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The motion of the Earth's layers due to internal pressures is simulated in this research with an efficient mathematical model. The Earth, which revolves around its axis of rotation and is under internal pressure, will change the shape and displacement of the internal layers and tectonic plates. Applied mathematical models are based on a new approach to shell theory involving both two and three-dimensional approaches. It is the first time studying all necessary measures that increase the accuracy of the obtained results. These parameters are essential to perform a completely nonlinear analysis and consider the effects of the Earth's rotation around its axis. Unlike most modeling of nonlinear partial differential equations in applied mechanics that only considers nonlinear effects in a particular direction, the general nonlinear terms are considered in the present study, which increases the accuracy of the amount of displacement of the Earth's inner layers. Also, the fully nonlinear and dynamic differential equations are solved by a semi-analytical polynomial method which is an innovative and efficient method. Determining the amount of critical pressure at the fault location that will cause phenomena such as earthquakes is one of the useful results that can be obtained from the mathematical modeling in this research. [ABSTRACT FROM AUTHOR]

We discuss experimental and numerical studies of the deformation and destruction of fine-grained concrete B22.5 under dynamic loading. The experiments were carried out using the Kolsky (or split-Hopkinson pressure bar) method, and its modifications in the strain rate range from 400 to 2000 s−1. The rate dependences of ultimate stresses and fracture energy in tension and compression are obtained. Based on experimental data, the identification of the dynamic component of two models from the LS-DYNA computational complex was carried out: *MAT_CONCRETE_DAMAGE and *MAT_CSCM. The results of a comparative analysis of the identified models based on single-element modeling and comparison with experimental data are presented. It is shown that the obtained experimental strain rate dependences of the fracture characteristics can significantly improve the predictive ability of the model compared to the default parameter set. Information about the rate dependence of the fracture energy in *MAT_CSCM model makes it possible to more realistically simulate the behavior of the material beyond the ultimate stress. [ABSTRACT FROM AUTHOR]

We formulate a series of strong ellipticity inequalities for equilibrium equations of the gradient elasticity up to the Nth order. Within this model of a continuum, there exists a deformation energy introduced as an objective function of deformation gradients up to the Nth order. As a result, the equilibrium equations constitute a system of 2 N -order nonlinear partial differential equations (PDEs). Using these inequalities for a boundary-value problem with the Dirichlet boundary conditions, we prove the positive definiteness of the second variation of the functional of the total energy. In other words, we establish sufficient conditions for infinitesimal instability. Here, we restrict ourselves to a particular class of deformations which includes affine deformations. [ABSTRACT FROM AUTHOR]

We discuss new type of surface waves which exist in elastic media with surface energy. Here we present the model of a coating made of polymeric brush. From the physical point of view the considered model of surface elasticity describes a highly anisotropic surface coating. Here the surface energy model could be treated as 2D reduced strain gradient continuum as surface strain energy depends on few second spatial derivatives of displacements. From the mechanical point of view the proposed model relates to 2D coating made of long fibers undergoing stretching and bending deformations. We consider here anti-plane surface waves. The dispersion relation is derived and its dependence on the material parameters is analysed.

We discuss conservation laws for thin structures which could be modeled as a material minimal surface, i.e., a surface with zero mean curvatures. The models of an elastic membrane and micropolar (six-parameter) shell undergoing finite deformations are considered. We show that for a minimal surface, it is possible to formulate a conservation law similar to three-dimensional non-linear elasticity. It brings us a path-independent J -integral which could be used in mechanics of fracture. So, the class of minimal surfaces extends significantly a possible geometry of two-dimensional structures which possess conservation laws. [ABSTRACT FROM AUTHOR]

It was with Maugin that Francesco dell'Isola met Paul Germain for the first time. It is in this perspective that some of dell'Isola's works in the field of Economics, such as dell'Isola and del Monte [[18]] or his recent book [[19]], should be framed. Graph We are very pleased to dedicate this Special Issue of I Mathematics and Mechanics of Solids i to Francesco dell'Isola for celebrating his 60th birthday. Francesco dell'Isola's academic formation can be ascribed to a double matrix: the primary Italian one, for which it is even possible to draw up a genealogy, and the French one, mediated by his numerous collaborations with French scientists and inspired by his fruitful contact, as a young scholar, with Paul Germain. [Extracted from the article]

