Anisotropic Damage Effect Tensors for the Symmetrization of the Effective Stress Tensor

Abstract

Based on the concept of the effective stress and on the description of anisotropic damage deformation within the framework of continuum damage mechanics, a fourth order damage effective tensor is properly defined. For a general state of deformation and damage, it is seen that the effective stress tensor is usually asymmetric. Its symmetrization is necessary for a continuum theory to be valid in the classical sense. In order to transform the current stress tensor to a symmetric effective stress tensor, a fourth order damage effect tensor should be defined such that it follows the rules of tensor algebra and maintains a physical description of damage. Moreover, an explicit expression of the damage effect tensor is of particular importance in order to obtain the constitutive relation in the damaged material. The damage effect tensor in this work is explicitly characterized in terms of a kinematic measure of damage through a second-order damage tensor. In this work, tensorial forms are used for the derivation of such a linear transformation tensor which is then converted to a matrix form.

Non-linear material modeling of fiber-reinforced polymers based on coupled viscoelasticity–viscoplasticity with anisotropic continuous damage mechanics

“…Different types of M tensor and their symmetrization problem have been discussed in Voyiadjis and Park [25]. It is accepted following Voigt notation to write M tensor as:…”

Modelling Damage Growth and Failure in Elastic Materials with Random Defect Distributions

“…Several symmetry preserving transformations have been proposed based on the values of a second-rank damage tensor, d [28]. This allows a relatively simple and physically intuitive method of representing anisotropic damage.…”

“…This allows a relatively simple and physically intuitive method of representing anisotropic damage. In the current model the damage effect tensor is assumed to take the following form [28]:…”

Postbuckling analysis of composite panels with imperfection damage

“…The reduction of post-buckling strength due to delamination has been studied by Naganarayana and Atluri (1995), and Chow and Atluri (1995) using the criterion of mixed-mode stress intensity factors. In order to characterize the buckling behavior of laminated composite shells which have both material and delamination damage, the kinematics of material and delamination damage deformation by Park (1997 and1999), and Park and Voyiadjis (1998) will be incorporated within the frame work of continuum damage mechanics. The results of Fig.…”

“…As a result of ignoring the quadratic terms, the submatrix of the geometric element stiffness, K e 22 G is zero in Eq. (25). The geometric element stiffness matrix obtained by transformation to the global frame has the following submatrices (Kim 1992):…”