## No 1 (2018)

**Year:**2018**Articles:**10**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/14**DOI:**https://doi.org/10.15593/perm.mech/2018.1

A mathematical model for hydraulic fracture propagation in three dimensional poroelastic medium

#### Abstract

Currently hydraulic fracturing (HF) is a stimulation technique which is most widely used during industrial development of gas and oil reservoirs. At the same time widely used mathematical models of hydraulic fracturing are often oversimplified, as fracture geometry is assumed to be planar and predefined, a comprehensive treatment of geomechanical effects is seldom considered, and the fracture growth is often assumed using empirical criteria. Despite their successful applications, their possibilities are not sufficient for solving a number of important problems of reservoir development. This paper considers a complete three dimensional self-consistent mathematical model for large scale hydraulic fracture development. The model consist of several groups of equations including non-isothermal Biot poroelastic model to describe reservoir behavior, Reynold’s lubrication equations to describe flow inside fracture and corresponding “reservoir”/”fracture” interface conditions. The fracture’s geometric model assumes that it is an arbitrary smooth surface with a boundary. The fracture’s surface evolution is governed by the physically-sound criteria based on J-integral of Rice and Cherepanov in the vector from. The model is suitable for describing hydraulic fracture development as well as for the analysis of flow and geomechanical effects induced by normal operations of a fractured well. The main purpose of the suggested model is a consistent description of hydraulic fracture development in a general setting with a minimal number of a-priory assumptions and, at the same time, useful for solution of applied reservoir development problems using advanced numerical simulation techniques. Thus, we have also given a short review of the computational algorithms suitable for the model’s implementation.

**PNRPU Mechanics Bulletin**. 2018;(1):5-17

Numerical study of a thermomechanical behavior of a crystallizing polymer medium with regard to finite deformations

#### Abstract

The paper presents phenomenological constitutive relations for a crystallizing polymer medium, obtained in the framework of nonlinear solid mechanics. The relationships are based on the representation of the medium in the form of a composition of a molten and fully crystallized material, taking into account the history of a continuous nucleation and deformation of a new phase in the temperature range of phase transformations. A general formulation is carried out regarding the evolutionary boundary-value problem of the nonlinear mechanics of polymer materials under phase transitions with the use of the proposed constitutive relations. Algorithms are considered that are aimed at a numerical realization of the thermokinetic problem and determining the stress-strain state of the solidifying system for the case of a plane deformed state. A linearization procedure is developed - which is convenient for constructing numerical algorithms to solve the set evolutionary boundary value problems - using the assumption of the proximity of each intermediate configuration to the current one, which corresponds to the procedure for superimposing small deformations of crystallized particles on finite deformations of a crystallizing medium. The linearization procedure for the initial formulation of mechanics problem is realized taking into account temperature and structural deformations. A numerical algorithm aimed at solving a flat boundary value problem is developed and implemented with the aim of investigating the evolution of the stress-strain state in a polymer structure. A method is proposed and implemented to construct a discrete analogue of boundary value problems, based on the use of the Galerkin method with a selection basis functions with a compact support by the finite element method. In this case, the increments of the displacement functions at the current time step are taken as nodal unknowns. The regularities of a shell type defect’s formation in the crystallizing polymer cylinder are established.

