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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 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]

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]

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]