Mechanics of Heterogeneous Materials

We discuss the non-linear stress–strain behavior of microcracked polycrystalline ceramics under uniaxial tension and compression (displacement control). Micromechanics explanation and modeling of its basic features, such as non-linearity and hysteresis in stress–strain curves, are developed, with stable microcrack propagation and “roughness” of intergranular cracks playing critical roles in tension and crack sliding playing a critical role in compression. Experiments involving complex loading histories are explained, and the model is shown to reproduce the basic features of the observed stress–strain curves.

This study presents the refined theory of elastic plates developed by A. D. Shamrovskii. This theory has some important features that differ it from well-known Uflyand-Mindlin theory.

Charge propagation through the films or fibers of ferroelectric and piezoelectric polymers leads to their (ultra)acoustic signal generation and micro(electro)mechanical dynamics. Surface acoustic waves can be generated in piezoelectric polymer materials under the electron beam. Therefore, SEM methods can provide not only metastable surface acoustic wave potential contrasts, but also excitation/induction of micromechanical local movements in piezosamples associated with the propagation of the surface acoustic waves. The dynamic phenomena of piezoelectric charging, charge traveling/wandering over their surface, charge gating and leakage, and micro(electro)mechanical movements under the electron beam can be studied by stroboscopic electron microscopy. However, it can provide only visualization, but not quantification of the structural effects, and does not reveal their reversibility/irreversibility. Therefore, we propose to combine time-resolved or stroboscopic SEM methods with multifractal analysis methods. This paper provides examples of using this approach to analyze the behavior of ferroelectric polymer networks under the electron beam. It has been shown that, when the effect is reversible, the multifractal spectra also change reversibly and soon return to the initial profiles. In the case of an incomplete relaxation or “tetanic contraction” of the polymer fiber, the multifractal spectra profiles do not completely return to their initial state. Also, it has been shown that the graphs of generalized fractal dimension D(q) as a function of q-order moment for the polymer fibers do not change significantly, so they cannot serve as an indicator of the equilibrium shifts in the fiber. At the same time, the shape of the multifractal spectrum f(α) as the function of Lipschitz-Holder exponent graph changes significantly, and hence can be used as a characteristic descriptor and predictor of the fiber state. Such techniques of analysis and prediction can be useful for the design of “artificial muscles” based on piezoelectric polymers, as well as for the design of thread microfluidics using capillarity and (electron beam-driven) electrocapillarity effects. In general, the above principles can be applied to create engineering/bioengineering systems with controlled electrowetting based on ferroelectric polymer fiber materials.

Hematopoietic stem cells (HSCs) enable hematopoiesis throughout the lifespan of an organism. The process of transformation of these cells from endothelial ones of the embryonic aorta is known as an endothelial-to-hematopoietic transition (EHT) and was first discovered in embryos of zebrafish, which remains a primary model to study EHT. In the present paper, the mechanisms driving shape changes in zebrafish dorsal aorta during the main wave of HSCs production were studied. A minimalistic coarse-grained model of the aorta, which takes into account inhomogeneous stress in the vessel caused by the cell growth and migration processes was developed. We compared the results of our numerical simulations with experimental images of the aorta and demonstrated that the morphological features developed during the EHT phase can result from mechanical instability arising due to stress accumulation and inhomogeneous elasticity of the surrounding tissue matrix. The role of such a shape transition was discussed as well.

The homogenisation of the fracture toughness is considered in the context of a propagating hydraulic fracture. The radial (penny-shape) model is utilised, in order to incorporate the impact of the viscosity-toughness regime transition over time. A homogenisation strategy based on temporal averaging is investigated. This approach incorporates the instantaneous fracture velocity, meaning that it should remain effective in the case of step-wise crack advancement. The effectiveness of the approach is demonstrated for periodic toughness distributions, including those which are unbalanced, utilising a highly accurate solver.

This report describes novel methods of multi-angle multi-channel optical porometry (both in lensless and non-lensless versions) and scanning electron microscopic porometry, indicating the feasibility of their technical and algorithmic integration within the framework of CLEM (Correlative Light and Electron Microscopy) by creating correlative light and electron porometry. The data on the reconstruction of the surface texture of a complex pore using an incoherent light source and the Sobel–Feldman operator is presented. Examples of optical density and luminance isolines obtained for spectrozonal multiplexing of porometric images using Orekhov's technology are given. The image processing methods described in this report reconstruct and visualize the fillable volume of pores and the heterogeneity of their surface, which makes it possible to use the above methods as complementary data sources for volumetric porosimetry.