In this paper, we describe Laplace domain boundary element method (BEM) for transient dynamic problems of three-dimensional finite homogeneous anisotropic linearly elastic solids. The employed boundary integral equations for displacements are regularized using the static traction fundamental solution. Modified integral expressions for the dynamic parts of anisotropic fundamental solutions and their first derivatives are obtained. Boundary elements with mixed approximation of geometry and field variables with the standard nodal collocation procedure are used for spatial discretization. In order to obtain time-domain solutions, the classic Durbin's method is applied for numerical inversion of Laplace transform. Problem of alleviating Gibbs oscillations is addressed. Dynamic boundary element analysis of the model problem involving trigonal material is performed to test presented formulation. Obtained results are compared with finite element solutions. [ABSTRACT FROM AUTHOR]

The problem of dynamics of a linear micropolar shell with a finite set of rigid inclusions is considered. The equations of motion consist of the system of partial differential equations (PDEs) describing small deformations of an elastic shell and ordinary differential equations (ODEs) describing the motions of inclusions. Few types of the contact of the shell with inclusions are considered. The weak setup of the problem is formulated and studied. It is proved a theorem of existence and uniqueness of a weak solution for the problem under consideration. [ABSTRACT FROM AUTHOR]

The paper presents the results of an experimental study and numerical simulation of dynamic deformation of dry clay at strain rates of ~103 s-1. The main physical and mechanical characteristics of the clay were determined using the modified Split Hopkinson Pressure Bar method for testing of lowly cohesive media in a rigid cage. Three series of experiments were carried out at strain rates of 1400 s-1, 1800 s-1 and 2500 s-1. The maximum values of the realized in the experiment axial stresses in clay were about 400 MPa and maximum pressures were 250 MPa. Based on the results of the experiments, the dependences of axial stresses on axial deformations σx-Σx, shear stresses on pressure ?-P and pressure on volumetric deformation P-e (curves of volumetric compressibility) were plotted. The shear resistance of clay is noted to be well described by the Mohr-Coulomb law. The obtained deformation diagrams are found to be practically independent of deformation rate. The clay behavior under dynamic loads is shown to be essentially nonlinear. On the basis of the obtained experimental data, a parametric identification of the clay deformation model in the form of Grigoryan's constitutive relation was carried out, which was implemented in the framework of the LS-DYNA software in the form of MAT_SOIL_AND_FOAM model. Using the LS-DYNA computational complex, a numerical simulation of the deformation process of a sample under real experimental conditions was carried out. In the computational experiment, the clay behavior was described by the identified model. Good agreement was obtained between numerical and experimental results. [ABSTRACT FROM AUTHOR]

This work is devoted to the direct impact method for determining the deformation diagrams of viscoplastic materials at high strain rates. As the conventional Split Hopkinson Pressure Bar method, the direct impact method is based on the measuring bar technique. The description of the experimental scheme and the traditional experimental data proceeding method are given. The description and the results of numerical analysis of the direct impact scheme are presented. A modified procedure for processing experimental information is proposed which allows to expand the area of correct calculation of strains in the specimen according to the experimental data obtained by the direct impact method. As an illustration the deformation diagrams of copper S101 and aluminum alloy D16T in the strain rate range from 1000 to 10000 s-1 have been obtained using the Split Hopkinson Pressure Bar method and the direct impact method. The use of the direct impact method made it possible to obtain deformation curves at strain rates an order of magnitude higher than the conventional SHPB method. The range of studied plastic deformations is increased by 4 times for the case of copper and 3 times for aluminum alloy. [ABSTRACT FROM AUTHOR]

We provide the proof of an existence and uniqueness theorem for weak solutions of the equilibrium problem in linear dilatational strain gradient elasticity for bodies occupying, in the reference configuration, Lipschitz domains with edges. The considered elastic model belongs to the class of so-called incomplete strain gradient continua whose potential energy density depends quadratically on linear strains and on the gradient of dilatation only. Such a model has many applications, e.g., to describe phenomena of interest in poroelasticity or in some situations where media with scalar microstructure are necessary. We present an extension of the previous results by Eremeyev et al. (2020 Z angew Math Phys 71 (6): 1–16) to the case of domains with edges and when external line forces are applied. Let us note that the interest paid to Lipschitz polyhedra-type domains is at least twofold. First, it is known that geometrical singularity of the boundary may essentially influence singularity of solutions. On the other hand, the analysis of weak solutions in polyhedral domains is of great significance for design of optimal computations using a finite-element method and for the analysis of convergence of numerical solutions. [ABSTRACT FROM AUTHOR]

