In this work various new proposed damage variables are introduced, examined and compared. Only the scalar case pertaining to isotropic damage is investigated here. Several types of new damage variables are proposed as follows: (1) damage variables that are defined in terms of cross-sectional area, (2) damage variables that are defined in terms of the elastic modulus or stiffness, and (3) composite damage variables that are defined in terms of two parameters relating to both cross-sectional area and stiffness. However, the generalization to tensors and general states of deformation and damage may be a straightforward process but is beyond the scope of this work. The damage variables introduced in this work can be applied to elastic materials including homogeneous materials like metals and heterogeneous materials like composite laminates. In the second part of this work, higher-order strain energy forms are proposed. It is seen that a specific nonlinear stress-strain relationship is associated with each higher-order strain energy form. These higher order strain energy forms along with some of the proposed damage variables are used in trying to lay the theoretical groundwork for the design of undamageable materials, i.e., materials that cannot be damaged where the value of the damage variable remains zero throughout the deformation process.