- Email: [email protected]
composite bonded joints
Computational Materials Science 45 (2009) 735–738
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Prediction of the failure metal/composite bonded joints Agnieszka Derewon´ko * Department of Mechanics and Applied Computer Science, Military University of Technology, 2 Kaliskiego Street, 00-908 Warsaw, Poland
a r t i c l e
i n f o
Article history: Received 28 November 2007 Received in revised form 15 July 2008 Accepted 17 July 2008 Available online 26 August 2008 PACS: 61.43.Bn 62.20.Fe 68.35.Np Keywords: Adhesive bonded joint FEM Nonlinearity
a b s t r a c t Bonded joints became very popular in aircraft, automotive and marine applications. Loading is transferred by adhesion and cohesion forces. These forces are very hard to determine due to a small dimension of bonded joints. Composite/metal double lap joint is investigated. A global model of the double lap joint is presented. A laminated composite structure is manufactured by stacking multiple layers of prepreg. The composite and metal are jointed by an adhesive. Ten prepreg layers and interlaminar interfaces are modelled in the composite substrate. Two thin three-dimensional solids are used in modelling an adhesive bonding composite and metal substrates. Adhesive/substrate and interlaminar interfaces are modelled as two sets of nodes. Interaction stresses (adhesion stress) are determined for each stage of loading. Normal stress in the function of the loading percentage is allowed to show. The Gurson criterion that includes hydrostatic stress sensitivity is used to describe ductile fracture as relation to the nucleation voids in the adhesive. Numerical method is proposed to three-dimensional model interlaminar interface for prediction of delamination initiation as for the static loading. Proposed approach allows to simulate the failure of the joint and composite substrate. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Bonding is a very comfortable and widespread method of joining two materials with different stiffness like metal and composite. But the strength of this type of joint strongly depends on manufacturing process such as: surface moisture and roughness [1,9]. Numerical analysis is useful to solve this problem but roughness and moisture of surface are not included directly in the global model. A non-ideally uniform adhesive material strength is limited due to a large number of flaws which will cause local increases in stress when loaded [2,8]. If those local stresses exceed the bond strength, the material fails. The Gurson–Tvergaard–Needleman model is used to describe damaging behaviour of the adhesive [5]. The use of composites brings other changes in the design approach. In order to get required strength and stiffness, several plies must be used, each in different shape and fibre (woven fabric) orientation. Usually, in the finite element model, composite layers are converted into one layer with equivalent stiffness behaviours [4,7]. Only one integration point is needed across the thickness. It is particularly attractive when the number of composite layers is large and all the layers of the laminated composite have the same length, because analysis of these smeared shell structures uses less computer time and storage space. When structure consists of lami-
* Tel.: +48 022 683 79 06; fax: +48 022 683 93 55. E-mail addresses: [email protected], [email protected] 0927-0256/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2008.07.018
nated composite layers with different length other modelling techniques are necessary. Practical engineering approach to the model of the double lap metal–composite joint is presented in this paper. Each body is meshed separately. Contact problem is introduced to model adhesive/metal and adhesive/composite sticking interface. The same technique is used for modelling interface between laminate composite layers. Professional engineering software is used for the numerical analysis. 2. Theoretical background Numerical analysis is performed with engineering software MSC.Marc [5]. Composite material is modelled as the stack of layers of different materials with various layer thickness. From one layer to the next one the orientation of the fibre can be changed. For each individual layer, all of the constitutive laws can be used. The interlaminar shear and normal stresses are calculated by averaging the stresses in the stacked two layers. The stresses are transformed into a component tangent to the interface and a component normal to the interface. Contact between deformable bodies in MSC.Marc is automatically solved. This procedure is based on constraint minimization problem. The augmented lagrangian method is a procedure of numerical implementation of the contact constraints. The surfaces of the two bodies is given in discrete form for a set of nodes. When contact are automatically detected, the degrees of freedom are
