Skip to main content
Springer Nature Link
Log in

A practical model to predict the time-dependent behaviour of angle-ply laminates from limited creep data

  • Published:
Mechanics of Time-Dependent Materials Aims and scope Submit manuscript

Abstract

Modelling non-linear behaviour of polymers and polymer fibre-reinforced composites retains significant interest due to its extensive application in primary structures. Simple models are helpful in the design, preferably calibrated from a few short-term mechanical tests. Creep tests are simple to implement, and their duration may depend on the period of interest to analyse. The Norton–Bailey law complies with simplicity and effectiveness when representing the creep response. The time-dependent mechanical response of polymers under arbitrary loading conditions may follow two different theories; time- and strain-hardening. The formal demonstration developed here allowed us to link these two theories to specific time-integration schemes employed in the past. However, both formulations lead to the same expression for the creep-loading condition, which implies the same model parameters in this case. Few research works have applied these theories to polymer-based composites restricted to study the creep-loading condition. Here, a novel approach proposes to model the time-dependent behaviour through the time-hardening and strain-hardening theories with a single viscoplastic element governed by a Norton–Bailey or a Singh–Mitchell law. Experimental data collected from the literature of carbon-reinforced epoxy angle-ply laminates supported the validation process under different loading conditions. The calibration of the model parameters proceeds from creep data at two different stress levels. Thus, time- and strain-hardening theories made predictions for distinct loading conditions, i.e. for multiple creep and creep-recovery cycles, constant strain rate and multi-step stress relaxation. The strain-hardening formulation proved capable of predicting the time-dependent response under different loading conditions from a unique set of model parameters. The current methodology reduces the effort needed to characterise engineering materials when considering the time-dependent behaviour, particularly for polymer-based composites.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

Data Availability

Not applicable.

Materials Availability

Not applicable.

Code Availability

Not applicable.

References

  • Al-Haik, M., Vaghar, M.R., Garmestani, H., Shahawy, M.: Viscoplastic analysis of structural polymer composites using stress relaxation and creep data. Composites, Part B, Eng. 32, 165–170 (2001)

    Google Scholar 

  • Bailey, R.W.: The utilization of creep test data in engineering design. Proc. Inst. Mech. Eng. 131, 131–349 (1935)

    Google Scholar 

  • Barbero, E.J., Lonetti, P.: Damage model for composites defined in terms of available data. Mech. Compos. Mater. Struct. 8, 299–315 (2001)

    Google Scholar 

  • Barbero, E.J., Tomblin, J.: A damage mechanics model for compression strength of composites. Int. J. Solids Struct. 33, 4379–4393 (1996)

    MATH  Google Scholar 

  • Behera, A., Thawre, M.M., Ballal, A.: Failure analysis of CFRP multidirectional laminates using the probabilistic Weibull distribution model under static loading. Fiber Polym. 20, 2390–2399 (2019)

    Google Scholar 

  • Berezvai, S., Kossa, A.: Performance of a parallel viscoelastic-viscoplastic model for a microcellular thermoplastic foam on wide temperature range. Polym. Test. 84, 106395 (2020)

    Google Scholar 

  • Betten, J.: Creep Mechanics. Springer, Berlin Heidelberg (2008)

    MATH  Google Scholar 

  • Chen, J.L., Sun, C.T.: Modeling creep behavior of fiber composites. J. Thermoplast. Compos. Mater. 2, 19–33 (1989)

    Google Scholar 

  • Christensen, R.M.: A probabilistic treatment of creep rupture behavior for polymers and other materials. Mech. Time-Depend. Mater. 8, 1–15 (2004)

    Google Scholar 

  • Christensen, R., Miyano, Y.: Stress intensity controlled kinetic crack growth and stress history dependent life prediction with statistical variability. Int. J. Fract. 137, 77–87 (2006)

    MATH  Google Scholar 

  • Christensen, R., Miyano, Y.: Deterministic and probabilistic lifetimes from kinetic crack growth – generalized forms. Int. J. Fract. 143, 35–39 (2007)

    MATH  Google Scholar 

  • Christensen, R., Miyano, Y., Nakada, M.: The size dependence of tensile strength for brittle isotropic materials and carbon fiber composite materials. Compos. Sci. Technol. 106, 9–14 (2015)

