Issue 32

Frattura ed Integrità Strutturale, 32 (2015); International Journal of the Italian Group of Fracture

Table of Contents

N. Bisht, P. C. Gope, K. Panwar Influence of crack offset distance on the interaction of multiple cracks on the same side in a rectangular plate ………….………………………………………………………………………... 1 N. Golinelli, A. Spaggiari Design of a novel magnetorheological damper with internal pressure control …………….................. 13 I. Telichev Development of an engineering methodology for non-linear fracture analysis of impact-damaged pressurized spacecraft structures ……………………………………………………......…... 24

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Fracture and Structural Integrity, 32 (2015); ISSN 1971-9883

Editor-in-Chief Francesco Iacoviello Associate Editors Alfredo Navarro

(Università di Cassino e del Lazio Meridionale, Italy)

(Escuela Superior de Ingenieros, Universidad de Sevilla, Spain) (Ecole Nationale Supérieure d'Arts et Métiers, Paris, France)

Thierry Palin-Luc

Luca Susmel John Yates

(University of Sheffield, UK) (University of Manchester, UK)

Advisory Editorial Board Harm Askes

(University of Sheffield, Italy) (Politecnico di Torino, Italy) (Università di Parma, Italy) (Politecnico di Torino, Italy) (University of Plymouth, UK)

Alberto Carpinteri Andrea Carpinteri Donato Firrao M. Neil James Gary Marquis Ashok Saxena Darrell F. Socie Shouwen Yu Ramesh Talreja David Taylor Robert O. Ritchie Cetin Morris Sonsino Editorial Board Stefano Beretta Elisabeth Bowman Roberto Citarella Claudio Dalle Donne Manuel de Freitas Vittorio Di Cocco Giuseppe Ferro Eugenio Giner Tommaso Ghidini Daniele Dini Nicola Bonora

(Helsinki University of Technology, Finland)

(University of California, USA)

(Galgotias University, Greater Noida, UP, India; University of Arkansas, USA)

(University of Illinois at Urbana-Champaign, USA)

(Tsinghua University, China) (Fraunhofer LBF, Germany) (Texas A&M University, USA) (University of Dublin, Ireland)

(Politecnico di Milano, Italy)

(Università di Cassino e del Lazio Meridionale, Italy)

(University of Sheffield) (Università di Salerno, Italy) (EADS, Munich, Germany) (EDAM MIT, Portugal)

(Università di Cassino e del Lazio Meridionale, Italy)

(Imperial College, UK)

(Politecnico di Torino, Italy)

(Universitat Politecnica de Valencia, Spain) (European Space Agency - ESA-ESRIN)

Paolo Leonetti Carmine Maletta Liviu Marsavina

(Università della Calabria, Italy) (Università della Calabria, Italy) (University of Timisoara, Romania) (University of Porto, Portugal)

Lucas Filipe Martins da Silva

Hisao Matsunaga Mahmoud Mostafavi

(Kyushu University, Japan) (University of Sheffield, UK) (Politecnico di Torino, Italy)

Marco Paggi Oleg Plekhov

(Russian Academy of Sciences, Ural Section, Moscow Russian Federation)

Alessandro Pirondi

(Università di Parma, Italy)

Luis Reis

(Instituto Superior Técnico, Portugal)

Giacomo Risitano Roberto Roberti

(Università di Messina, Italy) (Università di Brescia, Italy) (Università di Bologna, Italy) (Università di Parma, Italy)

Marco Savoia

Andrea Spagnoli Charles V. White

(Kettering University, Michigan,USA)

II

Frattura ed Integrità Strutturale, 32 (2015); International Journal of the Italian Group of Fracture

Journal description and aims Frattura ed Integrità Strutturale (Fracture and Structural Integrity) is the official Journal of the Italian Group of Fracture. It is an open-access Journal published on-line every three months (July, October, January, April). Frattura ed Integrità Strutturale encompasses the broad topic of structural integrity, which is based on the mechanics of fatigue and fracture, and is concerned with the reliability and effectiveness of structural components. The aim of the Journal is to promote works and researches on fracture phenomena, as well as the development of new materials and new standards for structural integrity assessment. The Journal is interdisciplinary and accepts contributions from engineers, metallurgists, materials scientists, physicists, chemists, and mathematicians. Contributions Frattura ed Integrità Strutturale is a medium for rapid dissemination of original analytical, numerical and experimental contributions on fracture mechanics and structural integrity. Research works which provide improved understanding of the fracture behaviour of conventional and innovative engineering material systems are welcome. Technical notes, letters and review papers may also be accepted depending on their quality. Special issues containing full-length papers presented during selected conferences or symposia are also solicited by the Editorial Board. Manuscript submission Manuscripts have to be written using a standard word file without any specific format and submitted via e-mail to iacoviello@unicas.it. The paper may be written in English or Italian (with an English 1000 words abstract). A confirmation of reception will be sent within 48 hours. The review and the on-line publication process will be concluded within three months from the date of submission. Peer review process Frattura ed Integrità Strutturale adopts a single blind reviewing procedure. The Editor in Chief receives the manuscript and, considering the paper’s main topics, the paper is remitted to a panel of referees involved in those research areas. They can be either external or members of the Editorial Board. Each paper is reviewed by two referees. After evaluation, the referees produce reports about the paper, by which the paper can be: a) accepted without modifications; the Editor in Chief forwards to the corresponding author the result of the reviewing process and the paper is directly submitted to the publishing procedure; b) accepted with minor modifications or corrections (a second review process of the modified paper is not mandatory); the Editor in Chief returns the manuscript to the corresponding author, together with the referees’ reports and all the suggestions, recommendations and comments therein. c) accepted with major modifications or corrections (a second review process of the modified paper is mandatory); the Editor in Chief returns the manuscript to the corresponding author, together with the referees’ reports and all the suggestions, recommendations and comments therein. d) rejected. The final decision concerning the papers publication belongs to the Editor in Chief and to the Associate Editors. The reviewing process is completed within three months. The paper is published in the first issue that is available after the end of the reviewing process.