´ ko / Computational Materials Science 45 (2009) 735–738 A. Derewon
736
transformed to a local system and displacement constraints are imposed as presented below
Table 1 Laminate composite parameters
Dunormal ¼ v n
ð1Þ
E11 (MPa)
E22 (MPa)
E33 (MPa)
G12 (MPa)
G23 (MPa)
G31 (MPa)
m12
m23
m31
ð2Þ
58,093
58,093
9759
3545
2564
2564
0.0154
0.5356
0.1575
Dutangential ¼ v t;
t¼
v kvk
where v, the prescribed relative velocity; n, normal vector; t, tangential vector in the direction of relative velocity. Interface values such as contact normal and friction stress for each step loading are determined during the iterative procedure. The von Mises yield condition is the default condition for isotropic materials. When the actual material model changed due to a failure an other material model is necessary [10]. The yield surface is modified based upon the Gurson model for void damage. A modified von Mises criterion, with hydrostatic stress and void dependency, is used [5]. The yield criterion is given by
F¼
r ry
2
þ 2q1 f cosh
h i q2 rkk 1 þ ðq1 f Þ2 ¼ 0 2ry
ð3Þ
where q1, first yield surface multiplier; q2, second yield surface multiplier; f*, modified void volume fraction. Void nucleation and void growth are based on the model by Tvergaard and Needleman and are controlled by mean stress [11]. The existing value of the void volume fraction changes due to the growth of existing voids and due to the nucleation of new voids according to relation:
P fN X 1 rn f_ ¼ f_ growth þ f_ nucleation ¼ ð1 f Þ_epkk þ pffiffiffiffiffiffiffi exp 2 S S 2p 1 þ rkk R¼r 3 ð4Þ where fN, volume fraction of void forming particles; rn, mean stress for void nucleation, S, standard deviation. 3. Double lap joint Only half a double lap joint is considered due to its symmetry and simple construction. Laminated composite is sticked between
two aluminium adherends (Fig. 1). Aluminium and laminated composite are attached by an adhesive model of which is created as a set of brick elements. Laminate composite consists of ten plies, which are modelled as five sets of the three-dimensional finite element with composite property. Continuity conditions are ensured by contact equations imposed between sticking plies. The 3D composite model includes layer thickness, the ply angle and an orthotropic material with nine material constants: Young’s moduli, Poisson ratios, shear moduli in the three perpendicular directions [3]. Mechanical properties of the composite were obtained from an experimental investigation and are shown in Table 1. Materials’ parameters and stress–strain relationship for adhesive were obtained during investigations (Fig. 2a). The parameters q1 were introduced by Tvergaard to improve the Gurson model at small values of the void volume fraction. For solids with periodically spaced voids numerical studies [12] showed that the values of q1 = 1.5, q2 = 1 were accurate [5]. According to literature [6] initial void volume is a 1.1% of the adhesive thickness, which is 0.18 mm. Failure void volume fraction equals thickness of the adhesive. Aluminium is modelled as elastic–plastic material. The true strain–stress relationship, Young moduli and Poisson ratio for aluminium is shown in Fig. 2b. Inelastic behaviour (piecewise linear representation) of aluminium material is simulated. All sticking interfaces (adhesive/metal, adhesive/laminated composite) are modelled as a contact problem with glue option, which impose that there is no relative tangential motion of contacting bodies. This method is used for stick layers in the laminated composite model. This modelling technique allows to determine a contact friction stress in the interface. As an example, contact friction stress versus loading for interface between second and third laminated composite layers, in the overlap region, is shown in Fig. 3.
Fig. 1. Scheme of the investigated joint.
´ ko / Computational Materials Science 45 (2009) 735–738 A. Derewon
737
b
a 80
600
70
500
60
400 [MPa]
[MPa]
50 40 30
E = 2083 MPa v = 0.35
20
300
E = 78900 MPa v = 0.365
200 100
10
0
0 0
0.02
0.04 ε
0.06
0.08
0
0.05
0.1 ε
0.15
0.2
y = 7E-16x5 - 2E-11x4 + 2E-07x3 - 0,0004x2 + 0,2364x + 32,319
25 20 15 10 5 0
4000
0
10 5 Force [kN]
Contact Stress
Contact friction stress [MPa]
Fig. 2. True stress–strain relationship for (a) adhesive and (b) aluminium.
15
Fig. 3. Contact friction stress versus loading.
