The Peak Stress Method Applied to Bi-Material Corners

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The Peak Stress Method Applied to Bi-Material Corners The Peak Stress Method Applied to Bi-Material Corners

XV Portuguese Conference on Fracture, PCF 2016, Paço de Arcos,aPortugal a b 2016, 10-12 February a

Mauro Ricottaa*, Michele Zappalortob, Mattia Marchioria, Alberto Campagnoloa, Mauro Ricotta *, Michele Zappalorto Mattia Marchiori , Alberto Campagnolo , a Giovanni, Meneghetti a Thermo-mechanical modeling a high pressure turbine blade of GiovanniofMeneghetti Department of Industrial Engineering, University of Padova, via Venezia, 1, 35131 Padova, Italy a b Department of Industrial Engineering,University UniversityofofPadova, Padova,Stradella via Venezia, 351313,Padova, Italy Department of Management and Engineering, San 1, Nicola, 36100 Vicenza, Italy b Department of Management and Engineering, University of Padova, Stradella San Nicola, 3, 36100 Vicenza, Italy a

airplane gas turbine engine a

b

an

c

P. Brandão , V. Infante , A.M. Deus * Abstract Abstracta Department of Mechanical Engineering, Superior Técnico,applications, Universidade such de Lisboa, Av. Rovisco Pais, adhesive 1, 1049-001 Lisboa, Bi-material interfaces are unavoidably presentInstituto in many engineering as microelectronics, joints, fiberPortugal Bi-material interfaces and are unavoidably present in many applications, such asmaterial microelectronics, adhesive joints,field fiberreinforced composites thermal barrier coatings. Underengineering the hypothesis of linear elastic behaviour, the local stress at b IDMEC, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, reinforced composites thermalofbarrier coatings. interface Under thehas hypothesis ofbehaviour, linear elastic material behaviour, the local stress field ata the point located at theand free-edge the bi-material a singular of which the intensity can be quantified by Portugal c the point located at the free-edge of the bi-material interface has a singular behaviour, of which the intensity can be quantified by generalized intensity factor, H. Engineering, However, the numerical evaluation of H usuallyderequires very accurate and Lisboa, largea CeFEMA,stress Department of Mechanical Instituto Superior Técnico, Universidade Lisboa, Av. Rovisco Pais,meshes 1, 1049-001 generalized stress intensity factor,the H. use However, the numerical evaluation of H practice. usually requires very meshes work and large computational efforts, hampering of H-based criteria inPortugal the engineering The main aimaccurate of the present is to computational efforts, hampering the use of H-based criteria corners in the engineering practice. The(PSM), main aim the present work is to overcome this limitation by extending to isotropic bi-material the Peak Stress Method firstofproposed by Meneghetti overcome this limitation bythe extending to isotropic the Peaksingular Stress Method (PSM), first coarse proposed by patterns. Meneghetti and co-workers to estimate stress intensity factorbi-material at the tip ofcorners a geometrical point with relatively mesh Abstract and co-workers to estimate the stress intensity factor at the tip of a geometrical singular point with relatively coarse mesh patterns. © 2018 The Authors. Published by Elsevier B.V. © 2018 Thetheir Authors. Publishedmodern by Elsevier B.V. engine components are subjected to increasingly demanding operating conditions, operation, aircraft © During 2018 The under Authors. Published by B.V. Peer-review responsibility of Elsevier the ECF22 organizers. Peer-review under responsibility ofturbine the ECF22 organizers. especially the high pressure (HPT) blades. Such conditions cause these parts to undergo different types of time-dependent Peer-review under responsibility of the ECF22 organizers. degradation, one of which is creep. A model using the finite element method (FEM) was developed, in order to be able to predict Keywords: bi-material corners, generalised stress intensity factor, Peak Stress Method the creep behaviour of HPT blades. Flight datafactor, records for a specific aircraft, provided by a commercial aviation Keywords: bi-material corners, generalised stress intensity Peak(FDR) Stress Method company, were used to obtain thermal and mechanical data for three different flight cycles. In order to create the 3D model for the FEM analysis, a HPT blade scrap was scanned, and its chemical composition and material properties were 1. needed Introduction obtained. The data that was gathered was fed into the FEM model and different simulations were run, first with a simplified 3D 1. Introduction rectangular block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The For several have put effort the problem of finding stressedge distributions pointssuch of a overall expecteddecades, behaviourscientists in terms of displacement wasinto observed, in particular at the trailing of the blade.near Therefore For several decades, scientists have put effort into the problem of finding stress distributions near points of singularity, basically with the aim to develop engineering strength criteria implicitly, or explicitly, based on local stress model can be useful in the goal of predicting turbine blade life, given a set of FDR data.

