On the Theory of Elasticity of Microinhomogeneous Media with Account for Stochastic Changes in the Connectivity of Constituent Components

The paper proposes a mathematical model aimed at calculating the effective elastic moduli of a micro-inhomogeneous two-component isotropic composite material, which components are connected randomly depending on the level of their relative volumetric contents. A stochastic equation is formulated for the connectivity parameter of the constituent components, according to which, with an increase in the volumetric content of the filler, individual inclusions build the structures of the matrix mixture in the form of interpenetrating frameworks, and then turn into a new binding matrix with individual inclusions from the material of the rest of the old matrix. The algorithm for the numerical solution of this stochastic differential equation is constructed in accordance with the Euler-Maruyama method. For each implementation of this algorithm, the corresponding stochastic trajectories are constructed for the random connectivity function of the constituent components of the composite material. A variant of the method aimed at calculating the mathematical expectation of a random connectivity function of the constituent components has been developed and the corresponding differential equation has been obtained for it. It is shown that the numerical solution of this equation and the average value of the production factor function calculated for all realizations of stochastic trajectories give close numerical values. New macroscopic constitutive relations are found for microinhomogeneous materials with a variable microstructure and their effective elastic moduli are calculated. It is noted that the formulas for these effective elastic moduli generalize the known results for isotropic composite materials. The values of the effective elastic moduli, constructed according to the expressions obtained in the paper, lie within the Khashin-Shtrikman range for the lower and upper bounds of the effective elastic moduli of the composite materials. The numerical analysis of the developed models showed a good agreement with the known experimental data.

Studying Deformation Behaviors in Austenitic Stainless Steels within a Temperature Range of 143 K < T < 420 K

This work deals with studying staging and macroscopic strain localization in austenitic stainless steel 12Kh18N9T within a temperature range of 143 K < T < 420 K. The visualization and evolution of macroscopic localized plastic deformation bands at different stages of work hardening were carried out by the method of the double-exposure speckle photography (DESP), which allows registering displacement fields with a high accuracy by tracing changes on the surface of the material under study and then comparing the specklograms recorded during uniaxial tension. The shape of the tensile curves σ(ε) undergoes a significant change with a decreasing temperature due to the γ-α'-phase transformation induced by plastic deformation. The processing of the deformation curves of the steel samples made it possible to distinguish the following stages of strain hardening, i.e. the stage of linear hardening and jerky flow stage. A comparative analysis of the design diagrams (with the introduction of additional parameters of the Ludwigson equation) and experimental diagrams of tension of steel 12Kh18N9T for different temperatures is carried out. The analysis of local strains distributions showed that at the stage of linear work hardening, a mobile system of plastic strain localization centers is observed. The temperature dependence of the parameters of plastic deformation localization at the stages of linear work hardening has been established. Unlike the linear hardening, the jerky flow possesses the propagation of single plastic strain fronts that occur one after another through the sample due to the γ-α' phase transition and the Portevin-Le Chatelier effect. It was found that at the jerky flow stage, which is the final stage before the destruction of the sample, the centers of deformation localization do not merge, leading to the neck formation.

Influence of preliminary thermal aging on the residual interlayer strength and staging of damage accumulation in structural carbon plastic

Aging of composites is a pervasive problem that leads to mechanical properties degradation, reduced design life of a structure and premature accidental failure. The work is devoted to an experimental study of the preliminary temperature aging effect on the residual mechanical properties of structural CFRP. The joint use of test systems and systems for registration and analysis of acoustic emission signals was applied. The Short Beam Shear Test of CFRP specimens were carried out using the short beam method. The tests were carried out on universal electromechanical systems Instron 5882 and Instron 5965 in accordance with the recommendations of ASTM D2344. In the process of loading the samples were continuously recorded by using the acoustic emission signals system AMSY-6. A piezoelectric sensor with a frequency range of 300-800 kHz was used. The test and diagnostic systems were synchronized during the tests. In the course of the work the values of the interlayer shear strength were determined for the samples of CFRP. Typical types of the sample destruction are illustrated. When analyzing the change in the mechanical properties of the carbon fiber reinforced plastic from a temperature increase the critical values of temperatures were established in which a sharp decline in the strength and elastic characteristics of materials occurs due to an active destruction of the binder. The graphs of the energy parameter dependence and frequency characteristics of acoustic emission signals on time have been constructed and analyzed. The estimate of the processes of damage accumulation in composites is carried out. The change of the damage accumulation mechanisms was illustrated. The obtained results illustrate the effect of elevated temperatures and the duration of their impact on the mechanical behavior of structural CFRP specimens during the static tests for the interlayer shear.