Publisher Gruppo Italiano Frattura (IGF) http://www.gruppofrattura.it ISSN 1971-8993 Reg. Trib. di Cassino n. 729/07, 30/07/2007

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Fracture and Structural Integrity, 32 (2015); ISSN 1971-9883

IGF events in 2015

D

ear Friend, 2015 will be a very exciting year for IGF. Two consecutive workshops will be held in Urbino on two important topics:

- Characterization of Crack Tip Fields , april 20-22, 2015 - Challenges in Multiaxial Fatigue , april 22-24, 2015

In June, IGF will organize the XXIII National IGF Conference - 1 st International Edition (June 22-24, 2015). All our friends from all over the world are warmly invited to join us and participate to this conference that will be held in the little but wonderful island of Favignana (near Sicily). The official language of the event will be English and proceedings will be published on Procedia Engineering (new abstract deadline: 05/04/2015. Submission procedure is published in the IGF website). … and, obviously, do not forget the European Conference on Fracture (ECF21) that will be held in Catania in June 2016 (www.ecf21.eu). Looking forward to meeting you soon in Urbino and/or in Favignana,

Francesco Iacoviello F&IS Chief Editor

Frattura ed Integrità Strutturale (Fracture and Structural Integrity) is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0)

IV

N. Bisht et alii, Frattura ed Integrità Strutturale, 32 (2015) 1-12; DOI: 10.3221/IGF-ESIS.32.01

Influence of crack offset distance on the interaction of multiple cracks on the same side in a rectangular plate

Neeraj Bisht, P. C. Gope, Kuldeep Panwar College of Technology, Govind Ballabh Pant University of Agriculture and Technology, Pantnagar (263145), Uttarakhand, India. neerajbisht30@gmail.com

A BSTRACT . In the present work finite element method has been employed to study the interaction of multiple cracks in a finite rectangular plate of unit thickness with cracks on the same side under uniaxial loading conditions. The variation of the stress intensity factor and stress distribution around the crack tip with crack offset distance has been studied. Due to the presence of a neighbouring crack, two types of interactions viz. intensification and shielding effect have been observed. The interaction between the cracks is seen to be dependent on the crack offset distance. It is seen that the presence of a neighbouring crack results in the appearance of mode II stress intensity factor which was otherwise absent for a single edge crack. It can be said that the proximity of cracks is non-desirable for structural integrity. The von-Mises stress for different crack orientations has been computed. Linear elastic analysis of state of stress around the crack tip has also been done. K EYWORDS . Finite element method; Crack interaction; Von-Mises stress. he fracture mechanics theory can be used to analyse structures and machine components with cracks and to obtain an efficient design. The basic principles of fracture mechanics developed from studies of [1-3] are based on the concepts of linear elasticity. The interaction between multiple cracks has a major influence on crack growth behaviours. This influence is particularly significant in stress corrosion cracking (SCC), welding, riveting etc. because of the relatively large number of cracks initiated due to environmental effects. Pseudo – traction –electric – displacement –magnetic –induction method has been proposed [4] to solve the multiple crack interaction problems in the magneto elastic material. Most of the real life situations have the problem of multiple cracks and so it becomes imperative to study this interaction for an array of cracks, keeping this in mind interaction between two parallel cracks has been studied and a detailed analysis has been done in this regard. Since today, there have been over 20 approaches to calculate stress intensity factors. Some of these are the integral transform method [5], the Westergaard method [6],the complex variable function method [7], the singular equation integral method [8], conformal mapping [9], the Laurent series expansion [10], boundary collocation method [11], Green’s function method [12], the continuous distribution dislocation method [13], the finite element method [14], the boundary element method [15], the body force method [16] and the displacement discontinuity method [17]. The solutions of many of the fracture mechanics problems have been compiled in data hand books for stress intensity factors [18] and [19]. The configuration of multiple cracks is so complicated that a solution may not be available from the handbooks and literatures. The above mentioned methods with analytical features, which are usually suitable for special cases or very simple crack configurations, are not sufficient to obtain reasonable results for general orientations due to the multiple I NTRODUCTION

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N. Bisht et alii, Frattura ed Integrità Strutturale, 32 (2015) 1-12; DOI: 10.3221/IGF-ESIS.32.01

restrictions. In these cases numerical approaches are usually employed. In the numerical approaches proposed so far finite element method provides a very simple, effective and accurate technique for evaluation of fracture parameters. The historical development of computational fracture mechanics is found in the works of Ingraffea et al. [20] and Sinclair [21]. Sinclair presented an extensive review of numerical prediction models to determine stress intensity factors. The advantages and disadvantages of using finite element in computational fracture mechanics have been well addressed by Ingraffea [22]. The aspect of mesh refinement and associated error in computing stress intensity factors using finite element method has also been studied by Miranda et al. [23]. It has been reported that excessive mesh refinement may significantly degrade the calculation accuracy in crack problems. They also pointed out that the ratio between the longest and shortest element edge lengths should be kept below 1600 to avoid calculation errors in SIF calculations. For meshes with length ratios higher than 1600, improved numerical methods to deal with ill conditioned matrices would be necessary to not compromise the calculation accuracy of the calculated SIF. Many works on mesh generation algorithms and new methods to improve the numerical computation of SIF values have been found in the works of Miranda et al. [24, 25]. Recent studies have also shown that the coefficients of higher order terms can also play an important role in the fracture process in notched or cracked structures. It has been observed that in addition to the singular term, the higher order terms, in particular, the first non-singular stress term ( known as the T stress) may also have significant effects on the near notch tip stress field. The T-stress is considered in some studies as an auxiliary parameter for increasing the accuracy of the results for K I . Kim et al. [26], for instance, showed that this non-singular term has noticeable effects on the size and shape of plastic zone near the notch tip. It has been demonstrated that the first non-singular term may have considerable contributions to the stress components around the notch tip and also on the fracture resistance of notched components under mode I loading [27-29].

