Investigation On Fracture Behaviors And Damage Constitutive Relation Of Concrete Considering Fractal Effects

Cracking behavior of concrete material has been concerned by more and more researchers in recent years. Many researchers studied the crack development through the 3-point bending tests of the plain concrete beams. Their theorical foundation comes from classic Griffith fracture mechanics, which asumes the fracture surface and crack of concrete are smooth. They got amount of important conclusions. But imprecise result may be obtained by applying the classic fracture theory to studying the fracture behavior of concrete for the roughness of its fracture surface and cracks. In order to take this feature into account, fractal geometry theory is introduced in this work to discribe its fracture surface and crack. Fracture and damage behaviors of concrete are studied by theorical analysis and numerical calculation. The main innovative work completed in this paper is included as follows:1. By constructing the cracking boundary of concrete with a standard Koch fractal curve, the mathematic expressions of the elastic strain energy and the surface energy in the fracture zone are derived based on the area-perimeter relation given in fractal geometry. And the classic Griffith fracture theory is extended to the fractal space in which the non-smooth crack can be discribed. Behaviors of the critical cracking stress, the fracture toughness, the driving force and the fracture energy are analyzed.2. Considering the quasi-brittle feature of concrete material, the fracture surface is simulated by a Brownian fractal surface. The multi-fractal size effect law of Carpinteri is expressed as a function of Brownian index. The size effect of the fracture toughness is obtained from the fracture energy. The size effect laws of Ba?ant and Carpinteri are applied in the formula of stress strength factor; and the relation between the fracture toughness and size of the specimen and crack. The statistical size coefficients between concrete specimens given in the Standard for test method of mechanical properties on ordinary concrete are proofed by using the strength size effect law of the size effect law by Carpinteri.3. By simulating the pore structure of cement paste with Menger sponge, porosity can be exoressed as a function of fractal dimension. Based on this, the relationships between strength of cement paste and the construction parameters are analyzed quantitatively considering the relation between the porosity and strength. The expression of concrete strength considering the pore distribution is obtained based on the strength relations of cement paste and concrete.4. A three-phase micro fracture spherical model for concrete considering the influence of temperature is established based on a rule that volumes of each component are the same as those of a real material. The components of concrete is simplified as aggregate, water and cement paste, and each part is replaced by a spherical layer. The stress states of the aggregate and cement paste layers are studied. For the smooth cracking case, the relation of the fracture toughness and temperature is given based on the Griffith fracture criterion. For the fractal cracking case, the surface energy and stress strength factor of a 3-dimension crack is defined by the dimension analysis. The updated fracture criterion is employed in this model and the expression between the fracture toughness and temperature is got therefore. The presented model inherits an advantage that only fracture toughness at two different temperatures are needed to be given to predict other temperature fracture toughness for concrete.5. The fractal dimension and stress of concrete coresponding to the initial damage and critical fracture state are determined by using the relation between the fracture toughness and the fractal dimension of quasi-brittle material. And the damage variable is written in term of fractal dimension. The Helmholtz free energy including isotropic damage, anisotropic damage and irreversible deformation parts is defined and thus a damage constitutive model of concrete considering the fractal feature is derived with a corresponding damage evlution law followed. Numerical computation is conducted for concrete under the uniaxial and the biaxial compression which shows the validation of the presented model is proofed by comparing results with the experimental data. This model provides an approach to link the macro properties of a material with its micro-structure change.6. The perimeter-area relation is adopted to derive the transformation rule between damage variables in the fractal space and Euclidean space. A plastic damage constitutive model for concrete in the Euclidean space is generalized to fractal case according to the transformation rule of damage variables. The present model considering the fractal effect is used to analyze a notched plain concrete beam under 3-point bending. The numerical results show the efficiency and validation of the present model for structural analysis.