The boundary element method is used to analyze the problem of dynamic loading acting inside a cubic cavity located in a partially saturated poroelastic halfspace. Defining relations of a Biot's porous medium are used, which are written in Laplace representations for unknown functions of displacements of the skeleton and pore pressures of the fillers. Solutions in time are obtained using the stepped method of numerical inversion of Laplace transforms. Dynamic responses of displacements and pore pressures at points on the surface of the halfspace and the cavity have been constructed. The effect of the values of the saturation coefficient and of the depth of the location of the cavity on dynamic responses has been studied. [ABSTRACT FROM AUTHOR]

The results of dynamic compression tests of aspen under elevated temperature up to +60°C are presented. The tests were carried out based on the Kolsky method using the split Hopkinson pressure bar. To study the anisotropy of properties, aspen samples were fabricated and tested by cutting along and across the fibers direction. Dynamic stress-strain curves were obtained as well as the average values of modulus of active loading sites. The greatest steepness of the loading branches and the highest breaking stresses are observed for the samples loaded along the fiber direction, while the smallest values are noted under loading across the fiber direction. Also the effect of elevated temperature on strength and deformation properties of aspen is estimated. [ABSTRACT FROM AUTHOR]

Using an unified approach based on the local material symmetry group introduced for general first- and second-order strain gradient elastic media, we analyze the constitutive equations of strain gradient fluids. For the strain gradient medium there exists a strain energy density dependent on first- and higher-order gradients of placement vector, whereas for fluids a strain energy depends on a current mass density and its gradients. Both models found applications to modeling of materials with complex inner structure such as beam-lattice metamaterials and fluids at small scales. The local material symmetry group is formed through such transformations of a reference placement which cannot be experimentally detected within the considered material model. We show that considering maximal symmetry group, i.e. material with strain energy that is independent of the choice of a reference placement, one comes to the constitutive equations of gradient fluids introduced independently on general strain gradient continua. [ABSTRACT FROM AUTHOR]

In this work, the propagation behaviour of a surface wave in a micropolar elastic half-space with surface strain and kinetic energies localized at the surface and the propagation behaviour of an interfacial anti-plane wave between two micropolar elastic half-spaces with interfacial strain and kinetic energies localized at the interface have been studied. The Gurtin–Murdoch model has been adopted for surface and interfacial elasticity. Dispersion equations for both models have been obtained in algebraic form for two types of anti-plane wave, i.e. a Love-type wave and a new type of surface wave (due to micropolarity). The angular frequency and phase velocity of anti-plane waves have been analysed through a numerical study within cut-off frequencies. The obtained results may find suitable applications in thin film technology, non-destructive analysis or biomechanics, where the models discussed here may serve as theoretical frameworks for similar types of phenomena. [ABSTRACT FROM AUTHOR]

As at the nanoscale the surface-to-volume ratio may be comparable with any characteristic length, while the material properties may essentially depend on surface/interface energy properties. In order to get effective material properties at the nanoscale, one can use various generalized models of continuum. In particular, within the framework of continuum mechanics, the surface elasticity is applied to the modelling of surface-related phenomena. In this paper, we derive an expression for the effective bending stiffness of a laminate plate, considering the Steigmann–Ogden surface elasticity. To this end, we consider plane bending deformations and utilize the through-the-thickness integration procedure. As a result, the calculated elastic bending stiffness depends on lamina thickness and on bulk and surface elastic moduli. The obtained expression could be useful for the description of the bending of multilayered thin films. [ABSTRACT FROM AUTHOR]

Here we discuss the similarities and differences in anti-plane surface wave propagation in an elastic half-space within the framework of the theories of Gurtin–Murdoch surface elasticity and Toupin–Mindlin strain-gradient elasticity. The qualitative behaviour of the dispersion curves and the decay of the obtained solutions are quite similar. On the other hand, we show that the solutions relating to the surface elasticity model are more localised near the free surface. For the strain-gradient elasticity model there is a range of wavenumbers where the amplitude of displacements decays very slowly. [ABSTRACT FROM AUTHOR]

