Abstract
The fracture toughness, the driving force and the fracture energy for an infinite plate with a fractal crack are investigated in the fractal space in this work. The perimeter-area relation is adopted to derive the transforma-tion rule between damage variables in the fractal space and Euclidean space. A plasticity yield criterion is introduced and a damage variable tensor is decomposed into tensile and compressive components to describe the distinct behaviors in tension and compression. A plastic damage constitutive model for concrete in the Euclidean space is developed and generalized to fractal case according to the transformation rule of damage variables. Numerical calculations of the present model with and without fractal are conducted and compared with experimental data to verify the efficiency of this model and show the necessity of considering the fractal effect in the constitutive model of concrete. The structural response and mesh sensitivity of a notched unre-inforced concrete beam under 3-point bending test are theoretical studied and show good agreement with the experimental data.
Highlights
Concrete has been widely used in civil engineering for its good in-situ casting and molding abilities
Fractal geometry is established by Mandelbrot in 1970s [1], which plays an important role in the development of fracture mechanics theory
The fracture toughness, the driving force and the fracture energy of a material with fractal cracks are investigated and their theoretical expresses in the fractal space are derived based on fracture mechanics and fractal geometry
Summary
Concrete has been widely used in civil engineering for its good in-situ casting and molding abilities. Researches show that the fracture zone of metal, rock and concrete has fractal characteristics [2,3,4] This leads a widely use of fractal geometry in many fields of material science, for instances, the Sierpinski carpet was adopted by Carpinteri et al [5] to simulate the composition of concrete cross section, and the fractal effect was introduced into the cohesive crack model. Another remarkable application of fractal geometry is to describe the roughness of cracks quantitatively. A notched plain concrete beam under 3-point bending test is simulated to verify the efficiency of the model
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