modelled by continuous distributions of dislocation acCeFEMA, structure Department and the application of the dislocation Cracks andUniversidade boundaries are modelled by continuous distributions of Mechanical Engineering, Superior Técnico, de Lisboa, Av. Rovisco Pais, Lisboa, of densities. This approach gives a set Instituto of technique. singular integral equations with Cauchy kernels, which can1,be1049-001 solved using of dislocation densities. This approach gives a set of singular integral equations with Cauchy kernels, which can be solved using Portugal Gauss-Chebyshev numerical quadrature [Erdogan et al. (1973)]. Once knowing the dislocation densities, the strain at an arbitrary Gauss-Chebyshev numerical quadrature [Erdogan et al. Once are knowing the dislocation thebystrain at an point can be calculated. The crack parameters as well as (1973)]. external loads parameters which aredensities, determined solving thearbitrary inverse point canwith be calculated. The crack parameters as well as external loads are parameters which determined by solving the inverse problem a genetic algorithm. Once knowing loading and crack parameters, the SIFs canare be calculated. Abstract problem with a genetic algorithm. Once knowing loading and crack parameters, the SIFs can be calculated. c 2018 � 2018The TheAuthors. Authors. Published by Elsevier © Published by Elsevier B.V. B.V. their operation, modern aircraft engine components are subjected to increasingly demanding operating conditions, c During � 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the ECF22 organizers. Peer-review under responsibility of the ECF22 organizers. especially under the high pressure turbine Such conditions cause these parts to undergo different types of time-dependent Peer-review responsibility of the(HPT) ECF22blades. organizers. degradation, one of which is creep. A model the finite element method (FEM) was developed, in order to be able to predict Keywords: Crack detection; distributed dislocations;using inverse problem; finite plate; strain gauges detection; dislocations; problem;(FDR) finite plate; gaugesaircraft, provided by a commercial aviation Keywords: the creepCrack behaviour of distributed HPT blades. Flight inverse data records for strain a specific company, were used to obtain thermal and mechanical data for three different flight cycles. In order to create the 3D model needed for the FEM analysis, a HPT blade scrap was scanned, and its chemical composition and material properties were obtained. The data that was gathered was fed into the FEM model and different simulations were run, first with a simplified 3D block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The 1. rectangular Introduction 1. overall Introduction expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a model can be useful in the goal turbine blade life, given a set of FDR data. Engineering structures are of in predicting general exposed to cyclic or stochastic mechanical loading. Exhibiting incipient
Engineering structures are in general exposed to cyclic or stochastic mechanical loading. Exhibiting incipient cracks, particularly light-weight shell and plate structures suffer from fatigue crack growth, limiting the life time of © 2016particularly The Authors. Published byshell Elsevier cracks, light-weight andB.V. plate structures suffer from fatigue crack growth, limiting the life time of thePeer-review structure and supplying theofrisk a fatalCommittee failure. Due to the uncertainty of loading boundary conditions and responsibility the of Scientific of PCF 2016. the structure under and supplying the risk of a fatal failure. Due to the uncertainty of loading boundary conditions and the geometrical complexity of many engineering structures, numerical predictions of fatigue crack growth rates and the geometrical complexity of many engineering structures, numerical predictions of