Rigid Spherical Inclusion Research Articles

Application of artificial neural networks in atomic force microscopy

The paper describes a method of artificial neural network application to build a computer database for numerical simulation of the process of indentation of an atomic force microscope probe into an elastomeric composite with a granular filler (nonlinear elastic medium with rigid spherical inclusions). The use of such a base makes it possible to significantly improve the speed and quality of interpretation of the results of nanoindentation for structurally inhomogeneous materials. In this case, information becomes available not only about what is done on the surface of the sample but also in the near-surface layer inside it. An algorithm has been developed with the help of which an artificial neural network was built and “trained,” designed to obtain indentation curves depending on the size of the filler particles and its localization in the near-surface layer of the composite (depth and horizontal distance from the top of the AFM probe). It is shown that the speed of constructing indentation curves increases by several orders of magnitude compared to conventional approaches based on the numerical solution of the corresponding boundary value problems for each specific case. Accordingly, computer costs are also significantly reduced, that is, in the presence of an already built and “trained” neural network, powerful and high-speed computers are not needed.

Proximity Effects in Matrix-Inclusion Composites: Elastic Effective Behavior, Phase Moments, and Full-Field Computational Analysis

This work focuses on the effects of inclusion proximity on the elastic behavior of dilute matrix-inclusion composites. Rigid or soft monodisperse spherical inclusions are considered with moderate volume fractions. To conduct this study, Representative Volume Elements (RVE) with an effective local minimum distance between inclusions varying between the sphere’s radius and one-tenth of the radius are built. Numerical finite element calculations on the RVE are performed. The obtained homogenized elastic properties, as well as the phase stress moments (first and second), are compared to Mori–Tanaka estimates, which are well established for this kind of composite. The behavior of local fields (stresses) in the microstructure with respect to inclusion proximity is also analyzed. It follows that the effective properties and phase stress moments converge asymptotically to the Mori–Tanaka estimates when the minimal distance between spheres increases. The asymptote seems to be reached around a distance equal to the sphere’s radius. Effective and phase behaviors show a deviation that can achieve and even exceed (for the second moments) ten percent when the inclusions are close. The impact of the inclusions’ proximities is even more important on local stress fields. The maximum stress values (hydrostatic or equivalent) can be more than twice as high locally.

Effective thermal conductivity of composite materials made of a randomly packed densified spherical phase

The prediction of the effective thermal conductivity of composite materials is of paramount importance for many applications. While there exist correlations for bi-phase materials made of rigid spherical inclusions in a continuum matrix, there is lack of correlations for composites whose spherical phase is densely packed, as in densified granular materials or in foams with overlapping bubbles. In this study we combine a packing algorithm with a finite difference solver to determine the effective thermal conductivity when the spherical phase is highly overlapping, with volume fraction larger than 55%, for all the conductivity ratios of the two phases. Three fit relations are proposed to encompass the whole range of conductivities, showing good accuracy for both simulated and experimental data and outperforming existing relations commonly used for the estimation of the effective thermal conductivity. Our relations can also be used for estimating the effective electric conductivity, magnetic permeability and mass diffusivity in continuum regime.

Characterizing in situ poroelastic properties of cytoplasm by the translation of a rigid spherical inclusion

Poroelasticity of cytoplasm is a rate- and size-dependent biphasic material behavior that reflects the normal activities and pathological states of cells, mainly caused by the migration of fluid molecules and the deformation of porous solid skeleton (protein scaffold). While micro/nano-indentation tests have been extensively used to characterize the poroelasticity of a cell, characterizing the in situ poroelasticity of cytoplasm remains elusive. In this study, based on the theory of the translation of a rigid spherical inclusion, we proposed a new method to characterize the in situ poroelasticity of cytoplasm. Based on data from optical/magnetic tweezers tests, we estimated three key poroelasticity parameters—shear modulus, Poisson ratio and diffusion coefficient—of cytoplasm for a variety of cells, including cardiomyocytes, endothelial cells of bovine capillary, and fibroblasts. The proposed method provides a powerful tool for in situ measurement of poroelastic properties of cytoplasm via optical/magnetic tweezers.

