Fictitious crack length and effective process-zone size versus aggregate microstructure in size and boundary effect models for concrete fracture

Quasi-brittle fracture models used to extract the tensile strength $f_t$ and fracture toughness $K_c$ of concrete from laboratory fracture tests — Bažant's Type-2 size effect law (SEL) and the Hu–Guan boundary effect model (BEM) family — each depend on a length parameter that ties the fracture process zone to the concrete microstructure: the multiplier $\beta_1$ in the fictitious crack length $\Delta a_{fic}=\beta_1 d_{max}$ of the non-local Hu–Guan formulations, and Bažant's effective process-zone size $c_f$ (or the ratio $\gamma_1=c_f/l_{ch}$) of the SEL. A 2019 comparison of four such identification schemes (Gao et al., Applied Sciences) showed that the identified parameter pairs move jointly with the assumed value of these length parameters; it called for further study of $\beta_1$ and for an approximate relation between $c_f$ and the aggregate sizes $d_{max}$, $d_{av}$, noted that the non-local stress definition is restricted to a fictitious crack length shorter than the uncracked ligament, and reported that fracture data for fully graded dam concrete were then still being collected. Since then the large-aggregate dam-concrete experimental gap has been largely filled by the Baihetan arch-dam test campaigns (2022–2026) and grading-based rules for $\Delta a_{fic}$ have been developed within the BEM school, but no validated cross-mix law for $\beta_1$ or $c_f$ exists, the mapping $c_f(d_{max},d_{av})$ remains unpublished, and the boundary-effect-versus-size-effect controversy over the status of $\beta$ is unsettled. The problem — making the process-zone length parameters of size and boundary effect models predictable from aggregate statistics and delimiting their validity — remains open.

Background

Concrete fails in tension in a quasi-brittle way: a fracture process zone (FPZ) — the region ahead of a traction-free crack tip over which aggregates bridge the crack and transmit decreasing cohesive stress — develops over a length comparable to the aggregate scale before the crack becomes traction-free, so neither pure strength criteria nor linear elastic fracture mechanics (LEFM) alone describes failure of ordinary laboratory specimens. In the fictitious (cohesive) crack model of Hillerborg, Modéer and Petersson (1976), a softening traction–separation law acts across the FPZ, and the material is characterized by the tensile strength $f_t$ and the fracture energy, from which a characteristic length $l_{ch}=E\,G_f/f_t^2$ (with $E$ the elastic modulus and $G_f$ the fracture energy) can be formed. Bažant's Type-2 size effect law (SEL; Bažant 1984) describes how the nominal stress at failure $\sigma_n$ of a geometrically similar notched structure falls between the plasticity/strength limit $\sigma_n\propto D^0$ and the LEFM limit $\sigma_n\propto D^{-1/2}$ as the structure size $D$ grows, because the FPZ has a finite size $c_f$ — the effective (extrapolated) process-zone length — which is treated as a material constant in the size-effect method for measuring fracture energy (Bažant and Kazemi 1990). The boundary effect model (BEM) of Hu, Duan, Guan and co-workers (Duan, Hu and Wittmann 2006; Guan, Hu, Xie, Li and Wu 2018) attacks the same problem from the crack-length side: for a specimen of depth $W$ with initial crack $a_0$, it defines an equivalent crack length $a_e$ that interpolates between the strength-controlled limit of a very short crack and the LEFM-controlled limit of a long crack, so that a linear regression of $1/\sigma_n^2$ against $a_e$ yields $f_t$ from the intercept and the fracture toughness $K_c$ from the slope. The non-local Hu–Guan versions (so called because the nominal stress is evaluated on the reduced ligament $W-a_0-\Delta a_{fic}$ rather than from the elastic notch field alone) model the partially developed FPZ at peak load by a fictitious crack length $\Delta a_{fic}$ carrying a uniform bridging stress, with the empirical scaling $\Delta a_{fic}=\beta_1 d_{max}$ where $d_{max}$ is the maximum aggregate size and $\beta_1$ a dimensionless constant (commonly taken as 1, reported to be a good approximation when $d_{max}$ is the dominant aggregate size of the mix and $W/d_{max}$ is below 20 or even 50); the assumed stress profile is only admissible while $\Delta a_{fic}$ is shorter than the uncracked ligament $W-a_0$. Gao, Koval and Chazallon (2016, 2017) introduced a local-energetic size-and-boundary model (the Koval–Gao model) that avoids any assumed process-zone length by deriving the strength contribution from the derivative of the energy release rate, giving a purely geometry-dependent equivalent crack length. Gao et al. (2019, Applied Sciences 9(7):1337) then compared four linear identification schemes — the original and a modified Hu–Guan BEM, the Koval–Gao model, and a boundary-effect transform of Bažant's Type-2 SEL sharing the same equivalent crack length — on four published wedge-splitting (a stable fracture test in which a wedge assembly pushed into a groove splits a notched block) and compact-tension series with $d_{max}=10$–$20$ mm. They found that the Koval–Gao and transformed-Bažant schemes give mutually consistent $K_c$ (with $f_t$ from the latter requiring an assumed ratio $\gamma_1=c_f/l_{ch}$, taken as 0.28), while both Hu–Guan variants yield $(f_t,K_c)$ pairs that move jointly with the choice of $\beta_1$; they also showed numerically that the modified equivalent crack length increasingly departs from the geometry-only definition as $d_{max}$ grows toward dam-concrete scales (80–150 mm). Their paper leaves these issues unresolved: the value or rule for $\beta_1$ across concrete mixes and specimen geometries is unknown, and the paper states only that this issue "should be further studied" (§3.1.2), recommending meanwhile that one fracture parameter be fixed by tests or given a narrow range in advance so that the other, and hence a reasonable $\beta_1$, follows; the relation of Bažant's $c_f$ (or $\gamma_1$) to the aggregate microstructure ($d_{max}$, average aggregate size $d_{av}$) is not known, so that using the transformed Type-2 SEL to infer $f_t$ carries an uncontrolled uncertainty — the paper calls for future theoretical and experimental work on the approximate relation between $c_f$ and $d_{max}$ or $d_{av}$ (§2.3) and notes that $\gamma_1\approx0.28$ should be better verified by more experimental results as $d_{max}$ grows toward dam-concrete values (§3.1.1); the non-local Hu–Guan formulations are defined only while $\Delta a_{fic}$ is shorter than the ligament, a shortcoming that the paper says can be rarely encountered in the analysis of usual fracture test results (§2.4); and fracture data for fully graded dam concrete — mass concrete cast with its complete coarse-aggregate gradation, $d_{max}$ of 80–150 mm, unlike the wet-sieved laboratory concrete from which the largest aggregates are removed — were then unavailable, the paper reporting only a planned test program (three geometrically similar sizes plus one size with three notch ratios) with fully graded wedge-splitting tests "currently undergoing", leaving model recommendations provisional. Independently, Carloni, Cusatis, Salviato, Le, Hoover and Bažant (2019) criticized the BEM from the cohesive-crack/SEL standpoint, arguing that $\beta$ is an empirical fitting parameter with no known size-dependence law and that the assumed ligament stress profile conflicts with cohesive-zone simulations and digital image correlation (full-field optical strain) measurements — a controversy that frames how any future rule for these length parameters must be validated.