F INITE ELEMENT MODELLING

T

he numerical simulations were run by means of the finite element (FE) software ANSYS to determine the stress intensity factors of two edge cracks on the same side of the specimen. The specimen is schematized by a 2D model. The specimen thickness in FE analysis was kept 1.0 mm. The other dimensions are length L= 200 mm, width W= 80 mm and the model was studied in plane strain condition, the specimen geometry and detailed dimensions are shown in Fig 1(a). The simulations have been run on full model. The stresses are applied at the two extremes of the specimen in the direction perpendicular to the crack plane (Fig. 1(b)).

Figure 1(a) : Specimen Geometry.

Figure 1(b) : Specimen with boundary conditions.

Isoparametric quadrilateral element (PLANE 82) having 8 nodes with singularity elements at and around the crack tip has been used throughout the analysis and is shown in Fig. 2. The singularity phenomenon at the vicinity of the crack tips has been addressed by applying triangular element option of the PLANE 82 element. The radius of second row of elements is taken as a /8, where a is the half crack length and the radius ratio (second row/first row) is adjusted

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N. Bisht et alii, Frattura ed Integrità Strutturale, 32 (2015) 1-12; DOI: 10.3221/IGF-ESIS.32.01

automatically. The number of elements around the circumference is taken as 32. The FE modelling parameters are optimized on the basis of the error analysis presented in Fig. 3 for two edge cracks.

Figure 2 : Plane 82 element with 8-nodes.

Figure 3(a) : Variation of stress intensity factor with number of crack tip elements.

Figure 3(b) : Variation of stress intensity factor with radius of first row of elements, K I,FEM and K I, THE are finite element and theoretical based computations of mode I stress intensity factor

The error analysis has been done by varying the radius of the first row of the crack tip element and number of elements in the first row and computing the stress intensity factor for various parameters. The density of the FE mesh is modified by varying the number of the elements of the first row as 16, 20, 24, 32 and 40, keeping the radius of the first row as a /8 where a has been taken as 10 mm. Also, the radius of the first row ( a/n ) around the crack tip is varied, taking n= 8, 10, 12, 16 and 20 with 32 number of elements. For comparison K I is calculated theoretically from the relation given for two collinear edge cracks [30]. It is found that the K I calculation errors stay below 0.8% for all meshing strategies. Fig. 3(a-b) show variation of the normalized K I (FE based computation to the actual (theoretical) Mode I stress intensity factor) for different mesh configurations. It can be observed from Fig. 3 that, on average, the calculation with the radius of first element as 1.25 mm (i.e. a / 8=1.25) and number of elements around the crack tip as 32 yields least error i.e. closest to unity amongst the other meshing strategies. It is mentioned by Miranda et al. [32] that calculation error and associated standard deviation tend to increase for higher mesh density. These increasing errors are a result of an ill conditioned numerical problem [32]. The increasing error due to the decrease in first crack tip radius may be due to the stress singularity that is present at the crack tip and the linear elastic fracture mechanics concept fails to predict the stress intensity factor. The other FE modelling parameters are shown in Fig. 4 and 5. In Fig. 5 node 1 is the crack tip node and the nodes 2, 3, 4 and 5 are used to represent the crack path for evaluating the fracture parameters. The nodes are essentially taken in this order when a full crack model is employed. For the numerical simulation, a uniform pressure intensity of 1.0 MPa (80 N) is applied to the upper and lower edges in the vertical direction (y axis). The material properties are Young’s Modulus E=70 GPa, Poisson’s ratio=0.33. The mode I and mode II stress intensity factors are computed from the following relations using the displacement extrapolation method.

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N. Bisht et alii, Frattura ed Integrità Strutturale, 32 (2015) 1-12; DOI: 10.3221/IGF-ESIS.32.01

Figure 4 : Crack tip meshing and different parameters.

Figure 5 : Nodes used for defining crack path.

The analysis uses a fit of the nodal displacements in the vicinity of the crack. The actual displacements at and near a crack for linear elastic materials are given by Paris [31] as:     3 3 2 1 cos cos  2 3 sin sin 4 2 2 2 4 2 2 2 I II K K r r u k k G G                         (1)







K

K

r

r

θ 2

3

3

  

 

  

  











2 1 sin sin k  



 2 3 cos cos k  

v

(2)

I

II





4 2 G

2 4 2 G 

2

2



III K r

2



w

(3)

 sin 



G

2

2

where: u, v, w = displacements in a local Cartesian coordinate system, shown in Fig.6. r, θ = coordinates in a local cylindrical coordinate system, shown in Fig.6. G = shear modulus K I , K II , K III = stress intensity factors relating to deformation shapes, shown in Fig.6. ν = Poisson’s ratio

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N. Bisht et alii, Frattura ed Integrità Strutturale, 32 (2015) 1-12; DOI: 10.3221/IGF-ESIS.32.01

Figure 6 : Representation of crack with coordinate system.

