Advanced Materials Modelling for Mechanical, Medical and Biological Applications

Experimental observations reveal that abnormal biological tissues, such as tumors, found in the breast and prostate, tend to be stiffer than healthy biological tissues. Classical linear elasticity is then used to model the responses of these tissues under small strains. In particular, soft tissues are modeled as linearly elastic, isotropic, and incompressible materials. For a particular class of plane problems, it has been shown that if the shear elastic modulus, \(\mu \), is known at four different points in a sample, then it can be uniquely determined from the knowledge of two displacement fields obtained from two distinct experiments performed on the same sample. We use this result to propose a non-iterative numerical procedure to determine \(\mu \) in a sample of soft tissue, which is subjected to two quasi-static experiments that are possible to reproduce in laboratory and are simulated numerically using the finite element method. No a priori knowledge of the shear elastic modulus is required, but it is assumed that resultant forces are known on complementary parts of the boundary of the sample. Results for the distribution of \(\mu \) in a long cylinder of rectangular cross section containing an eccentric circular inclusion and an inclusion with a complex geometry are presented. The methodology yields numerical results that are both in very good agreement with analytical results and more accurate than numerical results obtained from another methodology presented in the literature. This work is of great interest in the detection of cancerous tumors and in the differential diagnosis of biological tissues.

An efficient method for describing plane strain bending of wide rigid viscoplastic sheets at large strains is developed. Both pure bending and bending under tension are considered. The approach is based on the transformation equations between Eulerian and Lagrangian coordinates. In addition, it is shown that it is advantageous to use the equivalent strain rate as an independent variable instead of the space coordinate. Due to this change in the independent variable, a uniform treatment of an arbitrary dependence of the yield stress on the equivalent strain rate is possible. The solution reduces to an ordinary differential equation that should be solved numerically. However, one term of this equation reduces to the expression 0/0 at the initial instant. Therefore, analytic treatment of the differential equation is required to describe an initial stage of the process. An illustrative example is provided. This example illustrates the effect of the parameter involved in the Bingham model on the through-thickness distribution of stresses and the bending moment. It is shown that the thickness of the sheet decreases as the deformation proceeds.

We proposed a new method for measuring the dynamic Poisson’s ratio of both isotropic and anisotropic materials. The specimen was loaded with a compressive pulsed load generated by gas gun and a split Hopkinson bar. The longitudinal compression deformation of the specimen was determined from the signals recorded with the use of small-base strain gauges glued to the measuring pressure bars. To measure the temporal progress of the radial components of the specimen deformation, a millimeter-wave radio interferometer was used. To assess the possible asymmetry of the radial expansion, measurements were carried out using two independent channels irradiating diametrically opposite zones of the lateral surface of the specimen. The tests were carried out on a pine specimen with air humidity in the form of a cylinder 54 mm in diameter and 30 mm in height. A compressive pulsed load was applied along the fibers by using the split Hopkinson pressure bar (SHPB). The separate movement of the lateral surfaces of the sample was recorded both along and across the annual layers by using the two-channel radio interferometer. It was determined that the opposite displacements of the regions of the lateral surface of the specimen during its expansion along the annual layers are quite close; while during expansion across the annual layers, they are very different. The relative transverse deformation of the sample in both cases was determined as the sum of lateral opposite displacements relative to the specimen’s diameter. As a result, two components of the dynamic Poisson’s ratio were obtained, which amounted to ~0.2 (in the direction along the annual layers) and ~0.24 (in the direction across the annual layers).

An application of thermal analogy for the solution of active control problems is demonstrated for quasi-static beam testing and investigation of the active twist behaviour of smart helicopter rotor blade. To obtain accurate finite element solutions, rigorous convergence study using different finite element types and modelling approaches is carried out. The methodology for optimal placement of MFC actuators on lightweight structures is developed to control the structural vibrations in the low-frequency range. The results of numerical study are successfully confirmed by the results of physical experiments.

Investigation of the behaviour of structural materials in a wide range of the strain rates is an urgent task. The sources of dynamic loads originate from explosion, shock, and earthquake vibration. The loading rate is an important index corresponding to different velocities of impacts for construction materials. The strain rate of structures under impact loading is in the range of 0.1–200 s⁻¹, but it is more than 200 s⁻¹ under blast loading. The rock engineering often involves the dynamic loading scenarios, such as excavation engineering, civil engineering, blasting engineering, projectile impact, seismic events, and rock collapse. There is still limited data in strain rate regimes of relevance, specifically for drop shock applications. This paper describes a pneumo-dynamic experimental setup created in the dynamic testing laboratory of the Research Institute for Mechanics of Nizhny Novgorod State University, to study the dynamic behaviour of structural materials at strain rates of the order of 100 s⁻¹. The results of approbation are given on the example of dynamic tests of plate specimens made of steel HX220BD.