    Google Scholar 

  • Chung, I., Sun, C.T., Chang, I.Y.: Modeling creep in thermoplastic composites. J. Compos. Mater. 27, 1009–1029 (1993)

    Google Scholar 

  • Cui, H., Thomson, D., Pellegrino, A., Wiegand, J., Petrinic, N.: Effect of strain rate and fibre rotation on the in-plane shear response of ±45° laminates in tension and compression tests. Compos. Sci. Technol. 135, 106–115 (2016)

    Google Scholar 

  • Dillard, D.A., Morris, D.H., Brinson, H.F.: Creep and Creep Rupture of Laminated Graphite/Epoxy Composites. In, Blacksburg, VA, United States (1981). Virginia Polytechnic Inst. and State University

    Google Scholar 

  • Fallahi, H., Taheri-Behrooz, F., Asadi, A.: Nonlinear mechanical response of polymer matrix composites: a review. Polym. Rev. 60, 42–85 (2020)

    Google Scholar 

  • Faria, H., Guedes, R.M.: Long-term behaviour of GFRP pipes: reducing the prediction test duration. Polym. Test. 29, 337–345 (2010)

    Google Scholar 

  • Findley, W.N.: 26-Year creep and recovery of poly(vinyl chloride) and polyethylene. Polym. Eng. Sci. 27, 582–585 (1987)

    Google Scholar 

  • Findley, W.N., Lai, J.S., Onaran, K.: Creep and Relaxation of Nonlinear Viscoelastic Materials: With an Introduction to Linear Viscoelasticity. Dover, New York (1989)

    MATH  Google Scholar 

  • Fuller, J.D., Wisnom, M.R.: Exploration of the potential for pseudo-ductility in thin ply CFRP angle-ply laminates via an analytical method. Compos. Sci. Technol. 112, 8–15 (2015a)

    Google Scholar 

  • Fuller, J.D., Wisnom, M.R.: Pseudo-ductility and damage suppression in thin ply CFRP angle-ply laminates. Composites, Part A, Appl. Sci. Manuf. 69, 64–71 (2015b)

    Google Scholar 

  • Garber, D.P.: Tensile stress-strain behavior of graphite/epoxy laminates. In: NASA Contractor Report 3592 (1982). Langley Research Center

    Google Scholar 

  • Gates, T.S., Sun, C.T.: Elastic/viscoplastic constitutive model for fiber reinforced thermoplastic composites. AIAA J. 29, 457–463 (1991)

    Google Scholar 

  • Giannadakis, K., Varna, J.: Analysis of nonlinear shear stress-strain response of unidirectional GF/EP composite. Composites, Part A, Appl. Sci. Manuf. 62, 67–76 (2014)

    Google Scholar 

  • Giannadakis, K., Mannberg, P., Joffe, R., Varna, J.: The sources of inelastic behavior of Glass Fibre/Vinylester non-crimp fabric [±45]s laminates. J. Reinf. Plast. Compos. 30, 1015–1028 (2011)

    Google Scholar 

  • Guedes, R.M.: Lifetime predictions of polymer matrix composites under constant or monotonic load. Composites, Part A, Appl. Sci. Manuf. 37, 703–715 (2006)

    Google Scholar 

  • Guedes, R.M.: Creep and fatigue lifetime prediction of polymer matrix composites based on simple cumulative damage laws. Composites, Part A, Appl. Sci. Manuf. 39, 1716–1725 (2008)

    Google Scholar 

  • Guedes, R.M.: Relationship between lifetime under creep and constant stress rate for polymer-matrix composites. Compos. Sci. Technol. 69, 1200–1205 (2009a)

    Google Scholar 

  • Guedes, R.M.: Viscoplastic analysis of fiber reinforced polymer matrix composites under various loading conditions. Polym. Compos. 30, 1601–1610 (2009b)

    Google Scholar 

  • Guedes, R.M., Marques, A.T., Cardon, A.: Analytical and experimental evaluation of nonlinear viscoelastic-viscoplastic composite laminates under creep, creep-recovery, relaxation and ramp loading. Mech. Time-Depend. Mater. 2, 113–128 (1998)

    Google Scholar 

  • Herakovich, C.T., Schroedter Iii, R.D., Gasser, A., Guitard, L.: Damage evolution in [±45](s) laminates with fiber rotation. Compos. Sci. Technol. 60, 2781–2789 (2000)