0 180

  

Evaluating Eq. (1-3) at

and dropping the higher order terms Eq. (1-3) yields:

 II K r u 1 k 2G 2π     2 2 I K r v k G  



(4)

(5)

III K r

2 



w

(6)



G

2

For full crack models Eq. (4-6) can be reorganized to | | 2 I v G K k r   

(7)

| | 

u G





K

(8)

2

II



k r

1

| | w 





K

G

(9)

2

III

r

where Δv, Δu and Δw are the motions of one crack face with respect to the other. k= 3−4ν, for plane strain or axisymmetric; (3−ν)/ (1+ ν) for plane stress; where ν is Poisson's ratio. The final factor v r 

is evaluated based on the nodal displacements and locations. For practical purposes the value of



 by simply evaluating the following expression for a small fixed value

v r

v r

is approximated by limiting the value of

of r (small in relation to the size of the crack) as: | |  v A Br r  

(10)

At point I shown in Fig.7, v=0 Hence, in the limiting condition

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N. Bisht et alii, Frattura ed Integrità Strutturale, 32 (2015) 1-12; DOI: 10.3221/IGF-ESIS.32.01

v A r 

| |  lim

(11)



r

0

Figure 7 : Representation of nodes on crack face used for crack analysis.

The Eq. (7) becomes 2 2 I K  

GA

(12)

k

Similarly

| |  u C Dr r  

(13)

At point I of Fig. 7 u=0 Hence, in the limiting condition

u C r 

| | lim

(14)



r

0

and Eq. (8) becomes,

GC





K

(15)

2

II



k

1

Similarly for evaluating K III

, using

| | w E Fr r  

(16)

and at point I of Fig. 7 w=0 and Eq. 9 in the limiting condition becomes,

0 | | lim  w E r 

(17)



r

Thus, Eq. 9 becomes 2 III K 

GE 

(18)

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N. Bisht et alii, Frattura ed Integrità Strutturale, 32 (2015) 1-12; DOI: 10.3221/IGF-ESIS.32.01

Mode III stress intensity factor in the present investigation is not considered because the thickness of the plate in FE analysis is taken as unity. However, the three dimensional effect or plate thickness effect on the stress intensity factors can be seen in the work of Kotousov et al. [33, 34]. In the present FE analysis, out of balance convergence and degree of freedom increment convergence criteria have been applied. The tolerance level has been taken as 0.001. These criteria are well documented in the ANSYS manual [35].

R ESULTS AND DISCUSSION

T

Stress Intensity Factor

he stress intensity factors computed were normalized by K 0

given for single edge cracked plate under uniform

tension [30].

  a W



 

K

a f

(19)

0

  a f W is the geometric correction factor given as:

where σ is the applied stress, a is the crack length and

2

3

4

  a f W

a      

a      

a       W

a       W

 







1.12 0.23

10.56

21.74

30.42

(20)

W W

In the present investigation, a = 10 mm, W = 80 mm. Thus geometric correction factor becomes   a f w  1.13 = 400.57 MPa mm  Fig. 8 shows the effect of crack offset distance H on the normalized stress intensity factors. From Fig. 8 it can be seen that normalized mode I SIF (K I /K 0 ) for H=0.5 mm reduces drastically to about 66% value as compared to single edge crack. As the crack offset distance increases beyond H=0.5 mm the normalised mode I stress intensity factor starts increasing, and its value becomes 95% of single edge crack when H becomes 20 mm. So, it can be said that there is a shielding effect due to proximity of cracks due to which mode I SIF decreases as compared to single edge crack, however this shielding effect ceases to exist when the cracks move farther away. (21) and K 0

Figure 8 : Variation of normalized stress intensity factors with crack offset distance H.

The variation of normalized mode II SIF K II /K 0 shown in Fig.8 is just opposite to that of mode I. Fig. 8 shows that when normalised mode I SIF attains the minimum value, associated mode II attains the maximum value which is almost 19% of mode I SIF for single edge crack. This behaviour is seen at H= 0.5 mm. Thereafter mode II SIF decreases with increase

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N. Bisht et alii, Frattura ed Integrità Strutturale, 32 (2015) 1-12; DOI: 10.3221/IGF-ESIS.32.01

of crack offset distance H. There is an amplification effect as far as mode II SIF is concerned which also becomes insignificant for remotely placed cracks. This indicates that when two edge cracks are extremely close to each other (H ≤ 0.5 mm), shielding effect occurs and the normal stress components σ xx and σ yy reduces causing the shear component of the stress to increase. As a result, mode I SIF decreases and mode II SIF increases. Now, as the crack offset distance H increases from H=0.5 mm to higher value, mode I SIF increases and mode II SIF decreases. It is found that for about H ≥ 20 mm, the crack interaction becomes negligible and both cracks behave as a single crack. The variation of normalised K II /K I with H is shown in Fig. 9. This figure reveals that when the cracks are very close to each other mode II SIF plays a significant role in crack growth and contributes about 27% of mode I SIF. Beyond H= 20 mm, the contribution of mode II SIF remains below 2% only. These results indicate that special care should be taken while predicting growth from stress intensity factor, particularly when two or more edge cracks are very close to each other.

Figure 9 : Variation of K II

/K I

with crack offset distance H.