3000 2000
Friction
0 -1000
0
4000
6000
8000
10000
12000
14000
Force [N]
Two main reasons of difference in experimental and simulation results can be: aluminium material model and insufficient joint manufacturing. The true stress–strain curve for aluminium was obtained from quasi–static tensile test. The specimens were taken in three different directions with respect to the rolling direction of the target plate in order to reveal the plastic anisotropy. Laboratory tests show anisotropic material behaviour. However, the isotropic material model is used for aluminium in the numerical analyses. Handmade method at manufacturing composite can produces bigger voids than was assumed.
d Longitudinal Strain
2000
Fig. 5. Distribution of contact friction and normal stress.
Shear test was performed to verify results of the numerical analysis (Fig. 4a). Six gauges were glued on the bonded double lap joint in the symmetric axis, in the position which is indicated in Fig. 4b. Deformation of the structure which was obtained from numerical analysis is shown in Fig. 4c. Longitudinal strain values versus force from experimental and numerical investigations for two gauges are shown in Fig. 4d and e, respectively. ‘Middle’ and ‘Metal’ gauges are glued on the composite and aluminium parts, respectively. Results are compared. Difference of longitudinal strain equals 24.2% for ‘Middle’ gauge and does not depend on a void fraction in the adhesive. Differences for ‘Metal’ gauge are: 12.3% for model without void and 12.1% for model with void.
Middle
0.005 0.004 0.003 with void
0.002
without void
0.001
exp
0 0
5000 10000 Force [N]
e Longitudinal Strain
Normal
1000
15000
Metal 0.0012 0.001 0.0008 0.0006 0.0004 0.0002 0
with void without void exp
0
5000 10000 Force [N]
Fig. 4. Experimental verification.
15000
738
´ ko / Computational Materials Science 45 (2009) 735–738 A. Derewon
4. Conclusion
References
Application of two surfaces in the contact region allows to determine internal forces (stress, strain) on each of them. Assumption of the void coalescence and the growth in the adhesive model causes change of the stress level in the layer of the laminated composite. Distribution of contact friction stress in the bonded interface can be presented as a function of joint load (Fig. 5). Contact friction stresses along aluminium/adhesive bondline in the overlap region are shown in Fig. 5. Those stresses are about 90% higher than stress along adhesive/composite bondline. This function can be used as an initial stress for bonded materials in the overlap region. Therefore coarse mesh may be applied for global model without an adhesive to save the computer time.
[1] W.R. Broughton, L.E. Crocker, M.R.L. Gower, Design Requirements for Bonded and Bolted Composite Structure, National Physical Laboratory Materials Centre, UK, 2002. [2] H.B. Chew, T.F. Guo, L. Cheng, Thin Solid Films 504 (2006) 325–330. [3] A. Derewonko, T. Niezgoda, J. Godzimirski, 3D Numerical Investigation of Tensile Loaded Lap Bonded Joint of Aircraft Structure, 32nd International Scientific Congress on Powertrain and Transport Means European Kones 2006, Warsaw-Lublin-Naleczow, Poland, 10–13 September 2006. [4] J. German, Podstawy Mechaniki Kompozytów Włóknistych, Politechnika Krakowska, Kraków, 1996. [5] MSC.Marc Volume A: Theory and User Information, Version 2005. [6] E.Yu. Maeva, I. Seviaryna, G.B. Chapman, F.M. Severin, Monitoring of Adhesive Cure Process and Following Evaluation of Adhesive Joint Structure by Acoustic Techniques, http://www.ndt.net/article/ecndt2006/doc/We.2.2.4.pdf. [7] A. Muc, Mechanika Kompozytów Włóknistych, Ksie˛garnia Akademicka, Kraków, 2003. [8] V. Tvergaard, International Journal of Mechanical Sciences 42 (2000) 381–395. [9] V. Tvergaard, J.W. Hutchinson, Journal of the Mechanics and Physics of Solids 44 (5) (1996) 789. [10] V. Tvergaard, A. Needleman, International Journal of Solids and Structures 32 (8/9) (1995) 1063–1077. [11] Z.L. Zhang, C. Thulow, J. Odegard, Engineering Fracture Mechanics 67 (2000) 155–168. [12] Current Recommended Constitutive Equations for Inelastic Design Analysis of FFTF Components, ORLN-TM-360Z, October 1971.
Acknowledgements This work has been made possible through the financial support of Polish Scientific Research Committee (KBN) under research Grant No. 4 T12C 010 27.
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Prediction of the failure metal/composite bonded joints Agnieszka Derewon´ko * Department of Mechanics and Applied Computer Science, Military University of Technology, 2 Kaliskiego Street, 00-908 Warsaw, Poland
a r t i c l e
i n f o
Article history: Received 28 November 2007 Received in revised form 15 July 2008 Accepted 17 July 2008 Available online 26 August 2008 PACS: 61.43.Bn 62.20.Fe 68.35.Np Keywords: Adhesive bonded joint FEM Nonlinearity
a b s t r a c t Bonded joints became very popular in aircraft, automotive and marine applications. Loading is transferred by adhesion and cohesion forces. These forces are very hard to determine due to a small dimension of bonded joints. Composite/metal double lap joint is investigated. A global model of the double lap joint is presented. A laminated composite structure is manufactured by stacking multiple layers of prepreg. The composite and metal are jointed by an adhesive. Ten prepreg layers and interlaminar interfaces are modelled in the composite substrate. Two thin three-dimensional solids are used in modelling an adhesive bonding composite and metal substrates. Adhesive/substrate and interlaminar interfaces are modelled as two sets of nodes. Interaction stresses (adhesion stress) are determined for each stage of loading. Normal stress in the function of the loading percentage is allowed to show. The Gurson criterion that includes hydrostatic stress sensitivity is used to describe ductile fracture as relation to the nucleation voids in the adhesive. Numerical method is proposed to three-dimensional model interlaminar interface for prediction of delamination initiation as for the static loading. Proposed approach allows to simulate the failure of the joint and composite substrate. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Bonding is a very comfortable and widespread method of joining two materials with different stiffness like metal and composite. But the strength of this type of joint strongly depends on manufacturing process such as: surface moisture and roughness [1,9]. Numerical analysis is useful to solve this problem but roughness and moisture of surface are not included directly in the global model. A non-ideally uniform adhesive material strength is limited due to a large number of flaws which will cause local increases in stress when loaded [2,8]. If those local stresses exceed the bond strength, the material fails. The Gurson–Tvergaard–Needleman model is used to describe damaging behaviour of the adhesive [5]. The use of composites brings other changes in the design approach. In order to get required strength and stiffness, several plies must be used, each in different shape and fibre (woven fabric) orientation. Usually, in the finite element model, composite layers are converted into one layer with equivalent stiffness behaviours [4,7]. Only one integration point is needed across the thickness. It is particularly attractive when the number of composite layers is large and all the layers of the laminated composite have the same length, because analysis of these smeared shell structures uses less computer time and storage space. When structure consists of lami-
* Tel.: +48 022 683 79 06; fax: +48 022 683 93 55. E-mail addresses: [email protected], [email protected] 0927-0256/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2008.07.018
nated composite layers with different length other modelling techniques are necessary. Practical engineering approach to the model of the double lap metal–composite joint is presented in this paper. Each body is meshed separately. Contact problem is introduced to model adhesive/metal and adhesive/composite sticking interface. The same technique is used for modelling interface between laminate composite layers. Professional engineering software is used for the numerical analysis. 2. Theoretical background Numerical analysis is performed with engineering software MSC.Marc [5]. Composite material is modelled as the stack of layers of different materials with various layer thickness. From one layer to the next one the orientation of the fibre can be changed. For each individual layer, all of the constitutive laws can be used. The interlaminar shear and normal stresses are calculated by averaging the stresses in the stacked two layers. The stresses are transformed into a component tangent to the interface and a component normal to the interface. Contact between deformable bodies in MSC.Marc is automatically solved. This procedure is based on constraint minimization problem. The augmented lagrangian method is a procedure of numerical implementation of the contact constraints. The surfaces of the two bodies is given in discrete form for a set of nodes. When contact are automatically detected, the degrees of freedom are