    Google Scholar 

  • Hu, J., Chen, W., Li, Y., Qu, Y., Zhao, B., Yang, D.: Temperature-stress-time methodology for flat-patterning ETFE cushions in use for large-span building structures. Eng. Struct. 204, 109607 (2020)

    Google Scholar 

  • Huang, Y., Talreja, R.: Statistical analysis of oblique crack evolution in composite laminates. Composites, Part B, Eng. 65, 34–39 (2014)

    Google Scholar 

  • Kachanov, L.M.: Time of the rupture process under creep conditions. Izv. Akad. Nauk SSSR 8, 26–31 (1958)

    Google Scholar 

  • Kazanci, M., Schwartz, P., Phoenix, S.L.: The effect of matrix stiffness on the creep-rupture lifetime of carbon fiber/epoxy composites. Compos. Struct. 54, 221–223 (2001)

    Google Scholar 

  • Kibler, K.G.: Effects of temperature and moisture on the creep compliance of graphite-epoxy composites. In: 50th Meeting of the AGARD Structures and Materials Panel, 8.1-8.8, AGARD Conference Proceedings No. 288, 14–17 April 1980, Athens, Greece (1980a)

    Google Scholar 

  • Kibler, K.G.: Time-Dependent Environmental Behavior of Graphite/Epoxy Composites. Technical Report AFWAL-TR-80-4052. Materials laboratory, air force Wright Aeronautical Laboratories, Air Force Systems Command, Wright-Patterson Air Force Base, Ohio 45433 (1980b)

  • Kim, W., Sun, C.T.: Modeling relaxation of a polymeric composite during loading and unloading. J. Compos. Mater. 36, 745–755 (2002)

    Google Scholar 

  • Lafarie-Frenot, M.C., Touchard, F.: Comparative in-plane shear behaviour of long-carbon-fibre composites with thermoset or thermoplastic matrix. Compos. Sci. Technol. 52, 417–425 (1994)

    Google Scholar 

  • Lai, J., Bakker, A.: An integral constitutive equation for nonlinear plasto-viscoelastic behavior of high-density polyethylene. Polym. Eng. Sci. 35, 1339–1347 (1995)

    Google Scholar 

  • Lekou, D.J., Philippidis, T.P.: Mechanical property variability in FRP laminates and its effect on failure prediction. Composites, Part B, Eng. 39, 1247–1256 (2008)

    Google Scholar 

  • Lou, Y.C., Schapery, R.A.: Viscoelastic characterization of a nonlinear fiber-reinforced plastic. J. Compos. Mater. 5, 208–234 (1971)

    Google Scholar 

  • Mandel, U., Taubert, R., Hinterhölzl, R.: Mechanism based nonlinear constitutive model for composite laminates subjected to large deformations. Compos. Struct. 132, 98–108 (2015)

    Google Scholar 

  • Mandel, U., Taubert, R., Hinterhölzl, R.: Three-dimensional nonlinear constitutive model for composites. Compos. Struct. 142, 78–86 (2016)

    Google Scholar 

  • Mauget, B.R., Minnetyan, L., Chamis, C.C.: Large deformation nonlinear response of soft composite structures via laminate analogy. J. Adv. Mater. 33, 21–26 (2001)

    Google Scholar 

  • May, J.A.: Development of an Experimentally Validated Non-linear Viscoelastic Viscoplastic Model for a Novel Fuel Cell Membrane Material (2014). Virginia Tech.

    Google Scholar 

  • May, D.L., Gordon, A.P., Segletes, D.S.: The application of the Norton-Bailey law for creep prediction through power law regression. In: Proceedings of the ASME Turbo Expo (2013)

    Google Scholar 

  • May, J.A., Ellis, M.W., Dillard, D.A., Case, S.W., Moore, R.B., Li, Y., Lai, Y.H., Gittleman, C.A.: Development and validation of a uniaxial nonlinear viscoelastic viscoplastic stress model for a fuel cell membrane. J. Fuel Cell Sci. Technol. 12, 061011 (2015)

    Google Scholar 

  • Merry, S.M., Bray, J.D.: Time-dependent mechanical response of HDPE geomembranes. J. Geotech. Eng. 123, 57–65 (1997)

