Skip to main content
SpringerLink
Log in

Energy Release Rate and Mode Partitioning of Moment-Loaded Fracture Tests on Layered Beams with Bending–Extension Coupling and Hygrothermal Stresses

  • Chapter
  • First Online:
Fracture Analysis of Layered Beams With an Elastically Coupled Behavior and Hygrothermal Stresses

Part of the book series: Springer Theses ((Springer Theses))

  • 125 Accesses

Abstract

In the present chapter, the analytical framework proposed in Chap. 2 is enriched, and the study of the fracture toughness of the titanium-to-CFRP joint presented in Chaps. 3 and 4 is extended. New closed-form expressions are derived for the ERR and MM of interfacial fracture tests on layered beams with BEC and RHTS, loaded with uneven bending moments. The analytical model uses Timoshenko beam kinematics, a semi-rigid interface model, and the crack closure integral for mode partitioning. The proposed expressions are validated via the finite element method in two typical examples of metal-to-composite joints. In addition, experiments employing the DCB-UBM test setup are performed in a titanium-to-CFRP adhesive joint, and the new analytical expressions are used to achieve data reduction. We demonstrate that the effect of RTS on the ERR and MM of the metal/composite interfaces can be non-negligible and, thus, should be considered in the design and analysis of such structures. The proposed formulae serve as a DRS to determine fracture toughness and MM of moment-loaded tests on arbitrarily layered beams with BEC and hygrothermal stresses.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Nevertheless, when loading with pure bending moments, where no shear forces are present, shear deformability cannot influence the test results.

  2. 2.

    The term crack is used to describe both delamination and interfacial disbonding phenomena, depending on whether the beam structure under consideration is, for example, a composite laminate or an adhesive joint, respectively.

  3. 3.

    In the present chapter, we chose to define the directions of the applied moments, \({M}_{1}\) and \({M}_{2}\), as shown in Fig. 5.2a for consistency with most of the relevant literature.

  4. 4.

    As discussed in Chap. 2, it is hard to extract analytical, closed-form expressions for the contact force when the effects of BEC and hygrothermal stresses are simultaneously present. This could be the topic of a future work.

  5. 5.

    Based on the definitions given in Sect. 4.3.3.1, this is the VIS criterion.

  6. 6.

    This observation was first reported by Yokozeki et al. [32].

References

  1. Berggreen C, Saseendran V, Carlsson LA (2018) A modified DCB-UBM test method for interfacial fracture toughness characterization of sandwich composites. Eng Fract Mech 203:208–223. https://doi.org/10.1016/j.engfracmech.2018.06.036

    Article  Google Scholar 

  2. Bhat S, Narayanan S (2014) Quantification of fibre bridging in mode I cracked Glare without delaminations. Eur J Mech A Solids 43:152–170. https://doi.org/10.1016/j.euromechsol.2013.09.005

    Article  Google Scholar 

  3. Dimitri R, Tornabene F, Zavarise G (2018) Analytical and numerical modeling of the mixed-mode delamination process for composite moment-loaded double cantilever beams. Compos Struct 187:535–553. https://doi.org/10.1016/j.compstruct.2017.11.039

    Article  Google Scholar 

  4. Freiman SW, Mulville DR, Mast PW (1973) Crack propagation studies in brittle materials. J Mater Sci 8(11):1527–1533. https://doi.org/10.1007/bf00754886

    Article  Google Scholar 

  5. Harvey CM, Wang S (2012) Experimental assessment of mixed-mode partition theories. Compos Struct 94(6):2057–2067. https://doi.org/10.1016/j.compstruct.2012.02.007

    Article  Google Scholar 

  6. Harvey CM, Eplett MR, Wang S (2015) Experimental assessment of mixed-mode partition theories for generally laminated composite beams. Compos Struct 124:10–18. https://doi.org/10.1016/j.compstruct.2014.12.064

    Article  Google Scholar 

  7. Irwin GR (1958) Fracture. In: Flugge S (ed) Handbuch der physik, vol VI. Springer, pp 551–590

    Google Scholar 

  8. Krueger R (2004) Virtual crack closure technique: history, approach, and applications. Appl Mech Rev 57(2):109–143. https://doi.org/10.1115/1.1595677