We discuss the Lamé problem on large deformations of an elastic hollow circular cylinder made of an anisotropic material with continuously distributed dislocations. For the distribution of edge dislocations, an exact solution is found in an explicit analytical form. The interaction of dislocations with external and internal hydrostatic pressures is studied. It is shown that scalar densities of the dislocation density tensor can be arbitrary, including the Dirac delta function. This density is used to model dislocations concentrated on a cylindrical surface inside a cylinder.

This work is devoted to the memory of our friend and colleague Igor Sevostianov, who did a great contribution to micromechanics of inhomogeneous solids. Elaboration on new models of elastic inclusions with non-ellipsoidal shapes was one of the main topics in his activity. Working under related problems in micromechanics of stress relaxation in various misfitting heterostructures, we suggest for this memorial book our recent work dealing with an elastic model of a finite-length tubular inclusion in an infinite matrix with the matrix inside its hole. We give and study an analytical solution for the stress fields of the inclusion and suggest a mechanism for the relaxation of these stresses through the formation of small rectangular prismatic dislocation loops in different places of the inclusion axial section. We determine and study the energy barriers and the critical conditions for the generation of the loops with special attention to the most preferred place of their nucleation within the inclusion wall and the most preferred shape of the loops. We show that (i) the most preferred place for loop generation is the region in the middle of the wall section at its inner boundary, (ii) the most preferable loops are elongated along this boundary, and (iii) the critical misfit value at which the generation of these loops becomes energetically favorable, decreases with the increase in the outer radius of the tubular inclusion and the ratio of its inner and outer radii and with a decrease in the inclusion height. Thus, the plane-ring-shaped inclusions of relatively larger radii and thinner walls are shown to be the least stable in case of this way of misfit stress relaxation.

The interactions between inhomogeneities and dislocations may affect the mechanical properties of heterogenous materials remarkably. However, the previous analytical solutions accounting for the interactions are mainly limited to inhomogeneities with idealized geometries. In this chapter, an efficient iterative computational scheme is presented for evaluating the elastic fields and consequently the interaction energy due to a screw/edge dislocation interacting with an arbitrarily shaped inhomogeneous inclusion, by employing the numerical equivalent inclusion method (NEIM) in conjunction with the two-dimensional fast Fourier transform (2D-FFT) technique. The effectiveness of the proposed method is illustrated by several examples, including an inhomogeneity with complex boundary, layered inhomogeneities of various shapes, and multiple inhomogeneities.

Stress-induced phase transformations in elastic solids with circular or elliptical holes as stress concentrators are considered. The evolution of the interface is described by a kinetic equation that relates the velocity of the interface with a configurational force equal to the jump of the normal component of the Eshelby stress tensor. Kinetics of propagation of a planar interface is studied analytically and numerically to verify the developed numerical procedure. Then the interface propagation in the vicinity of elliptical holes with various ratios of the semi-axes is analyzed basing on numerical modeling. It is studied how the shape of the hole and the thickness of the new phase layer affect the distribution of the configurational force along the interface. It is demonstrated how the stress concentration generated by the hole may induce a phase transformation even at external stress, at which no phase transformations occur in the absence of a stress concentrator.

This paper presents new work on gravity-induced wave motion of gyroscopic systems composed of gyropendulums connected by elastic springs. Classification of trajectories of a single gyropendulum is given, followed by the Floquet-Bloch analysis of the dispersion and localisation for waves within a periodic gyroscopic chain. We construct Green’s matrices to identify regimes of propagating modes and wave localisation, which correspond to elliptical motions of the nodal elements. The waveforms subjected to gravity for localised defect modes are discussed in addition to the effect of no pre-tension along the chiral chain. The analytical results are accompanied by illustrative examples.