The mathematical models of the pressurized eyeball are studied. The eyeball is represented as two joint segments (the cornea and the sclera) of different geometric and mechanical properties. Both the cornea and sclera are considered as spherical segments of nonuniform thicknesses. Finite element simulation is performed by means of the engineering simulation software ANSYS Inc. Short time effect of fluid (medication) administration into the eye is investigated. Numerical results of two models (with and without separating strip) are compared. Results obtained with the finite element models of a composed eyeball shell are compered with the shell-based model of a pressurized spherical shell. If the strip separating the inner volume of the eye is taken into account, the difference of the pressures inside subcorneal and subscleral volume is about 1–2 mm Hg. If a single inner volume is considered, the inner pressure elevates to the average value for two subspaces from the model with the strip.

The classical formulation of the boundary integral equation method is successfully implemented for solving three-dimensional isotropic problems of the dynamic theory of elasticity, viscoelasticity, and poroelasticity. The extension of this method for solving dynamic anisotropic problems requires the development of special new schemes. At present, numerical schemes are being constructed based on the double application of the reciprocity theorem, which goes back to the work of D. Nardini and K. A. Brebbia or based on the integral representation of Green’s matrices. The use of regular Fredholm integral equations of the first kind (integral equations on a plane wave) is an alternative to the classical formulation of the boundary integral equation method. The construction of such boundary integral equations is based on the structure of the dynamic fundamental solution. The approach employs explicit boundary integral equations and goes back to the work of Babeshko. The paper considers the application of the non-classical approach of the boundary integral equation method in combination with the integral Laplace transform in time to modeling wave processes in anisotropic elastic bodies. In this case, the inverse Laplace transform is constructed numerically using the Durbin method. A numerical solution of the dynamic problem of anisotropic elasticity theory by the boundary integral equation method in non-classical formulation is given. The boundary element scheme of the boundary integral equation method is constructed on the basis of a regular integral equation of the first kind. Numerical results prove the efficiency of using boundary integral equations on a single plane wave for solving three-dimensional anisotropic dynamic problems of elasticity theory. The achieved accuracy of calculation is not inferior to the accuracy of boundary element schemes for classical boundary integral equations.

In this paper, we describe qualitatively material deformations and fracture phenomena using a discrete 2D kinematic model. This model is based on the swarm intelligence approach, where simple laws among the agents can lead the whole group to achieve complex behaviors; it seems to be promising because it is able to not only qualitatively reproduce standard deformations and fracture phenomena but also a lot of exotically ones that other methods in literature cannot reproduce or their computational costs are too high. Instead, in our model the fracture arises more naturally. Furthermore, it has a low computational cost and it is parallelizable, allowing us to take profit of CUDA\(^{\circledR }\) architecture. Some numerical simulations are provided and discussed using two different kind of lattices and changing some model’s parameters.

This study deals with four incorrectly posed boundary value problems for an elastic strip. The formulations assume that three scalar conditions are given on one side of the strip and one on the other side. The Fourier transform is used to find analytical solutions in all four problems. However, the direct inverse transform is not applicable, which necessitates the reduction to integral equations of the Fredholm type of the first kind. Numerical approaches associated with non-stable solutions of the derived integral equation are discussed.

Contact problems for functionally graded (FG) materials of complex structure are of traditional research interest due to various practical applications in the chemical industry, materials science, etc. In this article, we consider the dynamic response of such materials with FG or stepwise layered coatings. Numerical analysis is carried out based on the explicit integral and asymptotic representations obtained in terms of Green’s matrix for elastic half-spaces with periodic coverages. Their soft interlayers form internal waveguide channels, which leads to unusual wave and energy transfer effects in response to dynamic surface loading.

This paper analyzes an experimental study of high-rate deformation and fracture of fine-grained concrete under tensile stresses. A number of sources of foreign and domestic authors are analyzed. Based on the analysis, it was concluded that some properties of concrete have not been fully investigated. In connection with the above, an experimental study of the dynamic properties of concrete materials is relevant today. Experimental studies were carried out on the basis of modifications of the Kolsky method. These modifications make it possible to determine the strength and time characteristics of concrete deformation under high-speed loading. To analyze the nature and time of the destruction, experiments were carried out using high-speed photography. Splitting (Brazilian test) and straight tensile experiments have two speed modes. On the basis of the performed experimental work, tensile strain diagrams and stress versus time diagrams during splitting were obtained. The experimental data indicate the effect of the strain rate on the ultimate tensile strength characteristics. The dynamic tensile strength of fine-grained concrete was about 8 MPa. The value of the coefficient of dynamic tensile hardening is in the range from 4 to 6 MPa. This factor depends on the strain rate. The opposite effect of the influence of the loading rate was obtained both during splitting and stretching. This means that with an increase in the strain rate, the maximum breaking stresses decrease. On the basis of high-speed photography, the features of high-speed destruction of fine-grained concrete under tensile stresses are revealed. The found characteristics can be used to create mathematical models necessary to determine the strength of concrete structures subjected to dynamic effects.