    Article  Google Scholar 

  9. Lindgaard E, Bak BLV (2019) Experimental characterization of delamination in off-axis GFRP laminates during mode I loading. Compos Struct 220:953–960. https://doi.org/10.1016/j.compstruct.2019.04.022

    Article  Google Scholar 

  10. Lindhagen JE, Berglund LA (2000) Application of bridging-law concepts to short-fibre composites. Part 1: DCB test procedures for bridging law and fracture energy. Comp Sci Technol 60(6):871–883. https://doi.org/10.1016/s0266-3538(00)00004-x

    Article  Google Scholar 

  11. Loutas T, Tsokanas P, Kostopoulos V, Nijhuis P, van den Brink WM (2021) Mode I fracture toughness of asymmetric metal-composite adhesive joints. Mater Today Proc 34(1):250–259. https://doi.org/10.1016/j.matpr.2020.03.075

    Article  Google Scholar 

  12. Lundsgaard-Larsen C, Sørensen BF, Berggreen C, Østergaard RC (2008) A modified DCB sandwich specimen for measuring mixed-mode cohesive laws. Eng Fract Mech 75(8):2514–2530. https://doi.org/10.1016/j.engfracmech.2007.07.020

    Article  Google Scholar 

  13. Østergaard RC, Sørensen BF (2007) Interface crack in sandwich specimen. Int J Fract 143(4):301–316. https://doi.org/10.1007/s10704-007-9059-4

    Article  MATH  Google Scholar 

  14. Pappas GA, Botsis J (2020) Variations on R-curves and traction-separation relations in DCB specimens loaded under end opening forces or pure moments. Int J Solids Struct 191–192:42–55. https://doi.org/10.1016/j.ijsolstr.2019.11.022

    Article  Google Scholar 

  15. Plausinis D, Spelt JK (1995) Application of a new constant G load-jig to creep crack growth in adhesive joints. Int J Adhes Adhes 15(4):225–232. https://doi.org/10.1016/0143-7496(96)83703-1

    Article  Google Scholar 

  16. Qiao P, Chen F (2011) On the compliance and energy release rate of generally-unified beam-type fracture specimens. J Compos Mater 45(1):65–101. https://doi.org/10.1177/0021998310371545

    Article  Google Scholar 

  17. Raju IS, Crews JH Jr, Aminpour MA (1988) Convergence of strain energy release rate components for edge-delaminated composite laminates. Eng Fract Mech 30(3):383–396. https://doi.org/10.1016/0013-7944(88)90196-8

    Article  Google Scholar 

  18. Rice JR (1968) A path independent integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech 35(2):379–386. https://doi.org/10.1115/1.3601206

    Article  Google Scholar 

  19. Saseendran V, Berggreen C, Carlsson LA (2018) Fracture mechanics analysis of reinforced DCB sandwich deboned specimen loaded by moments. AIAA J 56(1):413–422. https://doi.org/10.2514/1.j056039

    Article  Google Scholar 

  20. Sørensen BF, Brethe P, Skov-Hansen P (1996) Controlled crack growth in ceramics: the DCB specimen loaded with pure moments. J Eur Ceram Soc 16(9):1021–1025. https://doi.org/10.1016/0955-2219(96)00021-0

    Article  Google Scholar 

  21. Sørensen BF, Jørgensen K, Jacobsen TK, Østergaard RC (2006) DCB-specimen loaded with uneven bending moments. Int J Fract 141(1–2):163–176. https://doi.org/10.1007/s10704-006-0071-x

    Article  Google Scholar 

  22. Szekrényes A (2007) Improved analysis of unidirectional composite delamination specimens. Mech Mater 39(10):953–974. https://doi.org/10.1016/j.mechmat.2007.04.002

    Article  Google Scholar 

  23. Tsokanas P, Loutas T (2019) Hygrothermal effect on the strain energy release rates and mode mixity of asymmetric delaminations in generally layered beams. Eng Fract Mech 214:390–409. https://doi.org/10.1016/j.engfracmech.2019.03.006

    Article  Google Scholar 

  24. Tsokanas P, Loutas T, Kotsinis G, Kostopoulos V, van den Brink WM, Martin de la Escalera F (2020) On the fracture toughness of metal-composite adhesive joints with bending–extension coupling and residual thermal stresses effect. Compos B Eng 185:107694. https://doi.org/10.1016/j.compositesb.2019.107694

