## Abstract

Recent optimum fitting of the data from eleven different types of distinctive fracture tests of quasibrittle materials, including the gap tests and the size effect tests, revealed that the crack band model (CBM) with the M7 damage constitutive law outperforms all the other existing computational fracture models for concrete. This is attributed to physically correct boundary conditions, especially those at the crack faces,to the crack front of finite width which allows using a triaxial tensorial damage constitutive law, and to choosing for this law a realistic material behavior description, the M7. Nevertheless the CBM has limitations—the width of the crack band cannot be varied, the statistically smooth damage distribution across the band cannot be resolved, and the propagation direction is biased by the mesh orientation when a regular mesh is used. These limitations have been overcome by the smooth crack band model (sCBM), which uses unconventional homogenization of damage mechanics to yield an additional Helmholtz free energy density, called the “sprain energy”. This energy is a function of the curvature tensor (or a hessian) of the vectorial displacement field, called the “sprain curvature tensor” (or sprain tensor for brevity), which is a sum of the strain gradient tensor and, importantly, the rotation gradient tensor. But, in a previous study, difficulties arose in computational implementation—the use of constant-strain elements required the curvature to be resisted by sprain forces that represented the derivatives of sprain energy with respect to nodal displacements and had to be applied on the nodes of adjacent elements. The need for adjacent nodes is here avoided by a variational derivation of the Isogeometric Smooth Crack-Band Model (isCBM). The microplane damage constitutive model M7 for concrete is adopted, which is made possible by establishing the relations of the sprain stress (or spress, for brevity) and sprain on microplanes of arbitrary orientations. Using Isogeometric Analysis (IGA) based on non-uniform rational B-Splines (NURBS), the excessive displacement curvature and its resistance (spress) can be described by the shape functions with C^{1}-continuity. Computational results show that the orientation bias of regular meshes is eliminated while keeping the constitutive law intact, which is important for compatibility with the existing material laws in general. In addition, comparisons with experimental data reveal that isCBM with M7 can reproduce the correspondence between the size of the fracture process zone and the fracture energy, as indicated by gap tests. Finally, the isCBM does not compromise the strain-path-dependence of material response.

Original language | English (US) |
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Article number | 105470 |

Journal | Journal of the Mechanics and Physics of Solids |

Volume | 181 |

DOIs | |

State | Published - Dec 2023 |

### Funding

This work was supported through the Office of Naval Research (ONR) Grant No. N00014-21-1-2670 to Brown University. Additional partial support was also obtained under National Science Foundation (NSF) Grant CMMI-2029641 to Northwestern University.

## Keywords

- Displacement curvature
- Isogeometric analysis
- Localization limiter
- Material characteristic length
- Material rotation gradient
- Quasibrittle fracture
- Smooth crack band model
- Softening damage
- Sprain tensor
- Sprain-displacement matrix
- Spress–sprain relations

## ASJC Scopus subject areas

- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering