Skip to main content
Log in

Size effect on double-K fracture parameters of concrete based on fracture extreme theory

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

Based on fracture extreme theory (FET), the size effect on initial fracture toughness \(K_{\mathrm{I}}^{\mathrm{ini}}\) and unstable fracture toughness \(K_{\mathrm{I}}^{\mathrm{un}}\) of concrete for three-point bending beam was investigated. Nine groups of geometrically similar specimen were simulated to obtain peak load and critical crack mouth opening displacement, of which specimen depth was from 200 to 1000 mm and initial crack length-to-depth ratios were from 0.1 to 0.6. The \(K_{\mathrm{I}}^{\mathrm{ini}}\) and \(K_{\mathrm{I}}^{\mathrm{un}}\) were calculated by FET and double-K method, in which FET adopted the linear, bilinear, and trilinear cohesive stress distribution assumptions and double-K method only used the linear cohesive stress distribution assumption. With linear cohesive stress distribution assumption, \(K_{\mathrm{I}}^{\mathrm{ini}}\) and \(K_{\mathrm{I}}^{\mathrm{un}}\) determined by FET and double- K method were compared. Then, the influence of specimen depth on \(K_{\mathrm{I}}^{\mathrm{ini}}\) and \(K_{\mathrm{I}}^{\mathrm{un}}\) was discussed. In addition, \(K_{\mathrm{I}}^{\mathrm{ini}}/K_{\mathrm{I}}^{\mathrm{un}}\) calculated via FET using different cohesive stress distribution assumptions were analyzed.

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

Access this article

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

Similar content being viewed by others

References

  1. Bažant, Z.P., Planas, J.: Fracture and Size Effect in Concrete and Other Quasibrittle Materials. CRC Press, Boca Raton (1998)

    Google Scholar 

  2. Hillerborg, A., Modeer, M., Petersson, P.E.: Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem. Concr. Res. 6(6), 773–81 (1976)

    Article  Google Scholar 

  3. Bažant, Z.P., Oh, B.H.: Crack band theory for fracture of concrete. Mater. Struct. 16(3), 155–77 (1983)

    Google Scholar 

  4. Jenq, Y.S., Shah, S.P.: Two-parameter fracture model for concrete. J. Eng. Mech. 111(10), 1227–41 (1985)

    Article  Google Scholar 

  5. Bažant, Z.P.: Size effect in blunt fracture: concrete, rock, metal. J. Eng. Mech. 110(4), 518–35 (1984)

    Article  Google Scholar 

  6. Swartz, S.E., Go, C.G.: Validity of compliance calibration to cracked concrete beams in bending. Exp. Mech. 24(2), 129–34 (1984)

    Article  Google Scholar 

  7. Karihaloo, B.L., Nallathambi, P.: Effective crack model for the determination of fracture toughness \(K_{{\rm IC}}^{{\rm s}}\) of concrete. Eng. Fract. Mech. 35(4/5), 637–45 (1990)

    Google Scholar 

  8. Xu, S.L., Reinhardt, H.W.: Determination of double-\(K\) criterion for crack propagation in quasi-brittle materials, part I: experimental investigation of crack propagation. Int. J. Fract. 98, 111–49 (1999)

    Google Scholar 

  9. Xu, S.L., Reinhardt, H.W.: A simplified method for determining double-\(K\) fracture parameters for three-point bending tests. Int. J. Fract. 104(2), 181–209 (2000)

    Google Scholar 

  10. Kumar, S., Barai, S.V.: Determining double-\(K\) fracture parameters of concrete for compact tension and wedge splitting tests using weight function. Eng. Fract. Mech. 76(7), 935–48 (2009)

    Google Scholar 

  11. Kumar, S., Barai, S.V.: Determining the double-\(K\) fracture parameters for three-point bending notched concrete beams using weight function. Fatigue Fract. Eng. Mater. Struct. 33, 645–60 (2010)

    Google Scholar 

  12. Qing, L.B., Li, Q.B.: A theoretical method for determining initiation toughness based on experimental peak load. Eng. Fract. Mech. 99(1), 295–305 (2013)

    Article  Google Scholar 

  13. Qing, L.B., Nie, Y.T., Wang, J., et al.: A simplified extreme method for determining double-\(K\) fracture parameters of concrete using experimental peak load. Fatigue Fract. Eng. Mater. Struct. 40(2), 254–66 (2017)

    Google Scholar 

  14. Qing, L.B., Dong, M.W., Guan, J.F.: Determining initial fracture toughness of concrete for split-tension specimens based on the extreme theory. Eng. Fract. Mech. 189, 427–38 (2018)

    Article  Google Scholar 

  15. Alexander, M.G., Blight, G.E.: A comparative study of fracture parameters in notched concrete beams. Mag. Concr. Res. 40(142), 50–8 (1988)

    Article  Google Scholar 

  16. Issa Mohsen, A., Issa Mahmoud, A., Islam Mohammad, S., et al.: Size effect in concrete fracture, part II: analysis of test results. Int. J. Fract. 102, 25–42 (2000)

    Article  Google Scholar 

  17. Nallthambi, P., Karihaloo, B., Heaton, B.: Effect of specimen and crack sizes, water/cement ratio and coarse aggregate texture upon fracture toughness of concrete. Mag. Concr. Res. 36(129), 227–36 (1984)

    Article  Google Scholar 

  18. Perdikaris, P., Calomino, A., Chudnovsky, A.: Effect of fatigue on fracture toughness of concrete. J. Eng. Mech. 112(8), 776–91 (1986)

    Article  Google Scholar 

  19. Kumar, S., Barai, S.V.: Size-effect prediction from the double-\(K\) fracture model for notched concrete beam. Int. J. Damage Mech. 19(4), 473–97 (2010)

    Google Scholar 

  20. Planas, J., Elices, M.: Fracture criteria for concrete: mathematical approximations and experimental validation. Eng. Fract. Mech. 35(1), 87–94 (1990)

    Article  Google Scholar 

  21. Choubey, R.K., Kumar, S., Rao, M.C.: Numerical evaluation of simplified extreme peak load method for determining the double-\(K\) fracture parameters of concrete. Int. J. Eng. Technol. 9(3), 2097–110 (2017)

    Google Scholar 

  22. Reinhardt, H.W., Cornelissen, H.A.W., Hordijk, D.A.: Tensile tests and failure analysis of concrete. J. Struct. Eng. ASCE 112, 2462–77 (1986)

    Article  Google Scholar 

  23. Tada, H., Paris, P.C., Irwin, G.R.: The Stress Analysis of Cracks Handbook. Paris Productions Incorporated, St. Louis (1973)

    Google Scholar 

  24. Dong, W., Wu, Z.M., Zhou, X.M.: Calculating crack extension resistance of concrete based on a new crack propagation criterion. Constr. Build. Mater. 38, 879–89 (2013)

    Article  Google Scholar 

  25. Qing, L.B., Shi, X.Y., Mu, R., et al.: Determining tensile strength of concrete based on experimental loads in fracture test. Eng. Fract. Mech. 202, 87–102 (2018)

    Article  Google Scholar 

  26. Refai, T.M.E., Swartz, S.E.: Fracture behavior of concrete beams in three–point bending considering the influence of size effects. Engineering Experiment Station, Kansas State University. Report No. 190 (1987)

  27. Yon, J.H., Hawkins, N.M., Kobayashi, A.S.: S-FPZ model for concrete SEN specimen. In: Bazant, Z.P. (ed.) Fracture Mechanics of Concrete Structure, pp. 208–13. Elsevier, London (1992)

    Google Scholar 

  28. Li, Y., Qing, L.B., Cheng, Y.H., et al.: A general framework for determining fracture parameters of concrete based on fracture extreme theory. Theor. Appl. Fract. Mech. 103, 102259 (2019)

    Article  Google Scholar 

Download references

Acknowledgements

The work presented in the paper was supported by the National Natural Science Foundation of China (No. 51779069); Key Project of University Science and Technology Research of Hebei Province (Nos. ZD2019072, ZD2020190); Hebei Province Graduate Innovation Funding Project (No. CXZZSS2019028).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Longbang Qing.