    Article  Google Scholar 

  25. Tsokanas P, Loutas T, Kotsinis G, van den Brink WM, Nijhuis P (2021) Strain energy release rate and mode partitioning of moment-loaded elastically coupled laminated beams with hygrothermal stresses. Compos Struct 259:113237. https://doi.org/10.1016/j.compstruct.2020.113237

    Article  Google Scholar 

  26. Tsokanas P, Loutas T, Nijhuis P (2020) Interfacial fracture toughness assessment of a new titanium-CFRP adhesive joint: an experimental comparative study. Metals 10(5):699. https://doi.org/10.3390/met10050699

    Article  Google Scholar 

  27. Valvo PS (2016) On the calculation of energy release rate and mode mixity in delaminated laminated beams. Eng Fract Mech 165:114–139. https://doi.org/10.1016/j.engfracmech.2016.08.010

    Article  Google Scholar 

  28. Valvo PS, Sørensen BF, Toftegaard HL (2015) Modelling the double cantilever beam test with bending moments by using bilinear discontinuous cohesive laws [Paper presentation]. In: 20th International conference on composite materials, Copenhagen, Denmark

    Google Scholar 

  29. Wang J, Qiao P (2004) On the energy release rate and mode mix of delaminated shear deformable composite plates. Int J Solids Struct 41(9–10):2757–2779. https://doi.org/10.1016/j.ijsolstr.2003.11.039

    Article  MATH  Google Scholar 

  30. Williams JG (1988) On the calculation of energy release rates for cracked laminates. Int J Fract 36(2):101–119. https://doi.org/10.1007/bf00017790

    Article  Google Scholar 

  31. Yokozeki T (2010) Energy release rates of bi-material interface crack including residual thermal stresses: application of crack tip element method. Eng Fract Mech 77(1):84–93. https://doi.org/10.1016/j.engfracmech.2009.09.018

    Article  Google Scholar 

  32. Yokozeki T, Ogasawara T, Aoki T (2008) Correction method for evaluation of interfacial fracture toughness of DCB, ENF and MMB specimens with residual thermal stresses. Compos Sci Technol 68(3–4):760–767. https://doi.org/10.1016/j.compscitech.2007.08.025

    Article  Google Scholar 

Download references

Acknowledgements

A significant part of the work was supported financially by the Clean Sky 2 Joint Undertaking under the European Union’s Horizon 2020 research and innovation program TICOAJO (grant agreement number: 737785). The author appreciates this support. The author also appreciates the financial support provided by the General Secretariat for Research and Technology (GSRT) and the Hellenic Foundation for Research and Innovation (HFRI) in the context of the action “1st Proclamation of Scholarships from ELIDEK for PhD Candidates” (scholarship code: 1688). Furthermore, the author thanks his TICOAJO partners from the Structures Technology Department, Royal Netherlands Aerospace Centre, the Netherlands, and especially Mr. Wouter M. van den Brink, Mr. Peter Nijhuis, and Mr. Marcelo Müller, for assisting with the execution of the experiments. Lastly, the author thanks his TICOAJO partners from the Structural Integrity and Composites research group, Delft University of Technology, the Netherlands, and especially Dr. Wandong Wang, Prof. Johannes A. Poulis, Prof. Sofia Teixeira de Freitas, and Prof. Dimitrios Zarouchas, for performing the surface pre-treatment studies.

Author information

Authors and Affiliations

  1. Applied Mechanics Laboratory, Department of Mechanical Engineering and Aeronautics, University of Patras, Patras, Greece

    Panayiotis Tsokanas

Authors
  1. Panayiotis Tsokanas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Panayiotis Tsokanas .

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Tsokanas, P. (2023). Energy Release Rate and Mode Partitioning of Moment-Loaded Fracture Tests on Layered Beams with Bending–Extension Coupling and Hygrothermal Stresses. In: Fracture Analysis of Layered Beams With an Elastically Coupled Behavior and Hygrothermal Stresses. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-031-17621-0_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-17621-0_5

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-17620-3

  • Online ISBN: 978-3-031-17621-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation