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  • Open Access

Size effects in fluctuation-based damage mechanics in the semigrand canonical ensemble

Ariel Attias1, Zdeněk P. Bažant2, and Franz-Josef Ulm1,*

  • *Contact author: ulm@mit.edu
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Phys. Rev. E 114, 045504 – Published 15 October, 2026

DOI: https://doi.org/10.1103/nxng-f58v

Abstract

We present results of a fluctuation-based investigation of size effects on damage and fracture of quasibrittle materials and structures. Set within the semigrand canonical (ΔμVT) ensemble, we find that size effects of geometrically similar structures are a system property rather than a material property. We reach this conclusion by considering damage and fracture as thermodynamic equilibrium states accessible through Monte Carlo simulations, and develop stress-strain curves and damage isotherms based on ensemble averaging, considering both short-range and long-range bond interactions. We suggest that, to preserve the probability distribution of damage and fracture states for geometrically similar structures, a scaling of the thermodynamic variables of the ΔμVT- ensemble needs to be considered, including a scaling of the bath potential, Δμ∝λ−d−1, for the system's size dilation λ, considering the geometric dimension d of the fracture surface. Derived from mere thermodynamic equilibrium conditions in the fluctuation-based damage approach, we show that such scaling allows the transition between strength theories (d=0) and linear elastic fracture mechanics (LEFM), for which the strength scales as σN∝λ−1/2, for all geometries and types of bond interactions. The rationalization of size effects within the realm of statistical mechanics opens venues toward a large range of engineering applications involving damage-prone materials and structures, from concrete bridges and high-rise buildings to hydraulic fracturing of gas shale and damage of hard and soft biological tissues.

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Outline

Information

Phys. Rev. E 114, 045504– Published 15 October, 2026
  • Received 18 September 2025
  • Revised 26 May 2026
  • Accepted 8 July 2026
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