Von-Mises Stresses Distribution of von-Mises stresses around the crack tip for various values of H at uniform stress σ = 1 MPa are shown in Fig. 10-11. von-Mises stresses are computed taking midpoint or element stresses. Identical boundary conditions, crack tip element size and full mesh model have been used for all crack configurations. The distribution of equivalent von Mises stresses for two edge cracks configuration for H = 2 mm and H = 20 mm have been shown in Fig.10 and 11. From Fig. 10 and 11 it can be observed that with increasing crack offset distance H the equivalent von Mises stresses increases. This indicates that crack offset distance have greater affect on the state of stress ahead of the crack tip. Analysis of the State of Stress around the crack tip The two dimensional finite element model presented in this work has been used to investigate the state of stress around the crack tip in plane strain condition. The state of stress at a radial distance r from the crack tip is schematically shown in Fig. 12. The results obtained for different crack configurations are presented in Fig. 13 to 15. The stresses computed for double edge cracks are normalized with σ y0 which is the FE solution for single edge crack corresponding to θ=0 0 and for single edge crack= 731.57 MPa) for two edge cracks on the same side of a rectangular plate and separated by offset distance H= 0, 2, 10 and 16 mm are shown in Fig. 13 – 15. The variation of these stresses around the crack tip at a radius of 0.5 mm from the crack tip is shown for the assumed plane strain condition. Identical loading, boundary conditions and mesh arrangements were used for all crack offset conditions. The stresses are taken at the midpoint of each crack tip element. It has been observed that symmetrical r=0.5 mm along the crack plane. The variation of normalised σ xx , σ yy and τ xy (normalised by σ yy

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N. Bisht et alii, Frattura ed Integrità Strutturale, 32 (2015) 1-12; DOI: 10.3221/IGF-ESIS.32.01

distribution of σ yy around the crack tip.

occurred on the upper and lower side of the crack, whereas σ xx

and τ xy

show asymmetrical distribution

Figure 10 : Von Mises stress distribution for two offset edge crack geometry for H=2 mm.

Figure 11 : Von Mises stress distribution for two offset edge crack geometry for H=20.

Figure 12 : Schematic representation of state of stress ahead of the crack tip, r and θ are the polar coordinates.

The σ xx

component is about 0.1 to 0.75 times of σ yy

at the crack tip element. Fig. 13 and 14 also reveal that σ xx

and σ yy

are

increases with increasing offset is least when two cracks are very

lower in magnitude compared to a single edge crack. The normal stress component σ xx distance, but its magnitude remains less than a single edge crack. It is also seen that σ xx close to each other (H ≤ 2 mm). It is well established that mode I SIF mainly depends upon variation of σ yy

at the crack

is higher for single edge crack as compared to multiple cracks with different offset for H = 0 mm is found to be higher as compared to other multiple crack configurations. It is minimum for H = 2 mm. Thus mode I SIF shown in Fig. 8 is found to attain minimum

tip. The present analysis shows that σ yy distance H. Hence, in Fig. 9 K I is also seen in Fig.13 that σ yy

value at H = 2 mm, and thereafter it increases but remains less than a single edge crack. The variation of τ xy for different crack offset distance H = 0, 2, 10, 16 mm is shown in Fig.15. The figure shows that shearing stress is zero for single edge crack (H = 0 mm) and maximum for H = 2 mm. It is seen that as H increases, the shearing stress decreases. Thus, mode II SIF which is mainly due to shear component attains maximum value at H = 2 mm and thereafter it reduces as H increases. The results shown in fig. 8 also confirms that for single edge crack (H = 0 mm), mode II SIF is zero. This is also in conformity with the theoretical results for single edge crack.

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N. Bisht et alii, Frattura ed Integrità Strutturale, 32 (2015) 1-12; DOI: 10.3221/IGF-ESIS.32.01

Figure 13 : Variation of σ xx

/σ y

with H for specimen geometry A1.

Figure 14 : Variation of σ yy

/σ y

with H for specimen geometry A1.

Figure 15 : Variation of τ xy

/σ y

with H for specimen geometry A1.

C ONCLUSIONS

1. There is a shielding effect on mode I SIF due to the presence of a neighbouring crack. 2. The shielding becomes more pronounced as the cracks move towards each other.

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N. Bisht et alii, Frattura ed Integrità Strutturale, 32 (2015) 1-12; DOI: 10.3221/IGF-ESIS.32.01

3. For mode II SIF there is an amplification effect. Mode II SIF which is otherwise absent for single cracks develops due to the presence of neighbouring cracks. 4. The amplification is more significant for closer cracks. 5. Both the amplification and shielding effects cease to exist as cracks move farther away from each other.

