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.
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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).
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Appendix
Appendix
The \(g\prime (a)\) and \(k\prime (\alpha )\) in Eq. (22) are shown as follows:
where
when \(i =1\), 3, \(a_{{i}} = a_{\mathrm{s}}\); when \(i =2\), \(a_{{i}} = a_{0}\).
where
where
The partial derivatives of \(M_{{j}}\) to a are expressed as follows:
when \(j = 1\) or 3,
when \(j \quad =\) 2,
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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
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DOI: https://doi.org/10.1007/s00419-020-01781-5