**PNRPU Mechanics Bulletin**. 2018;(1):18-28

Plastic deformation of materials sensitive to a type of stress state

#### Abstract

The paper considers main principles and equations of the plasticity theory for materials sensitive to stress state, i.e. materials which have different plastic deformation curves under uniaxial tension, compression and torsion (shear). Thus, such materials don’t have a unified plastic deformation curve under beam (simple) loading processes. The considered theory of plasticity refers to the plastic flow theory under combined hardening, in which yield surface radius is taken to be dependent on the first stress tensor invariant and parameter of the active stress state type. In this case the defining functions of the evolution equation for yield surface displacement are dependent on the parameter of the additional stress state type (state of microstresses). The parameter of the stress state type is determined as a ratio between the third and second invariants in the power of 3/2 of the corresponding deviators, and under compression it is equal to 1, under tension it is equal to +1, under shear it is equal to 0. Plastic volume change (loosening) is considered within this theory in case of dependence between the yield surface on the first stress tensor invariant. For damage accumulation processes, the kinetic equation is presented based on the work of microstresses on the field of plastic deformations. In this equation the destruction energy is considered to be dependent on the first stress tensor invariant and type parameter of microstresses’ state. Material functions, completing the theory and their determination method are presented. Results of theoretical and experimental researches have been analyzed regarding the elastoplastic deformation of aluminum alloy D16T samples along double-part deformation trajectories, as well as samples made of 30HGSA steel under loading along double-part orthogonal stress trajectories. A satisfactory compliance between the calculation and experimental results has been obtained. The effect of “deformations splitting” was explored that lead to the fact that beam trajectories of deformation (stresses) may correspond to non-beam trajectories of stresses (deformations), and plain trajectories may correspond to non-plain ones.

**PNRPU Mechanics Bulletin**. 2018;(1):29-39

Experimental and theoretical study of the relation between phase and structural deformations in shape memory alloys

#### Abstract

The structure of the oriented martensite, which determines the macroscopic deformation of shape memory alloys (SMA), can be formed in two ways: directly from the austenitic phase as a result of direct phase transformation under load, and also from the chaotic martensite under its isothermal structural transformation. The deformation produced by the first method is called the phase deformation, the deformation produced by the second method is called the structural deformation, but the difference in the terms used reflects only the difference in the mechanisms of their initiation, whereas the final product - the oriented martensite - is the same for both types of deformation. Therefore, some SMA phenomenological models take into account the uniformity of these two deformation components by determining their interrelation through the direct transformation diagrams F1 (for phase deformation) and the diagrams of martensitic inelasticity F2 (for structural deformation). The theoretical framework of these models is based on the hypothesis that the process of further deformation of the oriented martensite does not depend on the mechanism of its formation and involves three material functions: F1, F2, and also the function of their interrelation f . The experimental study of wire samples of TiNi, described in this paper, was performed with the aim to substantiate the developed hypothesis and establish three material functions used in the suggested theoretical description. A new method for determining the function f , which can be used as a verifying experiment, is proposed. The range of validity of the hypothesis has been determined both for the succeeding isothermal deformation of the samples with initial phase and structural deformations, and for the processes associated with their subsequent heating. The experiments demonstrated that the further-orientation diagrams for such samples coincide, which is indicative of the cross- hardening effect.

**PNRPU Mechanics Bulletin**. 2018;(1):40-57

INHOMOGENEITIES IN GRAINS OF POLYCRYSTALLINE MATERIALS AND ESHELBY PROBLEM

#### Abstract

The paper presents the method aimed at calculating inhomogeneous strain fields in grains of polycrystalline materials. The calculations are based on the earlier developed method of solving boundary values problem for inhomogeneous polycrystalline bodies by means of the original perturbation theory variant based on analogies with the quantum fields theory. The boundary value problem for inhomogeneous strain fields in a differential form transforms into the integral equation for strains tensor. The solution of the integral equation is formed as a series upon the intensity of strains interaction. This allows interpreting inhomogeneous strain at any point in a grain as a superposition of macrostrain, caused by boundary conditions and two components conditioned by intragrain and intergrain interaction. It is shown that in untextured polycrystals, despite the long range type of elastic interaction, one can take into account the interaction only with the nearest and second neighbor grains to evaluate how the intergrain interaction influences the inhomogeneity in the given grain. The contributions of interactions with farther grains mutually annihilate each other. The strain field inhomogeneous within one grain is approximated by the step-wise constant function. For that, each grain is divided into a great quantity of small subgrains, where subgrain strain fields are supposed to be homogeneous. This approximation reduces the integral equations for local strains into linear algebraic ones, which are solved numerically. The application of this method to a classical problem related to calculating strains in a spherical inclusion embedded into the infinite matrix gives Eshelby solution. The numerical evaluation of strain inhomogeneities is made using model zinc polycrystals. Close to boundaries in spherical grains the extreme strain values, caused by intergrain interaction, surpass mean strain values by 30 percent. The strains concentration is much higher in materials with a lower elastic symmetry of grains.

