Mechanics

1999, No. 4

V.I. Astafev, E.V. Lomakin, A.N. Polilov, S.A. Shesterikov

ACADEMICIAN Y.N. RABOTNOV SCIENTIFIC SCHOOL (TO THE 85^th ANNIVERSARY)

V.I. Astafev, A.N. Krutov

STRESS DISTRIBUTION NEAR THE TOP OF INCLINED CRACK

In this paper the stress distribution near the top of inclined crack in a nonlinear mechanics of destruction is calculated. The determining relations are set as power dependence. For it the compatibility equation in stress is use, in this one the Airy stress function is introduced and two-point boundary value problem is formulated. The obtained boundary value problem is solved with the help of false position method. The solution is reduced for the plain stress and the plain strain.

Y.N. Radayev

ON THE CANONICAL PRINCIPAL LINE INVARIANTS OF THE PLASTICITY-DAMAGE COUPLED EQUATIONS

A general analysis of three-dimensional static and kinematic equations of the strain-damage coupled model for ductile deformation processes is presented in an attempt to find approaches for analytic study of the damage effect on the plastic flow. The damage is represented by a symmetric second-rank damage tensor, known from a number of our previous discussions devoted to description of anisotropic damage state. The modified by anisotropic damage effect Tresca yielding criterion, associated flow and damage rules are used to formulate the strain-damage coupled constitutive equations, thus allowing to take account of effect of anisotropic damage on the plastic flow and vice versa. The separate analysis for the loading paths going along the facets and edges the Tresca prism is given. In the present study the elastic strains are assumed not significantly affect a damage state that, as it is shown, implies the principal axes of damage are being co-oriented while loading to those of the Cauchy stress tensor. The principal axes of stresses are chosen as a local frame for representing the static, kinematic and constitutive equations. It is expedient for stress and damage analysis, but requires much effort to convert the kinematic equations. The closed system of equations represented in the local principal frame needed for formulation of the strain-damage coupled model is obtained. The relations along the principal stress-damage lines are derived for the following discriminated cases: 1) three-dimensional elastic-plastic state of deformation and damage, and loading paths on the Tresca prism facet; 2) three-dimensional elastic-plastic state of deformation and damage, and loading paths going along the Tresca prism edge; 3) axially-symmetric state of elastic-plastic deformation and damage; 4) plane state of elastic-plastic deformation and damage. All the situations when these relations can be integrated along the stress-damage principal lines are then determined. The analysis in the principal local frame is extremely effective for the paths going along the Tresca prism edges and for the plane state due to invariants (in the form of ratios involving principal damages, stresses and curvatures of the principal lines) along the principal stress-damage lines can be obtained. The invariants establish a balance of principal stresses, damages and the internal geometry parameters of the principal lines inside a damaged media. The principal line invariants provide also a new technique for the analysis of the coupled equations by canonical transformations of the plastic flow zone. A canonical transformation has an intrinsic invariant - the volume (or area in the two dimensions) of a spatial (or plane) element, which value is not affected by the transformation - thus being applicable and usable in many fields of mechanics. The canonical transformation can be determined by its generating function that affords a new formulation of the strain-damage coupled problem. In the plane and axial symmetric case non-linear equations for generating functions of the canonical transformations are derived. Those are proved be the invariants of the Legendre (plane problem) and Ampere (axially-symmetric problem) transformations.