The chapter includes three parts. The first one presents the research results on the microstructure and properties of a metal-matrix composite fabricated by the fusion of the Ti–6Al–4V wire and the TiC powder using electron beam additive manufacturing. The TiC_x/Ti–6Al–4V composites were characterized by the uniform distribution of TiC globular eutectic particles in the titanium matrix. The segregation of the TiC eutectic phase particles along the boundaries of primary β grains caused reducing their dimensions with rising the TiC volume fraction. In the TiC_8%/Ti–6Al–4V and TiC_20%/Ti–6Al–4V samples, primary β-phase grain sizes ranged from 30 to 100 µm. Inside them, martensitic α-phase plates were observed in addition to the TiC eutectic phase particles, which distribution density rose as the TiC volume fraction increased. The main phases in the TiC_x/Ti–6Al–4V composites were the α-Ti, β-Ti and TiC ones. The β-Ti volume fraction varied within 3–5% regardless of the TiC contents. According to the X-ray diffraction analysis data, rising the TiC volume fraction was accompanied by increasing the lattice parameters of the α-Ti solid solution due to the presence of carbon atoms. At the TiC contents less than 10%, the levels of residual compressive stresses varied in the range from 0.7 to 0.9 GPa, weakly depending on its volume fraction. In the TiC_20%/Ti–6Al–4V sample, the valued of residual compressive stresses was 1.5 GPa. Enhancing the TiC volume fraction in the TiCx/Ti–6Al–4V composites caused rising the microhardness values of both matrix and eutectic particles. Their maximum levels values (6400 and 8400 MPa, respectively) were found in the TiC_20%/Ti–6Al–4V sample. The TiC_x/Ti–6Al–4V composites were also characterized by the greater tensile strength values but lower ductility compared to those of the Ti–6Al–4V alloy sample, fabricated by the same EBAM method. The TiC_5%/Ti–6Al–4V sample possessed the maximum ultimate tensile strength of 1040 MPa. At the TiC volume fractions of 8% and more, the TiC_x/Ti–6Al–4V composites experienced almost no plastic strains and brittle fracture occurred when applied stresses exceeded their yield points. The second part reports patterns of the structure formation and their effect on the tribological properties of 3D-printed composites based on polyetheretherketone (PEEK) filled with nanoparticles of hydroxyapatite (HA) and polytetrafluoroethylene (PTFE). Compared with neat PEEK, its simultaneous loading with HA and PTFE deteriorated the composite structure to some extent. However, the wear rate level was greatly reduced and microabrasive damages to both steel and ceramic counterparts were eliminated by facilitating the transfer film formation. In addition to the self-lubricating effect of the formed composite structure, another (probable) reason for such a protection of the steel counterpart was the shielding effect of a transfer film from the standpoint of suppressing tribological oxidative processes during its interaction with PEEK. The slight lowering of the physical and mechanical properties of the composite fabricated by 3D printing, compared with hot-compressed one, was associated with the specifics of the additive manufacturing process. In this case, the interlayer adhesion had been reduced and the complete internal space filling had not been provided during the layer-by-layer formation of the composite macrostructure due to the decrease in the melt flow rate after loading PEEK with HA nanoparticles. Finally, the influence of forming intermediate phases between a matrix and inclusions on the evolution of the functional properties of composites is shown. The methods of micromechanics and the reactive diffusion theory have been applied for assessing changes in the functional properties of both Fe- and Ti–Al–C composites during their synthesis.

We discuss the periodicity cells problem of the homogenization theory for the fiber-reinforced plate – elastic and thermoelastic, both linear and nonlinear. A specific feature of the plate periodicity cells is that they have free surfaces, corresponding to the top and bottom surfaces of the plate. We carry out numerical analysis of the plate periodicity cells for linear elasticity, linear thermoelasticity, and nonlinear elasticity problems. As follows from our computations, the boundary layers appear at the top/bottom surfaces of the plates in all the mentioned problems. We investigate the characteristics of the boundary layers for the unidirectional and cross reinforced plates. We introduce the notion of the representative plate and demonstrate that a three-layer plate is the representative plate for a plate with arbitrary number of layers. Furthermore, we use the notion of the representative plate to construct the homogenized strength criterion for the reinforced plates.

We deal with a layered composite formed from “thick” layers of unidirectional fibers (layers containing many unidirectional fibers). It is demonstrated that the local stress–strain state in such composite consists of local stress–strain states in “thick” layers and local stress–strain states at the layer interfaces. We theoretically predict and numerically confirm the presence of a boundary layer at the junction of adjacent fiber layers. This asymptotic phenomenon, to the best of our knowledge, has not been described earlier.

Over last couple decades, various industries use more and more coatings to improve contact performance of different machine parts. Coatings are used to reduce contact friction, reduce energy losses, improve wear and pitting resistance, etc. In some cases, coating applications are limited by the coating endurance. In most cases, it occurs due to coating delamination. The paper is focused on two things: (a) modeling and analyzing the behavior of the interface cracks between a coating and a substrate and (b) a subsequent statistical modeling of coating delamination life. To do the first part, a contact problem for a flat rigid punch indented in a double layered elastic half-plane with a crack parallel to its surface and located in the middle of the intermediate coating is considered. The punch is indented with friction while crack surfaces are free from friction and can partially open/close or completely open/close. The punch slowly moves along the surface. The elastic materials of the coatings and the substrate are homogeneous but different and coatings are firmly bonded between each other and to the substrate. The second part of the paper considers how initial statistical distribution of small interfacial cracks grows under the action of cycling loading and how to effectively determine the delamination time of the upper coating. Some formulas for the coating delamination life are obtained and analyzed.