A paper deals with describing a microstructural material on the base of a non-Euclidean model of a continuous medium. The solution for the stress field is presented as a sum of elastic and self-balanced fields, the parameterization of which is given through non-Euclidean characteristics: the scalar curvature of the Ricci tensor. The requirement of the absence of singular contributions in the constructed field leads to the fact that the singularities of the elastic field are compensated by the singularities of the self-balanced stress fields, and a linear transformation relates the coefficients at the singularities of both fields. Thus, the coefficients at the singularities of the classical theory of elasticity are “hidden” parameters of the non-Euclidean model. The compensating role of self-balanced stress fields makes it possible to construct in equilibrium a nonsingular distribution for the stress field of a plane-deformed and spherically symmetric state of a continuous medium. The compensating role of self-balanced stress fields enables to construct in equilibrium a nonsingular distribution for the stress field of a plane-deformed and spherically symmetric state of a continuous medium.

We study an inflation of an inhomogeneous thin-walled hyperelastic tube. Either the thickness of the tube or the material properties of tube the change on its cross section. The inhomogeneity leads to a bending of tube subjected by an internal uniform pressure. We analyse the effect of the magnitude of the inhomogeneity, the size of its area and the pressure on the bending of tube.

The paper considers models of foam materials in the form of Gibson–Ashby cell arrays. The method of effective moduli, based on the equality of the potential energies of the porous and homogeneous material, is described. Six boundary value problems of the linear static elasticity theory for a representative volume are given. These problems together allow determining all the coefficients of the effective stiffness matrix for any anisotropy class of the frame material and geometric asymmetry. The finite element package ANSYS and the capabilities of its command language APDL are used to construct representative volumes and to solve homogenization problems numerically. The procedures for creating solid and finite element models of arrays composed of open Gibson–Ashby cells with regular and irregular structure are described in detail. Two different algorithms for regular lattices of low and high porosity are offered. For an irregular lattice, the sizes of the cube frames are randomly generated with a uniform and normal distribution. The results of numerical calculations for stainless steel lattices in a wide range of porosity are presented. The dependencies of the effective elastic moduli on porosity for a single Gibson–Ashby cell and for regular and irregular lattices with uniform and normal distribution are analyzed. It is shown that the applied Gibson–Ashby model predicts the elastic properties of highly porous materials quite well. But the prediction for lattices with porosity less than 75% gives a sufficiently large error. It is noted that regular and irregular lattices with a large number of cells give similar results for effective elastic stiffness moduli. Meanwhile, individual irregular structures with strongly different sizes of cubic cell frames can take extreme values of effective moduli with pronounced anisotropy in different directions. These effects depend on the geometric asymmetry of the irregular lattices and on the stress concentrations. Examples of the stress–strain state in strongly irregular Gibson–Ashby lattices are given. The analysis of the value scatter of various effective elastic moduli, demonstrating the anisotropy of strongly irregular Gibson–Ashby lattices, is given.

Zr-based coatings are multifunctional materials and very perspective in macro- and microtechnics as protective and tribological coatings. ZrN and ZrCN coatings were deposited on steel substrates by reactive magnetron sputtering at the constant magnetron power of 700 W and at the substrate bias voltage of − 10 V with different nitrogen or acetylene flow rate. The surface investigations were carried out using scanning electron microscopy, atomic force microscopy, nanoindentation and glancing angle X-ray diffraction. Tribological tests in the condition of microcontact were carried out using atomic force microscopy under ambient temperature and humidity controlled conditions. It was found that the mechanical and tribological properties and the size of grains in the polycrystalline ZrN and ZrCN coatings can be controlled using different gas flows.

Functionally graded structures have shown the perspective of materials in a higher efficient and consistent manner. This study reports a short investigation by concentrating on the flexomagnetic response of a functionally graded piezomagnetic nano-actuator, keeping in mind that the converse magnetic effect is only taken into evaluation. The rule of mixture assuming exponential composition of properties along with the thickness is developed for the ferromagnetic bulk. Nonlocal effects are assigned to the model, respecting Eringen’s hypothesis. The derived equations deserve to be analytically solved. Therefore, numerical results are generated for fully fixed ends. It is denoted that the functionality grading feature of a magnetic nanobeam can magnify the flexomagnetic effect leading to high-performance nanosensors/actuators.