**PNRPU Mechanics Bulletin**. 2018;(1):58-72

Mathematical model of pulse scanning of pressure along a piezo-electro-luminescent fiber-optical sensor

#### Abstract

The mathematical model is developed aimed at locating pressure non-uniformity along a fiber optic-piezo-electro-luminescent sensor using the location scanning electrical video pulse with a step by step change of its value. The algorithm of finding the function of pressure distribution along a local section and the whole length of the sensor is developed based on the results of the intensity of light proceeding from a fiber optic phase measured on the edge section of the sensor for a case of nonlinear “luminescence function”, which is a dependence between the intensity of light and voltage acting on the luminescent element. The problem is reduced to the solution of the Fredholm integral equation of the 1st kind with the differential kernel depending on the operating and informative transfer coefficients of the sensor and on the set luminescence function of the element. Analytical solutions for the pressure distribution functions along the sensor are obtained for special cases, when the kernel or the function of distribution density is expressed via delta function, and the Fredholm integral equation becomes algebraic. Domains of admissible values of the operating voltage for various modes of diagnostics of pressure distribution are defined. Results of numerical solutions of direct and reverse problems for non-uniform distribution of pressure by means of “pointed” scanning of this pressure are presented by an extremely narrow impulse of the operating voltage. Luminescence functions at the exit from the optical fiber at various time points and values of the impulse of the operating voltage taking into account the set luminescence function of an electroluminescent element are found in the direct problem. The distribution of pressure depending on the values of the luminescence intensity function at various time points is found in the reverse problem.

**PNRPU Mechanics Bulletin**. 2018;(1):73-82

A model of crystallization process taking into account phase changes in formation of a metallic material by laser fusion method

#### Abstract

Theoretical principles of models of crystallizing bodies being developed are now applied to model the technological process of selective laser welding regarding the description of interaction between “melt - solid body” and determining a body’s buckling owing to emergence of residual stresses in a cooling down work piece. At the same time one of the relevant problems of the research is creating the defining relations which make it possible to describe the interaction between “melt - solid body”. This work considers the application of the known defining relations of the viscoelastic growing body for the hardening metal alloy. The crystallization process takes place in a wide temperature range and is followed by structural changes. A crystallization process is a transition of metal material from a liquid state into the hard one. Within the research, the general statement of the boundary problem of mechanics of the crystallizing body is considered. Realization of the defining relations describing the crystallization process is executed using two problems, i.e. the problem related to the nonuniform controllable cooling of the rod and plate’s cooling from the temperature that is higher than the melting point of the studied material. Thus, numerical models of crystallization of isotropic bodies taking into account phase transfers for linear and two-dimensional statements based on the finite element method have been obtained. The results of applying the finite element method based on the new defining relations do not contradict the physics of the crystallization process and can be applied, when modeling the selective laser welding of metal materials taking into account phase transfer. Images of stress-strain states of the structure have been obtained for each model problem: fields of distribution, stresses and strains. Also, the analysis of the numerical solution convergence is performed, the fulfillment of the natural boundary conditions is analyzed, the temperature fields and the degree of crystallization are obtained within the framework of the study.

**PNRPU Mechanics Bulletin**. 2018;(1):83-92

ESTIMATION OF THE CYLINDRICAL COMPOSITE SHELL STIFFNESS AT THE INITIAL STAGE OF CURING DURING DEPLOYMENT BY INTERNAL PRESSURE

