Kinetic equations of creep and damage for description of materials with non-monotonic dependence of fracture strain on stress

OBRABOTKAMETALLOV MATERIAL SCIENCE Vol. 23 No. 3 2021 Detailed reviews of creep models that take into account the accumulation of damage in the material are carried out in [1–4]. Models of damage accumulation are divided into phenomenological and physically based ones. The founder of the phenomenological approach is L. M. Kachanov [5]. He introduced the concepts of “continuity” or “cracking”, describing the state of the material with one structural parameter ( ) t ψ ( 0 1 ≤ ψ ≤ , t – time). The mechanisms of damage and the physical nature of this damage parameter are not specified. Later Yu. N. Rabotnov introduced the parameter ( 0 1 q ≤ ≤ ) “quite conditionally”, assuming that when 0 q = the material is considered not damaged, and when 1 q = microscopic cracks begin to form, which actually means its destruction [6]. Even later Yu. N. Rabotnov generalizes the model by introducing several such damage parameters without giving it a specific physical meaning. Such parameters can describe various aspects of damage accumulation, for example, consider the aggressiveness of the environment [2]. Physically substantiated models take into account the microstructure of the material, the density of pores or dislocations in the process of damage accumulation [7–9]. Since most materials have anisotropic properties, the damage, as a rule, has a tensor or vector form [2, 4, 10]. However, until now, the introduction of vectors and damage tensors into the models is limited, since the calculations are significantly complicated. Various creep models with a scalar damage parameter are actively used to this day, and the introduction of the corresponding equivalent stress into the equations in some cases makes it possible to take into account the presence of anisotropy properties. According to the model of Yu. N. Rabotnov, the constitutive relations for the uniaxial stress state have the following form [11]: 1 2 1 2 ( , , , , ..., ), ( , , , , , , ..., ) c c i c n c n dq d f T q q q T t q q q dt dt ε = σ = ϕ σ ε , (1) where c ε – irreversible creep strains, T – temperature, t – time, i q – structural parameters. In the case of a single damage parameter q , the system (1) can be concretized in the following form [6]: 1 2 , (1 ) (1 ) n g c B B d dq dt dt q q ε ω κ κ σ σ ε = = − − . Here the parameters , , , B B n g ε ω , 1 2 , κ κ are determined on the basis of experimental data and generally depend on the temperature. It should be noted that this system of equations has arbitrariness, since it is impossible to determine the parameters of the equations from experimental data independently of each other [6]. There is no uniform method for determining the parameters. There is no unified method for determining the parameters, and when choosing it, researchers, as a rule, are guided by the desire to describe the experimental data as best as possible. To describe creep and damage accumulation, authors [12] introduce the value of the dissipation power c A ij ij W = ε σ  , where c ij ε , ij σ are the components of the creep strain and stress tensors (symbol “point” denotes the derivative with respect to time t ), while it is assumed that the work of dissipation at the time of fracture is constant * const A = (energy approach in the version of O. V. Sosnin). Using the phenomenological approach of Yu. N. Rabotnov to describe deformation, authors [13] generalize the application of the energy approach of the kinetic equations on the case in which the creep strain at fracture time isn’t constant value * const c ε ≠ ( * const A ≠ ). This paper demonstrates the possibility of describing deformation processes using the Sosnin - Gorev model [13] in the case when the function * ( ) c ε σ on the “strain – time” creep diagrams at stresses const σ = is non-monotonic. The method of determining the parameters of the constitutive equations of creep and damage is described. The choice of deformation modes in order to reduce the level of damage accumulation to increase the product life during production and operation is an urgent task. The papers of I. Y. Tsvelodub and