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.

Nanoindentation is a well-stablished experiment to study the mechanical properties of materials at the small length scales of micro and nano. Unlike the conventional indentation experiments, the nanoindentation response of the material depends on the corresponding length scales, such as indentation depth, which is commonly termed the size effect. In the current work, first, the conventional experimental observations and theoretical models of the size effect during nanoindentation are reviewed in the case of crystalline metals, which are the focus of the current work. Next, the recent advancements in the visualization of the dislocation structure during the nanoindentation experiment is discussed, and the observed underlying mechanisms of the size effect are addressed. Finally, the recent computer simulations using molecular dynamics are reviewed as a powerful tool to investigate the nanoindentation experiment and its governing mechanisms of the size effect.

In this study, damage and healing mechanisms are discussed in detail with special emphasis given on two new healing variables. They are introduced in terms of healing variable based on area and healing variable based on elastic stiffness. The main objective of the introduced healing variables is to formulate the mechanical behavior of the self-healing materials and compensate the lack of physically consistent healing variables. Different healing mechanisms, existing in current self-healing systems, are characterized and for a uniaxial case of loading and isotropic material, the scalar healing variables are proposed. The performance of these new healing variables is examined for different damage-healing processes as well as coupled and uncoupled healing systems. The state of damage and healing is assumed to be isotropic for both damaged and healed states. The results show consistency with the basic concepts of the continuum damage mechanics in which introduced healing variables facilitate the self-healing materials characterization.

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.