singularity, basicallythe with the aim develop engineering criteria implicitly, explicitly, of based onengineering local stress fields. In particular, analysis ofto bi-material corners is astrength basic issue for the strength or assessments many fields. In particular, analysisby ofElsevier bi-material a basic issue forthermal the strength assessments manythis engineering © 2016 The Authors. Published B.V.corners iscomposites components, such astheadhesive joints, fiber-reinforced and barrier coatings. of Within context, Peer-review under responsibility ofWilliams the Scientific Committee of 2016. components, suchisas adhesive joints, fiber-reinforced composites and thermal barrier coatings. Within context, worth of mention the paper by (1959), where thePCF eigen-function expansion method was usedthis to obtain an worth of mention the the paper by Williams where the eigen-function expansion method materials, was used toand obtain an analytical solutionis for stress fields in(1959), the neighborhood of a crack between dissimilar it was Keywords:solution High Pressure Blade;fields Creep; in Finite Method; 3Dof Model; Simulation. analytical for Turbine the stress theElement neighborhood a crack between dissimilar materials, and it was

* Corresponding author. Tel.: +39 049 827 6762; fax: +39 049 827 6785. * Corresponding Tel.: +39 049 827 6762; fax: +39 049 827 6785. E-mail address:author. [email protected] E-mail address: [email protected] 2452-3216 © 2018 The Authors. Published by Elsevier B.V. 2452-3216 © 2018 Authors. Published Elsevier B.V. Peer-review underThe responsibility of theby ECF22 organizers. * Corresponding Tel.: +351of218419991. Peer-review underauthor. responsibility the ECF22 organizers. E-mail address: [email protected] 2452-3216 © 2016 The Authors. Published by Elsevier B.V.

Peer-review under responsibility of the Scientific Committee of PCF 2016. 2452-3216  2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the ECF22 organizers. 10.1016/j.prostr.2018.12.318

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discovered for first time that local stresses can possess an oscillatory nature. Williams analysis was later extended to bending loads by Sih and Rice (1964), and to any opening angle in the presence of in-plane loadings by Bogy (1971), Demsey and Sinclair (1981), amongst the others. In Bogy (1971) and Demsey and Sinclair (1981) it is proved that the near tip stress fields are governed by a singular term rs, where sdepends on the V-notch opening angle and the elastic properties of the dissimilar materials, whereas r is the distance from the point of singularity. Differently, the intensity of the local stress fields can be quantified by a Generalized Stress Intensity Factor, H, thought of as an extension of Gross and Mendelson parameter (Gross and Mandelson 1972) to bi-material problems (Lazzarin et al (2002)). Therefore, it can be adopted as a fracture parameter controlling the local failure in mechanical components made of dissimilar materials (see for example Lazzarin et al (2002) and references reported therein). However, the numerical evaluation of H and of the associate strength of singularity requires very accurate meshes and large computational efforts, thus commonly hampering the adoption of this criterion in the engineering practice. The main aim of the present work is to overcome this limitation by extending to isotropic bi-material corners the Peak Stress Method (PSM), which was proposed by Meneghetti and Lazzarin (2007) to estimate the stress intensity factor at the tip of a geometrical singular point with relatively coarse mesh patterns. 2. Theoretical background Consider a bi-material corner between two elastic materials (Fig. 1a). Generally speaking, the stress distributions in the very close neighborhood of the corner can be given as a one term asymptotic expansion according to Eq. (1) (Lazzarin et al 2002):

 ij (r, ) H  r s fij ( )

(1)

In Eq. (1), fij ( ) are the stress angular functions, H is the Generalised Stress Intensity Factor associated to the leading order term of the stress distribution and -s (with s<0) is the singularity degree of the stress field, depending on the notch angle, the plane hypothesis used (plane stress or plane strain) and Dundurs’ parameters (Dundurs 1969), defined as (under plane strain): 