It is worth mentioning stress and strain gradient elasticity, surface elasticity, media with internal degrees of freedom, and nonlocal continua among others. 32967152 10 Tocci Monaco G., Fantuzzi N., Fabbrocino F., Luciano R. Critical temperatures for vibrations and buckling of magneto-electro-elastic nonlocal strain gradient plates. Recent advances in technologies of design, manufacturing and further studies of new materials and structures result in an essential extension of classic models of continuum and structural mechanics. [Extracted from the article]

Within the linear micropolar elasticity we discuss the development of new finite element and its implementation in commercial software. Here we implement the developed 8-node hybrid isoparametric element into ABAQUS and perform solutions of contact problems. We consider the contact of polymeric stamp modelled within the micropolar elasticity with an elastic substrate. The peculiarities of modelling of contact problems with a user defined finite element in ABAQUS are discussed. The provided comparison of solutions obtained within the micropolar and classical elasticity shows the influence of micropolar properties on stress concentration in the vicinity of contact area. [ABSTRACT FROM AUTHOR]

We discuss the material symmetry group of the micropolar continuum and related consistently simplified constitutive equations. Following Eremeyev and Pietraszkiewicz (Int J Solid Struct 2012; 49: 1993–2005; Generalized continua as models for materials, Heidelberg: Springer, 2013, 77–90) we extend the definition of the group proposed by Eringen and Kafadar (Continuum physics, vol. 4, New York, NY: Academic Press, 1976, 1–75) by taking into account the microstructure curvature tensor as well as different transformation properties of polar (true) and axial (pseudo) tensors. Our material symmetry group consists of ordered triples of tensors which make the strain energy density of the micropolar continuum invariant under change of the reference placement. Within micropolar solids we discuss the isotropic, hemitropic, orthotropic, transversely isotropic and clinotropic materials and give explicitly the consistently reduced representations of the strain energy density. [ABSTRACT FROM AUTHOR]

Courant’s minimax variational principle is considered in application to the six-parameter theory of prestressed shells. The equations of a prestressed micropolar shell are deduced in detail. Courant’s principle is used to study the dependence of the least and higher eigenfrequencies on shell parameters and boundary conditions. Cases involving boundary reinforcements and shell junctions are also treated. [ABSTRACT FROM AUTHOR]

The article reports that the special issue is dedicated to Professor Dr Eron L. Aero on the occasion of his 80th birthday.

In this paper a direct non-local peridynamics theory for thin plates is developed. Peridynamic points are assumed to behave like rigid bodies with independent translation and finite rotation degrees of freedom. The non-local mechanical interaction between points is characterized by force and moment vectors. The balance equations including the linear momentum, the angular momentum and the energy are presented. Peridynamic deformation states of the plate are introduced including the actual bond vector (relative translation of two points within the bond) and the relative finite rotation tensor (actual relative orientation of two points in the bond). The corresponding power-conjugate bond force and bond moment states are derived. The framework to develop constitutive equations for the plate peridynamic states is addressed. Special cases of the theory including plates with zero drilling moments, membranes, soft interlayers as well as stiff plate layers with small relative rotations are considered. [ABSTRACT FROM AUTHOR]

Dynamic tests of fine-grained fired dioxide-zirconia ceramics under compression under uniaxial stress conditions were carried out. The influence of the specimen length on the obtained strength and deformation properties of ceramics is investigated. The thickness of the specimen has a significant impact on the course of the obtained dynamic stress–strain diagrams: short specimens have a much more sloping area of active loading branch. The main contribution to the modulus of the load branch resulting from tests of brittle porous media is made by the geometry of the specimens and the porosity of the material. When choosing the length of specimens for dynamic tests, the optimal geometry of the tested specimens is preferable in accordance with the Davies–Hunter criterion, when the contributions of axial and radial inertia are mutually compensated, and the contribution of the effects of friction in the resulting diagram is minimal. When choosing the geometry of specimens of brittle porous media, the structure of the material should be taken into account so that the size of the specimen (both length and diameter) exceeds the size of the internal fractions of the material by at least five times. [ABSTRACT FROM AUTHOR]