Additional information

Publisher's Note

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

Appendix

Appendix

The \(g\prime (a)\) and \(k\prime (\alpha )\) in Eq. (22) are shown as follows:

$$\begin{aligned} k^{'}\left( \alpha \right)= & {} \frac{\text{1 }}{D(1+2\alpha ^{2})(1-\alpha )^{3}} \times \left\{ {\left( {-2.15+12.16\alpha -19.89\alpha ^{2} +10.8\alpha ^{3}} \right) } \right. \times \left( {1+2\alpha } \right) \left( {1-\alpha }\right) ^{3/2} \nonumber \\&-\left( {1.99-2.15\alpha +6.08\alpha ^{2}-6.63\alpha ^{3}+2.7\alpha ^{4}}\right) \left. {\times \left[ {2(1-\alpha )^{3/2}-\frac{3}{2}(1+2\alpha ) (1-\alpha )^{1/2}} \right] } \right\} \qquad \end{aligned}$$
(A1)
$$\begin{aligned} {g}'(a)= & {} {g}'_{1} (a)+{g}'_{2} (a)-{g}'_{3} (a) \end{aligned}$$
(A2)

where

$$\begin{aligned} {g}'_{{i}} (a)= & {} \left( {A_{1}^{{i}}+{A_{1}^{{i}}}'a} \right) \left( {2s_{{i}}^{\mathrm{1/2}} +M_{1} s_{{i}} +\frac{2}{3}M_{2} s_{{i}}^{3/2} +\frac{1}{2}M_{3} s_{{i}}^{2} } \right) +A_{1}^{{i}}a\left[ {s_{{i}}^{\mathrm{-1/2}} s_{{i}}'+M_{1} s_{{i}} '+M_{2}'s_{{i}} +M_{3} s_{{i}} ^{1/2}s_{{i}}'} \right] \nonumber \\&+\,A_{1}^{{i}}a\left[ {\frac{2}{3}M_{2}'s_{{i}} ^{3/2}+M_{3} s_{{i}} s_{{i}}'+\frac{1}{2}M_{3}' s_{{i}}^{2}} \right] +A_{2}^{{i}}a^{2}\left[ {2s_{{i}} ^{1/2}s_{{i}}'+M_{1} s_{{i}} s_{{i}}' +\frac{M_{1}'}{2}s_{{i}}^{2}+\frac{2}{3}M_{2} s_{{i}}^{\mathrm{3/2}} s_{{i}}'} \right] \nonumber \\&+\,A_{2}^{{i}}a^{2}\left[ {\frac{4}{15}M_{2}' s_{{i}}^{\mathrm{5/2}} +\frac{M_{3}'}{6} \left[ {1-\left( {\frac{a_{{i}} }{a}} \right) ^{3}-3s_{{i}} \frac{a_{{i}} }{a}} \right] } \right] +A_{2}^{{i}} a^{2}f_{{i}}(a) \nonumber \\&+\,\left( {2A_{2}^{{i}}a+A_{2}'a^{2}} \right) \times \left[ {\frac{4}{3}s_{{i}}^{\mathrm{3/2}} +\frac{M_{1}}{2}s_{{i}}^{2} +\frac{4}{15}M_{2} s_{{i}}^{\mathrm{5/2}} +\frac{M_{3} }{6}\left\{ {1-\left( {\frac{a_{{i}} }{a}} \right) ^{3}-3s_{1} \frac{a_{{i}} }{a }} \right\} } \right] \end{aligned}$$
(A3)

when \(i =1\), 3, \(a_{{i}} = a_{\mathrm{s}}\); when \(i =2\), \(a_{{i}} = a_{0}\).

$$\begin{aligned} f_{1,3} (a)= & {} \frac{M_{3} }{2}\left( {-\left( {\frac{a_{\mathrm{s}} }{a}} \right) ^{2}\frac{a-2a_{\mathrm{s}} }{2a^{2}}-s_{{i}}' \frac{a_{\mathrm{s}} }{a}-s_{{i}} \frac{a-2a_{\mathrm{s}} }{2a^{2}}}\right) \end{aligned}$$
(A4)
$$\begin{aligned} f_{2} (a)= & {} \frac{M_{3} }{2}\left( {\frac{a_{0}^{3}}{a^{4} }-s_{{i}} '\frac{a_{0} }{a}+s_{{i}} \frac{a_{0} }{a^{2}}} \right) \end{aligned}$$
(A5)