´ ko / Computational Materials Science 45 (2009) 735–738 A. Derewon
736
transformed to a local system and displacement constraints are imposed as presented below
Table 1 Laminate composite parameters
Dunormal ¼ v n
ð1Þ
E11 (MPa)
E22 (MPa)
E33 (MPa)
G12 (MPa)
G23 (MPa)
G31 (MPa)
m12
m23
m31
ð2Þ
58,093
58,093
9759
3545
2564
2564
0.0154
0.5356
0.1575
Dutangential ¼ v t;
t¼
v kvk
where v, the prescribed relative velocity; n, normal vector; t, tangential vector in the direction of relative velocity. Interface values such as contact normal and friction stress for each step loading are determined during the iterative procedure. The von Mises yield condition is the default condition for isotropic materials. When the actual material model changed due to a failure an other material model is necessary [10]. The yield surface is modified based upon the Gurson model for void damage. A modified von Mises criterion, with hydrostatic stress and void dependency, is used [5]. The yield criterion is given by
F¼
r ry
2
þ 2q1 f cosh
h i q2 rkk 1 þ ðq1 f Þ2 ¼ 0 2ry
ð3Þ
where q1, first yield surface multiplier; q2, second yield surface multiplier; f*, modified void volume fraction. Void nucleation and void growth are based on the model by Tvergaard and Needleman and are controlled by mean stress [11]. The existing value of the void volume fraction changes due to the growth of existing voids and due to the nucleation of new voids according to relation:
P fN X 1 rn f_ ¼ f_ growth þ f_ nucleation ¼ ð1 f Þ_epkk þ pffiffiffiffiffiffiffi exp 2 S S 2p 1 þ rkk R¼r 3 ð4Þ where fN, volume fraction of void forming particles; rn, mean stress for void nucleation, S, standard deviation. 3. Double lap joint Only half a double lap joint is considered due to its symmetry and simple construction. Laminated composite is sticked between
two aluminium adherends (Fig. 1). Aluminium and laminated composite are attached by an adhesive model of which is created as a set of brick elements. Laminate composite consists of ten plies, which are modelled as five sets of the three-dimensional finite element with composite property. Continuity conditions are ensured by contact equations imposed between sticking plies. The 3D composite model includes layer thickness, the ply angle and an orthotropic material with nine material constants: Young’s moduli, Poisson ratios, shear moduli in the three perpendicular directions [3]. Mechanical properties of the composite were obtained from an experimental investigation and are shown in Table 1. Materials’ parameters and stress–strain relationship for adhesive were obtained during investigations (Fig. 2a). The parameters q1 were introduced by Tvergaard to improve the Gurson model at small values of the void volume fraction. For solids with periodically spaced voids numerical studies [12] showed that the values of q1 = 1.5, q2 = 1 were accurate [5]. According to literature [6] initial void volume is a 1.1% of the adhesive thickness, which is 0.18 mm. Failure void volume fraction equals thickness of the adhesive. Aluminium is modelled as elastic–plastic material. The true strain–stress relationship, Young moduli and Poisson ratio for aluminium is shown in Fig. 2b. Inelastic behaviour (piecewise linear representation) of aluminium material is simulated. All sticking interfaces (adhesive/metal, adhesive/laminated composite) are modelled as a contact problem with glue option, which impose that there is no relative tangential motion of contacting bodies. This method is used for stick layers in the laminated composite model. This modelling technique allows to determine a contact friction stress in the interface. As an example, contact friction stress versus loading for interface between second and third laminated composite layers, in the overlap region, is shown in Fig. 3.
Fig. 1. Scheme of the investigated joint.
´ ko / Computational Materials Science 45 (2009) 735–738 A. Derewon
737
b
a 80
600
70
500
60
400 [MPa]
[MPa]
50 40 30
E = 2083 MPa v = 0.35
20
300
E = 78900 MPa v = 0.365
200 100
10
0
0 0
0.02
0.04 ε
0.06
0.08
0
0.05
0.1 ε
0.15
0.2
y = 7E-16x5 - 2E-11x4 + 2E-07x3 - 0,0004x2 + 0,2364x + 32,319
25 20 15 10 5 0
4000
0
10 5 Force [kN]
Contact Stress
Contact friction stress [MPa]
Fig. 2. True stress–strain relationship for (a) adhesive and (b) aluminium.
15
Fig. 3. Contact friction stress versus loading.