fatigue crack growth rates and residual strength are not reliable. monitoring techniques, Keywords: High Pressure Turbine Blade;Most Creep;experimental Finite Element Method; 3D Model; Simulation.nowadays, are based on the principle residual strength are not reliable. Most experimental monitoring techniques, nowadays, are based on the principle of wave scattering at the free surfaces of cracks. Many of them are working well, supplying information about the of wave scattering at the free surfaces of cracks. Many of them are working well, supplying information about the position of cracks. One disadvantage is that those methods do not provide any information on the loading of the crack position of cracks. One disadvantage is that those methods do not provide any information on the loading of the crack tip. In this work, the development of a concept for the detection of cracks in finite and semi-infinite plate structures tip. In this work, the development of a concept for the detection of cracks in finite and semi-infinite plate structures under mixed mode loading conditions is presented. The concept is based on the measurement of remote strain fields under mixed mode loading conditions is presented. The concept is based on the measurement of remote strain fields by using strain gauges and the application of the distributed dislocation technique of linear elasticity [Bilby et al. by using strain gauges and the application of the distributed dislocation technique of linear elasticity [Bilby et al. (1968); Weertman (1996)]. As an approach, different from the FEM, cracks are modelled with a collocation of discrete (1968); Weertman (1996)]. As an approach, different from the FEM, cracks are modelled with a collocation of discrete Corresponding author. Tel.: +351 218419991. ∗*Corresponding author. Tel.: +49 561 804 2505 ; ∗ Corresponding E-mail address: [email protected] author. Tel.: +49 561 804 2505 ;
fax: +49 561 804 2720. fax: +49 561 804 2720. E-mail address: [email protected] E-mail address: [email protected] 2452-3216 2016 The Authors. Published Elsevier B.V. c © 2210-7843 � 2018 The Authors. Published byby Elsevier B.V. cunder 2210-7843 � 2018 The responsibility Authors.of Published by organizers. Elsevier B.V. Peer-review under of the Scientific Committee of PCF 2016. Peer-review responsibility the ECF22 Peer-review under responsibility of the ECF22 organizers. 2452-3216 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the ECF22 organizers. 10.1016/j.prostr.2018.12.015
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dislocations. Within a continuum mechanics framework, these dislocations are no lattice defects but displacement discontinuities describing the local crack opening displacement. Thus, it is not necessary to discretize the domain around the crack, considerably saving computation time and data, which is crucial for an efficient solution of the inverse problem. Solving the inverse problem, e.g. with a genetic algorithm [Holland (1992)] or an adaptive simulated annealing [Metropolis et al. (1953)], this allows the identification of external loading, crack position parameters, such as length, location or inclination angles and the calculation of stress intensity factors for straight and even for curved cracks in partly or fully bounded plane structures. 2. Theoretical Background The stress field in a body with cracks can be calculated by superposition of the stresses σiDj due to dislocations, which are distributed at the positions of the cracks. The total stress field σ∗i j ( xˆ, yˆ) in local coordinates ( xˆ, yˆ ) is as follows:
σ∗i j ( xˆ, yˆ) =
M �
σiD;I ˆ ) + σ∞ ij , j ( xˆ, y
(1)
I=1