The analysis of flutter characteristics based on generalized parameters of eigen modes of vibrations

Flutter numerical analysis of a dynamically scaled model (DSM) of a high aspect ratio wing was performed using experimentally obtained generalized parameters of eigen modes of vibrations. The DSM is made of polymer composite materials and is designed for aeroelastic studies in a high-speed wind tunnel. As a result of the analysis, safe operation conditions (flutter limits) of the DSM were determined. The input data to develop the flutter mathematical model are DSM modal test results, i.e. eigen frequencies, mode shapes, modal damping coefficients, and generalized masses obtained from the experiment. The known methods to determine generalized masses have experimental errors. In this work some of the most practical methods to get generalized masses are used: mechanical loading, quadrature component addition and the complex power method. Errors of the above methods were analyzed, and the most reliable methods were selected for flutter analysis. Comparison was made between the flutter analysis using generalized parameters and a pure theoretical one based on developing the mathematical model from the DSM design specifications. According to the design specifications, the mathematical model utilizes the beam-like schematization of the wing. The analysis was performed for Mach numbers from 0.2 to 0.8 and relative air densities of 0.5, 1, 1.5. Comparison of the two methods showed the difference in critical flutter dynamic pressure no more than 6%, which indicates good prospects of the flutter analysis based on generalized parameters of eigen modes.

Numerical Algorithms for Simulation of a Fluid-Filed Fracture Evolution in a Poroelastic Medium

The paper deals with the computational framework for the numerical simulation of the three dimensional fluid-filled fracture evolution in a poroelastic medium. The model consists of several groups of equations including the Biot poroelastic model to describe a bulk medium behavior, Reynold’s lubrication equations to describe a flow inside fracture and corresponding bulk/fracture interface conditions. The geometric model of the fracture assumes that it is described as an arbitrary sufficiently smooth surface with a boundary. Main attention is paid to describing numerical algorithms for particular problems (poroelasticity, fracture fluid flow, fracture evolution) as well as an algorithm for the coupled problem solution. An implicit fracture mid-surface representation approach based on the closest point projection operator is a particular feature of the proposed algorithms. Such a representation is used to describe the fracture mid-surface in the poroelastic solver, Reynold’s lubrication equation solver and for simulation of fracture evolutions. The poroelastic solver is based on a special variant of X-FEM algorithms, which uses the closest point representation of the fracture. To solve Reynold’s lubrication equations, which model the fluid flow in fracture, a finite element version of the closet point projection method for PDEs surface is used. As a result, the algorithm for the coupled problem is purely Eulerian and uses the same finite element mesh to solve equations defined in the bulk and on the fracture mid-surface. Finally, we present results of the numerical simulations which demonstrate possibilities of the proposed numerical techniques, in particular, a problem in a media with a heterogeneous distribution of transport, elastic and toughness properties.

The Image-Based Finite Element Evaluation of the Deformed State

The article presents one of the possible approaches to modeling objects with anisotropic properties based on images of the study area. Data from such images are taken into account when building a numerical model. In this case, material inhomogeneity can be included by integrating the local stiffness matrix of each finite element with a certain weight function. The purpose of the presented work is to develop a finite element for the formation of a computational ensemble and simulation of mechanical behavior taking into account the data of two-dimensional medical images. To implement the proposed approach, we used the assumption that there is a correlation between the values in the image pixels and the elastic properties of the material. Meshing was based on a four-node plane finite element. This approach allows using the quantitative phase or scanning electronic images, as well as computed tomography data. A number of test problems for compression of elementary geometry samples were calculated. The distal part of the rat femur was considered as a model problem. A computed tomography scan of the sample was used to construct a numerical model taking into account the inhomogeneity of the material distribution inside the organ. The distribution field of the nodal displacements based on data obtained from the images of the study area is presented. Within the framework of a model problem, we considered how a computer tomograph resolution influences the quality of the obtained results. For this purpose, calculations were carried out based on compressed input medical images.

Modeling the Effective Conductive Properties of Polymer Nanocomposites with a Random Arrangement of Graphene Oxide Particles

Сomposite materials are widely used in various industrial sectors, for example, in the aviation, marine and automotive industries, civil engineering and others. Methods based on measuring the electrical conductivity of a composite material have been actively developed to detect internal damage in polymer composite materials, such as matrix cracking, delamination, and other types of defects, which make it possible to monitor a composite’s state during its entire service life. Polymers are often used as matrices in composite materials. However, almost always pure polymers are dielectrics. The addition of nanofillers, such as graphene and its derivatives, has been successfully used to create conductive composites based on insulating polymers. The final properties of nanomodified composites can be influenced by many factors, including the type and intrinsic properties of nanoscale objects, their dispersion in the polymer matrix, and interphase interactions. The work deals with modeling of effective electric conductive properties of the representative volume elements of nanoscale composites based on a polymer matrix with graphene oxide particles distributed in it. In particular, methods for evaluating effective, electrically conductive properties have been studied, finite element modelling of representative volumes of polymer matrices with graphene oxide particles have been performed, and the influence of the tunneling effect and the orientation of inclusions on the conductive properties of materials have been investigated. The possibility of using models of resistive strain gauges operating on the principle of the tunneling effect is studied. Based on the finite-element modeling and graph theory tools, we created approaches for estimating changes in the conductive properties of the representative volume elements of a nanomodified matrix subjected to mechanical loading.

