On fibre bundle approach to a damage analysis

In this paper the geometric structure of a damaged state in the medium is investigated based on the fibre bundle technique. The damage (a failure) in some region of the medium under loads is regarded as breakdown of the holonomicity in this region. A measure of reduction of load carrying area elements caused by the development of microcracks or microvoids in the damaged medium is identified with the damage tensor. Multiplicative representations of the damage tensor formulated both on the tangent bundle with an affine connection and a fibre bundle with a non-linear connection over the damaged medium are obtained. In the former case the generalization of the damage tensor to the tangent bundle rests on the technique of prolonging the deformation vector to include not only the independent variables and dependent variables appearing in the damage tensor, but also the derivatives of the dependent variables. The so-called lifting technique is used to express the damage tensor by initial, additional and direct (deformation-induced) damages or by direct and transferred ones. In the latter one the higher-order contact geometry approach identified with the Finslerian geometry is employed to analyze the influences of elastic, inelastic, direct, transferred and initial damages on the total damage tensor. Time-dependent relationships of the damage tensor are then presented.