where M is the number of cracks and i j = xˆ xˆ, yˆyˆ , xˆyˆ . Using the principle of superposition, the induced stresses in particular at the points on the I-th crack line due to the distribution of dislocations of all the M cracks, can be calculated by: D;I IJ IJ ˆ ˆ � a σxˆ xˆ ( xˆI ) M � J G ˆ xˆ xˆ ( xˆ I ; ξ J ) G yˆ xˆ xˆ ( xˆ I ; ξ J ) B (ξˆ )� 2µ � xIJ D;I xˆ J IJ ˆ ˆ ( x ˆ ) σ d ξˆJ , | xˆI | < aI , yˆ yˆ I = G xˆyˆ yˆ ( xˆI ; ξ J ) Gyˆ yˆ yˆ ( xˆI ; ξ J ) D;I IJ π (κ + 1) J=1 Byˆ (ξˆJ ) IJ ˆ ˆ σ xˆyˆ ( xˆI ) −a J G xˆ xˆyˆ ( xˆ I ; ξ J ) G yˆ xˆyˆ ( xˆ I ; ξ J )
(2)
where κ is Kolosov’s constant and µ is the shear modulus. Eq. (2) is applied to incorporate boundary conditions at crack faces. The dislocation influence functions Gki j are given in Hills et al. (1996); Dundurs et al. (1964). These describe stresses at a field point ( xˆ, yˆ) due to a unit Burgers vector at the source point ξˆ J . The index k = xˆ, yˆ in the influence function indicates the direction of dislocations whereas i j = xˆ xˆ, yˆ yˆ , xˆyˆ denote the components of induced stresses. The Eq. (2) gives a set of singular integral equations with Cauchy kernels, which can be solved using Gauss-Chebyshev numerical quadrature. The dislocation densities Bk (ξˆJ ) are determined accounting for boundary conditions. The first condition is that the crack surfaces are traction free. Secondly, the stresses on the external boundaries are equal to the subjected boundary loads. Finely, the displacement jumps at the crack tips are equal to zero and the gradient fields at this points are singular. Once having computed Bk (ξˆJ ), the strain at arbitrary points is calculated assuming plane stress conditions. 3. Crack detection and parameter identification 3.1. Finite and semi-infinite plate with inclined cracks at arbitrary locations First verifications have been carried out numerically. The strain εi j (Pm ) emerging from the distributed dislocation technique is ”measured” at points Pm representing the positions of virtual strain gauges aligned along the edges of a rectangle with corner coordinates ( x¯, y¯) and ( x˜, y˜ ). First example is a finite plate (20 mm x 20 mm) with two inclined cracks and Pm (m = 1, ..., 12) measuring points, ( x¯(a) ; y¯ (a) ) = (1; 19) mm, ( x˜(a) ; y˜ (a) ) = (19; 1) mm, as shown in Fig. 1(a). The second example is a semi-infinite plate with inclined interior crack and Pm (m = 1, ..., 4) measuring points, ( x¯(b) ; y¯ (b) ) = (2; 20) mm, ( x˜(b) ; y˜ (b) ) = (18; −20) mm, w = 20 mm, as shown in Fig. 1(b). The third example is a semi-infinite plane with inclined edge crack and Pm (m = 1, ..., 4) measuring points, ( x¯(c) ; y¯ (c) ) = (2; 20) mm,
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( x˜(c) ; y˜(c) ) = (18; −20) mm, as shown in Fig. 1(c). The ”unknown” parameters from the inverse problem solution based on a genetic algorithm [Holland (1992)] are given in Tables 1, 2, 3. The parameters have successfully been determined by the solution of the inverse problem. The SIFs KI and KII have been calculated subsequently based on the identified parameters. σ∞ yy
σ∞ xy
σ¯ yy
σ¯ xy
σ∞ yy
σ∞ xy
σ¯ xx ( x¯(b) ; y¯ (b) )
( x¯(a) ; y¯(a) )
2a
α2
2a
measuring point Pm
2
(x2 ; y2 )
α1 2a 1 (x1 ; y1 ) ( x˜(a) ; y˜(a) )
(+
( x¯(c) ; x¯(c) )
)
α (xa ; ya ) y
y x
(
−)
α x
a
w ( x˜(c) ; x˜(c) )
( x˜(b) ; y˜ (b) )
y x (b)
(a)
(c)
Fig. 1: (a): Finite plate with two inclined cracks and Pm (m = 1, ..., 12) measuring points; (b): Semi-infinite plate with inclined interior crack and Pm (m = 1, ..., 4) measuring points; (c): Semi-infinite plane with inclined edge crack and Pm (m = 1, ..., 4) measuring points.
Table 1: Results of the crack detection and parameter identification, see Fig. 1(a). parameters
given
identified
σ ¯ xx [MPa] σ ¯ yy [MPa] σ ¯ xy [MPa] a1 [mm] a2 [mm] α1 [◦ ] α2 [◦ ] x1 [mm] x2 [mm] y1 [mm] y2 [mm]
25 50 10 3 2 30 150 8 13 6 13
24, 99 49, 99 10, 0 3, 0 2, 0 30, 0 150, 0 7, 99 13, 0 6, 0 12, 99
Table 2: Results of the crack detection and parameter identification, see Fig. 1(b); √ KI , KII [MPa mm].
Table 3: Results of the parameter identification, see Fig. 1(c); √ KI , KII [MPa mm].