R EFERENCES

[1] Inglis, C.E., Stresses in a plate due to the presence of cracks and sharp corners, Transactions-Institute of Naval Architect, 55 (1913) 219-230. [2] Griffith, A.A., The phenomenon of rupture and flow in solids, Trans. Royal Soci. London, 221 (1920) 163-198. [3] Westgaard, H. M., Bearing pressure and cracks, J. Appl. Maths Mech., 6 (1939) 49-53. [4] Wen-Ye, T., Gabbert, U., Multiple crack interaction problems in magnetoelectrostatic solids, Europ. J. Mech. A/Solids, 23 (2004) 599. [5] Sneddon, I. N., Lowengrub, M., Crack Problems in Classical Theory of Elasticity, 1 st Edn., Wiley, New York, (1969). [6] Westergaard, H. M., Bearing pressure and cracks, ASME J. Appl. Mech., 6 (1939) 49-53. [7] Muskhelishvili, N. I., Some basic problems of mathematical theory of elasticity, 4 th Edn. Holland, Noordhoff, (1977). [8] Erdogan, F., Stress Intensity Factors, ASME J. Appl. Mech., 50 (1983) 992-1002. [9] Bowie, O. L., Analysis of an infinite plate containing radial cracks originating at the boundary of an internal circular hole, J. Math. Phys., 25 (1956) 60-71. [10] Isida, M., Stress intensity factors for the tension of an eccentrically cracked strip, ASME J. Appl. Mech., 33 (1966) 674-675. [11] Williams, M. L., On the stress distribution at the base of a stationary crack, ASME J Appl. Mech., 24 (1957) 104-114. [12] Sih, G. C., Paris, P. C., Erdogan, F., Crack tip stress intensity factors for plane extension and plate bending problems, ASME J. Appl. Mech., 29 (1962) 306-314. [13] Eshelby, J. D., Franck, F. C., Nabarro, F. R. N., The equilibrium of linear arrays of dislocations, Phil. Mag., 42 (1951) 351-364. [14] Watwood, V. B., The finite element method for prediction of crack behaviour, Nucl. Engg. Des., 11 (1969) 323-332. [15] Cruse, T. A., Numerical solutions in three dimensional elastostatics, Int. J. Solids Struct., 5 (1969) 1259-1274. [16] Nisitani, H., Solutions of notch problems by the body force method in fracture mechanics, Edited by G. C. Sih, fifth ed. Noordhoff International, (1978) 1-68. [17] Crouch, S. L., Starfield, A. M., Boundary element method in solid mechanics, with application in rock mechanics and geological mechanics, 1 st Edn., George Allen & Unwin, (1983). [18] Pook, L. P., Stress intensity factor expressions for regular crack arrays in pressurised cylinders, Fatig. Fract. Eng. Mater. Struct., 13 (1990) 135–143. [19] Rooke, D. P., Cartwright, D. J., Compendium of stress intensity factors Great Britain, Ministry of Defence, Procurement Executive, (1976). [20] Ingraffea, A.R., Wawrzynek, P.A., Comprehensive structural integrity, 1st Edn., R.d.B.a.H. Mang (Ed.), Elsevier Science Ltd., Oxford, (2003). [21] Sinclair, G., Stress singularities in classical elasticity—II: Asymptotic identification, App. Mech. Rev., 57 (2004) 251– 298. [22] Ingraffea, A.R., Encyclopaedia of computational mechanics, John Wiley and Sons, (2004). [23] de Oliveira Miranda, A. C., Meggiolaro, M. A., Martha, L. F., de Castro, J. T. P., Stress intensity factor predictions: Comparison and roundoff error, Comput. Mater. Sci., 53 (2012) 354–358. [24] Miranda, A.C.O., Meggiolaro, M.A., Castro, J.T.P., Martha, L.F., Bittencourt, T.N., Fatigue life and crack path predictions in generic 2D structural components, Eng. Fract. Mech., 70 (2003) 1259–1279. [25] de Oliveira Miranda, A.C., Meggiolaro, M. A., de Castro, J. T. P., Martha, L. F., Fatigue life prediction of complex 2D components under mixed-mode variable amplitude loading, Int. J. Fatig., 25 (2003) 1157–1167. [26] Kim, J. K., Cho, S. B., Effect of second non-singular term of mode I near the tip of a V notched crack, Fatig. Fract. Eng. Mater. Struct., 2 (2009) 346–356. [27] Ayatollahi, M. R., Nejati, M., Determination of NSIFs and coefficients of higher order terms for sharp notches using finite element method, Int. J. Mech. Sci., 53 (2011) 164-177. [28] Ayatollahi, M. R., Dehghany, M., On T-stresses near V-notches, Int. J. Fract., 165 (2010) 121-126.

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[29] Ayatollahi, M. R., Nejati, M., Experimental evaluation of stress field around the sharp notches using photoelasticity, Mater. Des., 32 (2011) 561-569. [30] Prashant, K., Elements of Fracture Mechanics, First Ed., McGraw-Hill Book Company, New-Delhi, (2009). [31] Paris, P. C., Sih, G. C., In ASTM Spec. Tech. Publ., STP 381, ASTM Philadelphia, STP 381 (1965) 30-83. [32] Rice, J. R., Matrix Computations and Mathematical software, 1st Edn., McGraw-Hill Computer Science Series, McGraw-Hill, (1981). [33] Kotousov, A., Berto, F., Lazzarin, P., Pegorin, F. Three dimensional finite element mixed fracture mode under anti- plane loading of a crack, Theo. App. Fract. Mech., 62 (2012) 26–33. [34] Kotousov, A., Effect of plate thickness on stress state at sharp notches and the strength paradox of thick plates, Int. J. Solids Struct., 47 (2010) 1916–1923. [35] ANSYS Manual Release 8.0.

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N. Golinelli et alii, Frattura ed Integrità Strutturale, 32 (2015) 13-23; DOI: 10.3221/IGF-ESIS.32.02

Design of a novel magnetorheological damper with internal pressure control

Nicola Golinelli, Andrea Spaggiari Department of Sciences and Methods for Engineering University of Modena and Reggio Emilia, Italy nicola.golinelli@unimore.it, andrea.spaggiari@unimore.it

A BSTRACT . In this work we designed and manufactured a novel magnetorheological (MR) fluid damper with internal pressure control. Previous authors’ works showed that the yield stress τB of MR fluids depends both on the magnetic field intensity and on the working pressure. Since the increase of the magnetic field intensity is limited by considerations like power consumption and magnetic saturation, an active pressure control leads to a simple and efficient enhancement of the performances of these systems. There are three main design topics covered in this paper about the MR damper design. First, the design of the magnetic circuit; second the design of the hydraulic system and third the development of an innovative pressure control apparatus. The design approach adopted is mainly analytical and provides the equations needed for system design, taking into account the desired force and stroke as well as the maximum external dimensions. K EYWORDS . Magnetorheological damper; Design and manufacturing; Squeeze-strengthen effect.

