viiiThe material appearing in this text is limited to plastic deformation and damage in ductile materials (e.g. metals and metal matrix composites). The authors elect to exclude many of the recent advances made in creep, brittle fracture, and temperature effects. The authors feel that these topics require a separate volume for this presentation. Furthermore, the applications presented in the book are the simplest possible ones and are mainly based on the uniaxial tension test. The presentation of more challenging examples is left to the researchers in this field. This book does not claim to revolutionize the way in which research is done in this area, but it does advance a few new ideas and it does have several noteworthy features:

Self-healing smart materials have emerged into the research arena and have been deployed in industrial and biomedical applications, in which the modelling techniques and predicting schemes are crucial for designers to optimize these smart materials. In practice, plastic deformation is coupled with damage and healing in these systems, which necessitates a coupled formulation for characterization. The thermodynamics of inelastic deformation, damage and healing processes are incorporated here to establish the coupled constitutive equations for healing materials. This thermodynamic consistent formulation provides the designers with the ability to predict the irregular inelastic deformation of glassy polymers and damage and healing patterns for a highly anisotropic self-healing system. Moreover, the lack of a physically consistent method to measure and calibrate the healing process in the literature is addressed here. Within the continuum damage mechanics (CDM) framework, the physics of damage and healing processes is used to introduce the healing effect into the CDM concept and a set of two new anisotropic damage-healing variables are derived. These novel damage-healing variables together with the proposed thermodynamic consistent coupled theory constitute a well-structured method for accurately predicting the degradation and healing mechanisms in material systems. The inelastic and damage response for a shape memory polymer-based self-healing system is captured herein. While the healing experimental results are limited in the literature, the proposed theory provides the mathematical competency to capture the most nonlinear responses.

In this work, various definitions of the damage variables are examined and compared. In particular, special emphasis is given to a new damage variable that is defined in terms of the elastic stiffness of the material. Both the scalar and tensorial cases are investigated. The scalar definition of the new damage variable was used recently by many researchers. However, the generalization to tensors and general states of deformation and damage is new and appears here for the first time. In addition, transformation laws for various elastic constants are derived. Finally, the cases of plane stress, plane strain, and isotropic elasticity are examined in detail. In these cases it is shown that only two independent damage parameters are needed to describe the complete state of damage in the material. In this work, a physical basis is sought for the damage tensor [M] that is used to link the damage state of the material with effective undamaged configuration. The authors and numerous other researchers have used different paths including fabric tensors (Voyiadjis and Kattan, 2006a; Voyiadjis et al., 2007) to connect the two configurations. However, the approach presented here provides for a strong physical basis for this missing link.

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