where

$$\begin{aligned}&{s}'_{1,3} =-\frac{a-2a_{\mathrm{s}} }{2a^{2}} \end{aligned}$$
(A6)
$$\begin{aligned}&{s}'_{2} =\frac{a_{0} }{a^{2}} \end{aligned}$$
(A7)
$$\begin{aligned}&{A_{1}^{1}}'=\frac{\partial \sigma _{\mathrm{s}} \left( {\omega _{\mathrm{s}} } \right) }{\partial a}=\frac{\partial \sigma _{\mathrm{s}} \left( {\omega _{\mathrm{s}} } \right) }{\partial \omega _{\mathrm{s}} } \frac{\partial \omega _{\mathrm{s}} }{\partial a} \end{aligned}$$
(A8)
$$\begin{aligned}&{A_{2}^{1}}'=-\frac{{A_{1}^{2}}' \left( {a-a_{\mathrm{s}} } \right) +\left( {f_{\mathrm{t}} -A_{1}^{2} } \right) /2}{\left( {a-a_{\mathrm{s}} } \right) ^{2}} \end{aligned}$$
(A9)
$$\begin{aligned}&{A_{1}^{2}}'=\frac{\partial \sigma _{\mathrm{s}} \left( {\hbox {CTOD}} \right) }{\partial a}=\frac{\partial \sigma _{\mathrm{s}} \left( {\hbox {CTOD}} \right) }{\partial \hbox {CTOD}}\frac{\partial \hbox {CTOD}}{\partial a} \end{aligned}$$
(A10)
$$\begin{aligned}&{A_{2}^{2}}'=\frac{\left( {{\sigma }'_{\mathrm{s}} (\omega _{\mathrm{t}} ) -{A_{1}^{2}}'} \right) \left( {a-a_{0} } \right) -\left( {\sigma _{\mathrm{s}} (\omega _{\mathrm{t}} )-A_{1}^{2} } \right) }{\left( {a-a_{0} } \right) ^{2}} \end{aligned}$$
(A11)
$$\begin{aligned}&{A_{1}^{3}}'= {A_{1}^{1}}' \end{aligned}$$
(A12)
$$\begin{aligned}&{A_{2}^{3}}'=\frac{\left( {{\sigma }'_{\mathrm{s}} (\omega _{\mathrm{t}}) -{A_{1}^{1}}'} \right) \left( {a-a_{\mathrm{s}} }\right) -\left( {\sigma _{\mathrm{s}} (\omega _{\mathrm{t}}) -A_{1}^{1} } \right) /2}{\left( {a-a_{\mathrm{s}} }\right) ^{2}} \end{aligned}$$
(A13)

