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# Meisam Asgari

**Grade: ** Master

**Thesis Title:**

Damage Modeling of Quasi-Brittle Materials with Micro-Plane Theory (Co-supervised by Dr. Kadkhodaei)

**Year:** Sept. 2008- Feb. 2011.

**Abstract:**

The aim of this thesis is to extend a damage model for quasi-brittle materials based on the so-called Micro-plane theory with the V-D split. Micro-plane theory yields a simple constitutive equation relating the macroscopic stress or strain tensor to their components on different planes at each material point. Traditional approaches of microplane theory do not satisfy the second law of thermodynamics. Among all approaches of microplane theory, the formulation with V-D split is consistent with the second law of thermodynamics. In this research, the thermodynamically consistency of the V-D micro-plane model is clearly shown by extracting the same Material Jacobian Matrix as that of the linear elasticity. Damage mechanics is a new branch in mechanical engineering that deals with the science of growth and increasing the size of microcracks and microvoids in engineering material in order to predict the material response in a more precise way. Because damage is naturally an anisotropic characteristic of material, continuum damage mechanics which considers damage as an isotropic property of the material, can not formulate the anisotropy of damage, unless damage be considered as a complex forth or eighth order tensor at each material point. These assumptions in formulating anisotropic damage cause several complexities in dealing with difficult mathematical calculations. The aim to use Micro-plane theory to formulate the damage is its ability to incorporate anisotropic behavior of damage in a natural and simple way. In this research, the damage parameters of each Micro-plane have been considered as dependent parameters to the normal vector of the relevant Micro-plane. In previous researches, the damage parameters have been simply assumed to be the same for all of the micro-planes. In those researches, the results seemed reasonable; however, in the present work, not only the results are rational, but also the model is a more reliable continuum-based one. The presented model is then implemented in a VUMAT subroutine in ABAQUS and is applied to solve some definite problems like simple shear and uniaxial tension. In order to appraise the results of the extracted model, a macroscopic damage model based on linear elasticity and continuum damage mechanics is developed and implemented via a VUMAT subroutine in ABAQUS. The results of these two models are compared and analyzed. It is obvious from the results that the response of Microplane damage formulation is very similar to the results of macroscopic damage formulation. Furthermore, after progressive loading, it is observed that the response of microplane damage model converges to the response of the macroscopic model.

**Keywords: **Quasi-brittle materials,Microplane theory, Damage mechanics.