    Google Scholar 

  • Miyano, Y., Nakada, M., Kageta, S.: Statistical assessment of tensile static, creep and fatigue strengths for unidirectional CFRP. Exp. Mech. 61, 1171–1179 (2021)

    Google Scholar 

  • Naresh, K., Shankar, K., Velmurugan, R.: Reliability analysis of tensile strengths using Weibull distribution in glass/epoxy and carbon/epoxy composites. Composites, Part B, Eng. 133, 129–144 (2018)

    Google Scholar 

  • Norton, F.H.: The creep of steel at high temperatures (1929). McGraw-Hill book company, Inc.

    Google Scholar 

  • Paimushin, V.N., Kholmogorov, S.A.: Physical-mechanical properties of a fiber-reinforced composite based on an ELUR-P carbon tape and XT-118 binder. Mech. Compos. Mater. 54, 2–12 (2018)

    Google Scholar 

  • Paimushin, V.N., Kayumov, R.A., Kholmogorov, S.A.: Deformation features and models of [±45]2 s cross-ply fiber-reinforced plastics in tension. Mech. Compos. Mater. 55, 141–154 (2019)

    Google Scholar 

  • Pierik, E.R., Grouve, W.J.B., van Drongelen, M., Akkerman, R.: The influence of physical ageing on the in-plane shear creep compliance of 5HS C/PPS. Mech. Time-Depend. Mater. 24, 197–220 (2020)

    Google Scholar 

  • Robinson, D.N., Binienda, W.K., Ruggles, M.B.: Creep of polymer matrix composites I: Norton/Bailey creep law for transverse isotropy. J. Eng. Mech. 129, 310–317 (2003)

    Google Scholar 

  • Rosen, B.W.: A simple procedure for experimental determination of the longitudinal shear modulus of unidirectional composites. J. Compos. Mater. 6, 552–554 (1972)

    Google Scholar 

  • Schapery, R.A.: Nonlinear viscoelastic and viscoplastic constitutive equations based on thermodynamics. Mech. Time-Depend. Mater. 1, 209–240 (1997)

    Google Scholar 

  • Schuecker, C., Pettermann, H.E.: Combining elastic brittle damage with plasticity to model the non-linear behavior of fiber reinforced laminates. In: Computational Methods in Applied Sciences, pp. 99–117 (2008)

    Google Scholar 

  • Scott, D.W., Lai, J.S., Zureick, A.H.: Creep behavior of fiber-reinforced polymeric composites: a review of the technical literature. J. Reinf. Plast. Compos. 14, 588–617 (1995)

    Google Scholar 

  • Shokrieh, M.M., Omidi, M.J.: Investigation of strain rate effects on in-plane shear properties of glass/epoxy composites. Compos. Struct. 91, 95–102 (2009)

    Google Scholar 

  • Simulia, D.S.: Abaqus 2018 documentation. In: ABAQUS 2018 Documentation (2018)

    Google Scholar 

  • Singh, A., Mitchell, J.K.: General stress-strain-time function for soils. J. Soil Mech. Found. Div. 94, 21–46 (1968)

    Google Scholar 

  • Sket, F., Enfedaque, A., Alton, C., González, C., Molina-Aldareguia, J.M., Llorca, J.: Automatic quantification of matrix cracking and fiber rotation by X-ray computed tomography in shear-deformed carbon fiber-reinforced laminates. Compos. Sci. Technol. 90, 129–138 (2014)

    Google Scholar 

  • Sun, C.T., Chen, J.L.: A simple flow rule for characterizing nonlinear behavior of fiber composites. J. Compos. Mater. 23, 1009–1020 (1989)

    Google Scholar 

  • Sun, C.T., Zhu, C.: The effect of deformation-induced change of fiber orientation on the non-linear behavior of polymeric composite laminates. Compos. Sci. Technol. 60, 2337–2345 (2000)

    Google Scholar 

  • Tang, W., Li, M., Wen, M.P., Zhang, Q., Zhao, X.D.: PBX creep model based on modified time hardening theory. Chin. J. Energ. Mater. 16, 34–36 (2008)

    Google Scholar 

  • Thiruppukuzhi, S.V., Sun, C.T.: Testing and modeling high strain rate behavior of polymeric composites. Composites, Part B, Eng. 29, 535–546 (1998)