In recent years, the static and dynamic response of micro/nanobeams made of hyperelasticity materials received great attention. In the majority of studies in this area, the strain-stiffing effect that plays a major role in many hyperelastic materials has not been investigated deeply. Moreover, the influence of the size effect and large rotation for such a beam that is important for the large deformation was not addressed. This paper attempts to explore the free and forced vibrations of a micro/nanobeam made of a hyperelastic material incorporating strain-stiffening, size effect, and moderate rotation. The beam is modelled based on the Euler–Bernoulli beam theory, and strains are obtained via an extended von Kármán theory. Boundary conditions and governing equations are derived by way of Hamilton's principle. The multiple scales method is applied to obtain the frequency response equation, and Hamilton's technique is utilized to obtain the free undamped nonlinear frequency. The influence of important system parameters such as the stiffening parameter, damping coefficient, length of the beam, length-scale parameter, and forcing amplitude on the frequency response, force response, and nonlinear frequency is analyzed. Results show that the hyperelastic microbeam shows a nonlinear hardening behavior, which this type of nonlinearity gets stronger by increasing the strain-stiffening effect. Conversely, as the strain-stiffening effect is decreased, the nonlinear frequency is decreased accordingly. The evidence from this study suggests that incorporating strain-stiffening in hyperelastic beams could improve their vibrational performance. The model proposed in this paper is mathematically simple and can be utilized for other kinds of micro/nanobeams with different boundary conditions. [ABSTRACT FROM AUTHOR]

In the present research, wood flour reinforced polyethylene polymer composites with a coupling agent were prepared by injection molding. The effects of wood flour size, aspect ratios, and mold injection temperature on the composites' mechanical properties were investigated. For the preparation of the polymer composites, five different formulations were created. The mechanical properties including tensile strength and the modulus, flexural strength and the modulus, and impact energy were measured. To investigate the changes in the properties resulting from different compositions, mechanical static and impact testing was performed. The obtained results indicate that by reducing the flour size, the tensile strength and modulus, flexural strength, and impact energy were reduced. In contrast, the flexural modulus increased. Furthermore, with the increment of injection molding temperature, the tensile strength and the modulus and the impact energy of the specimens were reduced. On the other hand, the flexural strength and the modulus increased. Thus, an optimized amount of injection molding temperature can provide improvements in the mechanical properties of the composite. [ABSTRACT FROM AUTHOR]

In this study, the effect of wood powder and nanoclay particle content on composites' mechanical behavior made with polyethylene matrix has been investigated. The wood flour as a reinforcer made of wood powder was at levels of 30, 40, and 50 wt.%, and additional reinforcement with nanoclay at 0, 1, 3, and 5 wt.%. Furthermore, to make a composite matrix, high-density polyethylene was used at levels of 70, 60, and 50% by weight. Wood-plastic composite (WPC) specimens were manufactured in injection molding. After preparing the specimens, tensile and bending tests were performed on samples. The mechanical properties such as tensile and flexural strength and flexural modulus were measured. Results showed that nanoclay particle content increases flexural modulus, flexural strength, modulus of elasticity, and tensile strength. The experimental test results show that Young's moduli increased with the volume of wood flour. The biggest modulus of elasticity was achieved in the samples having 50 wt.% of wood powder. Furthermore, the highest value of tensile strength was achieved at the level of 30 wt.%. The highest flexural strength was for the sample containing 50% wood powder by weight. Additionally, a numerical model was made utilizing the Abaqus software using the finite element method (FEM). Comparing the numerical and experimental results, it was found that they are compatible in the linear-elastic and plastic state of the material. There are no crucial differences between experiment and FEM. [ABSTRACT FROM AUTHOR]

The concept of a spherically symmetric second-rank tensor field is formulated. A general representation of such a tensor field is derived. Results related to tensor analysis of spherically symmetric fields and their geometric properties are presented. Using these results, a formulation of the spherically symmetric problem of the nonlinear theory of dislocations is given. For an isotropic nonlinear elastic material with an arbitrary spherically symmetric distribution of dislocations, this problem is reduced to a nonlinear boundary value problem for a system of ordinary differential equations. In the case of an incompressible isotropic material and a spherically symmetric distribution of screw dislocations in the radial direction, an exact analytical solution is found for the equilibrium of a hollow sphere loaded from the outside and from the inside by hydrostatic pressures. This solution is suitable for any models of an isotropic incompressible body, i.e., universal in the specified class of materials. Based on the obtained solution, numerical calculations on the effect of dislocations on the stress state of an elastic hollow sphere at large deformations are carried out. [ABSTRACT FROM AUTHOR]