parameters
given
identified
parameters
given
identified
σ∞ yy [MPa] σ∞ xy [MPa] xa [mm] ya [mm] a [mm] α [◦ ] KI (+) KI (−) KII (+) KII (−)
50 30 12 4 6 40 176, 54 168, 55 133, 85 114, 40
50, 00 29, 99 11, 99 3, 99 6, 00 39, 99 176, 62 168, 62 133, 86 114, 40
σ∞ yy [MPa] σ∞ xy [MPa] a [mm] α [◦ ] KI KII
50 30 6 40 −21, 95 156, 71
50, 00 29, 99 5, 99 39, 99 −21, 86 156, 60
3.2. Sensitivity analysis for a finite plate 3.2.1. Crack detection and parameter identification depending on number of strain gauges To study the influence of the number of strain gauges on the accuracy of parameter identification for an inclined center crack in a finite plate (20 mm x 20 mm), the configurations depicted in Fig. 2 have been investigated with ( x¯; y¯ ) = (1; 19) mm, ( x˜; y˜ ) = (19; 1) mm. Four, eight and twelve virtual strain gauges are used to solve the inverse problem, see Fig. 2(a), (b), (c). Tables 4, 5, 6 show the error between the given problem and obtained results by using four gauges, eight gauges and twelve gauges for different crack lengths. Tab. 7 shows the error between the given problem and obtained results for different inclination angles of a center crack a = 6 mm in a plate with twelve strain gauges, see Fig. 2(c). From these results, it appears that four strain gauges may be sufficient for the identification of crack parameters only for the smallest crack. Eight strain gauges are enough to identify the small and medium crack lengths. With twelve strain gauges, all parameters could be identified very well. Also for different inclinations, the parameters were successfully determined by using twelve strain gauges, see, Tab. 7.
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σ¯ xy
σ¯ xy
σ¯ yy
σ¯ yy
σ¯ xx
σ¯ xy
σ¯ yy
(xc ; yc )
σ¯ xx
( x¯ ; y¯ )
(xa ; ya ) measuring point Pm
y
(+) α
2a
α
2a
2a
α
(xa ; ya )
(xa ; ya )
( x˜ ; y˜ ) y
x
y x
x (b)
(a)
(c)
Fig. 2: Inclined center crack in finite plate (xc ; yc ) = (20; 20) mm under boundary loads σ ¯ xx = 30 MPa, σ¯ yy = 90 MPa, σ¯ xy = 20 MPa with: (a): Pm (m = 1, ..., 4) measuring points; (b): Pm (m = 1, ..., 8) measuring points; (c): Pm (m = 1, ..., 12) measuring points. Table 4: Results of the crack detection and parameter identification by using four strain gauges (α = 60◦ ), see Fig. 2(a).
Table 5: Results of the crack detection and parameter identification by using eight strain gauges (α = 60◦ ), see Fig. 2(b).
parameters
error: given problem/obtained results (4 gauges) crack half-length a [mm] 1 3 5 7 9
parameters
error: given problem/obtained results (8 gauges) crack half-length a [mm] 1 3 5 7 9
σ ¯ xx [%] σ ¯ yy [%] σ ¯ xy [%] a [%] α [%] x0 [%] y0 [%]
0, 03 0, 011 0 8, 5 12, 11 0, 1 0, 1
σ ¯ xx [%] σ ¯ yy [%] σ ¯ xy [%] a [%] α [%] x0 [%] y0 [%]
0, 03 0, 011 0 0, 1 0, 05 0 0, 1
0, 23 0, 077 2 69, 8 53, 81 92, 14 12, 4
0, 06 0, 011 0, 15 11 15, 21 0, 1 0
1, 76 0, 188 9, 15 63, 42 8, 98 73, 8 91, 8
2, 83 0, 255 14, 45 79, 33 132, 25 81, 7 45.9
0, 03 0, 011 0, 05 0 0 0 0, 1
0 0 0 0, 2 0 0 0, 1
0 0, 011 0 0 0 0 0, 1
7, 13 5, 41 16, 55 9, 22 34, 85 0 0, 1
Table 6: Results of the crack detection and parameter identification by using twelve strain gauges (α = 60◦ ), see Fig. 2(c).
Table 7: Results of crack detection and parameter identification for different inclination angles α and a = 6 mm , see Fig. 2(c).