G1 (k 2  1)  (k1  1)G 2 G1 (k 2  1)  (k1  1)G 2   G1 (k 2  1)  (k1  1)G 2 G1 (k 2  1)  (k1  1)G 2

(2)

where subscript i = 1 or 2 specifies the material, Gi is the shear elastic modulus and, finally, ki is equal to 3 – 4i. The numerical evaluation of H and of the associate strength of singularity requires very accurate meshes and large computational efforts, thus commonly hampering the adoption of this criterion in the engineering practice.

Fig. 1. (a) schematic view of the singular zone showing the Cartesian and polar coordinate system and (b) a pointed V-notch.

The intensity of mode I asymptotic stress field in the close neighborhood of a pointed V-notch can be quantified by the Notch Stress Intensity Factor (NSIF), defined as (Gross and Mendelson 1972):

K1 2 lim ( )0  r11 r 0

(3)

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where (r,) is a polar coordinate system centred at the notch tip (Fig.1b), is the local stress and 1 is Williams eigen-value (Williams 1952) for symmetric loadings. It is well known that the direct evaluation of NSIFs by means of Eq. (3) through FE analyses requires very refined meshes. In order to overcome this limitation and to provide a sound approach for the rapid estimation of the mode I NSIF in isotropic materials, the following relation was proposed by Meneghetti and Lazzarin (2007): (4) K1  K*FE  peak  d11 where:  K1 is the exact mode I NSIF of the analysed V-notched geometry, as given by Eq. (3); typically, K1 is thought of as ‘exact’ if calculated using definition (3) applied to the results of a very refined FE analysis;  σpeak is the opening peak0) in Fig. 1a), linear elastic peak stress, as calculated with the FE method at the node located at the V-notch tip using a fixed pattern of elements having a fixed average size, i.e. a fixed value of d;  K *FE is a coefficient equal to 1.38 for quadrilateral, four-node elements with linear shape functions, as implemented in Ansys software package. The range of applicability of Eq. (4) is soundly discussed in Meneghetti and Lazzarin (2007). It is here recalled that the minimum mesh density ratio a/d required is equal to 3, a being the notch depth. Equation (4) allows K1 to be estimated with an accuracy of approximately 3%. The use of element types or FE patterns different from those discussed in Meneghetti and Lazzarin (2007) would lead to a K *FE coefficient different from 1.38. An extension to different FE software has been recently performed in Meneghetti et al. (2018) 3. Finite element results With the aim to extend the Peak Stress Method to bi-material corners, assuming that fij ( 0) 1 , Eq. (4) through Eq.(1) becomes: (5)

H  H*FE peak  ds

where s is calculated according to Bogy (1971). To evaluate the H*FE coefficient, two different geometries were considered, namely a butt joint and a notched joint, the latter characterized by a notch depth, a, equal to 7.05 mm, as shown in Fig. 2c. Concerning the butt geometry, different elastic properties of materials 1 and 2 were considered according to the data listed in Table 1, while for the notched joint, the notch opening angle 2 was increased from 0° to 165° with steps of 15°. The material elastic properties were equal to E1=330 MPa, 1=0.49, E2=2200 MPa and 2=0.41 (i.e. =0.7179 and =-0.0047). A total number of 45 different conditions were analysed. 2D-plain strain linear elastic FE analyses were performed in Ansys ® 18.1 commercial software, by using quadrilateral 8-node PLANE 183 elements with element option “pure displacement element formulation” and by switching the i,B i “Full Graphycs” option on, to average the nodal stresses at the interface: i,A    / 2    at the i-th node. After





setting the global element size parameter d, the free mesh generation algorithm was run. a

bb

c

d

Fig. 2. (a) butt joint and (b) relevant mesh; (c) notched joint and (d) relevant mesh.