    Google Scholar 

  • Thiruppukuzhi, S.V., Sun, C.T.: Models for the strain-rate-dependent behavior of polymer composites. Compos. Sci. Technol. 61, 1–12 (2001)

    Google Scholar 

  • Tsai, J., Sun, C.T.: Constitutive model for high strain rate response of polymeric composites. Compos. Sci. Technol. 62, 1289–1297 (2002)

    Google Scholar 

  • Tsai, J.L., Sun, C.T.: Strain rate effect on in-plane shear strength of unidirectional polymeric composites. Compos. Sci. Technol. 65, 1941–1947 (2005)

    Google Scholar 

  • Tsugiyuki, O., Nakada, M., Miyano, Y.: Temperature dependence of longitudinal tensile strength in unidirectional carbon fiber-reinforced plastic. J. Compos. Mater. 48, 3569–3573 (2014)

    Google Scholar 

  • Tuttle, M.E., Brinson, H.F.: Prediction of the long-term creep compliance of general composite laminates. Exp. Mech. 26, 89–102 (1986)

    Google Scholar 

  • Tuttle, M.E., Pasricha, A., Emery, A.F.: The nonlinear viscoelastic-viscoplastic behavior of IM7/5260 composites subjected to cyclic loading. J. Compos. Mater. 29, 2025–2046 (1995)

    Google Scholar 

  • Weeks, C.A., Sun, C.T.: Modeling non-linear rate-dependent behavior in fiber-reinforced composites. Compos. Sci. Technol. 58, 603–611 (1998)

    Google Scholar 

  • Weibull, W.: A statistical distribution function of wide applicability. J. Appl. Mech. 18, 293–297 (1951)

    MATH  Google Scholar 

  • Weng, F., Fang, Y., Ren, M., Sun, J., Feng, L.: Effect of high strain rate on shear properties of carbon fiber reinforced composites. Compos. Sci. Technol. 203, 108599 (2021)

    Google Scholar 

  • Wisnom, M.R.: The effect of fibre rotation in ±45° tension tests on measured shear properties. Composites 26, 25–32 (1995)

    Google Scholar 

  • Wu, E.M., Moore, R.L., Nguyen, N.Q.: Matrix-dominated Time-dependent Deformation and Damage of Graphite/epoxy Composite Experimental Data under Ramp, AFWAL-TR-82-3076. Lawrence Livermore National Laboratory (1982)

  • Wu, E.M., Moore, R.L., Nguyen, N.Q.: Matrix-dominated Time-dependent Deformation and Damage of Graphite/epoxy Composite Experimental Data under Creep and Recovery, AFWAL-TR-83-3066. Lawrence Livermore National Laboratory (1983a)

  • Wu, E.M., Moore, R.L., Nguyen, N.Q.: Matrix-dominated Time-dependent Deformation and Damage of Graphite/epoxy Composite Experimental Data under Multiple-step Relaxation, AFWAL-TR-83-3056. Lawrence Livermore National Laboratory (1983b)

  • Xiao, Y., Lv, J., Ben Jar, P.Y.: A stress-relaxation approach to determine onset of delamination in angle ply laminates. J. Compos. Mater. 54, 2521–2527 (2020)

    Google Scholar 

  • Zapas, L.J., Crissman, J.M.: Creep and recovery behaviour of ultra-high molecular weight polyethylene in the region of small uniaxial deformations. Polymer 25, 57–62 (1984)

    Google Scholar 

Download references

Funding

Not applicable.

Author information

Authors and Affiliations

  1. LAETA, UMAI-INEGI, Department of Mechanical Engineering (DEMec), Faculty of Engineering of the University of Porto (FEUP), Rua Dr. Roberto Frias s/n, 4200-465, Porto, Portugal

    Rui Miranda Guedes

Authors
  1. Rui Miranda Guedes
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Rui Miranda Guedes.

Ethics declarations

Conflicts of interest/Competing interests

Not applicable.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Miranda Guedes, R. A practical model to predict the time-dependent behaviour of angle-ply laminates from limited creep data. Mech Time-Depend Mater 27, 1043–1067 (2023). https://doi.org/10.1007/s11043-022-09544-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11043-022-09544-1

Keywords