parameters
error: given problem/obtained results (12 gauges) crack half-length a [mm] 1 3 5 7 9
parameters
0, 033 0, 011 0 0 0 0 0 0, 036 0
σ ¯ xx [%] σ ¯ yy [%] σ ¯ xy [%] a [%] α [%] x0 [%] y0 [%] KI (+) [%] KII (+) [%]
σ ¯ xx [%] σ ¯ yy [%] σ ¯ xy [%] a [%] α [%] x0 [%] y0 [%] KI (+) [%] KII (+) [%]
0 0 0, 05 0, 033 0, 016 0, 01 0 0, 187 0, 11
0 0 0, 05 0 0, 016 0, 1 0 0, 046 0, 07
0 0 0, 05 0, 14 0, 016 0 0 0, 184 0, 07
1, 93 0, 56 4, 55 0, 07 0, 5 0, 1 0 1, 66 4, 55
0
error: given problem/obtained results inclination α [◦ ] 40 80 120 160
0 0, 02 0, 1 0 1 0 0 0, 030 0, 050
0 0 0, 05 0, 16 0, 025 0 0 0, 16 0, 159
0, 033 0 0, 05 0 0 0 0, 1 0, 03 0, 150
0, 033 0, 011 0, 05 0 0 0 0 0, 029 0, 013
0, 033 0, 011 0 0 0 0, 1 0 0, 008 0.018
3.2.2. Parameter identification depending on locations of strain gauges To study the influence of the distance of strain gauges on the accuracy of the solution of the inverse problem for an inclined center crack (a = 1 mm) in a finite plate (30 mm x 30 mm), the configurations depicted in Fig. 3 have been investigated. Five different squares indicate the positions of virtual strain gauges. The dimensions of these squares are given in the caption of Fig. 3. Along each of the squares, 12 gauges are distributed just as shown in Fig. 3. Tab. 8 shows the given problem and results of parameter identification depending on locations of strain gauges. The parameters have been determined very accurately at all strain gauge positions except for the smallest square (1).
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σ ¯ xy
σ ¯ yy
(3)
(5) (4)
(30;30) σ ¯ xx
(2) 2a α (1) (xa , ya )
y x Fig. 3: Inclined center crack (xa ; ya ) = (15; 15) mm, 2a = 2 mm, α = 60◦ in finite plate 30 mm x 30 mm under boundary loads σ¯ xx = 30 MPa, σ¯ yy = 90 MPa, σ¯ xy = 20 MPa. Different locations of strain gauges: square (1): 6 mm x 6 mm; square (2): 12 mm x 12 mm; square (3): 18 mm x 18 mm; square (4): 24 mm x 24 mm; square (5): 28 mm x 28 mm. Table 8: Results of parameter identification depending on locations of strain gauges, see Fig. 3. parameters
given
σ¯ xx [MPa] σ¯ yy [MPa] σ¯ xy [MPa] a [mm] α [◦ ] x0 [mm] y0 [mm]
30 90 20 1 60 15 15
square (1) 29, 94 89, 973 20, 0 1, 006 66, 337 5, 59 11, 87
square (2) 30, 0 90, 0 19, 99 0, 999 60, 0 14, 99 14, 99
identified square (3) 29, 99 89, 99 20, 0 1, 0 59, 986 15, 0 15, 0
square (4) 29, 99 89, 99 20, 0 0, 999 59, 986 15, 0 14, 99
square (5) 29, 99 90, 0 20, 0 0, 99 60, 0 15, 0 15.0
3.2.3. Parameter identification depending on the size of the plate and related gauge positions To determine the maximum size (xc , yc ) of the plate at which a small crack is detected accurately, different plate sizes have been examined, where the strain gauges are always close to the edges. Hereby a center crack length of 2a = 2 mm was assumed, see Fig. 2(c). Tab. 9 shows the given problems and results of parameter identification depending on the dimensions of the plate. It is very interesting to see that the small crack and all parameters could be identified on large plates up to 60 mm x 60 mm. Table 9: Results of parameter identification depending on the size of the plate and gauge positions. parameters (xc , yc ) [mm] σ¯ xx [MPa] σ¯ yy [MPa] σ¯ xy [MPa] α [◦ ] a [mm] x0 [mm] y0 [mm]
given 30 90 20 60 1 10 10
identified (20; 20) 29, 99 89, 99 20, 0 60, 0 1, 0 10, 0 10, 0
given 30 90 20 60 1 15 15
identified (30; 30) 29, 99 90, 0 20, 0 60, 0 0, 99 15, 00 15.00
given 30 90 20 60 1 20 20
identified (40; 40) 29, 99 89, 99 20, 00 59, 99 0, 999 19, 99 19, 99
given 30 90 20 60 1 25 25
identified (50; 50) 29, 99 89, 99 19, 99 60, 35 1, 000 24, 99 25, 00
given 30 90 20 60 1 30 30
identified (60; 60) 30, 00 90, 00 20, 00 59, 92 0, 999 29, 98 29, 99
given 30 90 20 60 1 35 35
identified (70; 70) − − − − − − −
