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# Neda Latifi

**Grade: ** Master

**Thesis Title:**

Micro-Plane Modeling and Finite Element Analysis of Nonlocal Damage (Co-supervised by Dr. Kadkhodaei).

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

**Abstract:**

In many structures subjected to extreme loading conditions, the initially smooth distribution of strain changes into a highly localized one. Continuum damage description of failure mechanisms generally exhibits strain softening. In numerical simulations employing standard damage theory, strain softening causes a severe dependence on the size and orientation of the FE discretization. As a consequence, upon mesh refinement, the energy dissipated by the numerical model decreases and tends to extremely low values. On the other hand, damage is an anisotropic material characteristic; however, it is considered to be isotropic in conventional damage models. The contribution of this research is to present an anisotropic damage model based on the so-called microplane concept with its most reliable and thermodynamically consistent approach, the V-D split. The previous research on this topic has been done by the use of another approach of microplane theory which is called the V-D-T split. Although the results of that research seem to be correct, the model can not be considered as a thermodynamically consistent approach of the theory of microplane. The aim to use microplane theory in damage modeling is its ability to incorporate anisotropic behavior of damage in a natural and simple way. Choosing the implicit gradient-enhanced approach as for nonlocal damage modeling, the strain tensor has been considered as the nonlocal parameter. A second partial differential equation has to be solved in addition to the equations of equilibrium. In practical simulations, the differential equations governing the material response are solved numerically. Additional nodal degrees of freedom are introduced which lead to a modified element formulation. Firstly, finite element analysis of the extended formulation has been carried out. In order to do the finite element analysis, the weak forms of equilibrium equations accompanied by the implicit gradient-enhanced equations have been simultaneously solved through discretization and linearization of the corresponding equations. The result of this solution is to extend the stiffness matrix and the internal and external force vectors. Solving this set of nonlinear equations, a consistent incremental-iterative Newton-Raphson solution procedure has been applied. A UEL, User element, subroutine has been extended. For a two-dimensional plane strain example, numerical solutions are presented. They show the effectiveness of the gradient dependence as a measure against mesh sensitivity. Upon mesh refinement, the numerical solution shows a consistent material response for different numbers of meshes. Besides, by increasing the characteristic length the material shows a less flexible behavior which seems to be more similar to the real material behavior.

**Keywords:** Nonlocal continuum,Gradient-enhanced, Microplane theory,Damage mechanics,Finite element method.