Scattering of plane compressional waves in a poroelastic medium having two rigid spherical inclusions

ABSTRACT In this paper, scattering of plane compressional waves by two rigid inclusions in an infinite poroelastic medium is investigated. Laplace transform and Erdelyi's formula are used to satisfy the planar boundary conditions, and three infinite series equations are obtained by matching the boundary conditions on the surface of the sphere. The solution of these infinite series equations is obtained by successive approximations, and the existence and uniqueness of the solution are discussed.

Displacement field around a rigid sphere in a compressible elastic environment, corresponding higher-order Faxén relations, as well as higher-order displaceability and rotateability matrices.

An efficient route to the displacement field around a rigid spherical inclusion in an infinitely extended homogeneous elastic medium is presented in a slightly alternative way when compared to some common textbook methods. Moreover, two Faxén relations of next-higher order beyond the stresslet are calculated explicitly for compressible media. They quantify higher-order moments involving the force distribution on a rigid spherical particle in a deformed elastic medium. As a consequence, additional contributions to the distortions of the deformed elastic medium are identified that are absent to lower order. Furthermore, the displaceability and rotateability matrices for an ensemble of rigid spheres are calculated up to (including) sixth order in inverse particle separation distance. These matrices describe the interactions mediated between the rigid embedded particles by the elastic environment. In this way, additional coupling effects are identified that are absent to lower order, particularly when rotations and torques are involved. All methods and results can formally be transferred to the corresponding case of incompressible hydrodynamic low-Reynolds-number Stokes flow by considering the limit of an incompressible environment. The roles of compressibility of the embedding medium and of the here additionally derived higher-order contributions are highlighted by some selected example configurations.

Translation of a Coated Rigid Spherical Inclusion in an Elastic Matrix: Exact Solution, and Implications for Mechanobiology.

The displacement of relatively rigid beads within a relatively compliant, elastic matrix can be used to measure the mechanical properties of the matrix. For example, in mechanobiological studies, magnetic or reflective beads can be displaced with a known external force to estimate the matrix modulus. Although such beads are generally rigid compared to the matrix, the material surrounding the beads typically differs from the matrix in one or two ways. The first case, as is common in mechanobiological experimentation, is the situation in which the bead must be coated with materials such as protein ligands that enable adhesion to the matrix. These layers typically differ in stiffness relative to the matrix material. The second case, common for uncoated beads, is the situation in which the beads disrupt the structure of the hydrogel or polymer, leading to a region of enhanced or reduced stiffness in the neighborhood of the bead. To address both cases, we developed the first analytical solution of the problem of translation of a coated, rigid spherical inclusion displaced within an isotropic elastic matrix by a remotely applied force. The solution is applicable to cases of arbitrary coating stiffness and size of the coating. We conclude by discussing applications of the solution to mechanobiology.

Influence of heterogeneities on mechanical properties: Virtual material concept

The present article is connected to the study of a bituminous mixture (BM) material, used on modern railway track as sub-ballast layer. Influence of size and volume fraction of rigid monodisperse spherical inclusions, randomly packed into a hyperelastic matrix, on the mechanical behavior of an obtained composite structure was investigated using “Virtual Material” approach. 3D geometry of inclusions was defined and used for 3D printing and 3D finite element modeling. This approach allows a numerical study of a theoretical case without losing connection with real experiment (by means of direct geometrical correspondence). Parameters of 7 specimens were chosen in accordance with a Doehlert experimental design. Analysis performed on the basis of the “response surfaces” approach has shown that mechanical parameters of studied specimens have a strong dependence on the value of inclusions volume fraction and almost no dependence on the value of the size of inclusions. Stress/strain concentrations were analyzed using FE method in order to find and to visualize load-bearing chains going through the matrix. It was found that Von Mises stress in load-bearing chains is almost 8 times higher than the average in the matrix.

On the yield strength of a ductile material reinforced with rigid spherical inclusions

A micromechanics model for predicting the mechanical properties of a particulate composite is developed. The reinforcing particles are assumed to be rigid and spherical in shape, while the matrix is elastic–perfectly plastic. The interactions among the inclusions are taken into consideration by Mori–Tanaka’s approach. The elastic modulus, Poisson’s ratio and yield strength of the composite can be conveniently predicted and systematically investigated. The predictions of the model are discussed and compared with other theories. It is concluded that the yield strength of the composite can be improved by increasing the volume fraction of inclusions for most of loading conditions but not for certain loading paths such as spherical (hydrostatic) stress loading. The open-ended circular cylinder of v. Mises yield surface about the spherical stress axis in the principal stress space for the pure matrix changes to a more complicate yield surface ending at two apexes in the spherical stress axis for the composite.