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Table 1. Elastic properties of material 1 and 2 considered in the case of butt geometry ν1 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.3675 0.245 0.1225 0.3075 0.205 0.1025 0.49 0.49 0.49 0.49 0.49 0.49

E1 [MPa] 330 165 660 990 1650 2750 4400 330 330 330 330 330 330 330 330 330 330 330 330 330 330 330 330 330 330

ν2 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.3075 0.205 0.1025 0.3675 0.245 0.1225

E2 [MPa] 2200 2200 2200 2200 2200 2200 2200 1100 4400 6600 245.5 412.5 660 2200 2200 2200 2200 2200 2200 2200 2200 2200 2200 2200 2200

β - 0.0047 0.0065 - 0.0230 - 0.0372 - 0.0580 - 0.0799 - 0.0986 - 0.023 0.007 0.011 - 0.083 - 0.061 - 0.041 0.164 0.280 0.365 0.226 0.311 0.376 - 0.026 - 0.043 - 0.056 - 0.014 - 0.036 - 0.054

 0.7179 0.8482 0.5055 0.3399 0.0983 - 0.1556 - 0.3729 0.506 0.848 0.896 - 0.191 0.066 0.293 0.748 0.766 0.775 0.758 0.770 0.776 0.697 0.682 0.673 0.708 0.687 0.674

s -0.254 -0.319 -0.156 -0.091 -0.016 0 -0.0498 -0.158 -0.319 -0.344 -0.004 -0.0097 -0.073 -0.198 -0.128 -0.039 -0.166 -0.102 -0.021 -0.25 -0.248 -0.248 -0.252 -0.249 -0.248

Table 2. Elastic properties of material 1 and 2 considered in the case of notched joint with 2=120° E1 [MPa]

ν1

E2 [MPa]

ν2

α

β

s

330 330 330 330

0.49 0.49 0.49 0.245

2200 1100 4400 2200

0.41 0.41 0.41 0.41

0.7179 0.506 0.848 0.766

- 0.0047 - 0.023 0.007 0.280

-0.436 -0.414 -0.452 -0.349

As stated above, the Peak Stress Method requires a/d greater than 3. In the case of butt joints, once assumed the mesh pattern shown in Fig. 2b, the specimen width was assumed as reference dimension, a. It was kept constant and equal to 4.9 mm (Fig. 2a), while the element size, d, was varied from 0.125 to 1 mm, using steps of 0.125 mm. The H*FE values obtained considering the material properties listed in the first raw of Table 1 are shown in Fig. 3a, along with their mean value (H*FE=0.46) and the ±3% scatter band, that can well rationalise all the H*FE values, as shown in Meneghetti and Lazzarin (2007). 0.5

+3%

0.4

-3%

0.3

s = -0.255 H*FE=0.46 0.46

0.2

(b)

-3% 2=60 60° s=-0.493

0.2

H*FE=0.38 0.38

0.1

0.1 0

+3%

0.4 0.3

H*FE

H*FE

(a)

0 1

1/d [mm-1]

10

1

10 a/d

Fig. 3. (a) H*FE values vs 1/d for butt joint and s=-0.255 and (b) H*FE values vs a/d for 2=60° and s=-0.493.

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Concerning the notched joint, d was changed from 0.125 to 1.5 mm (with steps of 0.125 mm). In addition, two further analyses were carried out with d=2.55 mm and d=5.1 mm. In all cases, the standard mesh pattern shown in Fig. 2d was adopted. Fig. 3b shows the H*FE values calculated in the case of 2=60°. It is seen that, for a/d ranging from 4.7 to 56, all the data fall in the ±3% scatter with respect to the mean value equal to 0.38. For each analysed s value, the mean value of H*FE was calculated and the results are summarised in Fig. 4a, by plotting H*FE vs s. While in Meneghetti and Lazzarin (2007) a constant value of K*FE was found for -s ranging from 0.5 (2=0°) to 0.326 (2=135°), it is seen that, by using 8-node elements and considering bi-material problems, the H*FE value depends on s. (a) 1.0 0.8

0.6 0.4 0.2

0.0

Butt joint 2>120 Notched (2 elements) 2≤120 Notched (4 elements)

0.8 H*FE

H*FE

(b) 1.0

Butt joint Notched (2 elements) Notched (4 elements) 120° (4 elements) 120° (2 elements)

0.6

H*FE = 1.44 s + 1

H*FE = - 1.44 s + 0.29

0.4

H*FE = 2.34 s2 + 2.71 s + 1

0.2 -0.5

-0.4

-0.3

s

-0.2

-0.1

0.0

0.0

-0.5

-0.4

-0.3 s -0.2

-0.1

0.0

Fig. 4. (a) H*FE values vs s for different mesh patterns and (b) best fitting curves to evaluate H*FE having s and the notch opening angle.

