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## Mechanics of Damage, Healing, Damageability, and Integrity of Materials: A Conceptual Framework

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by George Z. Voyiadjis and Peter I. Kattan

In this work several new and fundamental concepts are proposed within the framework of continuum damage mechanics. These concepts deal primarily with the nature of the two processes of damage and healing along with introducing a consistent and systematic definition for the concepts of damageability and integrity of materials. Toward this end, seven sections are presented as follows: “The logarithmic damage variable” section introduces the logarithmic and exponential damage variables and makes comparisons with the classical damage variable. In “Integrity and damageability of materials” section a new formulation for damage mechanics is presented in which the two angles of damage–integrity and healing–damageability are introduced. It is shown that both the damage variable and the integrity variable can be derived from the damage–integrity angle while the healing variable and damageability variable are derived from the healing–damageability angle. “The integrity field” section introduces the new concept of the integrity field while “The healing field” section introduces the new concept of the healing field. These two fields are introduced as a generalization of the classical concepts of damage and integrity. “Unhealable damage and nondamageable integrity” section introduces the new and necessary concept of unrecoverable damage or unhealable damage. In this section the concept of permanent integrity or nondamageable integrity is also presented. In “Generalized nonlinear healing” section generalized healing is presented where a distinction is clearly made between linear healing and nonlinear healing. As an example of nonlinear healing the equations of quadratic healing are derived. Finally in “Dissection of the healing process” section a complete and logical/mathematical dissection is made of the healing process. It is hoped that these new and fundamental concepts will pave the way for new, consistent, and holistic avenues in research in damage mechanics and characterization of materials.

## Decomposition of Damage Tensor in Continuum Damage Mechanics

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by Peter I. Kattan and George Z. Voyiadjis

The principles of continuum damage mechanics are reviewed first for the case of uniaxial tension. The damage variable is then decomposed into two variables called crack damage variables and void damage variables. A consistent mathematical formulation is presented to decompose the damage tensor into two parts: one caused by voids and the other caused by cracks. In the first part of this work, isotropic damage in the uniaxial tension case is assumed. However, the generalization to three-dimensional states of damage is presented in the second part of this work, using tensorial damage variables. It is shown that the components of tensorial crack damage variables and void damage variables are not independent of each other, implying a coupling between the two damage mechanisms. This coupling may be obvious based on the physics of the problem, but a rigorous mathematical proof is given for it. Also, explicit relations governing the components of the crack and void damage variables are derived.

## A Comparative Study of Damage Variables in Continuum Damage Mechanics

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by George Z. Voyiadjis and Peter I. Kattan

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.