In this paper, iterative procedure for partial fraction approximation of vector functions in Laplace domain is presented. First, the essentials of the original formulation of vector fitting methodology for strictly proper rational approximations are outlined. Then a modification is presented for more efficient application to vector functions. Proposed modification reduces the problem of fitting all elements of vector function to fitting only some of them and sum of the rest. After the set of common poles is obtained, the residues are calculated for all functions as usual. Next, the algorithm for adaptive rational approximation of vector functions is thoroughly presented. Proposed technique consists of successive determination of partial fraction approximations with increasing orders until the specified convergence criterion is satisfied. Detailed results of the numerical example are provided to assess accuracy and efficiency of the proposed procedure.

Within the framework of the linearized model of a prestressed elastic body, we study some statements of two-dimensional inverse problems on the prestressed state restoration. We consider inverse problems of 2 types, in the presence of additional data on the measured displacement field: (1) in a set of points of the region at a fixed vibration frequency, and (2) on a boundary part in some frequency range. We investigate the questions of solution uniqueness for the 1st type inverse problem. We propose and discuss some techniques for solving the stated inverse problems.

In this paper, three-dimensional dynamic response of a cylindrical cavity in a half-space of soil under uniform normal traction applied at the cavity surface is studied numerically. The surrounding soil is treated as a partially saturated porous medium, and the governing equations for the dynamic problem are presented. Analysis of the problem is preformed in the Laplace domain using the direct boundary element technique. The technique is employed together with the Laplace inverse transform by the stepped method to obtain the time-domain solutions. A parametric study is presented to demonstrate the effect of poroelastic material parameters and geometrical parameters on the response of the cavity.

Caries in the stage of brown spot lesion is an advanced phase of the disease thus causing serious problems for patients. Having passed the dentine-enamel junction, carieogenic bacteria begin demineralization of dentine in its vicinity. If not treated promptly, the affected area may reach the pulp of the tooth, creating a gradient of strength characteristics in dentine. In the present work, the mechanical properties and mineral density in the direction from the dentine-enamel junction to the dental pulp were obtained using nanoindentation and X-ray computed micro-tomography and correlated. The results show that the mechanical properties of dentine are dependent on its mineral content. The approach can be used as control technique for the further research of remineralization procedures in clinical practice.

The overall properties—thermal, electric, elastic, etc.—of any porous material are strongly dependent on their three-dimensional (3D) microstructures, which include the porosity, pore sizes, and shapes and connectivity of the porous space. These microstructural parameters can be collectively described as the “tortuosity of the porous space” (see, e.g., Chen et al., J Power Sources 273:486–494 2013). We propose to use the tortuosity parameter to characterize the overall material properties of a porous material with interconnected porous space. In the text to follow, we discuss the concept of this parameter and its application to characterize, elastic, electric, and mass transport properties as well as the cross-property connections.

A class of hierarchically chiral metamaterials was analyzed with the finite element method for their effective linear viscoelastic properties under low frequency, uniaxial straining. The viscoelastic property of the solid phase was assumed to be of the standard linear solid. It is found that from Rank-1 to Rank-2 hierarchy effective Young’s modulus is enhanced. From Rank-2 to Rank-3, effective modulus may be roughly the same, slightly increased or decreased, depending on the ligament thickness. When the ligament thickness is less than 1.5 mm, increasing hierarchy decreases the overall damping. For larger thickness, overall damping may be slightly enhanced by hierarchy. Gradient lateral deformation is observed in the metamaterials under uniaxial straining, indicating the existence of chiral effects.

The basis of the method for determining the parameters of the depth-variable shear modulus of the functional-gradient half-space is the indentation method or the method of shear by the strip indenter of the elastic functional-gradient half-space. Mathematical modeling or description of the indentation process consists in solving the corresponding contact problem of shear by a strip indenter of the surface of an elastic functional-gradient half-space with an exponential shear modulus in depth. The solution of the contact problem is reduced to the solution of an integral equation of the first kind of the Fourier convolution type with a difference kernel. The analytical solution of the integral equation, obtained by asymptotic methods, contains the parameters of the exponential modulus of the half-space shear. To unambiguously determine the parameters of the shear modulus, the theoretical and experimental characteristics of the contact are compared, such as contact stresses, surface displacements outside the contact area and others, and the static condition on the contact is also used. In the space of parameters of the contact problem, methods are developed for studying the boundaries of the regions of existence and uniqueness of the values of the parameters of the exponential shear modulus. Formulas for the effective shear modulus of a functional-gradient half-space are determined. Its dependence on the parameters of the exponential shear modulus is investigated. In conclusion, the integral equations of contact problems for the homogeneous and functional-gradient half-space, as well as their solutions, are compared.