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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 study a time-dependent thermoelastic coupling within free vibrations of piezomagnetic (PM) microbeams considering the flexomagnetic (FM) phenomenon. The flexomagneticity relates to a magnetic field with a gradient of strains. Here, we use the generalized thermoelasticity theory of Lord–Shulman to analyze the interaction between elastic deformation and thermal conductivity. The uniform magnetic field is permeated in line with the transverse axis. Using the strain gradient approach, the beam yields microstructural properties. The analytical solving process has been gotten via applying sine Fourier technique on displacements. Graphical illustrations are assigned to shape numerical examples concerning variations in essential physical quantities. It was observed that the flexomagnetic effect could be extraordinary if the thermal conductivity of the material is higher or the thermal relaxation time of the heat source is lesser. This theoretical study will provide the way of starting studies on magneto-thermoelastic small-scale piezo-flexomagnetic structures based on the heat conduction models. [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]

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]

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]

On the basis of a theoretical study, this research incorporates an eccentricity into a system of compressed double-walled carbon nanotubes (DWCNTs). In order to formulate the stability equations, a kinematic displacement with reference to the classical beam hypothesis is utilized. Furthermore, the influence of nanoscale size is taken into account with regard to the nonlocal approach of strain gradient, and the van der Waals interaction for both inner and outer tubes is also considered based on the Lennard–Jones model. Galerkin decomposition is employed to numerically deal with the governing equations. It is evidently demonstrated that the geometrical eccentricity remarkably affects the stability threshold and its impact is to increase the static stability of DWCNTs. [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]

Abstract: We extend the material symmetry group of the non-linear polar-elastic continuum by taking into account microstructure curvature tensors as well as different transformation properties of polar and axial tensors. The group consists of an ordered triple of tensors which makes the strain energy density of polar-elastic continuum invariant under change of reference placement. An analog of the Noll rule is established. Four simple specific cases of the group with corresponding reduced forms of the strain energy density are discussed. Definitions of polar-elastic fluids, solids, liquid crystals and subfluids are given in terms of members of the symmetry group. Within polar-elastic solids we discuss in more detail isotropic, hemitropic, cubic-symmetric, transversely isotropic, and orthotropic materials and give explicitly corresponding reduced representations of the strain energy density. For physically linear polar-elastic solids, when the density becomes a quadratic function of strain measures, reduced representations of the density are established for monoclinic, orthotropic, cubic-symmetric, hemitropic and isotropic materials in terms of appropriate joint scalar invariants of stretch, wryness and undeformed structure curvature tensors. [Copyright &y& Elsevier]

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]

Our analysis incorporates the geometrically nonlinear bending of the Euler-Bernoulli ferromagnetic nanobeam accounting for a size-dependent model through assuming surface effects. In the framework of the flexomagnetic phenomenon, the large deflections are investigated referring to von-Kármán nonlinearity. Employing the nonlocal effects of stress coupled to the gradient of strain generates a scale-dependent Hookean stress–strain scheme related to the small scale. Taking into account the supports of the nanobeam in two cases, that is, totally fixed and hinged, the deformations are predicted. A constant static lateral load is postulated uniformly along the length of the beam, which forces the deformation. As the analysis is based on the one-dimensional media, the electrodes are embedded so that they give off a transverse magnetic field creating a longitudinal force. The newly developed mathematical model is computed by means of the differential quadrature method together with the Newton-Raphson technique. The computational section discusses and reveals the numerical results in detail for the characteristics and parameters involved in the design of beam-like magnetic nanosensors. As shown later, the conducted research presents that there is a strong linkage between the surface effect and the flexomagneticity behavior of the bulk. [ABSTRACT FROM AUTHOR]

In this paper, the equation governing the buckling of a magnetic composite nanoplate under the influence of an in-plane one-dimensional magnetic field, assuming the concept of flexomagnetic and considering the resulting flexural force and moment, is investigated for the first time by different analytical boundary conditions. To determine the equation governing the stability of the nanoplate, the nonlocal strain gradient theory has been used by taking into account the classical plate theory. The axial magnetic force, which is originated from the magnetic field, is investigated. After extracting the governing differential equation, the critical buckling load is obtained for different support conditions. The effect of nonlocal parameter, sheet aspect ratio and the effect of one-dimensional magnetic field on critical load are discussed. It was earned that if the nanoplate is rectangular so that the value of aspect ratio is less than one, the flexomagnetic response will be more noticeable. [ABSTRACT FROM AUTHOR]

We investigated the stability of an axially loaded Euler–Bernoulli porous nanobeam considering the flexomagnetic material properties. The flexomagneticity relates to the magnetization with strain gradients. Here we assume both piezomagnetic and flexomagnetic phenomena are coupled simultaneously with elastic relations in an inverse magnetization. Similar to flexoelectricity, the flexomagneticity is a size-dependent property. Therefore, its effect is more pronounced at small scales. We merge the stability equation with a nonlocal model of the strain gradient elasticity. The Navier sinusoidal transverse deflection is employed to attain the critical buckling load. Furthermore, different types of axial symmetric and asymmetric porosity distributions are studied. It was revealed that regardless of the high magnetic field, one can realize the flexomagnetic effect at a small scale. We demonstrate as well that for the larger thicknesses a difference between responses of piezomagnetic and piezo-flexomagnetic nanobeams would not be significant. [ABSTRACT FROM AUTHOR]

Large deformations and stress analyses in two types of space structures that are intended for people to live in space have been studied in this research. The structure under analysis is assumed to rotate around the central axis to create artificial gravitational acceleration equal to the gravity on the Earth's surface. The analysis is fully dynamic, which is formulated based on the energy method by using the first-order shear deformation shell theory in two systems, cylindrical and torus. Also, the nonlinear von Kármán strain field has been assumed. The obtained set of partial differential equations has been solved using the semi-analytical polynomial solution method (SAPM). The main purpose of this paper is to study the effects of unusual conditions in the space outside the Earth's atmosphere (which is a complete vacuum environment without pressure) on the strength of the analyzed structure. The numerical results of the governing equations have been evaluated using those of other studies and the simulation efficiency performed in this research has been proven. Finally, the effect of important parameters on the numerical results, including the angular velocity of the structure (which causes artificial gravity), the amount of imposed mechanical and hygro-thermal loads, the structure size and material specifications have been investigated in more detail. • Large dynamic deformations of torus-shaped and cylindrical space structures are studied. • Artificial gravity due to rotations is taken into account. • Hygro-thermal effects are also considered. • Functionally graded structures are analyzed. [ABSTRACT FROM AUTHOR]