Forces and torques on rigid inclusions in an elastic environment: Resulting matrix-mediated interactions, displacements, and rotations.

Embedding rigid inclusions into elastic matrix materials is a procedure of high practical relevance, for instance, for the fabrication of elastic composite materials. We theoretically analyze the following situation. Rigid spherical inclusions are enclosed by a homogeneous elastic medium under stick boundary conditions. Forces and torques are directly imposed from outside onto the inclusions or are externally induced between them. The inclusions respond to these forces and torques by translations and rotations against the surrounding elastic matrix. This leads to elastic matrix deformations, and in turn results in mutual long-ranged matrix-mediated interactions between the inclusions. Adapting a well-known approach from low-Reynolds-number hydrodynamics, we explicitly calculate the displacements and rotations of the inclusions from the externally imposed or induced forces and torques. Analytical expressions are presented as a function of the inclusion configuration in terms of displaceability and rotateability matrices. The role of the elastic environment is implicitly included in these relations. That is, the resulting expressions allow a calculation of the induced displacements and rotations directly from the inclusion configuration, without having to explicitly determine the deformations of the elastic environment. In contrast to the hydrodynamic case, compressibility of the surrounding medium is readily taken into account. We present the complete derivation based on the underlying equations of linear elasticity theory. In the future, the method will, for example, be helpful to characterize the behavior of externally tunable elastic composite materials, to accelerate numerical approaches, as well as to improve the quantitative interpretation of microrheological results.

A nanosized rigid spherical inclusion with interfacial diffusion and sliding

We examine the time-dependent deformations around a nanosized rigid spherical inclusion in an infinite elastic matrix under uniaxial tension at infinity. The elastic matrix is first endowed with separate Gurtin–Murdoch surface elasticity. Furthermore, interfacial diffusion and sliding both occur on the inclusion–matrix interface. Closed-form expressions of the time-dependent displacements and stresses in the matrix are derived by using Papkovich–Neuber displacement potentials. A concise and elegant expression of the steady-state normal stress on the surface of the inclusion is also obtained. It is seen that the displacements and stresses in the matrix evolve with two relaxation times which are reliant on three size-dependent parameters, one from surface elasticity and the other two from interfacial diffusion and sliding. Numerical results are presented to demonstrate the influence of surface elasticity on the relaxation times and on the stress distribution near the inclusion. It is observed that the surface elasticity can alter the nature of the steady state normal stress on the surface of the inclusion from tension to compression. When the radius of the inclusion is not greater than the ratio of residual surface tension to remote tension, the steady state normal stress on the surface of the inclusion is always compressive. The related problem of a nanosized rigid spherical inclusion with a spring-type imperfect interface is also solved. We find that it is feasible to design a neutral spherical inclusion that does not disturb a prescribed uniform uniaxial stress field or even any uniform stress field outside the inclusion through a judicious choice of the four imperfect interface parameters.

Permeability of textile fabrics with spherical inclusions

We investigated the effect of rigid spherical inclusions such as microcapsules and fillers on the permeability of fabrics by using glass beads as model inclusions. Beads with a range of diameters (40–800μm) and volume fractions (2.5–10%) were sieved between layers of woven E-glass fabric targeting a fiber volume fraction of 40%. Permeability measurements were completed by X-ray microtomography to analyze the samples pore structure and estimate their permeability using Computational Fluid Dynamics simulations. Experimental and numerical estimates were also fitted with a Kozeny model accounting for porosity and specific surface of samples. We identified a threshold curve related to bead diameter and volume fraction below which the permeability follows that of a plain packed textile, and above which a strong departure from this trend is observed, induced by strong distortions of the textile. This behavior was closely correlated to the internal features of the textile.

Indentation of a spherical cavity in an elastic body by a rigid spherical inclusion: influence of non-classical interface conditions

This paper examines the indentation of an elastic body by a rigid spherical inclusion. In contrast to conventional treatments where the contact between a rigid inclusion and the elastic medium is regarded as being perfectly bonded, we examine the influence of non-classical interface conditions including frictionless bilateral contact, separation and Coulomb friction on the load–displacement behaviour of the spherical rigid inclusion. Both analytical methods and boundary element techniques are used to examine the inclusion/elastic medium interaction problems. This paper also provides a comprehensive review of non-classical interface conditions between inclusions and the surrounding elastic media.