3000 2000
Friction
0 -1000
0
4000
6000
8000
10000
12000
14000
Force [N]
Two main reasons of difference in experimental and simulation results can be: aluminium material model and insufficient joint manufacturing. The true stress–strain curve for aluminium was obtained from quasi–static tensile test. The specimens were taken in three different directions with respect to the rolling direction of the target plate in order to reveal the plastic anisotropy. Laboratory tests show anisotropic material behaviour. However, the isotropic material model is used for aluminium in the numerical analyses. Handmade method at manufacturing composite can produces bigger voids than was assumed.
d Longitudinal Strain
2000
Fig. 5. Distribution of contact friction and normal stress.
Shear test was performed to verify results of the numerical analysis (Fig. 4a). Six gauges were glued on the bonded double lap joint in the symmetric axis, in the position which is indicated in Fig. 4b. Deformation of the structure which was obtained from numerical analysis is shown in Fig. 4c. Longitudinal strain values versus force from experimental and numerical investigations for two gauges are shown in Fig. 4d and e, respectively. ‘Middle’ and ‘Metal’ gauges are glued on the composite and aluminium parts, respectively. Results are compared. Difference of longitudinal strain equals 24.2% for ‘Middle’ gauge and does not depend on a void fraction in the adhesive. Differences for ‘Metal’ gauge are: 12.3% for model without void and 12.1% for model with void.
Middle
0.005 0.004 0.003 with void
0.002
without void
0.001
exp
0 0
5000 10000 Force [N]
e Longitudinal Strain
Normal
1000
15000
Metal 0.0012 0.001 0.0008 0.0006 0.0004 0.0002 0
with void without void exp
0
5000 10000 Force [N]
Fig. 4. Experimental verification.
15000
738
´ ko / Computational Materials Science 45 (2009) 735–738 A. Derewon
4. Conclusion
References
Application of two surfaces in the contact region allows to determine internal forces (stress, strain) on each of them. Assumption of the void coalescence and the growth in the adhesive model causes change of the stress level in the layer of the laminated composite. Distribution of contact friction stress in the bonded interface can be presented as a function of joint load (Fig. 5). Contact friction stresses along aluminium/adhesive bondline in the overlap region are shown in Fig. 5. Those stresses are about 90% higher than stress along adhesive/composite bondline. This function can be used as an initial stress for bonded materials in the overlap region. Therefore coarse mesh may be applied for global model without an adhesive to save the computer time.
[1] W.R. Broughton, L.E. Crocker, M.R.L. Gower, Design Requirements for Bonded and Bolted Composite Structure, National Physical Laboratory Materials Centre, UK, 2002. [2] H.B. Chew, T.F. Guo, L. Cheng, Thin Solid Films 504 (2006) 325–330. [3] A. Derewonko, T. Niezgoda, J. Godzimirski, 3D Numerical Investigation of Tensile Loaded Lap Bonded Joint of Aircraft Structure, 32nd International Scientific Congress on Powertrain and Transport Means European Kones 2006, Warsaw-Lublin-Naleczow, Poland, 10–13 September 2006. [4] J. German, Podstawy Mechaniki Kompozytów Włóknistych, Politechnika Krakowska, Kraków, 1996. [5] MSC.Marc Volume A: Theory and User Information, Version 2005. [6] E.Yu. Maeva, I. Seviaryna, G.B. Chapman, F.M. Severin, Monitoring of Adhesive Cure Process and Following Evaluation of Adhesive Joint Structure by Acoustic Techniques, http://www.ndt.net/article/ecndt2006/doc/We.2.2.4.pdf. [7] A. Muc, Mechanika Kompozytów Włóknistych, Ksie˛garnia Akademicka, Kraków, 2003. [8] V. Tvergaard, International Journal of Mechanical Sciences 42 (2000) 381–395. [9] V. Tvergaard, J.W. Hutchinson, Journal of the Mechanics and Physics of Solids 44 (5) (1996) 789. [10] V. Tvergaard, A. Needleman, International Journal of Solids and Structures 32 (8/9) (1995) 1063–1077. [11] Z.L. Zhang, C. Thulow, J. Odegard, Engineering Fracture Mechanics 67 (2000) 155–168. [12] Current Recommended Constitutive Equations for Inelastic Design Analysis of FFTF Components, ORLN-TM-360Z, October 1971.
Acknowledgements This work has been made possible through the financial support of Polish Scientific Research Committee (KBN) under research Grant No. 4 T12C 010 27.