I NTRODUCTION

N

owadays, there are several industrial applications in which magnetorheological fluids (MRFs) are used [1-3]. In particular, this paper focuses on the optimal design methodology for magnetorheological dampers (MRDs). The purpose of traditional dampers, or so-called shock absorbers, is to dissipate energy. MRDs compared to traditional dampers, exploit the change in the rheological behavior of MR fluids in order to achieve variable damping properties. The changing of the properties of MR fluids occurs when a magnetic field is applied. The magnetic field is typically generated by an axial coil, for which connecting leads are usually brought out through the hollow piston rod [4]. The main classification for MRDs concerns the methods by which the insertion volume of the rod is accommodated. This is a major design problem because the oil itself is nowhere near compressible enough to accept the internal volume reduction of 10% or more associated with the full stroke insertion. The aim of this work consists in exploiting the effect of pressure on MRFs to generate further controllable damping force, so accommodating the change in volume is very important. Clearly, a static pressure can be applied only when nearly incompressible material are used in the system, so no air or gases are allowed in the design. Several studies have been carried out in order to comprehend the influence of pressure on the properties of MRFs. In [5], a novel compressible MR fluid has been synthesized with additives that provide compressibility to the fluid. MR fluids are influenced by the presence of internal pressure [6-11]. In combined squeeze-shear mode, with a magnetic field of 300 mT, passing from 0 to 30 bar the yield shear stress to doubles its value. In flow mode instead, with a magnetic field intensity of 800 mT, the yield stress τ B increments its value by nearly ten times. There are three basic MRDs architectures [4], as is shown in Fig. 1: single-tube, double-tube and through-rod. The single-tube architecture (Fig. 1a) is based on a single-rod cylinder structure, in which the piston head divides the damper into extension and compression chamber. During piston movement, MR fluid passes through the control valve which is obtained into the piston head. A floating piston separates the MRF from the accumulator filled with compressed

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N. Golinelli et alii, Frattura ed Integrità Strutturale, 32 (2015) 13-23; DOI: 10.3221/IGF-ESIS.32.02

gas. The accumulator is used to compensate the volume change due to the piston rod moving inward the cylinder. To eliminate the floating piston, emulsified oil may be used, distributing the expansion and rod-accommodation volume throughout the main oil volume. The gas separates, but quickly re-emulsifies on action. The valves must be rated to allow for the passage of emulsion rather than liquid oil. Mineral damper oil has long chain hydrocarbon molecules which do not pack efficiently together. This allows a higher compressibility than a liquid such as water because the long molecular chains can distort. In the double-tube type of telescopic (Fig. 1b), a pair of concentric cylinder is used. The external one contains some gas to accommodate the rod displacement volume. The through-rod telescopic (Fig. 1c) avoids the displacement volume issue by having a passing-through rod which causes no volume variation. However this has several disadvantages; there are external seals at both ends subject to high pressures that causes additional friction, the protruding free end may be inconvenient or dangerous, and there is still no provision for thermal expansion of the oil. However it is a simple solution which is used for example in some seismic application. Even though this architecture has proved impractical for suspension damping, it is sometimes used for damping of the steering.

(a) (b) (c) Figure 1 : Telescopic architectures. Single-tube (a) , double-tube (b) and through-rod (c) .

M ATERIALS AND METHODS

S

ince the active control of the pressure is needed, no flexible diaphragms or compressible gases are allowed. This is because flexible parts would absorb the change in pressure. Hence, it is necessary an architecture without volume compensation. Fig. 2 shows the conceptual scheme of the damper presented in this paper. We used a bottom-rod fixed to the end plug and coupled with the piston head. The bottom-rod has the same diameter of the upper-rod so that there is no volume variation. During piston movement, the bottom-rod is moving inward the chamber obtained into the piston head. The chamber is also directly connected to the canal through the upper-rod in order to bring out the coil’s wire. Thereby, overpressure or depression within the chamber will not occur. It is worth noting that two coils were adopted. In this way, the longer axial length of the piston head is exploited to maximize the concatenated magnetic flux. The main dimensions of the damper are shown in Tab. 1.

Cylinder length, (mm) Cylinder diameter, (mm) Rod diameter, (mm)

192

50 20

Figure 2 : Conceptual scheme of the MR damper.

Table 1 : Main system dimensions.

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N. Golinelli et alii, Frattura ed Integrità Strutturale, 32 (2015) 13-23; DOI: 10.3221/IGF-ESIS.32.02

The design of the MR damper consisted in two main parts: the hydraulic-mechanical design [12, 13] and the magnetic circuit design. The specifications of the damper we developed are listed in Tab. 2.

Maximum force, (N)

2000

Maximum cylinder diameter, (mm) Maximum working current, (A)

40

2

Maximum pressure, (bar)

40 50

Stroke, (mm)

Maximum velocity, (mm/s)

100

Table 2 : Damper specifications. In order to keep the manufacturing of the damper as simple as possible, a commercial hydraulic cylinder and the associated cylinder head were chosen [14]. Hence, knowing the outer diameter of the cylinder (50 mm) and the wall thickness (5 mm), even the inner diameter of the cylinder was also fixed (40 mm) (Fig. 3c, d). The axial length of the hydraulic cylinder is 192 mm. The commercial cylinder head is arranged for a piston rod diameter of 20 mm (Fig. 3a) and it has its own system of seals (Fig. 3b). The minimum axial length of the piston head was also fixed and had to be at least L = 90 mm. That is because we decided to compensate for the piston volume using the piston head, so it has to host a compensating bottom rod (50 mm), as presented in Fig. 2.