Research and Modeling of Viscoelastic Behavior of Elastomeric Nanocomposites

The paper presents results of studying mechanical properties of polymer composites depending on types of filler particles (granular - carbon black, nanodiamonds; layered - graphene plates; fibrous - single-walled nanotubes). These nanofillers differ greatly from each other in their structure and geometry. A significant difference in behavior of nanocomposites was revealed even with little introduction of particles into the elastomer. The highest level of reinforcement of the matrix was obtained when single-wall nanotubes and detonation nanodiamonds were used as fillers. The viscoelastic properties and the Mullins softening effect [1-4] were investigated in experiments performed with material samples subjected to complex uniaxial cyclic deformation. In these experiments, the amplitude of deformations was changed step by step; and at each step a time delay was specified to complete rearrangement processes of the material structure. It was found that a pronounced softening effect after the first cycle of deformation and significant hysteresis losses occur in the material filled with single-walled nanotubes. These characteristics are insignificant for the rest of nanocomposites until elongation increases twofold. In accordance with the obtained results, a new version of the mathematical model to describe properties of the viscoelastic polymer materials was proposed. The constants of the constitutive relations were calculated for each material; the theoretical and experimental load curves were compared. As a result, the introduced model is able to describe the behavior of elastomeric nanocomposites with a high accuracy. Moreover, this model is relatively easy to use, suitable for a wide range of strain rates and stretch ratios and does not require the entire history of deformation as needed for integral models of viscoelasticity.

Interpretation of the generalized parameter of the probability of failure through the plastic stress intensity factor

The main purpose of this work is to statistically analyze the fracture toughness of compact specimens made of S55C steel in terms of elastic and plastic stress intensity factors. The fracture toughness tests results at three-point bending were used for a comparative statistical analysis of the fracture parameters. Five type of specimen configuration with various thicknesses were tested at a constant ratio between crack length and specimen width. The critical loads were obtained as a tests result for various combinations of crack length and specimen thickness. In addition, uniaxial tensile tests were carried out to determine the main mechanical properties of the material. Obtained material properties were used in numerical calculations. Numerical calculations were carried out to determine the elastic and plastic stress intensity factors. Three-dimensional finite element analysis was performed on the basis of the experimental data on curvilinear crack front positions in tested specimens. The crack tip stress-strain fields were obtained for each of the tested samples as a result of numerical calculations. These fields were used to calculate the values of the plastic intensity factors along the curvilinear crack fronts. A statistical analysis of the fracture toughness of compact specimens made of S55C steel was carried out based on the obtained critical values of elastic and plastic stress intensity factors. The advantages of using the plastic stress intensity factor as a generalized parameter for the fracture probability are demonstrated. In addition, the sensitivity of the plastic stress intensity factor to constraint effects avoids the introduction of additional parameters into the statistical models.

Experimental determination and finite element analysis of coefficients of the multi-parameter Williams series expansion in the vicinity of the crack tip in linear elastic materials. Part II

In this study coefficients of the multi-parameter Williams power series expansion for the stress field in the vicinity of the central crack in the rectangular plate and in the semi-circular notched disk under bending are obtained by the use of the finite element analysis. In SIMULIA Abaqus, the finite element analysis software, the numerical solutions for these two cracked geometries are found. The rectangular plate with the central crack has the geometry similar to the geometry used in the digital photoelasticity. Numerical simulations of the same cracked specimen as in the experimental photoelasticity method are performed. The numerical solutions obtained are utilized for the determination of the coefficients of the Williams series expansion. The higher-order coefficients are extracted from the finite element method calculations implemented in Simulia Abaqus software package and the outcomes are compared to experimental values. Determination of the coefficients of the terms of this series is performed using the least squares-based regression technique known as the over-deterministic method, for which stresses data obtained numerically in SIMULIA Abaqus software are taken as inputs. The plate with a small central crack has been considered either. This kind of the cracked specimen has been utilized for comparison of coefficients of the Williams series expansion obtained from the finite element analysis with the coefficients known from the theoretical solution based on the complex variable theory in plane elasticity. It is shown that the coefficients of the Williams series expansion match with good accuracy. The higher-order terms in the Williams series expansion for the semi-circular notch disk are found.