Problem Statement

Establish, theoretically and/or experimentally, how the free length parameters of concrete size- and boundary-effect fracture models depend on the aggregate microstructure, and over what range the resulting extrapolations are valid, so that these models yield unique and physically consistent values of the tensile strength $f_t$ and fracture toughness $K_c$ — including for fully graded dam concretes with $d_{max}$ up to 150 mm. Specifically:

(1) $\beta_1$ rule. Determine the value of, or a predictive rule for, the multiplier $\beta_1$ in the fictitious crack length $\Delta a_{fic}=\beta_1 d_{max}$ used by the non-local Hu–Guan boundary effect models (original and modified), together with the range of concrete mixes (aggregate gradation, average and maximum aggregate size, mix proportions, strength) and specimen geometries over which it holds, such that the $(f_t,K_c)$ pair identified from the corresponding linear regression is unique and physically plausible without independent re-calibration for every dataset. The regression is the variant-specific linear form of Gao et al. (2019, §2.1–2.2), of the general shape $1/\sigma_n^2=1/f_t^2+(1.12^2\pi/K_c^2)a_e$ (where the factor 1.12 is the standard free-surface correction for an edge crack): for the original Hu–Guan model it is $1/\sigma_{n2}^2$ against the geometry-only equivalent crack length $a_{e1}$, so that $\beta_1$ enters only through the non-local stress $\sigma_{n2}$ evaluated on the reduced ligament $W-a_0-\Delta a_{fic}$, whereas for the modified model it is $1/\sigma_{n2}^2$ against $a_{e2}$, in which $\beta_1$ enters both that stress and the equivalent crack length.

(2) $c_f$ mapping. Determine an approximate mapping $c_f(d_{max},d_{av})$ — equivalently a rule for the ratio $\gamma_1=c_f/l_{ch}$ — for Bažant's effective process-zone size, so that the boundary-effect transform of the Type-2 size effect law can deliver $f_t$ without assuming a fixed ratio (such as 0.28).

References

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