Controlled accuracy finite element estimates for the effective stiffness of composites with spherical inclusions

Mixed form finite element method was used to study the microstructural effects in the effective stiffness of composites reinforced or weakened by regular and random dispersions of non-overlapping identical spheres. Controlled accuracy stiffness estimates were achieved by using the unstructured mesh polynomial extension and asymptotic nested grid extension of the finite element method. Periodic regular and random microstructure computer models with up to a hundred spheres were considered. The significance of the elastostatic bridges separating near-to-touch spheres was studied and their impact on the effective stiffness was assessed. It was found that for a variety of practically relevant composites with moderate stiffness contrast between the phases, the effective stiffness estimates were remarkably insensitive to the underlying composite microstructure. The obtained finite element estimates were compared with predictions of the generalized self-consistent (GSC) model ( Christensen and Lo, 1979 ) and it was shown that the model was accurate over the entire range of practically relevant sphere fractions. We studied a high stiffness contrast composite consisting of a near incompressible rubber matrix filled with rigid spherical inclusions. In this case strong microstructural effects were observed but, nevertheless, for the random composite obeying the hard sphere Percus–Yevick statistics the GSC model remained strikingly accurate over the whole range of practically relevant inclusion fractions. The asymptotic Einstein limit GSC solution was derived and it was demonstrated that it gave accurate micromechanics-based solution of the classical Einstein homogenization problem of a random dispersion of perfectly rigid spherical inclusions in a fully incompressible matrix.

The nonlinear elastic response of suspensions of rigid inclusions in rubber: I—An exact result for dilute suspensions

A solution is constructed for the problem of the overall elastic response of ideal (Gaussian or, equivalently, Neo-Hookean) rubber reinforced by a dilute isotropic distribution of rigid particles under arbitrarily large deformations. The derivation makes use of a novel iterative homogenization technique in finite elasticity that allows to construct exact solutions for the homogenization problem of two-phase nonlinear elastic composites with particulate microstructures. The solution is fully explicit for axisymmetric loading, but is otherwise given in terms of an Eikonal partial differential equation in two variables for general loading conditions. In the limit of small deformations, it reduces to the classical Einstein–Smallwood result for dilute suspensions of rigid spherical particles. The solution is further confronted to 3D finite-element simulations for the large-deformation response of a rubber block containing a single rigid spherical inclusion of infinitesimal size. The two results are found to be in good agreement for all loading conditions. We conclude this work by devising a closed-form approximation to the constructed solution which is remarkably accurate and — as elaborated in Part II — proves particularly amenable as a fundamental building block to generate approximate solutions for suspensions with finite concentration of particles.

Spherical inflation of a class of compressible elastic bodies

Using a special model that belongs to a new class of elastic bodies wherein the Cauchy–Green stretch is given in terms of the Cauchy stress and its invariants, within the context of the spherical inflation of a spherical annulus, we show that interesting phenomena like the development of “stress boundary layers” manifest themselves. We consider two cases of boundary value problems, one in which there is a cavity in a sphere and the other in which there is a rigid spherical inclusion in a sphere. We show that in the case of a rigid inclusion, it is possible for a pronounced “stress boundary” layer to develop, in that the values of the stresses within this boundary layer that is adjacent to a spherical inclusion are much larger than external to it. We also show that in the case of both the cavity and a rigid inclusion, the stress concentration is an order of magnitude higher than the increase in the deformation gradient, that is, the stress and the stretch do not scale in a similar manner. While the stress adjacent to a rigid inclusion can be 2500 times the applied radial stress, the maximum stretch, which occurs at the rigid inclusion is about 10. While the variation in the stresses are linear in thin walled annular regions, we find that in thick walled annular regions, the variation of the stresses is non-linear.

