# Continuum Damage Mechanics And Numerical Applications (Advanced Topics In Science And Technology In

Over the last decade, there has been significant progress in using entropy for damage and fracture mechanics of materials and interfaces. This symposium will focus on experimental and computational damage, degradation and fracture mechanics of materials and interfaces using entropy as the key criterion. The symposium is open to all engineering applications of damage and fracture, including but not limited to processes involving thermo-mechanical, electrical current, corrosion, radiation, thermal gradient, and chemical degradation.

## Continuum Damage Mechanics And Numerical Applications (Advanced Topics In Science And Technology In

Over the last two decades, enormous progress has been made in damage healing of thermoset polymers and polymer composites, be it intrinsic healing or extrinsic healing. This symposium will focus on the recent progress in understanding the damage healing either using theoretical modeling, numerical simulation, or experimental testing. The symposium is open to all aspects of damage healing in thermoset polymers and polymer composites, including, but not limited to molecular dynamic simulation, continuum mechanics modeling, and damage/healing testing.

"Continuum Damage Mechanics and Numerical Applications" presents a systematic development of the theory of Continuum Damage Mechanics and its numerical engineering applications using a unified form of the mathematical formulations in anisotropic and isotropic damage models. The theoretical framework is based on the thermodynamic theory of energy and material dissipation and is described by a set of fundamental formulations of constitutive equations of damaged materials, development equations of the damaged state, and evolution equations of micro-structures. According to concepts of damage-dissipation of the material state and effective evolution of material properties, all these advanced equations, which take nonsymmetrized effects of damage aspects into account, are developed and modified from the traditional general failure models so they are more easily applied and verified in a wide range of engineering practices by experimental testing.

Abstract:Damage mechanics play an important role in the analysis of rock deformation and failure. Numerous damage variables have been proposed and the corresponding continuum damage models were suggested. Knowing how to apply these theoretical models appropriately in numerical simulations is the key to whether they can be adopted to solve practical problems. The continuum damage models were grouped into empirical damage models, statistical damage models, and elastoplastic damage models in this article. Their applicability and limitations were studied according to some numerical simulations of the most basic uniaxial compression test of a cylinder rock sample. Three representative damage models were chosen from the literature and applied to FEM numerical simulations by introducing a self-developed program. The stress-strain curves due to damage were obtained from the numerical simulation results and compared to those from the experimental results. The damage distribution and evolution of different damage models were investigated to evaluate their influences on rock deformation. It can be concluded that strain-softening stages presented by both the empirical damage models and the statistical damage models are caused by subtracting the elastic modulus gradually while those presented by the elastoplastic damage models are caused by reducing plastic yield stress gradually. Damage-elastic coupling cannot well reflect the irreversibility of damage. The elastoplastic damage models combine damage with plastic history, and thus the irreversibility of damage can be represented. Furthermore, the compulsory reduction of the elastic modulus can probably lead to extreme element distortion and even an unreasonable negative modulus when damage is very serious, which inevitably causes the numerical simulation to fail prematurely under complex stress states. Although the elastoplastic damage models are recommended at present rather than the other models, a more appropriate definition of the damage variable can be expected that should track the whole deformation and failure process. Knowing how to treat the adverse effect of local deterioration due to damage is the challenge numerical simulations have to face when they are applied in the actual project with complex stress states.Keywords: rock damage; numerical simulation; damage variable; stress-strain curve; strain softening

This book presents a systematic development of the theory of Continuum Damage Mechanics and its numerical engineering applications with a unified form of mathematical formulations in anisotropic and isotropic damage models. The areas studied in this book are (1) Review of damage mechanics; (2) Basis of isotropic damage mechanics; (3) Brittle damage mechanics of rock mass; (4) Theory of isotropic elasto-plastic damage mechanics; (5) Basis of anisotropic damage mechanics; (6) Theory of anisotropic elasto-plastic damage mechanics; (7) Theory of elasto-visco-plastic damage mechanics; (8) Dynamics of damage problems; (9) Fatigue damage of dynamic structures; (10) Micro-damage mechanics; (11) Random damage mechanics; (12) Numerical method in continuum damage mechanics; (13) Application of damage mechanics to problems coupled with multiphase medium.

The theoretical framework of continuum damage mechanics presented in this book is based on the thermodynamic theory of energy and material dissipation, and is described by employing a group of internal state variables as a set of fundamental formulations of constitutive equations of damaged materials, development equations of the damaged state, and evolution equations of micro-structures. According to concepts of damage-dissipation of the material state and effective evolution of material properties, all these advanced equations, which take damage aspects into account, are developed and modified from the traditional general failure models, because they are more easily applied and verified in a very wide range of engineering practices by experimental tests, either macroscopically or microscopically.The most practical applications of the theory developed in this book are presented in different engineering topics analyzed by a specified numerical method. Some essential programs of the continuum damage mechanics are listed in the appendices.

MAE 09. C/C++ Programming (4) C/C++ computer programming under the UNIX environment with applications to numerical problems fundamental to computational mechanics. Arithmetic operations, branches, arrays, data structures, and use of pointers are introduced. Programming ethics are discussed. Priority enrollment given to pre-engineering and engineering majors.

MAE 135. Computational Mechanics (4) Mathematical modeling in terms of systems of algebraic and differential equations. Overview of numerical methods. Problem statement, boundary, and initial conditions. Overview of commerical packages for solving the equations of Mathematical and Engineering Physics. Numerical solutions of selected examples drawn from real-life applications of fluid flow, solid mechanics, and heat transfer with emphasis on design. Prerequisite: consent of instructor.

205. Graduate Seminar (1) Each graduate student in MAE is expected to attend one seminar per quarter, of his or her choice, dealing with current topics in fluid mechanics, solid mechanics, applied plasma physics and fusion, chemical engineering, applied ocean sciences, energy and combustion, environmental engineering, or materials science, and dynamics and controls. Topics will vary. (S/U grades only)

209. Continuum Mechanics Applied to Medicine/Biology (4) (Cross-listed with BENG 209.) Introduction to the basic definitions of continuum mechanics and their mathematical formulation at the graduate level with applications to problems in medicine and biology. This course is intended for students with little or no background in mechanics; it is an introduction to the Biomechanics courses BENG 250 A-B in the Department of Bioengineering and to Solid and Fluid Mechanics courses MAE 210A and MAE 231A in the Department of Mechanical and Aerospace Engineering. This course should NOT be taken concurrently with MAE 210A or MAE 231A. Prerequisite: consent of instructor.

221A. Heat Transfer (4) (Cross-listed with CENG 221A.) Conduction, convection, and radiation heat transfer. Development of energy conservation equations. Analytical and numerical solutions to transport problems. Specific topics and applications vary. Prerequisite: MAE 101A-B-C or CENG 101A-B-C, or consent of instructor.

221B. Mass Transfer (4) (Cross-listed with CENG 221B.) Fundamentals of diffusive and convective mass transfer and mass transfer with chemical reaction. Development of mass conservation equations. Analytical and numerical solutions to mass transport problems. Specific topics and applications will vary. Prerequisite: MAE 101A-B-C or CENG 101A-B-C, or consent of instructor.