#### Abstract

An experimental-calculation method aimed at evaluating the stiffness properties of a cylindrical shell structure made of a composite material with a polymer matrix at the initial stage of its curing is considered. Evaluation of the stiffness composition parameters at this stage of polymerization due to the different physical state of the reinforcing elements and the binder by the methods of composite materials mechanics leads to poorly determined stiffness matrices that are not suitable for a reliable description of the mechanical behavior of the structure. The research relevance is related to the study of the manufacturing large-sized pneumatic structures technology based on compositions subjected to curing in space conditions. In the proposed method, the cylinder deployment pressure (the pressure at which the cylinder diameter assumes a nominal value) corresponding to the current binder polymerization degree is determined experimentally. The degree of polymerization is characterized by viscosity and the dynamic polymer module, measured with the viscosimeter. The structure specification as well as devices used in the experiments, instruments that fix the measured state parameters and the test procedure description are provided. The effective modulus of the cylinder’s elasticity of the material has to be corresponding to its stiffness characteristics during the loading by internal pressure to the deployment pressure. It is determined by the method of successive approximations based on a geometrically nonlinear elastic model. The deployment time is much shorter, than the total binder curing time. By comparing the experimental and calculated data, the dependence of the effective elasticity modulus on the binder curing parameters is established. The almost linear dependence of the cylinder deployment pressure on the effective modulus of elasticity is revealed. It allows to extrapolate the results of the study to the values of binder parameters that are not backed up by the experience. These results allow us to evaluate the internal pressure necessary for the deployment of composite cylindrical shells with a partially cured binder, by solving the problems of solid mechanics.

**PNRPU Mechanics Bulletin**. 2018;(1):93-99

Modelling the formation of new material surfaces during adhesive delamination of a composite

#### Abstract

The model of a composite material adhesive delamination is developed. The stress state of an adhesive bound varies to nil, when the bonds with the connected body are broken in the thermodynamic process which represents the delamination. The interaction between the part of the composite including the adhesive layer and the rest of the body is terminated as a result of delamination. We have obtained a system of two variational rate equations of the equilibrium flow of the process to describe the subcritical deformation and delamination. The averaging of the stress-strain state in the adhesive layer allows us to avoid singularity in the dead-end of the formed mathematical cut. The motion along the layer’s bounds of the delamination surface does not lead to singularity uprising. When solving the problem of the subcritical deformation, we have distinguished a small δ-surface on the bound of the adhesive, where the delamination criterion is reached. The load (node forces) distribution on the δ-surface is determined by a repeated solving of the subcritical deformation problem. But the law of motion of the adhesive layer bound at a current stage is known from the initial solution. The problems about simple unloading of the -surface of a body and keeping the external loading value on the level of the delamination start are solved. As a result, the body’s stress strain in the beginning of the local unloading differs from its state, when δ-unloading ends. For the linear elastic problem, we have performed a comparison between the results of the problem solving within the framework of the current model and the results for the model of a cohesive delamination, where a complete destruction of the cohesive layer is assumed. A substantial difference in boundary displacements of main composite layers during the destruction is established after the discontinuity surface’s growth between the adhesive layer and primary material.

**PNRPU Mechanics Bulletin**. 2018;(1):100-109

Applied grinding model of a solid particle with a simple shape on impact with a hard surface

#### Abstract

The complexity of phenomena caused by grinding and fracturing of solid particles makes it difficult to provide the theoretical description of this process. In this case, it is important to establish relationships between the parameters that determine the characteristics of the grinding process, determine the degree of their influence on each other, create and analyze the grinding process model taking into account crusher’s parameters, physical and mechanical properties of the material. Consequently, the improvement of formal calculation methods and justification of rational parameters of crushers ensure the effectiveness of their use during operation. By analyzing real materials’ state, a large group of scientists have created a number of theories explaining fracture conditions and mechanisms in solid materials. However, it is quite difficult to apply the existing theories for calculation of grinding processes. Therefore, there is a need to develop a new simple and convenient theory for practical application. Authors offer a new method aimed at theoretical description of a material’s fracture. Based on the simplified energy hypothesis and applied technical theory of wave spreading in elastic continuous medium, we have obtained a new refined solution of the fundamental dynamic mechanical problem of an elastically deformable rigid body about a longitudinal collision of a beam having a constant arbitrary cross section (simulating the material’s particle) with an absolutely rigid surface (simulating the working body of the crusher), taking into account the time parameter and linear dimension of the moving rod element (particle). The developed refined mechanical and mathematical model, which has been reduced to applicable calculated analytical dependencies and illustrated by typical numerical examples, allows to quantify the strength of the solid particle under destruction and grinding, makes it possible to implement a comprehensive descriptive approach regarding the dynamic process of material particles’ grinding by regulating and selecting the optimal physical and geometric characteristics, which provide the required grinding quality and predict the particles’ grinding process depending on the process parameters.

**PNRPU Mechanics Bulletin**. 2018;(1):110-120