where

$$\begin{aligned}&\frac{\partial \sigma _{\mathrm{s}} \left( \omega \right) }{\partial \omega } =f_{\mathrm{t}} \left[ {\exp \left( {-\frac{c_{2} \omega }{w_{0} }} \right) } \right. \left. {\left[ {\frac{3c_{1} }{w_{0} } \left( {\frac{c_{1} \omega }{w_{0} }} \right) ^{2} -\frac{c_{2} }{w_{0} }\left( {1+\left( {\frac{c_{1} \omega }{w_{0} }} \right) ^{3}} \right) } \right] -\frac{1}{w_{0} } \left( {1+c_{1}^{3} } \right) \exp \left( {-c_{2} } \right) } \right] \qquad \end{aligned}$$
(A14)
$$\begin{aligned}&\frac{\partial \hbox {CTOD}}{\partial a}=\left\{ {\frac{6PS}{BD^{2}E}} \right. \times \left. {\left[ {0.76-4.56\alpha +11.61\alpha ^{2}-8.16\alpha ^{3} +\frac{0.66}{\left( {1-\alpha } \right) ^{2}}+\frac{1.32\alpha }{\left( {1-\alpha } \right) ^{3}}} \right] } \right\} \nonumber \\&\qquad \qquad \qquad \times \, \left\{ {\left( {1-\frac{a_{0} }{a}} \right) ^{2} +\left( {1.081-1.149\alpha } \right) \left[ {\frac{a_{0} }{a} -\left( {\frac{a_{0}}{a}} \right) ^{2}} \right] } \right\} ^{1/2} \nonumber \\&\qquad \qquad \qquad +\,\frac{3PSa}{BD^{2}E}\times \left( {0.76-2.28\alpha +3.87\alpha ^{2} -2.04\alpha ^{3}+\frac{0.66}{\left( {1-\alpha } \right) ^{2}}} \right) \nonumber \\&\qquad \qquad \qquad \times \, \left\{ {\left( {1-\frac{a_{0} }{a}} \right) ^{2} +\left( {1.081-1.149\alpha } \right) \left[ {\frac{a_{0} }{a} -\left( {\frac{a_{0}}{a}} \right) ^{2}} \right] } \right\} ^{-1/2} \nonumber \\&\qquad \qquad \qquad \times \, \left\{ {2\left( {1-\frac{a_{0} }{a}} \right) \frac{a_{0}}{a^{2}} -\frac{1.149}{D}\left[ {\frac{a_{0} }{a}-\left( {\frac{a_{0} }{a}} \right) ^{2}} \right] } \right. -\left. {\left( {1.081-1.149\alpha } \right) \left( {\frac{a_{0} }{a^{2}}-2\frac{a_{0}^{2} }{a^{3}}} \right) }\right\} \end{aligned}$$
(A15)
$$\begin{aligned}&\frac{\partial \omega _{\mathrm{s}} }{\partial a} =\left\{ {\frac{6PS}{BD^{2}E}} \right. \times \left. {\left[ {0.76-4.56\alpha +11.61\alpha ^{2}-8.16\alpha ^{3}\text{+ }\frac{0.66}{\left( {1-\alpha } \right) ^{2}}+\frac{1.32\alpha }{\left( {1-\alpha } \right) ^{3}}} \right] }\right\} \nonumber \\&\qquad \quad \times \, \left\{ {\left( {1-\frac{a_{\mathrm{s}} }{a}} \right) ^{2} +\left( {1.081-1.149\alpha } \right) \left( {\frac{a_{\mathrm{s}} }{a} -\left( {\frac{a_{\mathrm{s}} }{a}} \right) ^{2}} \right) } \right\} ^{\frac{1}{2}} \nonumber \\&\qquad \quad +\,\frac{3PSa}{BD^{2}E}\times \left( {0.76-2.28\alpha +3.87\alpha ^{2} -2.04\alpha ^{3}\text{+ }\frac{0.66}{\left( {1-\alpha } \right) ^{2}}}\right) \nonumber \\&\qquad \quad \times \, \left( {\left( {1-\frac{a_{\mathrm{s}} }{a}} \right) ^{2}}\right. +\left( {1.081-1.149\alpha } \right) \left( {\frac{a_{\mathrm{s}}}{a} -\left( {\frac{a_{\mathrm{s}} }{a}} \right) ^{2}} \right) ^{-\frac{1}{2}} \nonumber \\&\qquad \quad \times \, \left\{ {2\left( {1-\frac{a_{\mathrm{s}} }{a}} \right) \frac{a_{0}}{2a^{2}}} \right. +\left( {-\frac{1.149}{D}} \right) \left[ {\frac{a_{\mathrm{s}} }{a}-\left( {\frac{a_{\mathrm{s}} }{a}} \right) ^{2}}\right] -\left. {\left( {1.081-1.149\alpha } \right) \left[ {\frac{a_{0}}{2a^{2}} -\frac{a_{0} a_{\mathrm{s}} }{a^{3}}} \right] } \right\} \qquad \quad \end{aligned}$$
(A16)
$$\begin{aligned}&{\sigma }'_{\mathrm{s}} (\omega _{\mathrm{t}})= 2{A_{1}^{1}}'- {A_{3}^{1}}' \end{aligned}$$
(A17)

The partial derivatives of \(M_{{j}}\) to a are expressed as follows:

when \(j = 1\) or 3,

$$\begin{aligned} M_{{j}}'= & {} \frac{1}{\left( {1-a/D} \right) ^{3/2}}\times \left[ {\frac{b_{{j}} }{D}+2c_{{j}} \frac{a}{D^{2}}+3d_{{j}} \frac{a^{2}}{D^{3}}+4e_{{j}} \frac{a^{3}}{D^{4}}+5f_{{j}} \frac{a^{4}}{D^{5}}} \right] \nonumber \\&+\,\frac{3}{2D}\left( {1-\frac{a}{D}} \right) ^{-5/2}\times \left[ {a_{{j}} +b_{{j}} \frac{a}{D}+c_{{j}} \left( {\frac{a}{D}} \right) ^{2}+d_{{j}} \left( {\frac{a}{D}} \right) ^{3}+e_{{j}} \left( {\frac{a}{D}} \right) ^{4}+f_{{j}} \left( {\frac{a}{D}}\right) ^{5}} \right] \end{aligned}$$
(A18)

when \(j \quad =\) 2,

$$\begin{aligned} M_{{j}}'=\frac{b_{{j}} }{D} \end{aligned}$$
(A19)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qing, L., Su, Y., Dong, M. et al. Size effect on double-K fracture parameters of concrete based on fracture extreme theory. Arch Appl Mech 91, 427–442 (2021). https://doi.org/10.1007/s00419-020-01781-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-020-01781-5

Keywords

Navigation