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The major goal of this book is to present the implementation of some damage models with finite elements. The damage models are based on the principles of continuum damage mechanics and the effective stress concept. Several books have appeared recently on damage mechanics but are mostly theoretical in nature. Alternatively, this book provides a complete finite element program that includes the effects of damage. The book consists of two parts. Part I includes two chapters mainly review­ ing topics from finite element analysis and continuum damage mechanics. The reader is cautioned that the material contained in this part is introductor- other references must be consulted for the theoretical aspects of these topics. For a complete theoretical treatment of the subject, the reader is referred to the book Advances in Damage Mechanics: Metals and Metal Matrix Composites by Voyiadjis and Kattan, published in 1999. In Part II the finite element program DNA is introduced in three chapters. DNA stands for "Da­ mage Nonlinear Analysis". The program can be used for the analysis of elasto­ plastic material behavior including the effects of damage within the frame­ work of damage mechanics. Two versions of DNA are presented - one for small strain analysis and one for finite strain analysis. The program makes extensive calls to a library of tensor operations developed by the authors. The tensor library is extensively outlined in the last chapter of the book.



1. Damage Mechanics

Kachanov (1958) pioneered the subject of continuum damage mechanics by introducing the concept of effective stress. This concept is based on considering a fictitious undamaged configuration of a body and comparing it with the actual damaged configuration. He originally formulated his theory using simple uniaxial tension. Following Kachanov's work researchers in different fields applied continuum damage mechanics to their areas in fields like brittle materials (Krajcinovic and Foneska, 1981; Krajcinovic, 1988) and ductile materials (Lemaitre, 1984, 1985, 1986; Kachanov, 1986; Murakami, 1988). In the 1990's applications of continuum damage mechanics to plasticity and composite materials have appeared (Voyiadjis and Kattan, 1990, 1993, 1999; Kattan and Voyiadjis, 1990, 1993a, 1993b, 1996; Voyiadjis and Venson, 1995; Voyiadjis and Thiagarajan, 1996; Voyiadjis and Park, 1997a, 1997b).
P. I. Kattan, G. Z. Voyiadjis

2. Finite Element Damage Analysis of Plate Bending

A ductile material is capable of undergoing large plastic deformations. The accumulated plastic deformation can induce the changes of microstructures of the material through, for example, the nucleation, growth and coalescence of microvoids. These changes in material microstructures are the irreversible thermodynamic processes and result in a progressive degradation on the material properties. The process of the initiation and growth of microvoids and other microdefects induced by plastic deformations in ductile solids is called the ductile plastic damage. The primary interest of the ductile plastic damage is to study the influence of microvoids resulting from plastic deformations on the degradation of material properties. The changes on material properties can be studied by either a phenomenological damage model or a micromechanical damage model. A number of damage definitions and measures were proposed for both the models (vide the review papers of Krajcinovic, 1984, 1989; Chaboche, 1988; among others). Within the framework of phenomenological damage model, the damage of a material can be measured in macroscale by the deduction of mechanical properties, such as the elasticity constants (Lemaitre et al., 1979). Moreover, the changes of the macro-mechanical properties can be characterized by the damage effect parameters which are able to be determined from experiments (Lemaitre, 1985). These damage parameters are the internal state variables in thermodynamics.
P. I. Kattan, G. Z. Voyiadjis

3. Using DNA

In this chapter we cover the basic steps needed to install and run the program DNA correctly and obtain the results. A complete list of DNA commands and supported finite elements are given in Chapter Chapter 4. A detailed outline of the tensor library is given in Chapter 5.
P. I. Kattan, G. Z. Voyiadjis

4. DNA Commands

DNA commands must be entered in the input file for any problem to be solved with this program. The following rules apply to DNA commands:
Real numbers can be entered using an “F” or “E” format with “D” or “E” exponent notation.
Many commands have numeric arguments but there are some commands that have no arguments.
Commands must be separated from their numeric arguments with white space (blank space or TAB character).
Numeric arguments can be separated by white space or comma.
Additionally, commands must be separated from previous commands (with or without arguments) with white space.
Anything following the characters /* on the line is considered a comment and will not be processed by DNA.
Only the first four characters of each command are read by DNA. The remaining part of the command is not processed.
P. I. Kattan, G. Z. Voyiadjis

5. The Tensor Library

The tensor library is a collection of 72 Fortran subroutines that handle several tensor operations useful in Damage Mechanics. The tensor library is called extensively by the DNA program to handle all the tensor operations. All tensors are assumed to have an index range of 1, 2, 3. All matrices are square of size 3×3 or 6×6 if full. The tensor library is available on the accompanying CD-ROM in three files, namely TENSOR.FOR, PRTL.FOR, and SXM.FOR. The subroutines are available for both single precision and double precision calculations. The following is a list of all the subroutines in the tensor library. Remove the letter “D” from the beginning of the name of each subroutine to use the single precision version; otherwise the double precision version will be used.
P. I. Kattan, G. Z. Voyiadjis


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