Microstructural $$\eta$$: chemical–mechanical bridge for anchorage design in alkali–activated concretes

Abstract The anchor design for Low-Carbon Concretes still lacks parameters that link microchemistry to structural performance. This work introduces a microstructural coefficient $$\eta$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>η</mml:mi> </mml:math> , defined from gel density and interfacial porosity of the N-A - S-H / C - A - S-H networks, to extend the classical Tepfers. The analytical derivation shows that $$\eta$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>η</mml:mi> </mml:math> multiplies the circumferential stress solution without altering its geometric scale, predicting a linear increase in initial stiffness and anchor energy. Three-dimensional bar–matrix contact simulations ( FEniCS ) for $$\eta =1.0-1.6$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>η</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1.0</mml:mn> <mml:mo>-</mml:mo> <mml:mn>1.6</mml:mn> </mml:mrow> </mml:math> and two cover ratios $$\lambda = d/c = 0.20,\;0.14$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>d</mml:mi> <mml:mo>/</mml:mo> <mml:mi>c</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.20</mml:mn> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mn>0.14</mml:mn> </mml:mrow> </mml:math> confirmed this prediction within 3 % for peak hoop stress and 2 % for secant stiffness. A $$2\times 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>×</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> factorial pull-out program on fly-ash concretes (two covers, three Si / Al ratios) experimentally validated the proportionality $$K_{\text {exp}} = \eta K_{0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mtext>exp</mml:mtext> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>η</mml:mi> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> experimentally; two-way ANOVA ranked gel chemistry as the dominant factor, explaining up to 93 % of stiffness variance at 28 days, while residual errors remained negligible. The metric sweeps of the fracture number $$\kappa$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>κ</mml:mi> </mml:math> and the friction number $$\textrm{Re}_{\mu }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>Re</mml:mtext> <mml:mi>μ</mml:mi> </mml:msub> </mml:math> confirmed that $$\eta$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>η</mml:mi> </mml:math> governs the stiffness, $$\kappa$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>κ</mml:mi> </mml:math> controls the post-peak toughness and $$\textrm{Re}_{\mu }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>Re</mml:mtext> <mml:mi>μ</mml:mi> </mml:msub> </mml:math> scales the residual plateau, delineating a domain of validity of $$\eta \le 1.6$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>η</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>1.6</mml:mn> </mml:mrow> </mml:math> , $$\kappa \ge 0.004$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>κ</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0.004</mml:mn> </mml:mrow> </mml:math> , $$0.12\le \lambda \le 0.25$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0.12</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>0.25</mml:mn> </mml:mrow> </mml:math> . These convergent results demonstrate that a single parameter captures the chemomechanical enhancement of alkali-activated concretes, providing a direct calibration rule that can be incorporated into future revisions of the ACI 323 Low-Carbon Concrete code.