S.A Naeini, - Full Professor in Department of Civil Engineering, Imam Khomeini International University
M. Mortezaee - Ph.D candidate in Department of Civil Engineering, Imam Khomeini International University,

A numerical approach to the nonlinear/damage analysis of concrete micro-cracks is developed and proposed to present crack initiation and growth. The crack geometry effect is encountered in compliance matrix as a tensor containing the stress/strain components interrelation presenting material behavior aspects that may be activated during material deformation. The components of compliance matrix, also, vary due to the effects of directional fabric behavior of material created by induced/inherent anisotropy that can be different in any of multi direction through material. Mathematically, changes in multi directional material behavior are numerically integrated and are included in material compliance matrix. Upon this numerical integration facility is made to sum up multi directional behavior aspects to present a complete compliance matrix. However, to satisfy equilibrium and compatibility condition through cracked medium as a continuum, the discretized matrix form equations must be lead to satisfy both sets of equilibrium and compatibility differential equations. The results of employed numerical integration also, must be able to satisfy both sets of these differential equations.A certain discussion showing the capability of proposed damage model is presented as proves of equilibrium and compatibility equation that are satisfied. This paper deals partly with the problem exists in the present multi-laminate models built on a numerical integration upon 13 sampling points, which both compatibility and equilibrium conditions aren’t simultaneously satisfied. The reason of updating 13 sampling planes in to 17 to complete proposed model is presented. To verify the new numerical integration method upon a critical condition for crack growth on damaged sampling planes lead to comply overall compatibility conditions, the proposed damage formulation has been built on the basis of five fundamental types of stress/strain combinations, which essentially may occur on any of numerical integration sampling planes. To show the capability of model, a few fractured concrete test results under different loading stress/strain paths has been examined. The advantages of the proposed multi-laminate model are the capability of presenting pre-failure history of stress/strain progress on different predefined sampling planes lead to the final failure mechanism

multi-laminate model, concrete crack growth, damage, fabric, anisotropy