(a)

(b)

(c) (d) Figure 3 : Commercial components. Cylinder head (a) and sealing system (b) . Commercial hydraulic cylinder (c) with welded boss. Bottom part of the cylinder (d) . Optimal design of magnetorheological devices requires the knowledge and the characterization of the properties of the materials involved. Firstly, the knowledge of the yield shear stress of the fluid as function of the magnetic field is necessary. The yield stress τ B ሺH mrf ሻ of MRF 140-CG [15] is given by the experimentally-derived equation from [16, 17] and depends on the magnetic field intensity and the particle volume fraction φ :        1.5239 6 271700 tanh(6.33 10 ) B mrf mrf H C H (1)

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N. Golinelli et alii, Frattura ed Integrità Strutturale, 32 (2015) 13-23; DOI: 10.3221/IGF-ESIS.32.02

φ is 0.4 and C is a coefficient dependent on the carrier fluid of the MR fluid (C = 1 for hydrocarbons), according to [17] The magnetic B-H relationship of a MR fluid can be defined as [18]:               0 10.97 1.133 0 0 1.91 1  mrf H mrf B e H (2) Therefore, a MR fluid’s relative permeability can be defined as:            0 10.97 1.133 0 0 1.91 10.97 mrf H r mrf dB e dH (3) where, B is in Tesla, H mrf is in A/m, and μ 0 = 1.25x10 -6 H/m is the permeability of free space. Fig. 4 shows the graphs of the B-H relationship of the MRF 140-CG, the values of yield stress τ B and the relative magnetic permeability as a function of the magnetic field intensity H mrf . In order to reach the best performances, the material which composes the magnetic circuit should have high magnetic permeability and high magnetic saturation. A material with such properties is the AISI 1010, which is a low-carbon steel (C% < 0.10). This material though, is hardly available because of is being used for niche applications. Hence the AISI 430 was used. AISI 430 is a ferritic stainless steel with a high relative magnetic permeability, of about 600.

(a)

(b)

(c)

Figure 4 : MRF 140 CG [15] properties: B-H relationship (a) , yield stress τ B

vs H (b) and relative permeability vs H (c) .

Analytical design of the MR damper Fig. 5 shows the forces developed by a magnetorheological damper [19, 20]. Considering the parallel-plate Bingham model, the forces can be decomposed into three contributions [21]. First, the controllable force F τ , Eq. (4), directly correlates with the magnetic field applied through the yield stress τ B .

 D L A F c sign V h   ( ) B P A

(4)

where L P

is the axial activation length of the piston head, A A

is the annular piston’s area, h is the fluid gap and c is a

coefficient that depends on the volumetric flow rate, the viscosity and the yield stress. Second, F η viscous forces and depends on the length of the orifice, the fluid’s viscosity and flow rate.

, Eq. (5), represents the



QLA

12

 F k (5) where Q is the flow rate, L is the total axial length of the piston head, w is the mean circumference of the damper’s annular flow path and k is a constant depending on the volumetric flow rate and the velocity. Third, F f that stands for the friction forces like those related to the seals system. Moreover we should also account for the force derived from the effect of pressure F P . Hence, the total force will be obtained by adding up all these contributions: (6) The Dynamic Range D, is also a fundamental parameter which provides an estimate of the influence of the control variable on the system behavior. D can be calculated as the controllable forces divided by the uncontrollable forces (Eq. 7).  3 A wh       f P F F F F F tot

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N. Golinelli et alii, Frattura ed Integrità Strutturale, 32 (2015) 13-23; DOI: 10.3221/IGF-ESIS.32.02

 F F F F D F F F     c un f 

f

 

(7)

Figure 5 : The total force of MRDs can be obtained by summing: friction (dotted blue line), viscous (dashed blue line), magnetic (dash- dotted blue line) and pressure (solid red line) driven forces. Eq. (4)-(7) were manipulated taking into account the geometrical constraints and the design parameter of the remaining components were determined. In particular, considering that a fluid gap h=1 mm was chosen, the annular area A A is 819.24 mm 2 and the viscous forces can be calculated as follow:

  

  

    7 122.26 1 100 12 3 10 81954 90 819.24    

 A whV QLA 12 D

  

  

  1

  1



F

N

(8)

191



3

3

 2 81954

 122.26 1

Q wh

2

In which the velocity V D

=100 mm/s and the viscosity η=0.3 Pa·s. Assuming that the friction forces F f

= 250 N and the

total force F tot

= 2000 N, the required controllable force F τ is:              2000 2000 191 250 1559  f F F F N

(9)

and the dynamic range turns out to be:

  F F F

   1559 191 250 4.53 191 250 





f





D

(10)

 F F 

f

Once the controllable force was found, the yield stress of the fluid τ B current value I = 1 A. The total active pole length is obtainable by manipulating Eq.(4):

= 20 kPa was set, considering the nominal working

 1559 1

 F h







(11)

L

41.32mm

P

 c A 2.30 0.020 819.24   B A

tot

where the coefficient c = 2.30 [12]. The activation areas are four, which implies a single axial length   tot P P L L / 4 10mm. The chosen yield stress implies, by means of the Eq. (1)-(2), a magnetic field density B mrf along the active pole of 0.35 T. After that, the piston head (Fig. 6a) along with the flange (Fig. 6b, c), the rod (Fig. 6d) and the bottom-rod (Fig. 6e) were manufactured. The piston head and the flange were made of AISI 430. Conversely, the rod and the bottom-rod were made of brass because they do not have to influence the magnetic flux during operations. The flange is coupled with the rod by a drilled screw in order to let the coil’s wire passing through it (Fig. 6f).

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