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3.2.4. Parameter identification depending on strain measurements with presumed errors In order to investigate the sensitivity of the results to disturbances in the measured strains, random errors in the ”measured” strain components have been simulated. An inclined center crack in a finite plate (20 mm x 20 mm) is used for this investigation, see Fig. 2(c). Tab. 10 shows the results of the sensitivity analysis based on presumed measurement errors and demonstrates the sensitivity towards inaccurate strain determination. The results show that the external loads are not sensitive to inaccurate strain measurements. In contrast, crack length, inclination and position show non-negligible deviations starting from ±5% assumed error in strains. Stress intensity factors are well reproduced up to ±10% of assumed errors, then particularly KII becomes useless. Table 10: Results of parameter identification depending on strain measurement with presumed errors. parameters
given 0%
±1%
identified presumed error in measured strain ±3% ±5% ±7% ±9%
30 90 19, 999 59, 999 2, 99 9, 99 10 94, 63 52, 59
29, 87 89, 83 20, 03 59, 30 3, 06 10, 05 10, 04 96, 90 55, 90
30, 32 90, 81 19, 98 61, 18 3, 03 9, 99 10, 46 95, 01 49, 18
(xc , yc ) = (20; 20) mm σ ¯ xx [MPa] σ ¯ yy [MPa] σ ¯ xy [MPa] α [◦ ] a [mm] x0 [mm] y0 [mm] KI (+) [%] KII (+) [%]
30 90 20 60 3 10 10 94, 63 52, 60
29, 38 88, 83 19, 65 61, 47 3, 5 10, 85 9, 21 101, 42 52, 55
29, 25 87, 62 19, 61 62, 8 3, 42 11, 52 9, 62 97, 55 46, 26
28, 89 87, 09 19, 46 59, 17 3, 38 12, 35 9, 21 102, 94 60, 14
±11%
±13%
28, 83 85, 48 19, 35 67, 39 4, 07 11, 35 10, 04 103, 97 27, 19
30, 41 89, 14 19, 96 92, 60 0, 52 5.97 6, 0 41, 61 −28, 96
4. Conclusions In this work, the concept of distributed dislocations is applied for the detection of cracks and the calculation of SIFs in finite and semi-infinite plane structures. The crack and loading parameters could be successfully determined by the solution of the inverse problem. The sensitivity analysis has shown that the positions of the strain gauges have no significant influence on the solution of the inverse problem, as long as they are not too close to the crack. The deviation of the results when the strain gauges are close to the crack may be due i.a. to the genetic algorithm, since the number of collocation points in the genetic algorithm is chosen as small as possible to minimize the time for identification. On the other hand, increasing the number of measurement points could improve the results. The assumed errors in the strains had almost no influence on the identified boundary loads. In contrast, the crack parameters are more sensitive. The quality of parameter identification thus depends on the accuracy of the measured strains where errors should not considerably exceed ±5%. For this reason it is very important in real experiments to reduce errors in measurements of strains. It is interesting to see in the presented approach that even very small cracks in large plates with remote gauges can be identified accurately. References Bilby, B. A. , Eshelby, J. D., 1968. Dislocations and the Theory of Fracture, in Fracture, Ed. H. Liebowitz. Academic Press, New York, 1968. Dundurs, J., Mura, T., 1964. Interaction between an edge dislocation and a circular inclusion. Journal of the Mechanics and Physics of Solids 12, 177–189. Erdogan, F., Gupta, G. D., Cook, T. S., 1973. Numerical solution of singular integral equations. In Methods of Analysis and Solutions of Crack Problems, (1973, Ed. G.C. Sih). Noordhoff, Leyden, 1973. Hills, D. A., Kelly, P. A., Dai, D. N., Korsunsky, A. M., 1996. Solution of Crack Problems. Kluwer Academic, Netherlands, 1996. Holland, J. H., 1992, Adaptation in Natural and Artificial Systems. MIT Press, 1992. Metropolis, N. Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., Teller, E., 1953. Equation of state calculations by fast computing machines, Journal of Chemical Physics, 21(6), 1087–1092. Weertman, J., 1996. Dislocation based fracture mechanics. World scientific publishing Co. Pte. Ltd, 1996.