The mechanics of atherosclerotic plaque rupture by inclusion/matrix interfacial decohesion

Histological investigation along with finite element analysis of arterial wall/atherosclerotic plaque geometries indicates the paradoxical result that ruptures often occur at sites with predicted stresses of half the plaque cap strength. Recent experiments have revealed calcified cells within the cap suggesting that these inclusions, situated close to the cap/luminal blood surface, precipitate rupture at low nominal loads by concentrating stress. In this paper, we investigate the proposition that rupture at low nominal loads occurs by (possibly brittle) decohesion of the calcification/cap interface followed by tearing of cap tissue. A novel boundary value problem is analyzed consisting of a remotely loaded linear elastic layer (extracellular matrix cap) containing a rigid spherical inclusion (calcified cell) that interacts with it through a nonlinear structural interface which models the binding of the calcified cell to the extracellular matrix via integrin receptor proteins. Equilibrium solutions are obtained from equations derived from the Boussinesq potentials for spherical domains. Results indicate a brittle character to the rupture process with the size of the domains between the inclusion center and the matrix surfaces determining the concentration of stress. For an inclusion close to a surface the abrupt unloading of the interface during brittle decohesion produces a sharp spike in circumferential stress. We conjecture that when this dynamic stress exceeds the cap strength, tearing occurs followed by thrombus formation and possibly infarction.

An original force–displacement relationship for spherical inclusions in multilayered viscoelastic finite media

This paper presents original solutions of the force–displacement relationships for a rigid spherical bead embedded in a composite medium made of n-isotropic linearly viscoelastic finite layers. Analytical solutions were provided for both compressible and incompressible elastic and viscoelastic solids, assuming no-slip conditions between the rigid spherical inclusion and its adjacent medium as well as between each layer of the composite medium. Thanks to these general formulas, we investigated the effect of finite size media on the force-bead displacement response and derived the exact relationship linking apparent and intrinsic elastic moduli of the medium. Such theoretical solutions can be interestingly applied to identify layer’s heterogeneities and to characterize accurately the mechanical properties of living material like cells when using translational microrheology assays. This point is especially illustrated by modeling animal cell cytoskeleton as a bilayer composite medium probed by magnetic tweezers. Interestingly, our results highlighted the influence of finite cell size effects, while allowing to distinguish viscoelastic properties of deep cell cytoskeleton from those of cellular cortex. Moreover, we established that translational microrheology experiments are well suited to characterize locally the viscoelasticity properties of the layer in contact with the probe as soon this layer thickness is larger than ten bead diameters.

Onset of residual stress fields near solitary spherical inclusions in a perfectly elastoplastic medium

When computing residual stresses in deformable solids, one has to use the theory of elastoplastic solids, because the final level and distribution of residual stresses is determined exactly by the accumulated reversible strains. In turn, to compute the elastic strains, one needs to determine the displacement field. The problem of determining displacements in statically determinate problems of the theory of perfect elastoplastic solids was considered for the first time in [1, 2]. The techniques proposed there permitted solving the problem of finding the residual stresses near a cylindrical cavity in a perfectly elastoplastic medium [3]. It was shown that secondary plastic flow [4] may arise in the unloading processes, which significantly redistributes the final residual stresses. In the present paper, we consider the loading and unloading problems for a ball with a rigid or elastic spherical inclusion. We study the onset of secondary plastic flow under unloading and compute the residual stresses. Thus, we model the onset of the residual stress field near a more rigid inhomogeneity. The case of a softer inhomogeneity was essentially considered in [3], where the onset of the residual stress field near a continuity flaw was studied.

Numerical simulation of the rheological properties of granular composites by using a structural approach

Composites with an elastomeric matrix containing rigid particles of diameter 10–1000 µm are studied. One of possible mechanisms of the rheological behavior of such filled systems, related to the origination and growth of vacuoles near the rigid inclusions in a viscous matrix, is considered. For simulating the mechanism of formation of rheological properties of the filled elastomers, we use a structural cell in the form of an elastomeric cylinder, whose height and diameter are equal in magnitude, with a rigid spherical inclusion at its center. Deformation of the cells is examined with the observance of boundary conditions providing the preservation of their close packing. The inclusion is assumed to be rigid, and the matrix properties are described by equations of the linear hereditary viscoelasticity theory. The formation of vacuoles is described by using the approach suggesting that an initial debonding begins to propagate when the energy accumulated in the extended matrix reaches a value sufficient to create a new interface. The heterogeneity of the composite is simulated by taking into account the variability of the local filler concentration. Creep curves obtained for composite cells with different content of the solid phase are presented. Comparisons between the numerical and experimental results show a satisfactory agreement.