TY - JOUR

T1 - Variational damage model

T2 - A new paradigm for fractures

AU - Ren, Huilong

AU - Rabczuk, Timon

AU - Zhuang, Xiaoying

N1 - Publisher Copyright: © The Author(s) 2025.

PY - 2025/1

Y1 - 2025/1

N2 - The computational modeling of fracture in solids using damage mechanics faces challenges with complex crack topology. This can be addressed by using a variational framework to reformulate the damage mechanics. In this paper, we propose several mathematically elegant variational damage models (VDMs) for fracture mechanics without explicitly using damage variables. Based on the energy density ϕ, the fracture energy density is formulated as ϕ^=ϕ(1+ℓϕ/Gc) and the damage variable is expressed as s = ϕ/(ϕ + Gc/ℓ), which satisfy ϕ^∣ϕ→∞=Gc/ℓ and s∣ϕ→∞ = 1 as ϕ approaches infinity. These limits demonstrate that the new energy density converges to the Griffith energy release rate at full-damage state. The VDM profoundly modified the energy functional, implicitly incorporating the damage field. As a generalization of previous model, we propose a family of VDMs of varying orders. Additionally, we develop a multi-damage model to account for different types of energy densities, such as elastic thin plate and gradient elasticity. Using this functional, it is straightforward to deduce the governing equation for automatically evolving fractures. These formulations can be employed in conventional finite element method or other numerical methods with minimal modifications. Compared to the phase field method with the same mesh density, a sharper crack interface can be achieved. We demonstrate the capabilities of the proposed variational damage formulations using representative numerical examples.

AB - The computational modeling of fracture in solids using damage mechanics faces challenges with complex crack topology. This can be addressed by using a variational framework to reformulate the damage mechanics. In this paper, we propose several mathematically elegant variational damage models (VDMs) for fracture mechanics without explicitly using damage variables. Based on the energy density ϕ, the fracture energy density is formulated as ϕ^=ϕ(1+ℓϕ/Gc) and the damage variable is expressed as s = ϕ/(ϕ + Gc/ℓ), which satisfy ϕ^∣ϕ→∞=Gc/ℓ and s∣ϕ→∞ = 1 as ϕ approaches infinity. These limits demonstrate that the new energy density converges to the Griffith energy release rate at full-damage state. The VDM profoundly modified the energy functional, implicitly incorporating the damage field. As a generalization of previous model, we propose a family of VDMs of varying orders. Additionally, we develop a multi-damage model to account for different types of energy densities, such as elastic thin plate and gradient elasticity. Using this functional, it is straightforward to deduce the governing equation for automatically evolving fractures. These formulations can be employed in conventional finite element method or other numerical methods with minimal modifications. Compared to the phase field method with the same mesh density, a sharper crack interface can be achieved. We demonstrate the capabilities of the proposed variational damage formulations using representative numerical examples.

KW - damage mechanics

KW - fracture

KW - multi-damage

KW - variational principle

UR - http://www.scopus.com/inward/record.url?scp=85214202439&partnerID=8YFLogxK

U2 - 10.1007/s11709-025-1144-0

DO - 10.1007/s11709-025-1144-0

M3 - Article

AN - SCOPUS:85214202439

VL - 19

SP - 1

EP - 21

JO - Frontiers of Structural and Civil Engineering

JF - Frontiers of Structural and Civil Engineering

SN - 2095-2430

IS - 1

ER -