Generalization and fracture-mechanics grounding of the energy-based whole-process brittleness index for high-strength concrete under confinement

Zhou et al. (2020) proposed an energy-based brittleness index $B=B_{\mathrm{pre}}B_{\mathrm{post}}$ for high-strength concrete, built from the partition of input work into elastic strain energy and dissipative energy over the complete triaxial stress–strain curve, and fitted exponential laws $B(\sigma_3)$ for C60 and C70 concrete at confining pressures of 0–20 MPa. The index was validated only on those two strength grades, one mix design family, one specimen geometry, five discrete confining pressures, and quasi-static monotonic loading. The open problem is to establish the validity regime of this index as a material characterization: whether $B$ and its confining-pressure dependence generalize across strength grades, mixes, specimen sizes and shapes, wider confinement ranges, and other loading paths; how sensitive the energy partitions and $B$ are to the construction assumptions (no heat exchange, initial elastic modulus, constant Poisson's ratio) and to neglected rate, temperature, and unloading-nonlinearity effects; whether the fitted exponential forms have predictive validity outside the fitted range; and how $B$ scales with specimen size and relates to fracture-process-zone and size-effect-law parameters.

Background

In a triaxial compression test, a cylindrical concrete specimen is loaded axially while a constant confining pressure $\sigma_3$ acts on its lateral surface; this is the standard experiment for simulating deep underground structural service conditions, such as mine shaft linings. High-strength concrete (grades C60 and C70, nominal cubic compressive strength of 60 and 70 MPa) fails in a quasi-brittle manner: it is stiffer and more brittle than ordinary concrete, and its failure drops are strongly suppressed by confinement.

A common way to quantify brittleness is energy bookkeeping. Assuming no heat exchange between the specimen and its environment during a slow test, the work $W$ input by the testing machine equals the recoverable elastic strain energy $W_E$ plus the dissipative energy $W_D$ spent on cracking, friction, and plasticity (first law of thermodynamics). $W_E$ is computed from elasticity theory using the initial elastic modulus $E$ and Poisson's ratio $\mu$; both axial and radial strains must be measured. Existing brittleness indices before 2020 mostly used only pre-peak information (e.g., ratios of elastic to total energy at peak) or strength ratios, and therefore did not describe the whole failure process including the post-peak stress drop.

Zhou et al. (2020, Crystals 10:1099) proposed a whole-process index. They define a pre-peak brittleness index $B_{\mathrm{pre}}=W_E(B)/(W_E(B)+W_D)$, where $W_E(B)$ denotes the elastic strain energy stored at the peak point and $W_D$ the dissipative energy accumulated before the peak, so that $B_{\mathrm{pre}}$ is the fraction of pre-peak input energy still stored elastically at the peak, and a post-peak index $B_{\mathrm{post}}$ comparing the residual elastic energy released, the fracture energy, and the additional energy $W_{F(\mathrm{post})}$ that the testing machine must supply after the peak to drive the specimen to failure; both lie in (0,1), with 1 ideally brittle and 0 ideally plastic. The overall index is the product $B=B_{\mathrm{pre}}B_{\mathrm{post}}$. In their tests on C60 and C70 mixes (basalt gravel aggregate; mix proportions set by an orthogonal, i.e. fractional-factorial, experimental design) at $\sigma_3=0,5,10,15,20$ MPa, $B$ decreased monotonically with confinement (C60: 0.329 at 0 MPa to 0.005 at 20 MPa; C70: 0.402 to 0.006), confirming that confinement suppresses brittleness, and $B$ was larger for C70 than C60 at equal confinement. They fitted, from their own dataset, exponential laws $B=0.008+0.32\,e^{-\sigma_3/2.36}$ for C60 ($R^2=0.999$) and $B=0.008+0.39\,e^{-\sigma_3/2.81}$ for C70. The paper notes that brittleness performance is affected by specimen shape, size effect, and loading conditions, but does not analyze any of these factors.

The resulting open question is the standard one for a new characterization method: is $B$ a robust material descriptor, or a construct of the particular mixes, specimen geometry, confinement levels, and loading protocol used to calibrate it? The energy partition itself rests on assumptions — no heat exchange, a constant initial elastic modulus, and a single Poisson's ratio in the elastic-energy expressions — whose consequences under inelastic stiffness evolution, non-proportional unloading, temperature, or strain-rate effects were not examined. Quasi-brittle materials are also known to exhibit size effects governed by the fracture process zone (the damaged region ahead of a crack tip in which stress is transferred through aggregate interlock and bridging), described for instance by Bažant's size-effect law, an empirical–theoretical relation for how the nominal strength of a quasi-brittle structure decreases as the structure size increases, bridging the plastic-strength and linear-elastic-fracture-mechanics limits; no relation between the energy-based index $B$ and such fracture-mechanics length scales has been established.

Problem Statement

Establish the validity regime of the energy-based whole-process brittleness index $B=B_{\mathrm{pre}}B_{\mathrm{post}}$ for high-strength concrete under confinement, as defined by Zhou et al. (2020). The overarching objective is to determine whether $B$ is a robust material characterization rather than a dataset-specific construct. Address each of the following principal questions:

1. Generalization tests. Determine whether the monotone decrease of $B$ with confining pressure $\sigma_3$, the ranking of $B$ across strength grades, and the overall behavior of the index persist beyond the validated regime — for other high-strength grades and mix designs, other specimen sizes and shapes, confining pressures beyond 20 MPa, and loading paths other than quasi-static monotonic compression (e.g., cyclic loading, unloading paths, dynamic loading).
2. Sensitivity of the construction. Quantify how the computed energy partitions and $B$ are affected by the index's assumptions — negligible heat exchange, use of the initial elastic modulus, and a single constant Poisson's ratio — and by the neglected effects of inelastic stiffness evolution, nonlinearity of the unloading modulus, temperature, and strain rate.
3. Predictive validity of the fitted laws. Test whether the fitted exponential forms $B(\sigma_3)$ (for C60, $B=0.008+0.32\,e^{-\sigma_3/2.36}$; for C70, $B=0.008+0.39\,e^{-\sigma_3/2.81}$) predict $B$ on conditions not used in the fit (other confinement levels within and beyond 0–20 MPa), and establish their sensitivity to measurement-protocol variations such as displacement-control rate, preload, and axial/radial strain measurement fidelity.
4. Size effect and fracture-mechanics link. Determine how $B$ scales with specimen size, and relate (or fail to relate) the index to fracture-mechanics descriptors — the fracture process zone size and size-effect-law parameters — so that energy-based brittleness and quasi-brittle fracture mechanics describe a consistent material property.

The grades, mixes, specimen sizes and shapes, confining pressures, and loading paths listed under Question 1, together with the assumptions and neglected effects listed under Question 2, define the scope of the validity regime to be assessed rather than a set of conditions each requiring separate treatment.

References

1. Ruihe Zhou, Hua Cheng, Mingjing Li, Liangliang Zhang, Rongbao Hong (2020). Energy Evolution Analysis and Brittleness Evaluation of High-Strength Concrete Considering the Whole Failure Process. Crystals 10(12):1099. DOI: 10.3390/cryst10121099. https://doi.org/10.3390/cryst10121099
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