Crack band theory for fracture of concrete

A fracture theory for a heterogeneous aggregate material which exhibits a gradual strainsoftening due to microcracking and contains aggregate pieces that are not necessarily small compared to struttural dimensions is developed. Only Mode I is considered. The fracture is modeled as a blunt smeared crack band, which is justified by the random nature of the microstructure. Simple triaxial stress-strain relations which model the strain-softening and describe the effect of gradual microcracking in the crack band are derived. It is shown that it is easier to use compliance rather than stiffness matrices and that it suffices to adjust a single diagonal term of the compliance matrix. The limiting case of this matrix for complete (continuous) cracking is shown to be identical to the inverse of the well-known stiffness matrix for a perfectly cracked material. The material fracture properties are characterized by only three paPlameters -fracture energy, uniaxial strength limit and width of the crack band (fracture Process zone), while the strain-softening modulus is a function of these parameters. A m~thod of determining the fracture energy from measured complete stressstrain relations is' also given. Triaxial stress effects on fracture can be taken into account. The theory is verljied by comparisons with numerous experimental data from the literature. Satisfactory fits of maximum load data as well as resistance curves are achieved and values of the three matetial parameters involved, namely the fracture energy, the strength, and the width of crack b~nd front, are determined from test data. The optimum value of the latter width is found to be about 3 aggregate sizes, which is also justified as the minimum acceptable for a homogeneous continuum modeling. The method of implementing the theory in a finite element code is al$o indicated, and rules for achieving objectivity of results with regard to the analyst's choice of element size are given. Finally, a simple formula is derived to predict from the tensile strength and aggregate size the fracture energy, as well as the strain-softening modulus. A statistical analysis of the errors reveals a drastic improvement compared to the linear fracture th~ory as well as the strength theory. The applicability of fracture mechanics to concrete is thz4 solidly established.