Electromagnetism in electronic components becomes relevant in high frequencies, where induced currents cause a deviation of the response from “slow” operation conditions. As electric field and magnetic flux alter in time, dissipative effects occur due to different mechanisms such as viscous deformation and Joule’s heating. Indeed, dissipation is an energy loss and thus the industry tries to minimize this effect by amending engineering design. For a better prediction of an electronic system with many components, a detailed multiphysics simulation fails to be feasible. Hence, by using a reduced order modeling, we discuss how to involve energy dissipation in electromagnetism computations with the finite element method (FEM) and propose a connection with an industrial quality measurement out of such computations.

Classical Eshelby’s formulas are generalized assuming that the stored energy density of the media depends on the arbitrary number of internal variables and their first and second gradients. Clapeyron’s theorem and the standard Eshelby approach are used to derive the relation between the total energies stored in the media with and without inhomogeneities. Derived relations can be used in the energy-based homogenization methods of micromechanics.

This chapter presents a brief introduction to the background and development of contact modeling and analyses for heterogeneous material with the numerical equivalent inclusion method (NEIM). Contact theory and some detailed computational advancements are discussed to shed light upon the fundamentals of the interdisciplinary research on computational contact micromechanics. The NEIM is capable of dealing with any distribution of arbitrarily shaped material inhomogeneities, which constitute the computational domain that is discretized into hundred thousands of elements. The computational efficiency is tremendously enhanced with the assistance of the conjugate gradient based iterative schemes. Some numerical examples are surveyed to show the effects of material heterogeneity on the contact behaviors and mechanical responses of solids.

In this work, the homogenization theory is applied within the framework of three-dimensional linear micropolar media. The fundamental results derived by the asymptotic homogenization method to compute the effective engineering moduli for a laminated micropolar elastic composite with centro-symmetric constituents are summarized, in which the interface between the layer phases is considered imperfect spring type. The layers are considered with isotropic symmetry. Non-uniform and, as a particular case, uniform imperfections are assumed, where different imperfection parameters and cell lengths in the \(y_3\)-direction are assigned for the analysis. The analytical expressions of the engineering constants related to the stiffness and torque are given as functions of the imperfection parameters. The behavior of the engineering coefficients depending on the imperfection is studied. The influence of the imperfection and the cell length in the direction of the imperfection is observed. The present study allows validating other models and experimental results, as well as the investigation of fracture prediction in laminated composite materials.

In this paper, the exact solution of the poroelasticity problem for finite and semi-infinite rectangular regions in terms of Biot’s model is derived. To solve the problem, the analytical method based on the formulation of a mathematical physics’ boundary problem with mixed boundary conditions is used. It allowed to construct the exact solutions of the problems for permeable and impermeable boundary surfaces in an explicit form. The formulas for displacements, stress and pore pressure are used not only for quantitative, but also for qualitative analysis: the influence of different poroelastic constants, loading types and boundary conditions on the stress state of rectangular domains was investigated. The obtained formulas can serve as an etalon for the verification of effective numeric solving methods in poroelasticity problems.

Filtration of fluid from a fluid-saturated poroelastic medium by pressing is widely used in the processing industries of agriculture, medical, chemical, oil, gas and many other industries. The need for mathematical modeling of such processes is associated primarily with the selection of power pressing modes, the selection of materials for a power filtration unit, etc. Mathematical modeling of the pressing process consists in solving the quasi-static contact problem of upsetting a rigid indenter with a flat base shape into a fluid-saturated poroelastic medium. The base of the indenter is transparent to the pore fluid, which, when the base of the indenter settles, can accumulate both inside and outside the indenter. With the help of integral transformations, the posed contact problem is reduced to solving a two-dimensional integral equation of the first kind with a two-dimensional kernel depending both on the difference in coordinate variables and on the difference in time variables. Solving a two-dimensional integral equation using the Laplace transform reduces to solving the corresponding one-dimensional integral equation. The solution of the two-dimensional integral equation is obtained by inverting the solution of the one-dimensional integral equation using the inverse Laplace transform. The resulting solution makes it possible to obtain formulas for the velocities of the pore fluid, both at the contact and outside the contact, as well as the fluid flow rate through the base of the indenter.

The elastic equilibrium of a thin elastic beam is studied using asymptotic analysis starting from a 2D formulation within the context of plane elasticity. The aim of the paper is to elucidate the influence of density and hence small volume strain of Young’s modulus on the response of the beam. The adopted scaling at leading order supports the classical 1D Euler-Bernoulli beam approximation. The effect of strain-dependent Young’s modulus arises at the next order resulting in weak coupling between bending and extension deformations due to the asymmetry of the problem. The refined bending equation of interest is reduced to the form involving an explicit asymptotic correction to the prescribed transverse loading, similar to the previous considerations on the subject.