Moreover, it is worth noting that the standard free mesh algorithm adopted by ANSYS code uses 4 elements around the notch tip for 2≤120° (see as an example Fig. 2d) and 2 elements for 2>120°. Consequently, the value of the peak stress is influenced by the different mesh patterns, as shown in Fig. 4a, where triangular open and filled symbols refer to notch tip modelled by 4 and 2 elements, respectively. Dedicated FE analyses were carried out in the case of 2=120° to confirm this result, by using mapped meshes to enforce 2 elements as well as free mesh with 4 elements and the different combinations of material properties listed in Table 2. The results are shown in Fig. 4a with circular filled and open symbols, respectively, and it can be seen that they fall in the relevant group of data. Therefore, with the aim to provide designers with a simple tool to evaluate H*FE, once calculated s according to Bogy (1971), different curves have been defined by fitting the FE results shown in Fig 4b, by using the least square method. Concerning butt joints and notches having 2>120°, a quadratic curve has been adopted by imposing that H*FE=1 for s=0 (i.e. plain material). Conversely, in the case of 2≤120 two linear curves have been proposed depending on s, as shown in Fig 4b. It is worth noting that a mesh density ratio a/d equal or greater than 5 is required to adopt the best fitting curve shown in Fig.4b. 4. Conclusions In this paper, the numerical evaluation by means of coarse meshes of the intensity of the linear elastic stress fields close to the singularity point at the interface of bi-material corner is analysed. In particular, the concept of Generalized Stress Intensity Factor H, thought of an extension of Gross and Mendelson parameter (Gross and Mandelson 1972), was considered. In view of this, the Peak Stress Method proposed by Meneghetti and Lazzarin (2007) to estimate the Stress Intensity Factor at the tip of a geometrical singular point was extended to isotropic bi-material corners, by using 2D plain strain quadrilateral 8-node elements of ANSYS code. A design rule was proposed to estimate H as a function of the opening angle, either lower or greater than 120°, and the singularity exponent s calculated according to the open literature and dependent on the notch-opening angle and elastic material properties.

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References 1.

Williams, M.L., 1959. The stresses around a fault or crack in dissimilar media. Bulletin of the Seismological Society of America 49, 199204. 2. Sih, G.C., Rice J.R., 1964. The bending of plates of dissimilar materials with cracks. Journal of Applied Mechanics 31, 477-482. 3. Bogy, D.B., 1971. Two edge-bonded elastic wedges of different materials and wedge angles under surface tractions. Journal of Applied Mechanics 38, 377-386. 4. Dempsey, J.P., Sinclair, G.B., 1981. On the singular behavior at the vertex of a bi-material wedge. Journal of Elasticity 11, 317-327. 5. Gross, R., Mendelson, A., (1972) Plane elastostatic analysis of V-notched plates. International Journal of Fracture Mechanics8, 267-276. 6. Lazzarin, P., Quaresimin, M., Ferro, P., 2002. A two-term stress function approach to evaluate stress distributions in bonded joints of different geometries. Journal of Strain Analysis for Engineering Design 37, 385-39. 7. Meneghetti, G., Lazzarin, P., (2007). Significance of the elastic peak stress evaluated by FE analyses at the point of singularity of sharp V-notched components. Fatigue and Fracture of Engineering Materials and Structures 30, 95-106. 8. Dundurs, J., 1969. Mathematical theory of dislocations. American Society of Mechanical Engineers, New York 9. Williams, M.L., (1952). Stress singularities resulting from various boundary conditions in angular corners of plates in extension. Journal of Applied Mechanics 19, 526–528. 10. Meneghetti, G., Campagnolo, A., et al., (2018). Rapid evaluation of notch stress intensity factors using the peak stress method: Comparison of commercial finite element codes for a range of mesh patterns. Fatigue and Fracture of Engineering Materials and Structures 30, 95-106.